Runtime Complexity (innermost) proof of /tmp/tmpQqV3vb/Ex49_GM04

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 CdtProblem

↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CdtProblem

↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳8 CdtProblem

↳9 CdtUsableRulesProof (⇔, 110 ms)

↳10 CdtProblem

↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳12 CdtProblem

↳13 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳14 CdtProblem

↳15 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳16 CdtProblem

↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳18 CdtProblem

↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳20 CdtProblem

↳21 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳22 CdtProblem

↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳24 CdtProblem

↳25 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳26 CdtProblem

↳27 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳28 CdtProblem

↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳30 CdtProblem

↳31 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳32 CdtProblem

↳33 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳34 CdtProblem

↳35 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳36 CdtProblem

↳37 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 4 ms)

↳38 CdtProblem

↳39 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳40 CdtProblem

↳41 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳42 CdtProblem

↳43 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳44 CdtProblem

↳45 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳46 CdtProblem

↳47 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳48 CdtProblem

↳49 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳50 CdtProblem

↳51 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳52 CdtProblem

↳53 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 2475 ms)

↳54 CdtProblem

↳55 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 1091 ms)

↳56 CdtProblem

↳57 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 1788 ms)

↳58 CdtProblem

↳59 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳60 CdtProblem

↳61 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳62 CdtProblem

↳63 CdtUnreachableProof (⇔, 0 ms)

↳64 CdtProblem

↳65 CdtUsableRulesProof (⇔, 0 ms)

↳66 CdtProblem

↳67 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳68 CdtProblem

↳69 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳70 CdtProblem

↳71 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)

↳72 BOUNDS(O(1), O(1))

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
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(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(minus(0, z0)) → c
ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(z0, 0)) → c2
ACTIVE(geq(0, s(z0))) → c3
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(div(0, s(z0))) → c5
ACTIVE(div(s(z0), s(z1))) → c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1), S(div(minus(z0, z1), s(z1))), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1), S(z1))
ACTIVE(if(true, z0, z1)) → c7
ACTIVE(if(false, z0, z1)) → c8
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c19
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(true) → c22
PROPER(false) → c23
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(minus(0, z0)) → c
ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(z0, 0)) → c2
ACTIVE(geq(0, s(z0))) → c3
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(div(0, s(z0))) → c5
ACTIVE(div(s(z0), s(z1))) → c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1), S(div(minus(z0, z1), s(z1))), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1), S(z1))
ACTIVE(if(true, z0, z1)) → c7
ACTIVE(if(false, z0, z1)) → c8
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c19
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(true) → c22
PROPER(false) → c23
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, 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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 9 trailing nodes:

ACTIVE(minus(0, z0)) → c
ACTIVE(div(0, s(z0))) → c5
PROPER(false) → c23
PROPER(true) → c22
ACTIVE(geq(z0, 0)) → c2
ACTIVE(geq(0, s(z0))) → c3
PROPER(0) → c19
ACTIVE(if(true, z0, z1)) → c7
ACTIVE(if(false, z0, z1)) → c8

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1), S(div(minus(z0, z1), s(z1))), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1), S(z1))
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1), S(div(minus(z0, z1), s(z1))), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1), S(z1))
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c6, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c6(GEQ(z0, z1), MINUS(z0, z1), S(z1))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c6(GEQ(z0, z1), MINUS(z0, z1), S(z1))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c6

(7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c

(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(minus(0, z0))) → c9(S(mark(0)), ACTIVE(minus(0, z0)))
ACTIVE(s(minus(s(z0), s(z1)))) → c9(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)), ACTIVE(geq(z0, 0)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)), ACTIVE(geq(0, s(z0))))
ACTIVE(s(geq(s(z0), s(z1)))) → c9(S(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0))))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)), ACTIVE(minus(0, z0)))
ACTIVE(s(minus(s(z0), s(z1)))) → c9(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)), ACTIVE(geq(z0, 0)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)), ACTIVE(geq(0, s(z0))))
ACTIVE(s(geq(s(z0), s(z1)))) → c9(S(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0))))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)), ACTIVE(minus(0, z0)))
ACTIVE(s(minus(s(z0), s(z1)))) → c9(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)), ACTIVE(geq(z0, 0)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)), ACTIVE(geq(0, s(z0))))
ACTIVE(s(geq(s(z0), s(z1)))) → c9(S(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0))))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9

