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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 291 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 42 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, b_7HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, b_7HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && b_7HAT0 = b_7HATpost && Result_4HAT0 = Result_4HATpost && 0 <= -1 - x_5HAT0 + y_6HAT0
l2(x, x1, x2, x3) -> l3(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 && 1 + x1 <= 0
l2(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && 1 <= x9
l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x18 = x22 && x16 = x20 && x23 = -1 + x19 && x21 = 0
l0(x24, x25, x26, x27) -> l4(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x28 = x28 && -1 * x26 + x27 <= 0
l5(x32, x33, x34, x35) -> l1(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x32 = x36 && x37 = 0
l1(x40, x41, x42, x43) -> l0(x44, x45, x46, x47) :|: x43 = x47 && x40 = x44 && x46 = 1 + x42 && x45 = 1 && 0 <= x41 && x41 <= 0 && 0 <= -1 - x42 + x43
l1(x48, x49, x50, x51) -> l4(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x52 = x52 && -1 * x50 + x51 <= 0
l6(x56, x57, x58, x59) -> l5(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60
Start term: l6(Result_4HAT0, 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, b_7HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, b_7HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && b_7HAT0 = b_7HATpost && Result_4HAT0 = Result_4HATpost && 0 <= -1 - x_5HAT0 + y_6HAT0
l2(x, x1, x2, x3) -> l3(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 && 1 + x1 <= 0
l2(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && 1 <= x9
l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x18 = x22 && x16 = x20 && x23 = -1 + x19 && x21 = 0
l0(x24, x25, x26, x27) -> l4(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x28 = x28 && -1 * x26 + x27 <= 0
l5(x32, x33, x34, x35) -> l1(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x32 = x36 && x37 = 0
l1(x40, x41, x42, x43) -> l0(x44, x45, x46, x47) :|: x43 = x47 && x40 = x44 && x46 = 1 + x42 && x45 = 1 && 0 <= x41 && x41 <= 0 && 0 <= -1 - x42 + x43
l1(x48, x49, x50, x51) -> l4(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x52 = x52 && -1 * x50 + x51 <= 0
l6(x56, x57, x58, x59) -> l5(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60
Start term: l6(Result_4HAT0, b_7HAT0, x_5HAT0, y_6HAT0)

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

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(Result_4HAT0, b_7HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, b_7HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && b_7HAT0 = b_7HATpost && Result_4HAT0 = Result_4HATpost && 0 <= -1 - x_5HAT0 + y_6HAT0
(2) l1(x40, x41, x42, x43) -> l0(x44, x45, x46, x47) :|: x43 = x47 && x40 = x44 && x46 = 1 + x42 && x45 = 1 && 0 <= x41 && x41 <= 0 && 0 <= -1 - x42 + x43
(3) l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x18 = x22 && x16 = x20 && x23 = -1 + x19 && x21 = 0
(4) l2(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && 1 <= x9
(5) l2(x, x1, x2, x3) -> l3(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 && 1 + x1 <= 0

Arcs:
(1) -> (4), (5)
(2) -> (1)
(3) -> (2)
(4) -> (3)
(5) -> (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, x17:0, x18:0, x19:0) -> l3(Result_4HATpost:0, 1, 1 + x18:0, -1 + x19:0) :|: 0 <= -1 - x18:0 + (-1 + x19:0) && 0 <= -1 - (1 + x18:0) + (-1 + x19:0)

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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) -> l3(x3, x4)

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(8)
Obligation:
Rules:
l3(x18:0, x19:0) -> l3(1 + x18:0, -1 + x19:0) :|: 0 <= -1 - x18:0 + (-1 + x19:0) && 0 <= -1 - (1 + x18:0) + (-1 + x19:0)

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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(x18:0, x19:0) -> l3(c, c1) :|: c1 = -1 + x19:0 && c = 1 + x18:0 && (0 <= -1 - x18:0 + (-1 + x19:0) && 0 <= -1 - (1 + x18:0) + (-1 + x19:0))

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

The following rules are decreasing:
l3(x18:0, x19:0) -> l3(c, c1) :|: c1 = -1 + x19:0 && c = 1 + x18:0 && (0 <= -1 - x18:0 + (-1 + x19:0) && 0 <= -1 - (1 + x18:0) + (-1 + x19:0))
The following rules are bounded:
l3(x18:0, x19:0) -> l3(c, c1) :|: c1 = -1 + x19:0 && c = 1 + x18:0 && (0 <= -1 - x18:0 + (-1 + x19:0) && 0 <= -1 - (1 + x18:0) + (-1 + x19:0))

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(12)
YES