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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 138 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 25 ms]
(6) IRSwT
(7) FilterProof [EQUIVALENT, 0 ms]
(8) IntTRS
(9) TerminationGraphProcessor [EQUIVALENT, 0 ms]
(10) YES


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

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

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHATpost = yHAT0 && 0 <= xHAT0 && xHAT0 <= 0 && 1 <= yHAT0
(2) l2(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
(3) l0(x4, x5) -> l2(x6, x7) :|: x5 = x7 && x6 = x5 && 0 <= x4 && x4 <= 0 && 1 + x5 <= 0
(4) l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2

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:
l0(x4:0, x10:0) -> l0(x10:0, x10:0) :|: x4:0 < 1 && x4:0 > -1 && x10:0 < 0
l0(xHAT0:0, x2:0) -> l0(x2:0, x2:0) :|: xHAT0:0 < 1 && xHAT0:0 > -1 && x2:0 > 0

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(7) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
l0(x4:0, x10:0) -> l0(x10:0, x10:0) :|: x4:0 < 1 && x4:0 > -1 && x10:0 < 0
l0(xHAT0:0, x2:0) -> l0(x2:0, x2:0) :|: xHAT0:0 < 1 && xHAT0:0 > -1 && x2:0 > 0

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(9) TerminationGraphProcessor (EQUIVALENT)
Constructed the termination graph and obtained no non-trivial SCC(s).

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(10)
YES