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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 212 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 48 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 56 ms]
(10) IntTRS
(11) RankingReductionPairProof [EQUIVALENT, 0 ms]
(12) IntTRS
(13) RankingReductionPairProof [EQUIVALENT, 0 ms]
(14) YES


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(0)
Obligation:
Rules:
l0(Result_4HAT0, w_5HAT0, x_6HAT0) -> l1(Result_4HATpost, w_5HATpost, x_6HATpost) :|: x_6HAT0 = x_6HATpost && w_5HAT0 = w_5HATpost && Result_4HATpost = Result_4HATpost && 3 - w_5HAT0 <= 0 && 2 - x_6HAT0 <= 0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x = x3 && x4 = 1 + x1 && x5 = 1 + x2 && 0 <= 2 - x1 && 2 - x2 <= 0
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
l0(x12, x13, x14) -> l3(x15, x16, x17) :|: x12 = x15 && x16 = 1 + x13 && x17 = 1 + x14 && 0 <= 1 - x14
l3(x18, x19, x20) -> l0(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
l4(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
l5(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33
Start term: l5(Result_4HAT0, w_5HAT0, x_6HAT0)

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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(Result_4HAT0, w_5HAT0, x_6HAT0) -> l1(Result_4HATpost, w_5HATpost, x_6HATpost) :|: x_6HAT0 = x_6HATpost && w_5HAT0 = w_5HATpost && Result_4HATpost = Result_4HATpost && 3 - w_5HAT0 <= 0 && 2 - x_6HAT0 <= 0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x = x3 && x4 = 1 + x1 && x5 = 1 + x2 && 0 <= 2 - x1 && 2 - x2 <= 0
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
l0(x12, x13, x14) -> l3(x15, x16, x17) :|: x12 = x15 && x16 = 1 + x13 && x17 = 1 + x14 && 0 <= 1 - x14
l3(x18, x19, x20) -> l0(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
l4(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
l5(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33
Start term: l5(Result_4HAT0, w_5HAT0, x_6HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(Result_4HAT0, w_5HAT0, x_6HAT0) -> l1(Result_4HATpost, w_5HATpost, x_6HATpost) :|: x_6HAT0 = x_6HATpost && w_5HAT0 = w_5HATpost && Result_4HATpost = Result_4HATpost && 3 - w_5HAT0 <= 0 && 2 - x_6HAT0 <= 0
(2) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x = x3 && x4 = 1 + x1 && x5 = 1 + x2 && 0 <= 2 - x1 && 2 - x2 <= 0
(3) l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
(4) l0(x12, x13, x14) -> l3(x15, x16, x17) :|: x12 = x15 && x16 = 1 + x13 && x17 = 1 + x14 && 0 <= 1 - x14
(5) l3(x18, x19, x20) -> l0(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
(6) l4(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
(7) l5(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x = x3 && x4 = 1 + x1 && x5 = 1 + x2 && 0 <= 2 - x1 && 2 - x2 <= 0
(2) l3(x18, x19, x20) -> l0(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
(3) l0(x12, x13, x14) -> l3(x15, x16, x17) :|: x12 = x15 && x16 = 1 + x13 && x17 = 1 + x14 && 0 <= 1 - x14
(4) l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l0(x12:0, x13:0, x14:0) -> l0(x12:0, 1 + x13:0, 1 + x14:0) :|: x14:0 < 2
l0(x3:0, x1:0, x2:0) -> l0(x3:0, 1 + x1:0, 1 + x2:0) :|: x2:0 > 1 && x1:0 < 3

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

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

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(8)
Obligation:
Rules:
l0(x13:0, x14:0) -> l0(1 + x13:0, 1 + x14:0) :|: x14:0 < 2
l0(x1:0, x2:0) -> l0(1 + x1:0, 1 + x2:0) :|: x2:0 > 1 && x1:0 < 3

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(VARIABLE, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l0(x13:0, x14:0) -> l0(c, c1) :|: c1 = 1 + x14:0 && c = 1 + x13:0 && x14:0 < 2
l0(x1:0, x2:0) -> l0(c2, c3) :|: c3 = 1 + x2:0 && c2 = 1 + x1:0 && (x2:0 > 1 && x1:0 < 3)

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(11) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l0 ] = l0_2 + -2*l0_1

The following rules are decreasing:
l0(x13:0, x14:0) -> l0(c, c1) :|: c1 = 1 + x14:0 && c = 1 + x13:0 && x14:0 < 2
l0(x1:0, x2:0) -> l0(c2, c3) :|: c3 = 1 + x2:0 && c2 = 1 + x1:0 && (x2:0 > 1 && x1:0 < 3)

The following rules are bounded:
l0(x1:0, x2:0) -> l0(c2, c3) :|: c3 = 1 + x2:0 && c2 = 1 + x1:0 && (x2:0 > 1 && x1:0 < 3)


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(12)
Obligation:
Rules:
l0(x13:0, x14:0) -> l0(c, c1) :|: c1 = 1 + x14:0 && c = 1 + x13:0 && x14:0 < 2

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(13) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l0 ] = -1*l0_2

The following rules are decreasing:
l0(x13:0, x14:0) -> l0(c, c1) :|: c1 = 1 + x14:0 && c = 1 + x13:0 && x14:0 < 2

The following rules are bounded:
l0(x13:0, x14:0) -> l0(c, c1) :|: c1 = 1 + x14:0 && c = 1 + x13:0 && x14:0 < 2


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(14)
YES