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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 148 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 8 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) FilterProof [EQUIVALENT, 0 ms]
        (11) IntTRS
        (12) IntTRSPeriodicNontermProof [COMPLETE, 0 ms]
        (13) NO
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 20 ms]
        (16) IRSwT
        (17) FilterProof [EQUIVALENT, 0 ms]
        (18) IntTRS
        (19) IntTRSPeriodicNontermProof [COMPLETE, 0 ms]
        (20) NO


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(0)
Obligation:
Rules:
l0(tmp2HAT0, xHAT0) -> l1(tmp2HATpost, xHATpost) :|: xHAT0 = xHATpost && tmp2HAT0 = tmp2HATpost
l2(x, x1) -> l3(x2, x3) :|: x <= 0 && 0 <= x && x4 = 0 && x3 = 1 && x = x2
l2(x5, x6) -> l0(x7, x8) :|: x6 = x8 && x5 = x7 && 1 <= x5
l2(x9, x10) -> l0(x11, x12) :|: x10 = x12 && x9 = x11 && 1 + x9 <= 0
l1(x13, x14) -> l2(x15, x16) :|: x14 = x16 && x15 = x15
l3(x17, x18) -> l4(x19, x20) :|: x18 = x20 && x17 = x19
l4(x21, x22) -> l3(x23, x24) :|: x22 = x24 && x21 = x23
l5(x25, x26) -> l6(x27, x28) :|: x26 = x28 && x25 = x27
l7(x29, x30) -> l1(x31, x32) :|: x29 = x31 && x32 = 1
l8(x33, x34) -> l7(x35, x36) :|: x34 = x36 && x33 = x35
Start term: l8(tmp2HAT0, 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(tmp2HAT0, xHAT0) -> l1(tmp2HATpost, xHATpost) :|: xHAT0 = xHATpost && tmp2HAT0 = tmp2HATpost
l2(x, x1) -> l3(x2, x3) :|: x <= 0 && 0 <= x && x4 = 0 && x3 = 1 && x = x2
l2(x5, x6) -> l0(x7, x8) :|: x6 = x8 && x5 = x7 && 1 <= x5
l2(x9, x10) -> l0(x11, x12) :|: x10 = x12 && x9 = x11 && 1 + x9 <= 0
l1(x13, x14) -> l2(x15, x16) :|: x14 = x16 && x15 = x15
l3(x17, x18) -> l4(x19, x20) :|: x18 = x20 && x17 = x19
l4(x21, x22) -> l3(x23, x24) :|: x22 = x24 && x21 = x23
l5(x25, x26) -> l6(x27, x28) :|: x26 = x28 && x25 = x27
l7(x29, x30) -> l1(x31, x32) :|: x29 = x31 && x32 = 1
l8(x33, x34) -> l7(x35, x36) :|: x34 = x36 && x33 = x35
Start term: l8(tmp2HAT0, xHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(tmp2HAT0, xHAT0) -> l1(tmp2HATpost, xHATpost) :|: xHAT0 = xHATpost && tmp2HAT0 = tmp2HATpost
(2) l2(x, x1) -> l3(x2, x3) :|: x <= 0 && 0 <= x && x4 = 0 && x3 = 1 && x = x2
(3) l2(x5, x6) -> l0(x7, x8) :|: x6 = x8 && x5 = x7 && 1 <= x5
(4) l2(x9, x10) -> l0(x11, x12) :|: x10 = x12 && x9 = x11 && 1 + x9 <= 0
(5) l1(x13, x14) -> l2(x15, x16) :|: x14 = x16 && x15 = x15
(6) l3(x17, x18) -> l4(x19, x20) :|: x18 = x20 && x17 = x19
(7) l4(x21, x22) -> l3(x23, x24) :|: x22 = x24 && x21 = x23
(8) l5(x25, x26) -> l6(x27, x28) :|: x26 = x28 && x25 = x27
(9) l7(x29, x30) -> l1(x31, x32) :|: x29 = x31 && x32 = 1
(10) l8(x33, x34) -> l7(x35, x36) :|: x34 = x36 && x33 = x35

Arcs:
(1) -> (5)
(2) -> (6)
(3) -> (1)
(4) -> (1)
(5) -> (2), (3), (4)
(6) -> (7)
(7) -> (6)
(9) -> (5)
(10) -> (9)

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(tmp2HAT0, xHAT0) -> l1(tmp2HATpost, xHATpost) :|: xHAT0 = xHATpost && tmp2HAT0 = tmp2HATpost
(2) l2(x9, x10) -> l0(x11, x12) :|: x10 = x12 && x9 = x11 && 1 + x9 <= 0
(3) l2(x5, x6) -> l0(x7, x8) :|: x6 = x8 && x5 = x7 && 1 <= x5
(4) l1(x13, x14) -> l2(x15, x16) :|: x14 = x16 && x15 = x15

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

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
l2(tmp2HATpost:0, x10:0) -> l2(x15:0, x10:0) :|: tmp2HATpost:0 < 0
l2(x, x1) -> l2(x2, x1) :|: x > 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:

   l2(x1, x2) -> l2(x1)

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(9)
Obligation:
Rules:
l2(tmp2HATpost:0) -> l2(x15:0) :|: tmp2HATpost:0 < 0
l2(x) -> l2(x2) :|: x > 0

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(10) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l2(VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
l2(tmp2HATpost:0) -> l2(x15:0) :|: tmp2HATpost:0 < 0
l2(x) -> l2(x2) :|: x > 0

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(12) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, tmp2HATpost:0) -> f(1, x15:0) :|: pc = 1 && tmp2HATpost:0 < 0
f(pc, x) -> f(1, x2) :|: pc = 1 && x > 0
Witness term starting non-terminating reduction: f(1, 4)
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(13)
NO

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(14)
Obligation:

Termination digraph:
Nodes:
(1) l3(x17, x18) -> l4(x19, x20) :|: x18 = x20 && x17 = x19
(2) l4(x21, x22) -> l3(x23, x24) :|: x22 = x24 && x21 = x23

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:
l3(x17:0, x18:0) -> l3(x17:0, x18:0) :|: TRUE

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(17) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l3(VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(18)
Obligation:
Rules:
l3(x17:0, x18:0) -> l3(x17:0, x18:0) :|: TRUE

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(19) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x17:0, x18:0) -> f(1, x17:0, x18:0) :|: pc = 1 && TRUE
Witness term starting non-terminating reduction: f(1, -8, -8)
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(20)
NO