WORST_CASE(Omega(n^1),O(n^2))
proof of input_f1muAch4nB.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 318 ms]
(2) CpxRelTRS
(3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(4) CdtProblem
    (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 87 ms]
    (14) CdtProblem
    (15) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CdtProblem
    (17) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 231 ms]
    (20) CdtProblem
    (21) CdtKnowledgeProof [FINISHED, 0 ms]
    (22) BOUNDS(1, 1)
(23) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CdtProblem
    (25) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRelTRS
    (27) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CpxRelTRS
    (29) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) typed CpxTrs
    (31) OrderProof [LOWER BOUND(ID), 0 ms]
    (32) typed CpxTrs
    (33) RewriteLemmaProof [LOWER BOUND(ID), 590 ms]
    (34) BEST
        (35) proven lower bound
            (36) LowerBoundPropagationProof [FINISHED, 0 ms]
            (37) BOUNDS(n^1, INF)
        (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 42 ms]
            (40) typed CpxTrs
            (41) RewriteLemmaProof [LOWER BOUND(ID), 56 ms]
            (42) typed CpxTrs
            (43) RewriteLemmaProof [LOWER BOUND(ID), 175 ms]
            (44) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   monus(S(x'), S(x)) -> monus(x', x)
   gcd(x, y) -> gcd[Ite](equal0(x, y), x, y)
   equal0(a, b) -> equal0[Ite](<(a, b), a, b)

The (relative) TRS S consists of the following rules:

   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   gcd[Ite](False, x, y) -> gcd[False][Ite](<(x, y), x, y)
   gcd[Ite](True, x, y) -> x
   gcd[False][Ite](False, x, y) -> gcd(y, monus(y, x))
   gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y)
   equal0[Ite](False, a, b) -> False
   equal0[Ite](True, a, b) -> equal0[True][Ite](<(b, a), a, b)
   equal0[True][Ite](False, a, b) -> False
   equal0[True][Ite](True, a, b) -> True

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   monus(S(x'), S(x)) -> monus(x', x)
   gcd(x, y) -> gcd[Ite](equal0(x, y), x, y)
   equal0(a, b) -> equal0[Ite](<(a, b), a, b)

The (relative) TRS S consists of the following rules:

   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   gcd[Ite](False, x, y) -> gcd[False][Ite](<(x, y), x, y)
   gcd[Ite](True, x, y) -> x
   gcd[False][Ite](False, x, y) -> gcd(y, monus(y, x))
   gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y)
   equal0[Ite](False, a, b) -> False
   equal0[Ite](True, a, b) -> equal0[True][Ite](<(b, a), a, b)
   equal0[True][Ite](False, a, b) -> False
   equal0[True][Ite](True, a, b) -> True

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
   gcd[Ite](True, z0, z1) -> z0
   gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
   gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
   equal0[Ite](False, z0, z1) -> False
   equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
   equal0[True][Ite](False, z0, z1) -> False
   equal0[True][Ite](True, z0, z1) -> True
   monus(S(z0), S(z1)) -> monus(z0, z1)
   gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
   equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   <'(0, S(z0)) -> c1
   <'(z0, 0) -> c2
   GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
   GCD[ITE](True, z0, z1) -> c4
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   EQUAL0[ITE](False, z0, z1) -> c7
   EQUAL0[ITE](True, z0, z1) -> c8(EQUAL0[TRUE][ITE](<(z1, z0), z0, z1), <'(z1, z0))
   EQUAL0[TRUE][ITE](False, z0, z1) -> c9
   EQUAL0[TRUE][ITE](True, z0, z1) -> c10
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
S tuples:
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
K tuples:none
Defined Rule Symbols:   monus_2, gcd_2, equal0_2, <_2, gcd[Ite]_3, gcd[False][Ite]_3, equal0[Ite]_3, equal0[True][Ite]_3

Defined Pair Symbols:   <'_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, EQUAL0[ITE]_3, EQUAL0[TRUE][ITE]_3, MONUS_2, GCD_2, EQUAL0_2

Compound Symbols:   c_1, c1, c2, c3_2, c4, c5_2, c6_2, c7, c8_2, c9, c10, c11_1, c12_2, c13_2


----------------------------------------

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   EQUAL0[ITE](False, z0, z1) -> c7
   <'(z0, 0) -> c2
   GCD[ITE](True, z0, z1) -> c4
   <'(0, S(z0)) -> c1
   EQUAL0[TRUE][ITE](True, z0, z1) -> c10
   EQUAL0[TRUE][ITE](False, z0, z1) -> c9

----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
   gcd[Ite](True, z0, z1) -> z0
   gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
   gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
   equal0[Ite](False, z0, z1) -> False
   equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
   equal0[True][Ite](False, z0, z1) -> False
   equal0[True][Ite](True, z0, z1) -> True
   monus(S(z0), S(z1)) -> monus(z0, z1)
   gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
   equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   EQUAL0[ITE](True, z0, z1) -> c8(EQUAL0[TRUE][ITE](<(z1, z0), z0, z1), <'(z1, z0))
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
S tuples:
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
K tuples:none
Defined Rule Symbols:   monus_2, gcd_2, equal0_2, <_2, gcd[Ite]_3, gcd[False][Ite]_3, equal0[Ite]_3, equal0[True][Ite]_3

Defined Pair Symbols:   <'_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, EQUAL0[ITE]_3, MONUS_2, GCD_2, EQUAL0_2

Compound Symbols:   c_1, c3_2, c5_2, c6_2, c8_2, c11_1, c12_2, c13_2


----------------------------------------

(7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
   gcd[Ite](True, z0, z1) -> z0
   gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
   gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
   equal0[Ite](False, z0, z1) -> False
   equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
   equal0[True][Ite](False, z0, z1) -> False
   equal0[True][Ite](True, z0, z1) -> True
   monus(S(z0), S(z1)) -> monus(z0, z1)
   gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
   equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
   EQUAL0[ITE](True, z0, z1) -> c8(<'(z1, z0))
S tuples:
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
K tuples:none
Defined Rule Symbols:   monus_2, gcd_2, equal0_2, <_2, gcd[Ite]_3, gcd[False][Ite]_3, equal0[Ite]_3, equal0[True][Ite]_3

Defined Pair Symbols:   <'_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, MONUS_2, GCD_2, EQUAL0_2, EQUAL0[ITE]_3

