/export/starexec/sandbox2/solver/bin/starexec_run_c /export/starexec/sandbox2/benchmark/theBenchmark.c /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.c
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given C Problem could be proven:

(0) C Problem
(1) CToIRSProof [EQUIVALENT, 0 ms]
(2) IntTRS
(3) TerminationGraphProcessor [SOUND, 72 ms]
(4) IntTRS
(5) IntTRSCompressionProof [EQUIVALENT, 31 ms]
(6) IntTRS
(7) CaseAnalysis [EQUIVALENT, 21 ms]
(8) AND
    (9) IntTRS
        (10) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (11) IntTRS
        (12) CaseAnalysis [EQUIVALENT, 14 ms]
        (13) AND
            (14) IntTRS
                (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
                (16) IntTRS
                (17) PolynomialOrderProcessor [EQUIVALENT, 3 ms]
                (18) IntTRS
                (19) IntTRSCompressionProof [EQUIVALENT, 0 ms]
                (20) IntTRS
                (21) TerminationGraphProcessor [EQUIVALENT, 0 ms]
                (22) IntTRS
                (23) IntTRSCompressionProof [EQUIVALENT, 0 ms]
                (24) IntTRS
                (25) PolynomialOrderProcessor [EQUIVALENT, 5 ms]
                (26) YES
            (27) IntTRS
                (28) IntTRSCompressionProof [EQUIVALENT, 0 ms]
                (29) IntTRS
                (30) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
                (31) IntTRS
                (32) IntTRSCompressionProof [EQUIVALENT, 0 ms]
                (33) IntTRS
                (34) RankingReductionPairProof [EQUIVALENT, 0 ms]
                (35) YES
    (36) IntTRS
        (37) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (38) IntTRS
        (39) RankingReductionPairProof [EQUIVALENT, 9 ms]
        (40) IntTRS
        (41) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (42) IntTRS
        (43) CaseAnalysis [EQUIVALENT, 11 ms]
        (44) AND
            (45) IntTRS
                (46) IntTRSCompressionProof [EQUIVALENT, 0 ms]
                (47) IntTRS
                (48) RankingReductionPairProof [EQUIVALENT, 0 ms]
                (49) YES
            (50) IntTRS
                (51) IntTRSCompressionProof [EQUIVALENT, 0 ms]
                (52) IntTRS
                (53) PolynomialOrderProcessor [EQUIVALENT, 4 ms]
                (54) YES


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

(0)
Obligation:
c file /export/starexec/sandbox2/benchmark/theBenchmark.c
----------------------------------------

(1) CToIRSProof (EQUIVALENT)
Parsed C Integer Program as IRS.
----------------------------------------

(2)
Obligation:
Rules:
f1(x, y, z) -> f2(x_1, y, z) :|: TRUE
f2(x1, x2, x3) -> f3(x1, x4, x3) :|: TRUE
f3(x5, x6, x7) -> f4(x5, x6, x8) :|: TRUE
f6(x9, x10, x11) -> f9(arith, x10, x11) :|: TRUE && arith = x9 + 1
f9(x47, x48, x49) -> f10(x47, x50, x49) :|: TRUE && x50 = x48 + x47
f7(x51, x52, x53) -> f11(x54, x52, x53) :|: TRUE && x54 = x51 - x53
f11(x55, x56, x57) -> f12(x55, x58, x57) :|: TRUE && x58 = x56 + x57 * x57
f12(x59, x60, x61) -> f13(x59, x60, x62) :|: TRUE && x62 = x61 - 1
f5(x24, x25, x26) -> f6(x24, x25, x26) :|: x27 < 0
f5(x63, x64, x65) -> f6(x63, x64, x65) :|: x66 > 0
f5(x28, x29, x30) -> f7(x28, x29, x30) :|: x31 = 0
f10(x32, x33, x34) -> f8(x32, x33, x34) :|: TRUE
f13(x35, x36, x37) -> f8(x35, x36, x37) :|: TRUE
f4(x38, x39, x40) -> f5(x38, x39, x40) :|: x38 >= x39
f8(x41, x42, x43) -> f4(x41, x42, x43) :|: TRUE
f4(x44, x45, x46) -> f14(x44, x45, x46) :|: x44 < x45
Start term: f1(x, y, z)

