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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 69 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 8 ms]
        (7) IRSwT
        (8) FilterProof [EQUIVALENT, 0 ms]
        (9) IntTRS
        (10) IntTRSNonPeriodicNontermProof [COMPLETE, 4 ms]
        (11) NO
    (12) IRSwT
        (13) IntTRSCompressionProof [EQUIVALENT, 10 ms]
        (14) IRSwT
        (15) FilterProof [EQUIVALENT, 0 ms]
        (16) IntTRS
        (17) IntTRSNonPeriodicNontermProof [COMPLETE, 1 ms]
        (18) NO


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(0)
Obligation:
Rules:
l0(xHAT0) -> l1(xHATpost) :|: xHAT0 = xHATpost && xHAT0 <= xHAT0 && xHAT0 <= xHAT0
l1(x) -> l0(x1) :|: x = x1
l2(x2) -> l3(x3) :|: x2 = x3
l3(x4) -> l2(x5) :|: x4 = x5
l4(x6) -> l0(x7) :|: x6 = x7
l5(x8) -> l4(x9) :|: x8 = x9
Start term: l5(xHAT0)

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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) -> l1(xHATpost) :|: xHAT0 = xHATpost && xHAT0 <= xHAT0 && xHAT0 <= xHAT0
l1(x) -> l0(x1) :|: x = x1
l2(x2) -> l3(x3) :|: x2 = x3
l3(x4) -> l2(x5) :|: x4 = x5
l4(x6) -> l0(x7) :|: x6 = x7
l5(x8) -> l4(x9) :|: x8 = x9
Start term: l5(xHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(xHAT0) -> l1(xHATpost) :|: xHAT0 = xHATpost && xHAT0 <= xHAT0 && xHAT0 <= xHAT0
(2) l1(x) -> l0(x1) :|: x = x1
(3) l2(x2) -> l3(x3) :|: x2 = x3
(4) l3(x4) -> l2(x5) :|: x4 = x5
(5) l4(x6) -> l0(x7) :|: x6 = x7
(6) l5(x8) -> l4(x9) :|: x8 = x9

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

This digraph is fully evaluated!
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(4)
Complex Obligation (AND)

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(5)
Obligation:

Termination digraph:
Nodes:
(1) l2(x2) -> l3(x3) :|: x2 = x3
(2) l3(x4) -> l2(x5) :|: x4 = x5

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

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
l2(x2:0) -> l2(x2:0) :|: TRUE

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(8) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l2(VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(9)
Obligation:
Rules:
l2(x2:0) -> l2(x2:0) :|: TRUE

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(10) IntTRSNonPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x2:0) -> f(1, x2:0) :|: pc = 1 && TRUE
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 T)) and !(((run2_0 * 1)) = ((1 * 1)) and T))
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 T))

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(11)
NO

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(12)
Obligation:

Termination digraph:
Nodes:
(1) l0(xHAT0) -> l1(xHATpost) :|: xHAT0 = xHATpost && xHAT0 <= xHAT0 && xHAT0 <= xHAT0
(2) l1(x) -> l0(x1) :|: x = x1

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

This digraph is fully evaluated!

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(13) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(14)
Obligation:
Rules:
l0(x1:0) -> l0(x1:0) :|: TRUE

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(15) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l0(VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(16)
Obligation:
Rules:
l0(x1:0) -> l0(x1:0) :|: TRUE

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(17) IntTRSNonPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x1:0) -> f(1, x1:0) :|: pc = 1 && TRUE
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 T)) and !(((run2_0 * 1)) = ((1 * 1)) and T))
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 T))

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(18)
NO