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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 189 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 16 ms]
        (11) IntTRS
        (12) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 15 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (22) YES


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(0)
Obligation:
Rules:
l0(iHAT0, jHAT0) -> l1(iHATpost, jHATpost) :|: iHAT0 = iHATpost && jHATpost = 0 && 2 <= iHAT0
l0(x, x1) -> l2(x2, x3) :|: x1 = x3 && x2 = 1 + x && 1 + x <= 2
l2(x4, x5) -> l0(x6, x7) :|: x5 = x7 && x4 = x6
l1(x8, x9) -> l3(x10, x11) :|: x9 = x11 && x8 = x10
l4(x12, x13) -> l5(x14, x15) :|: x13 = x15 && x12 = x14
l6(x16, x17) -> l4(x18, x19) :|: x17 = x19 && x16 = x18
l3(x20, x21) -> l6(x22, x23) :|: x21 = x23 && x20 = x22 && 2 <= x21
l3(x24, x25) -> l1(x26, x27) :|: x24 = x26 && x27 = 1 + x25 && 1 + x25 <= 2
l7(x28, x29) -> l2(x30, x31) :|: x29 = x31 && x30 = 0
l8(x32, x33) -> l7(x34, x35) :|: x33 = x35 && x32 = x34
Start term: l8(iHAT0, jHAT0)

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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(iHAT0, jHAT0) -> l1(iHATpost, jHATpost) :|: iHAT0 = iHATpost && jHATpost = 0 && 2 <= iHAT0
l0(x, x1) -> l2(x2, x3) :|: x1 = x3 && x2 = 1 + x && 1 + x <= 2
l2(x4, x5) -> l0(x6, x7) :|: x5 = x7 && x4 = x6
l1(x8, x9) -> l3(x10, x11) :|: x9 = x11 && x8 = x10
l4(x12, x13) -> l5(x14, x15) :|: x13 = x15 && x12 = x14
l6(x16, x17) -> l4(x18, x19) :|: x17 = x19 && x16 = x18
l3(x20, x21) -> l6(x22, x23) :|: x21 = x23 && x20 = x22 && 2 <= x21
l3(x24, x25) -> l1(x26, x27) :|: x24 = x26 && x27 = 1 + x25 && 1 + x25 <= 2
l7(x28, x29) -> l2(x30, x31) :|: x29 = x31 && x30 = 0
l8(x32, x33) -> l7(x34, x35) :|: x33 = x35 && x32 = x34
Start term: l8(iHAT0, jHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(iHAT0, jHAT0) -> l1(iHATpost, jHATpost) :|: iHAT0 = iHATpost && jHATpost = 0 && 2 <= iHAT0
(2) l0(x, x1) -> l2(x2, x3) :|: x1 = x3 && x2 = 1 + x && 1 + x <= 2
(3) l2(x4, x5) -> l0(x6, x7) :|: x5 = x7 && x4 = x6
(4) l1(x8, x9) -> l3(x10, x11) :|: x9 = x11 && x8 = x10
(5) l4(x12, x13) -> l5(x14, x15) :|: x13 = x15 && x12 = x14
(6) l6(x16, x17) -> l4(x18, x19) :|: x17 = x19 && x16 = x18
(7) l3(x20, x21) -> l6(x22, x23) :|: x21 = x23 && x20 = x22 && 2 <= x21
(8) l3(x24, x25) -> l1(x26, x27) :|: x24 = x26 && x27 = 1 + x25 && 1 + x25 <= 2
(9) l7(x28, x29) -> l2(x30, x31) :|: x29 = x31 && x30 = 0
(10) l8(x32, x33) -> l7(x34, x35) :|: x33 = x35 && x32 = x34

Arcs:
(1) -> (4)
(2) -> (3)
(3) -> (1), (2)
(4) -> (7), (8)
(6) -> (5)
(7) -> (6)
(8) -> (4)
(9) -> (3)
(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(x, x1) -> l2(x2, x3) :|: x1 = x3 && x2 = 1 + x && 1 + x <= 2
(2) l2(x4, x5) -> l0(x6, x7) :|: x5 = x7 && x4 = x6

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, x1:0) -> l0(1 + x:0, x1:0) :|: x:0 < 2

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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) -> l0(x1)

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(9)
Obligation:
Rules:
l0(x:0) -> l0(1 + x:0) :|: x:0 < 2

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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 < 2

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(12) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l0 ] = -1*l0_1

The following rules are decreasing:
l0(x:0) -> l0(c) :|: c = 1 + x:0 && x:0 < 2

The following rules are bounded:
l0(x:0) -> l0(c) :|: c = 1 + x:0 && x:0 < 2


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(13)
YES

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(14)
Obligation:

Termination digraph:
Nodes:
(1) l1(x8, x9) -> l3(x10, x11) :|: x9 = x11 && x8 = x10
(2) l3(x24, x25) -> l1(x26, x27) :|: x24 = x26 && x27 = 1 + x25 && 1 + x25 <= 2

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(x10:0, x11:0) -> l1(x10:0, 1 + x11:0) :|: x11:0 < 2

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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) -> l1(x2)

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(18)
Obligation:
Rules:
l1(x11:0) -> l1(1 + x11:0) :|: x11:0 < 2

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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(x11:0) -> l1(c) :|: c = 1 + x11:0 && x11:0 < 2

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(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x)] = 1 - x

The following rules are decreasing:
l1(x11:0) -> l1(c) :|: c = 1 + x11:0 && x11:0 < 2
The following rules are bounded:
l1(x11:0) -> l1(c) :|: c = 1 + x11:0 && x11:0 < 2

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(22)
YES