(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(fib(X)) → fib(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(fib1(X1, X2)) → fib1(active(X1), X2)
active(fib1(X1, X2)) → fib1(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fib(mark(X)) → mark(fib(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fib(X)) → fib(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) → fib1(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
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(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(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))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)), S(0), S(0))
ACTIVE(fib1(z0, z1)) → c1(CONS(z0, fib1(z1, add(z0, z1))), FIB1(z1, add(z0, z1)), ADD(z0, z1))
ACTIVE(add(0, z0)) → c2
ACTIVE(add(s(z0), z1)) → c3(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(sel(0, cons(z0, z1))) → c4
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(0) → c34
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
S tuples:
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)), S(0), S(0))
ACTIVE(fib1(z0, z1)) → c1(CONS(z0, fib1(z1, add(z0, z1))), FIB1(z1, add(z0, z1)), ADD(z0, z1))
ACTIVE(add(0, z0)) → c2
ACTIVE(add(s(z0), z1)) → c3(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(sel(0, cons(z0, z1))) → c4
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(0) → c34
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:
active, fib, sel, fib1, s, cons, add, proper, top
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, 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, c37, c38
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
PROPER(0) → c34
ACTIVE(add(0, z0)) → c2
ACTIVE(sel(0, cons(z0, z1))) → c4
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(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))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)), S(0), S(0))
ACTIVE(fib1(z0, z1)) → c1(CONS(z0, fib1(z1, add(z0, z1))), FIB1(z1, add(z0, z1)), ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
S tuples:
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)), S(0), S(0))
ACTIVE(fib1(z0, z1)) → c1(CONS(z0, fib1(z1, add(z0, z1))), FIB1(z1, add(z0, z1)), ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:
active, fib, sel, fib1, s, cons, add, proper, top
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c, c1, c3, 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, c35, c36, c37, c38
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(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))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
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(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
K tuples:none
Defined Rule Symbols:
active, fib, sel, fib1, s, cons, add, proper, top
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
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, c35, c36, c37, c38, c, c1, c3
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
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, c35, c36, c37, c38, c, c1, c3
(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
fib(
z0)) →
c6(
FIB(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, 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, c35, c36, c37, c38, c, c1, c3, c6
(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
s(
z0)) →
c11(
S(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c, c1, c3, c6, c11
(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
cons(
z0,
z1)) →
c12(
CONS(
active(
z0),
z1),
ACTIVE(
z0)) by
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c, c1, c3, c6, c11, c12
(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
fib(
z0)) →
c30(
FIB(
proper(
z0)),
PROPER(
z0)) by
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(0)) → c30(FIB(ok(0)), PROPER(0))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(0)) → c30(FIB(ok(0)), PROPER(0))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(0)) → c30(FIB(ok(0)), PROPER(0))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c31, c32, c33, c35, c36, c37, c38, c, c1, c3, c6, c11, c12, c30
(17) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c31, c32, c33, c35, c36, c37, c38, c, c1, c3, c6, c11, c12, c30, c30
(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
sel(
z0,
z1)) →
c31(
SEL(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
(20) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c32, c33, c35, c36, c37, c38, c, c1, c3, c6, c11, c12, c30, c30, c31
(21) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(22) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c32, c33, c35, c36, c37, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31
(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
fib1(
z0,
z1)) →
c32(
FIB1(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
(24) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c33, c35, c36, c37, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32
(25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(26) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c33, c35, c36, c37, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32, c32
(27) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
s(
z0)) →
c33(
S(
proper(
z0)),
PROPER(
z0)) by
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(0)) → c33(S(ok(0)), PROPER(0))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
(28) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(0)) → c33(S(ok(0)), PROPER(0))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(0)) → c33(S(ok(0)), PROPER(0))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c35, c36, c37, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32, c32, c33
