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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 139 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 39 ms]
(6) IRSwT
(7) IRSwTChainingProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
(10) IRSwT
(11) TempFilterProof [SOUND, 7 ms]
(12) IntTRS
(13) RankingReductionPairProof [EQUIVALENT, 0 ms]
(14) YES


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(0)
Obligation:
Rules:
l0(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHATpost = -2 + yHAT0 && xHATpost = 1 + xHAT0 && 2 <= yHAT0 && 0 <= xHAT0
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l0(x4, x5) -> l2(x6, x7) :|: x7 = -1 + x6 && x6 = 0 && x5 <= 1 && 1 <= x4
l2(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
l3(x12, x13) -> l0(x14, x15) :|: x13 = x15 && x12 = x14
l4(x16, x17) -> l3(x18, x19) :|: x17 = x19 && x16 = x18
Start term: l4(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) :|: yHATpost = -2 + yHAT0 && xHATpost = 1 + xHAT0 && 2 <= yHAT0 && 0 <= xHAT0
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l0(x4, x5) -> l2(x6, x7) :|: x7 = -1 + x6 && x6 = 0 && x5 <= 1 && 1 <= x4
l2(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
l3(x12, x13) -> l0(x14, x15) :|: x13 = x15 && x12 = x14
l4(x16, x17) -> l3(x18, x19) :|: x17 = x19 && x16 = x18
Start term: l4(xHAT0, yHAT0)

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

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

This digraph is fully evaluated!
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(4)
Obligation:

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

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(7) IRSwTChainingProof (EQUIVALENT)
Chaining!
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(8)
Obligation:
Rules:
l0(x, x1) -> l0(0, -1) :|: TRUE && x >= 1 && x1 <= 1 && 0 >= 1
l0(xHAT0:0, yHAT0:0) -> l0(1 + xHAT0:0, -2 + yHAT0:0) :|: xHAT0:0 > -1 && yHAT0:0 > 1

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(9) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(x, x1) -> l0(0, -1) :|: TRUE && x >= 1 && x1 <= 1 && 0 >= 1
(2) l0(xHAT0:0, yHAT0:0) -> l0(1 + xHAT0:0, -2 + yHAT0:0) :|: xHAT0:0 > -1 && yHAT0:0 > 1

Arcs:
(2) -> (2)

This digraph is fully evaluated!
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(10)
Obligation:

Termination digraph:
Nodes:
(1) l0(xHAT0:0, yHAT0:0) -> l0(1 + xHAT0:0, -2 + yHAT0:0) :|: xHAT0:0 > -1 && yHAT0:0 > 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(11) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(12)
Obligation:
Rules:
l0(xHAT0:0, yHAT0:0) -> l0(c, c1) :|: c1 = -2 + yHAT0:0 && c = 1 + xHAT0:0 && (xHAT0:0 > -1 && yHAT0:0 > 1)

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(13) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l0 ] = 1/2*l0_2

The following rules are decreasing:
l0(xHAT0:0, yHAT0:0) -> l0(c, c1) :|: c1 = -2 + yHAT0:0 && c = 1 + xHAT0:0 && (xHAT0:0 > -1 && yHAT0:0 > 1)

The following rules are bounded:
l0(xHAT0:0, yHAT0:0) -> l0(c, c1) :|: c1 = -2 + yHAT0:0 && c = 1 + xHAT0:0 && (xHAT0:0 > -1 && yHAT0:0 > 1)


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(14)
YES