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

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

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

(4)
Obligation:
Rules:
f4(x38, x39, x40) -> f5(x38, x39, x40) :|: x38 >= x39
f8(x41, x42, x43) -> f4(x41, x42, x43) :|: TRUE
f10(x32, x33, x34) -> f8(x32, x33, x34) :|: TRUE
f9(x47, x48, x49) -> f10(x47, x50, x49) :|: TRUE && x50 = x48 + x47
f6(x9, x10, x11) -> f9(arith, x10, x11) :|: TRUE && arith = x9 + 1
f5(x24, x25, x26) -> f6(x24, x25, x26) :|: x27 < 0
f5(x63, x64, x65) -> f6(x63, x64, x65) :|: x66 > 0
f13(x35, x36, x37) -> f8(x35, x36, x37) :|: TRUE
f12(x59, x60, x61) -> f13(x59, x60, x62) :|: TRUE && x62 = x61 - 1
f11(x55, x56, x57) -> f12(x55, x58, x57) :|: TRUE && x58 = x56 + x57 * x57
f7(x51, x52, x53) -> f11(x54, x52, x53) :|: TRUE && x54 = x51 - x53
f5(x28, x29, x30) -> f7(x28, x29, x30) :|: x31 = 0

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(6)
Obligation:
Rules:
f8(x41:0, x42:0, x43:0) -> f8(x41:0 + 1, x42:0 + (x41:0 + 1), x43:0) :|: x42:0 <= x41:0 && x66:0 > 0
f8(x, x1, x2) -> f8(x - x2, x1 + x2 * x2, x2 - 1) :|: x1 <= x
f8(x3, x4, x5) -> f8(x3 + 1, x4 + (x3 + 1), x5) :|: x4 <= x3 && x6 < 0

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(7) CaseAnalysis (EQUIVALENT)
Found the following inductive condition: 
f8(x0, x1, x2): -1 - x2>=0

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(8)
Complex Obligation (AND)

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

(9)
Obligation:
Rules:
f8(x41:0, x42:0, x43:0) -> f8(x41:0 + 1, x42:0 + (x41:0 + 1), x43:0) :|: x42:0 <= x41:0 && x66:0 > 0 && -1 * x43:0 + -1 >= 0
f8(x, x1, x2) -> f8(x - x2, x1 + x2 * x2, x2 - 1) :|: x1 <= x && -1 * x2 + -1 >= 0
f8(x3, x4, x5) -> f8(x3 + 1, x4 + (x3 + 1), x5) :|: x4 <= x3 && x6 < 0 && -1 * x5 + -1 >= 0

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(10) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(11)
Obligation:
Rules:
f8(x3:0, x4:0, x5:0) -> f8(x3:0 + 1, x4:0 + (x3:0 + 1), x5:0) :|: x4:0 <= x3:0 && x6:0 < 0 && 1 <= -1 * x5:0
f8(x41:0:0, x42:0:0, x43:0:0) -> f8(x41:0:0 + 1, x42:0:0 + (x41:0:0 + 1), x43:0:0) :|: x42:0:0 <= x41:0:0 && x66:0:0 > 0 && 1 <= -1 * x43:0:0
f8(x:0, x1:0, x2:0) -> f8(x:0 - x2:0, x1:0 + x2:0 * x2:0, x2:0 - 1) :|: x:0 >= x1:0 && 1 <= -1 * x2:0

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(12) CaseAnalysis (EQUIVALENT)
Found the following inductive condition: 
f8(x0, x1, x2): -1 + x0>=0

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

(13)
Complex Obligation (AND)

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

(14)
Obligation:
Rules:
f8(x3:0, x4:0, x5:0) -> f8(x3:0 + 1, x4:0 + (x3:0 + 1), x5:0) :|: x4:0 <= x3:0 && x6:0 < 0 && 1 <= -1 * x5:0 && x3:0 + -1 >= 0
f8(x41:0:0, x42:0:0, x43:0:0) -> f8(x41:0:0 + 1, x42:0:0 + (x41:0:0 + 1), x43:0:0) :|: x42:0:0 <= x41:0:0 && x66:0:0 > 0 && 1 <= -1 * x43:0:0 && x41:0:0 + -1 >= 0
f8(x:0, x1:0, x2:0) -> f8(x:0 - x2:0, x1:0 + x2:0 * x2:0, x2:0 - 1) :|: x:0 >= x1:0 && 1 <= -1 * x2:0 && x:0 + -1 >= 0

