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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 161 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 24 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(__const_10HAT0, i5HAT0, length4HAT0, sHAT0, tmpHAT0, tmp___08HAT0) -> l1(__const_10HATpost, i5HATpost, length4HATpost, sHATpost, tmpHATpost, tmp___08HATpost) :|: tmp___08HAT0 = tmp___08HATpost && tmpHAT0 = tmpHATpost && sHAT0 = sHATpost && length4HAT0 = length4HATpost && i5HAT0 = i5HATpost && __const_10HAT0 = __const_10HATpost && length4HAT0 <= i5HAT0
l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x4 = x10 && x3 = x9 && x2 = x8 && x = x6 && x7 = 1 + x1 && x11 = x11 && 1 + x1 <= x2
l2(x12, x13, x14, x15, x16, x17) -> l0(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x14 = x20 && x13 = x19 && x12 = x18
l3(x24, x25, x26, x27, x28, x29) -> l2(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x24 = x30 && x31 = 0 && x32 = x24 && x33 = x34 && x34 = x34
l4(x36, x37, x38, x39, x40, x41) -> l3(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x40 = x46 && x39 = x45 && x38 = x44 && x37 = x43 && x36 = x42
Start term: l4(__const_10HAT0, 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(__const_10HAT0, i5HAT0, length4HAT0, sHAT0, tmpHAT0, tmp___08HAT0) -> l1(__const_10HATpost, i5HATpost, length4HATpost, sHATpost, tmpHATpost, tmp___08HATpost) :|: tmp___08HAT0 = tmp___08HATpost && tmpHAT0 = tmpHATpost && sHAT0 = sHATpost && length4HAT0 = length4HATpost && i5HAT0 = i5HATpost && __const_10HAT0 = __const_10HATpost && length4HAT0 <= i5HAT0
l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x4 = x10 && x3 = x9 && x2 = x8 && x = x6 && x7 = 1 + x1 && x11 = x11 && 1 + x1 <= x2
l2(x12, x13, x14, x15, x16, x17) -> l0(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x14 = x20 && x13 = x19 && x12 = x18
l3(x24, x25, x26, x27, x28, x29) -> l2(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x24 = x30 && x31 = 0 && x32 = x24 && x33 = x34 && x34 = x34
l4(x36, x37, x38, x39, x40, x41) -> l3(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x40 = x46 && x39 = x45 && x38 = x44 && x37 = x43 && x36 = x42
Start term: l4(__const_10HAT0, i5HAT0, length4HAT0, sHAT0, tmpHAT0, tmp___08HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_10HAT0, i5HAT0, length4HAT0, sHAT0, tmpHAT0, tmp___08HAT0) -> l1(__const_10HATpost, i5HATpost, length4HATpost, sHATpost, tmpHATpost, tmp___08HATpost) :|: tmp___08HAT0 = tmp___08HATpost && tmpHAT0 = tmpHATpost && sHAT0 = sHATpost && length4HAT0 = length4HATpost && i5HAT0 = i5HATpost && __const_10HAT0 = __const_10HATpost && length4HAT0 <= i5HAT0
(2) l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x4 = x10 && x3 = x9 && x2 = x8 && x = x6 && x7 = 1 + x1 && x11 = x11 && 1 + x1 <= x2
(3) l2(x12, x13, x14, x15, x16, x17) -> l0(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x14 = x20 && x13 = x19 && x12 = x18
(4) l3(x24, x25, x26, x27, x28, x29) -> l2(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x24 = x30 && x31 = 0 && x32 = x24 && x33 = x34 && x34 = x34
(5) l4(x36, x37, x38, x39, x40, x41) -> l3(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x40 = x46 && x39 = x45 && x38 = x44 && x37 = x43 && x36 = x42

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, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x4 = x10 && x3 = x9 && x2 = x8 && x = x6 && x7 = 1 + x1 && x11 = x11 && 1 + x1 <= x2
(2) l2(x12, x13, x14, x15, x16, x17) -> l0(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x14 = x20 && x13 = x19 && x12 = x18

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(x18:0, x1:0, x20:0, x21:0, x10:0, x5:0) -> l0(x18:0, 1 + x1:0, x20:0, x21:0, x10:0, x11:0) :|: x20:0 >= 1 + x1:0

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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, x6) -> l0(x2, x3)

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(8)
Obligation:
Rules:
l0(x1:0, x20:0) -> l0(1 + x1:0, x20:0) :|: x20:0 >= 1 + x1:0

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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(x1:0, x20:0) -> l0(c, x20:0) :|: c = 1 + x1:0 && x20:0 >= 1 + x1:0

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

The following rules are decreasing:
l0(x1:0, x20:0) -> l0(c, x20:0) :|: c = 1 + x1:0 && x20:0 >= 1 + x1:0
The following rules are bounded:
l0(x1:0, x20:0) -> l0(c, x20:0) :|: c = 1 + x1:0 && x20:0 >= 1 + x1:0

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(12)
YES