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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 146 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 156 ms]
(8) IRSwT
(9) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
(10) IRSwT


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(0)
Obligation:
Rules:
l0(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost
l1(x, x1) -> l3(x2, x3) :|: x1 = x3 && x = x2 && 1 <= x
l3(x4, x5) -> l4(x6, x7) :|: x5 = x7 && x4 = x6 && 1 + x5 <= 0
l3(x8, x9) -> l4(x10, x11) :|: x9 = x11 && x8 = x10 && 1 <= x9
l4(x12, x13) -> l2(x14, x15) :|: x13 = x15 && x14 = x12 + x13
l2(x16, x17) -> l1(x18, x19) :|: x17 = x19 && x16 = x18
l5(x20, x21) -> l0(x22, x23) :|: x21 = x23 && x20 = x22
Start term: l5(xHAT0, yHAT0)

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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, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost
l1(x, x1) -> l3(x2, x3) :|: x1 = x3 && x = x2 && 1 <= x
l3(x4, x5) -> l4(x6, x7) :|: x5 = x7 && x4 = x6 && 1 + x5 <= 0
l3(x8, x9) -> l4(x10, x11) :|: x9 = x11 && x8 = x10 && 1 <= x9
l4(x12, x13) -> l2(x14, x15) :|: x13 = x15 && x14 = x12 + x13
l2(x16, x17) -> l1(x18, x19) :|: x17 = x19 && x16 = x18
l5(x20, x21) -> l0(x22, x23) :|: x21 = x23 && x20 = x22
Start term: l5(xHAT0, yHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost
(2) l1(x, x1) -> l3(x2, x3) :|: x1 = x3 && x = x2 && 1 <= x
(3) l3(x4, x5) -> l4(x6, x7) :|: x5 = x7 && x4 = x6 && 1 + x5 <= 0
(4) l3(x8, x9) -> l4(x10, x11) :|: x9 = x11 && x8 = x10 && 1 <= x9
(5) l4(x12, x13) -> l2(x14, x15) :|: x13 = x15 && x14 = x12 + x13
(6) l2(x16, x17) -> l1(x18, x19) :|: x17 = x19 && x16 = x18
(7) l5(x20, x21) -> l0(x22, x23) :|: x21 = x23 && x20 = x22

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l1(x, x1) -> l3(x2, x3) :|: x1 = x3 && x = x2 && 1 <= x
(2) l2(x16, x17) -> l1(x18, x19) :|: x17 = x19 && x16 = x18
(3) l4(x12, x13) -> l2(x14, x15) :|: x13 = x15 && x14 = x12 + x13
(4) l3(x8, x9) -> l4(x10, x11) :|: x9 = x11 && x8 = x10 && 1 <= x9
(5) l3(x4, x5) -> l4(x6, x7) :|: x5 = x7 && x4 = x6 && 1 + x5 <= 0

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l4(x12:0, x11:0) -> l4(x12:0 + x11:0, x11:0) :|: x12:0 + x11:0 > 0 && x11:0 > 0
l4(x, x1) -> l4(x + x1, x1) :|: x + x1 > 0 && x1 < 0

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l4(INTEGER, 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:
l4(x12:0, x11:0) -> l4(c, x11:0) :|: c = x12:0 + x11:0 && (x12:0 + x11:0 > 0 && x11:0 > 0)
l4(x, x1) -> l4(c1, x1) :|: c1 = x + x1 && (x + x1 > 0 && x1 < 0)

Found the following polynomial interpretation:
[l4(x, x1)] = -4 - x - x*x1 - 4*x1

The following rules are decreasing:
l4(x12:0, x11:0) -> l4(c, x11:0) :|: c = x12:0 + x11:0 && (x12:0 + x11:0 > 0 && x11:0 > 0)
The following rules are bounded:
l4(x, x1) -> l4(c1, x1) :|: c1 = x + x1 && (x + x1 > 0 && x1 < 0)



- IntTRS
  - PolynomialOrderProcessor
    - AND
      - IntTRS
      - IntTRS
        - RankingReductionPairProof
      - IntTRS

Rules:
l4(x, x1) -> l4(c1, x1) :|: c1 = x + x1 && (x + x1 > 0 && x1 < 0)

Interpretation:
[ l4 ] = l4_1 + l4_2

The following rules are decreasing:
l4(x, x1) -> l4(c1, x1) :|: c1 = x + x1 && (x + x1 > 0 && x1 < 0)

The following rules are bounded:
l4(x, x1) -> l4(c1, x1) :|: c1 = x + x1 && (x + x1 > 0 && x1 < 0)




- IntTRS
  - PolynomialOrderProcessor
    - AND
      - IntTRS
      - IntTRS
      - IntTRS

Rules:
l4(x12:0, x11:0) -> l4(c, x11:0) :|: c = x12:0 + x11:0 && (x12:0 + x11:0 > 0 && x11:0 > 0)



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(8)
Obligation:
Rules:
l4(x12:0, x11:0) -> l4(x12:0 + x11:0, x11:0) :|: x12:0 + x11:0 > 0 && x11:0 > 0

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(9) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l4(x12:0, x11:0) -> l4(x12:0 + x11:0, x11:0) :|: x12:0 + x11:0 > 0 && x11:0 > 0

Arcs:
(1) -> (1)

This digraph is fully evaluated!
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(10)
Obligation:

Termination digraph:
Nodes:
(1) l4(x12:0, x11:0) -> l4(x12:0 + x11:0, x11:0) :|: x12:0 + x11:0 > 0 && x11:0 > 0

Arcs:
(1) -> (1)

This digraph is fully evaluated!