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(15) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(16)
Obligation:
Rules:
f8(x41:0:0:0, x42:0:0:0, x43:0:0:0) -> f8(x41:0:0:0 + 1, x42:0:0:0 + (x41:0:0:0 + 1), x43:0:0:0) :|: 1 <= -1 * x43:0:0:0 && x41:0:0:0 > 0 && x66:0:0:0 > 0 && x42:0:0:0 <= x41:0:0:0
f8(x3:0:0, x4:0:0, x5:0:0) -> f8(x3:0:0 + 1, x4:0:0 + (x3:0:0 + 1), x5:0:0) :|: 1 <= -1 * x5:0:0 && x3:0:0 > 0 && x6:0:0 < 0 && x4:0:0 <= x3:0:0
f8(x:0:0, x1:0:0, x2:0:0) -> f8(x:0:0 - x2:0:0, x1:0:0 + x2:0:0 * x2:0:0, x2:0:0 - 1) :|: x:0:0 >= x1:0:0 && 1 <= -1 * x2:0:0 && x:0:0 > 0

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(17) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f8(x, x1, x2)] = x - x1

The following rules are decreasing:
f8(x41:0:0:0, x42:0:0:0, x43:0:0:0) -> f8(x41:0:0:0 + 1, x42:0:0:0 + (x41:0:0:0 + 1), x43:0:0:0) :|: 1 <= -1 * x43:0:0:0 && x41:0:0:0 > 0 && x66:0:0:0 > 0 && x42:0:0:0 <= x41:0:0:0
f8(x3:0:0, x4:0:0, x5:0:0) -> f8(x3:0:0 + 1, x4:0:0 + (x3:0:0 + 1), x5:0:0) :|: 1 <= -1 * x5:0:0 && x3:0:0 > 0 && x6:0:0 < 0 && x4:0:0 <= x3:0:0
The following rules are bounded:
f8(x41:0:0:0, x42:0:0:0, x43:0:0:0) -> f8(x41:0:0:0 + 1, x42:0:0:0 + (x41:0:0:0 + 1), x43:0:0:0) :|: 1 <= -1 * x43:0:0:0 && x41:0:0:0 > 0 && x66:0:0:0 > 0 && x42:0:0:0 <= x41:0:0:0
f8(x3:0:0, x4:0:0, x5:0:0) -> f8(x3:0:0 + 1, x4:0:0 + (x3:0:0 + 1), x5:0:0) :|: 1 <= -1 * x5:0:0 && x3:0:0 > 0 && x6:0:0 < 0 && x4:0:0 <= x3:0:0
f8(x:0:0, x1:0:0, x2:0:0) -> f8(x:0:0 - x2:0:0, x1:0:0 + x2:0:0 * x2:0:0, x2:0:0 - 1) :|: x:0:0 >= x1:0:0 && 1 <= -1 * x2:0:0 && x:0:0 > 0

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

(18)
Obligation:
Rules:
f8(x:0:0, x1:0:0, x2:0:0) -> f8(x:0:0 - x2:0:0, x1:0:0 + x2:0:0 * x2:0:0, x2:0:0 - 1) :|: x:0:0 >= x1:0:0 && 1 <= -1 * x2:0:0 && x:0:0 > 0

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

(19) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(20)
Obligation:
Rules:
f8(x:0:0:0, x1:0:0:0, x2:0:0:0) -> f8(x:0:0:0 - x2:0:0:0, x1:0:0:0 + x2:0:0:0 * x2:0:0:0, x2:0:0:0 - 1) :|: x:0:0:0 >= x1:0:0:0 && 1 <= -1 * x2:0:0:0 && x:0:0:0 > 0

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(21) TerminationGraphProcessor (EQUIVALENT)
Constructed the termination graph and obtained one non-trivial SCC.

f8(x:0:0:0, x1:0:0:0, x2:0:0:0) -> f8(x:0:0:0 - x2:0:0:0, x1:0:0:0 + x2:0:0:0 * x2:0:0:0, x2:0:0:0 - 1) :|: x:0:0:0 >= x1:0:0:0 && 1 <= -1 * x2:0:0:0 && x:0:0:0 > 0
has been transformed into
f8(x:0:0:0, x1:0:0:0, x2:0:0:0) -> f8(x:0:0:0 - x2:0:0:0, x1:0:0:0 + x2:0:0:0 * x2:0:0:0, x2:0:0:0 - 1) :|: x2:0:0:0 = x8 - 1 && (x1:0:0:0 = x7 + x8 * x8 && (x:0:0:0 = x6 - x8 && (x:0:0:0 >= x1:0:0:0 && 1 <= -1 * x2:0:0:0 && x:0:0:0 > 0))) && x6 >= x7 && 1 <= -1 * x8 && x6 > 0.


