Runtime Complexity (innermost) proof of /tmp/tmpQuJbxv/PEANO_nosorts

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS

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

↳2 CdtProblem

↳3 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

↳4 CdtProblem

↳5 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))

↳6 CdtProblem

↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

↳8 CdtProblem

↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

↳10 CdtProblem

↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

↳12 CdtProblem

↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

↳14 CdtProblem

↳15 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳16 CdtProblem

↳17 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

↳18 CdtProblem

↳19 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))

↳20 CdtProblem

↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

↳22 CdtProblem

↳23 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳24 CdtProblem

↳25 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

↳26 CdtProblem

↳27 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))

↳28 CdtProblem

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

↳30 CdtProblem

↳31 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳32 CdtProblem

↳33 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))

↳34 CdtProblem

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

↳36 CdtProblem

↳37 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))

↳38 CdtProblem

↳39 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

↳40 CdtProblem

↳41 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

↳42 CdtProblem

↳43 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))

↳44 CdtProblem

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

↳46 CdtProblem

↳47 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳48 CdtProblem

↳49 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))

↳50 CdtProblem

↳51 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

↳52 CdtProblem

↳53 CdtUnreachableProof (⇔)

↳54 CdtProblem

↳55 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳56 CdtProblem

↳57 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

↳58 CdtProblem

↳59 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

↳60 CdtProblem

↳61 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

↳62 CdtProblem

↳63 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

↳64 CdtProblem

↳65 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

↳66 CdtProblem

↳67 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

↳68 CdtProblem

↳69 SIsEmptyProof (⇔)

↳70 BOUNDS(O(1), O(1))

(0) Obligation:

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

active(and(tt, X)) → mark(X)
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))
active(and(X1, X2)) → and(active(X1), X2)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(s(X)) → s(active(X))
and(mark(X1), X2) → mark(and(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
s(mark(X)) → mark(s(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(ok(X)) → ok(s(X))
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(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, s(z1))) → c2(S(plus(z0, z1)), PLUS(z0, z1))
ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(plus(z0, s(z1))) → c2(S(plus(z0, z1)), PLUS(z0, z1))
ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20

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

Split RHS of tuples not part of any SCC

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(S(plus(z0, z1)))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(S(plus(z0, z1)))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20, c

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(plus(z0, s(z1))) → c

S tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(plus(z0, s(z1))) → c

K tuples:none
Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20, c, c

(7) 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)) → c19(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
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))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))

And the Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(plus(z0, s(z1))) → c

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

POL(0) = [2]
POL(ACTIVE(x₁)) = 0
POL(AND(x₁, x₂)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = [4]x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [4] + x₁ + x₂
POL(c) = 0
POL(c(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c16(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c18(x₁, x₂)) = x₁ + x₂
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c3(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)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [4] + x₁
POL(ok(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + [2]x₂
POL(proper(x₁)) = x₁
POL(s(x₁)) = [4] + x₁
POL(tt) = 0

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(plus(z0, s(z1))) → c

S tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(plus(z0, s(z1))) → c

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20, c, c

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

We considered the (Usable) Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
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))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))

And the Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(plus(z0, s(z1))) → c

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

POL(0) = [3]
POL(ACTIVE(x₁)) = 0
POL(AND(x₁, x₂)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(PROPER(x₁)) = [1] + [2]x₁
POL(S(x₁)) = 0
POL(TOP(x₁)) = [2]x₁²
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [1] + [2]x₁ + x₂
POL(c) = 0
POL(c(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c16(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c18(x₁, x₂)) = x₁ + x₂
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c3(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)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = x₁
POL(plus(x₁, x₂)) = [1] + [2]x₁ + x₂ + x₂² + [2]x₁·x₂
POL(proper(x₁)) = x₁
POL(s(x₁)) = [3] + [3]x₁
POL(tt) = [1]

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(plus(z0, s(z1))) → c

S tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(plus(z0, s(z1))) → c

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20, c, c

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
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))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))

And the Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(plus(z0, s(z1))) → c

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

POL(0) = [1]
POL(ACTIVE(x₁)) = 0
POL(AND(x₁, x₂)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(PROPER(x₁)) = x₁
POL(S(x₁)) = 0
POL(TOP(x₁)) = [2]x₁²
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(c) = 0
POL(c(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c16(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c18(x₁, x₂)) = x₁ + x₂
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c3(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)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁
POL(plus(x₁, x₂)) = [2]x₁ + [2]x₂
POL(proper(x₁)) = x₁
POL(s(x₁)) = [1] + x₁
POL(tt) = [1]

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(plus(z0, s(z1))) → c

S tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(plus(z0, s(z1))) → c

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20, c, c

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

Use narrowing to replace ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0)) by

ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(plus(z0, s(z1))) → c
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(plus(z0, s(z1))) → c
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20, c, c, c3

(15) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 24 dangling nodes:

ACTIVE(plus(z0, s(z1))) → c

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20, c, c3

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

Split RHS of tuples not part of any SCC

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c1(ACTIVE(and(tt, z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c1(ACTIVE(and(tt, z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20, c, c3, c1