Compound Symbols:   c_1, c3_2, c5_2, c6_2, c11_1, c12_2, c13_2, c8_1


----------------------------------------

(9) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
   gcd[Ite](True, z0, z1) -> z0
   gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
   gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
   equal0[Ite](False, z0, z1) -> False
   equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
   equal0[True][Ite](False, z0, z1) -> False
   equal0[True][Ite](True, z0, z1) -> True
   monus(S(z0), S(z1)) -> monus(z0, z1)
   gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
   equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0[ITE](True, z0, z1) -> c8(<'(z1, z0))
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
S tuples:
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
K tuples:none
Defined Rule Symbols:   monus_2, gcd_2, equal0_2, <_2, gcd[Ite]_3, gcd[False][Ite]_3, equal0[Ite]_3, equal0[True][Ite]_3

Defined Pair Symbols:   <'_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, MONUS_2, GCD_2, EQUAL0[ITE]_3, EQUAL0_2

Compound Symbols:   c_1, c3_2, c5_2, c6_2, c11_1, c12_2, c8_1, c1_1


----------------------------------------

(11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
   gcd[Ite](True, z0, z1) -> z0
   gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
   gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
   gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)

----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   monus(S(z0), S(z1)) -> monus(z0, z1)
   equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)
   equal0[Ite](False, z0, z1) -> False
   equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
   equal0[True][Ite](False, z0, z1) -> False
   equal0[True][Ite](True, z0, z1) -> True
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0[ITE](True, z0, z1) -> c8(<'(z1, z0))
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
S tuples:
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
K tuples:none
Defined Rule Symbols:   <_2, monus_2, equal0_2, equal0[Ite]_3, equal0[True][Ite]_3

Defined Pair Symbols:   <'_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, MONUS_2, GCD_2, EQUAL0[ITE]_3, EQUAL0_2

Compound Symbols:   c_1, c3_2, c5_2, c6_2, c11_1, c12_2, c8_1, c1_1


----------------------------------------

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
We considered the (Usable) Rules:
   monus(S(z0), S(z1)) -> monus(z0, z1)
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0[ITE](True, z0, z1) -> c8(<'(z1, z0))
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<(x_1, x_2)) = 0
   POL(<'(x_1, x_2)) = 0
   POL(EQUAL0(x_1, x_2)) = 0
   POL(EQUAL0[ITE](x_1, x_2, x_3)) = 0
   POL(False) = 0
   POL(GCD(x_1, x_2)) = x_1 + [2]x_2
   POL(GCD[FALSE][ITE](x_1, x_2, x_3)) = x_2 + [2]x_3
   POL(GCD[ITE](x_1, x_2, x_3)) = x_2 + [2]x_3
   POL(MONUS(x_1, x_2)) = x_1
   POL(S(x_1)) = [2] + x_1
   POL(True) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(equal0(x_1, x_2)) = [1] + x_1 + [3]x_2
   POL(equal0[Ite](x_1, x_2, x_3)) = [3] + [3]x_2 + [3]x_3
   POL(equal0[True][Ite](x_1, x_2, x_3)) = [3] + [3]x_2 + [3]x_3
   POL(monus(x_1, x_2)) = 0

----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   monus(S(z0), S(z1)) -> monus(z0, z1)
   equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)
   equal0[Ite](False, z0, z1) -> False
   equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
   equal0[True][Ite](False, z0, z1) -> False
   equal0[True][Ite](True, z0, z1) -> True
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0[ITE](True, z0, z1) -> c8(<'(z1, z0))
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
S tuples:
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
K tuples:
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
Defined Rule Symbols:   <_2, monus_2, equal0_2, equal0[Ite]_3, equal0[True][Ite]_3

Defined Pair Symbols:   <'_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, MONUS_2, GCD_2, EQUAL0[ITE]_3, EQUAL0_2

Compound Symbols:   c_1, c3_2, c5_2, c6_2, c11_1, c12_2, c8_1, c1_1


----------------------------------------

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1)) by 
   GCD[ITE](False, S(z0), S(z1)) -> c3(GCD[FALSE][ITE](<(z0, z1), S(z0), S(z1)), <'(S(z0), S(z1)))
   GCD[ITE](False, 0, S(z0)) -> c3(GCD[FALSE][ITE](True, 0, S(z0)), <'(0, S(z0)))
   GCD[ITE](False, z0, 0) -> c3(GCD[FALSE][ITE](False, z0, 0), <'(z0, 0))

----------------------------------------

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   monus(S(z0), S(z1)) -> monus(z0, z1)
   equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)
   equal0[Ite](False, z0, z1) -> False
   equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
   equal0[True][Ite](False, z0, z1) -> False
   equal0[True][Ite](True, z0, z1) -> True
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0[ITE](True, z0, z1) -> c8(<'(z1, z0))
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
   GCD[ITE](False, S(z0), S(z1)) -> c3(GCD[FALSE][ITE](<(z0, z1), S(z0), S(z1)), <'(S(z0), S(z1)))
   GCD[ITE](False, 0, S(z0)) -> c3(GCD[FALSE][ITE](True, 0, S(z0)), <'(0, S(z0)))
   GCD[ITE](False, z0, 0) -> c3(GCD[FALSE][ITE](False, z0, 0), <'(z0, 0))
S tuples:
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
K tuples:
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
Defined Rule Symbols:   <_2, monus_2, equal0_2, equal0[Ite]_3, equal0[True][Ite]_3

Defined Pair Symbols:   <'_2, GCD[FALSE][ITE]_3, MONUS_2, GCD_2, EQUAL0[ITE]_3, EQUAL0_2, GCD[ITE]_3

Compound Symbols:   c_1, c5_2, c6_2, c11_1, c12_2, c8_1, c1_1, c3_2


----------------------------------------

(17) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   monus(S(z0), S(z1)) -> monus(z0, z1)
   equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)
   equal0[Ite](False, z0, z1) -> False
   equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
   equal0[True][Ite](False, z0, z1) -> False
   equal0[True][Ite](True, z0, z1) -> True
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0[ITE](True, z0, z1) -> c8(<'(z1, z0))
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
   GCD[ITE](False, S(z0), S(z1)) -> c3(GCD[FALSE][ITE](<(z0, z1), S(z0), S(z1)), <'(S(z0), S(z1)))
   GCD[ITE](False, 0, S(z0)) -> c3(GCD[FALSE][ITE](True, 0, S(z0)))
   GCD[ITE](False, z0, 0) -> c3(GCD[FALSE][ITE](False, z0, 0))
S tuples:
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
K tuples:
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
Defined Rule Symbols:   <_2, monus_2, equal0_2, equal0[Ite]_3, equal0[True][Ite]_3

Defined Pair Symbols:   <'_2, GCD[FALSE][ITE]_3, MONUS_2, GCD_2, EQUAL0[ITE]_3, EQUAL0_2, GCD[ITE]_3

