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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 124 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 26 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 12 ms]
(10) YES


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(0)
Obligation:
Rules:
l0(i1HAT0) -> l1(i1HATpost) :|: i1HATpost = 1 + i1HAT0
l2(x) -> l0(x1) :|: x = x1
l3(x2) -> l2(x3) :|: x2 = x3
l3(x4) -> l0(x5) :|: x4 = x5
l4(x6) -> l5(x7) :|: x6 = x7 && 42 <= x6
l4(x8) -> l3(x9) :|: x8 = x9 && 1 + x8 <= 42
l1(x10) -> l4(x11) :|: x10 = x11
l6(x12) -> l1(x13) :|: x13 = 0
l7(x14) -> l6(x15) :|: x14 = x15
Start term: l7(i1HAT0)

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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(i1HAT0) -> l1(i1HATpost) :|: i1HATpost = 1 + i1HAT0
l2(x) -> l0(x1) :|: x = x1
l3(x2) -> l2(x3) :|: x2 = x3
l3(x4) -> l0(x5) :|: x4 = x5
l4(x6) -> l5(x7) :|: x6 = x7 && 42 <= x6
l4(x8) -> l3(x9) :|: x8 = x9 && 1 + x8 <= 42
l1(x10) -> l4(x11) :|: x10 = x11
l6(x12) -> l1(x13) :|: x13 = 0
l7(x14) -> l6(x15) :|: x14 = x15
Start term: l7(i1HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(i1HAT0) -> l1(i1HATpost) :|: i1HATpost = 1 + i1HAT0
(2) l2(x) -> l0(x1) :|: x = x1
(3) l3(x2) -> l2(x3) :|: x2 = x3
(4) l3(x4) -> l0(x5) :|: x4 = x5
(5) l4(x6) -> l5(x7) :|: x6 = x7 && 42 <= x6
(6) l4(x8) -> l3(x9) :|: x8 = x9 && 1 + x8 <= 42
(7) l1(x10) -> l4(x11) :|: x10 = x11
(8) l6(x12) -> l1(x13) :|: x13 = 0
(9) l7(x14) -> l6(x15) :|: x14 = x15

Arcs:
(1) -> (7)
(2) -> (1)
(3) -> (2)
(4) -> (1)
(6) -> (3), (4)
(7) -> (5), (6)
(8) -> (7)
(9) -> (8)

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(i1HAT0) -> l1(i1HATpost) :|: i1HATpost = 1 + i1HAT0
(2) l3(x4) -> l0(x5) :|: x4 = x5
(3) l2(x) -> l0(x1) :|: x = x1
(4) l3(x2) -> l2(x3) :|: x2 = x3
(5) l4(x8) -> l3(x9) :|: x8 = x9 && 1 + x8 <= 42
(6) l1(x10) -> l4(x11) :|: x10 = x11

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l4(x5:0) -> l4(1 + x5:0) :|: x5:0 < 42

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l4(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
l4(x5:0) -> l4(c) :|: c = 1 + x5:0 && x5:0 < 42

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(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l4(x)] = 41 - x

The following rules are decreasing:
l4(x5:0) -> l4(c) :|: c = 1 + x5:0 && x5:0 < 42
The following rules are bounded:
l4(x5:0) -> l4(c) :|: c = 1 + x5:0 && x5:0 < 42

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(10)
YES