(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(dbl(0)) → c
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(dbls(nil)) → c2
ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(0, cons(z0, z1))) → c4
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(nil, z0)) → c6
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(from(z0)) → c8(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(dbl(z0)) → c9(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(0) → c24
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(nil) → c27
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
S tuples:
ACTIVE(dbl(0)) → c
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(dbls(nil)) → c2
ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(0, cons(z0, z1))) → c4
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(nil, z0)) → c6
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(from(z0)) → c8(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(dbl(z0)) → c9(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(0) → c24
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(nil) → c27
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:
active, dbl, dbls, sel, indx, proper, s, cons, from, top
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing nodes:
ACTIVE(dbls(nil)) → c2
ACTIVE(indx(nil, z0)) → c6
PROPER(0) → c24
ACTIVE(dbl(0)) → c
PROPER(nil) → c27
ACTIVE(sel(0, cons(z0, z1))) → c4
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(from(z0)) → c8(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(dbl(z0)) → c9(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
S tuples:
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(from(z0)) → c8(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(dbl(z0)) → c9(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:
active, dbl, dbls, sel, indx, proper, s, cons, from, top
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c1, c3, c5, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl(z0)) → c9(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
S tuples:
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c3(CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c7(CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(dbl(z0)) → c9(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
K tuples:none
Defined Rule Symbols:
active, dbl, dbls, sel, indx, proper, s, cons, from, top
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c1, c3, c5, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8
(7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(ok(z0)) → ok(from(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbl(z0)) → c9(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbl(z0)) → c9(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
K tuples:none
Defined Rule Symbols:
active, dbl, dbls, sel, indx, proper, s, cons, from, top
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c
(9) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbl(z0)) → c9(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbl(z0)) → c9(DBL(active(z0)), ACTIVE(z0))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c
(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
dbl(
z0)) →
c9(
DBL(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)), ACTIVE(dbl(0)))
ACTIVE(dbl(dbl(s(z0)))) → c9(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)), ACTIVE(dbls(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c9(DBL(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(from(z0))) → c9(DBL(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)), ACTIVE(dbl(0)))
ACTIVE(dbl(dbl(s(z0)))) → c9(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)), ACTIVE(dbls(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c9(DBL(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(from(z0))) → c9(DBL(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)), ACTIVE(dbl(0)))
ACTIVE(dbl(dbl(s(z0)))) → c9(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)), ACTIVE(dbls(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c9(DBL(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(from(z0))) → c9(DBL(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9
(13) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(dbl(s(z0)))) → c9(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)), ACTIVE(dbls(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c9(DBL(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(from(z0))) → c9(DBL(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(dbl(s(z0)))) → c9(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)), ACTIVE(dbls(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c9(DBL(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(from(z0))) → c9(DBL(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9
(15) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)), ACTIVE(dbls(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c9(DBL(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(dbls(z0)) → c10(DBLS(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)), ACTIVE(dbls(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c9(DBL(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1
(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
dbls(
z0)) →
c10(
DBLS(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(dbls(dbl(s(z0)))) → c10(DBLS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)), ACTIVE(dbls(nil)))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c10(DBLS(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c10(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(from(z0))) → c10(DBLS(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)), ACTIVE(dbls(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c9(DBL(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(dbls(dbl(s(z0)))) → c10(DBLS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)), ACTIVE(dbls(nil)))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c10(DBLS(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c10(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(from(z0))) → c10(DBLS(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)), ACTIVE(dbls(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c9(DBL(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(dbls(dbl(s(z0)))) → c10(DBLS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)), ACTIVE(dbls(nil)))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c10(DBLS(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c10(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(from(z0))) → c10(DBLS(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10