Compound Symbols:   c_1, c5_2, c6_2, c11_1, c12_2, c8_1, c1_1, c3_2, c3_1


----------------------------------------

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
We considered the (Usable) Rules:
   monus(S(z0), S(z1)) -> monus(z0, z1)
   <(z0, 0) -> False
   <(0, S(z0)) -> True
   <(S(z0), S(z1)) -> <(z0, z1)
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0[ITE](True, z0, z1) -> c8(<'(z1, z0))
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
   GCD[ITE](False, S(z0), S(z1)) -> c3(GCD[FALSE][ITE](<(z0, z1), S(z0), S(z1)), <'(S(z0), S(z1)))
   GCD[ITE](False, 0, S(z0)) -> c3(GCD[FALSE][ITE](True, 0, S(z0)))
   GCD[ITE](False, z0, 0) -> c3(GCD[FALSE][ITE](False, z0, 0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [2]
   POL(<(x_1, x_2)) = x_1
   POL(<'(x_1, x_2)) = 0
   POL(EQUAL0(x_1, x_2)) = 0
   POL(EQUAL0[ITE](x_1, x_2, x_3)) = 0
   POL(False) = 0
   POL(GCD(x_1, x_2)) = [1] + x_2 + [2]x_1*x_2
   POL(GCD[FALSE][ITE](x_1, x_2, x_3)) = [1] + [2]x_1*x_3
   POL(GCD[ITE](x_1, x_2, x_3)) = x_3 + [2]x_2*x_3
   POL(MONUS(x_1, x_2)) = 0
   POL(S(x_1)) = [1] + x_1
   POL(True) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(equal0(x_1, x_2)) = 0
   POL(equal0[Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 + x_3^2 + x_2*x_3 + x_2^2
   POL(equal0[True][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 + x_3^2 + x_2*x_3 + x_2^2
   POL(monus(x_1, x_2)) = 0

----------------------------------------

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   monus(S(z0), S(z1)) -> monus(z0, z1)
   equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)
   equal0[Ite](False, z0, z1) -> False
   equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
   equal0[True][Ite](False, z0, z1) -> False
   equal0[True][Ite](True, z0, z1) -> True
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0[ITE](True, z0, z1) -> c8(<'(z1, z0))
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
   GCD[ITE](False, S(z0), S(z1)) -> c3(GCD[FALSE][ITE](<(z0, z1), S(z0), S(z1)), <'(S(z0), S(z1)))
   GCD[ITE](False, 0, S(z0)) -> c3(GCD[FALSE][ITE](True, 0, S(z0)))
   GCD[ITE](False, z0, 0) -> c3(GCD[FALSE][ITE](False, z0, 0))
S tuples:
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
K tuples:
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
Defined Rule Symbols:   <_2, monus_2, equal0_2, equal0[Ite]_3, equal0[True][Ite]_3

Defined Pair Symbols:   <'_2, GCD[FALSE][ITE]_3, MONUS_2, GCD_2, EQUAL0[ITE]_3, EQUAL0_2, GCD[ITE]_3

Compound Symbols:   c_1, c5_2, c6_2, c11_1, c12_2, c8_1, c1_1, c3_2, c3_1


----------------------------------------

(21) CdtKnowledgeProof (FINISHED)
The following tuples could be moved from S to K by knowledge propagation:
   EQUAL0(z0, z1) -> c1(EQUAL0[ITE](<(z0, z1), z0, z1))
   EQUAL0(z0, z1) -> c1(<'(z0, z1))
   EQUAL0[ITE](True, z0, z1) -> c8(<'(z1, z0))
Now S is empty
----------------------------------------

(22)
BOUNDS(1, 1)

----------------------------------------

(23) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
   gcd[Ite](True, z0, z1) -> z0
   gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
   gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
   equal0[Ite](False, z0, z1) -> False
   equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
   equal0[True][Ite](False, z0, z1) -> False
   equal0[True][Ite](True, z0, z1) -> True
   monus(S(z0), S(z1)) -> monus(z0, z1)
   gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
   equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   <'(0, S(z0)) -> c1
   <'(z0, 0) -> c2
   GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
   GCD[ITE](True, z0, z1) -> c4
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   EQUAL0[ITE](False, z0, z1) -> c7
   EQUAL0[ITE](True, z0, z1) -> c8(EQUAL0[TRUE][ITE](<(z1, z0), z0, z1), <'(z1, z0))
   EQUAL0[TRUE][ITE](False, z0, z1) -> c9
   EQUAL0[TRUE][ITE](True, z0, z1) -> c10
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
S tuples:
   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
K tuples:none
Defined Rule Symbols:   monus_2, gcd_2, equal0_2, <_2, gcd[Ite]_3, gcd[False][Ite]_3, equal0[Ite]_3, equal0[True][Ite]_3

Defined Pair Symbols:   <'_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, EQUAL0[ITE]_3, EQUAL0[TRUE][ITE]_3, MONUS_2, GCD_2, EQUAL0_2

Compound Symbols:   c_1, c1, c2, c3_2, c4, c5_2, c6_2, c7, c8_2, c9, c10, c11_1, c12_2, c13_2


----------------------------------------

(25) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(26)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))

The (relative) TRS S consists of the following rules:

   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   <'(0, S(z0)) -> c1
   <'(z0, 0) -> c2
   GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
   GCD[ITE](True, z0, z1) -> c4
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   EQUAL0[ITE](False, z0, z1) -> c7
   EQUAL0[ITE](True, z0, z1) -> c8(EQUAL0[TRUE][ITE](<(z1, z0), z0, z1), <'(z1, z0))
   EQUAL0[TRUE][ITE](False, z0, z1) -> c9
   EQUAL0[TRUE][ITE](True, z0, z1) -> c10
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
   gcd[Ite](True, z0, z1) -> z0
   gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
   gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
   equal0[Ite](False, z0, z1) -> False
   equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
   equal0[True][Ite](False, z0, z1) -> False
   equal0[True][Ite](True, z0, z1) -> True
   monus(S(z0), S(z1)) -> monus(z0, z1)
   gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
   equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(27) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(28)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
   GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
   EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))

The (relative) TRS S consists of the following rules:

   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   <'(0', S(z0)) -> c1
   <'(z0, 0') -> c2
   GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
   GCD[ITE](True, z0, z1) -> c4
   GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
   EQUAL0[ITE](False, z0, z1) -> c7
   EQUAL0[ITE](True, z0, z1) -> c8(EQUAL0[TRUE][ITE](<(z1, z0), z0, z1), <'(z1, z0))
   EQUAL0[TRUE][ITE](False, z0, z1) -> c9
   EQUAL0[TRUE][ITE](True, z0, z1) -> c10
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0', S(z0)) -> True
   <(z0, 0') -> False
   gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
   gcd[Ite](True, z0, z1) -> z0
   gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
   gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
   equal0[Ite](False, z0, z1) -> False
   equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
   equal0[True][Ite](False, z0, z1) -> False
   equal0[True][Ite](True, z0, z1) -> True
   monus(S(z0), S(z1)) -> monus(z0, z1)
   gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
   equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(29) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(30)
Obligation:
Innermost TRS:
Rules:
MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
<'(S(z0), S(z1)) -> c(<'(z0, z1))
<'(0', S(z0)) -> c1
<'(z0, 0') -> c2
GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
GCD[ITE](True, z0, z1) -> c4
GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
EQUAL0[ITE](False, z0, z1) -> c7
EQUAL0[ITE](True, z0, z1) -> c8(EQUAL0[TRUE][ITE](<(z1, z0), z0, z1), <'(z1, z0))
EQUAL0[TRUE][ITE](False, z0, z1) -> c9
EQUAL0[TRUE][ITE](True, z0, z1) -> c10
<(S(z0), S(z1)) -> <(z0, z1)
<(0', S(z0)) -> True
<(z0, 0') -> False
gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
gcd[Ite](True, z0, z1) -> z0
gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
equal0[Ite](False, z0, z1) -> False
equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
equal0[True][Ite](False, z0, z1) -> False
equal0[True][Ite](True, z0, z1) -> True
monus(S(z0), S(z1)) -> monus(z0, z1)
gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)

Types:
MONUS :: S:0' -> S:0' -> c11
S :: S:0' -> S:0'
c11 :: c11 -> c11
GCD :: S:0' -> S:0' -> c12
c12 :: c3:c4 -> c13 -> c12
GCD[ITE] :: False:True -> S:0' -> S:0' -> c3:c4
equal0 :: S:0' -> S:0' -> False:True
EQUAL0 :: S:0' -> S:0' -> c13
c13 :: c7:c8 -> c:c1:c2 -> c13
EQUAL0[ITE] :: False:True -> S:0' -> S:0' -> c7:c8
< :: S:0' -> S:0' -> False:True
<' :: S:0' -> S:0' -> c:c1:c2
c :: c:c1:c2 -> c:c1:c2
0' :: S:0'
c1 :: c:c1:c2
c2 :: c:c1:c2
False :: False:True
c3 :: c5:c6 -> c:c1:c2 -> c3:c4
GCD[FALSE][ITE] :: False:True -> S:0' -> S:0' -> c5:c6
True :: False:True
c4 :: c3:c4
c5 :: c12 -> c11 -> c5:c6
monus :: S:0' -> S:0' -> S:0'
c6 :: c12 -> c11 -> c5:c6
c7 :: c7:c8
c8 :: c9:c10 -> c:c1:c2 -> c7:c8
EQUAL0[TRUE][ITE] :: False:True -> S:0' -> S:0' -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
gcd[Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd[False][Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd :: S:0' -> S:0' -> S:0'
equal0[Ite] :: False:True -> S:0' -> S:0' -> False:True
equal0[True][Ite] :: False:True -> S:0' -> S:0' -> False:True
hole_c111_14 :: c11
hole_S:0'2_14 :: S:0'
hole_c123_14 :: c12
hole_c3:c44_14 :: c3:c4
hole_c135_14 :: c13
hole_False:True6_14 :: False:True
hole_c7:c87_14 :: c7:c8
hole_c:c1:c28_14 :: c:c1:c2
hole_c5:c69_14 :: c5:c6
hole_c9:c1010_14 :: c9:c10
gen_c1111_14 :: Nat -> c11
gen_S:0'12_14 :: Nat -> S:0'
gen_c:c1:c213_14 :: Nat -> c:c1:c2

----------------------------------------

(31) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MONUS, GCD, <, <', monus, gcd

They will be analysed ascendingly in the following order:
MONUS < GCD
< < GCD
<' < GCD
monus < GCD
< < gcd
monus < gcd

----------------------------------------

(32)
Obligation:
Innermost TRS:
Rules:
MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
<'(S(z0), S(z1)) -> c(<'(z0, z1))
<'(0', S(z0)) -> c1
<'(z0, 0') -> c2
GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
GCD[ITE](True, z0, z1) -> c4
GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
EQUAL0[ITE](False, z0, z1) -> c7
EQUAL0[ITE](True, z0, z1) -> c8(EQUAL0[TRUE][ITE](<(z1, z0), z0, z1), <'(z1, z0))
EQUAL0[TRUE][ITE](False, z0, z1) -> c9
EQUAL0[TRUE][ITE](True, z0, z1) -> c10
<(S(z0), S(z1)) -> <(z0, z1)
<(0', S(z0)) -> True
<(z0, 0') -> False
gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
gcd[Ite](True, z0, z1) -> z0
gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
equal0[Ite](False, z0, z1) -> False
equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
equal0[True][Ite](False, z0, z1) -> False
equal0[True][Ite](True, z0, z1) -> True
monus(S(z0), S(z1)) -> monus(z0, z1)
gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)

Types:
MONUS :: S:0' -> S:0' -> c11
S :: S:0' -> S:0'
c11 :: c11 -> c11
GCD :: S:0' -> S:0' -> c12
c12 :: c3:c4 -> c13 -> c12
GCD[ITE] :: False:True -> S:0' -> S:0' -> c3:c4
equal0 :: S:0' -> S:0' -> False:True
EQUAL0 :: S:0' -> S:0' -> c13
c13 :: c7:c8 -> c:c1:c2 -> c13
EQUAL0[ITE] :: False:True -> S:0' -> S:0' -> c7:c8
< :: S:0' -> S:0' -> False:True
<' :: S:0' -> S:0' -> c:c1:c2
c :: c:c1:c2 -> c:c1:c2
0' :: S:0'
c1 :: c:c1:c2
c2 :: c:c1:c2
False :: False:True
c3 :: c5:c6 -> c:c1:c2 -> c3:c4
GCD[FALSE][ITE] :: False:True -> S:0' -> S:0' -> c5:c6
True :: False:True
c4 :: c3:c4
c5 :: c12 -> c11 -> c5:c6
monus :: S:0' -> S:0' -> S:0'
c6 :: c12 -> c11 -> c5:c6
c7 :: c7:c8
c8 :: c9:c10 -> c:c1:c2 -> c7:c8
EQUAL0[TRUE][ITE] :: False:True -> S:0' -> S:0' -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
gcd[Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd[False][Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd :: S:0' -> S:0' -> S:0'
equal0[Ite] :: False:True -> S:0' -> S:0' -> False:True
equal0[True][Ite] :: False:True -> S:0' -> S:0' -> False:True
hole_c111_14 :: c11
hole_S:0'2_14 :: S:0'
hole_c123_14 :: c12
hole_c3:c44_14 :: c3:c4
hole_c135_14 :: c13
hole_False:True6_14 :: False:True
hole_c7:c87_14 :: c7:c8
hole_c:c1:c28_14 :: c:c1:c2
hole_c5:c69_14 :: c5:c6
hole_c9:c1010_14 :: c9:c10
gen_c1111_14 :: Nat -> c11
gen_S:0'12_14 :: Nat -> S:0'
gen_c:c1:c213_14 :: Nat -> c:c1:c2


