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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 77 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 31 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) FilterProof [EQUIVALENT, 0 ms]
(10) IntTRS
(11) IntTRSNonPeriodicNontermProof [COMPLETE, 0 ms]
(12) NO


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(0)
Obligation:
Rules:
l0(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && 0 <= xHAT0
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l2(x4, x5) -> l0(x6, x7) :|: x8 = 0 && x7 = -1 + x5 && x9 = 1 && x6 = 0
l3(x10, x11) -> l2(x12, x13) :|: x11 = x13 && x10 = x12
Start term: l3(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(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && 0 <= xHAT0
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l2(x4, x5) -> l0(x6, x7) :|: x8 = 0 && x7 = -1 + x5 && x9 = 1 && x6 = 0
l3(x10, x11) -> l2(x12, x13) :|: x11 = x13 && x10 = x12
Start term: l3(xHAT0, yHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && 0 <= xHAT0
(2) l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
(3) l2(x4, x5) -> l0(x6, x7) :|: x8 = 0 && x7 = -1 + x5 && x9 = 1 && x6 = 0
(4) l3(x10, x11) -> l2(x12, x13) :|: x11 = x13 && x10 = x12

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(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && 0 <= xHAT0
(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(x2:0, x3:0) -> l0(x2:0, x3:0) :|: x2:0 > -1

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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(x1)

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

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(9) FilterProof (EQUIVALENT)
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(x2:0) :|: x2:0 > -1

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(11) IntTRSNonPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x2:0) -> f(1, x2:0) :|: pc = 1 && x2:0 > -1
Proved unsatisfiability of the following formula, indicating that the system is never left after entering:
(((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and ((run1_1 * 1)) > ((1 * -1)))) and !(((run2_0 * 1)) = ((1 * 1)) and ((run2_1 * 1)) > ((1 * -1))))
Proved satisfiability of the following formula, indicating that the system is entered at least once:
((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and ((run1_1 * 1)) > ((1 * -1))))

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(12)
NO