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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 151 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 33 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 15 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(i5HAT0, length4HAT0, sHAT0, tmpHAT0, tmp___08HAT0) -> l1(i5HATpost, length4HATpost, sHATpost, tmpHATpost, tmp___08HATpost) :|: tmp___08HAT0 = tmp___08HATpost && tmpHAT0 = tmpHATpost && sHAT0 = sHATpost && length4HAT0 = length4HATpost && i5HAT0 = i5HATpost && length4HAT0 <= i5HAT0
l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x3 = x8 && x2 = x7 && x1 = x6 && x5 = 1 + x && x9 = x9 && 1 + x <= x1
l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15
l3(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x25 = 0 && x26 = 10 && x27 = x28 && x28 = x28
l4(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
Start term: l4(i5HAT0, length4HAT0, sHAT0, tmpHAT0, tmp___08HAT0)

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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(i5HAT0, length4HAT0, sHAT0, tmpHAT0, tmp___08HAT0) -> l1(i5HATpost, length4HATpost, sHATpost, tmpHATpost, tmp___08HATpost) :|: tmp___08HAT0 = tmp___08HATpost && tmpHAT0 = tmpHATpost && sHAT0 = sHATpost && length4HAT0 = length4HATpost && i5HAT0 = i5HATpost && length4HAT0 <= i5HAT0
l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x3 = x8 && x2 = x7 && x1 = x6 && x5 = 1 + x && x9 = x9 && 1 + x <= x1
l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15
l3(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x25 = 0 && x26 = 10 && x27 = x28 && x28 = x28
l4(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
Start term: l4(i5HAT0, length4HAT0, sHAT0, tmpHAT0, tmp___08HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(i5HAT0, length4HAT0, sHAT0, tmpHAT0, tmp___08HAT0) -> l1(i5HATpost, length4HATpost, sHATpost, tmpHATpost, tmp___08HATpost) :|: tmp___08HAT0 = tmp___08HATpost && tmpHAT0 = tmpHATpost && sHAT0 = sHATpost && length4HAT0 = length4HATpost && i5HAT0 = i5HATpost && length4HAT0 <= i5HAT0
(2) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x3 = x8 && x2 = x7 && x1 = x6 && x5 = 1 + x && x9 = x9 && 1 + x <= x1
(3) l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15
(4) l3(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x25 = 0 && x26 = 10 && x27 = x28 && x28 = x28
(5) l4(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x3 = x8 && x2 = x7 && x1 = x6 && x5 = 1 + x && x9 = x9 && 1 + x <= x1
(2) l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15

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(x:0, x16:0, x17:0, x18:0, x4:0) -> l0(1 + x:0, x16:0, x17:0, x18:0, x19:0) :|: x16:0 >= 1 + x: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, x5) -> l0(x1, x2)

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(8)
Obligation:
Rules:
l0(x:0, x16:0) -> l0(1 + x:0, x16:0) :|: x16:0 >= 1 + x:0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l0(x:0, x16:0) -> l0(c, x16:0) :|: c = 1 + x:0 && x16:0 >= 1 + x:0

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

The following rules are decreasing:
l0(x:0, x16:0) -> l0(c, x16:0) :|: c = 1 + x:0 && x16:0 >= 1 + x:0
The following rules are bounded:
l0(x:0, x16:0) -> l0(c, x16:0) :|: c = 1 + x:0 && x16:0 >= 1 + x:0

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(12)
YES