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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 76 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 36 ms]
(6) IRSwT
(7) IRSwTChainingProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
(10) IRSwT
(11) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(12) IRSwT
(13) IRSwTChainingProof [EQUIVALENT, 0 ms]
(14) IRSwT
(15) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
(16) IRSwT
(17) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(18) IRSwT
(19) TempFilterProof [SOUND, 32 ms]
(20) IntTRS
(21) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(22) YES


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

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

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHATpost = 1 + yHAT0 && xHATpost = xHAT0 - yHAT0 && 1 <= xHAT0
(2) l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2

Arcs:
(1) -> (2)
(2) -> (1)

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l0(xHAT0:0, yHAT0:0) -> l0(xHAT0:0 - yHAT0:0, 1 + yHAT0:0) :|: xHAT0:0 > 0

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

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

Arcs:
(1) -> (1)

This digraph is fully evaluated!
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(10)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1) -> l0(x + -2 * x1 + -1, 2 + x1) :|: TRUE && x >= 1 && x + -1 * x1 >= 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(12)
Obligation:
Rules:
l0(x:0, x1:0) -> l0(x:0 + -2 * x1:0 - 1, 2 + x1:0) :|: x:0 + -1 * x1:0 >= 1 && x:0 > 0

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(13) IRSwTChainingProof (EQUIVALENT)
Chaining!
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(14)
Obligation:
Rules:
l0(x, x1) -> l0(x + -4 * x1 + -6, 4 + x1) :|: TRUE && x + -1 * x1 >= 1 && x >= 1 && x + -3 * x1 >= 4 && x + -2 * x1 >= 2

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(15) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(x, x1) -> l0(x + -4 * x1 + -6, 4 + x1) :|: TRUE && x + -1 * x1 >= 1 && x >= 1 && x + -3 * x1 >= 4 && x + -2 * x1 >= 2

Arcs:
(1) -> (1)

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

(16)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1) -> l0(x + -4 * x1 + -6, 4 + x1) :|: TRUE && x + -1 * x1 >= 1 && x >= 1 && x + -3 * x1 >= 4 && x + -2 * x1 >= 2

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(17) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(18)
Obligation:
Rules:
l0(x:0, x1:0) -> l0(x:0 + -4 * x1:0 - 6, 4 + x1:0) :|: x:0 + -3 * x1:0 >= 4 && x:0 + -2 * x1:0 >= 2 && x:0 + -1 * x1:0 >= 1 && x:0 > 0

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(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(20)
Obligation:
Rules:
l0(x:0, x1:0) -> l0(c, c1) :|: c1 = 4 + x1:0 && c = x:0 + -4 * x1:0 - 6 && (x:0 + -3 * x1:0 >= 4 && x:0 + -2 * x1:0 >= 2 && x:0 + -1 * x1:0 >= 1 && x:0 > 0)

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(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1)] = -2 + 2*x - 2*x1 + x1^2

The following rules are decreasing:
l0(x:0, x1:0) -> l0(c, c1) :|: c1 = 4 + x1:0 && c = x:0 + -4 * x1:0 - 6 && (x:0 + -3 * x1:0 >= 4 && x:0 + -2 * x1:0 >= 2 && x:0 + -1 * x1:0 >= 1 && x:0 > 0)
The following rules are bounded:
l0(x:0, x1:0) -> l0(c, c1) :|: c1 = 4 + x1:0 && c = x:0 + -4 * x1:0 - 6 && (x:0 + -3 * x1:0 >= 4 && x:0 + -2 * x1:0 >= 2 && x:0 + -1 * x1:0 >= 1 && x:0 > 0)

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(22)
YES