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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 3706 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 37 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 13 ms]
(12) YES


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

(0)
Obligation:
Rules:
l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, oldX4HAT0, oldX5HAT0, x0HAT0, x1HAT0, x2HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, oldX4HATpost, oldX5HATpost, x0HATpost, x1HATpost, x2HATpost) :|: oldX5HAT0 = oldX5HATpost && oldX4HAT0 = oldX4HATpost && oldX3HAT0 = oldX3HATpost && x2HATpost = -1 * oldX1HATpost + oldX2HATpost && x1HATpost = oldX1HATpost && x0HATpost = oldX0HATpost && oldX2HATpost = x2HAT0 && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l1(x, x1, x2, x3, x4, x5, x6, x7, x8) -> l2(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x5 = x14 && x4 = x13 && x17 = x12 && x16 = x10 && x15 = x9 && x10 <= 0 && x12 = x12 && x11 = x8 && x10 = x7 && x9 = x6
l1(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l2(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x23 = x32 && x22 = x31 && x35 = x30 && x34 = x28 && x33 = x27 && x29 <= 0 && x30 = x30 && x29 = x26 && x28 = x25 && x27 = x24
l1(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x41 = x50 && x40 = x49 && x39 = x48 && x53 = x47 && x52 = x46 && x51 = x45 && 1 <= x46 && 1 <= x47 && x47 = x44 && x46 = x43 && x45 = x42
l2(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l3(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x71 = x68 && x70 = x67 && x69 = x66 && x68 = x68 && x67 = x67 && x66 = x66 && x65 = x62 && x64 = x61 && x63 = x60
l4(x72, x73, x74, x75, x76, x77, x78, x79, x80) -> l1(x81, x82, x83, x84, x85, x86, x87, x88, x89) :|: x77 = x86 && x76 = x85 && x75 = x84 && x89 = x81 && x88 = x82 && x87 = x81 && 1 <= x81 && x83 = x80 && x82 = x79 && x81 = x78
l4(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l1(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x95 = x104 && x94 = x103 && x93 = x102 && x107 = x99 && x106 = x100 && x105 = x99 && 1 + x99 <= 0 && x101 = x98 && x100 = x97 && x99 = x96
l4(x108, x109, x110, x111, x112, x113, x114, x115, x116) -> l2(x117, x118, x119, x120, x121, x122, x123, x124, x125) :|: x113 = x122 && x112 = x121 && x125 = x120 && x124 = x118 && x123 = x117 && 0 <= x117 && x117 <= 0 && x120 = x120 && x119 = x116 && x118 = x115 && x117 = x114
l5(x126, x127, x128, x129, x130, x131, x132, x133, x134) -> l4(x135, x136, x137, x138, x139, x140, x141, x142, x143) :|: x131 = x140 && x130 = x139 && x143 = x138 && x142 = x136 && x141 = x135 && x138 = x138 && x137 = x134 && x136 = x133 && x135 = x132
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) -> l0(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
l5(x180, x181, x182, x183, x184, x185, x186, x187, x188) -> l1(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
l5(x198, x199, x200, x201, x202, x203, x204, x205, x206) -> l2(x207, x208, x209, x210, x211, x212, x213, x214, x215) :|: x206 = x215 && x205 = x214 && x204 = x213 && x203 = x212 && x202 = x211 && x201 = x210 && x200 = x209 && x199 = x208 && x198 = x207
l5(x216, x217, x218, x219, x220, x221, x222, x223, x224) -> l4(x225, x226, x227, x228, x229, x230, x231, x232, x233) :|: x224 = x233 && x223 = x232 && x222 = x231 && x221 = x230 && x220 = x229 && x219 = x228 && x218 = x227 && x217 = x226 && x216 = x225
l6(x234, x235, x236, x237, x238, x239, x240, x241, x242) -> l5(x243, x244, x245, x246, x247, x248, x249, x250, x251) :|: x242 = x251 && x241 = x250 && x240 = x249 && x239 = x248 && x238 = x247 && x237 = x246 && x236 = x245 && x235 = x244 && x234 = x243
Start term: l6(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, oldX4HAT0, oldX5HAT0, x0HAT0, x1HAT0, x2HAT0)

