WORST_CASE(Omega(n^1),O(n^3))
proof of input_xGGTWvsH02.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxWeightedTrs
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedTrs
    (15) CompletionProof [UPPER BOUND(ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) CompleteCoflocoProof [FINISHED, 6414 ms]
    (20) BOUNDS(1, n^3)
(21) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CdtProblem
    (23) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRelTRS
    (25) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRelTRS
    (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) typed CpxTrs
    (29) OrderProof [LOWER BOUND(ID), 3 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 335 ms]
    (32) typed CpxTrs
    (33) RewriteLemmaProof [LOWER BOUND(ID), 153 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), 85 ms]
            (40) typed CpxTrs
            (41) RewriteLemmaProof [LOWER BOUND(ID), 35 ms]
            (42) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   minus(s(x), y) -> if(gt(s(x), y), x, y)
   if(true, x, y) -> s(minus(x, y))
   if(false, x, y) -> 0
   gcd(x, y) -> if1(ge(x, y), x, y)
   if1(true, x, y) -> if2(gt(y, 0), x, y)
   if1(false, x, y) -> if3(gt(x, 0), x, y)
   if2(true, x, y) -> gcd(minus(x, y), y)
   if2(false, x, y) -> x
   if3(true, x, y) -> gcd(x, minus(y, x))
   if3(false, x, y) -> y
   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)

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

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

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
   if(true, z0, z1) -> s(minus(z0, z1))
   if(false, z0, z1) -> 0
   gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
   if1(true, z0, z1) -> if2(gt(z1, 0), z0, z1)
   if1(false, z0, z1) -> if3(gt(z0, 0), z0, z1)
   if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
   if2(false, z0, z1) -> z0
   if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
   if3(false, z0, z1) -> z1
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   IF(false, z0, z1) -> c2
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1), GT(z1, 0))
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1), GT(z0, 0))
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
   IF2(false, z0, z1) -> c7
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
   IF3(false, z0, z1) -> c9
   GT(0, z0) -> c10
   GT(s(z0), 0) -> c11
   GT(s(z0), s(z1)) -> c12(GT(z0, z1))
   GE(z0, 0) -> c13
   GE(0, s(z0)) -> c14
   GE(s(z0), s(z1)) -> c15(GE(z0, z1))
S tuples:
   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   IF(false, z0, z1) -> c2
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1), GT(z1, 0))
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1), GT(z0, 0))
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
   IF2(false, z0, z1) -> c7
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
   IF3(false, z0, z1) -> c9
   GT(0, z0) -> c10
   GT(s(z0), 0) -> c11
   GT(s(z0), s(z1)) -> c12(GT(z0, z1))
   GE(z0, 0) -> c13
   GE(0, s(z0)) -> c14
   GE(s(z0), s(z1)) -> c15(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, if_3, gcd_2, if1_3, if2_3, if3_3, gt_2, ge_2

Defined Pair Symbols:   MINUS_2, IF_3, GCD_2, IF1_3, IF2_3, IF3_3, GT_2, GE_2

Compound Symbols:   c_2, c1_1, c2, c3_2, c4_2, c5_2, c6_2, c7, c8_2, c9, c10, c11, c12_1, c13, c14, c15_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   GE(0, s(z0)) -> c14
   GE(z0, 0) -> c13
   GT(s(z0), 0) -> c11
   IF2(false, z0, z1) -> c7
   GT(0, z0) -> c10
   IF(false, z0, z1) -> c2
   IF3(false, z0, z1) -> c9

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
   if(true, z0, z1) -> s(minus(z0, z1))
   if(false, z0, z1) -> 0
   gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
   if1(true, z0, z1) -> if2(gt(z1, 0), z0, z1)
   if1(false, z0, z1) -> if3(gt(z0, 0), z0, z1)
   if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
   if2(false, z0, z1) -> z0
   if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
   if3(false, z0, z1) -> z1
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1), GT(z1, 0))
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1), GT(z0, 0))
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
   GT(s(z0), s(z1)) -> c12(GT(z0, z1))
   GE(s(z0), s(z1)) -> c15(GE(z0, z1))
S tuples:
   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1), GT(z1, 0))
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1), GT(z0, 0))
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
   GT(s(z0), s(z1)) -> c12(GT(z0, z1))
   GE(s(z0), s(z1)) -> c15(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, if_3, gcd_2, if1_3, if2_3, if3_3, gt_2, ge_2

Defined Pair Symbols:   MINUS_2, IF_3, GCD_2, IF1_3, IF2_3, IF3_3, GT_2, GE_2

Compound Symbols:   c_2, c1_1, c3_2, c4_2, c5_2, c6_2, c8_2, c12_1, c15_1


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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
   if(true, z0, z1) -> s(minus(z0, z1))
   if(false, z0, z1) -> 0
   gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
   if1(true, z0, z1) -> if2(gt(z1, 0), z0, z1)
   if1(false, z0, z1) -> if3(gt(z0, 0), z0, z1)
   if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
   if2(false, z0, z1) -> z0
   if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
   if3(false, z0, z1) -> z1
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
   GT(s(z0), s(z1)) -> c12(GT(z0, z1))
   GE(s(z0), s(z1)) -> c15(GE(z0, z1))
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1))
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1))
S tuples:
   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
   GT(s(z0), s(z1)) -> c12(GT(z0, z1))
   GE(s(z0), s(z1)) -> c15(GE(z0, z1))
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1))
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, if_3, gcd_2, if1_3, if2_3, if3_3, gt_2, ge_2

Defined Pair Symbols:   MINUS_2, IF_3, GCD_2, IF2_3, IF3_3, GT_2, GE_2, IF1_3

Compound Symbols:   c_2, c1_1, c3_2, c6_2, c8_2, c12_1, c15_1, c4_1, c5_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
   if1(true, z0, z1) -> if2(gt(z1, 0), z0, z1)
   if1(false, z0, z1) -> if3(gt(z0, 0), z0, z1)
   if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
   if2(false, z0, z1) -> z0
   if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
   if3(false, z0, z1) -> z1

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
   gt(0, z0) -> false
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
   if(true, z0, z1) -> s(minus(z0, z1))
   if(false, z0, z1) -> 0
Tuples:
   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
   GT(s(z0), s(z1)) -> c12(GT(z0, z1))
   GE(s(z0), s(z1)) -> c15(GE(z0, z1))
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1))
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1))
S tuples:
   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
   GT(s(z0), s(z1)) -> c12(GT(z0, z1))
   GE(s(z0), s(z1)) -> c15(GE(z0, z1))
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1))
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1))
K tuples:none
Defined Rule Symbols:   gt_2, ge_2, minus_2, if_3

Defined Pair Symbols:   MINUS_2, IF_3, GCD_2, IF2_3, IF3_3, GT_2, GE_2, IF1_3

Compound Symbols:   c_2, c1_1, c3_2, c6_2, c8_2, c12_1, c15_1, c4_1, c5_1


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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
   GT(s(z0), s(z1)) -> c12(GT(z0, z1))
   GE(s(z0), s(z1)) -> c15(GE(z0, z1))
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1))
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1))

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

   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
   gt(0, z0) -> false
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
   if(true, z0, z1) -> s(minus(z0, z1))
   if(false, z0, z1) -> 0

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

(11) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1)) [1]
   IF(true, z0, z1) -> c1(MINUS(z0, z1)) [1]
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1)) [1]
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1)) [1]
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0)) [1]
   GT(s(z0), s(z1)) -> c12(GT(z0, z1)) [1]
   GE(s(z0), s(z1)) -> c15(GE(z0, z1)) [1]
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1)) [1]
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1)) [1]
   gt(s(z0), 0) -> true [0]
   gt(s(z0), s(z1)) -> gt(z0, z1) [0]
   gt(0, z0) -> false [0]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1) [0]
   if(true, z0, z1) -> s(minus(z0, z1)) [0]
   if(false, z0, z1) -> 0 [0]

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

(13) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(14)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1)) [1]
   IF(true, z0, z1) -> c1(MINUS(z0, z1)) [1]
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1)) [1]
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1)) [1]
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0)) [1]
   GT(s(z0), s(z1)) -> c12(GT(z0, z1)) [1]
   GE(s(z0), s(z1)) -> c15(GE(z0, z1)) [1]
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1)) [1]
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1)) [1]
   gt(s(z0), 0) -> true [0]
   gt(s(z0), s(z1)) -> gt(z0, z1) [0]
   gt(0, z0) -> false [0]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1) [0]
   if(true, z0, z1) -> s(minus(z0, z1)) [0]
   if(false, z0, z1) -> 0 [0]

The TRS has the following type information:
MINUS :: s:0 -> s:0 -> c
s :: s:0 -> s:0
c :: c1 -> c12 -> c
IF :: true:false -> s:0 -> s:0 -> c1
gt :: s:0 -> s:0 -> true:false
GT :: s:0 -> s:0 -> c12
true :: true:false
c1 :: c -> c1
GCD :: s:0 -> s:0 -> c3
c3 :: c4:c5 -> c15 -> c3
IF1 :: true:false -> s:0 -> s:0 -> c4:c5
ge :: s:0 -> s:0 -> true:false
GE :: s:0 -> s:0 -> c15
IF2 :: true:false -> s:0 -> s:0 -> c6
c6 :: c3 -> c -> c6
minus :: s:0 -> s:0 -> s:0
IF3 :: true:false -> s:0 -> s:0 -> c8
c8 :: c3 -> c -> c8
c12 :: c12 -> c12
c15 :: c15 -> c15
c4 :: c6 -> c4:c5
0 :: s:0
false :: true:false
c5 :: c8 -> c4:c5
if :: true:false -> s:0 -> s:0 -> s:0

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

(15) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   gt(v0, v1) -> null_gt [0]
   ge(v0, v1) -> null_ge [0]
   minus(v0, v1) -> null_minus [0]
   if(v0, v1, v2) -> null_if [0]
   MINUS(v0, v1) -> null_MINUS [0]
   IF(v0, v1, v2) -> null_IF [0]
   IF2(v0, v1, v2) -> null_IF2 [0]
   IF3(v0, v1, v2) -> null_IF3 [0]
   GT(v0, v1) -> null_GT [0]
   GE(v0, v1) -> null_GE [0]
   IF1(v0, v1, v2) -> null_IF1 [0]

And the following fresh constants:    null_gt, null_ge, null_minus, null_if, null_MINUS, null_IF, null_IF2, null_IF3, null_GT, null_GE, null_IF1, const

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

(16)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1)) [1]
   IF(true, z0, z1) -> c1(MINUS(z0, z1)) [1]
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1)) [1]
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1)) [1]
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0)) [1]
   GT(s(z0), s(z1)) -> c12(GT(z0, z1)) [1]
   GE(s(z0), s(z1)) -> c15(GE(z0, z1)) [1]
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1)) [1]
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1)) [1]
   gt(s(z0), 0) -> true [0]
   gt(s(z0), s(z1)) -> gt(z0, z1) [0]
   gt(0, z0) -> false [0]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1) [0]
   if(true, z0, z1) -> s(minus(z0, z1)) [0]
   if(false, z0, z1) -> 0 [0]
   gt(v0, v1) -> null_gt [0]
   ge(v0, v1) -> null_ge [0]
   minus(v0, v1) -> null_minus [0]
   if(v0, v1, v2) -> null_if [0]
   MINUS(v0, v1) -> null_MINUS [0]
   IF(v0, v1, v2) -> null_IF [0]
   IF2(v0, v1, v2) -> null_IF2 [0]
   IF3(v0, v1, v2) -> null_IF3 [0]
   GT(v0, v1) -> null_GT [0]
   GE(v0, v1) -> null_GE [0]
   IF1(v0, v1, v2) -> null_IF1 [0]

