YES
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could be proven:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 284 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 12 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 72 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) AND
    (13) IntTRS
        (14) RankingReductionPairProof [EQUIVALENT, 4 ms]
        (15) YES
    (16) IntTRS
        (17) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (18) YES


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(0)
Obligation:
Rules:
l0(ResultHAT0, xHAT0, yHAT0) -> l1(ResultHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && ResultHATpost = ResultHATpost && -1 * xHAT0 <= 0
l0(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && 0 <= -1 - x1
l3(x6, x7, x8) -> l4(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && 1 + x8 <= -1
l3(x12, x13, x14) -> l4(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 0 <= x14
l4(x18, x19, x20) -> l2(x21, x22, x23) :|: x18 = x21 && x23 = 1 + x20 && x22 = 1 + x19
l2(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
l0(x30, x31, x32) -> l5(x33, x34, x35) :|: x30 = x33 && x34 = -99 + x31 && x35 = 1 + x32 && -1 <= x32 && x32 <= -1 && 0 <= -1 - x31
l5(x36, x37, x38) -> l0(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39
l6(x42, x43, x44) -> l0(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
l7(x48, x49, x50) -> l6(x51, x52, x53) :|: x50 = x53 && x49 = x52 && x48 = x51
Start term: l7(ResultHAT0, xHAT0, yHAT0)

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(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
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(2)
Obligation:
Rules:
l0(ResultHAT0, xHAT0, yHAT0) -> l1(ResultHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && ResultHATpost = ResultHATpost && -1 * xHAT0 <= 0
l0(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && 0 <= -1 - x1
l3(x6, x7, x8) -> l4(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && 1 + x8 <= -1
l3(x12, x13, x14) -> l4(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 0 <= x14
l4(x18, x19, x20) -> l2(x21, x22, x23) :|: x18 = x21 && x23 = 1 + x20 && x22 = 1 + x19
l2(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
l0(x30, x31, x32) -> l5(x33, x34, x35) :|: x30 = x33 && x34 = -99 + x31 && x35 = 1 + x32 && -1 <= x32 && x32 <= -1 && 0 <= -1 - x31
l5(x36, x37, x38) -> l0(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39
l6(x42, x43, x44) -> l0(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
l7(x48, x49, x50) -> l6(x51, x52, x53) :|: x50 = x53 && x49 = x52 && x48 = x51
Start term: l7(ResultHAT0, xHAT0, yHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(ResultHAT0, xHAT0, yHAT0) -> l1(ResultHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && ResultHATpost = ResultHATpost && -1 * xHAT0 <= 0
(2) l0(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && 0 <= -1 - x1
(3) l3(x6, x7, x8) -> l4(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && 1 + x8 <= -1
(4) l3(x12, x13, x14) -> l4(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 0 <= x14
(5) l4(x18, x19, x20) -> l2(x21, x22, x23) :|: x18 = x21 && x23 = 1 + x20 && x22 = 1 + x19
(6) l2(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
(7) l0(x30, x31, x32) -> l5(x33, x34, x35) :|: x30 = x33 && x34 = -99 + x31 && x35 = 1 + x32 && -1 <= x32 && x32 <= -1 && 0 <= -1 - x31
(8) l5(x36, x37, x38) -> l0(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39
(9) l6(x42, x43, x44) -> l0(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
(10) l7(x48, x49, x50) -> l6(x51, x52, x53) :|: x50 = x53 && x49 = x52 && x48 = x51

Arcs:
(2) -> (3), (4)
(3) -> (5)
(4) -> (5)
(5) -> (6)
(6) -> (1), (2), (7)
(7) -> (8)
(8) -> (1), (2), (7)
(9) -> (1), (2), (7)
(10) -> (9)

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && 0 <= -1 - x1
(2) l5(x36, x37, x38) -> l0(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39
(3) l0(x30, x31, x32) -> l5(x33, x34, x35) :|: x30 = x33 && x34 = -99 + x31 && x35 = 1 + x32 && -1 <= x32 && x32 <= -1 && 0 <= -1 - x31
(4) l2(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
(5) l4(x18, x19, x20) -> l2(x21, x22, x23) :|: x18 = x21 && x23 = 1 + x20 && x22 = 1 + x19
(6) l3(x12, x13, x14) -> l4(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 0 <= x14
(7) l3(x6, x7, x8) -> l4(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && 1 + x8 <= -1

