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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 267 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 12 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 16 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 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_4HAT0 = Result_4HATpost
l1(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && -1 * x1 + x2 <= 0 && -1 * x1 + x2 <= 0
l3(x6, x7, x8) -> l4(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && 1 + x7 <= x8
l3(x12, x13, x14) -> l4(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 1 + x14 <= x13
l4(x18, x19, x20) -> l2(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x21 = x21
l1(x24, x25, x26) -> l5(x27, x28, x29) :|: x26 = x29 && x24 = x27 && x28 = 1 + x25 && x26 <= x25 && x25 <= x26 && -1 * x25 + x26 <= 0 && -1 * x25 + x26 <= 0
l5(x30, x31, x32) -> l1(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33
l1(x36, x37, x38) -> l6(x39, x40, x41) :|: x38 = x41 && x36 = x39 && x40 = 1 + x37 && 0 <= -1 - x37 + x38
l6(x42, x43, x44) -> l1(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
l7(x48, x49, x50) -> l0(x51, x52, x53) :|: x50 = x53 && x49 = x52 && x48 = x51
Start term: l7(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_4HAT0 = Result_4HATpost
l1(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && -1 * x1 + x2 <= 0 && -1 * x1 + x2 <= 0
l3(x6, x7, x8) -> l4(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && 1 + x7 <= x8
l3(x12, x13, x14) -> l4(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 1 + x14 <= x13
l4(x18, x19, x20) -> l2(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x21 = x21
l1(x24, x25, x26) -> l5(x27, x28, x29) :|: x26 = x29 && x24 = x27 && x28 = 1 + x25 && x26 <= x25 && x25 <= x26 && -1 * x25 + x26 <= 0 && -1 * x25 + x26 <= 0
l5(x30, x31, x32) -> l1(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33
l1(x36, x37, x38) -> l6(x39, x40, x41) :|: x38 = x41 && x36 = x39 && x40 = 1 + x37 && 0 <= -1 - x37 + x38
l6(x42, x43, x44) -> l1(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
l7(x48, x49, x50) -> l0(x51, x52, x53) :|: x50 = x53 && x49 = x52 && x48 = x51
Start term: l7(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_4HAT0 = Result_4HATpost
(2) l1(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && -1 * x1 + x2 <= 0 && -1 * x1 + x2 <= 0
(3) l3(x6, x7, x8) -> l4(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && 1 + x7 <= x8
(4) l3(x12, x13, x14) -> l4(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 1 + x14 <= x13
(5) l4(x18, x19, x20) -> l2(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x21 = x21
(6) l1(x24, x25, x26) -> l5(x27, x28, x29) :|: x26 = x29 && x24 = x27 && x28 = 1 + x25 && x26 <= x25 && x25 <= x26 && -1 * x25 + x26 <= 0 && -1 * x25 + x26 <= 0
(7) l5(x30, x31, x32) -> l1(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33
(8) l1(x36, x37, x38) -> l6(x39, x40, x41) :|: x38 = x41 && x36 = x39 && x40 = 1 + x37 && 0 <= -1 - x37 + x38
(9) l6(x42, x43, x44) -> l1(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
(10) l7(x48, x49, x50) -> l0(x51, x52, x53) :|: x50 = x53 && x49 = x52 && x48 = x51

Arcs:
(1) -> (2), (6), (8)
(2) -> (4)
(3) -> (5)
(4) -> (5)
(6) -> (7)
(7) -> (2), (6), (8)
(8) -> (9)
(9) -> (2), (6), (8)
(10) -> (1)

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l1(x24, x25, x26) -> l5(x27, x28, x29) :|: x26 = x29 && x24 = x27 && x28 = 1 + x25 && x26 <= x25 && x25 <= x26 && -1 * x25 + x26 <= 0 && -1 * x25 + x26 <= 0
(2) l6(x42, x43, x44) -> l1(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
(3) l1(x36, x37, x38) -> l6(x39, x40, x41) :|: x38 = x41 && x36 = x39 && x40 = 1 + x37 && 0 <= -1 - x37 + x38
(4) l5(x30, x31, x32) -> l1(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33

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:
l1(x36:0, x37:0, x38:0) -> l1(x36:0, 1 + x37:0, x38:0) :|: 0 <= -1 - x37:0 + x38:0
l1(x24:0, x25:0, x25:0) -> l1(x24:0, 1 + x25:0, x25:0) :|: 0 >= -1 * x25:0 + x25:0

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

   l1(x1, x2, x3) -> l1(x2, x3)

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(8)
Obligation:
Rules:
l1(x37:0, x38:0) -> l1(1 + x37:0, x38:0) :|: 0 <= -1 - x37:0 + x38:0
l1(x25:0, x25:0) -> l1(1 + x25:0, x25:0) :|: 0 >= -1 * x25:0 + x25:0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l1(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l1(x37:0, x38:0) -> l1(c, x38:0) :|: c = 1 + x37:0 && 0 <= -1 - x37:0 + x38:0
l1(x25:0, x25:0) -> l1(c1, x25:0) :|: c1 = 1 + x25:0 && 0 >= -1 * x25:0 + x25:0

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x, x1)] = -x + x1

The following rules are decreasing:
l1(x37:0, x38:0) -> l1(c, x38:0) :|: c = 1 + x37:0 && 0 <= -1 - x37:0 + x38:0
l1(x25:0, x25:0) -> l1(c1, x25:0) :|: c1 = 1 + x25:0 && 0 >= -1 * x25:0 + x25:0
The following rules are bounded:
l1(x37:0, x38:0) -> l1(c, x38:0) :|: c = 1 + x37:0 && 0 <= -1 - x37:0 + x38:0
l1(x25:0, x25:0) -> l1(c1, x25:0) :|: c1 = 1 + x25:0 && 0 >= -1 * x25:0 + x25:0

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(12)
YES