(29) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(30) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c35, c36, c37, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32, c32, c33, c33
(31) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
cons(
z0,
z1)) →
c35(
CONS(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
(32) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c36, c37, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32, c32, c33, c33, c35
(33) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(34) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
PROPER(add(z0, z1)) → c36(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c36, c37, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32, c32, c33, c33, c35, c35
(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
add(
z0,
z1)) →
c36(
ADD(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
(36) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, TOP, PROPER
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c37, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32, c32, c33, c33, c35, c35, c36
(37) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(38) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, TOP, PROPER
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c37, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32, c32, c33, c33, c35, c35, c36, c36
(39) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
TOP(
mark(
z0)) →
c37(
TOP(
proper(
z0)),
PROPER(
z0)) by
TOP(mark(fib(z0))) → c37(TOP(fib(proper(z0))), PROPER(fib(z0)))
TOP(mark(sel(z0, z1))) → c37(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(fib1(z0, z1))) → c37(TOP(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(0)) → c37(TOP(ok(0)), PROPER(0))
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
(40) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
TOP(mark(fib(z0))) → c37(TOP(fib(proper(z0))), PROPER(fib(z0)))
TOP(mark(sel(z0, z1))) → c37(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(fib1(z0, z1))) → c37(TOP(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(0)) → c37(TOP(ok(0)), PROPER(0))
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
TOP(mark(fib(z0))) → c37(TOP(fib(proper(z0))), PROPER(fib(z0)))
TOP(mark(sel(z0, z1))) → c37(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(fib1(z0, z1))) → c37(TOP(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(0)) → c37(TOP(ok(0)), PROPER(0))
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, TOP, PROPER
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32, c32, c33, c33, c35, c35, c36, c36, c37
(41) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(42) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
TOP(mark(fib(z0))) → c37(TOP(fib(proper(z0))), PROPER(fib(z0)))
TOP(mark(sel(z0, z1))) → c37(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(fib1(z0, z1))) → c37(TOP(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(0)) → c37(TOP(ok(0)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
TOP(mark(fib(z0))) → c37(TOP(fib(proper(z0))), PROPER(fib(z0)))
TOP(mark(sel(z0, z1))) → c37(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(fib1(z0, z1))) → c37(TOP(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(0)) → c37(TOP(ok(0)))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, TOP, PROPER
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32, c32, c33, c33, c35, c35, c36, c36, c37, c37
(43) 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)) → c37(TOP(ok(0)))
We considered the (Usable) Rules:
active(cons(z0, z1)) → cons(active(z0), z1)
add(mark(z0), z1) → mark(add(z0, z1))
active(sel(0, cons(z0, z1))) → mark(z0)
sel(z0, mark(z1)) → mark(sel(z0, z1))
s(mark(z0)) → mark(s(z0))
active(s(z0)) → s(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(add(z0, z1)) → add(active(z0), z1)
add(z0, mark(z1)) → mark(add(z0, z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
s(ok(z0)) → ok(s(z0))
active(add(0, z0)) → mark(z0)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib(ok(z0)) → ok(fib(z0))
fib(mark(z0)) → mark(fib(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
active(add(z0, z1)) → add(z0, active(z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
And the Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
TOP(mark(fib(z0))) → c37(TOP(fib(proper(z0))), PROPER(fib(z0)))
TOP(mark(sel(z0, z1))) → c37(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(fib1(z0, z1))) → c37(TOP(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(0)) → c37(TOP(ok(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(ADD(x1, x2)) = 0
POL(CONS(x1, x2)) = 0
POL(FIB(x1)) = 0
POL(FIB1(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(add(x1, x2)) = [1]
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c10(x1, x2)) = x1 + x2
POL(c11(x1, x2)) = x1 + x2
POL(c12(x1, x2)) = x1 + x2
POL(c13(x1, x2)) = x1 + x2
POL(c14(x1, x2)) = x1 + x2
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(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c3(x1)) = x1
POL(c30(x1)) = x1
POL(c30(x1, x2)) = x1 + x2
POL(c31(x1, x2)) = x1 + x2
POL(c31(x1, x2, x3)) = x1 + x2 + x3
POL(c32(x1, x2)) = x1 + x2
POL(c32(x1, x2, x3)) = x1 + x2 + x3
POL(c33(x1)) = x1
POL(c33(x1, x2)) = x1 + x2
POL(c35(x1, x2)) = x1 + x2
POL(c35(x1, x2, x3)) = x1 + x2 + x3
POL(c36(x1, x2)) = x1 + x2
POL(c36(x1, x2, x3)) = x1 + x2 + x3
POL(c37(x1)) = x1
POL(c37(x1, x2)) = x1 + x2
POL(c38(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c6(x1, x2)) = x1 + x2
POL(c7(x1, x2)) = x1 + x2
POL(c8(x1, x2)) = x1 + x2
POL(c9(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = [1]
POL(fib(x1)) = [1]
POL(fib1(x1, x2)) = [1]