The TRS has the following type information:
MINUS :: s:0:null_minus:null_if -> s:0:null_minus:null_if -> c:null_MINUS
s :: s:0:null_minus:null_if -> s:0:null_minus:null_if
c :: c1:null_IF -> c12:null_GT -> c:null_MINUS
IF :: true:false:null_gt:null_ge -> s:0:null_minus:null_if -> s:0:null_minus:null_if -> c1:null_IF
gt :: s:0:null_minus:null_if -> s:0:null_minus:null_if -> true:false:null_gt:null_ge
GT :: s:0:null_minus:null_if -> s:0:null_minus:null_if -> c12:null_GT
true :: true:false:null_gt:null_ge
c1 :: c:null_MINUS -> c1:null_IF
GCD :: s:0:null_minus:null_if -> s:0:null_minus:null_if -> c3
c3 :: c4:c5:null_IF1 -> c15:null_GE -> c3
IF1 :: true:false:null_gt:null_ge -> s:0:null_minus:null_if -> s:0:null_minus:null_if -> c4:c5:null_IF1
ge :: s:0:null_minus:null_if -> s:0:null_minus:null_if -> true:false:null_gt:null_ge
GE :: s:0:null_minus:null_if -> s:0:null_minus:null_if -> c15:null_GE
IF2 :: true:false:null_gt:null_ge -> s:0:null_minus:null_if -> s:0:null_minus:null_if -> c6:null_IF2
c6 :: c3 -> c:null_MINUS -> c6:null_IF2
minus :: s:0:null_minus:null_if -> s:0:null_minus:null_if -> s:0:null_minus:null_if
IF3 :: true:false:null_gt:null_ge -> s:0:null_minus:null_if -> s:0:null_minus:null_if -> c8:null_IF3
c8 :: c3 -> c:null_MINUS -> c8:null_IF3
c12 :: c12:null_GT -> c12:null_GT
c15 :: c15:null_GE -> c15:null_GE
c4 :: c6:null_IF2 -> c4:c5:null_IF1
0 :: s:0:null_minus:null_if
false :: true:false:null_gt:null_ge
c5 :: c8:null_IF3 -> c4:c5:null_IF1
if :: true:false:null_gt:null_ge -> s:0:null_minus:null_if -> s:0:null_minus:null_if -> s:0:null_minus:null_if
null_gt :: true:false:null_gt:null_ge
null_ge :: true:false:null_gt:null_ge
null_minus :: s:0:null_minus:null_if
null_if :: s:0:null_minus:null_if
null_MINUS :: c:null_MINUS
null_IF :: c1:null_IF
null_IF2 :: c6:null_IF2
null_IF3 :: c8:null_IF3
null_GT :: c12:null_GT
null_GE :: c15:null_GE
null_IF1 :: c4:c5:null_IF1
const :: c3

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

(17) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   true => 2
   0 => 0
   false => 1
   null_gt => 0
   null_ge => 0
   null_minus => 0
   null_if => 0
   null_MINUS => 0
   null_IF => 0
   null_IF2 => 0
   null_IF3 => 0
   null_GT => 0
   null_GE => 0
   null_IF1 => 0
   const => 0

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   GCD(z, z') -{ 1 }-> 1 + IF1(ge(z0, z1), z0, z1) + GE(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   GE(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   GE(z, z') -{ 1 }-> 1 + GE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   GT(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   GT(z, z') -{ 1 }-> 1 + GT(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   IF(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + MINUS(z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   IF1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   IF1(z, z', z'') -{ 1 }-> 1 + IF3(gt(z0, 0), z0, z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   IF1(z, z', z'') -{ 1 }-> 1 + IF2(gt(z1, 0), z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   IF2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   IF2(z, z', z'') -{ 1 }-> 1 + GCD(minus(z0, z1), z1) + MINUS(z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   IF3(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   IF3(z, z', z'') -{ 1 }-> 1 + GCD(z0, minus(z1, z0)) + MINUS(z1, z0) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   MINUS(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   MINUS(z, z') -{ 1 }-> 1 + IF(gt(1 + z0, z1), z0, z1) + GT(1 + z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   ge(z, z') -{ 0 }-> ge(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   ge(z, z') -{ 0 }-> 2 :|: z = z0, z0 >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   gt(z, z') -{ 0 }-> gt(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   gt(z, z') -{ 0 }-> 2 :|: z = 1 + z0, z0 >= 0, z' = 0
   gt(z, z') -{ 0 }-> 1 :|: z0 >= 0, z = 0, z' = z0
   gt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if(z, z', z'') -{ 0 }-> 1 + minus(z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   minus(z, z') -{ 0 }-> if(gt(1 + z0, z1), z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

(19) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V, V4),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[fun1(V1, V, V4, Out)],[V1 >= 0,V >= 0,V4 >= 0]).
eq(start(V1, V, V4),0,[fun3(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[fun6(V1, V, V4, Out)],[V1 >= 0,V >= 0,V4 >= 0]).
eq(start(V1, V, V4),0,[fun7(V1, V, V4, Out)],[V1 >= 0,V >= 0,V4 >= 0]).
eq(start(V1, V, V4),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[fun5(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[fun4(V1, V, V4, Out)],[V1 >= 0,V >= 0,V4 >= 0]).
eq(start(V1, V, V4),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[if(V1, V, V4, Out)],[V1 >= 0,V >= 0,V4 >= 0]).
eq(fun(V1, V, Out),1,[gt(1 + V3, V2, Ret010),fun1(Ret010, V3, V2, Ret01),fun2(1 + V3, V2, Ret1)],[Out = 1 + Ret01 + Ret1,V2 >= 0,V1 = 1 + V3,V = V2,V3 >= 0]).
eq(fun1(V1, V, V4, Out),1,[fun(V6, V5, Ret11)],[Out = 1 + Ret11,V1 = 2,V5 >= 0,V6 >= 0,V = V6,V4 = V5]).
eq(fun3(V1, V, Out),1,[ge(V8, V7, Ret0101),fun4(Ret0101, V8, V7, Ret011),fun5(V8, V7, Ret12)],[Out = 1 + Ret011 + Ret12,V1 = V8,V7 >= 0,V = V7,V8 >= 0]).
eq(fun6(V1, V, V4, Out),1,[minus(V9, V10, Ret0102),fun3(Ret0102, V10, Ret012),fun(V9, V10, Ret13)],[Out = 1 + Ret012 + Ret13,V1 = 2,V10 >= 0,V9 >= 0,V = V9,V4 = V10]).
eq(fun7(V1, V, V4, Out),1,[minus(V11, V12, Ret0111),fun3(V12, Ret0111, Ret013),fun(V11, V12, Ret14)],[Out = 1 + Ret013 + Ret14,V1 = 2,V11 >= 0,V12 >= 0,V = V12,V4 = V11]).
eq(fun2(V1, V, Out),1,[fun2(V14, V13, Ret15)],[Out = 1 + Ret15,V13 >= 0,V1 = 1 + V14,V14 >= 0,V = 1 + V13]).
eq(fun5(V1, V, Out),1,[fun5(V16, V15, Ret16)],[Out = 1 + Ret16,V15 >= 0,V1 = 1 + V16,V16 >= 0,V = 1 + V15]).
eq(fun4(V1, V, V4, Out),1,[gt(V17, 0, Ret10),fun6(Ret10, V18, V17, Ret17)],[Out = 1 + Ret17,V1 = 2,V17 >= 0,V18 >= 0,V = V18,V4 = V17]).
eq(fun4(V1, V, V4, Out),1,[gt(V20, 0, Ret101),fun7(Ret101, V20, V19, Ret18)],[Out = 1 + Ret18,V19 >= 0,V1 = 1,V20 >= 0,V = V20,V4 = V19]).
eq(gt(V1, V, Out),0,[],[Out = 2,V1 = 1 + V21,V21 >= 0,V = 0]).
eq(gt(V1, V, Out),0,[gt(V22, V23, Ret)],[Out = Ret,V23 >= 0,V1 = 1 + V22,V22 >= 0,V = 1 + V23]).
eq(gt(V1, V, Out),0,[],[Out = 1,V24 >= 0,V1 = 0,V = V24]).
eq(ge(V1, V, Out),0,[],[Out = 2,V1 = V25,V25 >= 0,V = 0]).
eq(ge(V1, V, Out),0,[],[Out = 1,V26 >= 0,V = 1 + V26,V1 = 0]).
eq(ge(V1, V, Out),0,[ge(V28, V27, Ret2)],[Out = Ret2,V27 >= 0,V1 = 1 + V28,V28 >= 0,V = 1 + V27]).
eq(minus(V1, V, Out),0,[gt(1 + V30, V29, Ret0),if(Ret0, V30, V29, Ret3)],[Out = Ret3,V29 >= 0,V1 = 1 + V30,V = V29,V30 >= 0]).
eq(if(V1, V, V4, Out),0,[minus(V32, V31, Ret19)],[Out = 1 + Ret19,V1 = 2,V31 >= 0,V32 >= 0,V = V32,V4 = V31]).
eq(if(V1, V, V4, Out),0,[],[Out = 0,V34 >= 0,V1 = 1,V33 >= 0,V = V33,V4 = V34]).
eq(gt(V1, V, Out),0,[],[Out = 0,V36 >= 0,V35 >= 0,V1 = V36,V = V35]).
eq(ge(V1, V, Out),0,[],[Out = 0,V38 >= 0,V37 >= 0,V1 = V38,V = V37]).
eq(minus(V1, V, Out),0,[],[Out = 0,V40 >= 0,V39 >= 0,V1 = V40,V = V39]).
eq(if(V1, V, V4, Out),0,[],[Out = 0,V41 >= 0,V4 = V43,V42 >= 0,V1 = V41,V = V42,V43 >= 0]).
eq(fun(V1, V, Out),0,[],[Out = 0,V45 >= 0,V44 >= 0,V1 = V45,V = V44]).
eq(fun1(V1, V, V4, Out),0,[],[Out = 0,V47 >= 0,V4 = V48,V46 >= 0,V1 = V47,V = V46,V48 >= 0]).
eq(fun6(V1, V, V4, Out),0,[],[Out = 0,V50 >= 0,V4 = V51,V49 >= 0,V1 = V50,V = V49,V51 >= 0]).
eq(fun7(V1, V, V4, Out),0,[],[Out = 0,V52 >= 0,V4 = V54,V53 >= 0,V1 = V52,V = V53,V54 >= 0]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V56 >= 0,V55 >= 0,V1 = V56,V = V55]).
eq(fun5(V1, V, Out),0,[],[Out = 0,V57 >= 0,V58 >= 0,V1 = V57,V = V58]).
eq(fun4(V1, V, V4, Out),0,[],[Out = 0,V59 >= 0,V4 = V61,V60 >= 0,V1 = V59,V = V60,V61 >= 0]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,V4,Out),[V1,V,V4],[Out]).
input_output_vars(fun3(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun6(V1,V,V4,Out),[V1,V,V4],[Out]).
input_output_vars(fun7(V1,V,V4,Out),[V1,V,V4],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun5(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun4(V1,V,V4,Out),[V1,V,V4],[Out]).
input_output_vars(gt(V1,V,Out),[V1,V],[Out]).
input_output_vars(ge(V1,V,Out),[V1,V],[Out]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(if(V1,V,V4,Out),[V1,V,V4],[Out]).


CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. recursive  : [fun2/3]
1. recursive  : [gt/3]
2. recursive [non_tail] : [fun/3,fun1/4]
3. recursive  : [if/4,minus/3]
4. recursive  : [fun5/3]
5. recursive  : [ge/3]
6. recursive [non_tail] : [fun3/3,fun4/4,fun6/4,fun7/4]
7. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun2/3
1. SCC is partially evaluated into gt/3
2. SCC is partially evaluated into fun/3
3. SCC is partially evaluated into minus/3
4. SCC is partially evaluated into fun5/3
5. SCC is partially evaluated into ge/3
6. SCC is partially evaluated into fun3/3
7. SCC is partially evaluated into start/3

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations fun2/3 
* CE 33 is refined into CE [40] 
* CE 32 is refined into CE [41] 


### Cost equations --> "Loop" of fun2/3 
* CEs [41] --> Loop 23 
* CEs [40] --> Loop 24 

### Ranking functions of CR fun2(V1,V,Out) 
* RF of phase [23]: [V,V1]

#### Partial ranking functions of CR fun2(V1,V,Out) 
* Partial RF of phase [23]:
  - RF of loop [23:1]:
    V
    V1


### Specialization of cost equations gt/3 
* CE 20 is refined into CE [42] 
* CE 17 is refined into CE [43] 
* CE 19 is refined into CE [44] 
* CE 18 is refined into CE [45] 


### Cost equations --> "Loop" of gt/3 
* CEs [45] --> Loop 25 
* CEs [42] --> Loop 26 
* CEs [43] --> Loop 27 
* CEs [44] --> Loop 28 

### Ranking functions of CR gt(V1,V,Out) 
* RF of phase [25]: [V,V1]

#### Partial ranking functions of CR gt(V1,V,Out) 
* Partial RF of phase [25]:
  - RF of loop [25:1]:
    V
    V1


### Specialization of cost equations fun/3 
* CE 29 is refined into CE [46,47,48,49,50,51,52] 
* CE 31 is refined into CE [53] 
* CE 30 is refined into CE [54,55,56] 


### Cost equations --> "Loop" of fun/3 
* CEs [56] --> Loop 29 
* CEs [55] --> Loop 30 
* CEs [54] --> Loop 31 
* CEs [52] --> Loop 32 
* CEs [48,50] --> Loop 33 
* CEs [53] --> Loop 34 
* CEs [46,47,49,51] --> Loop 35 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [29,30]: [V1-1,V1-V]
* RF of phase [31]: [V1]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [29,30]:
  - RF of loop [29:1,30:1]:
    V1-1
    V1-V
* Partial RF of phase [31]:
  - RF of loop [31:1]:
    V1