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

Removed 1 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c1

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c1

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20, c, c3, c1, c1

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

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

ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c1
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c1
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20, c, c3, c1, c1, c6

(23) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 30 dangling nodes:

ACTIVE(and(and(tt, z0), x1)) → c1

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20, c, c3, c1, c6

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

Split RHS of tuples not part of any SCC

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
ACTIVE(s(and(tt, z0))) → c2(ACTIVE(and(tt, z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
ACTIVE(s(and(tt, z0))) → c2(ACTIVE(and(tt, z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20, c, c3, c1, c6, c2

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

Removed 1 trailing tuple parts

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
ACTIVE(s(and(tt, z0))) → c2

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
ACTIVE(s(and(tt, z0))) → c2

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c20, c, c3, c1, c6, c2, c2

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

Use narrowing to replace PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
ACTIVE(s(and(tt, z0))) → c2
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
ACTIVE(s(and(tt, z0))) → c2

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c16, c18, c19, c20, c, c3, c1, c6, c2, c2, c14

(31) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 39 dangling nodes:

ACTIVE(s(and(tt, z0))) → c2

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c16, c18, c19, c20, c, c3, c1, c6, c2, c14

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

Removed 4 trailing tuple parts

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c16, c18, c19, c20, c, c3, c1, c6, c2, c14, c14

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

Use narrowing to replace PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c18, c19, c20, c, c3, c1, c6, c2, c14, c14, c16

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

Removed 4 trailing tuple parts

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c18, c19, c20, c, c3, c1, c6, c2, c14, c14, c16, c16

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

Use narrowing to replace PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0)) by

PROPER(s(and(z0, z1))) → c18(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(tt)) → c18(S(ok(tt)), PROPER(tt))
PROPER(s(plus(z0, z1))) → c18(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c18(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c18(S(s(proper(z0))), PROPER(s(z0)))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c18(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(tt)) → c18(S(ok(tt)), PROPER(tt))
PROPER(s(plus(z0, z1))) → c18(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c18(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c18(S(s(proper(z0))), PROPER(s(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, TOP, PROPER

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c19, c20, c, c3, c1, c6, c2, c14, c14, c16, c16, c18

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

Split RHS of tuples not part of any SCC

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c18(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c18(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c18(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(tt)) → c15(S(ok(tt)))
PROPER(s(tt)) → c15(PROPER(tt))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(s(0)) → c15(PROPER(0))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, TOP, PROPER

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c19, c20, c, c3, c1, c6, c2, c14, c14, c16, c16, c18, c15

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

Removed 2 trailing tuple parts

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c18(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c18(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c18(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(tt)) → c15(S(ok(tt)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(s(tt)) → c15
PROPER(s(0)) → c15

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, TOP, PROPER

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c19, c20, c, c3, c1, c6, c2, c14, c14, c16, c16, c18, c15, c15

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

Use narrowing to replace TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(and(z0, z1))) → c19(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(tt)) → c19(TOP(ok(tt)), PROPER(tt))
TOP(mark(plus(z0, z1))) → c19(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(0)) → c19(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c19(TOP(s(proper(z0))), PROPER(s(z0)))

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c18(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c18(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c18(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(tt)) → c15(S(ok(tt)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(s(tt)) → c15
PROPER(s(0)) → c15
TOP(mark(and(z0, z1))) → c19(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(tt)) → c19(TOP(ok(tt)), PROPER(tt))
TOP(mark(plus(z0, z1))) → c19(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(0)) → c19(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c19(TOP(s(proper(z0))), PROPER(s(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, TOP, PROPER

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c20, c, c3, c1, c6, c2, c14, c14, c16, c16, c18, c15, c15, c19

(47) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 57 dangling nodes:

PROPER(s(0)) → c15
PROPER(s(tt)) → c15

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c18(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c18(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c18(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(tt)) → c15(S(ok(tt)))
PROPER(s(0)) → c15(S(ok(0)))
TOP(mark(and(z0, z1))) → c19(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(tt)) → c19(TOP(ok(tt)), PROPER(tt))
TOP(mark(plus(z0, z1))) → c19(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(0)) → c19(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c19(TOP(s(proper(z0))), PROPER(s(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, TOP, PROPER

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c20, c, c3, c1, c6, c2, c14, c14, c16, c16, c18, c15, c19

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

Removed 2 trailing tuple parts

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c18(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c18(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c18(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(tt)) → c15(S(ok(tt)))
PROPER(s(0)) → c15(S(ok(0)))
TOP(mark(and(z0, z1))) → c19(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(plus(z0, z1))) → c19(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(s(z0))) → c19(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(tt)) → c19(TOP(ok(tt)))
TOP(mark(0)) → c19(TOP(ok(0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, TOP, PROPER

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c20, c, c3, c1, c6, c2, c14, c14, c16, c16, c18, c15, c19, c19

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

Use narrowing to replace TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(and(tt, z0))) → c20(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(plus(z0, 0))) → c20(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c20(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
TOP(ok(and(z0, z1))) → c20(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(plus(z0, z1))) → c20(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c20(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(s(z0))) → c20(TOP(s(active(z0))), ACTIVE(s(z0)))

