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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 292 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 39 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 35 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(Result_4HAT0, __disjvr_0HAT0, b_7HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, __disjvr_0HATpost, b_7HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && b_7HAT0 = b_7HATpost && __disjvr_0HAT0 = __disjvr_0HATpost && Result_4HAT0 = Result_4HATpost && 0 <= -1 - x_5HAT0 + y_6HAT0
l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && x6 = x1
l3(x10, x11, x12, x13, x14) -> l1(x15, x16, x17, x18, x19) :|: x13 = x18 && x11 = x16 && x10 = x15 && x19 = -1 + x14 && x17 = 0
l0(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x25 = x25 && -1 * x23 + x24 <= 0
l5(x30, x31, x32, x33, x34) -> l1(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x31 = x36 && x30 = x35 && x37 = 0
l1(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x41 = x46 && x40 = x45 && x48 = 1 + x43 && x47 = 1 && 0 <= x42 && x42 <= 0 && 0 <= -1 - x43 + x44
l1(x50, x51, x52, x53, x54) -> l4(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && -1 * x53 + x54 <= 0
l6(x60, x61, x62, x63, x64) -> l5(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65
Start term: l6(Result_4HAT0, __disjvr_0HAT0, b_7HAT0, 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, __disjvr_0HAT0, b_7HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, __disjvr_0HATpost, b_7HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && b_7HAT0 = b_7HATpost && __disjvr_0HAT0 = __disjvr_0HATpost && Result_4HAT0 = Result_4HATpost && 0 <= -1 - x_5HAT0 + y_6HAT0
l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && x6 = x1
l3(x10, x11, x12, x13, x14) -> l1(x15, x16, x17, x18, x19) :|: x13 = x18 && x11 = x16 && x10 = x15 && x19 = -1 + x14 && x17 = 0
l0(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x25 = x25 && -1 * x23 + x24 <= 0
l5(x30, x31, x32, x33, x34) -> l1(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x31 = x36 && x30 = x35 && x37 = 0
l1(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x41 = x46 && x40 = x45 && x48 = 1 + x43 && x47 = 1 && 0 <= x42 && x42 <= 0 && 0 <= -1 - x43 + x44
l1(x50, x51, x52, x53, x54) -> l4(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && -1 * x53 + x54 <= 0
l6(x60, x61, x62, x63, x64) -> l5(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65
Start term: l6(Result_4HAT0, __disjvr_0HAT0, b_7HAT0, x_5HAT0, y_6HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(Result_4HAT0, __disjvr_0HAT0, b_7HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, __disjvr_0HATpost, b_7HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && b_7HAT0 = b_7HATpost && __disjvr_0HAT0 = __disjvr_0HATpost && Result_4HAT0 = Result_4HATpost && 0 <= -1 - x_5HAT0 + y_6HAT0
(2) l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && x6 = x1
(3) l3(x10, x11, x12, x13, x14) -> l1(x15, x16, x17, x18, x19) :|: x13 = x18 && x11 = x16 && x10 = x15 && x19 = -1 + x14 && x17 = 0
(4) l0(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x25 = x25 && -1 * x23 + x24 <= 0
(5) l5(x30, x31, x32, x33, x34) -> l1(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x31 = x36 && x30 = x35 && x37 = 0
(6) l1(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x41 = x46 && x40 = x45 && x48 = 1 + x43 && x47 = 1 && 0 <= x42 && x42 <= 0 && 0 <= -1 - x43 + x44
(7) l1(x50, x51, x52, x53, x54) -> l4(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && -1 * x53 + x54 <= 0
(8) l6(x60, x61, x62, x63, x64) -> l5(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(Result_4HAT0, __disjvr_0HAT0, b_7HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, __disjvr_0HATpost, b_7HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && b_7HAT0 = b_7HATpost && __disjvr_0HAT0 = __disjvr_0HATpost && Result_4HAT0 = Result_4HATpost && 0 <= -1 - x_5HAT0 + y_6HAT0
(2) l1(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x41 = x46 && x40 = x45 && x48 = 1 + x43 && x47 = 1 && 0 <= x42 && x42 <= 0 && 0 <= -1 - x43 + x44
(3) l3(x10, x11, x12, x13, x14) -> l1(x15, x16, x17, x18, x19) :|: x13 = x18 && x11 = x16 && x10 = x15 && x19 = -1 + x14 && x17 = 0
(4) l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && x6 = x1

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l3(Result_4HATpost:0, __disjvr_0HATpost:0, x12:0, x13:0, x14:0) -> l3(Result_4HATpost:0, __disjvr_0HATpost:0, 1, 1 + x13:0, -1 + x14:0) :|: 0 <= -1 - x13:0 + (-1 + x14:0) && 0 <= -1 - (1 + x13:0) + (-1 + x14:0)

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

   l3(x1, x2, x3, x4, x5) -> l3(x4, x5)

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(8)
Obligation:
Rules:
l3(x13:0, x14:0) -> l3(1 + x13:0, -1 + x14:0) :|: 0 <= -1 - x13:0 + (-1 + x14:0) && 0 <= -1 - (1 + x13:0) + (-1 + x14:0)

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l3(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l3(x13:0, x14:0) -> l3(c, c1) :|: c1 = -1 + x14:0 && c = 1 + x13:0 && (0 <= -1 - x13:0 + (-1 + x14:0) && 0 <= -1 - (1 + x13:0) + (-1 + x14:0))

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l3(x, x1)] = -1 - x + x1

The following rules are decreasing:
l3(x13:0, x14:0) -> l3(c, c1) :|: c1 = -1 + x14:0 && c = 1 + x13:0 && (0 <= -1 - x13:0 + (-1 + x14:0) && 0 <= -1 - (1 + x13:0) + (-1 + x14:0))
The following rules are bounded:
l3(x13:0, x14:0) -> l3(c, c1) :|: c1 = -1 + x14:0 && c = 1 + x13:0 && (0 <= -1 - x13:0 + (-1 + x14:0) && 0 <= -1 - (1 + x13:0) + (-1 + x14:0))

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(12)
YES