YES
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could be proven:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 209 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 25 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 5 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 17 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (22) YES


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(0)
Obligation:
Rules:
l0(i5HAT0, iHAT0, tmpHAT0) -> l1(i5HATpost, iHATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && i5HAT0 = i5HATpost && iHATpost = 0 && 50 <= i5HAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x3 = 1 + x && 1 + x <= 50
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x7 = x10
l1(x12, x13, x14) -> l3(x15, x16, x17) :|: x14 = x17 && x12 = x15 && x13 = x16
l3(x18, x19, x20) -> l4(x21, x22, x23) :|: x20 = x23 && x18 = x21 && x19 = x22 && 50 <= x19
l3(x24, x25, x26) -> l1(x27, x28, x29) :|: x26 = x29 && x24 = x27 && x28 = 1 + x25 && 1 + x25 <= 50
l5(x30, x31, x32) -> l2(x33, x34, x35) :|: x34 = 0 && x35 = x35 && x36 = 0 && x33 = 0
l6(x37, x38, x39) -> l5(x40, x41, x42) :|: x39 = x42 && x37 = x40 && x38 = x41
Start term: l6(i5HAT0, iHAT0, tmpHAT0)

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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(i5HAT0, iHAT0, tmpHAT0) -> l1(i5HATpost, iHATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && i5HAT0 = i5HATpost && iHATpost = 0 && 50 <= i5HAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x3 = 1 + x && 1 + x <= 50
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x7 = x10
l1(x12, x13, x14) -> l3(x15, x16, x17) :|: x14 = x17 && x12 = x15 && x13 = x16
l3(x18, x19, x20) -> l4(x21, x22, x23) :|: x20 = x23 && x18 = x21 && x19 = x22 && 50 <= x19
l3(x24, x25, x26) -> l1(x27, x28, x29) :|: x26 = x29 && x24 = x27 && x28 = 1 + x25 && 1 + x25 <= 50
l5(x30, x31, x32) -> l2(x33, x34, x35) :|: x34 = 0 && x35 = x35 && x36 = 0 && x33 = 0
l6(x37, x38, x39) -> l5(x40, x41, x42) :|: x39 = x42 && x37 = x40 && x38 = x41
Start term: l6(i5HAT0, iHAT0, tmpHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(i5HAT0, iHAT0, tmpHAT0) -> l1(i5HATpost, iHATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && i5HAT0 = i5HATpost && iHATpost = 0 && 50 <= i5HAT0
(2) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x3 = 1 + x && 1 + x <= 50
(3) l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x7 = x10
(4) l1(x12, x13, x14) -> l3(x15, x16, x17) :|: x14 = x17 && x12 = x15 && x13 = x16
(5) l3(x18, x19, x20) -> l4(x21, x22, x23) :|: x20 = x23 && x18 = x21 && x19 = x22 && 50 <= x19
(6) l3(x24, x25, x26) -> l1(x27, x28, x29) :|: x26 = x29 && x24 = x27 && x28 = 1 + x25 && 1 + x25 <= 50
(7) l5(x30, x31, x32) -> l2(x33, x34, x35) :|: x34 = 0 && x35 = x35 && x36 = 0 && x33 = 0
(8) l6(x37, x38, x39) -> l5(x40, x41, x42) :|: x39 = x42 && x37 = x40 && x38 = x41

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

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x3 = 1 + x && 1 + x <= 50
(2) l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x7 = x10

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:
l0(x:0, x10:0, x11:0) -> l0(1 + x:0, x10:0, x11:0) :|: x:0 < 50

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(8) 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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(9)
Obligation:
Rules:
l0(x:0) -> l0(1 + x:0) :|: x:0 < 50

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
l0(x:0) -> l0(c) :|: c = 1 + x:0 && x:0 < 50

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x)] = 49 - x

The following rules are decreasing:
l0(x:0) -> l0(c) :|: c = 1 + x:0 && x:0 < 50
The following rules are bounded:
l0(x:0) -> l0(c) :|: c = 1 + x:0 && x:0 < 50

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(13)
YES

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(14)
Obligation:

Termination digraph:
Nodes:
(1) l1(x12, x13, x14) -> l3(x15, x16, x17) :|: x14 = x17 && x12 = x15 && x13 = x16
(2) l3(x24, x25, x26) -> l1(x27, x28, x29) :|: x26 = x29 && x24 = x27 && x28 = 1 + x25 && 1 + x25 <= 50

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:
l1(x12:0, x13:0, x14:0) -> l1(x12:0, 1 + x13:0, x14:0) :|: x13:0 < 50

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

   l1(x1, x2, x3) -> l1(x2)

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(18)
Obligation:
Rules:
l1(x13:0) -> l1(1 + x13:0) :|: x13:0 < 50

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(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l1(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(20)
Obligation:
Rules:
l1(x13:0) -> l1(c) :|: c = 1 + x13:0 && x13:0 < 50

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(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x)] = 49 - x

The following rules are decreasing:
l1(x13:0) -> l1(c) :|: c = 1 + x13:0 && x13:0 < 50
The following rules are bounded:
l1(x13:0) -> l1(c) :|: c = 1 + x13:0 && x13:0 < 50

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(22)
YES