f8(x:0:0:0, x1:0:0:0, x2:0:0:0) -> f8(x:0:0:0 - x2:0:0:0, x1:0:0:0 + x2:0:0:0 * x2:0:0:0, x2:0:0:0 - 1) :|: x2:0:0:0 = x8 - 1 && (x1:0:0:0 = x7 + x8 * x8 && (x:0:0:0 = x6 - x8 && (x:0:0:0 >= x1:0:0:0 && 1 <= -1 * x2:0:0:0 && x:0:0:0 > 0))) && x6 >= x7 && 1 <= -1 * x8 && x6 > 0 and 
f8(x:0:0:0, x1:0:0:0, x2:0:0:0) -> f8(x:0:0:0 - x2:0:0:0, x1:0:0:0 + x2:0:0:0 * x2:0:0:0, x2:0:0:0 - 1) :|: x2:0:0:0 = x8 - 1 && (x1:0:0:0 = x7 + x8 * x8 && (x:0:0:0 = x6 - x8 && (x:0:0:0 >= x1:0:0:0 && 1 <= -1 * x2:0:0:0 && x:0:0:0 > 0))) && x6 >= x7 && 1 <= -1 * x8 && x6 > 0
have been merged into the new rule
f8(x21, x22, x23) -> f8(x21 - x23 - (x23 - 1), x22 + x23 * x23 + (x23 - 1) * (x23 - 1), x23 - 1 - 1) :|: x23 = x24 - 1 && (x22 = x25 + x24 * x24 && (x21 = x26 - x24 && (x21 >= x22 && 1 <= -1 * x23 && x21 > 0))) && x26 >= x25 && 1 <= -1 * x24 && x26 > 0 && (x23 - 1 = x27 - 1 && (x22 + x23 * x23 = x28 + x27 * x27 && (x21 - x23 = x29 - x27 && (x21 - x23 >= x22 + x23 * x23 && 1 <= -1 * (x23 - 1) && x21 - x23 > 0))) && x29 >= x28 && 1 <= -1 * x27 && x29 > 0)


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(22)
Obligation:
Rules:
f8(x30, x31, x32) -> f8(x30 + -2 * x32 + 1, x31 + 2 * (x32 * x32) + -2 * x32 + 1, x32 + -2) :|: TRUE && x32 + -1 * x33 = -1 && x31 + -1 * x34 + -1 * (x33 * x33) = 0 && x30 + -1 * x35 + x33 = 0 && x30 + -1 * x31 >= 0 && x32 <= -1 && x30 >= 1 && x35 + -1 * x34 >= 0 && x33 <= -1 && x35 >= 1 && x32 + -1 * x36 = 0 && x31 + x32 * x32 + -1 * x37 + -1 * (x36 * x36) = 0 && x30 + -1 * x32 + -1 * x38 + x36 = 0 && x30 + -1 * x32 + -1 * x31 + -1 * (x32 * x32) >= 0 && x30 + -1 * x32 >= 1 && x38 + -1 * x37 >= 0 && x36 <= -1 && x38 >= 1

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(23) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(24)
Obligation:
Rules:
f8(x30:0, x31:0, x32:0) -> f8(x30:0 + -2 * x32:0 + 1, x31:0 + 2 * (x32:0 * x32:0) + -2 * x32:0 + 1, x32:0 - 2) :|: x36:0 < 0 && x38:0 > 0 && x38:0 + -1 * x37:0 >= 0 && x30:0 + -1 * x32:0 >= 1 && x30:0 + -1 * x32:0 + -1 * x31:0 + -1 * (x32:0 * x32:0) >= 0 && x30:0 + -1 * x32:0 + -1 * x38:0 + x36:0 = 0 && x31:0 + x32:0 * x32:0 + -1 * x37:0 + -1 * (x36:0 * x36:0) = 0 && x32:0 + -1 * x36:0 = 0 && x35:0 > 0 && x33:0 < 0 && x35:0 + -1 * x34:0 >= 0 && x30:0 > 0 && x32:0 < 0 && x30:0 + -1 * x31:0 >= 0 && x30:0 + -1 * x35:0 + x33:0 = 0 && x32:0 + -1 * x33:0 = -1 && x31:0 + -1 * x34:0 + -1 * (x33:0 * x33:0) = 0

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(25) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f8(x, x1, x2)] = -8 + 2*x - 2*x1 - 4*x2

