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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 264 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 46 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 14 ms]
        (13) IntTRS
        (14) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (15) YES
    (16) IRSwT
        (17) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (20) IRSwT
        (21) FilterProof [EQUIVALENT, 0 ms]
        (22) IntTRS
        (23) IntTRSPeriodicNontermProof [COMPLETE, 0 ms]
        (24) NO


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(0)
Obligation:
Rules:
l0(ndHAT0, pHAT0, xHAT0) -> l1(ndHATpost, pHATpost, xHATpost) :|: xHAT0 = xHATpost && ndHAT0 = ndHATpost && pHATpost = 2 && ndHAT0 <= 0
l0(x, x1, x2) -> l1(x3, x4, x5) :|: x2 = x5 && x = x3 && x4 = 1 && 1 <= x
l1(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x9 = x9
l2(x12, x13, x14) -> l3(x15, x16, x17) :|: x13 = x16 && x12 = x15 && x17 = -2 + x14 && x12 <= 0
l2(x18, x19, x20) -> l3(x21, x22, x23) :|: x19 = x22 && x18 = x21 && x23 = -1 + x20 && 1 <= x18
l3(x24, x25, x26) -> l1(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27 && x26 <= 0
l3(x30, x31, x32) -> l2(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x33 = x33 && 1 <= x32
l4(x36, x37, x38) -> l3(x39, x40, x41) :|: x38 = x41 && x36 = x39 && x40 = 0
l5(x42, x43, x44) -> l4(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
Start term: l5(ndHAT0, pHAT0, 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(ndHAT0, pHAT0, xHAT0) -> l1(ndHATpost, pHATpost, xHATpost) :|: xHAT0 = xHATpost && ndHAT0 = ndHATpost && pHATpost = 2 && ndHAT0 <= 0
l0(x, x1, x2) -> l1(x3, x4, x5) :|: x2 = x5 && x = x3 && x4 = 1 && 1 <= x
l1(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x9 = x9
l2(x12, x13, x14) -> l3(x15, x16, x17) :|: x13 = x16 && x12 = x15 && x17 = -2 + x14 && x12 <= 0
l2(x18, x19, x20) -> l3(x21, x22, x23) :|: x19 = x22 && x18 = x21 && x23 = -1 + x20 && 1 <= x18
l3(x24, x25, x26) -> l1(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27 && x26 <= 0
l3(x30, x31, x32) -> l2(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x33 = x33 && 1 <= x32
l4(x36, x37, x38) -> l3(x39, x40, x41) :|: x38 = x41 && x36 = x39 && x40 = 0
l5(x42, x43, x44) -> l4(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
Start term: l5(ndHAT0, pHAT0, xHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(ndHAT0, pHAT0, xHAT0) -> l1(ndHATpost, pHATpost, xHATpost) :|: xHAT0 = xHATpost && ndHAT0 = ndHATpost && pHATpost = 2 && ndHAT0 <= 0
(2) l0(x, x1, x2) -> l1(x3, x4, x5) :|: x2 = x5 && x = x3 && x4 = 1 && 1 <= x
(3) l1(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x9 = x9
(4) l2(x12, x13, x14) -> l3(x15, x16, x17) :|: x13 = x16 && x12 = x15 && x17 = -2 + x14 && x12 <= 0
(5) l2(x18, x19, x20) -> l3(x21, x22, x23) :|: x19 = x22 && x18 = x21 && x23 = -1 + x20 && 1 <= x18
(6) l3(x24, x25, x26) -> l1(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27 && x26 <= 0
(7) l3(x30, x31, x32) -> l2(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x33 = x33 && 1 <= x32
(8) l4(x36, x37, x38) -> l3(x39, x40, x41) :|: x38 = x41 && x36 = x39 && x40 = 0
(9) l5(x42, x43, x44) -> l4(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45

Arcs:
(1) -> (3)
(2) -> (3)
(3) -> (1), (2)
(4) -> (6), (7)
(5) -> (6), (7)
(6) -> (3)
(7) -> (4), (5)
(8) -> (6), (7)
(9) -> (8)

