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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 138 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 27 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 11 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 9 ms]
        (16) IRSwT
        (17) FilterProof [EQUIVALENT, 0 ms]
        (18) IntTRS
        (19) IntTRSPeriodicNontermProof [COMPLETE, 6 ms]
        (20) NO


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(0)
Obligation:
Rules:
l0(pHAT0, y_1HAT0) -> l1(pHATpost, y_1HATpost) :|: y_1HAT0 = y_1HATpost && pHAT0 = pHATpost
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l2(x4, x5) -> l3(x6, x7) :|: x5 = x7 && x6 = 0
l3(x8, x9) -> l4(x10, x11) :|: x8 = x10 && x11 = -1 + x9 && 1 <= x9
l4(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14
l3(x16, x17) -> l0(x18, x19) :|: x17 = x19 && x18 = -1 && x17 <= 0
l5(x20, x21) -> l2(x22, x23) :|: x21 = x23 && x20 = x22
Start term: l5(pHAT0, y_1HAT0)

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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(pHAT0, y_1HAT0) -> l1(pHATpost, y_1HATpost) :|: y_1HAT0 = y_1HATpost && pHAT0 = pHATpost
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l2(x4, x5) -> l3(x6, x7) :|: x5 = x7 && x6 = 0
l3(x8, x9) -> l4(x10, x11) :|: x8 = x10 && x11 = -1 + x9 && 1 <= x9
l4(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14
l3(x16, x17) -> l0(x18, x19) :|: x17 = x19 && x18 = -1 && x17 <= 0
l5(x20, x21) -> l2(x22, x23) :|: x21 = x23 && x20 = x22
Start term: l5(pHAT0, y_1HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(pHAT0, y_1HAT0) -> l1(pHATpost, y_1HATpost) :|: y_1HAT0 = y_1HATpost && pHAT0 = pHATpost
(2) l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
(3) l2(x4, x5) -> l3(x6, x7) :|: x5 = x7 && x6 = 0
(4) l3(x8, x9) -> l4(x10, x11) :|: x8 = x10 && x11 = -1 + x9 && 1 <= x9
(5) l4(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14
(6) l3(x16, x17) -> l0(x18, x19) :|: x17 = x19 && x18 = -1 && x17 <= 0
(7) l5(x20, x21) -> l2(x22, x23) :|: x21 = x23 && x20 = x22

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

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) l3(x8, x9) -> l4(x10, x11) :|: x8 = x10 && x11 = -1 + x9 && 1 <= x9
(2) l4(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14

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:
l3(x10:0, x9:0) -> l3(x10:0, -1 + x9:0) :|: x9:0 > 0

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

   l3(x1, x2) -> l3(x2)

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(9)
Obligation:
Rules:
l3(x9:0) -> l3(-1 + x9:0) :|: x9:0 > 0

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l3(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
l3(x9:0) -> l3(c) :|: c = -1 + x9:0 && x9:0 > 0

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l3(x)] = x

The following rules are decreasing:
l3(x9:0) -> l3(c) :|: c = -1 + x9:0 && x9:0 > 0
The following rules are bounded:
l3(x9:0) -> l3(c) :|: c = -1 + x9:0 && x9:0 > 0

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(13)
YES

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(14)
Obligation:

Termination digraph:
Nodes:
(1) l0(pHAT0, y_1HAT0) -> l1(pHATpost, y_1HATpost) :|: y_1HAT0 = y_1HATpost && pHAT0 = pHATpost
(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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(15) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(16)
Obligation:
Rules:
l0(pHAT0:0, x3:0) -> l0(pHAT0:0, x3:0) :|: TRUE

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(17) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l0(VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(18)
Obligation:
Rules:
l0(pHAT0:0, x3:0) -> l0(pHAT0:0, x3:0) :|: TRUE

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(19) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, pHAT0:0, x3:0) -> f(1, pHAT0:0, x3:0) :|: pc = 1 && TRUE
Witness term starting non-terminating reduction: f(1, -8, -8)
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(20)
NO