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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 234 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 40 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 33 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(__const_127HAT0, __const_36HAT0, counterHAT0, yHAT0, zHAT0) -> l1(__const_127HATpost, __const_36HATpost, counterHATpost, yHATpost, zHATpost) :|: zHAT0 = zHATpost && yHAT0 = yHATpost && counterHAT0 = counterHATpost && __const_36HAT0 = __const_36HATpost && __const_127HAT0 = __const_127HATpost && __const_36HAT0 <= counterHAT0
l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x3 = x8 && x1 = x6 && x = x5 && x7 = 1 + x2 && x9 = 1 + x4 && 1 + x2 <= x1
l3(x10, x11, x12, x13, x14) -> l2(x15, x16, x17, x18, x19) :|: x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && x19 = x19 && x13 <= x10
l3(x20, x21, x22, x23, x24) -> l1(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && 1 + x20 <= x23
l2(x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
l4(x40, x41, x42, x43, x44) -> l3(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x41 = x46 && x40 = x45 && x47 = 0
l5(x50, x51, x52, x53, x54) -> l4(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55
Start term: l5(__const_127HAT0, __const_36HAT0, counterHAT0, yHAT0, zHAT0)

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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_127HAT0, __const_36HAT0, counterHAT0, yHAT0, zHAT0) -> l1(__const_127HATpost, __const_36HATpost, counterHATpost, yHATpost, zHATpost) :|: zHAT0 = zHATpost && yHAT0 = yHATpost && counterHAT0 = counterHATpost && __const_36HAT0 = __const_36HATpost && __const_127HAT0 = __const_127HATpost && __const_36HAT0 <= counterHAT0
l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x3 = x8 && x1 = x6 && x = x5 && x7 = 1 + x2 && x9 = 1 + x4 && 1 + x2 <= x1
l3(x10, x11, x12, x13, x14) -> l2(x15, x16, x17, x18, x19) :|: x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && x19 = x19 && x13 <= x10
l3(x20, x21, x22, x23, x24) -> l1(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && 1 + x20 <= x23
l2(x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
l4(x40, x41, x42, x43, x44) -> l3(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x41 = x46 && x40 = x45 && x47 = 0
l5(x50, x51, x52, x53, x54) -> l4(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55
Start term: l5(__const_127HAT0, __const_36HAT0, counterHAT0, yHAT0, zHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_127HAT0, __const_36HAT0, counterHAT0, yHAT0, zHAT0) -> l1(__const_127HATpost, __const_36HATpost, counterHATpost, yHATpost, zHATpost) :|: zHAT0 = zHATpost && yHAT0 = yHATpost && counterHAT0 = counterHATpost && __const_36HAT0 = __const_36HATpost && __const_127HAT0 = __const_127HATpost && __const_36HAT0 <= counterHAT0
(2) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x3 = x8 && x1 = x6 && x = x5 && x7 = 1 + x2 && x9 = 1 + x4 && 1 + x2 <= x1
(3) l3(x10, x11, x12, x13, x14) -> l2(x15, x16, x17, x18, x19) :|: x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && x19 = x19 && x13 <= x10
(4) l3(x20, x21, x22, x23, x24) -> l1(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && 1 + x20 <= x23
(5) l2(x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
(6) l4(x40, x41, x42, x43, x44) -> l3(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x41 = x46 && x40 = x45 && x47 = 0
(7) l5(x50, x51, x52, x53, x54) -> l4(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55

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

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 && x1 = x6 && x = x5 && x7 = 1 + x2 && x9 = 1 + x4 && 1 + x2 <= x1
(2) l2(x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35

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(x35:0, x1:0, x2:0, x38:0, x4:0) -> l0(x35:0, x1:0, 1 + x2:0, x38:0, 1 + x4:0) :|: x1:0 >= 1 + x2:0

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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(x2, x3)

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(8)
Obligation:
Rules:
l0(x1:0, x2:0) -> l0(x1:0, 1 + x2:0) :|: x1:0 >= 1 + x2:0

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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, x2:0) -> l0(x1:0, c) :|: c = 1 + x2:0 && x1:0 >= 1 + x2:0

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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, x2:0) -> l0(x1:0, c) :|: c = 1 + x2:0 && x1:0 >= 1 + x2:0
The following rules are bounded:
l0(x1:0, x2:0) -> l0(x1:0, c) :|: c = 1 + x2:0 && x1:0 >= 1 + x2:0

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(12)
YES