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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 1830 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(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: x1HATpost = oldX3HATpost && x0HATpost = oldX2HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = -1 + x7 && x10 = -1 + x6 && x7 = x5 && x6 = x4
l3(x12, x13, x14, x15, x16, x17) -> l1(x18, x19, x20, x21, x22, x23) :|: x23 = x21 && x22 = x20 && 0 <= x18 && x18 <= 0 && x21 = x21 && x20 = x20 && x19 = x17 && x18 = x16
l3(x24, x25, x26, x27, x28, x29) -> l1(x30, x31, x32, x33, x34, x35) :|: x35 = x33 && x34 = x32 && 1 <= x30 && x33 = x33 && x32 = x32 && x31 = x29 && x30 = x28
l3(x36, x37, x38, x39, x40, x41) -> l1(x42, x43, x44, x45, x46, x47) :|: x47 = x45 && x46 = x44 && 1 + x42 <= 0 && x45 = x45 && x44 = x44 && x43 = x41 && x42 = x40
l4(x48, x49, x50, x51, x52, x53) -> l0(x54, x55, x56, x57, x58, x59) :|: x51 = x57 && x50 = x56 && x59 = x55 && x58 = x54 && 1 <= x55 && 1 <= x54 && x55 = x53 && x54 = x52
l4(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = x67 && x70 = x66 && x67 <= 0 && x67 = x65 && x66 = x64
l4(x72, x73, x74, x75, x76, x77) -> l3(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x74 = x80 && x83 = x79 && x82 = x78 && x78 <= 0 && x79 = x77 && x78 = x76
l2(x84, x85, x86, x87, x88, x89) -> l4(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && x91 = x89 && x90 = x88
l5(x96, x97, x98, x99, x100, x101) -> l1(x102, x103, x104, x105, x106, x107) :|: x101 = x107 && x100 = x106 && x99 = x105 && x98 = x104 && x97 = x103 && x96 = x102
l5(x108, x109, x110, x111, x112, x113) -> l0(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116 && x109 = x115 && x108 = x114
l5(x120, x121, x122, x123, x124, x125) -> l3(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126
l5(x132, x133, x134, x135, x136, x137) -> l4(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138
l5(x144, x145, x146, x147, x148, x149) -> l2(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150
l6(x156, x157, x158, x159, x160, x161) -> l5(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162
Start term: l6(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0)

