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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 163 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 47 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) IntTRS
(13) RankingReductionPairProof [EQUIVALENT, 4 ms]
(14) YES


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(0)
Obligation:
Rules:
l0(__const_5HAT0, xHAT0) -> l1(__const_5HATpost, xHATpost) :|: __const_5HAT0 = __const_5HATpost && xHATpost = 1 && xHAT0 <= 0
l0(x, x1) -> l1(x2, x3) :|: x = x2 && x3 = 1 + x1 && 1 <= x1
l2(x4, x5) -> l3(x6, x7) :|: x5 = x7 && x4 = x6 && 4 <= x5
l2(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10 && 1 + x9 <= 4
l1(x12, x13) -> l2(x14, x15) :|: x13 = x15 && x12 = x14
l4(x16, x17) -> l1(x18, x19) :|: x20 = x16 && x19 = x19 && x16 = x18
l5(x21, x22) -> l4(x23, x24) :|: x22 = x24 && x21 = x23
Start term: l5(__const_5HAT0, xHAT0)

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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(__const_5HAT0, xHAT0) -> l1(__const_5HATpost, xHATpost) :|: __const_5HAT0 = __const_5HATpost && xHATpost = 1 && xHAT0 <= 0
l0(x, x1) -> l1(x2, x3) :|: x = x2 && x3 = 1 + x1 && 1 <= x1
l2(x4, x5) -> l3(x6, x7) :|: x5 = x7 && x4 = x6 && 4 <= x5
l2(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10 && 1 + x9 <= 4
l1(x12, x13) -> l2(x14, x15) :|: x13 = x15 && x12 = x14
l4(x16, x17) -> l1(x18, x19) :|: x20 = x16 && x19 = x19 && x16 = x18
l5(x21, x22) -> l4(x23, x24) :|: x22 = x24 && x21 = x23
Start term: l5(__const_5HAT0, xHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_5HAT0, xHAT0) -> l1(__const_5HATpost, xHATpost) :|: __const_5HAT0 = __const_5HATpost && xHATpost = 1 && xHAT0 <= 0
(2) l0(x, x1) -> l1(x2, x3) :|: x = x2 && x3 = 1 + x1 && 1 <= x1
(3) l2(x4, x5) -> l3(x6, x7) :|: x5 = x7 && x4 = x6 && 4 <= x5
(4) l2(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10 && 1 + x9 <= 4
(5) l1(x12, x13) -> l2(x14, x15) :|: x13 = x15 && x12 = x14
(6) l4(x16, x17) -> l1(x18, x19) :|: x20 = x16 && x19 = x19 && x16 = x18
(7) l5(x21, x22) -> l4(x23, x24) :|: x22 = x24 && x21 = x23

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(__const_5HAT0, xHAT0) -> l1(__const_5HATpost, xHATpost) :|: __const_5HAT0 = __const_5HATpost && xHATpost = 1 && xHAT0 <= 0
(2) l2(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10 && 1 + x9 <= 4
(3) l1(x12, x13) -> l2(x14, x15) :|: x13 = x15 && x12 = x14
(4) l0(x, x1) -> l1(x2, x3) :|: x = x2 && x3 = 1 + x1 && 1 <= x1

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l2(__const_5HATpost:0, x11:0) -> l2(__const_5HATpost:0, 1) :|: x11:0 < 4 && x11:0 < 1
l2(x, x1) -> l2(x, 1 + x1) :|: x1 < 4 && x1 > 0

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

   l2(x1, x2) -> l2(x2)

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(8)
Obligation:
Rules:
l2(x11:0) -> l2(1) :|: x11:0 < 4 && x11:0 < 1
l2(x1) -> l2(1 + x1) :|: x1 < 4 && x1 > 0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l2(VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l2(x11:0) -> l2(c) :|: c = 1 && (x11:0 < 4 && x11:0 < 1)
l2(x1) -> l2(c1) :|: c1 = 1 + x1 && (x1 < 4 && x1 > 0)

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l2(x)] = -x

The following rules are decreasing:
l2(x11:0) -> l2(c) :|: c = 1 && (x11:0 < 4 && x11:0 < 1)
l2(x1) -> l2(c1) :|: c1 = 1 + x1 && (x1 < 4 && x1 > 0)
The following rules are bounded:
l2(x11:0) -> l2(c) :|: c = 1 && (x11:0 < 4 && x11:0 < 1)

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(12)
Obligation:
Rules:
l2(x1) -> l2(c1) :|: c1 = 1 + x1 && (x1 < 4 && x1 > 0)

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(13) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l2 ] = -1*l2_1

The following rules are decreasing:
l2(x1) -> l2(c1) :|: c1 = 1 + x1 && (x1 < 4 && x1 > 0)

The following rules are bounded:
l2(x1) -> l2(c1) :|: c1 = 1 + x1 && (x1 < 4 && x1 > 0)


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(14)
YES