### Specialization of cost equations minus/3 
* CE 21 is refined into CE [57,58,59,60] 
* CE 23 is refined into CE [61] 
* CE 22 is refined into CE [62,63] 


### Cost equations --> "Loop" of minus/3 
* CEs [63] --> Loop 36 
* CEs [62] --> Loop 37 
* CEs [57,58,59,60,61] --> Loop 38 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [36]: [V1-1,V1-V]
* RF of phase [37]: [V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [36]:
  - RF of loop [36:1]:
    V1-1
    V1-V
* Partial RF of phase [37]:
  - RF of loop [37:1]:
    V1


### Specialization of cost equations fun5/3 
* CE 35 is refined into CE [64] 
* CE 34 is refined into CE [65] 


### Cost equations --> "Loop" of fun5/3 
* CEs [65] --> Loop 39 
* CEs [64] --> Loop 40 

### Ranking functions of CR fun5(V1,V,Out) 
* RF of phase [39]: [V,V1]

#### Partial ranking functions of CR fun5(V1,V,Out) 
* Partial RF of phase [39]:
  - RF of loop [39:1]:
    V
    V1


### Specialization of cost equations ge/3 
* CE 39 is refined into CE [66] 
* CE 36 is refined into CE [67] 
* CE 37 is refined into CE [68] 
* CE 38 is refined into CE [69] 


### Cost equations --> "Loop" of ge/3 
* CEs [69] --> Loop 41 
* CEs [66] --> Loop 42 
* CEs [67] --> Loop 43 
* CEs [68] --> Loop 44 

### Ranking functions of CR ge(V1,V,Out) 
* RF of phase [41]: [V,V1]

#### Partial ranking functions of CR ge(V1,V,Out) 
* Partial RF of phase [41]:
  - RF of loop [41:1]:
    V
    V1


### Specialization of cost equations fun3/3 
* CE 24 is refined into CE [70,71,72,73,74,75,76,77] 
* CE 26 is refined into CE [78,79,80,81,82,83] 
* CE 28 is refined into CE [84,85,86,87,88,89] 
* CE 27 is refined into CE [90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105] 
* CE 25 is refined into CE [106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121] 


### Cost equations --> "Loop" of fun3/3 
* CEs [101,102,103,104,105] --> Loop 45 
* CEs [99,100] --> Loop 46 
* CEs [98] --> Loop 47 
* CEs [117,118,119,120,121] --> Loop 48 
* CEs [115,116] --> Loop 49 
* CEs [114] --> Loop 50 
* CEs [109,110,111,112,113] --> Loop 51 
* CEs [107,108] --> Loop 52 
* CEs [106] --> Loop 53 
* CEs [97] --> Loop 54 
* CEs [93,96] --> Loop 55 
* CEs [94,95] --> Loop 56 
* CEs [91,92] --> Loop 57 
* CEs [90] --> Loop 58 
* CEs [87,89] --> Loop 59 
* CEs [81,83] --> Loop 60 
* CEs [77,86,88] --> Loop 61 
* CEs [73,75,80,82] --> Loop 62 
* CEs [84,85] --> Loop 63 
* CEs [71] --> Loop 64 
* CEs [78,79] --> Loop 65 
* CEs [70,72,74,76] --> Loop 66 

### Ranking functions of CR fun3(V1,V,Out) 
* RF of phase [45,46,47,48,49,50]: [V1+V-2]

#### Partial ranking functions of CR fun3(V1,V,Out) 
* Partial RF of phase [45,46,47,48,49,50]:
  - RF of loop [45:1,46:1,47:1]:
    V1-1
    V1-V depends on loops [48:1,49:1,50:1] 
  - RF of loop [48:1,49:1,50:1]:
    V-1
    -V1+V depends on loops [45:1,46:1,47:1] 


### Specialization of cost equations start/3 
* CE 4 is refined into CE [122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173] 
* CE 5 is refined into CE [174,175,176] 
* CE 6 is refined into CE [177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238] 
* CE 7 is refined into CE [239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298] 
* CE 8 is refined into CE [299,300,301] 
* CE 9 is refined into CE [302,303,304,305,306] 
* CE 1 is refined into CE [307] 
* CE 2 is refined into CE [308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359] 
* CE 3 is refined into CE [360,361,362] 
* CE 10 is refined into CE [363,364,365,366,367] 
* CE 11 is refined into CE [368,369,370,371,372,373,374,375,376,377,378,379,380] 
* CE 12 is refined into CE [381,382] 
* CE 13 is refined into CE [383,384] 
* CE 14 is refined into CE [385,386,387,388,389] 
* CE 15 is refined into CE [390,391,392,393,394] 
* CE 16 is refined into CE [395,396,397] 


### Cost equations --> "Loop" of start/3 
* CEs [363,370,386,391,395] --> Loop 67 
* CEs [174,177,178,179,180,181,182,187,192,193,194,299,302] --> Loop 68 
* CEs [122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,175,176,183,184,185,186,188,189,190,191,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,300,301,303,304,305,306] --> Loop 69 
* CEs [308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362] --> Loop 70 
* CEs [307,364,365,366,367,368,369,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,387,388,389,390,392,393,394,396,397] --> Loop 71 

### Ranking functions of CR start(V1,V,V4) 

#### Partial ranking functions of CR start(V1,V,V4) 


Computing Bounds
=====================================

#### Cost of chains of fun2(V1,V,Out):
* Chain [[23],24]: 1*it(23)+0
  Such that:it(23) =< Out

  with precondition: [Out>=1,V1>=Out,V>=Out] 

* Chain [24]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of gt(V1,V,Out):
* Chain [[25],28]: 0
  with precondition: [Out=1,V1>=1,V>=V1] 

* Chain [[25],27]: 0
  with precondition: [Out=2,V>=1,V1>=V+1] 

* Chain [[25],26]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [28]: 0
  with precondition: [V1=0,Out=1,V>=0] 

* Chain [27]: 0
  with precondition: [V=0,Out=2,V1>=1] 

* Chain [26]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun(V1,V,Out):
* Chain [[31],35]: 2*it(31)+1
  Such that:it(31) =< Out/2

  with precondition: [V=0,Out>=3,2*V1>=Out+1] 

* Chain [[31],34]: 2*it(31)+0
  Such that:it(31) =< Out/2

  with precondition: [V=0,Out>=2,2*V1>=Out] 

* Chain [[29,30],35]: 4*it(29)+1*s(3)+1
  Such that:aux(2) =< V1
aux(1) =< V
aux(6) =< V1-V
it(29) =< aux(2)
it(29) =< aux(6)
s(3) =< it(29)*aux(1)

  with precondition: [V>=1,Out>=3,V1>=V+1] 

* Chain [[29,30],34]: 4*it(29)+1*s(3)+0
  Such that:aux(2) =< V1
aux(1) =< V
aux(7) =< V1-V
it(29) =< aux(2)
it(29) =< aux(7)
s(3) =< it(29)*aux(1)

  with precondition: [V>=1,Out>=2,V1>=V+1] 

* Chain [[29,30],33]: 4*it(29)+1*s(3)+1*s(4)+1*s(5)+1
  Such that:aux(3) =< V1-V
aux(8) =< V1
aux(9) =< V
s(5) =< aux(8)
s(4) =< aux(9)
it(29) =< aux(8)
it(29) =< aux(3)
s(3) =< it(29)*aux(9)

  with precondition: [V>=1,Out>=4,V1>=V+1] 

* Chain [[29,30],32]: 4*it(29)+1*s(3)+1*s(6)+1
  Such that:aux(2) =< V1
aux(10) =< V1-V
aux(11) =< V
s(6) =< aux(11)
it(29) =< aux(2)
it(29) =< aux(10)
s(3) =< it(29)*aux(11)

  with precondition: [V>=1,Out>=4,V1>=V+2] 

* Chain [35]: 1
  with precondition: [Out=1,V1>=1,V>=0] 

* Chain [34]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [33]: 1*s(4)+1*s(5)+1
  Such that:s(5) =< V1
s(4) =< V

  with precondition: [Out>=2,V1+1>=Out,V+1>=Out] 

* Chain [32]: 1*s(6)+1
  Such that:s(6) =< V

  with precondition: [Out>=2,V1>=V+1,V+1>=Out] 


#### Cost of chains of minus(V1,V,Out):
* Chain [[37],38]: 0
  with precondition: [V=0,Out>=1,V1>=Out] 

* Chain [[36],38]: 0
  with precondition: [V>=1,Out>=1,V1>=Out+V] 

* Chain [38]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun5(V1,V,Out):
* Chain [[39],40]: 1*it(39)+0
  Such that:it(39) =< Out

  with precondition: [Out>=1,V1>=Out,V>=Out] 

* Chain [40]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of ge(V1,V,Out):
* Chain [[41],44]: 0
  with precondition: [Out=1,V1>=1,V>=V1+1] 

* Chain [[41],43]: 0
  with precondition: [Out=2,V>=1,V1>=V] 

* Chain [[41],42]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [44]: 0
  with precondition: [V1=0,Out=1,V>=1] 

* Chain [43]: 0
  with precondition: [V=0,Out=2,V1>=0] 

* Chain [42]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun3(V1,V,Out):
* Chain [[45,46,47,48,49,50],66]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+12*s(103)+32*s(104)+8*s(105)+4*s(106)+1
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(51) =< V1
aux(52) =< V1+V
aux(53) =< V
it(45) =< aux(51)
s(95) =< aux(51)
it(45) =< aux(52)
it(48) =< aux(52)
it(48) =< aux(53)
s(103) =< aux(53)
aux(39) =< aux(53)
aux(31) =< aux(51)
it(48) =< aux(51)+aux(51)+aux(51)+aux(43)
it(45) =< aux(53)+aux(53)+aux(53)+aux(45)
s(108) =< it(48)*aux(53)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(51)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(51)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(53)

  with precondition: [V1>=1,V>=1,Out>=4,V+V1>=3] 

* Chain [[45,46,47,48,49,50],62]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+12*s(103)+32*s(104)+8*s(105)+4*s(106)+2*s(113)+2
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(54) =< V1
aux(55) =< V1+V
aux(56) =< V
s(113) =< aux(55)
it(45) =< aux(54)
s(95) =< aux(54)
it(45) =< aux(55)
it(48) =< aux(55)
it(48) =< aux(56)
s(103) =< aux(56)
aux(39) =< aux(56)
aux(31) =< aux(54)
it(48) =< aux(54)+aux(54)+aux(54)+aux(43)
it(45) =< aux(56)+aux(56)+aux(56)+aux(45)
s(108) =< it(48)*aux(56)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(54)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(54)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(56)

  with precondition: [V1>=1,V>=1,Out>=5,V+V1>=3] 

* Chain [[45,46,47,48,49,50],61]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+13*s(103)+32*s(104)+8*s(105)+4*s(106)+2
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(57) =< V1
aux(58) =< V1+V
aux(59) =< V
s(103) =< aux(59)
it(45) =< aux(57)
s(95) =< aux(57)
it(45) =< aux(58)
it(48) =< aux(58)
it(48) =< aux(59)
aux(39) =< aux(59)
aux(31) =< aux(57)
it(48) =< aux(57)+aux(57)+aux(57)+aux(43)
it(45) =< aux(59)+aux(59)+aux(59)+aux(45)
s(108) =< it(48)*aux(59)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(57)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(57)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(59)

  with precondition: [V1>=1,V>=1,Out>=5,V+V1>=3] 

* Chain [[45,46,47,48,49,50],60]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+14*s(103)+32*s(104)+8*s(105)+4*s(106)+2
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(61) =< V1
aux(62) =< V1+V
aux(63) =< V
s(103) =< aux(63)
it(45) =< aux(61)
s(95) =< aux(61)
it(45) =< aux(62)
it(48) =< aux(62)
it(48) =< aux(63)
aux(39) =< aux(63)
aux(31) =< aux(61)
it(48) =< aux(61)+aux(61)+aux(61)+aux(43)
it(45) =< aux(63)+aux(63)+aux(63)+aux(45)
s(108) =< it(48)*aux(63)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(61)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(61)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(63)

  with precondition: [V1>=1,V>=2,Out>=6,V1+2*V>=7] 

* Chain [[45,46,47,48,49,50],59]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+14*s(103)+32*s(104)+8*s(105)+4*s(106)+2
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(65) =< V1
aux(66) =< V1+V
aux(67) =< V
s(103) =< aux(67)
it(45) =< aux(65)
s(95) =< aux(65)
it(45) =< aux(66)
it(48) =< aux(66)
it(48) =< aux(67)
aux(39) =< aux(67)
aux(31) =< aux(65)
it(48) =< aux(65)+aux(65)+aux(65)+aux(43)
it(45) =< aux(67)+aux(67)+aux(67)+aux(45)
s(108) =< it(48)*aux(67)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(65)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(65)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(67)

  with precondition: [V1>=1,V>=1,Out>=6,V+V1>=3] 

