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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 540 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 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, oldX0HAT0, oldX1HAT0, x0HAT0) -> l1(__const_10HATpost, oldX0HATpost, oldX1HATpost, x0HATpost) :|: oldX1HAT0 = oldX1HATpost && __const_10HAT0 = __const_10HATpost && x0HATpost = 1 + oldX0HATpost && oldX0HATpost = x0HAT0
l2(x, x1, x2, x3) -> l3(x4, x5, x6, x7) :|: x = x4 && x7 = x6 && x6 = x6 && x5 = x3
l1(x8, x9, x10, x11) -> l0(x12, x13, x14, x15) :|: x10 = x14 && x8 = x12 && x15 = x13 && x13 <= x8 && x13 = x11
l1(x16, x17, x18, x19) -> l2(x20, x21, x22, x23) :|: x18 = x22 && x16 = x20 && x23 = x21 && 1 + x16 <= x21 && x21 = x19
l4(x24, x25, x26, x27) -> l1(x28, x29, x30, x31) :|: x26 = x30 && x24 = x28 && x31 = 0 && x29 = x27
l5(x32, x33, x34, x35) -> l4(x36, x37, x38, x39) :|: x32 = x36 && x39 = x38 && x38 = x38 && x37 = x35
l5(x40, x41, x42, x43) -> l3(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44
l5(x48, x49, x50, x51) -> l0(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52
l5(x56, x57, x58, x59) -> l2(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60
l5(x64, x65, x66, x67) -> l1(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68
l5(x72, x73, x74, x75) -> l4(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76
l6(x80, x81, x82, x83) -> l5(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x81 = x85 && x80 = x84
Start term: l6(__const_10HAT0, oldX0HAT0, oldX1HAT0, x0HAT0)

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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, oldX0HAT0, oldX1HAT0, x0HAT0) -> l1(__const_10HATpost, oldX0HATpost, oldX1HATpost, x0HATpost) :|: oldX1HAT0 = oldX1HATpost && __const_10HAT0 = __const_10HATpost && x0HATpost = 1 + oldX0HATpost && oldX0HATpost = x0HAT0
l2(x, x1, x2, x3) -> l3(x4, x5, x6, x7) :|: x = x4 && x7 = x6 && x6 = x6 && x5 = x3
l1(x8, x9, x10, x11) -> l0(x12, x13, x14, x15) :|: x10 = x14 && x8 = x12 && x15 = x13 && x13 <= x8 && x13 = x11
l1(x16, x17, x18, x19) -> l2(x20, x21, x22, x23) :|: x18 = x22 && x16 = x20 && x23 = x21 && 1 + x16 <= x21 && x21 = x19
l4(x24, x25, x26, x27) -> l1(x28, x29, x30, x31) :|: x26 = x30 && x24 = x28 && x31 = 0 && x29 = x27
l5(x32, x33, x34, x35) -> l4(x36, x37, x38, x39) :|: x32 = x36 && x39 = x38 && x38 = x38 && x37 = x35
l5(x40, x41, x42, x43) -> l3(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44
l5(x48, x49, x50, x51) -> l0(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52
l5(x56, x57, x58, x59) -> l2(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60
l5(x64, x65, x66, x67) -> l1(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68
l5(x72, x73, x74, x75) -> l4(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76
l6(x80, x81, x82, x83) -> l5(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x81 = x85 && x80 = x84
Start term: l6(__const_10HAT0, oldX0HAT0, oldX1HAT0, x0HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_10HAT0, oldX0HAT0, oldX1HAT0, x0HAT0) -> l1(__const_10HATpost, oldX0HATpost, oldX1HATpost, x0HATpost) :|: oldX1HAT0 = oldX1HATpost && __const_10HAT0 = __const_10HATpost && x0HATpost = 1 + oldX0HATpost && oldX0HATpost = x0HAT0
(2) l2(x, x1, x2, x3) -> l3(x4, x5, x6, x7) :|: x = x4 && x7 = x6 && x6 = x6 && x5 = x3
(3) l1(x8, x9, x10, x11) -> l0(x12, x13, x14, x15) :|: x10 = x14 && x8 = x12 && x15 = x13 && x13 <= x8 && x13 = x11
(4) l1(x16, x17, x18, x19) -> l2(x20, x21, x22, x23) :|: x18 = x22 && x16 = x20 && x23 = x21 && 1 + x16 <= x21 && x21 = x19
(5) l4(x24, x25, x26, x27) -> l1(x28, x29, x30, x31) :|: x26 = x30 && x24 = x28 && x31 = 0 && x29 = x27
(6) l5(x32, x33, x34, x35) -> l4(x36, x37, x38, x39) :|: x32 = x36 && x39 = x38 && x38 = x38 && x37 = x35
(7) l5(x40, x41, x42, x43) -> l3(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44
(8) l5(x48, x49, x50, x51) -> l0(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52
(9) l5(x56, x57, x58, x59) -> l2(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60
(10) l5(x64, x65, x66, x67) -> l1(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68
(11) l5(x72, x73, x74, x75) -> l4(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76
(12) l6(x80, x81, x82, x83) -> l5(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x81 = x85 && x80 = x84

Arcs:
(1) -> (3), (4)
(3) -> (1)
(4) -> (2)
(5) -> (3), (4)
(6) -> (5)
(8) -> (1)
(9) -> (2)
(10) -> (3), (4)
(11) -> (5)
(12) -> (6), (7), (8), (9), (10), (11)

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(__const_10HAT0, oldX0HAT0, oldX1HAT0, x0HAT0) -> l1(__const_10HATpost, oldX0HATpost, oldX1HATpost, x0HATpost) :|: oldX1HAT0 = oldX1HATpost && __const_10HAT0 = __const_10HATpost && x0HATpost = 1 + oldX0HATpost && oldX0HATpost = x0HAT0
(2) l1(x8, x9, x10, x11) -> l0(x12, x13, x14, x15) :|: x10 = x14 && x8 = x12 && x15 = x13 && x13 <= x8 && x13 = x11

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(__const_10HAT0:0, oldX0HAT0:0, oldX1HAT0:0, oldX0HATpost:0) -> l0(__const_10HAT0:0, 1 + oldX0HATpost:0, oldX1HAT0:0, 1 + oldX0HATpost:0) :|: __const_10HAT0:0 >= 1 + oldX0HATpost: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(x1, x4)

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(8)
Obligation:
Rules:
l0(__const_10HAT0:0, oldX0HATpost:0) -> l0(__const_10HAT0:0, 1 + oldX0HATpost:0) :|: __const_10HAT0:0 >= 1 + oldX0HATpost: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(__const_10HAT0:0, oldX0HATpost:0) -> l0(__const_10HAT0:0, c) :|: c = 1 + oldX0HATpost:0 && __const_10HAT0:0 >= 1 + oldX0HATpost:0

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

The following rules are decreasing:
l0(__const_10HAT0:0, oldX0HATpost:0) -> l0(__const_10HAT0:0, c) :|: c = 1 + oldX0HATpost:0 && __const_10HAT0:0 >= 1 + oldX0HATpost:0
The following rules are bounded:
l0(__const_10HAT0:0, oldX0HATpost:0) -> l0(__const_10HAT0:0, c) :|: c = 1 + oldX0HATpost:0 && __const_10HAT0:0 >= 1 + oldX0HATpost:0

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(12)
YES