(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing tuple parts
(20) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c9(DBL(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(dbl(s(z0)))) → c10(DBLS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c10(DBLS(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c10(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(from(z0))) → c10(DBLS(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c9(DBL(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(dbl(s(z0)))) → c10(DBLS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c10(DBLS(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c10(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(from(z0))) → c10(DBLS(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10
(21) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(22) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c10(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(indx(z0, z1)) → c13(INDX(active(z0), z1), ACTIVE(z0))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c10(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2
(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
indx(
z0,
z1)) →
c13(
INDX(
active(
z0),
z1),
ACTIVE(
z0)) by
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(indx(dbl(s(z0)), x1)) → c13(INDX(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1), ACTIVE(dbls(nil)))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c13(INDX(mark(cons(dbl(z0), dbls(z1))), x1), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1), ACTIVE(indx(nil, z0)))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c13(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c13(INDX(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
(24) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c10(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(indx(dbl(s(z0)), x1)) → c13(INDX(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1), ACTIVE(dbls(nil)))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c13(INDX(mark(cons(dbl(z0), dbls(z1))), x1), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1), ACTIVE(indx(nil, z0)))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c13(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c13(INDX(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)), ACTIVE(indx(nil, z0)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c10(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(indx(dbl(s(z0)), x1)) → c13(INDX(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1), ACTIVE(dbls(nil)))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c13(INDX(mark(cons(dbl(z0), dbls(z1))), x1), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1), ACTIVE(indx(nil, z0)))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c13(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c13(INDX(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13
(25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing tuple parts
(26) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c10(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(dbl(s(z0)), x1)) → c13(INDX(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c13(INDX(mark(cons(dbl(z0), dbls(z1))), x1), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c13(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c13(INDX(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c9(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c10(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(dbl(s(z0)), x1)) → c13(INDX(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c13(INDX(mark(cons(dbl(z0), dbls(z1))), x1), ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c13(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1), ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c13(INDX(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13
(27) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(28) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbl(z0)) → c23(DBL(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3
(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
dbl(
z0)) →
c23(
DBL(
proper(
z0)),
PROPER(
z0)) by
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)), PROPER(0))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)), PROPER(nil))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
(30) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)), PROPER(0))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)), PROPER(nil))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)), PROPER(0))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)), PROPER(nil))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23
(31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(32) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c25, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23
(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
s(
z0)) →
c25(
S(
proper(
z0)),
PROPER(
z0)) by
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(0)) → c25(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(nil)) → c25(S(ok(nil)), PROPER(nil))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