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c18(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c18(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c18(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(tt)) → c15(S(ok(tt)))
PROPER(s(0)) → c15(S(ok(0)))
TOP(mark(and(z0, z1))) → c19(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(plus(z0, z1))) → c19(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(s(z0))) → c19(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(tt)) → c19(TOP(ok(tt)))
TOP(mark(0)) → c19(TOP(ok(0)))
TOP(ok(and(tt, z0))) → c20(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(plus(z0, 0))) → c20(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c20(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
TOP(ok(and(z0, z1))) → c20(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(plus(z0, z1))) → c20(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c20(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(s(z0))) → c20(TOP(s(active(z0))), ACTIVE(s(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
TOP(ok(and(tt, z0))) → c20(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(plus(z0, 0))) → c20(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c20(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
TOP(ok(and(z0, z1))) → c20(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(plus(z0, z1))) → c20(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c20(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(s(z0))) → c20(TOP(s(active(z0))), ACTIVE(s(z0)))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

c4, c5, c7, c8, c9, c10, c11, c12, c13, c, c3, c1, c6, c2, c14, c14, c16, c16, c18, c15, c19, c19, c20

(53) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(plus(z0, s(z1))) → c(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c1(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
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(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c2(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c18(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c18(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c18(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(tt)) → c15(S(ok(tt)))
PROPER(s(0)) → c15(S(ok(0)))
TOP(mark(and(z0, z1))) → c19(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(plus(z0, z1))) → c19(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(s(z0))) → c19(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(ok(and(tt, z0))) → c20(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(plus(z0, 0))) → c20(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c20(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
TOP(ok(and(z0, z1))) → c20(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(plus(z0, z1))) → c20(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c20(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(s(z0))) → c20(TOP(s(active(z0))), ACTIVE(s(z0)))

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(mark(tt)) → c19(TOP(ok(tt)))
TOP(mark(0)) → c19(TOP(ok(0)))

S tuples:

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

AND, PLUS, S, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c19

(55) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 9 dangling nodes:

TOP(mark(tt)) → c19(TOP(ok(tt)))
TOP(mark(0)) → c19(TOP(ok(0)))

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

S tuples:

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

AND, PLUS, S

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13

(57) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

AND(ok(z0), ok(z1)) → c8(AND(z0, z1))

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

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

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

POL(AND(x₁, x₂)) = [3]x₂ + [3]x₂²
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [3] + x₁

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

S tuples:

AND(mark(z0), z1) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

K tuples:

AND(ok(z0), ok(z1)) → c8(AND(z0, z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

AND, PLUS, S

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13

(59) 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.

AND(mark(z0), z1) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))

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

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

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

POL(AND(x₁, x₂)) = x₁ + [3]x₂
POL(PLUS(x₁, x₂)) = x₁ + [4]x₂
POL(S(x₁)) = 0
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = x₁

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

S tuples:

PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

K tuples:

AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

AND, PLUS, S

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13

(61) 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.

PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))

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

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

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

POL(AND(x₁, x₂)) = [4]x₁ + [2]x₂
POL(PLUS(x₁, x₂)) = [3]x₁ + [3]x₂
POL(S(x₁)) = 0
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [4] + x₁
POL(ok(x₁)) = [4] + x₁

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

S tuples:

PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

K tuples:

AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

AND, PLUS, S

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13

(63) 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.

PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))

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

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

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

POL(AND(x₁, x₂)) = [3]x₂
POL(PLUS(x₁, x₂)) = [4]x₂
POL(S(x₁)) = 0
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [4] + x₁
POL(ok(x₁)) = x₁

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

S tuples:

S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

K tuples:

AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

AND, PLUS, S

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13

(65) 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.

S(mark(z0)) → c12(S(z0))

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

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

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

POL(AND(x₁, x₂)) = [2]x₂
POL(PLUS(x₁, x₂)) = [4]x₂
POL(S(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

S tuples:

S(ok(z0)) → c13(S(z0))

K tuples:

AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

AND, PLUS, S

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13

(67) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

S(ok(z0)) → c13(S(z0))

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

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

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

POL(AND(x₁, x₂)) = x₁ + [3]x₁·x₂
POL(PLUS(x₁, x₂)) = x₂² + x₁²
POL(S(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [1] + x₁

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
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))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

S tuples:none
K tuples:

AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

AND, PLUS, S

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13

(69) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(10) Obligation:

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(12) Obligation:

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

(14) Obligation:

(15) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

(23) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

(31) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

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

(44) Obligation:

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

(46) Obligation:

(47) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(48) Obligation:

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

(50) Obligation:

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

(52) Obligation:

(53) CdtUnreachableProof (EQUIVALENT transformation)

(54) Obligation:

(55) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(56) Obligation:

(57) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(58) Obligation:

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

(60) Obligation:

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

(62) Obligation:

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

(64) Obligation:

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

(66) Obligation:

(67) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(68) Obligation:

(69) SIsEmptyProof (EQUIVALENT transformation)

(70) BOUNDS(O(1), O(1))