YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.c
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given C Problem could be proven:

(0) C Problem
(1) CToIRSProof [EQUIVALENT, 0 ms]
(2) IntTRS
(3) TerminationGraphProcessor [SOUND, 16 ms]
(4) IntTRS
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IntTRS
(7) CaseAnalysis [EQUIVALENT, 13 ms]
(8) AND
    (9) IntTRS
        (10) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (11) IntTRS
        (12) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) IntTRS
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IntTRS
        (17) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (18) YES


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(0)
Obligation:
c file /export/starexec/sandbox/benchmark/theBenchmark.c
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(1) CToIRSProof (EQUIVALENT)
Parsed C Integer Program as IRS.
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(2)
Obligation:
Rules:
f1(x, y) -> f2(x_1, y) :|: TRUE
f2(x1, x2) -> f3(x1, x3) :|: TRUE
f4(x4, x5) -> f5(arith, x5) :|: TRUE && arith = x4 - 2 * x5
f5(x14, x15) -> f6(x14, x16) :|: TRUE && x16 = x15 + 1
f3(x8, x9) -> f4(x8, x9) :|: x8 > 0
f6(x10, x11) -> f3(x10, x11) :|: TRUE
f3(x12, x13) -> f7(x12, x13) :|: x12 <= 0
Start term: f1(x, y)

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(3) TerminationGraphProcessor (SOUND)
Constructed the termination graph and obtained one non-trivial SCC.

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(4)
Obligation:
Rules:
f3(x8, x9) -> f4(x8, x9) :|: x8 > 0
f6(x10, x11) -> f3(x10, x11) :|: TRUE
f5(x14, x15) -> f6(x14, x16) :|: TRUE && x16 = x15 + 1
f4(x4, x5) -> f5(arith, x5) :|: TRUE && arith = x4 - 2 * x5

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f5(x14:0, x15:0) -> f5(x14:0 - 2 * (x15:0 + 1), x15:0 + 1) :|: x14:0 > 0

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(7) CaseAnalysis (EQUIVALENT)
Found the following inductive condition: 
f5(x0, x1): 1 + 2*x1>=0

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(8)
Complex Obligation (AND)

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(9)
Obligation:
Rules:
f5(x14:0, x15:0) -> f5(x14:0 - 2 * (x15:0 + 1), x15:0 + 1) :|: x14:0 > 0 && 2 * x15:0 + 1 >= 0

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(10) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(11)
Obligation:
Rules:
f5(x14:0:0, x15:0:0) -> f5(x14:0:0 - 2 * (x15:0:0 + 1), x15:0:0 + 1) :|: x14:0:0 > 0 && 2 * x15:0:0 >= -1

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(12) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f5 ] = 1/2*f5_1

The following rules are decreasing:
f5(x14:0:0, x15:0:0) -> f5(x14:0:0 - 2 * (x15:0:0 + 1), x15:0:0 + 1) :|: x14:0:0 > 0 && 2 * x15:0:0 >= -1

The following rules are bounded:
f5(x14:0:0, x15:0:0) -> f5(x14:0:0 - 2 * (x15:0:0 + 1), x15:0:0 + 1) :|: x14:0:0 > 0 && 2 * x15:0:0 >= -1


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(13)
YES

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(14)
Obligation:
Rules:
f5(x14:0, x15:0) -> f5(x14:0 - 2 * (x15:0 + 1), x15:0 + 1) :|: x14:0 > 0 && 2 * x15:0 + 1 < 0

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(15) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(16)
Obligation:
Rules:
f5(x14:0:0, x15:0:0) -> f5(x14:0:0 - 2 * (x15:0:0 + 1), x15:0:0 + 1) :|: x14:0:0 > 0 && 2 * x15:0:0 < -1

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(17) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f5 ] = -1*f5_2

The following rules are decreasing:
f5(x14:0:0, x15:0:0) -> f5(x14:0:0 - 2 * (x15:0:0 + 1), x15:0:0 + 1) :|: x14:0:0 > 0 && 2 * x15:0:0 < -1

The following rules are bounded:
f5(x14:0:0, x15:0:0) -> f5(x14:0:0 - 2 * (x15:0:0 + 1), x15:0:0 + 1) :|: x14:0:0 > 0 && 2 * x15:0:0 < -1


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(18)
YES