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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 2718 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 15 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 49 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) IntTRS
(13) RankingReductionPairProof [EQUIVALENT, 3 ms]
(14) YES


----------------------------------------

(0)
Obligation:
Rules:
l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: oldX3HAT0 = oldX3HATpost && x1HATpost = oldX2HATpost && x0HATpost = 1 + oldX0HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = 1 + x7 && x10 = x6 && x7 = x5 && x6 = x4
l4(x12, x13, x14, x15, x16, x17) -> l5(x18, x19, x20, x21, x22, x23) :|: x23 = x21 && x22 = x20 && x21 = x21 && x20 = x20 && x19 = x17 && x18 = x16
l3(x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35) :|: x27 = x33 && x26 = x32 && x35 = x31 && x34 = x30 && 12 <= x31 && x31 = x29 && x30 = x28
l3(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = x43 && x46 = x42 && x43 <= 11 && x43 = x41 && x42 = x40
l1(x48, x49, x50, x51, x52, x53) -> l4(x54, x55, x56, x57, x58, x59) :|: x51 = x57 && x59 = x56 && x58 = x54 && 10 <= x54 && x56 = x56 && x55 = x53 && x54 = x52
l1(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = 3 && x70 = x66 && x66 <= 9 && x67 = x65 && x66 = x64
l6(x72, x73, x74, x75, x76, x77) -> l1(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x83 = x80 && x82 = 0 && x80 = x80 && x79 = x77 && x78 = x76
l7(x84, x85, x86, x87, x88, x89) -> l6(x90, x91, x92, x93, x94, x95) :|: x95 = x93 && x94 = x92 && x93 = x93 && x92 = x92 && x91 = x89 && x90 = x88
l7(x96, x97, x98, x99, x100, x101) -> l5(x102, x103, x104, x105, x106, x107) :|: x101 = x107 && x100 = x106 && x99 = x105 && x98 = x104 && x97 = x103 && x96 = x102
l7(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
l7(x120, x121, x122, x123, x124, x125) -> l2(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126
l7(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
l7(x144, x145, x146, x147, x148, x149) -> l3(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150
l7(x156, x157, x158, x159, x160, x161) -> l1(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162
l7(x168, x169, x170, x171, x172, x173) -> l6(x174, x175, x176, x177, x178, x179) :|: x173 = x179 && x172 = x178 && x171 = x177 && x170 = x176 && x169 = x175 && x168 = x174
l8(x180, x181, x182, x183, x184, x185) -> l7(x186, x187, x188, x189, x190, x191) :|: x185 = x191 && x184 = x190 && x183 = x189 && x182 = x188 && x181 = x187 && x180 = x186
Start term: l8(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0)

