/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 251 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)), 62 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)), 167 ms]
    (20) CdtProblem
    (21) CdtKnowledgeProof [FINISHED, 0 ms]
    (22) BOUNDS(1, 1)
(23) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(24) TRS for Loop Detection
    (25) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (26) BEST
        (27) proven lower bound
            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
            (29) BOUNDS(n^1, INF)
        (30) TRS for Loop Detection


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

(0)
Obligation:
The Runtime Complexity (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: INNERMOST
----------------------------------------

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

(2)
Obligation:
The Runtime Complexity (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: INNERMOST
----------------------------------------

(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS 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:
   <'(z0, 0) -> c2
   <'(0, S(z0)) -> c1
   EQUAL0[TRUE][ITE](False, z0, z1) -> c9
   EQUAL0[ITE](False, z0, z1) -> c7
   GCD[ITE](True, z0, z1) -> c4
   EQUAL0[TRUE][ITE](True, z0, z1) -> c10

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

(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) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(24)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (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: INNERMOST
----------------------------------------

(25) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

monus(S(x'), S(x)) ->^+ monus(x', x)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [x' / S(x'), x / S(x)].

The result substitution is [ ].




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

(26)
Complex Obligation (BEST)

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

(27)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (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: INNERMOST
----------------------------------------

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

(29)
BOUNDS(n^1, INF)

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

(30)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (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: INNERMOST