* Chain [[45,46,47,48,49,50],58,66]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+12*s(103)+32*s(104)+8*s(105)+4*s(106)+4
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(68) =< V1
aux(69) =< V1+V
aux(70) =< V
it(45) =< aux(68)
s(95) =< aux(68)
it(45) =< aux(69)
it(48) =< aux(69)
it(48) =< aux(70)
s(103) =< aux(70)
aux(39) =< aux(70)
aux(31) =< aux(68)
it(48) =< aux(68)+aux(68)+aux(68)+aux(43)
it(45) =< aux(70)+aux(70)+aux(70)+aux(45)
s(108) =< it(48)*aux(70)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(68)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(68)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(70)

  with precondition: [V1>=1,V>=1,Out>=7,V+V1>=3] 

* Chain [[45,46,47,48,49,50],58,65]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+12*s(103)+32*s(104)+8*s(105)+4*s(106)+5
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(71) =< V1
aux(72) =< V1+V
aux(73) =< V
it(45) =< aux(71)
s(95) =< aux(71)
it(45) =< aux(72)
it(48) =< aux(72)
it(48) =< aux(73)
s(103) =< aux(73)
aux(39) =< aux(73)
aux(31) =< aux(71)
it(48) =< aux(71)+aux(71)+aux(71)+aux(43)
it(45) =< aux(73)+aux(73)+aux(73)+aux(45)
s(108) =< it(48)*aux(73)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(71)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(71)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(73)

  with precondition: [V1>=1,V>=1,Out>=8,V+V1>=3] 

* Chain [[45,46,47,48,49,50],57,66]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+13*s(103)+32*s(104)+8*s(105)+4*s(106)+5
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(74) =< V1
aux(75) =< V1+V
aux(76) =< V
s(103) =< aux(76)
it(45) =< aux(74)
s(95) =< aux(74)
it(45) =< aux(75)
it(48) =< aux(75)
it(48) =< aux(76)
aux(39) =< aux(76)
aux(31) =< aux(74)
it(48) =< aux(74)+aux(74)+aux(74)+aux(43)
it(45) =< aux(76)+aux(76)+aux(76)+aux(45)
s(108) =< it(48)*aux(76)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(74)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(74)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(76)

  with precondition: [V1>=1,V>=1,Out>=8,V+V1>=3] 

* Chain [[45,46,47,48,49,50],57,65]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+13*s(103)+32*s(104)+8*s(105)+4*s(106)+6
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(77) =< V1
aux(78) =< V1+V
aux(79) =< V
s(103) =< aux(79)
it(45) =< aux(77)
s(95) =< aux(77)
it(45) =< aux(78)
it(48) =< aux(78)
it(48) =< aux(79)
aux(39) =< aux(79)
aux(31) =< aux(77)
it(48) =< aux(77)+aux(77)+aux(77)+aux(43)
it(45) =< aux(79)+aux(79)+aux(79)+aux(45)
s(108) =< it(48)*aux(79)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(77)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(77)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(79)

  with precondition: [V1>=1,V>=1,Out>=9,V+V1>=3] 

* Chain [[45,46,47,48,49,50],56,66]: 11*it(45)+11*it(48)+19*s(95)+32*s(96)+8*s(97)+4*s(98)+12*s(103)+32*s(104)+8*s(105)+4*s(106)+32*s(125)+8*s(126)+2*s(127)+5
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(84) =< V1
aux(85) =< V1+V
aux(86) =< V
s(95) =< aux(84)
s(125) =< aux(85)
s(125) =< aux(84)
s(126) =< s(125)*aux(84)
s(127) =< aux(85)
it(45) =< aux(84)
it(45) =< aux(85)
it(48) =< aux(85)
it(48) =< aux(86)
s(103) =< aux(86)
aux(39) =< aux(86)
aux(31) =< aux(84)
it(48) =< aux(84)+aux(84)+aux(84)+aux(43)
it(45) =< aux(86)+aux(86)+aux(86)+aux(45)
s(108) =< it(48)*aux(86)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(84)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(84)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(86)

  with precondition: [V1>=2,V>=1,Out>=9,V+2*V1>=7] 

* Chain [[45,46,47,48,49,50],56,65]: 11*it(45)+11*it(48)+19*s(95)+32*s(96)+8*s(97)+4*s(98)+12*s(103)+32*s(104)+8*s(105)+4*s(106)+32*s(125)+8*s(126)+2*s(127)+6
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(87) =< V1
aux(88) =< V1+V
aux(89) =< V
s(95) =< aux(87)
s(125) =< aux(88)
s(125) =< aux(87)
s(126) =< s(125)*aux(87)
s(127) =< aux(88)
it(45) =< aux(87)
it(45) =< aux(88)
it(48) =< aux(88)
it(48) =< aux(89)
s(103) =< aux(89)
aux(39) =< aux(89)
aux(31) =< aux(87)
it(48) =< aux(87)+aux(87)+aux(87)+aux(43)
it(45) =< aux(89)+aux(89)+aux(89)+aux(45)
s(108) =< it(48)*aux(89)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(87)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(87)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(89)

  with precondition: [V1>=2,V>=1,Out>=10,V+2*V1>=7] 

* Chain [[45,46,47,48,49,50],55,66]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+12*s(103)+32*s(104)+8*s(105)+4*s(106)+3*s(136)+5
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(91) =< V1
aux(92) =< V1+V
aux(93) =< V
s(136) =< aux(92)
it(45) =< aux(91)
s(95) =< aux(91)
it(45) =< aux(92)
it(48) =< aux(92)
it(48) =< aux(93)
s(103) =< aux(93)
aux(39) =< aux(93)
aux(31) =< aux(91)
it(48) =< aux(91)+aux(91)+aux(91)+aux(43)
it(45) =< aux(93)+aux(93)+aux(93)+aux(45)
s(108) =< it(48)*aux(93)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(91)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(91)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(93)

  with precondition: [V1>=1,V>=1,Out>=9,V+V1>=3] 

* Chain [[45,46,47,48,49,50],55,65]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+12*s(103)+32*s(104)+8*s(105)+4*s(106)+3*s(136)+6
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(94) =< V1
aux(95) =< V1+V
aux(96) =< V
s(136) =< aux(95)
it(45) =< aux(94)
s(95) =< aux(94)
it(45) =< aux(95)
it(48) =< aux(95)
it(48) =< aux(96)
s(103) =< aux(96)
aux(39) =< aux(96)
aux(31) =< aux(94)
it(48) =< aux(94)+aux(94)+aux(94)+aux(43)
it(45) =< aux(96)+aux(96)+aux(96)+aux(45)
s(108) =< it(48)*aux(96)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(94)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(94)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(96)

  with precondition: [V1>=1,V>=1,Out>=10,V+V1>=3] 

* Chain [[45,46,47,48,49,50],54,66]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+12*s(103)+32*s(104)+8*s(105)+4*s(106)+3*s(139)+5
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(98) =< V1
aux(99) =< V1+V
aux(100) =< V
s(139) =< aux(99)
it(45) =< aux(98)
s(95) =< aux(98)
it(45) =< aux(99)
it(48) =< aux(99)
it(48) =< aux(100)
s(103) =< aux(100)
aux(39) =< aux(100)
aux(31) =< aux(98)
it(48) =< aux(98)+aux(98)+aux(98)+aux(43)
it(45) =< aux(100)+aux(100)+aux(100)+aux(45)
s(108) =< it(48)*aux(100)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(98)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(98)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(100)

  with precondition: [V1>=1,V>=1,Out>=10,V+V1>=3] 

* Chain [[45,46,47,48,49,50],54,65]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+12*s(103)+32*s(104)+8*s(105)+4*s(106)+3*s(139)+6
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(101) =< V1
aux(102) =< V1+V
aux(103) =< V
s(139) =< aux(102)
it(45) =< aux(101)
s(95) =< aux(101)
it(45) =< aux(102)
it(48) =< aux(102)
it(48) =< aux(103)
s(103) =< aux(103)
aux(39) =< aux(103)
aux(31) =< aux(101)
it(48) =< aux(101)+aux(101)+aux(101)+aux(43)
it(45) =< aux(103)+aux(103)+aux(103)+aux(45)
s(108) =< it(48)*aux(103)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(101)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(101)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(103)

  with precondition: [V1>=1,V>=1,Out>=11,V+V1>=3] 

* Chain [[45,46,47,48,49,50],53,66]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+12*s(103)+32*s(104)+8*s(105)+4*s(106)+4
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(104) =< V1
aux(105) =< V1+V
aux(106) =< V
it(45) =< aux(104)
s(95) =< aux(104)
it(45) =< aux(105)
it(48) =< aux(105)
it(48) =< aux(106)
s(103) =< aux(106)
aux(39) =< aux(106)
aux(31) =< aux(104)
it(48) =< aux(104)+aux(104)+aux(104)+aux(43)
it(45) =< aux(106)+aux(106)+aux(106)+aux(45)
s(108) =< it(48)*aux(106)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(104)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(104)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(106)

  with precondition: [V1>=1,V>=2,Out>=7,V1+2*V>=7] 

* Chain [[45,46,47,48,49,50],53,64]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+12*s(103)+32*s(104)+8*s(105)+4*s(106)+4
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(107) =< V1
aux(108) =< V1+V
aux(109) =< V
it(45) =< aux(107)
s(95) =< aux(107)
it(45) =< aux(108)
it(48) =< aux(108)
it(48) =< aux(109)
s(103) =< aux(109)
aux(39) =< aux(109)
aux(31) =< aux(107)
it(48) =< aux(107)+aux(107)+aux(107)+aux(43)
it(45) =< aux(109)+aux(109)+aux(109)+aux(45)
s(108) =< it(48)*aux(109)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(107)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(107)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(109)

  with precondition: [V1>=1,V>=2,Out>=7,V1+2*V>=7] 

* Chain [[45,46,47,48,49,50],53,63]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+12*s(103)+32*s(104)+8*s(105)+4*s(106)+5
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(110) =< V1
aux(111) =< V1+V
aux(112) =< V
it(45) =< aux(110)
s(95) =< aux(110)
it(45) =< aux(111)
it(48) =< aux(111)
it(48) =< aux(112)
s(103) =< aux(112)
aux(39) =< aux(112)
aux(31) =< aux(110)
it(48) =< aux(110)+aux(110)+aux(110)+aux(43)
it(45) =< aux(112)+aux(112)+aux(112)+aux(45)
s(108) =< it(48)*aux(112)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(110)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(110)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(112)

  with precondition: [V1>=1,V>=2,Out>=8,V1+2*V>=7] 

* Chain [[45,46,47,48,49,50],52,66]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+13*s(103)+32*s(104)+8*s(105)+4*s(106)+5
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(113) =< V1
aux(114) =< V1+V
aux(115) =< V
s(103) =< aux(115)
it(45) =< aux(113)
s(95) =< aux(113)
it(45) =< aux(114)
it(48) =< aux(114)
it(48) =< aux(115)
aux(39) =< aux(115)
aux(31) =< aux(113)
it(48) =< aux(113)+aux(113)+aux(113)+aux(43)
it(45) =< aux(115)+aux(115)+aux(115)+aux(45)
s(108) =< it(48)*aux(115)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(113)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(113)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(115)

  with precondition: [V1>=1,V>=2,Out>=8,V1+2*V>=7] 

* Chain [[45,46,47,48,49,50],52,64]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+13*s(103)+32*s(104)+8*s(105)+4*s(106)+5
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(116) =< V1
aux(117) =< V1+V
aux(118) =< V
s(103) =< aux(118)
it(45) =< aux(116)
s(95) =< aux(116)
it(45) =< aux(117)
it(48) =< aux(117)
it(48) =< aux(118)
aux(39) =< aux(118)
aux(31) =< aux(116)
it(48) =< aux(116)+aux(116)+aux(116)+aux(43)
it(45) =< aux(118)+aux(118)+aux(118)+aux(45)
s(108) =< it(48)*aux(118)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(116)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(116)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(118)

  with precondition: [V1>=1,V>=2,Out>=8,V1+2*V>=7] 