(13) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(minus(s(z0), s(z1)))) → c9(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c9(S(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0))))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(minus(s(z0), s(z1)))) → c9(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c9(S(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0))))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9

(15) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0))))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0))))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) by

ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1), ACTIVE(minus(0, z0)))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c10(DIV(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1), ACTIVE(geq(z0, 0)))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1), ACTIVE(geq(0, s(z0))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c10(DIV(mark(geq(z0, z1)), x1), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1), ACTIVE(div(0, s(z0))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1)))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1)))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0))))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1), ACTIVE(minus(0, z0)))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c10(DIV(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1), ACTIVE(geq(z0, 0)))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1), ACTIVE(geq(0, s(z0))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c10(DIV(mark(geq(z0, z1)), x1), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1), ACTIVE(div(0, s(z0))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1)))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1)))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0))))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1), ACTIVE(minus(0, z0)))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c10(DIV(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1), ACTIVE(geq(z0, 0)))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1), ACTIVE(geq(0, s(z0))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c10(DIV(mark(geq(z0, z1)), x1), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1), ACTIVE(div(0, s(z0))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1)))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1)))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 5 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c10(DIV(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c10(DIV(mark(geq(z0, z1)), x1), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1)))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1)))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c10(DIV(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c10(DIV(mark(geq(z0, z1)), x1), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1)))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1)))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10

(21) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1)))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1)))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1)))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1)))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3

(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) by

ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(minus(0, z0)))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c11(IF(mark(minus(z0, z1)), x1, x2), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2), ACTIVE(geq(z0, 0)))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(geq(0, s(z0))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c11(IF(mark(geq(z0, z1)), x1, x2), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(div(0, s(z0))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1)))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1)))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1)))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1)))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(minus(0, z0)))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c11(IF(mark(minus(z0, z1)), x1, x2), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2), ACTIVE(geq(z0, 0)))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(geq(0, s(z0))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c11(IF(mark(geq(z0, z1)), x1, x2), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(div(0, s(z0))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1)))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1)))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1)))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1)))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(minus(0, z0)))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c11(IF(mark(minus(z0, z1)), x1, x2), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2), ACTIVE(geq(z0, 0)))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(geq(0, s(z0))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c11(IF(mark(geq(z0, z1)), x1, x2), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(div(0, s(z0))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1)))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1)))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11

(25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 10 trailing tuple parts

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c11(IF(mark(minus(z0, z1)), x1, x2), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c11(IF(mark(geq(z0, z1)), x1, x2), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c11(IF(mark(minus(z0, z1)), x1, x2), ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c11(IF(mark(geq(z0, z1)), x1, x2), ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11

(27) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18

(31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18

(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) by

PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(0)) → c20(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(true)) → c20(S(ok(true)), PROPER(true))
PROPER(s(false)) → c20(S(ok(false)), PROPER(false))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(0)) → c20(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(true)) → c20(S(ok(true)), PROPER(true))
PROPER(s(false)) → c20(S(ok(false)), PROPER(false))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(0)) → c20(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(true)) → c20(S(ok(true)), PROPER(true))
PROPER(s(false)) → c20(S(ok(false)), PROPER(false))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20

(35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20

(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21

(39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21

(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24

(43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, TOP, PROPER

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25

(47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 9 trailing tuple parts

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, TOP, PROPER

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(0)) → c28(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(true)) → c28(TOP(ok(true)), PROPER(true))
TOP(mark(false)) → c28(TOP(ok(false)), PROPER(false))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(0)) → c28(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(true)) → c28(TOP(ok(true)), PROPER(true))
TOP(mark(false)) → c28(TOP(ok(false)), PROPER(false))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(0)) → c28(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(true)) → c28(TOP(ok(true)), PROPER(true))
TOP(mark(false)) → c28(TOP(ok(false)), PROPER(false))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, TOP, PROPER

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25, c28

(51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(0)) → c28(TOP(ok(0)))
TOP(mark(true)) → c28(TOP(ok(true)))
TOP(mark(false)) → c28(TOP(ok(false)))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(0)) → c28(TOP(ok(0)))
TOP(mark(true)) → c28(TOP(ok(true)))
TOP(mark(false)) → c28(TOP(ok(false)))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, TOP, PROPER

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25, c28, c28

(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.

