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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 131 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IRSwTChainingProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
(10) IRSwT
(11) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(12) IRSwT
(13) TempFilterProof [SOUND, 15 ms]
(14) IntTRS
(15) RankingReductionPairProof [EQUIVALENT, 0 ms]
(16) YES


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(0)
Obligation:
Rules:
l0(cHAT0, pHAT0, sHAT0) -> l1(cHATpost, pHATpost, sHATpost) :|: cHAT0 = cHATpost && sHATpost = 1 + sHAT0 && pHATpost = sHAT0 && 1 + pHAT0 <= cHAT0
l1(x, x1, x2) -> l0(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x6 = x9 && x10 = 1 && x11 = 1
l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
Start term: l3(cHAT0, pHAT0, sHAT0)

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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(cHAT0, pHAT0, sHAT0) -> l1(cHATpost, pHATpost, sHATpost) :|: cHAT0 = cHATpost && sHATpost = 1 + sHAT0 && pHATpost = sHAT0 && 1 + pHAT0 <= cHAT0
l1(x, x1, x2) -> l0(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x6 = x9 && x10 = 1 && x11 = 1
l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
Start term: l3(cHAT0, pHAT0, sHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(cHAT0, pHAT0, sHAT0) -> l1(cHATpost, pHATpost, sHATpost) :|: cHAT0 = cHATpost && sHATpost = 1 + sHAT0 && pHATpost = sHAT0 && 1 + pHAT0 <= cHAT0
(2) l1(x, x1, x2) -> l0(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3
(3) l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x6 = x9 && x10 = 1 && x11 = 1
(4) l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(cHAT0, pHAT0, sHAT0) -> l1(cHATpost, pHATpost, sHATpost) :|: cHAT0 = cHATpost && sHATpost = 1 + sHAT0 && pHATpost = sHAT0 && 1 + pHAT0 <= cHAT0
(2) l1(x, x1, x2) -> l0(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3

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(cHAT0:0, pHAT0:0, pHATpost:0) -> l0(cHAT0:0, pHATpost:0, 1 + pHATpost:0) :|: cHAT0:0 >= 1 + pHAT0:0

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(7) IRSwTChainingProof (EQUIVALENT)
Chaining!
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(8)
Obligation:
Rules:
l0(x, x1, x2) -> l0(x, 1 + x2, 2 + x2) :|: TRUE && x + -1 * x1 >= 1 && x + -1 * x2 >= 1

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(9) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(x, x1, x2) -> l0(x, 1 + x2, 2 + x2) :|: TRUE && x + -1 * x1 >= 1 && x + -1 * x2 >= 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!
----------------------------------------

(10)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2) -> l0(x, 1 + x2, 2 + x2) :|: TRUE && x + -1 * x1 >= 1 && x + -1 * x2 >= 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(11) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(12)
Obligation:
Rules:
l0(x:0, x1:0, x2:0) -> l0(x:0, 1 + x2:0, 2 + x2:0) :|: x:0 + -1 * x2:0 >= 1 && x:0 + -1 * x1:0 >= 1

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(13) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(14)
Obligation:
Rules:
l0(x:0, x1:0, x2:0) -> l0(x:0, c, c1) :|: c1 = 2 + x2:0 && c = 1 + x2:0 && (x:0 + -1 * x2:0 >= 1 && x:0 + -1 * x1:0 >= 1)

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(15) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l0 ] = 1/2*l0_1 + -1/2*l0_3

The following rules are decreasing:
l0(x:0, x1:0, x2:0) -> l0(x:0, c, c1) :|: c1 = 2 + x2:0 && c = 1 + x2:0 && (x:0 + -1 * x2:0 >= 1 && x:0 + -1 * x1:0 >= 1)

The following rules are bounded:
l0(x:0, x1:0, x2:0) -> l0(x:0, c, c1) :|: c1 = 2 + x2:0 && c = 1 + x2:0 && (x:0 + -1 * x2:0 >= 1 && x:0 + -1 * x1:0 >= 1)


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(16)
YES