* Chain [[45,46,47,48,49,50],52,63]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+13*s(103)+32*s(104)+8*s(105)+4*s(106)+6
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(119) =< V1
aux(120) =< V1+V
aux(121) =< V
s(103) =< aux(121)
it(45) =< aux(119)
s(95) =< aux(119)
it(45) =< aux(120)
it(48) =< aux(120)
it(48) =< aux(121)
aux(39) =< aux(121)
aux(31) =< aux(119)
it(48) =< aux(119)+aux(119)+aux(119)+aux(43)
it(45) =< aux(121)+aux(121)+aux(121)+aux(45)
s(108) =< it(48)*aux(121)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(119)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(119)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(121)

  with precondition: [V1>=1,V>=2,Out>=9,V1+2*V>=7] 

* Chain [[45,46,47,48,49,50],51,66]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+23*s(103)+32*s(104)+8*s(105)+4*s(106)+32*s(148)+8*s(149)+4*s(150)+5
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(128) =< V1
aux(129) =< V1+V
aux(130) =< V
s(103) =< aux(130)
s(150) =< aux(129)
s(148) =< aux(129)
s(148) =< aux(130)
s(149) =< s(148)*aux(130)
it(45) =< aux(128)
s(95) =< aux(128)
it(45) =< aux(129)
it(48) =< aux(129)
it(48) =< aux(130)
aux(39) =< aux(130)
aux(31) =< aux(128)
it(48) =< aux(128)+aux(128)+aux(128)+aux(43)
it(45) =< aux(130)+aux(130)+aux(130)+aux(45)
s(108) =< it(48)*aux(130)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(128)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(128)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(130)

  with precondition: [V1>=1,V>=2,Out>=9,V1+2*V>=7] 

* Chain [[45,46,47,48,49,50],51,64]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+23*s(103)+32*s(104)+8*s(105)+4*s(106)+32*s(148)+8*s(149)+4*s(150)+5
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(132) =< V1
aux(133) =< V1+V
aux(134) =< V
s(103) =< aux(134)
s(150) =< aux(133)
s(148) =< aux(133)
s(148) =< aux(134)
s(149) =< s(148)*aux(134)
it(45) =< aux(132)
s(95) =< aux(132)
it(45) =< aux(133)
it(48) =< aux(133)
it(48) =< aux(134)
aux(39) =< aux(134)
aux(31) =< aux(132)
it(48) =< aux(132)+aux(132)+aux(132)+aux(43)
it(45) =< aux(134)+aux(134)+aux(134)+aux(45)
s(108) =< it(48)*aux(134)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(132)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(132)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(134)

  with precondition: [V1>=1,V>=2,Out>=9,V1+2*V>=7] 

* Chain [[45,46,47,48,49,50],51,63]: 11*it(45)+11*it(48)+12*s(95)+32*s(96)+8*s(97)+4*s(98)+23*s(103)+32*s(104)+8*s(105)+4*s(106)+32*s(148)+8*s(149)+4*s(150)+6
  Such that:aux(43) =< -V1+V
aux(45) =< V1-V
aux(136) =< V1
aux(137) =< V1+V
aux(138) =< V
s(103) =< aux(138)
s(150) =< aux(137)
s(148) =< aux(137)
s(148) =< aux(138)
s(149) =< s(148)*aux(138)
it(45) =< aux(136)
s(95) =< aux(136)
it(45) =< aux(137)
it(48) =< aux(137)
it(48) =< aux(138)
aux(39) =< aux(138)
aux(31) =< aux(136)
it(48) =< aux(136)+aux(136)+aux(136)+aux(43)
it(45) =< aux(138)+aux(138)+aux(138)+aux(45)
s(108) =< it(48)*aux(138)
s(109) =< it(48)*aux(39)
s(99) =< it(45)*aux(136)
s(100) =< it(45)*aux(31)
s(106) =< s(109)
s(104) =< s(109)
s(104) =< s(108)
s(105) =< s(104)*aux(136)
s(98) =< s(100)
s(96) =< s(100)
s(96) =< s(99)
s(97) =< s(96)*aux(138)

  with precondition: [V1>=1,V>=2,Out>=10,V1+2*V>=7] 

* Chain [66]: 1
  with precondition: [Out=1,V1>=0,V>=0] 

* Chain [65]: 2
  with precondition: [V1=0,Out=2,V>=1] 

* Chain [64]: 1
  with precondition: [V=0,Out=1,V1>=0] 

* Chain [63]: 2
  with precondition: [V=0,Out=2,V1>=0] 

* Chain [62]: 1*s(113)+1*s(114)+2
  Such that:s(114) =< V1
s(113) =< V

  with precondition: [Out>=2,V1+1>=Out,V+1>=Out] 

* Chain [61]: 1*s(115)+2
  Such that:s(115) =< V

  with precondition: [Out>=2,V1>=V,V+1>=Out] 

* Chain [60]: 2*s(116)+2
  Such that:aux(60) =< V1
s(116) =< aux(60)

  with precondition: [Out>=3,V>=V1+1,V1+2>=Out] 

* Chain [59]: 2*s(118)+2
  Such that:aux(64) =< V
s(118) =< aux(64)

  with precondition: [Out>=3,V1>=V,V+2>=Out] 

* Chain [58,66]: 4
  with precondition: [Out=4,V>=1,V1>=V] 

* Chain [58,65]: 5
  with precondition: [Out=5,V>=1,V1>=V] 

* Chain [57,66]: 1*s(120)+5
  Such that:s(120) =< V

  with precondition: [Out>=5,V1>=V,V+4>=Out] 

* Chain [57,65]: 1*s(120)+6
  Such that:s(120) =< V

  with precondition: [Out>=6,V1>=V,V+5>=Out] 

* Chain [56,66]: 7*s(124)+32*s(125)+8*s(126)+2*s(127)+5
  Such that:aux(81) =< V1
aux(82) =< V1-V
aux(83) =< V
s(124) =< aux(83)
s(125) =< aux(81)
s(125) =< aux(82)
s(126) =< s(125)*aux(83)
s(127) =< aux(81)

  with precondition: [V>=1,Out>=6,V1>=V+1] 

* Chain [56,65]: 7*s(124)+32*s(125)+8*s(126)+2*s(127)+6
  Such that:aux(81) =< V1
aux(82) =< V1-V
aux(83) =< V
s(124) =< aux(83)
s(125) =< aux(81)
s(125) =< aux(82)
s(126) =< s(125)*aux(83)
s(127) =< aux(81)

  with precondition: [V>=1,Out>=7,V1>=V+1] 

* Chain [55,66]: 2*s(136)+1*s(137)+5
  Such that:s(137) =< V1
aux(90) =< V
s(136) =< aux(90)

  with precondition: [Out>=6,V1>=V,V+5>=Out] 

* Chain [55,65]: 2*s(136)+1*s(137)+6
  Such that:s(137) =< V1
aux(90) =< V
s(136) =< aux(90)

  with precondition: [Out>=7,V1>=V,V+6>=Out] 

* Chain [54,66]: 1*s(139)+2*s(140)+5
  Such that:s(139) =< V1
aux(97) =< V
s(140) =< aux(97)

  with precondition: [Out>=7,V1>=V,2*V+5>=Out] 

* Chain [54,65]: 1*s(139)+2*s(140)+6
  Such that:s(139) =< V1
aux(97) =< V
s(140) =< aux(97)

  with precondition: [Out>=8,V1>=V,2*V+6>=Out] 

* Chain [53,66]: 4
  with precondition: [Out=4,V1>=1,V>=V1+1] 

* Chain [53,64]: 4
  with precondition: [Out=4,V1>=1,V>=V1+1] 

* Chain [53,63]: 5
  with precondition: [Out=5,V1>=1,V>=V1+1] 

* Chain [52,66]: 1*s(142)+5
  Such that:s(142) =< V1

  with precondition: [Out>=5,V>=V1+1,V1+4>=Out] 

* Chain [52,64]: 1*s(142)+5
  Such that:s(142) =< V1

  with precondition: [Out>=5,V>=V1+1,V1+4>=Out] 

* Chain [52,63]: 1*s(142)+6
  Such that:s(142) =< V1

  with precondition: [Out>=6,V>=V1+1,V1+5>=Out] 

* Chain [51,66]: 11*s(143)+32*s(148)+8*s(149)+4*s(150)+5
  Such that:aux(125) =< -V1+V
aux(124) =< V
aux(127) =< V1
s(143) =< aux(127)
s(150) =< aux(124)
s(148) =< aux(124)
s(148) =< aux(125)
s(149) =< s(148)*aux(127)

  with precondition: [V1>=1,Out>=6,V>=V1+1] 

* Chain [51,64]: 11*s(143)+32*s(148)+8*s(149)+4*s(150)+5
  Such that:aux(125) =< -V1+V
aux(124) =< V
aux(131) =< V1
s(143) =< aux(131)
s(150) =< aux(124)
s(148) =< aux(124)
s(148) =< aux(125)
s(149) =< s(148)*aux(131)

  with precondition: [V1>=1,Out>=6,V>=V1+1] 

* Chain [51,63]: 11*s(143)+32*s(148)+8*s(149)+4*s(150)+6
  Such that:aux(125) =< -V1+V
aux(124) =< V
aux(135) =< V1
s(143) =< aux(135)
s(150) =< aux(124)
s(148) =< aux(124)
s(148) =< aux(125)
s(149) =< s(148)*aux(135)

  with precondition: [V1>=1,Out>=7,V>=V1+1] 


#### Cost of chains of start(V1,V,V4):
* Chain [71]: 377*s(743)+80*s(744)+20*s(745)+351*s(746)+96*s(756)+24*s(757)+30*s(758)+96*s(759)+24*s(760)+264*s(761)+264*s(762)+96*s(769)+768*s(770)+192*s(771)+96*s(772)+768*s(773)+192*s(774)+64*s(775)+16*s(776)+6
  Such that:s(749) =< -V1+V
s(752) =< V1+V
aux(148) =< V1
aux(149) =< V1-V
aux(150) =< V
s(746) =< aux(148)
s(743) =< aux(150)
s(756) =< aux(150)
s(756) =< s(749)
s(757) =< s(756)*aux(148)
s(758) =< s(752)
s(759) =< s(752)
s(759) =< aux(150)
s(760) =< s(759)*aux(150)
s(761) =< aux(148)
s(761) =< s(752)
s(762) =< s(752)
s(762) =< aux(150)
s(763) =< aux(150)
s(764) =< aux(148)
s(762) =< aux(148)+aux(148)+aux(148)+s(749)
s(761) =< aux(150)+aux(150)+aux(150)+aux(149)
s(765) =< s(762)*aux(150)
s(766) =< s(762)*s(763)
s(767) =< s(761)*aux(148)
s(768) =< s(761)*s(764)
s(769) =< s(766)
s(770) =< s(766)
s(770) =< s(765)
s(771) =< s(770)*aux(148)
s(772) =< s(768)
s(773) =< s(768)
s(773) =< s(767)
s(774) =< s(773)*aux(150)
s(775) =< s(752)
s(775) =< aux(148)
s(776) =< s(775)*aux(148)
s(744) =< aux(148)
s(744) =< aux(149)
s(745) =< s(744)*aux(150)

  with precondition: [V1>=0,V>=0] 

* Chain [70]: 1731*s(802)+592*s(803)+52*s(804)+146*s(805)+1457*s(832)+384*s(833)+96*s(834)+96*s(837)+1056*s(838)+1056*s(839)+384*s(846)+3072*s(847)+768*s(848)+384*s(849)+3072*s(850)+768*s(851)+256*s(852)+64*s(853)+64*s(855)+9
  Such that:aux(180) =< -2*V+V4
aux(181) =< -V+V4
aux(182) =< V
aux(183) =< V4
s(832) =< aux(181)
s(802) =< aux(182)
s(805) =< aux(183)
s(833) =< aux(181)
s(833) =< aux(180)
s(834) =< s(833)*aux(182)
s(803) =< aux(183)
s(803) =< aux(181)
s(837) =< s(803)*aux(181)
s(838) =< aux(182)
s(838) =< aux(183)
s(839) =< aux(183)
s(839) =< aux(181)
s(840) =< aux(181)
s(841) =< aux(182)
s(839) =< aux(182)+aux(182)+aux(182)+aux(180)
s(838) =< aux(181)+aux(181)+aux(181)+aux(182)
s(842) =< s(839)*aux(181)
s(843) =< s(839)*s(840)
s(844) =< s(838)*aux(182)
s(845) =< s(838)*s(841)
s(846) =< s(843)
s(847) =< s(843)
s(847) =< s(842)
s(848) =< s(847)*aux(182)
s(849) =< s(845)
s(850) =< s(845)
s(850) =< s(844)
s(851) =< s(850)*aux(181)
s(852) =< aux(183)
s(852) =< aux(182)
s(853) =< s(852)*aux(182)
s(855) =< s(802)*aux(181)
s(804) =< s(803)*aux(182)

  with precondition: [V1=1,V>=0,V4>=0] 

