/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 168 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(8) CdtProblem
(9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
(13) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(14) CdtProblem
(15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
(17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 86 ms]
(18) CdtProblem
(19) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) BOUNDS(1, 1)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   gcd(x, 0) -> x
   gcd(0, y) -> y
   gcd(s(x), s(y)) -> if(<(x, y), gcd(s(x), -(y, x)), gcd(-(x, y), s(y)))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(<(x_1, x_2)) -> <(encArg(x_1), encArg(x_2))
   encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2))
   encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_<(x_1, x_2) -> <(encArg(x_1), encArg(x_2))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))


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

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


The TRS R consists of the following rules:

   gcd(x, 0) -> x
   gcd(0, y) -> y
   gcd(s(x), s(y)) -> if(<(x, y), gcd(s(x), -(y, x)), gcd(-(x, y), s(y)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(<(x_1, x_2)) -> <(encArg(x_1), encArg(x_2))
   encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2))
   encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_<(x_1, x_2) -> <(encArg(x_1), encArg(x_2))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   gcd(x, 0) -> x
   gcd(0, y) -> y
   gcd(s(x), s(y)) -> if(<(x, y), gcd(s(x), -(y, x)), gcd(-(x, y), s(y)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(<(x_1, x_2)) -> <(encArg(x_1), encArg(x_2))
   encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2))
   encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_<(x_1, x_2) -> <(encArg(x_1), encArg(x_2))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))

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

(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   encArg(<(z0, z1)) -> <(encArg(z0), encArg(z1))
   encArg(-(z0, z1)) -> -(encArg(z0), encArg(z1))
   encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1))
   encode_gcd(z0, z1) -> gcd(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_s(z0) -> s(encArg(z0))
   encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2))
   encode_<(z0, z1) -> <(encArg(z0), encArg(z1))
   encode_-(z0, z1) -> -(encArg(z0), encArg(z1))
   gcd(z0, 0) -> z0
   gcd(0, z0) -> z0
   gcd(s(z0), s(z1)) -> if(<(z0, z1), gcd(s(z0), -(z1, z0)), gcd(-(z0, z1), s(z1)))
Tuples:
   ENCARG(0) -> c
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(if(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(<(z0, z1)) -> c3(ENCARG(z0), ENCARG(z1))
   ENCARG(-(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_gcd(z0, z1)) -> c5(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_GCD(z0, z1) -> c6(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_0 -> c7
   ENCODE_S(z0) -> c8(ENCARG(z0))
   ENCODE_IF(z0, z1, z2) -> c9(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_<(z0, z1) -> c10(ENCARG(z0), ENCARG(z1))
   ENCODE_-(z0, z1) -> c11(ENCARG(z0), ENCARG(z1))
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14(GCD(s(z0), -(z1, z0)), GCD(-(z0, z1), s(z1)))
S tuples:
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14(GCD(s(z0), -(z1, z0)), GCD(-(z0, z1), s(z1)))
K tuples:none
Defined Rule Symbols:   gcd_2, encArg_1, encode_gcd_2, encode_0, encode_s_1, encode_if_3, encode_<_2, encode_-_2

Defined Pair Symbols:   ENCARG_1, ENCODE_GCD_2, ENCODE_0, ENCODE_S_1, ENCODE_IF_3, ENCODE_<_2, ENCODE_-_2, GCD_2

Compound Symbols:   c, c1_1, c2_3, c3_2, c4_2, c5_3, c6_3, c7, c8_1, c9_3, c10_2, c11_2, c12, c13, c14_2


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 4 leading nodes:
   ENCODE_S(z0) -> c8(ENCARG(z0))
   ENCODE_IF(z0, z1, z2) -> c9(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_<(z0, z1) -> c10(ENCARG(z0), ENCARG(z1))
   ENCODE_-(z0, z1) -> c11(ENCARG(z0), ENCARG(z1))
Removed 2 trailing nodes:
   ENCODE_0 -> c7
   ENCARG(0) -> c

