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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 1917 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 22 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 16 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, oldX4HAT0, oldX5HAT0, x0HAT0, x1HAT0, x2HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, oldX4HATpost, oldX5HATpost, x0HATpost, x1HATpost, x2HATpost) :|: x2HATpost = oldX5HATpost && x1HATpost = oldX4HATpost && x0HATpost = oldX3HATpost && oldX5HATpost = oldX5HATpost && oldX4HATpost = oldX4HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = x2HAT0 && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l2(x, x1, x2, x3, x4, x5, x6, x7, x8) -> l3(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x5 = x14 && x4 = x13 && x3 = x12 && x17 = 1 + x11 && x16 = 1 + x10 && x15 = x9 && x11 = x8 && x10 = x7 && x9 = x6
l3(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l0(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x23 = x32 && x22 = x31 && x21 = x30 && x35 = x29 && x34 = x28 && x33 = x27 && x27 <= x28 && x29 = x26 && x28 = x25 && x27 = x24
l3(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x41 = x50 && x40 = x49 && x39 = x48 && x53 = x47 && x52 = x46 && x51 = x45 && 1 + x46 <= x45 && x47 = x44 && x46 = x43 && x45 = x42
l4(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l3(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x59 = x68 && x71 = 0 && x70 = x67 && x69 = x66 && x67 = x67 && x66 = x66 && x65 = x62 && x64 = x61 && x63 = x60
l5(x72, x73, x74, x75, x76, x77, x78, x79, x80) -> l4(x81, x82, x83, x84, x85, x86, x87, x88, x89) :|: x89 = x86 && x88 = x85 && x87 = x84 && x86 = x86 && x85 = x85 && x84 = x84 && x83 = x80 && x82 = x79 && x81 = x78
l5(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l1(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x98 = x107 && x97 = x106 && x96 = x105 && x95 = x104 && x94 = x103 && x93 = x102 && x92 = x101 && x91 = x100 && x90 = x99
l5(x108, x109, x110, x111, x112, x113, x114, x115, x116) -> l0(x117, x118, x119, x120, x121, x122, x123, x124, x125) :|: x116 = x125 && x115 = x124 && x114 = x123 && x113 = x122 && x112 = x121 && x111 = x120 && x110 = x119 && x109 = x118 && x108 = x117
l5(x126, x127, x128, x129, x130, x131, x132, x133, x134) -> l2(x135, x136, x137, x138, x139, x140, x141, x142, x143) :|: x134 = x143 && x133 = x142 && x132 = x141 && x131 = x140 && x130 = x139 && x129 = x138 && x128 = x137 && x127 = x136 && x126 = x135
l5(x144, x145, x146, x147, x148, x149, x150, x151, x152) -> l3(x153, x154, x155, x156, x157, x158, x159, x160, x161) :|: x152 = x161 && x151 = x160 && x150 = x159 && x149 = x158 && x148 = x157 && x147 = x156 && x146 = x155 && x145 = x154 && x144 = x153
l5(x162, x163, x164, x165, x166, x167, x168, x169, x170) -> l4(x171, x172, x173, x174, x175, x176, x177, x178, x179) :|: x170 = x179 && x169 = x178 && x168 = x177 && x167 = x176 && x166 = x175 && x165 = x174 && x164 = x173 && x163 = x172 && x162 = x171
l6(x180, x181, x182, x183, x184, x185, x186, x187, x188) -> l5(x189, x190, x191, x192, x193, x194, x195, x196, x197) :|: x188 = x197 && x187 = x196 && x186 = x195 && x185 = x194 && x184 = x193 && x183 = x192 && x182 = x191 && x181 = x190 && x180 = x189
Start term: l6(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, oldX4HAT0, oldX5HAT0, x0HAT0, x1HAT0, x2HAT0)