TOP(mark(0)) → c28(TOP(ok(0)))

We considered the (Usable) Rules:

if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
active(geq(0, s(z0))) → mark(false)
s(mark(z0)) → mark(s(z0))
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(s(z0)) → s(active(z0))
div(mark(z0), z1) → mark(div(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
active(div(0, s(z0))) → mark(0)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
active(if(false, z0, z1)) → mark(z1)
active(minus(0, z0)) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(div(z0, z1)) → div(active(z0), z1)
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
active(geq(z0, 0)) → mark(true)
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
s(ok(z0)) → ok(s(z0))
active(if(true, z0, z1)) → mark(z0)

And the Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(0)) → c28(TOP(ok(0)))
TOP(mark(true)) → c28(TOP(ok(true)))
TOP(mark(false)) → c28(TOP(ok(false)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(DIV(x₁, x₂)) = 0
POL(GEQ(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃)) = 0
POL(MINUS(x₁, x₂)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = [2]x₁
POL(active(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁, x₂)) = x₁ + x₂
POL(c18(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c21(x₁, x₂)) = x₁ + x₂
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c24(x₁, x₂)) = x₁ + x₂
POL(c24(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c25(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c25(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c28(x₁, x₂)) = x₁ + x₂
POL(c29(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(c9(x₁, x₂)) = x₁ + x₂
POL(div(x₁, x₂)) = [1]
POL(false) = [1]
POL(geq(x₁, x₂)) = [1]
POL(if(x₁, x₂, x₃)) = [1]
POL(mark(x₁)) = [1]
POL(minus(x₁, x₂)) = [1]
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = [1]
POL(true) = [1]

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(0)) → c28(TOP(ok(0)))
TOP(mark(true)) → c28(TOP(ok(true)))
TOP(mark(false)) → c28(TOP(ok(false)))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(true)) → c28(TOP(ok(true)))
TOP(mark(false)) → c28(TOP(ok(false)))

K tuples:

TOP(mark(0)) → c28(TOP(ok(0)))

Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, TOP, PROPER

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25, c28, c28

(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.

TOP(mark(true)) → c28(TOP(ok(true)))

We considered the (Usable) Rules:

if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
active(geq(0, s(z0))) → mark(false)
s(mark(z0)) → mark(s(z0))
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(s(z0)) → s(active(z0))
div(mark(z0), z1) → mark(div(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
active(div(0, s(z0))) → mark(0)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
active(if(false, z0, z1)) → mark(z1)
active(minus(0, z0)) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(div(z0, z1)) → div(active(z0), z1)
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
active(geq(z0, 0)) → mark(true)
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
s(ok(z0)) → ok(s(z0))
active(if(true, z0, z1)) → mark(z0)

And the Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(0)) → c28(TOP(ok(0)))
TOP(mark(true)) → c28(TOP(ok(true)))
TOP(mark(false)) → c28(TOP(ok(false)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(DIV(x₁, x₂)) = 0
POL(GEQ(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃)) = 0
POL(MINUS(x₁, x₂)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = [2]x₁
POL(active(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁, x₂)) = x₁ + x₂
POL(c18(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c21(x₁, x₂)) = x₁ + x₂
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c24(x₁, x₂)) = x₁ + x₂
POL(c24(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c25(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c25(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c28(x₁, x₂)) = x₁ + x₂
POL(c29(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(c9(x₁, x₂)) = x₁ + x₂
POL(div(x₁, x₂)) = [2]
POL(false) = [2]
POL(geq(x₁, x₂)) = [2]
POL(if(x₁, x₂, x₃)) = [2]
POL(mark(x₁)) = [2]
POL(minus(x₁, x₂)) = [2]
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = [2]
POL(true) = 0

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(0)) → c28(TOP(ok(0)))
TOP(mark(true)) → c28(TOP(ok(true)))
TOP(mark(false)) → c28(TOP(ok(false)))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(false)) → c28(TOP(ok(false)))

K tuples:

TOP(mark(0)) → c28(TOP(ok(0)))
TOP(mark(true)) → c28(TOP(ok(true)))

Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, TOP, PROPER

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25, c28, c28

(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.