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

(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, oldX4HAT0, oldX5HAT0, x0HAT0, x1HAT0, x2HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, oldX4HATpost, oldX5HATpost, x0HATpost, x1HATpost, x2HATpost) :|: oldX5HAT0 = oldX5HATpost && oldX4HAT0 = oldX4HATpost && oldX3HAT0 = oldX3HATpost && x2HATpost = -1 * oldX1HATpost + oldX2HATpost && x1HATpost = oldX1HATpost && x0HATpost = oldX0HATpost && oldX2HATpost = x2HAT0 && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l1(x, x1, x2, x3, x4, x5, x6, x7, x8) -> l2(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x5 = x14 && x4 = x13 && x17 = x12 && x16 = x10 && x15 = x9 && x10 <= 0 && x12 = x12 && x11 = x8 && x10 = x7 && x9 = x6
l1(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l2(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x23 = x32 && x22 = x31 && x35 = x30 && x34 = x28 && x33 = x27 && x29 <= 0 && x30 = x30 && x29 = x26 && x28 = x25 && x27 = x24
l1(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x41 = x50 && x40 = x49 && x39 = x48 && x53 = x47 && x52 = x46 && x51 = x45 && 1 <= x46 && 1 <= x47 && x47 = x44 && x46 = x43 && x45 = x42
l2(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l3(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x71 = x68 && x70 = x67 && x69 = x66 && x68 = x68 && x67 = x67 && x66 = x66 && x65 = x62 && x64 = x61 && x63 = x60
l4(x72, x73, x74, x75, x76, x77, x78, x79, x80) -> l1(x81, x82, x83, x84, x85, x86, x87, x88, x89) :|: x77 = x86 && x76 = x85 && x75 = x84 && x89 = x81 && x88 = x82 && x87 = x81 && 1 <= x81 && x83 = x80 && x82 = x79 && x81 = x78
l4(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l1(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x95 = x104 && x94 = x103 && x93 = x102 && x107 = x99 && x106 = x100 && x105 = x99 && 1 + x99 <= 0 && x101 = x98 && x100 = x97 && x99 = x96
l4(x108, x109, x110, x111, x112, x113, x114, x115, x116) -> l2(x117, x118, x119, x120, x121, x122, x123, x124, x125) :|: x113 = x122 && x112 = x121 && x125 = x120 && x124 = x118 && x123 = x117 && 0 <= x117 && x117 <= 0 && x120 = x120 && x119 = x116 && x118 = x115 && x117 = x114
l5(x126, x127, x128, x129, x130, x131, x132, x133, x134) -> l4(x135, x136, x137, x138, x139, x140, x141, x142, x143) :|: x131 = x140 && x130 = x139 && x143 = x138 && x142 = x136 && x141 = x135 && x138 = x138 && x137 = x134 && x136 = x133 && x135 = x132
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) -> l0(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
l5(x180, x181, x182, x183, x184, x185, x186, x187, x188) -> l1(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
l5(x198, x199, x200, x201, x202, x203, x204, x205, x206) -> l2(x207, x208, x209, x210, x211, x212, x213, x214, x215) :|: x206 = x215 && x205 = x214 && x204 = x213 && x203 = x212 && x202 = x211 && x201 = x210 && x200 = x209 && x199 = x208 && x198 = x207
l5(x216, x217, x218, x219, x220, x221, x222, x223, x224) -> l4(x225, x226, x227, x228, x229, x230, x231, x232, x233) :|: x224 = x233 && x223 = x232 && x222 = x231 && x221 = x230 && x220 = x229 && x219 = x228 && x218 = x227 && x217 = x226 && x216 = x225
l6(x234, x235, x236, x237, x238, x239, x240, x241, x242) -> l5(x243, x244, x245, x246, x247, x248, x249, x250, x251) :|: x242 = x251 && x241 = x250 && x240 = x249 && x239 = x248 && x238 = x247 && x237 = x246 && x236 = x245 && x235 = x244 && x234 = x243
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) :|: oldX5HAT0 = oldX5HATpost && oldX4HAT0 = oldX4HATpost && oldX3HAT0 = oldX3HATpost && x2HATpost = -1 * oldX1HATpost + oldX2HATpost && x1HATpost = oldX1HATpost && x0HATpost = oldX0HATpost && oldX2HATpost = x2HAT0 && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
(2) l1(x, x1, x2, x3, x4, x5, x6, x7, x8) -> l2(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x5 = x14 && x4 = x13 && x17 = x12 && x16 = x10 && x15 = x9 && x10 <= 0 && x12 = x12 && x11 = x8 && x10 = x7 && x9 = x6
(3) l1(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l2(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x23 = x32 && x22 = x31 && x35 = x30 && x34 = x28 && x33 = x27 && x29 <= 0 && x30 = x30 && x29 = x26 && x28 = x25 && x27 = x24
(4) l1(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x41 = x50 && x40 = x49 && x39 = x48 && x53 = x47 && x52 = x46 && x51 = x45 && 1 <= x46 && 1 <= x47 && x47 = x44 && x46 = x43 && x45 = x42
(5) l2(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l3(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x71 = x68 && x70 = x67 && x69 = x66 && x68 = x68 && x67 = x67 && x66 = x66 && x65 = x62 && x64 = x61 && x63 = x60
(6) l4(x72, x73, x74, x75, x76, x77, x78, x79, x80) -> l1(x81, x82, x83, x84, x85, x86, x87, x88, x89) :|: x77 = x86 && x76 = x85 && x75 = x84 && x89 = x81 && x88 = x82 && x87 = x81 && 1 <= x81 && x83 = x80 && x82 = x79 && x81 = x78
(7) l4(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l1(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x95 = x104 && x94 = x103 && x93 = x102 && x107 = x99 && x106 = x100 && x105 = x99 && 1 + x99 <= 0 && x101 = x98 && x100 = x97 && x99 = x96
(8) l4(x108, x109, x110, x111, x112, x113, x114, x115, x116) -> l2(x117, x118, x119, x120, x121, x122, x123, x124, x125) :|: x113 = x122 && x112 = x121 && x125 = x120 && x124 = x118 && x123 = x117 && 0 <= x117 && x117 <= 0 && x120 = x120 && x119 = x116 && x118 = x115 && x117 = x114
(9) l5(x126, x127, x128, x129, x130, x131, x132, x133, x134) -> l4(x135, x136, x137, x138, x139, x140, x141, x142, x143) :|: x131 = x140 && x130 = x139 && x143 = x138 && x142 = x136 && x141 = x135 && x138 = x138 && x137 = x134 && x136 = x133 && x135 = x132
(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) -> l0(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) l5(x180, x181, x182, x183, x184, x185, x186, x187, x188) -> l1(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
(13) l5(x198, x199, x200, x201, x202, x203, x204, x205, x206) -> l2(x207, x208, x209, x210, x211, x212, x213, x214, x215) :|: x206 = x215 && x205 = x214 && x204 = x213 && x203 = x212 && x202 = x211 && x201 = x210 && x200 = x209 && x199 = x208 && x198 = x207
(14) l5(x216, x217, x218, x219, x220, x221, x222, x223, x224) -> l4(x225, x226, x227, x228, x229, x230, x231, x232, x233) :|: x224 = x233 && x223 = x232 && x222 = x231 && x221 = x230 && x220 = x229 && x219 = x228 && x218 = x227 && x217 = x226 && x216 = x225
(15) l6(x234, x235, x236, x237, x238, x239, x240, x241, x242) -> l5(x243, x244, x245, x246, x247, x248, x249, x250, x251) :|: x242 = x251 && x241 = x250 && x240 = x249 && x239 = x248 && x238 = x247 && x237 = x246 && x236 = x245 && x235 = x244 && x234 = x243