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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(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, oldX4HAT0, oldX5HAT0, x0HAT0, x1HAT0, x2HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, oldX4HATpost, oldX5HATpost, x0HATpost, x1HATpost, x2HATpost) :|: x2HATpost = oldX5HATpost && x1HATpost = oldX4HATpost && x0HATpost = oldX3HATpost && oldX5HATpost = oldX5HATpost && oldX4HATpost = oldX4HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = x2HAT0 && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l2(x, x1, x2, x3, x4, x5, x6, x7, x8) -> l3(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x5 = x14 && x4 = x13 && x3 = x12 && x17 = 1 + x11 && x16 = 1 + x10 && x15 = x9 && x11 = x8 && x10 = x7 && x9 = x6
l3(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l0(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x23 = x32 && x22 = x31 && x21 = x30 && x35 = x29 && x34 = x28 && x33 = x27 && x27 <= x28 && x29 = x26 && x28 = x25 && x27 = x24
l3(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x41 = x50 && x40 = x49 && x39 = x48 && x53 = x47 && x52 = x46 && x51 = x45 && 1 + x46 <= x45 && x47 = x44 && x46 = x43 && x45 = x42
l4(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l3(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x59 = x68 && x71 = 0 && x70 = x67 && x69 = x66 && x67 = x67 && x66 = x66 && x65 = x62 && x64 = x61 && x63 = x60
l5(x72, x73, x74, x75, x76, x77, x78, x79, x80) -> l4(x81, x82, x83, x84, x85, x86, x87, x88, x89) :|: x89 = x86 && x88 = x85 && x87 = x84 && x86 = x86 && x85 = x85 && x84 = x84 && x83 = x80 && x82 = x79 && x81 = x78
l5(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l1(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x98 = x107 && x97 = x106 && x96 = x105 && x95 = x104 && x94 = x103 && x93 = x102 && x92 = x101 && x91 = x100 && x90 = x99
l5(x108, x109, x110, x111, x112, x113, x114, x115, x116) -> l0(x117, x118, x119, x120, x121, x122, x123, x124, x125) :|: x116 = x125 && x115 = x124 && x114 = x123 && x113 = x122 && x112 = x121 && x111 = x120 && x110 = x119 && x109 = x118 && x108 = x117
l5(x126, x127, x128, x129, x130, x131, x132, x133, x134) -> l2(x135, x136, x137, x138, x139, x140, x141, x142, x143) :|: x134 = x143 && x133 = x142 && x132 = x141 && x131 = x140 && x130 = x139 && x129 = x138 && x128 = x137 && x127 = x136 && x126 = x135
l5(x144, x145, x146, x147, x148, x149, x150, x151, x152) -> l3(x153, x154, x155, x156, x157, x158, x159, x160, x161) :|: x152 = x161 && x151 = x160 && x150 = x159 && x149 = x158 && x148 = x157 && x147 = x156 && x146 = x155 && x145 = x154 && x144 = x153
l5(x162, x163, x164, x165, x166, x167, x168, x169, x170) -> l4(x171, x172, x173, x174, x175, x176, x177, x178, x179) :|: x170 = x179 && x169 = x178 && x168 = x177 && x167 = x176 && x166 = x175 && x165 = x174 && x164 = x173 && x163 = x172 && x162 = x171
l6(x180, x181, x182, x183, x184, x185, x186, x187, x188) -> l5(x189, x190, x191, x192, x193, x194, x195, x196, x197) :|: x188 = x197 && x187 = x196 && x186 = x195 && x185 = x194 && x184 = x193 && x183 = x192 && x182 = x191 && x181 = x190 && x180 = x189
Start term: l6(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, oldX4HAT0, oldX5HAT0, x0HAT0, x1HAT0, x2HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, oldX4HAT0, oldX5HAT0, x0HAT0, x1HAT0, x2HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, oldX4HATpost, oldX5HATpost, x0HATpost, x1HATpost, x2HATpost) :|: x2HATpost = oldX5HATpost && x1HATpost = oldX4HATpost && x0HATpost = oldX3HATpost && oldX5HATpost = oldX5HATpost && oldX4HATpost = oldX4HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = x2HAT0 && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
(2) l2(x, x1, x2, x3, x4, x5, x6, x7, x8) -> l3(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x5 = x14 && x4 = x13 && x3 = x12 && x17 = 1 + x11 && x16 = 1 + x10 && x15 = x9 && x11 = x8 && x10 = x7 && x9 = x6
(3) l3(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l0(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x23 = x32 && x22 = x31 && x21 = x30 && x35 = x29 && x34 = x28 && x33 = x27 && x27 <= x28 && x29 = x26 && x28 = x25 && x27 = x24
(4) l3(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x41 = x50 && x40 = x49 && x39 = x48 && x53 = x47 && x52 = x46 && x51 = x45 && 1 + x46 <= x45 && x47 = x44 && x46 = x43 && x45 = x42
(5) l4(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l3(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x59 = x68 && x71 = 0 && x70 = x67 && x69 = x66 && x67 = x67 && x66 = x66 && x65 = x62 && x64 = x61 && x63 = x60
(6) l5(x72, x73, x74, x75, x76, x77, x78, x79, x80) -> l4(x81, x82, x83, x84, x85, x86, x87, x88, x89) :|: x89 = x86 && x88 = x85 && x87 = x84 && x86 = x86 && x85 = x85 && x84 = x84 && x83 = x80 && x82 = x79 && x81 = x78
(7) l5(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l1(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x98 = x107 && x97 = x106 && x96 = x105 && x95 = x104 && x94 = x103 && x93 = x102 && x92 = x101 && x91 = x100 && x90 = x99
(8) l5(x108, x109, x110, x111, x112, x113, x114, x115, x116) -> l0(x117, x118, x119, x120, x121, x122, x123, x124, x125) :|: x116 = x125 && x115 = x124 && x114 = x123 && x113 = x122 && x112 = x121 && x111 = x120 && x110 = x119 && x109 = x118 && x108 = x117
(9) l5(x126, x127, x128, x129, x130, x131, x132, x133, x134) -> l2(x135, x136, x137, x138, x139, x140, x141, x142, x143) :|: x134 = x143 && x133 = x142 && x132 = x141 && x131 = x140 && x130 = x139 && x129 = x138 && x128 = x137 && x127 = x136 && x126 = x135
(10) l5(x144, x145, x146, x147, x148, x149, x150, x151, x152) -> l3(x153, x154, x155, x156, x157, x158, x159, x160, x161) :|: x152 = x161 && x151 = x160 && x150 = x159 && x149 = x158 && x148 = x157 && x147 = x156 && x146 = x155 && x145 = x154 && x144 = x153
(11) l5(x162, x163, x164, x165, x166, x167, x168, x169, x170) -> l4(x171, x172, x173, x174, x175, x176, x177, x178, x179) :|: x170 = x179 && x169 = x178 && x168 = x177 && x167 = x176 && x166 = x175 && x165 = x174 && x164 = x173 && x163 = x172 && x162 = x171
(12) l6(x180, x181, x182, x183, x184, x185, x186, x187, x188) -> l5(x189, x190, x191, x192, x193, x194, x195, x196, x197) :|: x188 = x197 && x187 = x196 && x186 = x195 && x185 = x194 && x184 = x193 && x183 = x192 && x182 = x191 && x181 = x190 && x180 = x189