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(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(2)
Obligation:
Rules:
l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: x1HATpost = oldX3HATpost && x0HATpost = oldX2HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = -1 + x7 && x10 = -1 + x6 && x7 = x5 && x6 = x4
l3(x12, x13, x14, x15, x16, x17) -> l1(x18, x19, x20, x21, x22, x23) :|: x23 = x21 && x22 = x20 && 0 <= x18 && x18 <= 0 && x21 = x21 && x20 = x20 && x19 = x17 && x18 = x16
l3(x24, x25, x26, x27, x28, x29) -> l1(x30, x31, x32, x33, x34, x35) :|: x35 = x33 && x34 = x32 && 1 <= x30 && x33 = x33 && x32 = x32 && x31 = x29 && x30 = x28
l3(x36, x37, x38, x39, x40, x41) -> l1(x42, x43, x44, x45, x46, x47) :|: x47 = x45 && x46 = x44 && 1 + x42 <= 0 && x45 = x45 && x44 = x44 && x43 = x41 && x42 = x40
l4(x48, x49, x50, x51, x52, x53) -> l0(x54, x55, x56, x57, x58, x59) :|: x51 = x57 && x50 = x56 && x59 = x55 && x58 = x54 && 1 <= x55 && 1 <= x54 && x55 = x53 && x54 = x52
l4(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = x67 && x70 = x66 && x67 <= 0 && x67 = x65 && x66 = x64
l4(x72, x73, x74, x75, x76, x77) -> l3(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x74 = x80 && x83 = x79 && x82 = x78 && x78 <= 0 && x79 = x77 && x78 = x76
l2(x84, x85, x86, x87, x88, x89) -> l4(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && x91 = x89 && x90 = x88
l5(x96, x97, x98, x99, x100, x101) -> l1(x102, x103, x104, x105, x106, x107) :|: x101 = x107 && x100 = x106 && x99 = x105 && x98 = x104 && x97 = x103 && x96 = x102
l5(x108, x109, x110, x111, x112, x113) -> l0(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116 && x109 = x115 && x108 = x114
l5(x120, x121, x122, x123, x124, x125) -> l3(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126
l5(x132, x133, x134, x135, x136, x137) -> l4(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138
l5(x144, x145, x146, x147, x148, x149) -> l2(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150
l6(x156, x157, x158, x159, x160, x161) -> l5(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162
Start term: l6(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: x1HATpost = oldX3HATpost && x0HATpost = oldX2HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
(2) l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = -1 + x7 && x10 = -1 + x6 && x7 = x5 && x6 = x4
(3) l3(x12, x13, x14, x15, x16, x17) -> l1(x18, x19, x20, x21, x22, x23) :|: x23 = x21 && x22 = x20 && 0 <= x18 && x18 <= 0 && x21 = x21 && x20 = x20 && x19 = x17 && x18 = x16
(4) l3(x24, x25, x26, x27, x28, x29) -> l1(x30, x31, x32, x33, x34, x35) :|: x35 = x33 && x34 = x32 && 1 <= x30 && x33 = x33 && x32 = x32 && x31 = x29 && x30 = x28
(5) l3(x36, x37, x38, x39, x40, x41) -> l1(x42, x43, x44, x45, x46, x47) :|: x47 = x45 && x46 = x44 && 1 + x42 <= 0 && x45 = x45 && x44 = x44 && x43 = x41 && x42 = x40
(6) l4(x48, x49, x50, x51, x52, x53) -> l0(x54, x55, x56, x57, x58, x59) :|: x51 = x57 && x50 = x56 && x59 = x55 && x58 = x54 && 1 <= x55 && 1 <= x54 && x55 = x53 && x54 = x52
(7) l4(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = x67 && x70 = x66 && x67 <= 0 && x67 = x65 && x66 = x64
(8) l4(x72, x73, x74, x75, x76, x77) -> l3(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x74 = x80 && x83 = x79 && x82 = x78 && x78 <= 0 && x79 = x77 && x78 = x76
(9) l2(x84, x85, x86, x87, x88, x89) -> l4(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && x91 = x89 && x90 = x88
(10) l5(x96, x97, x98, x99, x100, x101) -> l1(x102, x103, x104, x105, x106, x107) :|: x101 = x107 && x100 = x106 && x99 = x105 && x98 = x104 && x97 = x103 && x96 = x102
(11) l5(x108, x109, x110, x111, x112, x113) -> l0(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116 && x109 = x115 && x108 = x114
(12) l5(x120, x121, x122, x123, x124, x125) -> l3(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126
(13) l5(x132, x133, x134, x135, x136, x137) -> l4(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138
(14) l5(x144, x145, x146, x147, x148, x149) -> l2(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150
(15) l6(x156, x157, x158, x159, x160, x161) -> l5(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162

Arcs:
(2) -> (9)
(6) -> (1), (2)
(7) -> (3), (4), (5)
(8) -> (3), (5)
(9) -> (6), (7), (8)
(11) -> (1), (2)
(12) -> (3), (4), (5)
(13) -> (6), (7), (8)
(14) -> (9)
(15) -> (10), (11), (12), (13), (14)

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) :|: x3 = x9 && x2 = x8 && x11 = -1 + x7 && x10 = -1 + x6 && x7 = x5 && x6 = x4
(2) l4(x48, x49, x50, x51, x52, x53) -> l0(x54, x55, x56, x57, x58, x59) :|: x51 = x57 && x50 = x56 && x59 = x55 && x58 = x54 && 1 <= x55 && 1 <= x54 && x55 = x53 && x54 = x52
(3) l2(x84, x85, x86, x87, x88, x89) -> l4(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && x91 = x89 && x90 = x88

Arcs:
(1) -> (3)
(2) -> (1)
(3) -> (2)

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l4(x48:0, x49:0, x50:0, x51:0, x52:0, x53:0) -> l4(-1 + x52:0, -1 + x53:0, x50:0, x51:0, -1 + x52:0, -1 + x53:0) :|: x52:0 > 0 && x53:0 > 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:

   l4(x1, x2, x3, x4, x5, x6) -> l4(x5, x6)

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(8)
Obligation:
Rules:
l4(x52:0, x53:0) -> l4(-1 + x52:0, -1 + x53:0) :|: x52:0 > 0 && x53:0 > 0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l4(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(10)
Obligation:
Rules:
l4(x52:0, x53:0) -> l4(c, c1) :|: c1 = -1 + x53:0 && c = -1 + x52:0 && (x52:0 > 0 && x53:0 > 0)

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

The following rules are decreasing:
l4(x52:0, x53:0) -> l4(c, c1) :|: c1 = -1 + x53:0 && c = -1 + x52:0 && (x52:0 > 0 && x53:0 > 0)
The following rules are bounded:
l4(x52:0, x53:0) -> l4(c, c1) :|: c1 = -1 + x53:0 && c = -1 + x52:0 && (x52:0 > 0 && x53:0 > 0)

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(12)
YES