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

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, oldX4HAT0, oldX5HAT0, x0HAT0, x1HAT0, x2HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, oldX4HATpost, oldX5HATpost, x0HATpost, x1HATpost, x2HATpost) :|: oldX5HAT0 = oldX5HATpost && oldX4HAT0 = oldX4HATpost && oldX3HAT0 = oldX3HATpost && x2HATpost = -1 * oldX1HATpost + oldX2HATpost && x1HATpost = oldX1HATpost && x0HATpost = oldX0HATpost && oldX2HATpost = x2HAT0 && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
(2) l1(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x41 = x50 && x40 = x49 && x39 = x48 && x53 = x47 && x52 = x46 && x51 = x45 && 1 <= x46 && 1 <= x47 && 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.
----------------------------------------

(6)
Obligation:
Rules:
l0(oldX0HAT0:0, oldX1HAT0:0, oldX2HAT0:0, oldX3HAT0:0, oldX4HAT0:0, oldX5HAT0:0, oldX0HATpost:0, oldX1HATpost:0, oldX2HATpost:0) -> l0(oldX0HATpost:0, oldX1HATpost:0, -1 * oldX1HATpost:0 + oldX2HATpost:0, oldX3HAT0:0, oldX4HAT0:0, oldX5HAT0:0, oldX0HATpost:0, oldX1HATpost:0, -1 * oldX1HATpost:0 + oldX2HATpost:0) :|: 0 < -1 * oldX1HATpost:0 + oldX2HATpost:0 && oldX1HATpost: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:

   l0(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> l0(x8, x9)

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(8)
Obligation:
Rules:
l0(oldX1HATpost:0, oldX2HATpost:0) -> l0(oldX1HATpost:0, -1 * oldX1HATpost:0 + oldX2HATpost:0) :|: 0 < -1 * oldX1HATpost:0 + oldX2HATpost:0 && oldX1HATpost:0 > 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.
----------------------------------------

(10)
Obligation:
Rules:
l0(oldX1HATpost:0, oldX2HATpost:0) -> l0(oldX1HATpost:0, c) :|: c = -1 * oldX1HATpost:0 + oldX2HATpost:0 && (0 < -1 * oldX1HATpost:0 + oldX2HATpost:0 && oldX1HATpost:0 > 0)

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

The following rules are decreasing:
l0(oldX1HATpost:0, oldX2HATpost:0) -> l0(oldX1HATpost:0, c) :|: c = -1 * oldX1HATpost:0 + oldX2HATpost:0 && (0 < -1 * oldX1HATpost:0 + oldX2HATpost:0 && oldX1HATpost:0 > 0)
The following rules are bounded:
l0(oldX1HATpost:0, oldX2HATpost:0) -> l0(oldX1HATpost:0, c) :|: c = -1 * oldX1HATpost:0 + oldX2HATpost:0 && (0 < -1 * oldX1HATpost:0 + oldX2HATpost:0 && oldX1HATpost:0 > 0)

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

(12)
YES