TOP(mark(false)) → c28(TOP(ok(false)))

We considered the (Usable) Rules:

if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
active(geq(0, s(z0))) → mark(false)
s(mark(z0)) → mark(s(z0))
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(s(z0)) → s(active(z0))
div(mark(z0), z1) → mark(div(z0, z1))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
active(div(0, s(z0))) → mark(0)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
active(if(false, z0, z1)) → mark(z1)
active(minus(0, z0)) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(div(z0, z1)) → div(active(z0), z1)
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
active(geq(z0, 0)) → mark(true)
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
s(ok(z0)) → ok(s(z0))
active(if(true, z0, z1)) → mark(z0)

And the Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(0)) → c28(TOP(ok(0)))
TOP(mark(true)) → c28(TOP(ok(true)))
TOP(mark(false)) → c28(TOP(ok(false)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [2]
POL(ACTIVE(x₁)) = 0
POL(DIV(x₁, x₂)) = 0
POL(GEQ(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃)) = 0
POL(MINUS(x₁, x₂)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = [4]x₁
POL(active(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁, x₂)) = x₁ + x₂
POL(c18(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c21(x₁, x₂)) = x₁ + x₂
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c24(x₁, x₂)) = x₁ + x₂
POL(c24(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c25(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c25(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c28(x₁, x₂)) = x₁ + x₂
POL(c29(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(c9(x₁, x₂)) = x₁ + x₂
POL(div(x₁, x₂)) = [2]
POL(false) = 0
POL(geq(x₁, x₂)) = [2]
POL(if(x₁, x₂, x₃)) = [2]
POL(mark(x₁)) = [2]
POL(minus(x₁, x₂)) = [2]
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = [2]
POL(true) = [2]

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(0)) → c28(TOP(ok(0)))
TOP(mark(true)) → c28(TOP(ok(true)))
TOP(mark(false)) → c28(TOP(ok(false)))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))

K tuples:

TOP(mark(0)) → c28(TOP(ok(0)))
TOP(mark(true)) → c28(TOP(ok(true)))
TOP(mark(false)) → c28(TOP(ok(false)))

Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, TOP, PROPER

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25, c28, c28

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(minus(0, z0))) → c29(TOP(mark(0)), ACTIVE(minus(0, z0)))
TOP(ok(minus(s(z0), s(z1)))) → c29(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
TOP(ok(geq(z0, 0))) → c29(TOP(mark(true)), ACTIVE(geq(z0, 0)))
TOP(ok(geq(0, s(z0)))) → c29(TOP(mark(false)), ACTIVE(geq(0, s(z0))))
TOP(ok(geq(s(z0), s(z1)))) → c29(TOP(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1))))
TOP(ok(div(0, s(z0)))) → c29(TOP(mark(0)), ACTIVE(div(0, s(z0))))
TOP(ok(div(s(z0), s(z1)))) → c29(TOP(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
TOP(ok(if(true, z0, z1))) → c29(TOP(mark(z0)), ACTIVE(if(true, z0, z1)))
TOP(ok(if(false, z0, z1))) → c29(TOP(mark(z1)), ACTIVE(if(false, z0, z1)))
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(div(z0, z1))) → c29(TOP(div(active(z0), z1)), ACTIVE(div(z0, z1)))
TOP(ok(if(z0, z1, z2))) → c29(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(0)) → c28(TOP(ok(0)))
TOP(mark(true)) → c28(TOP(ok(true)))
TOP(mark(false)) → c28(TOP(ok(false)))
TOP(ok(minus(0, z0))) → c29(TOP(mark(0)), ACTIVE(minus(0, z0)))
TOP(ok(minus(s(z0), s(z1)))) → c29(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
TOP(ok(geq(z0, 0))) → c29(TOP(mark(true)), ACTIVE(geq(z0, 0)))
TOP(ok(geq(0, s(z0)))) → c29(TOP(mark(false)), ACTIVE(geq(0, s(z0))))
TOP(ok(geq(s(z0), s(z1)))) → c29(TOP(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1))))
TOP(ok(div(0, s(z0)))) → c29(TOP(mark(0)), ACTIVE(div(0, s(z0))))
TOP(ok(div(s(z0), s(z1)))) → c29(TOP(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
TOP(ok(if(true, z0, z1))) → c29(TOP(mark(z0)), ACTIVE(if(true, z0, z1)))
TOP(ok(if(false, z0, z1))) → c29(TOP(mark(z1)), ACTIVE(if(false, z0, z1)))
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(div(z0, z1))) → c29(TOP(div(active(z0), z1)), ACTIVE(div(z0, z1)))
TOP(ok(if(z0, z1, z2))) → c29(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(ok(minus(0, z0))) → c29(TOP(mark(0)), ACTIVE(minus(0, z0)))
TOP(ok(minus(s(z0), s(z1)))) → c29(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
TOP(ok(geq(z0, 0))) → c29(TOP(mark(true)), ACTIVE(geq(z0, 0)))
TOP(ok(geq(0, s(z0)))) → c29(TOP(mark(false)), ACTIVE(geq(0, s(z0))))
TOP(ok(geq(s(z0), s(z1)))) → c29(TOP(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1))))
TOP(ok(div(0, s(z0)))) → c29(TOP(mark(0)), ACTIVE(div(0, s(z0))))
TOP(ok(div(s(z0), s(z1)))) → c29(TOP(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
TOP(ok(if(true, z0, z1))) → c29(TOP(mark(z0)), ACTIVE(if(true, z0, z1)))
TOP(ok(if(false, z0, z1))) → c29(TOP(mark(z1)), ACTIVE(if(false, z0, z1)))
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(div(z0, z1))) → c29(TOP(div(active(z0), z1)), ACTIVE(div(z0, z1)))
TOP(ok(if(z0, z1, z2))) → c29(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))

