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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 221 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, 8 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(Result_4HAT0, b_7HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, b_7HATpost, x_5HATpost, y_6HATpost) :|: x_5HAT0 = x_5HATpost && Result_4HAT0 = Result_4HATpost && y_6HATpost = -1 + y_6HAT0 && b_7HATpost = 0 && 1 - b_7HAT0 <= 0 && 0 <= -1 - x_5HAT0 + y_6HAT0
l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x4 = x4 && -1 * x2 + x3 <= 0
l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x8 = x12 && x13 = 0
l1(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x16 = x20 && x22 = 1 + x18 && x21 = 1 && 0 <= -1 * x17 && 0 <= -1 - x18 + x19
l1(x24, x25, x26, x27) -> l2(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x28 = x28 && -1 * x26 + x27 <= 0
l4(x32, x33, x34, x35) -> l3(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36
Start term: l4(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) -> l1(Result_4HATpost, b_7HATpost, x_5HATpost, y_6HATpost) :|: x_5HAT0 = x_5HATpost && Result_4HAT0 = Result_4HATpost && y_6HATpost = -1 + y_6HAT0 && b_7HATpost = 0 && 1 - b_7HAT0 <= 0 && 0 <= -1 - x_5HAT0 + y_6HAT0
l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x4 = x4 && -1 * x2 + x3 <= 0
l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x8 = x12 && x13 = 0
l1(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x16 = x20 && x22 = 1 + x18 && x21 = 1 && 0 <= -1 * x17 && 0 <= -1 - x18 + x19
l1(x24, x25, x26, x27) -> l2(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x28 = x28 && -1 * x26 + x27 <= 0
l4(x32, x33, x34, x35) -> l3(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36
Start term: l4(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) -> l1(Result_4HATpost, b_7HATpost, x_5HATpost, y_6HATpost) :|: x_5HAT0 = x_5HATpost && Result_4HAT0 = Result_4HATpost && y_6HATpost = -1 + y_6HAT0 && b_7HATpost = 0 && 1 - b_7HAT0 <= 0 && 0 <= -1 - x_5HAT0 + y_6HAT0
(2) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x4 = x4 && -1 * x2 + x3 <= 0
(3) l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x8 = x12 && x13 = 0
(4) l1(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x16 = x20 && x22 = 1 + x18 && x21 = 1 && 0 <= -1 * x17 && 0 <= -1 - x18 + x19
(5) l1(x24, x25, x26, x27) -> l2(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x28 = x28 && -1 * x26 + x27 <= 0
(6) l4(x32, x33, x34, x35) -> l3(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(Result_4HAT0, b_7HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, b_7HATpost, x_5HATpost, y_6HATpost) :|: x_5HAT0 = x_5HATpost && Result_4HAT0 = Result_4HATpost && y_6HATpost = -1 + y_6HAT0 && b_7HATpost = 0 && 1 - b_7HAT0 <= 0 && 0 <= -1 - x_5HAT0 + y_6HAT0
(2) l1(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x16 = x20 && x22 = 1 + x18 && x21 = 1 && 0 <= -1 * x17 && 0 <= -1 - x18 + x19

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(Result_4HAT0:0, b_7HAT0:0, x_5HAT0:0, y_6HAT0:0) -> l0(Result_4HAT0:0, 1, 1 + x_5HAT0:0, -1 + y_6HAT0:0) :|: 0 <= -1 - x_5HAT0:0 + (-1 + y_6HAT0:0) && b_7HAT0:0 > 0 && 0 <= -1 - x_5HAT0:0 + y_6HAT0:0

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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, x4) -> l0(x2, x3, x4)

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(8)
Obligation:
Rules:
l0(b_7HAT0:0, x_5HAT0:0, y_6HAT0:0) -> l0(1, 1 + x_5HAT0:0, -1 + y_6HAT0:0) :|: 0 <= -1 - x_5HAT0:0 + (-1 + y_6HAT0:0) && b_7HAT0:0 > 0 && 0 <= -1 - x_5HAT0:0 + y_6HAT0:0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(VARIABLE, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l0(b_7HAT0:0, x_5HAT0:0, y_6HAT0:0) -> l0(c, c1, c2) :|: c2 = -1 + y_6HAT0:0 && (c1 = 1 + x_5HAT0:0 && c = 1) && (0 <= -1 - x_5HAT0:0 + (-1 + y_6HAT0:0) && b_7HAT0:0 > 0 && 0 <= -1 - x_5HAT0:0 + y_6HAT0:0)

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1, x2)] = -1 - x1 + x2

The following rules are decreasing:
l0(b_7HAT0:0, x_5HAT0:0, y_6HAT0:0) -> l0(c, c1, c2) :|: c2 = -1 + y_6HAT0:0 && (c1 = 1 + x_5HAT0:0 && c = 1) && (0 <= -1 - x_5HAT0:0 + (-1 + y_6HAT0:0) && b_7HAT0:0 > 0 && 0 <= -1 - x_5HAT0:0 + y_6HAT0:0)
The following rules are bounded:
l0(b_7HAT0:0, x_5HAT0:0, y_6HAT0:0) -> l0(c, c1, c2) :|: c2 = -1 + y_6HAT0:0 && (c1 = 1 + x_5HAT0:0 && c = 1) && (0 <= -1 - x_5HAT0:0 + (-1 + y_6HAT0:0) && b_7HAT0:0 > 0 && 0 <= -1 - x_5HAT0:0 + y_6HAT0:0)

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(12)
YES