The following rules are decreasing:
f8(x30:0, x31:0, x32:0) -> f8(x30:0 + -2 * x32:0 + 1, x31:0 + 2 * (x32:0 * x32:0) + -2 * x32:0 + 1, x32:0 - 2) :|: x36:0 < 0 && x38:0 > 0 && x38:0 + -1 * x37:0 >= 0 && x30:0 + -1 * x32:0 >= 1 && x30:0 + -1 * x32:0 + -1 * x31:0 + -1 * (x32:0 * x32:0) >= 0 && x30:0 + -1 * x32:0 + -1 * x38:0 + x36:0 = 0 && x31:0 + x32:0 * x32:0 + -1 * x37:0 + -1 * (x36:0 * x36:0) = 0 && x32:0 + -1 * x36:0 = 0 && x35:0 > 0 && x33:0 < 0 && x35:0 + -1 * x34:0 >= 0 && x30:0 > 0 && x32:0 < 0 && x30:0 + -1 * x31:0 >= 0 && x30:0 + -1 * x35:0 + x33:0 = 0 && x32:0 + -1 * x33:0 = -1 && x31:0 + -1 * x34:0 + -1 * (x33:0 * x33:0) = 0
The following rules are bounded:
f8(x30:0, x31:0, x32:0) -> f8(x30:0 + -2 * x32:0 + 1, x31:0 + 2 * (x32:0 * x32:0) + -2 * x32:0 + 1, x32:0 - 2) :|: x36:0 < 0 && x38:0 > 0 && x38:0 + -1 * x37:0 >= 0 && x30:0 + -1 * x32:0 >= 1 && x30:0 + -1 * x32:0 + -1 * x31:0 + -1 * (x32:0 * x32:0) >= 0 && x30:0 + -1 * x32:0 + -1 * x38:0 + x36:0 = 0 && x31:0 + x32:0 * x32:0 + -1 * x37:0 + -1 * (x36:0 * x36:0) = 0 && x32:0 + -1 * x36:0 = 0 && x35:0 > 0 && x33:0 < 0 && x35:0 + -1 * x34:0 >= 0 && x30:0 > 0 && x32:0 < 0 && x30:0 + -1 * x31:0 >= 0 && x30:0 + -1 * x35:0 + x33:0 = 0 && x32:0 + -1 * x33:0 = -1 && x31:0 + -1 * x34:0 + -1 * (x33:0 * x33:0) = 0

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

(26)
YES

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(27)
Obligation:
Rules:
f8(x3:0, x4:0, x5:0) -> f8(x3:0 + 1, x4:0 + (x3:0 + 1), x5:0) :|: x4:0 <= x3:0 && x6:0 < 0 && 1 <= -1 * x5:0 && x3:0 + -1 < 0
f8(x41:0:0, x42:0:0, x43:0:0) -> f8(x41:0:0 + 1, x42:0:0 + (x41:0:0 + 1), x43:0:0) :|: x42:0:0 <= x41:0:0 && x66:0:0 > 0 && 1 <= -1 * x43:0:0 && x41:0:0 + -1 < 0
f8(x:0, x1:0, x2:0) -> f8(x:0 - x2:0, x1:0 + x2:0 * x2:0, x2:0 - 1) :|: x:0 >= x1:0 && 1 <= -1 * x2:0 && x:0 + -1 < 0

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(28) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(29)
Obligation:
Rules:
f8(x41:0:0:0, x42:0:0:0, x43:0:0:0) -> f8(x41:0:0:0 + 1, x42:0:0:0 + (x41:0:0:0 + 1), x43:0:0:0) :|: 1 <= -1 * x43:0:0:0 && x41:0:0:0 < 1 && x66:0:0:0 > 0 && x42:0:0:0 <= x41:0:0:0
f8(x:0:0, x1:0:0, x2:0:0) -> f8(x:0:0 - x2:0:0, x1:0:0 + x2:0:0 * x2:0:0, x2:0:0 - 1) :|: x:0:0 >= x1:0:0 && 1 <= -1 * x2:0:0 && x:0:0 < 1
f8(x3:0:0, x4:0:0, x5:0:0) -> f8(x3:0:0 + 1, x4:0:0 + (x3:0:0 + 1), x5:0:0) :|: 1 <= -1 * x5:0:0 && x3:0:0 < 1 && x6:0:0 < 0 && x4:0:0 <= x3:0:0

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(30) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f8(x, x1, x2)] = -1 - x - x2

