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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 72 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 8 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 14 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(10) YES


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(0)
Obligation:
Rules:
l0(jHAT0) -> l1(jHATpost) :|: jHAT0 = jHATpost
l2(x) -> l0(x1) :|: x = x1 && 2 <= x
l2(x2) -> l3(x3) :|: x3 = 1 + x2 && 1 + x2 <= 2
l3(x4) -> l2(x5) :|: x4 = x5
l1(x6) -> l4(x7) :|: x6 = x7
l5(x8) -> l3(x9) :|: x9 = 0
l6(x10) -> l5(x11) :|: x10 = x11
Start term: l6(jHAT0)

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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(jHAT0) -> l1(jHATpost) :|: jHAT0 = jHATpost
l2(x) -> l0(x1) :|: x = x1 && 2 <= x
l2(x2) -> l3(x3) :|: x3 = 1 + x2 && 1 + x2 <= 2
l3(x4) -> l2(x5) :|: x4 = x5
l1(x6) -> l4(x7) :|: x6 = x7
l5(x8) -> l3(x9) :|: x9 = 0
l6(x10) -> l5(x11) :|: x10 = x11
Start term: l6(jHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(jHAT0) -> l1(jHATpost) :|: jHAT0 = jHATpost
(2) l2(x) -> l0(x1) :|: x = x1 && 2 <= x
(3) l2(x2) -> l3(x3) :|: x3 = 1 + x2 && 1 + x2 <= 2
(4) l3(x4) -> l2(x5) :|: x4 = x5
(5) l1(x6) -> l4(x7) :|: x6 = x7
(6) l5(x8) -> l3(x9) :|: x9 = 0
(7) l6(x10) -> l5(x11) :|: x10 = x11

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l2(x2) -> l3(x3) :|: x3 = 1 + x2 && 1 + x2 <= 2
(2) l3(x4) -> l2(x5) :|: x4 = x5

Arcs:
(1) -> (2)
(2) -> (1)

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l2(x2:0) -> l2(1 + x2:0) :|: x2:0 < 2

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l2(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
l2(x2:0) -> l2(c) :|: c = 1 + x2:0 && x2:0 < 2

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

The following rules are decreasing:
l2(x2:0) -> l2(c) :|: c = 1 + x2:0 && x2:0 < 2
The following rules are bounded:
l2(x2:0) -> l2(c) :|: c = 1 + x2:0 && x2:0 < 2

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(10)
YES