----------------------------------------

(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) :|: oldX3HAT0 = oldX3HATpost && x1HATpost = oldX2HATpost && x0HATpost = 1 + oldX0HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = 1 + x7 && x10 = x6 && x7 = x5 && x6 = x4
l4(x12, x13, x14, x15, x16, x17) -> l5(x18, x19, x20, x21, x22, x23) :|: x23 = x21 && x22 = x20 && x21 = x21 && x20 = x20 && x19 = x17 && x18 = x16
l3(x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35) :|: x27 = x33 && x26 = x32 && x35 = x31 && x34 = x30 && 12 <= x31 && x31 = x29 && x30 = x28
l3(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = x43 && x46 = x42 && x43 <= 11 && x43 = x41 && x42 = x40
l1(x48, x49, x50, x51, x52, x53) -> l4(x54, x55, x56, x57, x58, x59) :|: x51 = x57 && x59 = x56 && x58 = x54 && 10 <= x54 && x56 = x56 && x55 = x53 && x54 = x52
l1(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = 3 && x70 = x66 && x66 <= 9 && x67 = x65 && x66 = x64
l6(x72, x73, x74, x75, x76, x77) -> l1(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x83 = x80 && x82 = 0 && x80 = x80 && x79 = x77 && x78 = x76
l7(x84, x85, x86, x87, x88, x89) -> l6(x90, x91, x92, x93, x94, x95) :|: x95 = x93 && x94 = x92 && x93 = x93 && x92 = x92 && x91 = x89 && x90 = x88
l7(x96, x97, x98, x99, x100, x101) -> l5(x102, x103, x104, x105, x106, x107) :|: x101 = x107 && x100 = x106 && x99 = x105 && x98 = x104 && x97 = x103 && x96 = x102
l7(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
l7(x120, x121, x122, x123, x124, x125) -> l2(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126
l7(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
l7(x144, x145, x146, x147, x148, x149) -> l3(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150
l7(x156, x157, x158, x159, x160, x161) -> l1(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162
l7(x168, x169, x170, x171, x172, x173) -> l6(x174, x175, x176, x177, x178, x179) :|: x173 = x179 && x172 = x178 && x171 = x177 && x170 = x176 && x169 = x175 && x168 = x174
l8(x180, x181, x182, x183, x184, x185) -> l7(x186, x187, x188, x189, x190, x191) :|: x185 = x191 && x184 = x190 && x183 = x189 && x182 = x188 && x181 = x187 && x180 = x186
Start term: l8(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0)

----------------------------------------

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: oldX3HAT0 = oldX3HATpost && x1HATpost = oldX2HATpost && x0HATpost = 1 + oldX0HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
(2) l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = 1 + x7 && x10 = x6 && x7 = x5 && x6 = x4
(3) l4(x12, x13, x14, x15, x16, x17) -> l5(x18, x19, x20, x21, x22, x23) :|: x23 = x21 && x22 = x20 && x21 = x21 && x20 = x20 && x19 = x17 && x18 = x16
(4) l3(x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35) :|: x27 = x33 && x26 = x32 && x35 = x31 && x34 = x30 && 12 <= x31 && x31 = x29 && x30 = x28
(5) l3(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = x43 && x46 = x42 && x43 <= 11 && x43 = x41 && x42 = x40
(6) l1(x48, x49, x50, x51, x52, x53) -> l4(x54, x55, x56, x57, x58, x59) :|: x51 = x57 && x59 = x56 && x58 = x54 && 10 <= x54 && x56 = x56 && x55 = x53 && x54 = x52
(7) l1(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = 3 && x70 = x66 && x66 <= 9 && x67 = x65 && x66 = x64
(8) l6(x72, x73, x74, x75, x76, x77) -> l1(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x83 = x80 && x82 = 0 && x80 = x80 && x79 = x77 && x78 = x76
(9) l7(x84, x85, x86, x87, x88, x89) -> l6(x90, x91, x92, x93, x94, x95) :|: x95 = x93 && x94 = x92 && x93 = x93 && x92 = x92 && x91 = x89 && x90 = x88
(10) l7(x96, x97, x98, x99, x100, x101) -> l5(x102, x103, x104, x105, x106, x107) :|: x101 = x107 && x100 = x106 && x99 = x105 && x98 = x104 && x97 = x103 && x96 = x102
(11) l7(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) l7(x120, x121, x122, x123, x124, x125) -> l2(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126
(13) l7(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) l7(x144, x145, x146, x147, x148, x149) -> l3(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150
(15) l7(x156, x157, x158, x159, x160, x161) -> l1(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162
(16) l7(x168, x169, x170, x171, x172, x173) -> l6(x174, x175, x176, x177, x178, x179) :|: x173 = x179 && x172 = x178 && x171 = x177 && x170 = x176 && x169 = x175 && x168 = x174
(17) l8(x180, x181, x182, x183, x184, x185) -> l7(x186, x187, x188, x189, x190, x191) :|: x185 = x191 && x184 = x190 && x183 = x189 && x182 = x188 && x181 = x187 && x180 = x186