K tuples:

TOP(mark(0)) → c28(TOP(ok(0)))
TOP(mark(true)) → c28(TOP(ok(true)))
TOP(mark(false)) → c28(TOP(ok(false)))

Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, PROPER, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25, c28, c28, c29

(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 7 trailing nodes:

TOP(mark(0)) → c28(TOP(ok(0)))
TOP(ok(minus(0, z0))) → c29(TOP(mark(0)), ACTIVE(minus(0, z0)))
TOP(ok(geq(z0, 0))) → c29(TOP(mark(true)), ACTIVE(geq(z0, 0)))
TOP(ok(div(0, s(z0)))) → c29(TOP(mark(0)), ACTIVE(div(0, s(z0))))
TOP(ok(geq(0, s(z0)))) → c29(TOP(mark(false)), ACTIVE(geq(0, s(z0))))
TOP(mark(false)) → c28(TOP(ok(false)))
TOP(mark(true)) → c28(TOP(ok(true)))

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(ok(minus(s(z0), s(z1)))) → c29(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
TOP(ok(geq(s(z0), s(z1)))) → c29(TOP(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1))))
TOP(ok(div(s(z0), s(z1)))) → c29(TOP(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
TOP(ok(if(true, z0, z1))) → c29(TOP(mark(z0)), ACTIVE(if(true, z0, z1)))
TOP(ok(if(false, z0, z1))) → c29(TOP(mark(z1)), ACTIVE(if(false, z0, z1)))
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(div(z0, z1))) → c29(TOP(div(active(z0), z1)), ACTIVE(div(z0, z1)))
TOP(ok(if(z0, z1, z2))) → c29(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))

S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(ok(minus(s(z0), s(z1)))) → c29(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
TOP(ok(geq(s(z0), s(z1)))) → c29(TOP(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1))))
TOP(ok(div(s(z0), s(z1)))) → c29(TOP(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
TOP(ok(if(true, z0, z1))) → c29(TOP(mark(z0)), ACTIVE(if(true, z0, z1)))
TOP(ok(if(false, z0, z1))) → c29(TOP(mark(z1)), ACTIVE(if(false, z0, z1)))
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(div(z0, z1))) → c29(TOP(div(active(z0), z1)), ACTIVE(div(z0, z1)))
TOP(ok(if(z0, z1, z2))) → c29(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, PROPER, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25, c28, c29