The following rules are decreasing:
f8(x41:0:0:0, x42:0:0:0, x43:0:0:0) -> f8(x41:0:0:0 + 1, x42:0:0:0 + (x41:0:0:0 + 1), x43:0:0:0) :|: 1 <= -1 * x43:0:0:0 && x41:0:0:0 < 1 && x66:0:0:0 > 0 && x42:0:0:0 <= x41:0:0:0
f8(x3:0:0, x4:0:0, x5:0:0) -> f8(x3:0:0 + 1, x4:0:0 + (x3:0:0 + 1), x5:0:0) :|: 1 <= -1 * x5:0:0 && x3:0:0 < 1 && x6:0:0 < 0 && x4:0:0 <= x3:0:0
The following rules are bounded:
f8(x41:0:0:0, x42:0:0:0, x43:0:0:0) -> f8(x41:0:0:0 + 1, x42:0:0:0 + (x41:0:0:0 + 1), x43:0:0:0) :|: 1 <= -1 * x43:0:0:0 && x41:0:0:0 < 1 && x66:0:0:0 > 0 && x42:0:0:0 <= x41:0:0:0
f8(x:0:0, x1:0:0, x2:0:0) -> f8(x:0:0 - x2:0:0, x1:0:0 + x2:0:0 * x2:0:0, x2:0:0 - 1) :|: x:0:0 >= x1:0:0 && 1 <= -1 * x2:0:0 && x:0:0 < 1
f8(x3:0:0, x4:0:0, x5:0:0) -> f8(x3:0:0 + 1, x4:0:0 + (x3:0:0 + 1), x5:0:0) :|: 1 <= -1 * x5:0:0 && x3:0:0 < 1 && x6:0:0 < 0 && x4:0:0 <= x3:0:0

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(31)
Obligation:
Rules:
f8(x:0:0, x1:0:0, x2:0:0) -> f8(x:0:0 - x2:0:0, x1:0:0 + x2:0:0 * x2:0:0, x2:0:0 - 1) :|: x:0:0 >= x1:0:0 && 1 <= -1 * x2:0:0 && x:0:0 < 1

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(32) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(33)
Obligation:
Rules:
f8(x:0:0:0, x1:0:0:0, x2:0:0:0) -> f8(x:0:0:0 - x2:0:0:0, x1:0:0:0 + x2:0:0:0 * x2:0:0:0, x2:0:0:0 - 1) :|: x:0:0:0 >= x1:0:0:0 && 1 <= -1 * x2:0:0:0 && x:0:0:0 < 1

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(34) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f8 ] = -1*f8_1

The following rules are decreasing:
f8(x:0:0:0, x1:0:0:0, x2:0:0:0) -> f8(x:0:0:0 - x2:0:0:0, x1:0:0:0 + x2:0:0:0 * x2:0:0:0, x2:0:0:0 - 1) :|: x:0:0:0 >= x1:0:0:0 && 1 <= -1 * x2:0:0:0 && x:0:0:0 < 1

The following rules are bounded:
f8(x:0:0:0, x1:0:0:0, x2:0:0:0) -> f8(x:0:0:0 - x2:0:0:0, x1:0:0:0 + x2:0:0:0 * x2:0:0:0, x2:0:0:0 - 1) :|: x:0:0:0 >= x1:0:0:0 && 1 <= -1 * x2:0:0:0 && x:0:0:0 < 1


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(35)
YES

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(36)
Obligation:
Rules:
f8(x41:0, x42:0, x43:0) -> f8(x41:0 + 1, x42:0 + (x41:0 + 1), x43:0) :|: x42:0 <= x41:0 && x66:0 > 0 && -1 * x43:0 + -1 < 0
f8(x, x1, x2) -> f8(x - x2, x1 + x2 * x2, x2 - 1) :|: x1 <= x && -1 * x2 + -1 < 0
f8(x3, x4, x5) -> f8(x3 + 1, x4 + (x3 + 1), x5) :|: x4 <= x3 && x6 < 0 && -1 * x5 + -1 < 0

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(37) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(38)
Obligation:
Rules:
f8(x:0, x1:0, x2:0) -> f8(x:0 - x2:0, x1:0 + x2:0 * x2:0, x2:0 - 1) :|: x:0 >= x1:0 && 1 > -1 * x2:0
f8(x41:0:0, x42:0:0, x43:0:0) -> f8(x41:0:0 + 1, x42:0:0 + (x41:0:0 + 1), x43:0:0) :|: x42:0:0 <= x41:0:0 && x66:0:0 > 0 && 1 > -1 * x43:0:0
f8(x3:0, x4:0, x5:0) -> f8(x3:0 + 1, x4:0 + (x3:0 + 1), x5:0) :|: x4:0 <= x3:0 && x6:0 < 0 && 1 > -1 * x5:0

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(39) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f8 ] = f8_3