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   encArg(<(z0, z1)) -> <(encArg(z0), encArg(z1))
   encArg(-(z0, z1)) -> -(encArg(z0), encArg(z1))
   encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1))
   encode_gcd(z0, z1) -> gcd(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_s(z0) -> s(encArg(z0))
   encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2))
   encode_<(z0, z1) -> <(encArg(z0), encArg(z1))
   encode_-(z0, z1) -> -(encArg(z0), encArg(z1))
   gcd(z0, 0) -> z0
   gcd(0, z0) -> z0
   gcd(s(z0), s(z1)) -> if(<(z0, z1), gcd(s(z0), -(z1, z0)), gcd(-(z0, z1), s(z1)))
Tuples:
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(if(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(<(z0, z1)) -> c3(ENCARG(z0), ENCARG(z1))
   ENCARG(-(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_gcd(z0, z1)) -> c5(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_GCD(z0, z1) -> c6(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14(GCD(s(z0), -(z1, z0)), GCD(-(z0, z1), s(z1)))
S tuples:
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14(GCD(s(z0), -(z1, z0)), GCD(-(z0, z1), s(z1)))
K tuples:none
Defined Rule Symbols:   gcd_2, encArg_1, encode_gcd_2, encode_0, encode_s_1, encode_if_3, encode_<_2, encode_-_2

Defined Pair Symbols:   ENCARG_1, ENCODE_GCD_2, GCD_2

Compound Symbols:   c1_1, c2_3, c3_2, c4_2, c5_3, c6_3, c12, c13, c14_2


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   encArg(<(z0, z1)) -> <(encArg(z0), encArg(z1))
   encArg(-(z0, z1)) -> -(encArg(z0), encArg(z1))
   encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1))
   encode_gcd(z0, z1) -> gcd(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_s(z0) -> s(encArg(z0))
   encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2))
   encode_<(z0, z1) -> <(encArg(z0), encArg(z1))
   encode_-(z0, z1) -> -(encArg(z0), encArg(z1))
   gcd(z0, 0) -> z0
   gcd(0, z0) -> z0
   gcd(s(z0), s(z1)) -> if(<(z0, z1), gcd(s(z0), -(z1, z0)), gcd(-(z0, z1), s(z1)))
Tuples:
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(if(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(<(z0, z1)) -> c3(ENCARG(z0), ENCARG(z1))
   ENCARG(-(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_gcd(z0, z1)) -> c5(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_GCD(z0, z1) -> c6(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14
S tuples:
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14
K tuples:none
Defined Rule Symbols:   gcd_2, encArg_1, encode_gcd_2, encode_0, encode_s_1, encode_if_3, encode_<_2, encode_-_2

Defined Pair Symbols:   ENCARG_1, ENCODE_GCD_2, GCD_2

Compound Symbols:   c1_1, c2_3, c3_2, c4_2, c5_3, c6_3, c12, c13, c14


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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   encArg(<(z0, z1)) -> <(encArg(z0), encArg(z1))
   encArg(-(z0, z1)) -> -(encArg(z0), encArg(z1))
   encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1))
   encode_gcd(z0, z1) -> gcd(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_s(z0) -> s(encArg(z0))
   encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2))
   encode_<(z0, z1) -> <(encArg(z0), encArg(z1))
   encode_-(z0, z1) -> -(encArg(z0), encArg(z1))
   gcd(z0, 0) -> z0
   gcd(0, z0) -> z0
   gcd(s(z0), s(z1)) -> if(<(z0, z1), gcd(s(z0), -(z1, z0)), gcd(-(z0, z1), s(z1)))
Tuples:
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(if(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(<(z0, z1)) -> c3(ENCARG(z0), ENCARG(z1))
   ENCARG(-(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_gcd(z0, z1)) -> c5(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14
   ENCODE_GCD(z0, z1) -> c(GCD(encArg(z0), encArg(z1)))
   ENCODE_GCD(z0, z1) -> c(ENCARG(z0))
   ENCODE_GCD(z0, z1) -> c(ENCARG(z1))
S tuples:
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14
K tuples:none
Defined Rule Symbols:   gcd_2, encArg_1, encode_gcd_2, encode_0, encode_s_1, encode_if_3, encode_<_2, encode_-_2

