Runtime Complexity (innermost) proof of /tmp/tmpP3cZPI/PEANO_nosorts_noand

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 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1603 ms)

    ↳6 CdtProblem

    ↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 9 ms)

    ↳8 CdtProblem

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

    ↳10 CdtProblem

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

    ↳12 CdtProblem

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

    ↳14 CdtProblem

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

    ↳16 CdtProblem

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

    ↳18 CdtProblem

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

    ↳20 CdtProblem

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

    ↳22 CdtProblem

    ↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 24 ms)

    ↳24 CdtProblem

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

    ↳26 CdtProblem

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

    ↳28 CdtProblem

    ↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 25 ms)

    ↳30 CdtProblem

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

    ↳32 CdtProblem

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

    ↳34 CdtProblem

    ↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 18 ms)

    ↳36 CdtProblem

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

    ↳38 CdtProblem

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

    ↳40 CdtProblem

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

    ↳42 CdtProblem

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

    ↳44 CdtProblem

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

    ↳46 CdtProblem

↳47 CpxTrsMatchBoundsTAProof (⇔, 127 ms)

↳48 BOUNDS(O(1), O(n^1))

(0) Obligation:

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

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U11(tt, z0, z1)) → c(U12'(tt, z0, z1))
ACTIVE(U12(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(plus(z0, s(z1))) → c3(U11'(tt, z1, z0))
ACTIVE(U11(z0, z1, z2)) → c4(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(U11(tt, z0, z1)) → c(U12'(tt, z0, z1))
ACTIVE(U12(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(plus(z0, s(z1))) → c3(U11'(tt, z1, z0))
ACTIVE(U11(z0, z1, z2)) → c4(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c, c1, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25

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

Removed 2 trailing nodes:

ACTIVE(plus(z0, s(z1))) → c3(U11'(tt, z1, z0))
ACTIVE(U11(tt, z0, z1)) → c(U12'(tt, z0, z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U12(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U11(z0, z1, z2)) → c4(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(U12(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U11(z0, z1, z2)) → c4(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25

(5) CdtPolyRedPairProof (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(z0)) → c24(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

And the Tuples:

ACTIVE(U12(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U11(z0, z1, z2)) → c4(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))

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

POL(0) = [1]
POL(ACTIVE(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = [2]x₁
POL(U11(x₁, x₂, x₃)) = [3] + [5]x₁ + [4]x₂ + [2]x₃
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂, x₃)) = [5] + x₁ + [4]x₂ + [2]x₃
POL(U12'(x₁, x₂, x₃)) = 0
POL(active(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁)) = 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(c18(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c20(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c21(x₁, x₂)) = x₁ + x₂
POL(c22(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c24(x₁, x₂)) = x₁ + x₂
POL(c25(x₁, x₂)) = x₁ + x₂
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = x₁
POL(plus(x₁, x₂)) = [2]x₁ + [4]x₂
POL(proper(x₁)) = x₁
POL(s(x₁)) = [4] + x₁
POL(tt) = [1]

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U12(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U11(z0, z1, z2)) → c4(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(U12(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U11(z0, z1, z2)) → c4(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25

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

Use narrowing to replace ACTIVE(U12(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0)) by

ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U12(tt, x0, x1)) → c1

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U11(z0, z1, z2)) → c4(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U12(tt, x0, x1)) → c1

S tuples:

ACTIVE(U11(z0, z1, z2)) → c4(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U12(tt, x0, x1)) → c1

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25, c1, c1

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

Removed 1 trailing nodes:

ACTIVE(U12(tt, x0, x1)) → c1

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U11(z0, z1, z2)) → c4(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))

S tuples:

ACTIVE(U11(z0, z1, z2)) → c4(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25, c1

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

Use narrowing to replace ACTIVE(U11(z0, z1, z2)) → c4(U11'(active(z0), z1, z2), ACTIVE(z0)) by

ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(x0, x1, x2)) → c4

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(x0, x1, x2)) → c4

S tuples:

ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(x0, x1, x2)) → c4

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25, c1, c4, c4

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

Removed 1 trailing nodes:

ACTIVE(U11(x0, x1, x2)) → c4

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))

S tuples:

ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25, c1, c4

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

Removed 1 trailing tuple parts

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))

S tuples:

ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25, c1, c4, c4

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

Use narrowing to replace ACTIVE(U12(z0, z1, z2)) → c5(U12'(active(z0), z1, z2), ACTIVE(z0)) by

ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(x0, x1, x2)) → c5

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(x0, x1, x2)) → c5

S tuples:

ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(x0, x1, x2)) → c5

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25, c1, c4, c4, c5, c5

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

Removed 1 trailing nodes:

ACTIVE(U12(x0, x1, x2)) → c5

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))

S tuples:

ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25, c1, c4, c4, c5

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

Removed 1 trailing tuple parts

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))

S tuples:

ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25, c1, c4, c4, c5, c5

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

Use narrowing to replace ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(x0)) → c6

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(x0)) → c6

S tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(x0)) → c6

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25, c1, c4, c4, c5, c5, c6, c6

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

Removed 1 trailing nodes:

ACTIVE(s(x0)) → c6

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

S tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25, c1, c4, c4, c5, c5, c6

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

Removed 1 trailing tuple parts

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))

S tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c22, c24, c25, c1, c4, c4, c5, c5, c6, c6

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

Use narrowing to replace PROPER(U11(z0, z1, z2)) → c18(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, x2)) → c18

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, x2)) → c18

S tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, x2)) → c18

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c20, c21, c22, c24, c25, c1, c4, c4, c5, c5, c6, c6, c18, c18

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

Removed 1 trailing nodes:

PROPER(U11(x0, x1, x2)) → c18

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c20, c21, c22, c24, c25, c1, c4, c4, c5, c5, c6, c6, c18

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

Removed 6 trailing tuple parts

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c20, c21, c22, c24, c25, c1, c4, c4, c5, c5, c6, c6, c18, c18

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

Use narrowing to replace PROPER(U12(z0, z1, z2)) → c20(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(U12(x0, x1, U11(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, tt)) → c20(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c20(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c20(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, 0)) → c20(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c20(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, tt, x2)) → c20(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c20(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c20(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c20(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c20(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c20(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c20(U12'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c20(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c20(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c20(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c20(U12'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, x2)) → c20

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, tt)) → c20(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c20(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c20(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, 0)) → c20(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c20(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, tt, x2)) → c20(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c20(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c20(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c20(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c20(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c20(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c20(U12'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c20(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c20(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c20(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c20(U12'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, x2)) → c20

S tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, tt)) → c20(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c20(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c20(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, 0)) → c20(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c20(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, tt, x2)) → c20(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c20(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c20(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c20(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c20(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c20(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c20(U12'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c20(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c20(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c20(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c20(U12'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, x2)) → c20

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c21, c22, c24, c25, c1, c4, c4, c5, c5, c6, c6, c18, c18, c20, c20

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

Removed 1 trailing nodes:

PROPER(U12(x0, x1, x2)) → c20

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, tt)) → c20(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c20(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c20(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, 0)) → c20(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c20(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, tt, x2)) → c20(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c20(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c20(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c20(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c20(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c20(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c20(U12'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c20(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c20(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c20(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c20(U12'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, tt)) → c20(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c20(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c20(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, 0)) → c20(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c20(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, tt, x2)) → c20(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c20(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c20(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c20(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c20(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c20(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c20(U12'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c20(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c20(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c20(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c20(U12'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c21, c22, c24, c25, c1, c4, c4, c5, c5, c6, c6, c18, c18, c20

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

Removed 6 trailing tuple parts

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c20(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c20(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c20(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c20(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c20(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c20(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c20(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c20(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c20(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c20(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c20(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c20(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c20(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c20(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c20(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c20(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c20(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c20(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c20(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c20(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c20(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c20(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c20(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c20(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c20(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c20(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c20(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c20(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c20(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c20(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c20(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c20(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c21, c22, c24, c25, c1, c4, c4, c5, c5, c6, c6, c18, c18, c20, c20

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

Use narrowing to replace PROPER(s(z0)) → c21(S(proper(z0)), PROPER(z0)) by

PROPER(s(U11(z0, z1, z2))) → c21(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(tt)) → c21(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0, z1, z2))) → c21(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c21(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c21(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c21(S(ok(0)), PROPER(0))
PROPER(s(x0)) → c21

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c20(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c20(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c20(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c20(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c20(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c20(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c20(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c20(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c20(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c20(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c20(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c20(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c20(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c20(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c20(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c20(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c21(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(tt)) → c21(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0, z1, z2))) → c21(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c21(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c21(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c21(S(ok(0)), PROPER(0))
PROPER(s(x0)) → c21

S tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c20(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c20(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c20(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c20(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c20(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c20(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c20(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c20(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c20(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c20(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c20(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c20(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c20(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c20(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c20(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c20(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c21(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(tt)) → c21(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0, z1, z2))) → c21(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c21(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c21(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c21(S(ok(0)), PROPER(0))
PROPER(s(x0)) → c21