The following rules are decreasing:
f8(x:0, x1:0, x2:0) -> f8(x:0 - x2:0, x1:0 + x2:0 * x2:0, x2:0 - 1) :|: x:0 >= x1:0 && 1 > -1 * x2:0

The following rules are bounded:
f8(x:0, x1:0, x2:0) -> f8(x:0 - x2:0, x1:0 + x2:0 * x2:0, x2:0 - 1) :|: x:0 >= x1:0 && 1 > -1 * x2:0
f8(x41:0:0, x42:0:0, x43:0:0) -> f8(x41:0:0 + 1, x42:0:0 + (x41:0:0 + 1), x43:0:0) :|: x42:0:0 <= x41:0:0 && x66:0:0 > 0 && 1 > -1 * x43:0:0
f8(x3:0, x4:0, x5:0) -> f8(x3:0 + 1, x4:0 + (x3:0 + 1), x5:0) :|: x4:0 <= x3:0 && x6:0 < 0 && 1 > -1 * x5:0


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

(40)
Obligation:
Rules:
f8(x41:0:0, x42:0:0, x43:0:0) -> f8(x41:0:0 + 1, x42:0:0 + (x41:0:0 + 1), x43:0:0) :|: x42:0:0 <= x41:0:0 && x66:0:0 > 0 && 1 > -1 * x43:0:0
f8(x3:0, x4:0, x5:0) -> f8(x3:0 + 1, x4:0 + (x3:0 + 1), x5:0) :|: x4:0 <= x3:0 && x6:0 < 0 && 1 > -1 * x5:0

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(41) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(42)
Obligation:
Rules:
f8(x41:0:0:0, x42:0:0:0, x43:0:0:0) -> f8(x41:0:0:0 + 1, x42:0:0:0 + (x41:0:0:0 + 1), x43:0:0:0) :|: x42:0:0:0 <= x41:0:0:0 && x66:0:0:0 > 0 && 1 > -1 * x43:0:0:0
f8(x3:0:0, x4:0:0, x5:0:0) -> f8(x3:0:0 + 1, x4:0:0 + (x3:0:0 + 1), x5:0:0) :|: x4:0:0 <= x3:0:0 && x6:0:0 < 0 && 1 > -1 * x5:0:0

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(43) CaseAnalysis (EQUIVALENT)
Found the following inductive condition: 
f8(x0, x1, x2): -1 + x0>=0

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(44)
Complex Obligation (AND)

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

(45)
Obligation:
Rules:
f8(x41:0:0:0, x42:0:0:0, x43:0:0:0) -> f8(x41:0:0:0 + 1, x42:0:0:0 + (x41:0:0:0 + 1), x43:0:0:0) :|: x42:0:0:0 <= x41:0:0:0 && x66:0:0:0 > 0 && 1 > -1 * x43:0:0:0 && x41:0:0:0 + -1 >= 0
f8(x3:0:0, x4:0:0, x5:0:0) -> f8(x3:0:0 + 1, x4:0:0 + (x3:0:0 + 1), x5:0:0) :|: x4:0:0 <= x3:0:0 && x6:0:0 < 0 && 1 > -1 * x5:0:0 && x3:0:0 + -1 >= 0

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(46) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(47)
Obligation:
Rules:
f8(x3:0:0:0, x4:0:0:0, x5:0:0:0) -> f8(x3:0:0:0 + 1, x4:0:0:0 + (x3:0:0:0 + 1), x5:0:0:0) :|: 1 > -1 * x5:0:0:0 && x3:0:0:0 > 0 && x6:0:0:0 < 0 && x4:0:0:0 <= x3:0:0:0
f8(x41:0:0:0:0, x42:0:0:0:0, x43:0:0:0:0) -> f8(x41:0:0:0:0 + 1, x42:0:0:0:0 + (x41:0:0:0:0 + 1), x43:0:0:0:0) :|: 1 > -1 * x43:0:0:0:0 && x41:0:0:0:0 > 0 && x66:0:0:0:0 > 0 && x42:0:0:0:0 <= x41:0:0:0:0

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(48) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f8 ] = f8_1 + -1*f8_2

The following rules are decreasing:
f8(x3:0:0:0, x4:0:0:0, x5:0:0:0) -> f8(x3:0:0:0 + 1, x4:0:0:0 + (x3:0:0:0 + 1), x5:0:0:0) :|: 1 > -1 * x5:0:0:0 && x3:0:0:0 > 0 && x6:0:0:0 < 0 && x4:0:0:0 <= x3:0:0:0
f8(x41:0:0:0:0, x42:0:0:0:0, x43:0:0:0:0) -> f8(x41:0:0:0:0 + 1, x42:0:0:0:0 + (x41:0:0:0:0 + 1), x43:0:0:0:0) :|: 1 > -1 * x43:0:0:0:0 && x41:0:0:0:0 > 0 && x66:0:0:0:0 > 0 && x42:0:0:0:0 <= x41:0:0:0:0