POL(mark(x1)) = [1]
POL(ok(x1)) = x1
POL(proper(x1)) = 0
POL(s(x1)) = [1]
POL(sel(x1, x2)) = [1]
(44) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
TOP(mark(fib(z0))) → c37(TOP(fib(proper(z0))), PROPER(fib(z0)))
TOP(mark(sel(z0, z1))) → c37(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(fib1(z0, z1))) → c37(TOP(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(0)) → c37(TOP(ok(0)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
TOP(mark(fib(z0))) → c37(TOP(fib(proper(z0))), PROPER(fib(z0)))
TOP(mark(sel(z0, z1))) → c37(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(fib1(z0, z1))) → c37(TOP(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
K tuples:
TOP(mark(0)) → c37(TOP(ok(0)))
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, TOP, PROPER
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32, c32, c33, c33, c35, c35, c36, c36, c37, c37
(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
TOP(
ok(
z0)) →
c38(
TOP(
active(
z0)),
ACTIVE(
z0)) by
TOP(ok(fib(z0))) → c38(TOP(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
TOP(ok(fib1(z0, z1))) → c38(TOP(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
TOP(ok(add(0, z0))) → c38(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c38(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(sel(0, cons(z0, z1)))) → c38(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c38(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(fib(z0))) → c38(TOP(fib(active(z0))), ACTIVE(fib(z0)))
TOP(ok(sel(z0, z1))) → c38(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c38(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(fib1(z0, z1))) → c38(TOP(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
TOP(ok(fib1(z0, z1))) → c38(TOP(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
TOP(ok(s(z0))) → c38(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons(z0, z1))) → c38(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(add(z0, z1))) → c38(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c38(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
(46) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
TOP(mark(fib(z0))) → c37(TOP(fib(proper(z0))), PROPER(fib(z0)))
TOP(mark(sel(z0, z1))) → c37(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(fib1(z0, z1))) → c37(TOP(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(0)) → c37(TOP(ok(0)))
TOP(ok(fib(z0))) → c38(TOP(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
TOP(ok(fib1(z0, z1))) → c38(TOP(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
TOP(ok(add(0, z0))) → c38(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c38(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(sel(0, cons(z0, z1)))) → c38(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c38(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(fib(z0))) → c38(TOP(fib(active(z0))), ACTIVE(fib(z0)))
TOP(ok(sel(z0, z1))) → c38(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c38(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(fib1(z0, z1))) → c38(TOP(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
TOP(ok(fib1(z0, z1))) → c38(TOP(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
TOP(ok(s(z0))) → c38(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons(z0, z1))) → c38(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(add(z0, z1))) → c38(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c38(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
TOP(mark(fib(z0))) → c37(TOP(fib(proper(z0))), PROPER(fib(z0)))
TOP(mark(sel(z0, z1))) → c37(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(fib1(z0, z1))) → c37(TOP(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(ok(fib(z0))) → c38(TOP(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
TOP(ok(fib1(z0, z1))) → c38(TOP(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
TOP(ok(add(0, z0))) → c38(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c38(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(sel(0, cons(z0, z1)))) → c38(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c38(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(fib(z0))) → c38(TOP(fib(active(z0))), ACTIVE(fib(z0)))
TOP(ok(sel(z0, z1))) → c38(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c38(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(fib1(z0, z1))) → c38(TOP(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
TOP(ok(fib1(z0, z1))) → c38(TOP(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
TOP(ok(s(z0))) → c38(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons(z0, z1))) → c38(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(add(z0, z1))) → c38(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c38(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
K tuples:
TOP(mark(0)) → c37(TOP(ok(0)))
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32, c32, c33, c33, c35, c35, c36, c36, c37, c37, c38
(47) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
TOP(mark(0)) → c37(TOP(ok(0)))
(48) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
TOP(mark(fib(z0))) → c37(TOP(fib(proper(z0))), PROPER(fib(z0)))
TOP(mark(sel(z0, z1))) → c37(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(fib1(z0, z1))) → c37(TOP(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(ok(fib(z0))) → c38(TOP(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
TOP(ok(fib1(z0, z1))) → c38(TOP(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
TOP(ok(add(0, z0))) → c38(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c38(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(sel(0, cons(z0, z1)))) → c38(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c38(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(fib(z0))) → c38(TOP(fib(active(z0))), ACTIVE(fib(z0)))