* Chain [69]: 4006*s(1111)+944*s(1112)+108*s(1113)+2025*s(1114)+2792*s(1140)+192*s(1143)+1024*s(1145)+192*s(1146)+2112*s(1147)+2112*s(1148)+768*s(1155)+6144*s(1156)+1536*s(1157)+768*s(1158)+6144*s(1159)+1536*s(1160)+128*s(1162)+512*s(1163)+128*s(1164)+592*s(1736)+52*s(1737)+1457*s(1767)+384*s(1768)+96*s(1769)+96*s(1772)+1056*s(1773)+1056*s(1774)+384*s(1781)+3072*s(1782)+768*s(1783)+384*s(1784)+3072*s(1785)+768*s(1786)+64*s(1788)+64*s(1790)+9
  Such that:aux(261) =< -2*V+V4
aux(262) =< -V+V4
aux(263) =< V
aux(264) =< V-2*V4
aux(265) =< V-V4
aux(266) =< V4
s(1767) =< aux(262)
s(1114) =< aux(263)
s(1140) =< aux(265)
s(1111) =< aux(266)
s(1768) =< aux(262)
s(1768) =< aux(261)
s(1769) =< s(1768)*aux(263)
s(1736) =< aux(266)
s(1736) =< aux(262)
s(1772) =< s(1736)*aux(262)
s(1773) =< aux(263)
s(1773) =< aux(266)
s(1774) =< aux(266)
s(1774) =< aux(262)
s(1775) =< aux(262)
s(1776) =< aux(263)
s(1774) =< aux(263)+aux(263)+aux(263)+aux(261)
s(1773) =< aux(262)+aux(262)+aux(262)+aux(263)
s(1777) =< s(1774)*aux(262)
s(1778) =< s(1774)*s(1775)
s(1779) =< s(1773)*aux(263)
s(1780) =< s(1773)*s(1776)
s(1781) =< s(1778)
s(1782) =< s(1778)
s(1782) =< s(1777)
s(1783) =< s(1782)*aux(263)
s(1784) =< s(1780)
s(1785) =< s(1780)
s(1785) =< s(1779)
s(1786) =< s(1785)*aux(262)
s(1145) =< aux(266)
s(1145) =< aux(263)
s(1788) =< s(1145)*aux(263)
s(1790) =< s(1114)*aux(262)
s(1737) =< s(1736)*aux(263)
s(1112) =< aux(263)
s(1112) =< aux(265)
s(1113) =< s(1112)*aux(266)
s(1143) =< s(1111)*aux(265)
s(1146) =< s(1145)*aux(266)
s(1147) =< aux(265)
s(1147) =< aux(263)
s(1148) =< aux(263)
s(1148) =< aux(266)
s(1149) =< aux(266)
s(1150) =< aux(265)
s(1148) =< aux(265)+aux(265)+aux(265)+aux(266)
s(1147) =< aux(266)+aux(266)+aux(266)+aux(264)
s(1151) =< s(1148)*aux(266)
s(1152) =< s(1148)*s(1149)
s(1153) =< s(1147)*aux(265)
s(1154) =< s(1147)*s(1150)
s(1155) =< s(1152)
s(1156) =< s(1152)
s(1156) =< s(1151)
s(1157) =< s(1156)*aux(265)
s(1158) =< s(1154)
s(1159) =< s(1154)
s(1159) =< s(1153)
s(1160) =< s(1159)*aux(266)
s(1162) =< s(1112)*aux(265)
s(1163) =< aux(265)
s(1163) =< aux(264)
s(1164) =< s(1163)*aux(266)

  with precondition: [V1=2,V>=0,V4>=0] 

* Chain [68]: 20*s(2053)+4
  Such that:aux(267) =< V
s(2053) =< aux(267)

  with precondition: [V1=2,V4=0,V>=0] 

* Chain [67]: 4*s(2063)+2
  Such that:s(2062) =< V1
s(2063) =< s(2062)

  with precondition: [V=0,V1>=0] 


Closed-form bounds of start(V1,V,V4): 
-------------------------------------
* Chain [71] with precondition: [V1>=0,V>=0] 
    - Upper bound: 695*V1+6+864*V1*V1+192*V1*V1*V+44*V1*V+(V1+V)*(192*V1*V)+(V1+V)*(16*V1)+473*V+(V1+V)*(888*V)+(454*V1+454*V) 
    - Complexity: n^3 
* Chain [70] with precondition: [V1=1,V>=0,V4>=0] 
    - Upper bound: 2787*V+9+3456*V*V+768*V*V*nat(-V+V4)+116*V*V4+768*V*V4*nat(-V+V4)+160*V*nat(-V+V4)+2050*V4+3552*V4*nat(-V+V4)+nat(-V+V4)*1841 
    - Complexity: n^3 
* Chain [69] with precondition: [V1=2,V>=0,V4>=0] 
    - Upper bound: 6137*V+9+3456*V*V+768*V*V*nat(-V+V4)+7136*V*V4+768*V*V4*nat(-V+V4)+1536*V*V4*nat(V-V4)+160*V*nat(-V+V4)+128*V*nat(V-V4)+6678*V4+192*V4*V4+3552*V4*nat(-V+V4)+320*V4*nat(V-V4)+1536*V4*nat(V-V4)*nat(V-V4)+nat(-V+V4)*1841+nat(V-V4)*5416+nat(V-V4)*6912*nat(V-V4) 
    - Complexity: n^3 
* Chain [68] with precondition: [V1=2,V4=0,V>=0] 
    - Upper bound: 20*V+4 
    - Complexity: n 
* Chain [67] with precondition: [V=0,V1>=0] 
    - Upper bound: 4*V1+2 
    - Complexity: n 

### Maximum cost of start(V1,V,V4): max([4*V1,453*V+2+max([864*V1*V1+695*V1+192*V1*V1*V+44*V1*V+(V1+V)*(192*V1*V)+(V1+V)*(16*V1)+(V1+V)*(888*V)+(454*V1+454*V),7020*V*nat(V4)+3350*V+1536*V*nat(V4)*nat(V-V4)+128*V*nat(V-V4)+nat(V4)*4628+nat(V4)*192*nat(V4)+nat(V4)*320*nat(V-V4)+nat(V4)*1536*nat(V-V4)*nat(V-V4)+nat(V-V4)*5416+nat(V-V4)*6912*nat(V-V4)+(2314*V+3+3456*V*V+768*V*V*nat(-V+V4)+116*V*nat(V4)+768*V*nat(V4)*nat(-V+V4)+160*V*nat(-V+V4)+nat(V4)*2050+nat(V4)*3552*nat(-V+V4)+nat(-V+V4)*1841)])+(20*V+2)])+2 
Asymptotic class: n^3 
* Total analysis performed in 5686 ms.


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

(20)
BOUNDS(1, n^3)

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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
   if(true, z0, z1) -> s(minus(z0, z1))
   if(false, z0, z1) -> 0
   gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
   if1(true, z0, z1) -> if2(gt(z1, 0), z0, z1)
   if1(false, z0, z1) -> if3(gt(z0, 0), z0, z1)
   if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
   if2(false, z0, z1) -> z0
   if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
   if3(false, z0, z1) -> z1
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   IF(false, z0, z1) -> c2
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1), GT(z1, 0))
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1), GT(z0, 0))
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
   IF2(false, z0, z1) -> c7
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
   IF3(false, z0, z1) -> c9
   GT(0, z0) -> c10
   GT(s(z0), 0) -> c11
   GT(s(z0), s(z1)) -> c12(GT(z0, z1))
   GE(z0, 0) -> c13
   GE(0, s(z0)) -> c14
   GE(s(z0), s(z1)) -> c15(GE(z0, z1))
S tuples:
   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   IF(false, z0, z1) -> c2
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1), GT(z1, 0))
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1), GT(z0, 0))
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
   IF2(false, z0, z1) -> c7
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
   IF3(false, z0, z1) -> c9
   GT(0, z0) -> c10
   GT(s(z0), 0) -> c11
   GT(s(z0), s(z1)) -> c12(GT(z0, z1))
   GE(z0, 0) -> c13
   GE(0, s(z0)) -> c14
   GE(s(z0), s(z1)) -> c15(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, if_3, gcd_2, if1_3, if2_3, if3_3, gt_2, ge_2

Defined Pair Symbols:   MINUS_2, IF_3, GCD_2, IF1_3, IF2_3, IF3_3, GT_2, GE_2

Compound Symbols:   c_2, c1_1, c2, c3_2, c4_2, c5_2, c6_2, c7, c8_2, c9, c10, c11, c12_1, c13, c14, c15_1


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

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

(24)
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:

   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   IF(false, z0, z1) -> c2
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0), z0, z1), GT(z1, 0))
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0), z0, z1), GT(z0, 0))
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
   IF2(false, z0, z1) -> c7
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
   IF3(false, z0, z1) -> c9
   GT(0, z0) -> c10
   GT(s(z0), 0) -> c11
   GT(s(z0), s(z1)) -> c12(GT(z0, z1))
   GE(z0, 0) -> c13
   GE(0, s(z0)) -> c14
   GE(s(z0), s(z1)) -> c15(GE(z0, z1))

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

   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
   if(true, z0, z1) -> s(minus(z0, z1))
   if(false, z0, z1) -> 0
   gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
   if1(true, z0, z1) -> if2(gt(z1, 0), z0, z1)
   if1(false, z0, z1) -> if3(gt(z0, 0), z0, z1)
   if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
   if2(false, z0, z1) -> z0
   if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
   if3(false, z0, z1) -> z1
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)

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

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

(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:

   MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
   IF(true, z0, z1) -> c1(MINUS(z0, z1))
   IF(false, z0, z1) -> c2
   GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
   IF1(true, z0, z1) -> c4(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
   IF1(false, z0, z1) -> c5(IF3(gt(z0, 0'), z0, z1), GT(z0, 0'))
   IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
   IF2(false, z0, z1) -> c7
   IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
   IF3(false, z0, z1) -> c9
   GT(0', z0) -> c10
   GT(s(z0), 0') -> c11
   GT(s(z0), s(z1)) -> c12(GT(z0, z1))
   GE(z0, 0') -> c13
   GE(0', s(z0)) -> c14
   GE(s(z0), s(z1)) -> c15(GE(z0, z1))

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

   minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
   if(true, z0, z1) -> s(minus(z0, z1))
   if(false, z0, z1) -> 0'
   gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
   if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
   if1(false, z0, z1) -> if3(gt(z0, 0'), z0, z1)
   if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
   if2(false, z0, z1) -> z0
   if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
   if3(false, z0, z1) -> z1
   gt(0', z0) -> false
   gt(s(z0), 0') -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
   ge(z0, 0') -> true
   ge(0', s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)

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

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

(28)
Obligation:
Innermost TRS:
Rules:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c4(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c5(IF3(gt(z0, 0'), z0, z1), GT(z0, 0'))
IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c7
IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
IF3(false, z0, z1) -> c9
GT(0', z0) -> c10
GT(s(z0), 0') -> c11
GT(s(z0), s(z1)) -> c12(GT(z0, z1))
GE(z0, 0') -> c13
GE(0', s(z0)) -> c14
GE(s(z0), s(z1)) -> c15(GE(z0, z1))
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> if3(gt(z0, 0'), z0, z1)
if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
if2(false, z0, z1) -> z0
if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
if3(false, z0, z1) -> z1
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c10:c11:c12 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c10:c11:c12
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GCD :: s:0' -> s:0' -> c3
c3 :: c4:c5 -> c13:c14:c15 -> c3
IF1 :: true:false -> s:0' -> s:0' -> c4:c5
ge :: s:0' -> s:0' -> true:false
GE :: s:0' -> s:0' -> c13:c14:c15
c4 :: c6:c7 -> c10:c11:c12 -> c4:c5
IF2 :: true:false -> s:0' -> s:0' -> c6:c7
0' :: s:0'
c5 :: c8:c9 -> c10:c11:c12 -> c4:c5
IF3 :: true:false -> s:0' -> s:0' -> c8:c9
c6 :: c3 -> c -> c6:c7
minus :: s:0' -> s:0' -> s:0'
c7 :: c6:c7
c8 :: c3 -> c -> c8:c9
c9 :: c8:c9
c10 :: c10:c11:c12
c11 :: c10:c11:c12
c12 :: c10:c11:c12 -> c10:c11:c12
c13 :: c13:c14:c15
c14 :: c13:c14:c15
c15 :: c13:c14:c15 -> c13:c14:c15
if :: true:false -> s:0' -> s:0' -> s:0'
gcd :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
if3 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_16 :: c
hole_s:0'2_16 :: s:0'
hole_c1:c23_16 :: c1:c2
hole_c10:c11:c124_16 :: c10:c11:c12
hole_true:false5_16 :: true:false
hole_c36_16 :: c3
hole_c4:c57_16 :: c4:c5
hole_c13:c14:c158_16 :: c13:c14:c15
hole_c6:c79_16 :: c6:c7
hole_c8:c910_16 :: c8:c9
gen_s:0'11_16 :: Nat -> s:0'
gen_c10:c11:c1212_16 :: Nat -> c10:c11:c12
gen_c13:c14:c1513_16 :: Nat -> c13:c14:c15