Generator Equations:
gen_c1111_14(0) <=> hole_c111_14
gen_c1111_14(+(x, 1)) <=> c11(gen_c1111_14(x))
gen_S:0'12_14(0) <=> 0'
gen_S:0'12_14(+(x, 1)) <=> S(gen_S:0'12_14(x))
gen_c:c1:c213_14(0) <=> c1
gen_c:c1:c213_14(+(x, 1)) <=> c(gen_c:c1:c213_14(x))


The following defined symbols remain to be analysed:
MONUS, GCD, <, <', monus, gcd

They will be analysed ascendingly in the following order:
MONUS < GCD
< < GCD
<' < GCD
monus < GCD
< < gcd
monus < gcd

----------------------------------------

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MONUS(gen_S:0'12_14(+(1, n15_14)), gen_S:0'12_14(+(1, n15_14))) -> *14_14, rt in Omega(n15_14)

Induction Base:
MONUS(gen_S:0'12_14(+(1, 0)), gen_S:0'12_14(+(1, 0)))

Induction Step:
MONUS(gen_S:0'12_14(+(1, +(n15_14, 1))), gen_S:0'12_14(+(1, +(n15_14, 1)))) ->_R^Omega(1)
c11(MONUS(gen_S:0'12_14(+(1, n15_14)), gen_S:0'12_14(+(1, n15_14)))) ->_IH
c11(*14_14)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(34)
Complex Obligation (BEST)

----------------------------------------

(35)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
<'(S(z0), S(z1)) -> c(<'(z0, z1))
<'(0', S(z0)) -> c1
<'(z0, 0') -> c2
GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
GCD[ITE](True, z0, z1) -> c4
GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
EQUAL0[ITE](False, z0, z1) -> c7
EQUAL0[ITE](True, z0, z1) -> c8(EQUAL0[TRUE][ITE](<(z1, z0), z0, z1), <'(z1, z0))
EQUAL0[TRUE][ITE](False, z0, z1) -> c9
EQUAL0[TRUE][ITE](True, z0, z1) -> c10
<(S(z0), S(z1)) -> <(z0, z1)
<(0', S(z0)) -> True
<(z0, 0') -> False
gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
gcd[Ite](True, z0, z1) -> z0
gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
equal0[Ite](False, z0, z1) -> False
equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
equal0[True][Ite](False, z0, z1) -> False
equal0[True][Ite](True, z0, z1) -> True
monus(S(z0), S(z1)) -> monus(z0, z1)
gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)

Types:
MONUS :: S:0' -> S:0' -> c11
S :: S:0' -> S:0'
c11 :: c11 -> c11
GCD :: S:0' -> S:0' -> c12
c12 :: c3:c4 -> c13 -> c12
GCD[ITE] :: False:True -> S:0' -> S:0' -> c3:c4
equal0 :: S:0' -> S:0' -> False:True
EQUAL0 :: S:0' -> S:0' -> c13
c13 :: c7:c8 -> c:c1:c2 -> c13
EQUAL0[ITE] :: False:True -> S:0' -> S:0' -> c7:c8
< :: S:0' -> S:0' -> False:True
<' :: S:0' -> S:0' -> c:c1:c2
c :: c:c1:c2 -> c:c1:c2
0' :: S:0'
c1 :: c:c1:c2
c2 :: c:c1:c2
False :: False:True
c3 :: c5:c6 -> c:c1:c2 -> c3:c4
GCD[FALSE][ITE] :: False:True -> S:0' -> S:0' -> c5:c6
True :: False:True
c4 :: c3:c4
c5 :: c12 -> c11 -> c5:c6
monus :: S:0' -> S:0' -> S:0'
c6 :: c12 -> c11 -> c5:c6
c7 :: c7:c8
c8 :: c9:c10 -> c:c1:c2 -> c7:c8
EQUAL0[TRUE][ITE] :: False:True -> S:0' -> S:0' -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
gcd[Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd[False][Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd :: S:0' -> S:0' -> S:0'
equal0[Ite] :: False:True -> S:0' -> S:0' -> False:True
equal0[True][Ite] :: False:True -> S:0' -> S:0' -> False:True
hole_c111_14 :: c11
hole_S:0'2_14 :: S:0'
hole_c123_14 :: c12
hole_c3:c44_14 :: c3:c4
hole_c135_14 :: c13
hole_False:True6_14 :: False:True
hole_c7:c87_14 :: c7:c8
hole_c:c1:c28_14 :: c:c1:c2
hole_c5:c69_14 :: c5:c6
hole_c9:c1010_14 :: c9:c10
gen_c1111_14 :: Nat -> c11
gen_S:0'12_14 :: Nat -> S:0'
gen_c:c1:c213_14 :: Nat -> c:c1:c2


Generator Equations:
gen_c1111_14(0) <=> hole_c111_14
gen_c1111_14(+(x, 1)) <=> c11(gen_c1111_14(x))
gen_S:0'12_14(0) <=> 0'
gen_S:0'12_14(+(x, 1)) <=> S(gen_S:0'12_14(x))
gen_c:c1:c213_14(0) <=> c1
gen_c:c1:c213_14(+(x, 1)) <=> c(gen_c:c1:c213_14(x))


The following defined symbols remain to be analysed:
MONUS, GCD, <, <', monus, gcd

They will be analysed ascendingly in the following order:
MONUS < GCD
< < GCD
<' < GCD
monus < GCD
< < gcd
monus < gcd

----------------------------------------

(36) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(37)
BOUNDS(n^1, INF)

