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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 185 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 14 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(pHAT0, xHAT0, yHAT0) -> l1(pHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && pHATpost = 1 && xHAT0 <= 0 && yHAT0 <= 0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x1 = x4 && x = x3 && x5 = -1 + x2 && 1 <= x2
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && x13 <= 0
l3(x18, x19, x20) -> l4(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x21 = 1 + x18 && 1 <= x19
l5(x24, x25, x26) -> l3(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x27 = 0
l6(x30, x31, x32) -> l5(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33
Start term: l6(pHAT0, 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(pHAT0, xHAT0, yHAT0) -> l1(pHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && pHATpost = 1 && xHAT0 <= 0 && yHAT0 <= 0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x1 = x4 && x = x3 && x5 = -1 + x2 && 1 <= x2
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && x13 <= 0
l3(x18, x19, x20) -> l4(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x21 = 1 + x18 && 1 <= x19
l5(x24, x25, x26) -> l3(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x27 = 0
l6(x30, x31, x32) -> l5(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33
Start term: l6(pHAT0, xHAT0, yHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(pHAT0, xHAT0, yHAT0) -> l1(pHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && pHATpost = 1 && xHAT0 <= 0 && yHAT0 <= 0
(2) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x1 = x4 && x = x3 && x5 = -1 + x2 && 1 <= x2
(3) l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
(4) l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && x13 <= 0
(5) l3(x18, x19, x20) -> l4(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x21 = 1 + x18 && 1 <= x19
(6) l5(x24, x25, x26) -> l3(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x27 = 0
(7) l6(x30, x31, x32) -> l5(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x1 = x4 && x = x3 && x5 = -1 + x2 && 1 <= x2
(2) l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9

Arcs:
(1) -> (2)
(2) -> (1)

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l0(x3:0, x10:0, x2:0) -> l0(x3:0, x10:0, -1 + x2:0) :|: x2:0 > 0

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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(x3)

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(8)
Obligation:
Rules:
l0(x2:0) -> l0(-1 + x2:0) :|: x2:0 > 0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l0(x2:0) -> l0(c) :|: c = -1 + x2:0 && x2:0 > 0

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x)] = x

The following rules are decreasing:
l0(x2:0) -> l0(c) :|: c = -1 + x2:0 && x2:0 > 0
The following rules are bounded:
l0(x2:0) -> l0(c) :|: c = -1 + x2:0 && x2:0 > 0

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(12)
YES