(34) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(0)) → c25(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(nil)) → c25(S(ok(nil)), PROPER(nil))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(0)) → c25(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(nil)) → c25(S(ok(nil)), PROPER(nil))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25
(35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(36) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(dbls(z0)) → c26(DBLS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c26, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25
(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
dbls(
z0)) →
c26(
DBLS(
proper(
z0)),
PROPER(
z0)) by
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)), PROPER(0))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)), PROPER(nil))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
(38) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)), PROPER(0))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)), PROPER(nil))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)), PROPER(0))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)), PROPER(nil))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26
(39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(40) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(cons(z0, z1)) → c28(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c28, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26
(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
cons(
z0,
z1)) →
c28(
CONS(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
(42) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28
(43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(44) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(sel(z0, z1)) → c29(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c29, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28
(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
sel(
z0,
z1)) →
c29(
SEL(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
(46) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28, c29
(47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(48) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(indx(z0, z1)) → c30(INDX(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c30, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28, c29, c29
(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
indx(
z0,
z1)) →
c30(
INDX(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
(50) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28, c29, c29, c30
(51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(52) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, PROPER, S, CONS, FROM, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c31, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28, c29, c29, c30, c30
(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
from(
z0)) →
c31(
FROM(
proper(
z0)),
PROPER(
z0)) by
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(0)) → c31(FROM(ok(0)), PROPER(0))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(nil)) → c31(FROM(ok(nil)), PROPER(nil))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
(54) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(0)) → c31(FROM(ok(0)), PROPER(0))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(nil)) → c31(FROM(ok(nil)), PROPER(nil))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(0)) → c31(FROM(ok(0)), PROPER(0))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(nil)) → c31(FROM(ok(nil)), PROPER(nil))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, S, CONS, FROM, TOP, PROPER
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28, c29, c29, c30, c30, c31
(55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(56) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, S, CONS, FROM, TOP, PROPER
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28, c29, c29, c30, c30, c31, c31
(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
TOP(
mark(
z0)) →
c35(
TOP(
proper(
z0)),
PROPER(
z0)) by
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(0)) → c35(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(nil)) → c35(TOP(ok(nil)), PROPER(nil))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
(58) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(0)) → c35(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(nil)) → c35(TOP(ok(nil)), PROPER(nil))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(0)) → c35(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(nil)) → c35(TOP(ok(nil)), PROPER(nil))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, S, CONS, FROM, TOP, PROPER
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28, c29, c29, c30, c30, c31, c31, c35
(59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(60) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, S, CONS, FROM, TOP, PROPER
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28, c29, c29, c30, c30, c31, c31, c35, c35
(61) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
We considered the (Usable) Rules:
active(sel(0, cons(z0, z1))) → mark(z0)
sel(z0, mark(z1)) → mark(sel(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(indx(nil, z0)) → mark(nil)
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
active(dbls(nil)) → mark(nil)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbl(ok(z0)) → ok(dbl(z0))
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbls(z0)) → dbls(active(z0))
s(ok(z0)) → ok(s(z0))
active(sel(z0, z1)) → sel(z0, active(z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
dbl(mark(z0)) → mark(dbl(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
from(ok(z0)) → ok(from(z0))
active(dbl(0)) → mark(0)
And the Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = [4]
POL(ACTIVE(x1)) = 0
POL(CONS(x1, x2)) = 0
POL(DBL(x1)) = 0
POL(DBLS(x1)) = 0
POL(FROM(x1)) = 0
POL(INDX(x1, x2)) = 0
POL(PROPER(x1)) = 0
POL(S(x1)) = 0
POL(SEL(x1, x2)) = 0
POL(TOP(x1)) = [2]x1
POL(active(x1)) = x1