Arcs:
(2) -> (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) l2(x, x1, x2, x3, x4, x5, x6, x7, x8) -> l3(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x5 = x14 && x4 = x13 && x3 = x12 && x17 = 1 + x11 && x16 = 1 + x10 && x15 = x9 && x11 = x8 && x10 = x7 && x9 = x6
(2) l3(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x41 = x50 && x40 = x49 && x39 = x48 && x53 = x47 && x52 = x46 && x51 = x45 && 1 + x46 <= x45 && x47 = x44 && x46 = x43 && x45 = x42

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:
l2(x:0, x1:0, x2:0, x12:0, x13:0, x14:0, x15:0, x10:0, x11:0) -> l2(x15:0, 1 + x10:0, 1 + x11:0, x12:0, x13:0, x14:0, x15:0, 1 + x10:0, 1 + x11:0) :|: x15:0 >= 1 + (1 + x10:0)

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(7) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   l2(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> l2(x7, x8)

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(8)
Obligation:
Rules:
l2(x15:0, x10:0) -> l2(x15:0, 1 + x10:0) :|: x15:0 >= 1 + (1 + x10:0)

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l2(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l2(x15:0, x10:0) -> l2(x15:0, c) :|: c = 1 + x10:0 && x15:0 >= 1 + (1 + x10:0)

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l2(x, x1)] = x - x1

The following rules are decreasing:
l2(x15:0, x10:0) -> l2(x15:0, c) :|: c = 1 + x10:0 && x15:0 >= 1 + (1 + x10:0)
The following rules are bounded:
l2(x15:0, x10:0) -> l2(x15:0, c) :|: c = 1 + x10:0 && x15:0 >= 1 + (1 + x10:0)

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(12)
YES