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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 112 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 74 ms]
(8) IRSwT
(9) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
(10) IRSwT


----------------------------------------

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

----------------------------------------

(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(2)
Obligation:
Rules:
l0(xHAT0) -> l1(xHATpost) :|: xHATpost = -1 + xHAT0
l1(x) -> l0(x1) :|: x = x1
l0(x2) -> l2(x3) :|: 1 <= x3 && x3 = -1 + x2
l2(x4) -> l0(x5) :|: x4 = x5
l3(x6) -> l0(x7) :|: x6 = x7
l4(x8) -> l3(x9) :|: x8 = x9
Start term: l4(xHAT0)

----------------------------------------

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

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

This digraph is fully evaluated!
----------------------------------------

(4)
Obligation:

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

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

This digraph is fully evaluated!

----------------------------------------

(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(6)
Obligation:
Rules:
l0(x2:0) -> l0(-1 + x2:0) :|: x2:0 > 1
l0(xHAT0:0) -> l0(-1 + xHAT0:0) :|: TRUE

----------------------------------------

(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(VARIABLE)
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:
l0(x2:0) -> l0(c) :|: c = -1 + x2:0 && x2:0 > 1
l0(xHAT0:0) -> l0(c1) :|: c1 = -1 + xHAT0:0 && TRUE

Found the following polynomial interpretation:
[l0(x)] = -1 + x

The following rules are decreasing:
l0(x2:0) -> l0(c) :|: c = -1 + x2:0 && x2:0 > 1
l0(xHAT0:0) -> l0(c1) :|: c1 = -1 + xHAT0:0 && TRUE
The following rules are bounded:
l0(x2:0) -> l0(c) :|: c = -1 + x2:0 && x2:0 > 1


- IntTRS
  - PolynomialOrderProcessor
    - IntTRS

Rules:
l0(xHAT0:0) -> l0(c1) :|: c1 = -1 + xHAT0:0 && TRUE



----------------------------------------

(8)
Obligation:
Rules:
l0(xHAT0:0) -> l0(-1 + xHAT0:0) :|: TRUE

----------------------------------------

(9) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(xHAT0:0) -> l0(-1 + xHAT0:0) :|: TRUE

Arcs:
(1) -> (1)

This digraph is fully evaluated!
----------------------------------------

(10)
Obligation:

Termination digraph:
Nodes:
(1) l0(xHAT0:0) -> l0(-1 + xHAT0:0) :|: TRUE

Arcs:
(1) -> (1)

This digraph is fully evaluated!