Arcs:
(1) -> (6), (7)
(2) -> (4), (5)
(4) -> (1)
(5) -> (2)
(6) -> (3)
(7) -> (5)
(8) -> (7)
(9) -> (8)
(11) -> (1)
(12) -> (2)
(13) -> (3)
(14) -> (4), (5)
(15) -> (6), (7)
(16) -> (8)
(17) -> (9), (10), (11), (12), (13), (14), (15), (16)

This digraph is fully evaluated!
----------------------------------------

(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: oldX3HAT0 = oldX3HATpost && x1HATpost = oldX2HATpost && x0HATpost = 1 + oldX0HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
(2) l3(x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35) :|: x27 = x33 && x26 = x32 && x35 = x31 && x34 = x30 && 12 <= x31 && x31 = x29 && x30 = x28
(3) l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = 1 + x7 && x10 = x6 && x7 = x5 && x6 = x4
(4) l3(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = x43 && x46 = x42 && x43 <= 11 && x43 = x41 && x42 = x40
(5) l1(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = 3 && x70 = x66 && x66 <= 9 && x67 = x65 && x66 = x64

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

This digraph is fully evaluated!

----------------------------------------

(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(6)
Obligation:
Rules:
l3(x36:0, x37:0, x38:0, x39:0, x10:0, x41:0) -> l3(x10:0, x41:0, x38:0, x39:0, x10:0, 1 + x41:0) :|: x41:0 < 12
l3(x24:0, x25:0, x26:0, oldX3HATpost:0, oldX0HATpost:0, oldX1HATpost:0) -> l3(1 + oldX0HATpost:0, oldX2HATpost:0, oldX2HATpost:0, oldX3HATpost:0, 1 + oldX0HATpost:0, 3) :|: oldX1HATpost:0 > 11 && oldX0HATpost:0 < 9

----------------------------------------

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

   l3(x1, x2, x3, x4, x5, x6) -> l3(x5, x6)

----------------------------------------

(8)
Obligation:
Rules:
l3(x10:0, x41:0) -> l3(x10:0, 1 + x41:0) :|: x41:0 < 12
l3(oldX0HATpost:0, oldX1HATpost:0) -> l3(1 + oldX0HATpost:0, 3) :|: oldX1HATpost:0 > 11 && oldX0HATpost:0 < 9

----------------------------------------

(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l3(VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(10)
Obligation:
Rules:
l3(x10:0, x41:0) -> l3(x10:0, c) :|: c = 1 + x41:0 && x41:0 < 12
l3(oldX0HATpost:0, oldX1HATpost:0) -> l3(c1, c2) :|: c2 = 3 && c1 = 1 + oldX0HATpost:0 && (oldX1HATpost:0 > 11 && oldX0HATpost:0 < 9)

----------------------------------------

(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l3(x, x1)] = 8 - x

The following rules are decreasing:
l3(oldX0HATpost:0, oldX1HATpost:0) -> l3(c1, c2) :|: c2 = 3 && c1 = 1 + oldX0HATpost:0 && (oldX1HATpost:0 > 11 && oldX0HATpost:0 < 9)
The following rules are bounded:
l3(oldX0HATpost:0, oldX1HATpost:0) -> l3(c1, c2) :|: c2 = 3 && c1 = 1 + oldX0HATpost:0 && (oldX1HATpost:0 > 11 && oldX0HATpost:0 < 9)

----------------------------------------

(12)
Obligation:
Rules:
l3(x10:0, x41:0) -> l3(x10:0, c) :|: c = 1 + x41:0 && x41:0 < 12

----------------------------------------

(13) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l3 ] = -1*l3_2

The following rules are decreasing:
l3(x10:0, x41:0) -> l3(x10:0, c) :|: c = 1 + x41:0 && x41:0 < 12

The following rules are bounded:
l3(x10:0, x41:0) -> l3(x10:0, c) :|: c = 1 + x41:0 && x41:0 < 12


----------------------------------------

(14)
YES