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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 72 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 15 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 85 ms]
(8) IRSwT
(9) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
(10) IRSwT


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(0)
Obligation:
Rules:
l0(xHAT0) -> l2(xHATpost) :|: xHATpost = -1 + xHAT0
l2(x) -> l1(x1) :|: x = x1 && 1 <= x
l2(x2) -> l1(x3) :|: x2 = x3 && 1 + x2 <= 0
l1(x4) -> l0(x5) :|: x4 = x5
l3(x6) -> l0(x7) :|: x6 = x7 && 1 <= x6
l4(x8) -> l3(x9) :|: x8 = x9
Start term: l4(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(xHAT0) -> l2(xHATpost) :|: xHATpost = -1 + xHAT0
l2(x) -> l1(x1) :|: x = x1 && 1 <= x
l2(x2) -> l1(x3) :|: x2 = x3 && 1 + x2 <= 0
l1(x4) -> l0(x5) :|: x4 = x5
l3(x6) -> l0(x7) :|: x6 = x7 && 1 <= x6
l4(x8) -> l3(x9) :|: x8 = x9
Start term: l4(xHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(xHAT0) -> l2(xHATpost) :|: xHATpost = -1 + xHAT0
(2) l2(x) -> l1(x1) :|: x = x1 && 1 <= x
(3) l2(x2) -> l1(x3) :|: x2 = x3 && 1 + x2 <= 0
(4) l1(x4) -> l0(x5) :|: x4 = x5
(5) l3(x6) -> l0(x7) :|: x6 = x7 && 1 <= x6
(6) l4(x8) -> l3(x9) :|: x8 = x9

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(xHAT0) -> l2(xHATpost) :|: xHATpost = -1 + xHAT0
(2) l1(x4) -> l0(x5) :|: x4 = x5
(3) l2(x2) -> l1(x3) :|: x2 = x3 && 1 + x2 <= 0
(4) l2(x) -> l1(x1) :|: x = x1 && 1 <= x

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

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) -> l1(-1 + x) :|: x > 1

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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) -> l1(c1) :|: c1 = -1 + x && x > 1

Found the following polynomial interpretation:
[l1(x)] = -1 + x

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


- IntTRS
  - PolynomialOrderProcessor
    - IntTRS

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



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(8)
Obligation:
Rules:
l1(x4:0) -> l1(-1 + x4:0) :|: x4:0 < 1

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(9) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l1(x4:0) -> l1(-1 + x4:0) :|: x4:0 < 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!
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(10)
Obligation:

Termination digraph:
Nodes:
(1) l1(x4:0) -> l1(-1 + x4:0) :|: x4:0 < 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!