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

(29) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MINUS, gt, GT, GCD, ge, GE, minus, gcd

They will be analysed ascendingly in the following order:
gt < MINUS
GT < MINUS
MINUS < GCD
gt < GCD
gt < minus
gt < gcd
GT < GCD
ge < GCD
GE < GCD
minus < GCD
ge < gcd
minus < gcd

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

(30)
Obligation:
Innermost TRS:
Rules:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c4(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c5(IF3(gt(z0, 0'), z0, z1), GT(z0, 0'))
IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c7
IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
IF3(false, z0, z1) -> c9
GT(0', z0) -> c10
GT(s(z0), 0') -> c11
GT(s(z0), s(z1)) -> c12(GT(z0, z1))
GE(z0, 0') -> c13
GE(0', s(z0)) -> c14
GE(s(z0), s(z1)) -> c15(GE(z0, z1))
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> if3(gt(z0, 0'), z0, z1)
if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
if2(false, z0, z1) -> z0
if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
if3(false, z0, z1) -> z1
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c10:c11:c12 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c10:c11:c12
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GCD :: s:0' -> s:0' -> c3
c3 :: c4:c5 -> c13:c14:c15 -> c3
IF1 :: true:false -> s:0' -> s:0' -> c4:c5
ge :: s:0' -> s:0' -> true:false
GE :: s:0' -> s:0' -> c13:c14:c15
c4 :: c6:c7 -> c10:c11:c12 -> c4:c5
IF2 :: true:false -> s:0' -> s:0' -> c6:c7
0' :: s:0'
c5 :: c8:c9 -> c10:c11:c12 -> c4:c5
IF3 :: true:false -> s:0' -> s:0' -> c8:c9
c6 :: c3 -> c -> c6:c7
minus :: s:0' -> s:0' -> s:0'
c7 :: c6:c7
c8 :: c3 -> c -> c8:c9
c9 :: c8:c9
c10 :: c10:c11:c12
c11 :: c10:c11:c12
c12 :: c10:c11:c12 -> c10:c11:c12
c13 :: c13:c14:c15
c14 :: c13:c14:c15
c15 :: c13:c14:c15 -> c13:c14:c15
if :: true:false -> s:0' -> s:0' -> s:0'
gcd :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
if3 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_16 :: c
hole_s:0'2_16 :: s:0'
hole_c1:c23_16 :: c1:c2
hole_c10:c11:c124_16 :: c10:c11:c12
hole_true:false5_16 :: true:false
hole_c36_16 :: c3
hole_c4:c57_16 :: c4:c5
hole_c13:c14:c158_16 :: c13:c14:c15
hole_c6:c79_16 :: c6:c7
hole_c8:c910_16 :: c8:c9
gen_s:0'11_16 :: Nat -> s:0'
gen_c10:c11:c1212_16 :: Nat -> c10:c11:c12
gen_c13:c14:c1513_16 :: Nat -> c13:c14:c15


Generator Equations:
gen_s:0'11_16(0) <=> 0'
gen_s:0'11_16(+(x, 1)) <=> s(gen_s:0'11_16(x))
gen_c10:c11:c1212_16(0) <=> c10
gen_c10:c11:c1212_16(+(x, 1)) <=> c12(gen_c10:c11:c1212_16(x))
gen_c13:c14:c1513_16(0) <=> c13
gen_c13:c14:c1513_16(+(x, 1)) <=> c15(gen_c13:c14:c1513_16(x))


The following defined symbols remain to be analysed:
gt, MINUS, GT, GCD, ge, GE, minus, gcd

They will be analysed ascendingly in the following order:
gt < MINUS
GT < MINUS
MINUS < GCD
gt < GCD
gt < minus
gt < gcd
GT < GCD
ge < GCD
GE < GCD
minus < GCD
ge < gcd
minus < gcd

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_s:0'11_16(n15_16), gen_s:0'11_16(n15_16)) -> false, rt in Omega(0)

Induction Base:
gt(gen_s:0'11_16(0), gen_s:0'11_16(0)) ->_R^Omega(0)
false

Induction Step:
gt(gen_s:0'11_16(+(n15_16, 1)), gen_s:0'11_16(+(n15_16, 1))) ->_R^Omega(0)
gt(gen_s:0'11_16(n15_16), gen_s:0'11_16(n15_16)) ->_IH
false

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

(32)
Obligation:
Innermost TRS:
Rules:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c4(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c5(IF3(gt(z0, 0'), z0, z1), GT(z0, 0'))
IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c7
IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
IF3(false, z0, z1) -> c9
GT(0', z0) -> c10
GT(s(z0), 0') -> c11
GT(s(z0), s(z1)) -> c12(GT(z0, z1))
GE(z0, 0') -> c13
GE(0', s(z0)) -> c14
GE(s(z0), s(z1)) -> c15(GE(z0, z1))
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> if3(gt(z0, 0'), z0, z1)
if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
if2(false, z0, z1) -> z0
if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
if3(false, z0, z1) -> z1
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c10:c11:c12 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c10:c11:c12
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GCD :: s:0' -> s:0' -> c3
c3 :: c4:c5 -> c13:c14:c15 -> c3
IF1 :: true:false -> s:0' -> s:0' -> c4:c5
ge :: s:0' -> s:0' -> true:false
GE :: s:0' -> s:0' -> c13:c14:c15
c4 :: c6:c7 -> c10:c11:c12 -> c4:c5
IF2 :: true:false -> s:0' -> s:0' -> c6:c7
0' :: s:0'
c5 :: c8:c9 -> c10:c11:c12 -> c4:c5
IF3 :: true:false -> s:0' -> s:0' -> c8:c9
c6 :: c3 -> c -> c6:c7
minus :: s:0' -> s:0' -> s:0'
c7 :: c6:c7
c8 :: c3 -> c -> c8:c9
c9 :: c8:c9
c10 :: c10:c11:c12
c11 :: c10:c11:c12
c12 :: c10:c11:c12 -> c10:c11:c12
c13 :: c13:c14:c15
c14 :: c13:c14:c15
c15 :: c13:c14:c15 -> c13:c14:c15
if :: true:false -> s:0' -> s:0' -> s:0'
gcd :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
if3 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_16 :: c
hole_s:0'2_16 :: s:0'
hole_c1:c23_16 :: c1:c2
hole_c10:c11:c124_16 :: c10:c11:c12
hole_true:false5_16 :: true:false
hole_c36_16 :: c3
hole_c4:c57_16 :: c4:c5
hole_c13:c14:c158_16 :: c13:c14:c15
hole_c6:c79_16 :: c6:c7
hole_c8:c910_16 :: c8:c9
gen_s:0'11_16 :: Nat -> s:0'
gen_c10:c11:c1212_16 :: Nat -> c10:c11:c12
gen_c13:c14:c1513_16 :: Nat -> c13:c14:c15


Lemmas:
gt(gen_s:0'11_16(n15_16), gen_s:0'11_16(n15_16)) -> false, rt in Omega(0)


Generator Equations:
gen_s:0'11_16(0) <=> 0'
gen_s:0'11_16(+(x, 1)) <=> s(gen_s:0'11_16(x))
gen_c10:c11:c1212_16(0) <=> c10
gen_c10:c11:c1212_16(+(x, 1)) <=> c12(gen_c10:c11:c1212_16(x))
gen_c13:c14:c1513_16(0) <=> c13
gen_c13:c14:c1513_16(+(x, 1)) <=> c15(gen_c13:c14:c1513_16(x))


The following defined symbols remain to be analysed:
GT, MINUS, GCD, ge, GE, minus, gcd

They will be analysed ascendingly in the following order:
GT < MINUS
MINUS < GCD
GT < GCD
ge < GCD
GE < GCD
minus < GCD
ge < gcd
minus < gcd

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GT(gen_s:0'11_16(n462_16), gen_s:0'11_16(n462_16)) -> gen_c10:c11:c1212_16(n462_16), rt in Omega(1 + n462_16)

Induction Base:
GT(gen_s:0'11_16(0), gen_s:0'11_16(0)) ->_R^Omega(1)
c10

Induction Step:
GT(gen_s:0'11_16(+(n462_16, 1)), gen_s:0'11_16(+(n462_16, 1))) ->_R^Omega(1)
c12(GT(gen_s:0'11_16(n462_16), gen_s:0'11_16(n462_16))) ->_IH
c12(gen_c10:c11:c1212_16(c463_16))

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:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c4(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c5(IF3(gt(z0, 0'), z0, z1), GT(z0, 0'))
IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c7
IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
IF3(false, z0, z1) -> c9
GT(0', z0) -> c10
GT(s(z0), 0') -> c11
GT(s(z0), s(z1)) -> c12(GT(z0, z1))
GE(z0, 0') -> c13
GE(0', s(z0)) -> c14
GE(s(z0), s(z1)) -> c15(GE(z0, z1))
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> if3(gt(z0, 0'), z0, z1)
if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
if2(false, z0, z1) -> z0
if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
if3(false, z0, z1) -> z1
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c10:c11:c12 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c10:c11:c12
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GCD :: s:0' -> s:0' -> c3
c3 :: c4:c5 -> c13:c14:c15 -> c3
IF1 :: true:false -> s:0' -> s:0' -> c4:c5
ge :: s:0' -> s:0' -> true:false
GE :: s:0' -> s:0' -> c13:c14:c15
c4 :: c6:c7 -> c10:c11:c12 -> c4:c5
IF2 :: true:false -> s:0' -> s:0' -> c6:c7
0' :: s:0'
c5 :: c8:c9 -> c10:c11:c12 -> c4:c5
IF3 :: true:false -> s:0' -> s:0' -> c8:c9
c6 :: c3 -> c -> c6:c7
minus :: s:0' -> s:0' -> s:0'
c7 :: c6:c7
c8 :: c3 -> c -> c8:c9
c9 :: c8:c9
c10 :: c10:c11:c12
c11 :: c10:c11:c12
c12 :: c10:c11:c12 -> c10:c11:c12
c13 :: c13:c14:c15
c14 :: c13:c14:c15
c15 :: c13:c14:c15 -> c13:c14:c15
if :: true:false -> s:0' -> s:0' -> s:0'
gcd :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
if3 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_16 :: c
hole_s:0'2_16 :: s:0'
hole_c1:c23_16 :: c1:c2
hole_c10:c11:c124_16 :: c10:c11:c12
hole_true:false5_16 :: true:false
hole_c36_16 :: c3
hole_c4:c57_16 :: c4:c5
hole_c13:c14:c158_16 :: c13:c14:c15
hole_c6:c79_16 :: c6:c7
hole_c8:c910_16 :: c8:c9
gen_s:0'11_16 :: Nat -> s:0'
gen_c10:c11:c1212_16 :: Nat -> c10:c11:c12
gen_c13:c14:c1513_16 :: Nat -> c13:c14:c15


Lemmas:
gt(gen_s:0'11_16(n15_16), gen_s:0'11_16(n15_16)) -> false, rt in Omega(0)


Generator Equations:
gen_s:0'11_16(0) <=> 0'
gen_s:0'11_16(+(x, 1)) <=> s(gen_s:0'11_16(x))
gen_c10:c11:c1212_16(0) <=> c10
gen_c10:c11:c1212_16(+(x, 1)) <=> c12(gen_c10:c11:c1212_16(x))
gen_c13:c14:c1513_16(0) <=> c13
gen_c13:c14:c1513_16(+(x, 1)) <=> c15(gen_c13:c14:c1513_16(x))


The following defined symbols remain to be analysed:
GT, MINUS, GCD, ge, GE, minus, gcd

They will be analysed ascendingly in the following order:
GT < MINUS
MINUS < GCD
GT < GCD
ge < GCD
GE < GCD
minus < GCD
ge < gcd
minus < gcd

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

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

(37)
BOUNDS(n^1, INF)