Arcs:
(1) -> (6), (7)
(2) -> (1), (3)
(3) -> (2)
(4) -> (1), (3)
(5) -> (4)
(6) -> (5)
(7) -> (5)

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l0(x30:0, x31:0, x32:0) -> l0(x30:0, -99 + x31:0, 1 + x32:0) :|: x32:0 < 0 && x32:0 > -2 && x31:0 < 0
l0(x21:0, x10:0, x11:0) -> l0(x21:0, 1 + x10:0, 1 + x11:0) :|: x10:0 < 0 && x11:0 < -1
l0(x15:0, x16:0, x17:0) -> l0(x15:0, 1 + x16:0, 1 + x17:0) :|: x16:0 < 0 && x17:0 > -1

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(7) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   l0(x1, x2, x3) -> l0(x2, x3)

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(8)
Obligation:
Rules:
l0(x31:0, x32:0) -> l0(-99 + x31:0, 1 + x32:0) :|: x32:0 < 0 && x32:0 > -2 && x31:0 < 0
l0(x10:0, x11:0) -> l0(1 + x10:0, 1 + x11:0) :|: x10:0 < 0 && x11:0 < -1
l0(x16:0, x17:0) -> l0(1 + x16:0, 1 + x17:0) :|: x16:0 < 0 && x17:0 > -1

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l0(x31:0, x32:0) -> l0(c, c1) :|: c1 = 1 + x32:0 && c = -99 + x31:0 && (x32:0 < 0 && x32:0 > -2 && x31:0 < 0)
l0(x10:0, x11:0) -> l0(c2, c3) :|: c3 = 1 + x11:0 && c2 = 1 + x10:0 && (x10:0 < 0 && x11:0 < -1)
l0(x16:0, x17:0) -> l0(c4, c5) :|: c5 = 1 + x17:0 && c4 = 1 + x16:0 && (x16:0 < 0 && x17:0 > -1)

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1)] = -100 - x - 99*x1

The following rules are decreasing:
l0(x10:0, x11:0) -> l0(c2, c3) :|: c3 = 1 + x11:0 && c2 = 1 + x10:0 && (x10:0 < 0 && x11:0 < -1)
l0(x16:0, x17:0) -> l0(c4, c5) :|: c5 = 1 + x17:0 && c4 = 1 + x16:0 && (x16:0 < 0 && x17:0 > -1)
The following rules are bounded:
l0(x31:0, x32:0) -> l0(c, c1) :|: c1 = 1 + x32:0 && c = -99 + x31:0 && (x32:0 < 0 && x32:0 > -2 && x31:0 < 0)
l0(x10:0, x11:0) -> l0(c2, c3) :|: c3 = 1 + x11:0 && c2 = 1 + x10:0 && (x10:0 < 0 && x11:0 < -1)

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(12)
Complex Obligation (AND)

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(13)
Obligation:
Rules:
l0(x31:0, x32:0) -> l0(c, c1) :|: c1 = 1 + x32:0 && c = -99 + x31:0 && (x32:0 < 0 && x32:0 > -2 && x31:0 < 0)

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(14) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l0 ] = -1*l0_2

The following rules are decreasing:
l0(x31:0, x32:0) -> l0(c, c1) :|: c1 = 1 + x32:0 && c = -99 + x31:0 && (x32:0 < 0 && x32:0 > -2 && x31:0 < 0)

The following rules are bounded:
l0(x31:0, x32:0) -> l0(c, c1) :|: c1 = 1 + x32:0 && c = -99 + x31:0 && (x32:0 < 0 && x32:0 > -2 && x31:0 < 0)


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(15)
YES

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(16)
Obligation:
Rules:
l0(x16:0, x17:0) -> l0(c4, c5) :|: c5 = 1 + x17:0 && c4 = 1 + x16:0 && (x16:0 < 0 && x17:0 > -1)

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(17) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1)] = -x

The following rules are decreasing:
l0(x16:0, x17:0) -> l0(c4, c5) :|: c5 = 1 + x17:0 && c4 = 1 + x16:0 && (x16:0 < 0 && x17:0 > -1)
The following rules are bounded:
l0(x16:0, x17:0) -> l0(c4, c5) :|: c5 = 1 + x17:0 && c4 = 1 + x16:0 && (x16:0 < 0 && x17:0 > -1)

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(18)
YES