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) l2(x12, x13, x14) -> l3(x15, x16, x17) :|: x13 = x16 && x12 = x15 && x17 = -2 + x14 && x12 <= 0
(2) l3(x30, x31, x32) -> l2(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x33 = x33 && 1 <= x32
(3) l2(x18, x19, x20) -> l3(x21, x22, x23) :|: x19 = x22 && x18 = x21 && x23 = -1 + x20 && 1 <= x18

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

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
l2(x12:0, x13:0, x14:0) -> l2(x33:0, x13:0, -2 + x14:0) :|: x12:0 < 1 && x14:0 > 2
l2(x, x1, x2) -> l2(x3, x1, -1 + x2) :|: x > 0 && x2 > 1

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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, x3) -> l2(x1, x3)

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(9)
Obligation:
Rules:
l2(x12:0, x14:0) -> l2(x33:0, -2 + x14:0) :|: x12:0 < 1 && x14:0 > 2
l2(x, x2) -> l2(x3, -1 + x2) :|: x > 0 && x2 > 1

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l2(VARIABLE, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
l2(x12:0, x14:0) -> l2(x33:0, c) :|: c = -2 + x14:0 && (x12:0 < 1 && x14:0 > 2)
l2(x, x2) -> l2(x3, c1) :|: c1 = -1 + x2 && (x > 0 && x2 > 1)

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l2(x, x1)] = -3 + x1

The following rules are decreasing:
l2(x12:0, x14:0) -> l2(x33:0, c) :|: c = -2 + x14:0 && (x12:0 < 1 && x14:0 > 2)
l2(x, x2) -> l2(x3, c1) :|: c1 = -1 + x2 && (x > 0 && x2 > 1)
The following rules are bounded:
l2(x12:0, x14:0) -> l2(x33:0, c) :|: c = -2 + x14:0 && (x12:0 < 1 && x14:0 > 2)

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(13)
Obligation:
Rules:
l2(x, x2) -> l2(x3, c1) :|: c1 = -1 + x2 && (x > 0 && x2 > 1)

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(14) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l2(x, x1)] = x1

The following rules are decreasing:
l2(x, x2) -> l2(x3, c1) :|: c1 = -1 + x2 && (x > 0 && x2 > 1)
The following rules are bounded:
l2(x, x2) -> l2(x3, c1) :|: c1 = -1 + x2 && (x > 0 && x2 > 1)

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(15)
YES

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(16)
Obligation:

Termination digraph:
Nodes:
(1) l0(ndHAT0, pHAT0, xHAT0) -> l1(ndHATpost, pHATpost, xHATpost) :|: xHAT0 = xHATpost && ndHAT0 = ndHATpost && pHATpost = 2 && ndHAT0 <= 0
(2) l1(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x9 = x9
(3) l0(x, x1, x2) -> l1(x3, x4, x5) :|: x2 = x5 && x = x3 && x4 = 1 && 1 <= x

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

This digraph is fully evaluated!

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(17) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(18)
Obligation:
Rules:
l0(x3:0, x1:0, x11:0) -> l0(x9:0, 1, x11:0) :|: x3:0 > 0
l0(x, x1, x2) -> l0(x3, 2, x2) :|: x < 1

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

   l0(x1, x2, x3) -> l0(x1)

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(20)
Obligation:
Rules:
l0(x3:0) -> l0(x9:0) :|: x3:0 > 0
l0(x) -> l0(x3) :|: x < 1

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(21) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l0(VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(22)
Obligation:
Rules:
l0(x3:0) -> l0(x9:0) :|: x3:0 > 0
l0(x) -> l0(x3) :|: x < 1

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(23) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x3:0) -> f(1, x9:0) :|: pc = 1 && x3:0 > 0
f(pc, x) -> f(1, x3) :|: pc = 1 && x < 1
Witness term starting non-terminating reduction: f(1, -8)
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(24)
NO