(63) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1))
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1))
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1)))
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)))
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)))
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1))))
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1))))
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1))))
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1)))
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)))
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1))
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1))
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1))
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1))
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1))))
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1)))
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)))
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)))
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1))
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1))
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2))
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2))
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2))
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2))
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2))
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1))))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2))
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0))
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0))
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0))
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1))
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1))
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(true)) → c20(S(ok(true)))
PROPER(s(false)) → c20(S(ok(false)))
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0))
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0))
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0))
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1))
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1))
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1)))
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1)))
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1)))
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2)))
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1))
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1))
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1))
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0))
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0))
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0))
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1))
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1))
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1)))
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1)))
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2))
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2))
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1)))
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1)))
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(ok(minus(s(z0), s(z1)))) → c29(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1))))
TOP(ok(geq(s(z0), s(z1)))) → c29(TOP(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1))))
TOP(ok(div(s(z0), s(z1)))) → c29(TOP(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1))))
TOP(ok(if(true, z0, z1))) → c29(TOP(mark(z0)), ACTIVE(if(true, z0, z1)))
TOP(ok(if(false, z0, z1))) → c29(TOP(mark(z1)), ACTIVE(if(false, z0, z1)))
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(div(z0, z1))) → c29(TOP(div(active(z0), z1)), ACTIVE(div(z0, z1)))
TOP(ok(if(z0, z1, z2))) → c29(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

Tuples:

S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))

S tuples:

S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))

K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27

(65) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

active(minus(0, z0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(geq(z0, 0)) → mark(true)
active(geq(0, s(z0))) → mark(false)
active(geq(s(z0), s(z1))) → mark(geq(z0, z1))
active(div(0, s(z0))) → mark(0)
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
active(s(z0)) → s(active(z0))
active(div(z0, z1)) → div(active(z0), z1)
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
div(mark(z0), z1) → mark(div(z0, z1))
div(ok(z0), ok(z1)) → ok(div(z0, z1))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(minus(z0, z1)) → minus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(geq(z0, z1)) → geq(proper(z0), proper(z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(z0, z1)) → div(proper(z0), proper(z1))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
minus(ok(z0), ok(z1)) → ok(minus(z0, z1))
geq(ok(z0), ok(z1)) → ok(geq(z0, z1))

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))

S tuples:

S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27

(67) 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.

S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(DIV(x₁, x₂)) = x₁ + x₂
POL(GEQ(x₁, x₂)) = [4]x₂
POL(IF(x₁, x₂, x₃)) = [3]x₂
POL(MINUS(x₁, x₂)) = x₁ + [2]x₂
POL(S(x₁)) = [3]x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [1] + x₁

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))

S tuples:

IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))

K tuples:

S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27

(69) 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.

IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(DIV(x₁, x₂)) = [3]x₁ + [2]x₂
POL(GEQ(x₁, x₂)) = [3]x₁ + [2]x₂
POL(IF(x₁, x₂, x₃)) = [2]x₁ + [2]x₂
POL(MINUS(x₁, x₂)) = [3]x₁
POL(S(x₁)) = [4]x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))

S tuples:none
K tuples:

S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
DIV(mark(z0), z1) → c14(DIV(z0, z1))
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))

Defined Rule Symbols:none

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27

(71) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(0) Obligation:

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

(2) Obligation:

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(4) Obligation:

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(6) Obligation:

(7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

(8) Obligation:

(9) CdtUsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(12) Obligation:

(13) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(14) Obligation:

(15) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

(16) Obligation:

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(18) Obligation:

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(20) Obligation:

(21) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

(22) Obligation:

(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(24) Obligation:

(25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(26) Obligation:

(27) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

(28) Obligation:

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(30) Obligation:

(31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(32) Obligation:

(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(34) Obligation:

(35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(36) Obligation:

(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(38) Obligation:

(39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(40) Obligation:

(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(42) Obligation:

(43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(44) Obligation:

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(46) Obligation:

(47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(48) Obligation:

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(50) Obligation:

(51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(52) Obligation:

(53) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

(54) Obligation:

(55) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

(56) Obligation:

(57) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

(58) Obligation:

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(60) Obligation:

(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(62) Obligation:

(63) CdtUnreachableProof (EQUIVALENT transformation)

(64) Obligation:

(65) CdtUsableRulesProof (EQUIVALENT transformation)

(66) Obligation:

(67) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

(68) Obligation:

(69) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

(70) Obligation:

(71) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

(72) BOUNDS(O(1), O(1))