TOP(ok(sel(z0, z1))) → c38(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c38(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(fib1(z0, z1))) → c38(TOP(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
TOP(ok(fib1(z0, z1))) → c38(TOP(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
TOP(ok(s(z0))) → c38(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons(z0, z1))) → c38(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(add(z0, z1))) → c38(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c38(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
TOP(mark(fib(z0))) → c37(TOP(fib(proper(z0))), PROPER(fib(z0)))
TOP(mark(sel(z0, z1))) → c37(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(fib1(z0, z1))) → c37(TOP(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(ok(fib(z0))) → c38(TOP(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
TOP(ok(fib1(z0, z1))) → c38(TOP(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
TOP(ok(add(0, z0))) → c38(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c38(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(sel(0, cons(z0, z1)))) → c38(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c38(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(fib(z0))) → c38(TOP(fib(active(z0))), ACTIVE(fib(z0)))
TOP(ok(sel(z0, z1))) → c38(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c38(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(fib1(z0, z1))) → c38(TOP(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
TOP(ok(fib1(z0, z1))) → c38(TOP(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
TOP(ok(s(z0))) → c38(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons(z0, z1))) → c38(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(add(z0, z1))) → c38(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c38(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP
Compound Symbols:
c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32, c32, c33, c33, c35, c35, c36, c36, c37, c38
(49) 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)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib1(z0, z1)) → c1(ADD(z0, z1))
ACTIVE(add(s(z0), z1)) → c3(ADD(z0, z1))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(add(0, z0))) → c6(FIB(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(fib(add(s(z0), z1))) → c6(FIB(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(fib(add(z0, z1))) → c6(FIB(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(add(0, z0))) → c11(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c11(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c11(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(add(0, z0), x1)) → c12(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c12(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c12(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(add(z0, z1))) → c30(FIB(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, add(z0, z1))) → c31(SEL(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(add(z0, z1), x1)) → c31(SEL(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(x0, add(z0, z1))) → c32(FIB1(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(add(z0, z1), x1)) → c32(FIB1(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
PROPER(s(fib(z0))) → c33(S(fib(proper(z0))), PROPER(fib(z0)))
PROPER(s(sel(z0, z1))) → c33(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(fib1(z0, z1))) → c33(S(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(s(s(z0))) → c33(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c33(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(add(z0, z1))) → c33(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(0)) → c33(S(ok(0)))
PROPER(cons(x0, fib(z0))) → c35(CONS(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(cons(x0, sel(z0, z1))) → c35(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(x0, fib1(z0, z1))) → c35(CONS(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(cons(x0, s(z0))) → c35(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c35(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, add(z0, z1))) → c35(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(fib(z0), x1)) → c35(CONS(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c35(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(fib1(z0, z1), x1)) → c35(CONS(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c35(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c35(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c35(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c35(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c35(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(add(x0, fib(z0))) → c36(ADD(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(add(x0, sel(z0, z1))) → c36(ADD(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(add(x0, fib1(z0, z1))) → c36(ADD(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(add(x0, s(z0))) → c36(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, cons(z0, z1))) → c36(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, add(z0, z1))) → c36(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(fib(z0), x1)) → c36(ADD(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(add(sel(z0, z1), x1)) → c36(ADD(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(add(fib1(z0, z1), x1)) → c36(ADD(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(add(s(z0), x1)) → c36(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c36(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c36(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(x0, 0)) → c36(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(0, x1)) → c36(ADD(ok(0), proper(x1)), PROPER(x1))
