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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 286 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 23 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 44 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(Result_4HAT0, k_7HAT0, w_8HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, k_7HATpost, w_8HATpost, x_5HATpost, y_6HATpost) :|: w_8HAT0 = w_8HATpost && k_7HAT0 = k_7HATpost && Result_4HAT0 = Result_4HATpost && y_6HATpost = 1 + y_6HAT0 && x_5HATpost = 1 + x_5HAT0 && 0 <= -2 - x_5HAT0
l2(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 1 + x4 <= 0
l2(x10, x11, x12, x13, x14) -> l1(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && 1 <= x14
l1(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
l0(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x35 = x35 && -1 - x33 <= 0
l4(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && 0 <= -2 + x42 && 0 <= -2 + x41
l4(x50, x51, x52, x53, x54) -> l3(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && -1 + x51 <= 0
l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65
Start term: l5(Result_4HAT0, k_7HAT0, w_8HAT0, 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, k_7HAT0, w_8HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, k_7HATpost, w_8HATpost, x_5HATpost, y_6HATpost) :|: w_8HAT0 = w_8HATpost && k_7HAT0 = k_7HATpost && Result_4HAT0 = Result_4HATpost && y_6HATpost = 1 + y_6HAT0 && x_5HATpost = 1 + x_5HAT0 && 0 <= -2 - x_5HAT0
l2(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 1 + x4 <= 0
l2(x10, x11, x12, x13, x14) -> l1(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && 1 <= x14
l1(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
l0(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x35 = x35 && -1 - x33 <= 0
l4(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && 0 <= -2 + x42 && 0 <= -2 + x41
l4(x50, x51, x52, x53, x54) -> l3(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && -1 + x51 <= 0
l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65
Start term: l5(Result_4HAT0, k_7HAT0, w_8HAT0, x_5HAT0, y_6HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(Result_4HAT0, k_7HAT0, w_8HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, k_7HATpost, w_8HATpost, x_5HATpost, y_6HATpost) :|: w_8HAT0 = w_8HATpost && k_7HAT0 = k_7HATpost && Result_4HAT0 = Result_4HATpost && y_6HATpost = 1 + y_6HAT0 && x_5HATpost = 1 + x_5HAT0 && 0 <= -2 - x_5HAT0
(2) l2(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 1 + x4 <= 0
(3) l2(x10, x11, x12, x13, x14) -> l1(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && 1 <= x14
(4) l1(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
(5) l0(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x35 = x35 && -1 - x33 <= 0
(6) l4(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && 0 <= -2 + x42 && 0 <= -2 + x41
(7) l4(x50, x51, x52, x53, x54) -> l3(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && -1 + x51 <= 0
(8) l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(Result_4HAT0, k_7HAT0, w_8HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, k_7HATpost, w_8HATpost, x_5HATpost, y_6HATpost) :|: w_8HAT0 = w_8HATpost && k_7HAT0 = k_7HATpost && Result_4HAT0 = Result_4HATpost && y_6HATpost = 1 + y_6HAT0 && x_5HATpost = 1 + x_5HAT0 && 0 <= -2 - x_5HAT0
(2) l1(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25
(3) l2(x10, x11, x12, x13, x14) -> l1(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && 1 <= x14
(4) l2(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 1 + x4 <= 0

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l1(Result_4HATpost:0, k_7HATpost:0, w_8HATpost:0, x23:0, x24:0) -> l1(Result_4HATpost:0, k_7HATpost:0, w_8HATpost:0, 1 + x23:0, 1 + x24:0) :|: x23:0 < -1 && x24:0 < -1
l1(x, x1, x2, x3, x4) -> l1(x, x1, x2, 1 + x3, 1 + x4) :|: x3 < -1 && x4 > -1

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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, x4, x5) -> l1(x4, x5)

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(8)
Obligation:
Rules:
l1(x23:0, x24:0) -> l1(1 + x23:0, 1 + x24:0) :|: x23:0 < -1 && x24:0 < -1
l1(x3, x4) -> l1(1 + x3, 1 + x4) :|: x3 < -1 && x4 > -1

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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(x23:0, x24:0) -> l1(c, c1) :|: c1 = 1 + x24:0 && c = 1 + x23:0 && (x23:0 < -1 && x24:0 < -1)
l1(x3, x4) -> l1(c2, c3) :|: c3 = 1 + x4 && c2 = 1 + x3 && (x3 < -1 && x4 > -1)

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

The following rules are decreasing:
l1(x23:0, x24:0) -> l1(c, c1) :|: c1 = 1 + x24:0 && c = 1 + x23:0 && (x23:0 < -1 && x24:0 < -1)
l1(x3, x4) -> l1(c2, c3) :|: c3 = 1 + x4 && c2 = 1 + x3 && (x3 < -1 && x4 > -1)
The following rules are bounded:
l1(x23:0, x24:0) -> l1(c, c1) :|: c1 = 1 + x24:0 && c = 1 + x23:0 && (x23:0 < -1 && x24:0 < -1)
l1(x3, x4) -> l1(c2, c3) :|: c3 = 1 + x4 && c2 = 1 + x3 && (x3 < -1 && x4 > -1)

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(12)
YES