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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 146 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 25 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 10 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(Result_4HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && Result_4HATpost = Result_4HATpost && -1 * x_5HAT0 + y_6HAT0 <= 0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x = x3 && x4 = 1 + x1 && 0 <= -1 - x1 + x2
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
l3(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
l4(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
Start term: l4(Result_4HAT0, x_5HAT0, y_6HAT0)

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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(Result_4HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && Result_4HATpost = Result_4HATpost && -1 * x_5HAT0 + y_6HAT0 <= 0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x = x3 && x4 = 1 + x1 && 0 <= -1 - x1 + x2
l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
l3(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
l4(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21
Start term: l4(Result_4HAT0, x_5HAT0, y_6HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(Result_4HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && Result_4HATpost = Result_4HATpost && -1 * x_5HAT0 + y_6HAT0 <= 0
(2) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x = x3 && x4 = 1 + x1 && 0 <= -1 - x1 + x2
(3) l2(x6, x7, x8) -> l0(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
(4) l3(x12, x13, x14) -> l0(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
(5) l4(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21

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

This digraph is fully evaluated!
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(4)
Obligation:

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

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(x3:0, x1:0, x11:0) -> l0(x3:0, 1 + x1:0, x11:0) :|: 0 <= -1 - x1:0 + x11:0

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(7) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   l0(x1, x2, x3) -> l0(x2, x3)

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(8)
Obligation:
Rules:
l0(x1:0, x11:0) -> l0(1 + x1:0, x11:0) :|: 0 <= -1 - x1:0 + x11:0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l0(x1:0, x11:0) -> l0(c, x11:0) :|: c = 1 + x1:0 && 0 <= -1 - x1:0 + x11:0

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1)] = -x + x1

The following rules are decreasing:
l0(x1:0, x11:0) -> l0(c, x11:0) :|: c = 1 + x1:0 && 0 <= -1 - x1:0 + x11:0
The following rules are bounded:
l0(x1:0, x11:0) -> l0(c, x11:0) :|: c = 1 + x1:0 && 0 <= -1 - x1:0 + x11:0

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(12)
YES