The following rules are bounded:
f8(x3:0:0:0, x4:0:0:0, x5:0:0:0) -> f8(x3:0:0:0 + 1, x4:0:0:0 + (x3:0:0:0 + 1), x5:0:0:0) :|: 1 > -1 * x5:0:0:0 && x3:0:0:0 > 0 && x6:0:0:0 < 0 && x4:0:0:0 <= x3:0:0:0
f8(x41:0:0:0:0, x42:0:0:0:0, x43:0:0:0:0) -> f8(x41:0:0:0:0 + 1, x42:0:0:0:0 + (x41:0:0:0:0 + 1), x43:0:0:0:0) :|: 1 > -1 * x43:0:0:0:0 && x41:0:0:0:0 > 0 && x66:0:0:0:0 > 0 && x42:0:0:0:0 <= x41:0:0:0:0


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(49)
YES

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

(50)
Obligation:
Rules:
f8(x41:0:0:0, x42:0:0:0, x43:0:0:0) -> f8(x41:0:0:0 + 1, x42:0:0:0 + (x41:0:0:0 + 1), x43:0:0:0) :|: x42:0:0:0 <= x41:0:0:0 && x66:0:0:0 > 0 && 1 > -1 * x43:0:0:0 && x41:0:0:0 + -1 < 0
f8(x3:0:0, x4:0:0, x5:0:0) -> f8(x3:0:0 + 1, x4:0:0 + (x3:0:0 + 1), x5:0:0) :|: x4:0:0 <= x3:0:0 && x6:0:0 < 0 && 1 > -1 * x5:0:0 && x3:0:0 + -1 < 0

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(51) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(52)
Obligation:
Rules:
f8(x41:0:0:0:0, x42:0:0:0:0, x43:0:0:0:0) -> f8(x41:0:0:0:0 + 1, x42:0:0:0:0 + (x41:0:0:0:0 + 1), x43:0:0:0:0) :|: 1 > -1 * x43:0:0:0:0 && x41:0:0:0:0 < 1 && x66:0:0:0:0 > 0 && x42:0:0:0:0 <= x41:0:0:0:0
f8(x3:0:0:0, x4:0:0:0, x5:0:0:0) -> f8(x3:0:0:0 + 1, x4:0:0:0 + (x3:0:0:0 + 1), x5:0:0:0) :|: 1 > -1 * x5:0:0:0 && x3:0:0:0 < 1 && x6:0:0:0 < 0 && x4:0:0:0 <= x3:0:0:0

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(53) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f8(x, x1, x2)] = -x

The following rules are decreasing:
f8(x41:0:0:0:0, x42:0:0:0:0, x43:0:0:0:0) -> f8(x41:0:0:0:0 + 1, x42:0:0:0:0 + (x41:0:0:0:0 + 1), x43:0:0:0:0) :|: 1 > -1 * x43:0:0:0:0 && x41:0:0:0:0 < 1 && x66:0:0:0:0 > 0 && x42:0:0:0:0 <= x41:0:0:0:0
f8(x3:0:0:0, x4:0:0:0, x5:0:0:0) -> f8(x3:0:0:0 + 1, x4:0:0:0 + (x3:0:0:0 + 1), x5:0:0:0) :|: 1 > -1 * x5:0:0:0 && x3:0:0:0 < 1 && x6:0:0:0 < 0 && x4:0:0:0 <= x3:0:0:0
The following rules are bounded:
f8(x41:0:0:0:0, x42:0:0:0:0, x43:0:0:0:0) -> f8(x41:0:0:0:0 + 1, x42:0:0:0:0 + (x41:0:0:0:0 + 1), x43:0:0:0:0) :|: 1 > -1 * x43:0:0:0:0 && x41:0:0:0:0 < 1 && x66:0:0:0:0 > 0 && x42:0:0:0:0 <= x41:0:0:0:0
f8(x3:0:0:0, x4:0:0:0, x5:0:0:0) -> f8(x3:0:0:0 + 1, x4:0:0:0 + (x3:0:0:0 + 1), x5:0:0:0) :|: 1 > -1 * x5:0:0:0 && x3:0:0:0 < 1 && x6:0:0:0 < 0 && x4:0:0:0 <= x3:0:0:0

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(54)
YES