NO
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could be disproven:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 128 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) FilterProof [EQUIVALENT, 0 ms]
(10) IntTRS
(11) IntTRSPeriodicNontermProof [COMPLETE, 0 ms]
(12) NO


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(0)
Obligation:
Rules:
l0(Result_4HAT0, x_5HAT0) -> l1(Result_4HATpost, x_5HATpost) :|: x_5HAT0 = x_5HATpost && Result_4HAT0 = Result_4HATpost
l1(x, x1) -> l2(x2, x3) :|: x1 = x3 && x = x2 && -1 * x1 <= 0
l2(x4, x5) -> l1(x6, x7) :|: x5 = x7 && x4 = x6
l1(x8, x9) -> l3(x10, x11) :|: x9 = x11 && x10 = x10 && 0 <= -1 - x9
l4(x12, x13) -> l0(x14, x15) :|: x13 = x15 && x12 = x14
Start term: l4(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) :|: x_5HAT0 = x_5HATpost && Result_4HAT0 = Result_4HATpost
l1(x, x1) -> l2(x2, x3) :|: x1 = x3 && x = x2 && -1 * x1 <= 0
l2(x4, x5) -> l1(x6, x7) :|: x5 = x7 && x4 = x6
l1(x8, x9) -> l3(x10, x11) :|: x9 = x11 && x10 = x10 && 0 <= -1 - x9
l4(x12, x13) -> l0(x14, x15) :|: x13 = x15 && x12 = x14
Start term: l4(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) :|: x_5HAT0 = x_5HATpost && Result_4HAT0 = Result_4HATpost
(2) l1(x, x1) -> l2(x2, x3) :|: x1 = x3 && x = x2 && -1 * x1 <= 0
(3) l2(x4, x5) -> l1(x6, x7) :|: x5 = x7 && x4 = x6
(4) l1(x8, x9) -> l3(x10, x11) :|: x9 = x11 && x10 = x10 && 0 <= -1 - x9
(5) l4(x12, x13) -> l0(x14, x15) :|: x13 = x15 && x12 = x14

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l1(x, x1) -> l2(x2, x3) :|: x1 = x3 && x = x2 && -1 * x1 <= 0
(2) l2(x4, x5) -> l1(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:
l1(x2:0, x1:0) -> l1(x2:0, x1:0) :|: 0 >= -1 * x1:0

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

   l1(x1, x2) -> l1(x2)

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(8)
Obligation:
Rules:
l1(x1:0) -> l1(x1:0) :|: 0 >= -1 * x1:0

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(9) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l1(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l1(x1:0) -> l1(x1:0) :|: 0 >= -1 * x1:0

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(11) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x1:0) -> f(1, x1:0) :|: pc = 1 && 0 >= -1 * x1:0
Witness term starting non-terminating reduction: f(1, 0)
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(12)
NO