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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 143 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 22 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 51 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(10) IntTRS
(11) RankingReductionPairProof [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(iHAT0, jHAT0, nHAT0) -> l1(iHATpost, jHATpost, nHATpost) :|: nHAT0 = nHATpost && jHAT0 = jHATpost && iHATpost = 1 + iHAT0 && iHAT0 <= jHAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x = x3 && x4 = 1 + x1 && 1 + x1 <= x
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
l1(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x12 = x15 && x16 = 0 && 1 + x12 <= x14
l3(x18, x19, x20) -> l1(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x21 = 0
l4(x24, x25, x26) -> l3(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
Start term: l4(iHAT0, jHAT0, nHAT0)

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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, nHAT0) -> l1(iHATpost, jHATpost, nHATpost) :|: nHAT0 = nHATpost && jHAT0 = jHATpost && iHATpost = 1 + iHAT0 && iHAT0 <= jHAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x = x3 && x4 = 1 + x1 && 1 + x1 <= x
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
l1(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x12 = x15 && x16 = 0 && 1 + x12 <= x14
l3(x18, x19, x20) -> l1(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x21 = 0
l4(x24, x25, x26) -> l3(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
Start term: l4(iHAT0, jHAT0, nHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(iHAT0, jHAT0, nHAT0) -> l1(iHATpost, jHATpost, nHATpost) :|: nHAT0 = nHATpost && jHAT0 = jHATpost && iHATpost = 1 + iHAT0 && iHAT0 <= jHAT0
(2) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x = x3 && x4 = 1 + x1 && 1 + x1 <= x
(3) l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
(4) l1(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x12 = x15 && x16 = 0 && 1 + x12 <= x14
(5) l3(x18, x19, x20) -> l1(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x21 = 0
(6) l4(x24, x25, x26) -> l3(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(iHAT0, jHAT0, nHAT0) -> l1(iHATpost, jHATpost, nHATpost) :|: nHAT0 = nHATpost && jHAT0 = jHATpost && iHATpost = 1 + iHAT0 && iHAT0 <= jHAT0
(2) l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
(3) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x = x3 && x4 = 1 + x1 && 1 + x1 <= x
(4) l1(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x12 = x15 && x16 = 0 && 1 + x12 <= x14

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(x3:0, x1:0, x11:0) -> l0(x3:0, 1 + x1:0, x11:0) :|: x3:0 >= 1 + x1:0
l0(iHAT0:0, jHAT0:0, nHAT0:0) -> l0(1 + iHAT0:0, 0, nHAT0:0) :|: jHAT0:0 >= iHAT0:0 && nHAT0:0 >= 1 + (1 + iHAT0:0)

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
l0(x3:0, x1:0, x11:0) -> l0(x3:0, c, x11:0) :|: c = 1 + x1:0 && x3:0 >= 1 + x1:0
l0(iHAT0:0, jHAT0:0, nHAT0:0) -> l0(c1, c2, nHAT0:0) :|: c2 = 0 && c1 = 1 + iHAT0:0 && (jHAT0:0 >= iHAT0:0 && nHAT0:0 >= 1 + (1 + iHAT0:0))

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(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1, x2)] = -x + x2

The following rules are decreasing:
l0(iHAT0:0, jHAT0:0, nHAT0:0) -> l0(c1, c2, nHAT0:0) :|: c2 = 0 && c1 = 1 + iHAT0:0 && (jHAT0:0 >= iHAT0:0 && nHAT0:0 >= 1 + (1 + iHAT0:0))
The following rules are bounded:
l0(iHAT0:0, jHAT0:0, nHAT0:0) -> l0(c1, c2, nHAT0:0) :|: c2 = 0 && c1 = 1 + iHAT0:0 && (jHAT0:0 >= iHAT0:0 && nHAT0:0 >= 1 + (1 + iHAT0:0))

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(10)
Obligation:
Rules:
l0(x3:0, x1:0, x11:0) -> l0(x3:0, c, x11:0) :|: c = 1 + x1:0 && x3:0 >= 1 + x1:0

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(11) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l0 ] = l0_1 + -1*l0_2

The following rules are decreasing:
l0(x3:0, x1:0, x11:0) -> l0(x3:0, c, x11:0) :|: c = 1 + x1:0 && x3:0 >= 1 + x1:0

The following rules are bounded:
l0(x3:0, x1:0, x11:0) -> l0(x3:0, c, x11:0) :|: c = 1 + x1:0 && x3:0 >= 1 + x1:0


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(12)
YES