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c22, c24, c25, c1, c4, c4, c5, c5, c6, c6, c18, c18, c20, c20, c21, c21

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

Removed 1 trailing nodes:

PROPER(s(x0)) → c21

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c20(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c20(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c20(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c20(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c20(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c20(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c20(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c20(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c20(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c20(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c20(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c20(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c20(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c20(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c20(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c20(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c21(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(tt)) → c21(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0, z1, z2))) → c21(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c21(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c21(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c21(S(ok(0)), PROPER(0))

S tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c20(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c20(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c20(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c20(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c20(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c20(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c20(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c20(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c20(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c20(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c20(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c20(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c20(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c20(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c20(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c20(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c21(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(tt)) → c21(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0, z1, z2))) → c21(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c21(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c21(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c21(S(ok(0)), PROPER(0))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c22, c24, c25, c1, c4, c4, c5, c5, c6, c6, c18, c18, c20, c20, c21

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

Removed 2 trailing tuple parts

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c20(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c20(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c20(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c20(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c20(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c20(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c20(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c20(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c20(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c20(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c20(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c20(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c20(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c20(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c20(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c20(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c21(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c21(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c21(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c21(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c21(S(ok(tt)))
PROPER(s(0)) → c21(S(ok(0)))

S tuples:

ACTIVE(plus(z0, z1)) → c7(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c8(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c9(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c10(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c11(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c12(U12'(z0, z1, z2))
S(mark(z0)) → c13(S(z0))
S(ok(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c17(PLUS(z0, z1))
PROPER(plus(z0, z1)) → c22(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z1, mark(z0))) → c1(S(mark(plus(z0, z1))), PLUS(mark(z0), z1))
ACTIVE(U12(tt, mark(z1), z0)) → c1(S(mark(plus(z0, z1))), PLUS(z0, mark(z1)))
ACTIVE(U12(tt, ok(z1), ok(z0))) → c1(S(ok(plus(z0, z1))), PLUS(ok(z0), ok(z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c4(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c4(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c4(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c4(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c4(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c4(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c4(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c4(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c5(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c5(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c5(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c5(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c5(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c5(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c5(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c5(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(s(U12(tt, z0, z1))) → c6(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c6(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c6(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c6(S(mark(U12(tt, z0, z1))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c18(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c18(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c18(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c18(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c18(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c18(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c18(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c18(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c18(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c18(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c18(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c18(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c18(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c18(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c18(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c18(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c18(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c20(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c20(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c20(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c20(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c20(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c20(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c20(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c20(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c20(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c20(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c20(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c20(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c20(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c20(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c20(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c20(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c20(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c21(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c21(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c21(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c21(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c21(S(ok(tt)))
PROPER(s(0)) → c21(S(ok(0)))

K tuples:

TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, U11, U12, s, plus, proper, top

Defined Pair Symbols:

ACTIVE, U11', U12', S, PLUS, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c22, c24, c25, c1, c4, c4, c5, c5, c6, c6, c18, c18, c20, c20, c21, c21

(47) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5, 6, 7]
transitions:
tt0() → 0
mark0(0) → 0
00() → 0
ok0(0) → 0
active0(0) → 1
U110(0, 0, 0) → 2
U120(0, 0, 0) → 3
s0(0) → 4
plus0(0, 0) → 5
proper0(0) → 6
top0(0) → 7
U111(0, 0, 0) → 8
mark1(8) → 2
U121(0, 0, 0) → 9
mark1(9) → 3
s1(0) → 10
mark1(10) → 4
plus1(0, 0) → 11
mark1(11) → 5
tt1() → 12
ok1(12) → 6
01() → 13
ok1(13) → 6
U111(0, 0, 0) → 14
ok1(14) → 2
U121(0, 0, 0) → 15
ok1(15) → 3
s1(0) → 16
ok1(16) → 4
plus1(0, 0) → 17
ok1(17) → 5
proper1(0) → 18
top1(18) → 7
active1(0) → 19
top1(19) → 7
mark1(8) → 8
mark1(8) → 14
mark1(9) → 9
mark1(9) → 15
mark1(10) → 10
mark1(10) → 16
mark1(11) → 11
mark1(11) → 17
ok1(12) → 18
ok1(13) → 18
ok1(14) → 8
ok1(14) → 14
ok1(15) → 9
ok1(15) → 15
ok1(16) → 10
ok1(16) → 16
ok1(17) → 11
ok1(17) → 17
active2(12) → 20
top2(20) → 7
active2(13) → 20

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(44) Obligation:

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

(46) Obligation:

(47) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

(48) BOUNDS(O(1), O(n^1))