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c10(x1)) = x1
POL(c10(x1, x2)) = x1 + x2
POL(c11(x1, x2)) = x1 + x2
POL(c12(x1, x2)) = x1 + x2
POL(c13(x1)) = x1
POL(c13(x1, x2)) = x1 + x2
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c2(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c23(x1, x2)) = x1 + x2
POL(c25(x1)) = x1
POL(c25(x1, x2)) = x1 + x2
POL(c26(x1)) = x1
POL(c26(x1, x2)) = x1 + x2
POL(c28(x1, x2)) = x1 + x2
POL(c28(x1, x2, x3)) = x1 + x2 + x3
POL(c29(x1, x2)) = x1 + x2
POL(c29(x1, x2, x3)) = x1 + x2 + x3
POL(c3(x1)) = x1
POL(c30(x1, x2)) = x1 + x2
POL(c30(x1, x2, x3)) = x1 + x2 + x3
POL(c31(x1)) = x1
POL(c31(x1, x2)) = x1 + x2
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(c35(x1)) = x1
POL(c35(x1, x2)) = x1 + x2
POL(c36(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(c9(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = [4]
POL(dbl(x1)) = [4]
POL(dbls(x1)) = [4]
POL(from(x1)) = [4]
POL(indx(x1, x2)) = [4]
POL(mark(x1)) = [4]
POL(nil) = [4]
POL(ok(x1)) = x1
POL(proper(x1)) = 0
POL(s(x1)) = 0
POL(sel(x1, x2)) = [4]
(62) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
K tuples:
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, S, CONS, FROM, TOP, PROPER
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28, c29, c29, c30, c30, c31, c31, c35, c35
(63) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
We considered the (Usable) Rules:
active(sel(0, cons(z0, z1))) → mark(z0)
sel(z0, mark(z1)) → mark(sel(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(indx(nil, z0)) → mark(nil)
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
active(dbls(nil)) → mark(nil)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbl(ok(z0)) → ok(dbl(z0))
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbls(z0)) → dbls(active(z0))
s(ok(z0)) → ok(s(z0))
active(sel(z0, z1)) → sel(z0, active(z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
dbl(mark(z0)) → mark(dbl(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
from(ok(z0)) → ok(from(z0))
active(dbl(0)) → mark(0)
And the Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = [2]
POL(ACTIVE(x1)) = 0
POL(CONS(x1, x2)) = 0
POL(DBL(x1)) = 0
POL(DBLS(x1)) = 0
POL(FROM(x1)) = 0
POL(INDX(x1, x2)) = 0
POL(PROPER(x1)) = 0
POL(S(x1)) = 0
POL(SEL(x1, x2)) = 0
POL(TOP(x1)) = [4]x1
POL(active(x1)) = x1
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c10(x1)) = x1
POL(c10(x1, x2)) = x1 + x2
POL(c11(x1, x2)) = x1 + x2
POL(c12(x1, x2)) = x1 + x2
POL(c13(x1)) = x1
POL(c13(x1, x2)) = x1 + x2
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c2(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c23(x1, x2)) = x1 + x2
POL(c25(x1)) = x1
POL(c25(x1, x2)) = x1 + x2
POL(c26(x1)) = x1
POL(c26(x1, x2)) = x1 + x2
POL(c28(x1, x2)) = x1 + x2
POL(c28(x1, x2, x3)) = x1 + x2 + x3
POL(c29(x1, x2)) = x1 + x2
POL(c29(x1, x2, x3)) = x1 + x2 + x3
POL(c3(x1)) = x1
POL(c30(x1, x2)) = x1 + x2
POL(c30(x1, x2, x3)) = x1 + x2 + x3
POL(c31(x1)) = x1
POL(c31(x1, x2)) = x1 + x2
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(c35(x1)) = x1
POL(c35(x1, x2)) = x1 + x2
POL(c36(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(c9(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = 0
POL(dbl(x1)) = [2]
POL(dbls(x1)) = [2]
POL(from(x1)) = [2]
POL(indx(x1, x2)) = [2]
POL(mark(x1)) = [2]
POL(nil) = [2]
POL(ok(x1)) = x1
POL(proper(x1)) = 0
POL(s(x1)) = [2]
POL(sel(x1, x2)) = [2]
(64) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
K tuples:
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, S, CONS, FROM, TOP, PROPER
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28, c29, c29, c30, c30, c31, c31, c35, c35
(65) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
We considered the (Usable) Rules:
active(sel(0, cons(z0, z1))) → mark(z0)
sel(z0, mark(z1)) → mark(sel(z0, z1))
active(dbl(z0)) → dbl(active(z0))
active(indx(nil, z0)) → mark(nil)
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
active(dbls(nil)) → mark(nil)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbl(ok(z0)) → ok(dbl(z0))
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
active(indx(z0, z1)) → indx(active(z0), z1)
active(dbls(z0)) → dbls(active(z0))
s(ok(z0)) → ok(s(z0))
active(sel(z0, z1)) → sel(z0, active(z1))
active(sel(z0, z1)) → sel(active(z0), z1)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
dbl(mark(z0)) → mark(dbl(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
from(ok(z0)) → ok(from(z0))
active(dbl(0)) → mark(0)
And the Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(CONS(x1, x2)) = 0
POL(DBL(x1)) = 0
POL(DBLS(x1)) = 0
POL(FROM(x1)) = 0
POL(INDX(x1, x2)) = 0
POL(PROPER(x1)) = 0
POL(S(x1)) = 0
POL(SEL(x1, x2)) = 0
POL(TOP(x1)) = x1
POL(active(x1)) = x1
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c10(x1)) = x1
POL(c10(x1, x2)) = x1 + x2
POL(c11(x1, x2)) = x1 + x2
POL(c12(x1, x2)) = x1 + x2
POL(c13(x1)) = x1
POL(c13(x1, x2)) = x1 + x2
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c2(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c23(x1, x2)) = x1 + x2
POL(c25(x1)) = x1
POL(c25(x1, x2)) = x1 + x2
POL(c26(x1)) = x1
POL(c26(x1, x2)) = x1 + x2
POL(c28(x1, x2)) = x1 + x2
POL(c28(x1, x2, x3)) = x1 + x2 + x3
POL(c29(x1, x2)) = x1 + x2
POL(c29(x1, x2, x3)) = x1 + x2 + x3
POL(c3(x1)) = x1
POL(c30(x1, x2)) = x1 + x2
POL(c30(x1, x2, x3)) = x1 + x2 + x3
POL(c31(x1)) = x1
POL(c31(x1, x2)) = x1 + x2
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(c35(x1)) = x1
POL(c35(x1, x2)) = x1 + x2
POL(c36(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(c9(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = [1]
POL(dbl(x1)) = [1]
POL(dbls(x1)) = [1]
POL(from(x1)) = [1]
POL(indx(x1, x2)) = [1]
POL(mark(x1)) = [1]
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = 0
POL(s(x1)) = 0
POL(sel(x1, x2)) = [1]
(66) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
K tuples:
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, S, CONS, FROM, TOP, PROPER