----------------------------------------

(38)
Obligation:
Innermost TRS:
Rules:
MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
<'(S(z0), S(z1)) -> c(<'(z0, z1))
<'(0', S(z0)) -> c1
<'(z0, 0') -> c2
GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
GCD[ITE](True, z0, z1) -> c4
GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
EQUAL0[ITE](False, z0, z1) -> c7
EQUAL0[ITE](True, z0, z1) -> c8(EQUAL0[TRUE][ITE](<(z1, z0), z0, z1), <'(z1, z0))
EQUAL0[TRUE][ITE](False, z0, z1) -> c9
EQUAL0[TRUE][ITE](True, z0, z1) -> c10
<(S(z0), S(z1)) -> <(z0, z1)
<(0', S(z0)) -> True
<(z0, 0') -> False
gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
gcd[Ite](True, z0, z1) -> z0
gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
equal0[Ite](False, z0, z1) -> False
equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
equal0[True][Ite](False, z0, z1) -> False
equal0[True][Ite](True, z0, z1) -> True
monus(S(z0), S(z1)) -> monus(z0, z1)
gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)

Types:
MONUS :: S:0' -> S:0' -> c11
S :: S:0' -> S:0'
c11 :: c11 -> c11
GCD :: S:0' -> S:0' -> c12
c12 :: c3:c4 -> c13 -> c12
GCD[ITE] :: False:True -> S:0' -> S:0' -> c3:c4
equal0 :: S:0' -> S:0' -> False:True
EQUAL0 :: S:0' -> S:0' -> c13
c13 :: c7:c8 -> c:c1:c2 -> c13
EQUAL0[ITE] :: False:True -> S:0' -> S:0' -> c7:c8
< :: S:0' -> S:0' -> False:True
<' :: S:0' -> S:0' -> c:c1:c2
c :: c:c1:c2 -> c:c1:c2
0' :: S:0'
c1 :: c:c1:c2
c2 :: c:c1:c2
False :: False:True
c3 :: c5:c6 -> c:c1:c2 -> c3:c4
GCD[FALSE][ITE] :: False:True -> S:0' -> S:0' -> c5:c6
True :: False:True
c4 :: c3:c4
c5 :: c12 -> c11 -> c5:c6
monus :: S:0' -> S:0' -> S:0'
c6 :: c12 -> c11 -> c5:c6
c7 :: c7:c8
c8 :: c9:c10 -> c:c1:c2 -> c7:c8
EQUAL0[TRUE][ITE] :: False:True -> S:0' -> S:0' -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
gcd[Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd[False][Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd :: S:0' -> S:0' -> S:0'
equal0[Ite] :: False:True -> S:0' -> S:0' -> False:True
equal0[True][Ite] :: False:True -> S:0' -> S:0' -> False:True
hole_c111_14 :: c11
hole_S:0'2_14 :: S:0'
hole_c123_14 :: c12
hole_c3:c44_14 :: c3:c4
hole_c135_14 :: c13
hole_False:True6_14 :: False:True
hole_c7:c87_14 :: c7:c8
hole_c:c1:c28_14 :: c:c1:c2
hole_c5:c69_14 :: c5:c6
hole_c9:c1010_14 :: c9:c10
gen_c1111_14 :: Nat -> c11
gen_S:0'12_14 :: Nat -> S:0'
gen_c:c1:c213_14 :: Nat -> c:c1:c2


Lemmas:
MONUS(gen_S:0'12_14(+(1, n15_14)), gen_S:0'12_14(+(1, n15_14))) -> *14_14, rt in Omega(n15_14)


Generator Equations:
gen_c1111_14(0) <=> hole_c111_14
gen_c1111_14(+(x, 1)) <=> c11(gen_c1111_14(x))
gen_S:0'12_14(0) <=> 0'
gen_S:0'12_14(+(x, 1)) <=> S(gen_S:0'12_14(x))
gen_c:c1:c213_14(0) <=> c1
gen_c:c1:c213_14(+(x, 1)) <=> c(gen_c:c1:c213_14(x))


The following defined symbols remain to be analysed:
<, GCD, <', monus, gcd

They will be analysed ascendingly in the following order:
< < GCD
<' < GCD
monus < GCD
< < gcd
monus < gcd

----------------------------------------

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
<(gen_S:0'12_14(n492_14), gen_S:0'12_14(+(1, n492_14))) -> True, rt in Omega(0)

Induction Base:
<(gen_S:0'12_14(0), gen_S:0'12_14(+(1, 0))) ->_R^Omega(0)
True

Induction Step:
<(gen_S:0'12_14(+(n492_14, 1)), gen_S:0'12_14(+(1, +(n492_14, 1)))) ->_R^Omega(0)
<(gen_S:0'12_14(n492_14), gen_S:0'12_14(+(1, n492_14))) ->_IH
True

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(40)
Obligation:
Innermost TRS:
Rules:
MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
<'(S(z0), S(z1)) -> c(<'(z0, z1))
<'(0', S(z0)) -> c1
<'(z0, 0') -> c2
GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
GCD[ITE](True, z0, z1) -> c4
GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
EQUAL0[ITE](False, z0, z1) -> c7
EQUAL0[ITE](True, z0, z1) -> c8(EQUAL0[TRUE][ITE](<(z1, z0), z0, z1), <'(z1, z0))
EQUAL0[TRUE][ITE](False, z0, z1) -> c9
EQUAL0[TRUE][ITE](True, z0, z1) -> c10
<(S(z0), S(z1)) -> <(z0, z1)
<(0', S(z0)) -> True
<(z0, 0') -> False
gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
gcd[Ite](True, z0, z1) -> z0
gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
equal0[Ite](False, z0, z1) -> False
equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
equal0[True][Ite](False, z0, z1) -> False
equal0[True][Ite](True, z0, z1) -> True
monus(S(z0), S(z1)) -> monus(z0, z1)
gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)

Types:
MONUS :: S:0' -> S:0' -> c11
S :: S:0' -> S:0'
c11 :: c11 -> c11
GCD :: S:0' -> S:0' -> c12
c12 :: c3:c4 -> c13 -> c12
GCD[ITE] :: False:True -> S:0' -> S:0' -> c3:c4
equal0 :: S:0' -> S:0' -> False:True
EQUAL0 :: S:0' -> S:0' -> c13
c13 :: c7:c8 -> c:c1:c2 -> c13
EQUAL0[ITE] :: False:True -> S:0' -> S:0' -> c7:c8
< :: S:0' -> S:0' -> False:True
<' :: S:0' -> S:0' -> c:c1:c2
c :: c:c1:c2 -> c:c1:c2
0' :: S:0'
c1 :: c:c1:c2
c2 :: c:c1:c2
False :: False:True
c3 :: c5:c6 -> c:c1:c2 -> c3:c4
GCD[FALSE][ITE] :: False:True -> S:0' -> S:0' -> c5:c6
True :: False:True
c4 :: c3:c4
c5 :: c12 -> c11 -> c5:c6
monus :: S:0' -> S:0' -> S:0'
c6 :: c12 -> c11 -> c5:c6
c7 :: c7:c8
c8 :: c9:c10 -> c:c1:c2 -> c7:c8
EQUAL0[TRUE][ITE] :: False:True -> S:0' -> S:0' -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
gcd[Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd[False][Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd :: S:0' -> S:0' -> S:0'
equal0[Ite] :: False:True -> S:0' -> S:0' -> False:True
equal0[True][Ite] :: False:True -> S:0' -> S:0' -> False:True
hole_c111_14 :: c11
hole_S:0'2_14 :: S:0'
hole_c123_14 :: c12
hole_c3:c44_14 :: c3:c4
hole_c135_14 :: c13
hole_False:True6_14 :: False:True
hole_c7:c87_14 :: c7:c8
hole_c:c1:c28_14 :: c:c1:c2
hole_c5:c69_14 :: c5:c6
hole_c9:c1010_14 :: c9:c10
gen_c1111_14 :: Nat -> c11
gen_S:0'12_14 :: Nat -> S:0'
gen_c:c1:c213_14 :: Nat -> c:c1:c2