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

(38)
Obligation:
Innermost TRS:
Rules:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c4(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c5(IF3(gt(z0, 0'), z0, z1), GT(z0, 0'))
IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c7
IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
IF3(false, z0, z1) -> c9
GT(0', z0) -> c10
GT(s(z0), 0') -> c11
GT(s(z0), s(z1)) -> c12(GT(z0, z1))
GE(z0, 0') -> c13
GE(0', s(z0)) -> c14
GE(s(z0), s(z1)) -> c15(GE(z0, z1))
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> if3(gt(z0, 0'), z0, z1)
if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
if2(false, z0, z1) -> z0
if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
if3(false, z0, z1) -> z1
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c10:c11:c12 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c10:c11:c12
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GCD :: s:0' -> s:0' -> c3
c3 :: c4:c5 -> c13:c14:c15 -> c3
IF1 :: true:false -> s:0' -> s:0' -> c4:c5
ge :: s:0' -> s:0' -> true:false
GE :: s:0' -> s:0' -> c13:c14:c15
c4 :: c6:c7 -> c10:c11:c12 -> c4:c5
IF2 :: true:false -> s:0' -> s:0' -> c6:c7
0' :: s:0'
c5 :: c8:c9 -> c10:c11:c12 -> c4:c5
IF3 :: true:false -> s:0' -> s:0' -> c8:c9
c6 :: c3 -> c -> c6:c7
minus :: s:0' -> s:0' -> s:0'
c7 :: c6:c7
c8 :: c3 -> c -> c8:c9
c9 :: c8:c9
c10 :: c10:c11:c12
c11 :: c10:c11:c12
c12 :: c10:c11:c12 -> c10:c11:c12
c13 :: c13:c14:c15
c14 :: c13:c14:c15
c15 :: c13:c14:c15 -> c13:c14:c15
if :: true:false -> s:0' -> s:0' -> s:0'
gcd :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
if3 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_16 :: c
hole_s:0'2_16 :: s:0'
hole_c1:c23_16 :: c1:c2
hole_c10:c11:c124_16 :: c10:c11:c12
hole_true:false5_16 :: true:false
hole_c36_16 :: c3
hole_c4:c57_16 :: c4:c5
hole_c13:c14:c158_16 :: c13:c14:c15
hole_c6:c79_16 :: c6:c7
hole_c8:c910_16 :: c8:c9
gen_s:0'11_16 :: Nat -> s:0'
gen_c10:c11:c1212_16 :: Nat -> c10:c11:c12
gen_c13:c14:c1513_16 :: Nat -> c13:c14:c15


Lemmas:
gt(gen_s:0'11_16(n15_16), gen_s:0'11_16(n15_16)) -> false, rt in Omega(0)
GT(gen_s:0'11_16(n462_16), gen_s:0'11_16(n462_16)) -> gen_c10:c11:c1212_16(n462_16), rt in Omega(1 + n462_16)


Generator Equations:
gen_s:0'11_16(0) <=> 0'
gen_s:0'11_16(+(x, 1)) <=> s(gen_s:0'11_16(x))
gen_c10:c11:c1212_16(0) <=> c10
gen_c10:c11:c1212_16(+(x, 1)) <=> c12(gen_c10:c11:c1212_16(x))
gen_c13:c14:c1513_16(0) <=> c13
gen_c13:c14:c1513_16(+(x, 1)) <=> c15(gen_c13:c14:c1513_16(x))


The following defined symbols remain to be analysed:
MINUS, GCD, ge, GE, minus, gcd

They will be analysed ascendingly in the following order:
MINUS < GCD
ge < GCD
GE < GCD
minus < GCD
ge < gcd
minus < gcd

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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_s:0'11_16(n1584_16), gen_s:0'11_16(n1584_16)) -> true, rt in Omega(0)

Induction Base:
ge(gen_s:0'11_16(0), gen_s:0'11_16(0)) ->_R^Omega(0)
true

Induction Step:
ge(gen_s:0'11_16(+(n1584_16, 1)), gen_s:0'11_16(+(n1584_16, 1))) ->_R^Omega(0)
ge(gen_s:0'11_16(n1584_16), gen_s:0'11_16(n1584_16)) ->_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:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c4(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c5(IF3(gt(z0, 0'), z0, z1), GT(z0, 0'))
IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c7
IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
IF3(false, z0, z1) -> c9
GT(0', z0) -> c10
GT(s(z0), 0') -> c11
GT(s(z0), s(z1)) -> c12(GT(z0, z1))
GE(z0, 0') -> c13
GE(0', s(z0)) -> c14
GE(s(z0), s(z1)) -> c15(GE(z0, z1))
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> if3(gt(z0, 0'), z0, z1)
if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
if2(false, z0, z1) -> z0
if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
if3(false, z0, z1) -> z1
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c10:c11:c12 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c10:c11:c12
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GCD :: s:0' -> s:0' -> c3
c3 :: c4:c5 -> c13:c14:c15 -> c3
IF1 :: true:false -> s:0' -> s:0' -> c4:c5
ge :: s:0' -> s:0' -> true:false
GE :: s:0' -> s:0' -> c13:c14:c15
c4 :: c6:c7 -> c10:c11:c12 -> c4:c5
IF2 :: true:false -> s:0' -> s:0' -> c6:c7
0' :: s:0'
c5 :: c8:c9 -> c10:c11:c12 -> c4:c5
IF3 :: true:false -> s:0' -> s:0' -> c8:c9
c6 :: c3 -> c -> c6:c7
minus :: s:0' -> s:0' -> s:0'
c7 :: c6:c7
c8 :: c3 -> c -> c8:c9
c9 :: c8:c9
c10 :: c10:c11:c12
c11 :: c10:c11:c12
c12 :: c10:c11:c12 -> c10:c11:c12
c13 :: c13:c14:c15
c14 :: c13:c14:c15
c15 :: c13:c14:c15 -> c13:c14:c15
if :: true:false -> s:0' -> s:0' -> s:0'
gcd :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
if3 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_16 :: c
hole_s:0'2_16 :: s:0'
hole_c1:c23_16 :: c1:c2
hole_c10:c11:c124_16 :: c10:c11:c12
hole_true:false5_16 :: true:false
hole_c36_16 :: c3
hole_c4:c57_16 :: c4:c5
hole_c13:c14:c158_16 :: c13:c14:c15
hole_c6:c79_16 :: c6:c7
hole_c8:c910_16 :: c8:c9
gen_s:0'11_16 :: Nat -> s:0'
gen_c10:c11:c1212_16 :: Nat -> c10:c11:c12
gen_c13:c14:c1513_16 :: Nat -> c13:c14:c15


Lemmas:
gt(gen_s:0'11_16(n15_16), gen_s:0'11_16(n15_16)) -> false, rt in Omega(0)
GT(gen_s:0'11_16(n462_16), gen_s:0'11_16(n462_16)) -> gen_c10:c11:c1212_16(n462_16), rt in Omega(1 + n462_16)
ge(gen_s:0'11_16(n1584_16), gen_s:0'11_16(n1584_16)) -> true, rt in Omega(0)


Generator Equations:
gen_s:0'11_16(0) <=> 0'
gen_s:0'11_16(+(x, 1)) <=> s(gen_s:0'11_16(x))
gen_c10:c11:c1212_16(0) <=> c10
gen_c10:c11:c1212_16(+(x, 1)) <=> c12(gen_c10:c11:c1212_16(x))
gen_c13:c14:c1513_16(0) <=> c13
gen_c13:c14:c1513_16(+(x, 1)) <=> c15(gen_c13:c14:c1513_16(x))


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

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

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GE(gen_s:0'11_16(n2042_16), gen_s:0'11_16(n2042_16)) -> gen_c13:c14:c1513_16(n2042_16), rt in Omega(1 + n2042_16)

Induction Base:
GE(gen_s:0'11_16(0), gen_s:0'11_16(0)) ->_R^Omega(1)
c13

Induction Step:
GE(gen_s:0'11_16(+(n2042_16, 1)), gen_s:0'11_16(+(n2042_16, 1))) ->_R^Omega(1)
c15(GE(gen_s:0'11_16(n2042_16), gen_s:0'11_16(n2042_16))) ->_IH
c15(gen_c13:c14:c1513_16(c2043_16))

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

(42)
Obligation:
Innermost TRS:
Rules:
MINUS(s(z0), z1) -> c(IF(gt(s(z0), z1), z0, z1), GT(s(z0), z1))
IF(true, z0, z1) -> c1(MINUS(z0, z1))
IF(false, z0, z1) -> c2
GCD(z0, z1) -> c3(IF1(ge(z0, z1), z0, z1), GE(z0, z1))
IF1(true, z0, z1) -> c4(IF2(gt(z1, 0'), z0, z1), GT(z1, 0'))
IF1(false, z0, z1) -> c5(IF3(gt(z0, 0'), z0, z1), GT(z0, 0'))
IF2(true, z0, z1) -> c6(GCD(minus(z0, z1), z1), MINUS(z0, z1))
IF2(false, z0, z1) -> c7
IF3(true, z0, z1) -> c8(GCD(z0, minus(z1, z0)), MINUS(z1, z0))
IF3(false, z0, z1) -> c9
GT(0', z0) -> c10
GT(s(z0), 0') -> c11
GT(s(z0), s(z1)) -> c12(GT(z0, z1))
GE(z0, 0') -> c13
GE(0', s(z0)) -> c14
GE(s(z0), s(z1)) -> c15(GE(z0, z1))
minus(s(z0), z1) -> if(gt(s(z0), z1), z0, z1)
if(true, z0, z1) -> s(minus(z0, z1))
if(false, z0, z1) -> 0'
gcd(z0, z1) -> if1(ge(z0, z1), z0, z1)
if1(true, z0, z1) -> if2(gt(z1, 0'), z0, z1)
if1(false, z0, z1) -> if3(gt(z0, 0'), z0, z1)
if2(true, z0, z1) -> gcd(minus(z0, z1), z1)
if2(false, z0, z1) -> z0
if3(true, z0, z1) -> gcd(z0, minus(z1, z0))
if3(false, z0, z1) -> z1
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c10:c11:c12 -> c
IF :: true:false -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> true:false
GT :: s:0' -> s:0' -> c10:c11:c12
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c1:c2
GCD :: s:0' -> s:0' -> c3
c3 :: c4:c5 -> c13:c14:c15 -> c3
IF1 :: true:false -> s:0' -> s:0' -> c4:c5
ge :: s:0' -> s:0' -> true:false
GE :: s:0' -> s:0' -> c13:c14:c15
c4 :: c6:c7 -> c10:c11:c12 -> c4:c5
IF2 :: true:false -> s:0' -> s:0' -> c6:c7
0' :: s:0'
c5 :: c8:c9 -> c10:c11:c12 -> c4:c5
IF3 :: true:false -> s:0' -> s:0' -> c8:c9
c6 :: c3 -> c -> c6:c7
minus :: s:0' -> s:0' -> s:0'
c7 :: c6:c7
c8 :: c3 -> c -> c8:c9
c9 :: c8:c9
c10 :: c10:c11:c12
c11 :: c10:c11:c12
c12 :: c10:c11:c12 -> c10:c11:c12
c13 :: c13:c14:c15
c14 :: c13:c14:c15
c15 :: c13:c14:c15 -> c13:c14:c15
if :: true:false -> s:0' -> s:0' -> s:0'
gcd :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
if3 :: true:false -> s:0' -> s:0' -> s:0'
hole_c1_16 :: c
hole_s:0'2_16 :: s:0'
hole_c1:c23_16 :: c1:c2
hole_c10:c11:c124_16 :: c10:c11:c12
hole_true:false5_16 :: true:false
hole_c36_16 :: c3
hole_c4:c57_16 :: c4:c5
hole_c13:c14:c158_16 :: c13:c14:c15
hole_c6:c79_16 :: c6:c7
hole_c8:c910_16 :: c8:c9
gen_s:0'11_16 :: Nat -> s:0'
gen_c10:c11:c1212_16 :: Nat -> c10:c11:c12
gen_c13:c14:c1513_16 :: Nat -> c13:c14:c15


Lemmas:
gt(gen_s:0'11_16(n15_16), gen_s:0'11_16(n15_16)) -> false, rt in Omega(0)
GT(gen_s:0'11_16(n462_16), gen_s:0'11_16(n462_16)) -> gen_c10:c11:c1212_16(n462_16), rt in Omega(1 + n462_16)
ge(gen_s:0'11_16(n1584_16), gen_s:0'11_16(n1584_16)) -> true, rt in Omega(0)
GE(gen_s:0'11_16(n2042_16), gen_s:0'11_16(n2042_16)) -> gen_c13:c14:c1513_16(n2042_16), rt in Omega(1 + n2042_16)


Generator Equations:
gen_s:0'11_16(0) <=> 0'
gen_s:0'11_16(+(x, 1)) <=> s(gen_s:0'11_16(x))
gen_c10:c11:c1212_16(0) <=> c10
gen_c10:c11:c1212_16(+(x, 1)) <=> c12(gen_c10:c11:c1212_16(x))
gen_c13:c14:c1513_16(0) <=> c13
gen_c13:c14:c1513_16(+(x, 1)) <=> c15(gen_c13:c14:c1513_16(x))


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

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