MAYBE
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could not be shown:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 115 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 88 ms]
(8) IRSwT
(9) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
(10) IRSwT


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(0)
Obligation:
Rules:
l0(x_5HAT0) -> l1(x_5HATpost) :|: x_5HAT0 = x_5HATpost
l1(x) -> l2(x1) :|: x1 <= 0 && x1 = -1 + x
l2(x2) -> l1(x3) :|: x2 = x3
l1(x4) -> l3(x5) :|: 0 <= -1 + x5 && x5 = -1 + x4
l3(x6) -> l1(x7) :|: x6 = x7
l4(x8) -> l0(x9) :|: x8 = x9
Start term: l4(x_5HAT0)

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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(x_5HAT0) -> l1(x_5HATpost) :|: x_5HAT0 = x_5HATpost
l1(x) -> l2(x1) :|: x1 <= 0 && x1 = -1 + x
l2(x2) -> l1(x3) :|: x2 = x3
l1(x4) -> l3(x5) :|: 0 <= -1 + x5 && x5 = -1 + x4
l3(x6) -> l1(x7) :|: x6 = x7
l4(x8) -> l0(x9) :|: x8 = x9
Start term: l4(x_5HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(x_5HAT0) -> l1(x_5HATpost) :|: x_5HAT0 = x_5HATpost
(2) l1(x) -> l2(x1) :|: x1 <= 0 && x1 = -1 + x
(3) l2(x2) -> l1(x3) :|: x2 = x3
(4) l1(x4) -> l3(x5) :|: 0 <= -1 + x5 && x5 = -1 + x4
(5) l3(x6) -> l1(x7) :|: x6 = x7
(6) l4(x8) -> l0(x9) :|: x8 = x9

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l1(x) -> l2(x1) :|: x1 <= 0 && x1 = -1 + x
(2) l3(x6) -> l1(x7) :|: x6 = x7
(3) l1(x4) -> l3(x5) :|: 0 <= -1 + x5 && x5 = -1 + x4
(4) l2(x2) -> l1(x3) :|: x2 = x3

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:
l1(x4:0) -> l1(-1 + x4:0) :|: x4:0 > 1
l1(x:0) -> l1(-1 + x:0) :|: x:0 < 2

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l1(INTEGER)
Replaced non-predefined constructor symbols by 0.The following proof was generated: 
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IntTRS could not be shown:



- IntTRS
  - PolynomialOrderProcessor

Rules:
l1(x4:0) -> l1(c) :|: c = -1 + x4:0 && x4:0 > 1
l1(x:0) -> l1(c1) :|: c1 = -1 + x:0 && x:0 < 2

Found the following polynomial interpretation:
[l1(x)] = -2 + x

The following rules are decreasing:
l1(x4:0) -> l1(c) :|: c = -1 + x4:0 && x4:0 > 1
l1(x:0) -> l1(c1) :|: c1 = -1 + x:0 && x:0 < 2
The following rules are bounded:
l1(x4:0) -> l1(c) :|: c = -1 + x4:0 && x4:0 > 1


- IntTRS
  - PolynomialOrderProcessor
    - IntTRS

Rules:
l1(x:0) -> l1(c1) :|: c1 = -1 + x:0 && x:0 < 2



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(8)
Obligation:
Rules:
l1(x:0) -> l1(-1 + x:0) :|: x:0 < 2

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(9) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l1(x:0) -> l1(-1 + x:0) :|: x:0 < 2

Arcs:
(1) -> (1)

This digraph is fully evaluated!
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(10)
Obligation:

Termination digraph:
Nodes:
(1) l1(x:0) -> l1(-1 + x:0) :|: x:0 < 2

Arcs:
(1) -> (1)

This digraph is fully evaluated!