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34, c36, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28, c29, c29, c30, c30, c31, c31, c35, c35
(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
TOP(
ok(
z0)) →
c36(
TOP(
active(
z0)),
ACTIVE(
z0)) by
TOP(ok(dbl(0))) → c36(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c36(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(dbls(nil))) → c36(TOP(mark(nil)), ACTIVE(dbls(nil)))
TOP(ok(dbls(cons(z0, z1)))) → c36(TOP(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
TOP(ok(sel(0, cons(z0, z1)))) → c36(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c36(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(indx(nil, z0))) → c36(TOP(mark(nil)), ACTIVE(indx(nil, z0)))
TOP(ok(indx(cons(z0, z1), z2))) → c36(TOP(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
TOP(ok(from(z0))) → c36(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(dbl(z0))) → c36(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(dbls(z0))) → c36(TOP(dbls(active(z0))), ACTIVE(dbls(z0)))
TOP(ok(sel(z0, z1))) → c36(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c36(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(indx(z0, z1))) → c36(TOP(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
(68) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
TOP(ok(dbl(0))) → c36(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c36(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(dbls(nil))) → c36(TOP(mark(nil)), ACTIVE(dbls(nil)))
TOP(ok(dbls(cons(z0, z1)))) → c36(TOP(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
TOP(ok(sel(0, cons(z0, z1)))) → c36(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c36(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(indx(nil, z0))) → c36(TOP(mark(nil)), ACTIVE(indx(nil, z0)))
TOP(ok(indx(cons(z0, z1), z2))) → c36(TOP(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
TOP(ok(from(z0))) → c36(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(dbl(z0))) → c36(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(dbls(z0))) → c36(TOP(dbls(active(z0))), ACTIVE(dbls(z0)))
TOP(ok(sel(z0, z1))) → c36(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c36(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(indx(z0, z1))) → c36(TOP(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(ok(dbl(0))) → c36(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c36(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(dbls(nil))) → c36(TOP(mark(nil)), ACTIVE(dbls(nil)))
TOP(ok(dbls(cons(z0, z1)))) → c36(TOP(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
TOP(ok(sel(0, cons(z0, z1)))) → c36(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c36(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(indx(nil, z0))) → c36(TOP(mark(nil)), ACTIVE(indx(nil, z0)))
TOP(ok(indx(cons(z0, z1), z2))) → c36(TOP(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
TOP(ok(from(z0))) → c36(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(dbl(z0))) → c36(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(dbls(z0))) → c36(TOP(dbls(active(z0))), ACTIVE(dbls(z0)))
TOP(ok(sel(z0, z1))) → c36(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c36(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(indx(z0, z1))) → c36(TOP(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
K tuples:
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, S, CONS, FROM, PROPER, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28, c29, c29, c30, c30, c31, c31, c35, c35, c36
(69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing nodes:
TOP(ok(indx(nil, z0))) → c36(TOP(mark(nil)), ACTIVE(indx(nil, z0)))
TOP(ok(dbl(0))) → c36(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
TOP(ok(dbls(nil))) → c36(TOP(mark(nil)), ACTIVE(dbls(nil)))
TOP(mark(0)) → c35(TOP(ok(0)))
(70) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(ok(dbl(s(z0)))) → c36(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(dbls(cons(z0, z1)))) → c36(TOP(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
TOP(ok(sel(0, cons(z0, z1)))) → c36(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c36(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(indx(cons(z0, z1), z2))) → c36(TOP(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
TOP(ok(from(z0))) → c36(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(dbl(z0))) → c36(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(dbls(z0))) → c36(TOP(dbls(active(z0))), ACTIVE(dbls(z0)))
TOP(ok(sel(z0, z1))) → c36(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c36(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(indx(z0, z1))) → c36(TOP(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(ok(dbl(s(z0)))) → c36(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(dbls(cons(z0, z1)))) → c36(TOP(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
TOP(ok(sel(0, cons(z0, z1)))) → c36(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c36(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(indx(cons(z0, z1), z2))) → c36(TOP(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
TOP(ok(from(z0))) → c36(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(dbl(z0))) → c36(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(dbls(z0))) → c36(TOP(dbls(active(z0))), ACTIVE(dbls(z0)))
TOP(ok(sel(z0, z1))) → c36(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c36(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(indx(z0, z1))) → c36(TOP(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
K tuples:
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
ACTIVE, DBL, DBLS, SEL, INDX, S, CONS, FROM, PROPER, TOP
Compound Symbols:
c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34, c8, c, c9, c9, c1, c10, c10, c2, c13, c13, c3, c23, c23, c25, c25, c26, c26, c28, c28, c29, c29, c30, c30, c31, c31, c35, c36
(71) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c11(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c12(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c8(S(z0))