Lemmas:
MONUS(gen_S:0'12_14(+(1, n15_14)), gen_S:0'12_14(+(1, n15_14))) -> *14_14, rt in Omega(n15_14)
<(gen_S:0'12_14(n492_14), gen_S:0'12_14(+(1, n492_14))) -> True, rt in Omega(0)


Generator Equations:
gen_c1111_14(0) <=> hole_c111_14
gen_c1111_14(+(x, 1)) <=> c11(gen_c1111_14(x))
gen_S:0'12_14(0) <=> 0'
gen_S:0'12_14(+(x, 1)) <=> S(gen_S:0'12_14(x))
gen_c:c1:c213_14(0) <=> c1
gen_c:c1:c213_14(+(x, 1)) <=> c(gen_c:c1:c213_14(x))


The following defined symbols remain to be analysed:
<', GCD, monus, gcd

They will be analysed ascendingly in the following order:
<' < GCD
monus < GCD
monus < gcd

----------------------------------------

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
<'(gen_S:0'12_14(n910_14), gen_S:0'12_14(+(1, n910_14))) -> gen_c:c1:c213_14(n910_14), rt in Omega(0)

Induction Base:
<'(gen_S:0'12_14(0), gen_S:0'12_14(+(1, 0))) ->_R^Omega(0)
c1

Induction Step:
<'(gen_S:0'12_14(+(n910_14, 1)), gen_S:0'12_14(+(1, +(n910_14, 1)))) ->_R^Omega(0)
c(<'(gen_S:0'12_14(n910_14), gen_S:0'12_14(+(1, n910_14)))) ->_IH
c(gen_c:c1:c213_14(c911_14))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(42)
Obligation:
Innermost TRS:
Rules:
MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
<'(S(z0), S(z1)) -> c(<'(z0, z1))
<'(0', S(z0)) -> c1
<'(z0, 0') -> c2
GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
GCD[ITE](True, z0, z1) -> c4
GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
EQUAL0[ITE](False, z0, z1) -> c7
EQUAL0[ITE](True, z0, z1) -> c8(EQUAL0[TRUE][ITE](<(z1, z0), z0, z1), <'(z1, z0))
EQUAL0[TRUE][ITE](False, z0, z1) -> c9
EQUAL0[TRUE][ITE](True, z0, z1) -> c10
<(S(z0), S(z1)) -> <(z0, z1)
<(0', S(z0)) -> True
<(z0, 0') -> False
gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
gcd[Ite](True, z0, z1) -> z0
gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
equal0[Ite](False, z0, z1) -> False
equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
equal0[True][Ite](False, z0, z1) -> False
equal0[True][Ite](True, z0, z1) -> True
monus(S(z0), S(z1)) -> monus(z0, z1)
gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)

Types:
MONUS :: S:0' -> S:0' -> c11
S :: S:0' -> S:0'
c11 :: c11 -> c11
GCD :: S:0' -> S:0' -> c12
c12 :: c3:c4 -> c13 -> c12
GCD[ITE] :: False:True -> S:0' -> S:0' -> c3:c4
equal0 :: S:0' -> S:0' -> False:True
EQUAL0 :: S:0' -> S:0' -> c13
c13 :: c7:c8 -> c:c1:c2 -> c13
EQUAL0[ITE] :: False:True -> S:0' -> S:0' -> c7:c8
< :: S:0' -> S:0' -> False:True
<' :: S:0' -> S:0' -> c:c1:c2
c :: c:c1:c2 -> c:c1:c2
0' :: S:0'
c1 :: c:c1:c2
c2 :: c:c1:c2
False :: False:True
c3 :: c5:c6 -> c:c1:c2 -> c3:c4
GCD[FALSE][ITE] :: False:True -> S:0' -> S:0' -> c5:c6
True :: False:True
c4 :: c3:c4
c5 :: c12 -> c11 -> c5:c6
monus :: S:0' -> S:0' -> S:0'
c6 :: c12 -> c11 -> c5:c6
c7 :: c7:c8
c8 :: c9:c10 -> c:c1:c2 -> c7:c8
EQUAL0[TRUE][ITE] :: False:True -> S:0' -> S:0' -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
gcd[Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd[False][Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd :: S:0' -> S:0' -> S:0'
equal0[Ite] :: False:True -> S:0' -> S:0' -> False:True
equal0[True][Ite] :: False:True -> S:0' -> S:0' -> False:True
hole_c111_14 :: c11
hole_S:0'2_14 :: S:0'
hole_c123_14 :: c12
hole_c3:c44_14 :: c3:c4
hole_c135_14 :: c13
hole_False:True6_14 :: False:True
hole_c7:c87_14 :: c7:c8
hole_c:c1:c28_14 :: c:c1:c2
hole_c5:c69_14 :: c5:c6
hole_c9:c1010_14 :: c9:c10
gen_c1111_14 :: Nat -> c11
gen_S:0'12_14 :: Nat -> S:0'
gen_c:c1:c213_14 :: Nat -> c:c1:c2


Lemmas:
MONUS(gen_S:0'12_14(+(1, n15_14)), gen_S:0'12_14(+(1, n15_14))) -> *14_14, rt in Omega(n15_14)
<(gen_S:0'12_14(n492_14), gen_S:0'12_14(+(1, n492_14))) -> True, rt in Omega(0)
<'(gen_S:0'12_14(n910_14), gen_S:0'12_14(+(1, n910_14))) -> gen_c:c1:c213_14(n910_14), rt in Omega(0)


Generator Equations:
gen_c1111_14(0) <=> hole_c111_14
gen_c1111_14(+(x, 1)) <=> c11(gen_c1111_14(x))
gen_S:0'12_14(0) <=> 0'
gen_S:0'12_14(+(x, 1)) <=> S(gen_S:0'12_14(x))
gen_c:c1:c213_14(0) <=> c1
gen_c:c1:c213_14(+(x, 1)) <=> c(gen_c:c1:c213_14(x))