TOP(mark(fib(z0))) → c37(TOP(fib(proper(z0))), PROPER(fib(z0)))
TOP(mark(sel(z0, z1))) → c37(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(fib1(z0, z1))) → c37(TOP(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(ok(fib(z0))) → c38(TOP(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
TOP(ok(fib1(z0, z1))) → c38(TOP(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
TOP(ok(add(0, z0))) → c38(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c38(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(sel(0, cons(z0, z1)))) → c38(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c38(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(fib(z0))) → c38(TOP(fib(active(z0))), ACTIVE(fib(z0)))
TOP(ok(sel(z0, z1))) → c38(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c38(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
TOP(ok(fib1(z0, z1))) → c38(TOP(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
TOP(ok(fib1(z0, z1))) → c38(TOP(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
TOP(ok(s(z0))) → c38(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons(z0, z1))) → c38(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(add(z0, z1))) → c38(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c38(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
(50) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
Tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
S tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
K tuples:none
Defined Rule Symbols:
active, sel, fib, fib1, s, cons, add, proper
Defined Pair Symbols:
FIB, SEL, FIB1, S, CONS, ADD
Compound Symbols:
c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29
(51) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(add(z0, z1)) → add(proper(z0), proper(z1))
(52) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
S tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
FIB, SEL, FIB1, S, CONS, ADD
Compound Symbols:
c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29
(53) 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) → c17(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = [2]x1
POL(CONS(x1, x2)) = [2]x2
POL(FIB(x1)) = 0
POL(FIB1(x1, x2)) = x1 + [4]x2
POL(S(x1)) = [2]x1
POL(SEL(x1, x2)) = [2]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(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(mark(x1)) = [3] + x1
POL(ok(x1)) = [1] + x1
(54) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
S tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
K tuples:
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
FIB, SEL, FIB1, S, CONS, ADD
Compound Symbols:
c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29
(55) 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.
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
We considered the (Usable) Rules:none
And the Tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = 0
POL(CONS(x1, x2)) = [2]x2
POL(FIB(x1)) = [4]x1
POL(FIB1(x1, x2)) = [4]x2
POL(S(x1)) = 0
POL(SEL(x1, x2)) = 0
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(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(mark(x1)) = [4] + x1
POL(ok(x1)) = [4] + x1
(56) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
S tuples:
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
K tuples:
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
FIB, SEL, FIB1, S, CONS, ADD
Compound Symbols:
c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29
(57) 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.
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = x2
POL(CONS(x1, x2)) = 0
POL(FIB(x1)) = [2]x1
POL(FIB1(x1, x2)) = [3]x1 + [4]x2
POL(S(x1)) = 0
POL(SEL(x1, x2)) = 0
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(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(mark(x1)) = [1] + x1
POL(ok(x1)) = x1
(58) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
S tuples:
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
K tuples:
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
FIB, SEL, FIB1, S, CONS, ADD
Compound Symbols:
c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29
(59) 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.
CONS(mark(z0), z1) → c25(CONS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = 0
POL(CONS(x1, x2)) = [4]x1 + [2]x2
POL(FIB(x1)) = [3]x1
POL(FIB1(x1, x2)) = [4]x1
POL(S(x1)) = 0
POL(SEL(x1, x2)) = 0
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(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(mark(x1)) = [4] + x1
POL(ok(x1)) = x1
(60) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
S tuples:
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
K tuples:
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
FIB, SEL, FIB1, S, CONS, ADD
Compound Symbols:
c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29
(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.
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = [2]x1
POL(CONS(x1, x2)) = [4]x1 + [4]x2
POL(FIB(x1)) = 0
POL(FIB1(x1, x2)) = [4]x1 + [2]x2
POL(S(x1)) = 0
POL(SEL(x1, x2)) = x2
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(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(mark(x1)) = [1] + x1
POL(ok(x1)) = x1
(62) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
S tuples:none
K tuples:
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c29(ADD(z0, z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
FIB, SEL, FIB1, S, CONS, ADD
Compound Symbols:
c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29
(63) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(64) BOUNDS(O(1), O(1))