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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 136 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 1 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 32 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(Result_4HAT0, x_5HAT0) -> l1(Result_4HATpost, x_5HATpost) :|: Result_4HAT0 = Result_4HATpost && x_5HATpost = -1 + x_5HAT0 && 0 <= -1 + x_5HAT0
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l0(x4, x5) -> l2(x6, x7) :|: x5 <= 0 && x8 = x8 && x6 = x6 && x5 = x7
l3(x9, x10) -> l0(x11, x12) :|: x9 = x11 && x12 = -1 + x10 && 0 <= -1 + x10
l3(x13, x14) -> l2(x15, x16) :|: x14 <= 0 && x17 = x17 && x15 = x15 && x14 = x16
l4(x18, x19) -> l3(x20, x21) :|: x19 = x21 && x18 = x20
l5(x22, x23) -> l4(x24, x25) :|: x23 = x25 && x22 = x24
Start term: l5(Result_4HAT0, x_5HAT0)

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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(Result_4HAT0, x_5HAT0) -> l1(Result_4HATpost, x_5HATpost) :|: Result_4HAT0 = Result_4HATpost && x_5HATpost = -1 + x_5HAT0 && 0 <= -1 + x_5HAT0
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l0(x4, x5) -> l2(x6, x7) :|: x5 <= 0 && x8 = x8 && x6 = x6 && x5 = x7
l3(x9, x10) -> l0(x11, x12) :|: x9 = x11 && x12 = -1 + x10 && 0 <= -1 + x10
l3(x13, x14) -> l2(x15, x16) :|: x14 <= 0 && x17 = x17 && x15 = x15 && x14 = x16
l4(x18, x19) -> l3(x20, x21) :|: x19 = x21 && x18 = x20
l5(x22, x23) -> l4(x24, x25) :|: x23 = x25 && x22 = x24
Start term: l5(Result_4HAT0, x_5HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(Result_4HAT0, x_5HAT0) -> l1(Result_4HATpost, x_5HATpost) :|: Result_4HAT0 = Result_4HATpost && x_5HATpost = -1 + x_5HAT0 && 0 <= -1 + x_5HAT0
(2) l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
(3) l0(x4, x5) -> l2(x6, x7) :|: x5 <= 0 && x8 = x8 && x6 = x6 && x5 = x7
(4) l3(x9, x10) -> l0(x11, x12) :|: x9 = x11 && x12 = -1 + x10 && 0 <= -1 + x10
(5) l3(x13, x14) -> l2(x15, x16) :|: x14 <= 0 && x17 = x17 && x15 = x15 && x14 = x16
(6) l4(x18, x19) -> l3(x20, x21) :|: x19 = x21 && x18 = x20
(7) l5(x22, x23) -> l4(x24, x25) :|: x23 = x25 && x22 = x24

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(Result_4HAT0, x_5HAT0) -> l1(Result_4HATpost, x_5HATpost) :|: Result_4HAT0 = Result_4HATpost && x_5HATpost = -1 + x_5HAT0 && 0 <= -1 + x_5HAT0
(2) l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2

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(Result_4HAT0:0, x_5HAT0:0) -> l0(Result_4HAT0:0, -1 + x_5HAT0:0) :|: x_5HAT0:0 > 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(x_5HAT0:0) -> l0(-1 + x_5HAT0:0) :|: x_5HAT0:0 > 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(x_5HAT0:0) -> l0(c) :|: c = -1 + x_5HAT0:0 && x_5HAT0:0 > 0

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

The following rules are decreasing:
l0(x_5HAT0:0) -> l0(c) :|: c = -1 + x_5HAT0:0 && x_5HAT0:0 > 0
The following rules are bounded:
l0(x_5HAT0:0) -> l0(c) :|: c = -1 + x_5HAT0:0 && x_5HAT0:0 > 0

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(12)
YES