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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 284 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 65 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 20 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(cHAT0, oxHAT0, xHAT0) -> l1(cHATpost, oxHATpost, xHATpost) :|: xHAT0 = xHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && 1 + xHAT0 <= oxHAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && x1 <= x2
l3(x6, x7, x8) -> l1(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && x6 <= 0
l3(x12, x13, x14) -> l1(x15, x16, x17) :|: x14 = x17 && x15 = 1 && x16 = x14
l4(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21 && x18 <= 0
l4(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27 && 1 <= x24
l1(x30, x31, x32) -> l4(x33, x34, x35) :|: x31 = x34 && x30 = x33 && x35 = -1 + x32 && 1 <= x32
l5(x36, x37, x38) -> l1(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39 && x36 <= 0
l6(x42, x43, x44) -> l5(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
Start term: l6(cHAT0, oxHAT0, xHAT0)

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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(cHAT0, oxHAT0, xHAT0) -> l1(cHATpost, oxHATpost, xHATpost) :|: xHAT0 = xHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && 1 + xHAT0 <= oxHAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && x1 <= x2
l3(x6, x7, x8) -> l1(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && x6 <= 0
l3(x12, x13, x14) -> l1(x15, x16, x17) :|: x14 = x17 && x15 = 1 && x16 = x14
l4(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21 && x18 <= 0
l4(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27 && 1 <= x24
l1(x30, x31, x32) -> l4(x33, x34, x35) :|: x31 = x34 && x30 = x33 && x35 = -1 + x32 && 1 <= x32
l5(x36, x37, x38) -> l1(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39 && x36 <= 0
l6(x42, x43, x44) -> l5(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
Start term: l6(cHAT0, oxHAT0, xHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(cHAT0, oxHAT0, xHAT0) -> l1(cHATpost, oxHATpost, xHATpost) :|: xHAT0 = xHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && 1 + xHAT0 <= oxHAT0
(2) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && x1 <= x2
(3) l3(x6, x7, x8) -> l1(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && x6 <= 0
(4) l3(x12, x13, x14) -> l1(x15, x16, x17) :|: x14 = x17 && x15 = 1 && x16 = x14
(5) l4(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21 && x18 <= 0
(6) l4(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27 && 1 <= x24
(7) l1(x30, x31, x32) -> l4(x33, x34, x35) :|: x31 = x34 && x30 = x33 && x35 = -1 + x32 && 1 <= x32
(8) l5(x36, x37, x38) -> l1(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39 && x36 <= 0
(9) l6(x42, x43, x44) -> l5(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(cHAT0, oxHAT0, xHAT0) -> l1(cHATpost, oxHATpost, xHATpost) :|: xHAT0 = xHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && 1 + xHAT0 <= oxHAT0
(2) l4(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27 && 1 <= x24
(3) l1(x30, x31, x32) -> l4(x33, x34, x35) :|: x31 = x34 && x30 = x33 && x35 = -1 + x32 && 1 <= x32
(4) l3(x12, x13, x14) -> l1(x15, x16, x17) :|: x14 = x17 && x15 = 1 && x16 = x14
(5) l3(x6, x7, x8) -> l1(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && x6 <= 0
(6) l4(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21 && x18 <= 0

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l4(cHATpost:0, oxHATpost:0, x26:0) -> l4(cHATpost:0, oxHATpost:0, -1 + x26:0) :|: oxHATpost:0 >= 1 + x26:0 && x26:0 > 0 && cHATpost:0 > 0
l4(x18:0, x19:0, x16:0) -> l4(1, x16:0, -1 + x16:0) :|: x18:0 < 1 && x16:0 > 0
l4(x, x1, x2) -> l4(x, x1, -1 + x2) :|: x2 > 0 && x < 1

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l4(VARIABLE, VARIABLE, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
l4(cHATpost:0, oxHATpost:0, x26:0) -> l4(cHATpost:0, oxHATpost:0, c) :|: c = -1 + x26:0 && (oxHATpost:0 >= 1 + x26:0 && x26:0 > 0 && cHATpost:0 > 0)
l4(x18:0, x19:0, x16:0) -> l4(c1, x16:0, c2) :|: c2 = -1 + x16:0 && c1 = 1 && (x18:0 < 1 && x16:0 > 0)
l4(x, x1, x2) -> l4(x, x1, c3) :|: c3 = -1 + x2 && (x2 > 0 && x < 1)

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(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l4(x, x1, x2)] = -x

The following rules are decreasing:
l4(x18:0, x19:0, x16:0) -> l4(c1, x16:0, c2) :|: c2 = -1 + x16:0 && c1 = 1 && (x18:0 < 1 && x16:0 > 0)
The following rules are bounded:
l4(x18:0, x19:0, x16:0) -> l4(c1, x16:0, c2) :|: c2 = -1 + x16:0 && c1 = 1 && (x18:0 < 1 && x16:0 > 0)
l4(x, x1, x2) -> l4(x, x1, c3) :|: c3 = -1 + x2 && (x2 > 0 && x < 1)

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(10)
Obligation:
Rules:
l4(cHATpost:0, oxHATpost:0, x26:0) -> l4(cHATpost:0, oxHATpost:0, c) :|: c = -1 + x26:0 && (oxHATpost:0 >= 1 + x26:0 && x26:0 > 0 && cHATpost:0 > 0)
l4(x, x1, x2) -> l4(x, x1, c3) :|: c3 = -1 + x2 && (x2 > 0 && x < 1)

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l4(x, x1, x2)] = x2

The following rules are decreasing:
l4(cHATpost:0, oxHATpost:0, x26:0) -> l4(cHATpost:0, oxHATpost:0, c) :|: c = -1 + x26:0 && (oxHATpost:0 >= 1 + x26:0 && x26:0 > 0 && cHATpost:0 > 0)
l4(x, x1, x2) -> l4(x, x1, c3) :|: c3 = -1 + x2 && (x2 > 0 && x < 1)
The following rules are bounded:
l4(cHATpost:0, oxHATpost:0, x26:0) -> l4(cHATpost:0, oxHATpost:0, c) :|: c = -1 + x26:0 && (oxHATpost:0 >= 1 + x26:0 && x26:0 > 0 && cHATpost:0 > 0)
l4(x, x1, x2) -> l4(x, x1, c3) :|: c3 = -1 + x2 && (x2 > 0 && x < 1)

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(12)
YES