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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 193 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 62 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(10) IntTRS
(11) RankingReductionPairProof [EQUIVALENT, 7 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(xHAT0, yHAT0, zHAT0) -> l1(xHATpost, yHATpost, zHATpost) :|: zHAT0 = zHATpost && yHAT0 = yHATpost && xHAT0 = xHATpost
l2(x, x1, x2) -> l1(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x3 = 1 + x && x2 <= x1
l2(x6, x7, x8) -> l3(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x10 = 1 + x7 && 1 + x7 <= x8
l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
l1(x18, x19, x20) -> l2(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21 && 1 + x18 <= x19
l4(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
Start term: l4(xHAT0, yHAT0, zHAT0)

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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(xHAT0, yHAT0, zHAT0) -> l1(xHATpost, yHATpost, zHATpost) :|: zHAT0 = zHATpost && yHAT0 = yHATpost && xHAT0 = xHATpost
l2(x, x1, x2) -> l1(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x3 = 1 + x && x2 <= x1
l2(x6, x7, x8) -> l3(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x10 = 1 + x7 && 1 + x7 <= x8
l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
l1(x18, x19, x20) -> l2(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21 && 1 + x18 <= x19
l4(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
Start term: l4(xHAT0, yHAT0, zHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(xHAT0, yHAT0, zHAT0) -> l1(xHATpost, yHATpost, zHATpost) :|: zHAT0 = zHATpost && yHAT0 = yHATpost && xHAT0 = xHATpost
(2) l2(x, x1, x2) -> l1(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x3 = 1 + x && x2 <= x1
(3) l2(x6, x7, x8) -> l3(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x10 = 1 + x7 && 1 + x7 <= x8
(4) l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
(5) l1(x18, x19, x20) -> l2(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21 && 1 + x18 <= x19
(6) l4(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l1(x18, x19, x20) -> l2(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21 && 1 + x18 <= x19
(2) l2(x, x1, x2) -> l1(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x3 = 1 + x && x2 <= x1
(3) l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
(4) l2(x6, x7, x8) -> l3(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x10 = 1 + x7 && 1 + x7 <= x8

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l2(x:0, x1:0, x23:0) -> l2(1 + x:0, x1:0, x23:0) :|: x23:0 <= x1:0 && x1:0 >= 1 + (1 + x:0)
l2(x15:0, x7:0, x11:0) -> l2(x15:0, 1 + x7:0, x11:0) :|: x11:0 >= 1 + x7:0

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l2(VARIABLE, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
l2(x:0, x1:0, x23:0) -> l2(c, x1:0, x23:0) :|: c = 1 + x:0 && (x23:0 <= x1:0 && x1:0 >= 1 + (1 + x:0))
l2(x15:0, x7:0, x11:0) -> l2(x15:0, c1, x11:0) :|: c1 = 1 + x7:0 && x11:0 >= 1 + x7:0

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(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l2(x, x1, x2)] = -1 - x1 + x2

The following rules are decreasing:
l2(x15:0, x7:0, x11:0) -> l2(x15:0, c1, x11:0) :|: c1 = 1 + x7:0 && x11:0 >= 1 + x7:0
The following rules are bounded:
l2(x15:0, x7:0, x11:0) -> l2(x15:0, c1, x11:0) :|: c1 = 1 + x7:0 && x11:0 >= 1 + x7:0

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(10)
Obligation:
Rules:
l2(x:0, x1:0, x23:0) -> l2(c, x1:0, x23:0) :|: c = 1 + x:0 && (x23:0 <= x1:0 && x1:0 >= 1 + (1 + x:0))

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(11) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l2 ] = l2_2 + -1*l2_1

The following rules are decreasing:
l2(x:0, x1:0, x23:0) -> l2(c, x1:0, x23:0) :|: c = 1 + x:0 && (x23:0 <= x1:0 && x1:0 >= 1 + (1 + x:0))

The following rules are bounded:
l2(x:0, x1:0, x23:0) -> l2(c, x1:0, x23:0) :|: c = 1 + x:0 && (x23:0 <= x1:0 && x1:0 >= 1 + (1 + x:0))


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(12)
YES