ACTIVE(dbl(s(z0))) → c(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(CONS(dbl(z0), dbls(z1)))
ACTIVE(dbls(cons(z0, z1))) → c(DBL(z0))
ACTIVE(dbls(cons(z0, z1))) → c(DBLS(z1))
ACTIVE(indx(cons(z0, z1), z2)) → c(CONS(sel(z0, z2), indx(z1, z2)))
ACTIVE(indx(cons(z0, z1), z2)) → c(SEL(z0, z2))
ACTIVE(indx(cons(z0, z1), z2)) → c(INDX(z1, z2))
ACTIVE(dbl(sel(0, cons(z0, z1)))) → c9(DBL(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbl(sel(s(z0), cons(z1, z2)))) → c9(DBL(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbl(dbl(z0))) → c9(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(dbls(z0))) → c9(DBL(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(sel(z0, z1))) → c9(DBL(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbl(indx(z0, z1))) → c9(DBL(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbl(0))) → c9(DBL(mark(0)))
ACTIVE(dbl(dbl(s(z0)))) → c1(DBL(mark(s(s(dbl(z0))))))
ACTIVE(dbl(dbl(s(z0)))) → c1(ACTIVE(dbl(s(z0))))
ACTIVE(dbl(from(z0))) → c1(DBL(mark(cons(z0, from(s(z0))))))
ACTIVE(dbl(from(z0))) → c1(ACTIVE(from(z0)))
ACTIVE(dbls(sel(0, cons(z0, z1)))) → c10(DBLS(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(dbls(sel(s(z0), cons(z1, z2)))) → c10(DBLS(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(dbls(dbl(z0))) → c10(DBLS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbls(dbls(z0))) → c10(DBLS(dbls(active(z0))), ACTIVE(dbls(z0)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(sel(z0, z1))) → c10(DBLS(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(dbls(indx(z0, z1))) → c10(DBLS(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(dbls(nil))) → c9(DBL(mark(nil)))
ACTIVE(dbls(dbl(0))) → c10(DBLS(mark(0)))
ACTIVE(dbls(dbls(nil))) → c10(DBLS(mark(nil)))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(DBL(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbl(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(dbl(s(z0)))) → c2(DBLS(mark(s(s(dbl(z0))))))
ACTIVE(dbls(dbl(s(z0)))) → c2(ACTIVE(dbl(s(z0))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(DBLS(mark(cons(dbl(z0), dbls(z1)))))
ACTIVE(dbls(dbls(cons(z0, z1)))) → c2(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(dbls(from(z0))) → c2(DBLS(mark(cons(z0, from(s(z0))))))
ACTIVE(dbls(from(z0))) → c2(ACTIVE(from(z0)))
ACTIVE(indx(sel(0, cons(z0, z1)), x1)) → c13(INDX(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(indx(sel(s(z0), cons(z1, z2)), x1)) → c13(INDX(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(indx(dbl(z0), x1)) → c13(INDX(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(indx(dbls(z0), x1)) → c13(INDX(dbls(active(z0)), x1), ACTIVE(dbls(z0)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(sel(z0, z1), x1)) → c13(INDX(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(indx(indx(z0, z1), x1)) → c13(INDX(indx(active(z0), z1), x1), ACTIVE(indx(z0, z1)))
ACTIVE(dbl(indx(nil, z0))) → c9(DBL(mark(nil)))
ACTIVE(dbls(indx(nil, z0))) → c10(DBLS(mark(nil)))
ACTIVE(indx(dbl(0), x1)) → c13(INDX(mark(0), x1))
ACTIVE(indx(dbls(nil), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(indx(indx(nil, z0), x1)) → c13(INDX(mark(nil), x1))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(DBL(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbl(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(DBLS(mark(cons(sel(z0, z2), indx(z1, z2)))))
ACTIVE(dbls(indx(cons(z0, z1), z2))) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(INDX(mark(s(s(dbl(z0)))), x1))
ACTIVE(indx(dbl(s(z0)), x1)) → c3(ACTIVE(dbl(s(z0))))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(INDX(mark(cons(dbl(z0), dbls(z1))), x1))
ACTIVE(indx(dbls(cons(z0, z1)), x1)) → c3(ACTIVE(dbls(cons(z0, z1))))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(INDX(mark(cons(sel(z0, z2), indx(z1, z2))), x1))
ACTIVE(indx(indx(cons(z0, z1), z2), x1)) → c3(ACTIVE(indx(cons(z0, z1), z2)))
ACTIVE(indx(from(z0), x1)) → c3(INDX(mark(cons(z0, from(s(z0)))), x1))
ACTIVE(indx(from(z0), x1)) → c3(ACTIVE(from(z0)))
PROPER(dbl(dbl(z0))) → c23(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(s(z0))) → c23(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(dbls(z0))) → c23(DBL(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbl(cons(z0, z1))) → c23(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(sel(z0, z1))) → c23(DBL(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbl(indx(z0, z1))) → c23(DBL(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbl(from(z0))) → c23(DBL(from(proper(z0))), PROPER(from(z0)))
PROPER(dbl(0)) → c23(DBL(ok(0)))
PROPER(dbl(nil)) → c23(DBL(ok(nil)))
PROPER(s(dbl(z0))) → c25(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(dbls(z0))) → c25(S(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(s(cons(z0, z1))) → c25(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(sel(z0, z1))) → c25(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(indx(z0, z1))) → c25(S(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(s(from(z0))) → c25(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(s(nil)) → c25(S(ok(nil)))
PROPER(dbls(dbl(z0))) → c26(DBLS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbls(s(z0))) → c26(DBLS(s(proper(z0))), PROPER(s(z0)))
PROPER(dbls(dbls(z0))) → c26(DBLS(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(dbls(cons(z0, z1))) → c26(DBLS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbls(sel(z0, z1))) → c26(DBLS(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(dbls(indx(z0, z1))) → c26(DBLS(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(dbls(from(z0))) → c26(DBLS(from(proper(z0))), PROPER(from(z0)))
PROPER(dbls(0)) → c26(DBLS(ok(0)))
PROPER(dbls(nil)) → c26(DBLS(ok(nil)))