Defined Pair Symbols:   ENCARG_1, GCD_2, ENCODE_GCD_2

Compound Symbols:   c1_1, c2_3, c3_2, c4_2, c5_3, c12, c13, c14, c_1


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

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_GCD(z0, z1) -> c(ENCARG(z0))
   ENCODE_GCD(z0, z1) -> c(ENCARG(z1))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   encArg(<(z0, z1)) -> <(encArg(z0), encArg(z1))
   encArg(-(z0, z1)) -> -(encArg(z0), encArg(z1))
   encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1))
   encode_gcd(z0, z1) -> gcd(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_s(z0) -> s(encArg(z0))
   encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2))
   encode_<(z0, z1) -> <(encArg(z0), encArg(z1))
   encode_-(z0, z1) -> -(encArg(z0), encArg(z1))
   gcd(z0, 0) -> z0
   gcd(0, z0) -> z0
   gcd(s(z0), s(z1)) -> if(<(z0, z1), gcd(s(z0), -(z1, z0)), gcd(-(z0, z1), s(z1)))
Tuples:
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(if(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(<(z0, z1)) -> c3(ENCARG(z0), ENCARG(z1))
   ENCARG(-(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_gcd(z0, z1)) -> c5(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14
   ENCODE_GCD(z0, z1) -> c(GCD(encArg(z0), encArg(z1)))
S tuples:
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14
K tuples:none
Defined Rule Symbols:   gcd_2, encArg_1, encode_gcd_2, encode_0, encode_s_1, encode_if_3, encode_<_2, encode_-_2

Defined Pair Symbols:   ENCARG_1, GCD_2, ENCODE_GCD_2

Compound Symbols:   c1_1, c2_3, c3_2, c4_2, c5_3, c12, c13, c14, c_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_gcd(z0, z1) -> gcd(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_s(z0) -> s(encArg(z0))
   encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2))
   encode_<(z0, z1) -> <(encArg(z0), encArg(z1))
   encode_-(z0, z1) -> -(encArg(z0), encArg(z1))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   encArg(<(z0, z1)) -> <(encArg(z0), encArg(z1))
   encArg(-(z0, z1)) -> -(encArg(z0), encArg(z1))
   encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1))
   gcd(z0, 0) -> z0
   gcd(0, z0) -> z0
   gcd(s(z0), s(z1)) -> if(<(z0, z1), gcd(s(z0), -(z1, z0)), gcd(-(z0, z1), s(z1)))
Tuples:
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(if(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(<(z0, z1)) -> c3(ENCARG(z0), ENCARG(z1))
   ENCARG(-(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_gcd(z0, z1)) -> c5(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14
   ENCODE_GCD(z0, z1) -> c(GCD(encArg(z0), encArg(z1)))
S tuples:
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14
K tuples:none
Defined Rule Symbols:   encArg_1, gcd_2

Defined Pair Symbols:   ENCARG_1, GCD_2, ENCODE_GCD_2

Compound Symbols:   c1_1, c2_3, c3_2, c4_2, c5_3, c12, c13, c14, c_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(if(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(<(z0, z1)) -> c3(ENCARG(z0), ENCARG(z1))
   ENCARG(-(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_gcd(z0, z1)) -> c5(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14
   ENCODE_GCD(z0, z1) -> c(GCD(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1 + x_2
   POL(0) = [1]
   POL(<(x_1, x_2)) = [1] + x_1 + x_2
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_GCD(x_1, x_2)) = [1]
   POL(GCD(x_1, x_2)) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c12) = 0
   POL(c13) = 0
   POL(c14) = 0
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(cons_gcd(x_1, x_2)) = [1] + x_1 + x_2
   POL(encArg(x_1)) = [1] + x_1
   POL(gcd(x_1, x_2)) = [1] + x_1 + x_2
   POL(if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(s(x_1)) = [1] + x_1

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(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   encArg(<(z0, z1)) -> <(encArg(z0), encArg(z1))
   encArg(-(z0, z1)) -> -(encArg(z0), encArg(z1))
   encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1))
   gcd(z0, 0) -> z0
   gcd(0, z0) -> z0
   gcd(s(z0), s(z1)) -> if(<(z0, z1), gcd(s(z0), -(z1, z0)), gcd(-(z0, z1), s(z1)))
Tuples:
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(if(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(<(z0, z1)) -> c3(ENCARG(z0), ENCARG(z1))
   ENCARG(-(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_gcd(z0, z1)) -> c5(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14
   ENCODE_GCD(z0, z1) -> c(GCD(encArg(z0), encArg(z1)))
S tuples:none
K tuples:
   GCD(z0, 0) -> c12
   GCD(0, z0) -> c13
   GCD(s(z0), s(z1)) -> c14
Defined Rule Symbols:   encArg_1, gcd_2

Defined Pair Symbols:   ENCARG_1, GCD_2, ENCODE_GCD_2

Compound Symbols:   c1_1, c2_3, c3_2, c4_2, c5_3, c12, c13, c14, c_1


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(19) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(20)
BOUNDS(1, 1)