The following defined symbols remain to be analysed:
monus, GCD, gcd

They will be analysed ascendingly in the following order:
monus < GCD
monus < gcd

----------------------------------------

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
monus(gen_S:0'12_14(+(1, n1689_14)), gen_S:0'12_14(+(1, n1689_14))) -> *14_14, rt in Omega(0)

Induction Base:
monus(gen_S:0'12_14(+(1, 0)), gen_S:0'12_14(+(1, 0)))

Induction Step:
monus(gen_S:0'12_14(+(1, +(n1689_14, 1))), gen_S:0'12_14(+(1, +(n1689_14, 1)))) ->_R^Omega(0)
monus(gen_S:0'12_14(+(1, n1689_14)), gen_S:0'12_14(+(1, n1689_14))) ->_IH
*14_14

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(44)
Obligation:
Innermost TRS:
Rules:
MONUS(S(z0), S(z1)) -> c11(MONUS(z0, z1))
GCD(z0, z1) -> c12(GCD[ITE](equal0(z0, z1), z0, z1), EQUAL0(z0, z1))
EQUAL0(z0, z1) -> c13(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
<'(S(z0), S(z1)) -> c(<'(z0, z1))
<'(0', S(z0)) -> c1
<'(z0, 0') -> c2
GCD[ITE](False, z0, z1) -> c3(GCD[FALSE][ITE](<(z0, z1), z0, z1), <'(z0, z1))
GCD[ITE](True, z0, z1) -> c4
GCD[FALSE][ITE](False, z0, z1) -> c5(GCD(z1, monus(z1, z0)), MONUS(z1, z0))
GCD[FALSE][ITE](True, z0, z1) -> c6(GCD(monus(z0, z1), z1), MONUS(z0, z1))
EQUAL0[ITE](False, z0, z1) -> c7
EQUAL0[ITE](True, z0, z1) -> c8(EQUAL0[TRUE][ITE](<(z1, z0), z0, z1), <'(z1, z0))
EQUAL0[TRUE][ITE](False, z0, z1) -> c9
EQUAL0[TRUE][ITE](True, z0, z1) -> c10
<(S(z0), S(z1)) -> <(z0, z1)
<(0', S(z0)) -> True
<(z0, 0') -> False
gcd[Ite](False, z0, z1) -> gcd[False][Ite](<(z0, z1), z0, z1)
gcd[Ite](True, z0, z1) -> z0
gcd[False][Ite](False, z0, z1) -> gcd(z1, monus(z1, z0))
gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1)
equal0[Ite](False, z0, z1) -> False
equal0[Ite](True, z0, z1) -> equal0[True][Ite](<(z1, z0), z0, z1)
equal0[True][Ite](False, z0, z1) -> False
equal0[True][Ite](True, z0, z1) -> True
monus(S(z0), S(z1)) -> monus(z0, z1)
gcd(z0, z1) -> gcd[Ite](equal0(z0, z1), z0, z1)
equal0(z0, z1) -> equal0[Ite](<(z0, z1), z0, z1)

Types:
MONUS :: S:0' -> S:0' -> c11
S :: S:0' -> S:0'
c11 :: c11 -> c11
GCD :: S:0' -> S:0' -> c12
c12 :: c3:c4 -> c13 -> c12
GCD[ITE] :: False:True -> S:0' -> S:0' -> c3:c4
equal0 :: S:0' -> S:0' -> False:True
EQUAL0 :: S:0' -> S:0' -> c13
c13 :: c7:c8 -> c:c1:c2 -> c13
EQUAL0[ITE] :: False:True -> S:0' -> S:0' -> c7:c8
< :: S:0' -> S:0' -> False:True
<' :: S:0' -> S:0' -> c:c1:c2
c :: c:c1:c2 -> c:c1:c2
0' :: S:0'
c1 :: c:c1:c2
c2 :: c:c1:c2
False :: False:True
c3 :: c5:c6 -> c:c1:c2 -> c3:c4
GCD[FALSE][ITE] :: False:True -> S:0' -> S:0' -> c5:c6
True :: False:True
c4 :: c3:c4
c5 :: c12 -> c11 -> c5:c6
monus :: S:0' -> S:0' -> S:0'
c6 :: c12 -> c11 -> c5:c6
c7 :: c7:c8
c8 :: c9:c10 -> c:c1:c2 -> c7:c8
EQUAL0[TRUE][ITE] :: False:True -> S:0' -> S:0' -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
gcd[Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd[False][Ite] :: False:True -> S:0' -> S:0' -> S:0'
gcd :: S:0' -> S:0' -> S:0'
equal0[Ite] :: False:True -> S:0' -> S:0' -> False:True
equal0[True][Ite] :: False:True -> S:0' -> S:0' -> False:True
hole_c111_14 :: c11
hole_S:0'2_14 :: S:0'
hole_c123_14 :: c12
hole_c3:c44_14 :: c3:c4
hole_c135_14 :: c13
hole_False:True6_14 :: False:True
hole_c7:c87_14 :: c7:c8
hole_c:c1:c28_14 :: c:c1:c2
hole_c5:c69_14 :: c5:c6
hole_c9:c1010_14 :: c9:c10
gen_c1111_14 :: Nat -> c11
gen_S:0'12_14 :: Nat -> S:0'
gen_c:c1:c213_14 :: Nat -> c:c1:c2


Lemmas:
MONUS(gen_S:0'12_14(+(1, n15_14)), gen_S:0'12_14(+(1, n15_14))) -> *14_14, rt in Omega(n15_14)
<(gen_S:0'12_14(n492_14), gen_S:0'12_14(+(1, n492_14))) -> True, rt in Omega(0)
<'(gen_S:0'12_14(n910_14), gen_S:0'12_14(+(1, n910_14))) -> gen_c:c1:c213_14(n910_14), rt in Omega(0)
monus(gen_S:0'12_14(+(1, n1689_14)), gen_S:0'12_14(+(1, n1689_14))) -> *14_14, rt in Omega(0)


Generator Equations:
gen_c1111_14(0) <=> hole_c111_14
gen_c1111_14(+(x, 1)) <=> c11(gen_c1111_14(x))
gen_S:0'12_14(0) <=> 0'
gen_S:0'12_14(+(x, 1)) <=> S(gen_S:0'12_14(x))
gen_c:c1:c213_14(0) <=> c1
gen_c:c1:c213_14(+(x, 1)) <=> c(gen_c:c1:c213_14(x))


The following defined symbols remain to be analysed:
GCD, gcd