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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 79 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 15 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, yHAT0) -> l1(pHATpost, yHATpost) :|: yHAT0 = yHATpost && pHATpost = 1 && yHAT0 <= 0
l0(x, x1) -> l2(x2, x3) :|: x = x2 && x3 = -1 + x1 && 1 <= x1
l2(x4, x5) -> l0(x6, x7) :|: x5 = x7 && x4 = x6
l3(x8, x9) -> l2(x10, x11) :|: x9 = x11 && x10 = 0
l4(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14
Start term: l4(pHAT0, 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, yHAT0) -> l1(pHATpost, yHATpost) :|: yHAT0 = yHATpost && pHATpost = 1 && yHAT0 <= 0
l0(x, x1) -> l2(x2, x3) :|: x = x2 && x3 = -1 + x1 && 1 <= x1
l2(x4, x5) -> l0(x6, x7) :|: x5 = x7 && x4 = x6
l3(x8, x9) -> l2(x10, x11) :|: x9 = x11 && x10 = 0
l4(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14
Start term: l4(pHAT0, yHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(pHAT0, yHAT0) -> l1(pHATpost, yHATpost) :|: yHAT0 = yHATpost && pHATpost = 1 && yHAT0 <= 0
(2) l0(x, x1) -> l2(x2, x3) :|: x = x2 && x3 = -1 + x1 && 1 <= x1
(3) l2(x4, x5) -> l0(x6, x7) :|: x5 = x7 && x4 = x6
(4) l3(x8, x9) -> l2(x10, x11) :|: x9 = x11 && x10 = 0
(5) l4(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14

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

This digraph is fully evaluated!
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(4)
Obligation:

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

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(x2:0, x1:0) -> l0(x2:0, -1 + x1:0) :|: x1:0 > 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) -> l0(x2)

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(8)
Obligation:
Rules:
l0(x1:0) -> l0(-1 + x1:0) :|: x1:0 > 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(x1:0) -> l0(c) :|: c = -1 + x1:0 && x1:0 > 0

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

The following rules are decreasing:
l0(x1:0) -> l0(c) :|: c = -1 + x1:0 && x1:0 > 0
The following rules are bounded:
l0(x1:0) -> l0(c) :|: c = -1 + x1:0 && x1:0 > 0

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(12)
YES