PROPER(cons(x0, dbl(z0))) → c28(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, s(z0))) → c28(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, dbls(z0))) → c28(CONS(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(cons(x0, cons(z0, z1))) → c28(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, sel(z0, z1))) → c28(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, indx(z0, z1))) → c28(CONS(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(cons(x0, from(z0))) → c28(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(dbl(z0), x1)) → c28(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c28(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(dbls(z0), x1)) → c28(CONS(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c28(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c28(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(indx(z0, z1), x1)) → c28(CONS(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c28(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c28(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c28(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c28(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c28(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(sel(x0, dbl(z0))) → c29(SEL(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(sel(x0, s(z0))) → c29(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, dbls(z0))) → c29(SEL(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(sel(x0, cons(z0, z1))) → c29(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, sel(z0, z1))) → c29(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, indx(z0, z1))) → c29(SEL(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(sel(x0, from(z0))) → c29(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(sel(dbl(z0), x1)) → c29(SEL(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c29(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(dbls(z0), x1)) → c29(SEL(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c29(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c29(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(indx(z0, z1), x1)) → c29(SEL(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(sel(from(z0), x1)) → c29(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(sel(x0, 0)) → c29(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(x0, nil)) → c29(SEL(proper(x0), ok(nil)), PROPER(x0))
PROPER(sel(0, x1)) → c29(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(sel(nil, x1)) → c29(SEL(ok(nil), proper(x1)), PROPER(x1))
PROPER(indx(x0, dbl(z0))) → c30(INDX(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(indx(x0, s(z0))) → c30(INDX(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(indx(x0, dbls(z0))) → c30(INDX(proper(x0), dbls(proper(z0))), PROPER(x0), PROPER(dbls(z0)))
PROPER(indx(x0, cons(z0, z1))) → c30(INDX(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(indx(x0, sel(z0, z1))) → c30(INDX(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(indx(x0, indx(z0, z1))) → c30(INDX(proper(x0), indx(proper(z0), proper(z1))), PROPER(x0), PROPER(indx(z0, z1)))
PROPER(indx(x0, from(z0))) → c30(INDX(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(indx(dbl(z0), x1)) → c30(INDX(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(indx(s(z0), x1)) → c30(INDX(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(indx(dbls(z0), x1)) → c30(INDX(dbls(proper(z0)), proper(x1)), PROPER(dbls(z0)), PROPER(x1))
PROPER(indx(cons(z0, z1), x1)) → c30(INDX(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(indx(sel(z0, z1), x1)) → c30(INDX(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(indx(indx(z0, z1), x1)) → c30(INDX(indx(proper(z0), proper(z1)), proper(x1)), PROPER(indx(z0, z1)), PROPER(x1))
PROPER(indx(from(z0), x1)) → c30(INDX(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(indx(x0, 0)) → c30(INDX(proper(x0), ok(0)), PROPER(x0))
PROPER(indx(x0, nil)) → c30(INDX(proper(x0), ok(nil)), PROPER(x0))
PROPER(indx(0, x1)) → c30(INDX(ok(0), proper(x1)), PROPER(x1))
PROPER(indx(nil, x1)) → c30(INDX(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(dbl(z0))) → c31(FROM(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(from(s(z0))) → c31(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(dbls(z0))) → c31(FROM(dbls(proper(z0))), PROPER(dbls(z0)))
PROPER(from(cons(z0, z1))) → c31(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(sel(z0, z1))) → c31(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(from(indx(z0, z1))) → c31(FROM(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
PROPER(from(from(z0))) → c31(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c31(FROM(ok(0)))
PROPER(from(nil)) → c31(FROM(ok(nil)))
TOP(mark(dbl(z0))) → c35(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(dbls(z0))) → c35(TOP(dbls(proper(z0))), PROPER(dbls(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(sel(z0, z1))) → c35(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(indx(z0, z1))) → c35(TOP(indx(proper(z0), proper(z1))), PROPER(indx(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(ok(dbl(s(z0)))) → c36(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(dbls(cons(z0, z1)))) → c36(TOP(mark(cons(dbl(z0), dbls(z1)))), ACTIVE(dbls(cons(z0, z1))))
TOP(ok(sel(0, cons(z0, z1)))) → c36(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c36(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(indx(cons(z0, z1), z2))) → c36(TOP(mark(cons(sel(z0, z2), indx(z1, z2)))), ACTIVE(indx(cons(z0, z1), z2)))
TOP(ok(from(z0))) → c36(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(dbl(z0))) → c36(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(dbls(z0))) → c36(TOP(dbls(active(z0))), ACTIVE(dbls(z0)))
TOP(ok(sel(z0, z1))) → c36(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c36(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(indx(z0, z1))) → c36(TOP(indx(active(z0), z1)), ACTIVE(indx(z0, z1)))
(72) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
S tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
K tuples:none
Defined Rule Symbols:
active, s, dbl, cons, dbls, sel, indx, proper, from
Defined Pair Symbols:
DBL, DBLS, SEL, INDX, S, CONS, FROM
Compound Symbols:
c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34
(73) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(dbl(z0)) → dbl(active(z0))
active(dbls(z0)) → dbls(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(indx(z0, z1)) → indx(active(z0), z1)
s(ok(z0)) → ok(s(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(dbls(z0)) → dbls(proper(z0))
proper(nil) → ok(nil)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(indx(z0, z1)) → indx(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
from(ok(z0)) → ok(from(z0))
(74) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
S tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
DBL, DBLS, SEL, INDX, S, CONS, FROM
Compound Symbols:
c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34
(75) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(CONS(x1, x2)) = x2
POL(DBL(x1)) = 0
POL(DBLS(x1)) = 0
POL(FROM(x1)) = 0
POL(INDX(x1, x2)) = 0
POL(S(x1)) = 0
POL(SEL(x1, x2)) = [4]x1
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(mark(x1)) = x1
POL(ok(x1)) = [1] + x1
(76) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
S tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
FROM(ok(z0)) → c34(FROM(z0))
K tuples:
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
DBL, DBLS, SEL, INDX, S, CONS, FROM
Compound Symbols:
c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34
(77) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
DBL(ok(z0)) → c15(DBL(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
FROM(ok(z0)) → c34(FROM(z0))
We considered the (Usable) Rules:none
And the Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(CONS(x1, x2)) = [2]x1 + [2]x2
POL(DBL(x1)) = [3]x1
POL(DBLS(x1)) = [4]x1
POL(FROM(x1)) = [4]x1
POL(INDX(x1, x2)) = [2]x2
POL(S(x1)) = [2]x1
POL(SEL(x1, x2)) = [2]x2
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(mark(x1)) = x1
POL(ok(x1)) = [2] + x1
(78) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
S tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
K tuples:
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
FROM(ok(z0)) → c34(FROM(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
DBL, DBLS, SEL, INDX, S, CONS, FROM
Compound Symbols:
c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34
(79) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
DBLS(mark(z0)) → c16(DBLS(z0))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(CONS(x1, x2)) = 0
POL(DBL(x1)) = 0
POL(DBLS(x1)) = [4]x1
POL(FROM(x1)) = 0
POL(INDX(x1, x2)) = [2]x2
POL(S(x1)) = [2]x1
POL(SEL(x1, x2)) = [2]x2
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(mark(x1)) = [4] + x1
POL(ok(x1)) = x1
(80) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
S tuples:
DBL(mark(z0)) → c14(DBL(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
K tuples:
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
FROM(ok(z0)) → c34(FROM(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
DBL, DBLS, SEL, INDX, S, CONS, FROM
Compound Symbols:
c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34
(81) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
SEL(mark(z0), z1) → c18(SEL(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(CONS(x1, x2)) = [5]x1
POL(DBL(x1)) = 0
POL(DBLS(x1)) = 0
POL(FROM(x1)) = [3]x1
POL(INDX(x1, x2)) = [4]x2
POL(S(x1)) = [4]x1
POL(SEL(x1, x2)) = [2]x1 + [2]x2
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(mark(x1)) = [4] + x1
POL(ok(x1)) = x1
(82) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
S tuples:
DBL(mark(z0)) → c14(DBL(z0))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
K tuples:
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
FROM(ok(z0)) → c34(FROM(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
DBL, DBLS, SEL, INDX, S, CONS, FROM
Compound Symbols:
c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34
(83) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
DBL(mark(z0)) → c14(DBL(z0))
We considered the (Usable) Rules:none
And the Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(CONS(x1, x2)) = [2]x1 + [2]x2
POL(DBL(x1)) = [4]x1
POL(DBLS(x1)) = 0
POL(FROM(x1)) = 0
POL(INDX(x1, x2)) = 0
POL(S(x1)) = [2]x1
POL(SEL(x1, x2)) = x1 + [2]x2
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(mark(x1)) = [4] + x1
POL(ok(x1)) = x1
(84) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
S tuples:
INDX(mark(z0), z1) → c21(INDX(z0, z1))
K tuples:
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
FROM(ok(z0)) → c34(FROM(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
DBL(mark(z0)) → c14(DBL(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
DBL, DBLS, SEL, INDX, S, CONS, FROM
Compound Symbols:
c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34
(85) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
INDX(mark(z0), z1) → c21(INDX(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(CONS(x1, x2)) = [2]x1
POL(DBL(x1)) = x1
POL(DBLS(x1)) = 0
POL(FROM(x1)) = [2]x1
POL(INDX(x1, x2)) = [2]x1 + [4]x2
POL(S(x1)) = 0
POL(SEL(x1, x2)) = 0
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(mark(x1)) = [1] + x1
POL(ok(x1)) = x1
(86) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
DBL(mark(z0)) → c14(DBL(z0))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
FROM(ok(z0)) → c34(FROM(z0))
S tuples:none
K tuples:
SEL(ok(z0), ok(z1)) → c20(SEL(z0, z1))
CONS(ok(z0), ok(z1)) → c33(CONS(z0, z1))
DBL(ok(z0)) → c15(DBL(z0))
DBLS(ok(z0)) → c17(DBLS(z0))
INDX(ok(z0), ok(z1)) → c22(INDX(z0, z1))
S(ok(z0)) → c32(S(z0))
FROM(ok(z0)) → c34(FROM(z0))
DBLS(mark(z0)) → c16(DBLS(z0))
SEL(z0, mark(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c18(SEL(z0, z1))
DBL(mark(z0)) → c14(DBL(z0))
INDX(mark(z0), z1) → c21(INDX(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
DBL, DBLS, SEL, INDX, S, CONS, FROM
Compound Symbols:
c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34
(87) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(88) BOUNDS(O(1), O(1))