YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 4583 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 48 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 13 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 0 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) :|: 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 && x17 = x12 && x16 = x11 && x15 = x9 && x12 = x12 && x11 = x8 && x10 = x7 && x9 = x6 l2(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l4(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x23 = x32 && x35 = x31 && x34 = x30 && x33 = x29 && x31 = x31 && x30 = x30 && x29 = x26 && x28 = x25 && x27 = x24 l3(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 = -1 + x46 && x52 = x46 && x51 = x45 && -1 + x46 <= 0 && x47 = x44 && x46 = x43 && x45 = x42 l3(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l2(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x59 = x68 && x58 = x67 && x57 = x66 && x71 = -1 + x64 && x70 = x64 && x69 = x63 && 1 <= -1 + x64 && x65 = x62 && x64 = x61 && x63 = x60 l5(x72, x73, x74, x75, x76, x77, x78, x79, x80) -> l3(x81, x82, x83, x84, x85, x86, x87, x88, x89) :|: x77 = x86 && x76 = x85 && x89 = x84 && x88 = x81 && x87 = x81 && x84 = x84 && x83 = x80 && x82 = x79 && x81 = x78 l4(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l5(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x95 = x104 && x107 = x103 && x106 = x102 && x105 = x99 && x103 = x103 && x102 = x102 && x101 = x98 && x100 = x97 && x99 = x96 l6(x108, x109, x110, x111, x112, x113, x114, x115, x116) -> l7(x117, x118, x119, x120, x121, x122, x123, x124, x125) :|: x125 = x122 && x124 = x121 && x123 = x120 && x122 = x122 && x121 = x121 && x120 = x120 && x119 = x116 && x118 = x115 && x117 = x114 l6(x126, x127, x128, x129, x130, x131, x132, x133, x134) -> l7(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 l6(x144, x145, x146, x147, x148, x149, x150, x151, x152) -> l1(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 l6(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 l6(x180, x181, x182, x183, x184, x185, x186, x187, x188) -> l2(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 l6(x198, x199, x200, x201, x202, x203, x204, x205, x206) -> l3(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 l6(x216, x217, x218, x219, x220, x221, x222, x223, x224) -> l5(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) -> l4(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 l8(x252, x253, x254, x255, x256, x257, x258, x259, x260) -> l6(x261, x262, x263, x264, x265, x266, x267, x268, x269) :|: x260 = x269 && x259 = x268 && x258 = x267 && x257 = x266 && x256 = x265 && x255 = x264 && x254 = x263 && x253 = x262 && x252 = x261 Start term: l8(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) :|: 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 && x17 = x12 && x16 = x11 && x15 = x9 && x12 = x12 && x11 = x8 && x10 = x7 && x9 = x6 l2(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l4(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x23 = x32 && x35 = x31 && x34 = x30 && x33 = x29 && x31 = x31 && x30 = x30 && x29 = x26 && x28 = x25 && x27 = x24 l3(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 = -1 + x46 && x52 = x46 && x51 = x45 && -1 + x46 <= 0 && x47 = x44 && x46 = x43 && x45 = x42 l3(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l2(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x59 = x68 && x58 = x67 && x57 = x66 && x71 = -1 + x64 && x70 = x64 && x69 = x63 && 1 <= -1 + x64 && x65 = x62 && x64 = x61 && x63 = x60 l5(x72, x73, x74, x75, x76, x77, x78, x79, x80) -> l3(x81, x82, x83, x84, x85, x86, x87, x88, x89) :|: x77 = x86 && x76 = x85 && x89 = x84 && x88 = x81 && x87 = x81 && x84 = x84 && x83 = x80 && x82 = x79 && x81 = x78 l4(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l5(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x95 = x104 && x107 = x103 && x106 = x102 && x105 = x99 && x103 = x103 && x102 = x102 && x101 = x98 && x100 = x97 && x99 = x96 l6(x108, x109, x110, x111, x112, x113, x114, x115, x116) -> l7(x117, x118, x119, x120, x121, x122, x123, x124, x125) :|: x125 = x122 && x124 = x121 && x123 = x120 && x122 = x122 && x121 = x121 && x120 = x120 && x119 = x116 && x118 = x115 && x117 = x114 l6(x126, x127, x128, x129, x130, x131, x132, x133, x134) -> l7(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 l6(x144, x145, x146, x147, x148, x149, x150, x151, x152) -> l1(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 l6(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 l6(x180, x181, x182, x183, x184, x185, x186, x187, x188) -> l2(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 l6(x198, x199, x200, x201, x202, x203, x204, x205, x206) -> l3(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 l6(x216, x217, x218, x219, x220, x221, x222, x223, x224) -> l5(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) -> l4(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 l8(x252, x253, x254, x255, x256, x257, x258, x259, x260) -> l6(x261, x262, x263, x264, x265, x266, x267, x268, x269) :|: x260 = x269 && x259 = x268 && x258 = x267 && x257 = x266 && x256 = x265 && x255 = x264 && x254 = x263 && x253 = x262 && x252 = x261 Start term: l8(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, oldX4HAT0, oldX5HAT0, x0HAT0, x1HAT0, x2HAT0) ---------------------------------------- (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 && x17 = x12 && x16 = x11 && x15 = x9 && x12 = x12 && x11 = x8 && x10 = x7 && x9 = x6 (3) l2(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l4(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x23 = x32 && x35 = x31 && x34 = x30 && x33 = x29 && x31 = x31 && x30 = x30 && x29 = x26 && x28 = x25 && x27 = x24 (4) l3(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 = -1 + x46 && x52 = x46 && x51 = x45 && -1 + x46 <= 0 && x47 = x44 && x46 = x43 && x45 = x42 (5) l3(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l2(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x59 = x68 && x58 = x67 && x57 = x66 && x71 = -1 + x64 && x70 = x64 && x69 = x63 && 1 <= -1 + x64 && x65 = x62 && x64 = x61 && x63 = x60 (6) l5(x72, x73, x74, x75, x76, x77, x78, x79, x80) -> l3(x81, x82, x83, x84, x85, x86, x87, x88, x89) :|: x77 = x86 && x76 = x85 && x89 = x84 && x88 = x81 && x87 = x81 && x84 = x84 && x83 = x80 && x82 = x79 && x81 = x78 (7) l4(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l5(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x95 = x104 && x107 = x103 && x106 = x102 && x105 = x99 && x103 = x103 && x102 = x102 && x101 = x98 && x100 = x97 && x99 = x96 (8) l6(x108, x109, x110, x111, x112, x113, x114, x115, x116) -> l7(x117, x118, x119, x120, x121, x122, x123, x124, x125) :|: x125 = x122 && x124 = x121 && x123 = x120 && x122 = x122 && x121 = x121 && x120 = x120 && x119 = x116 && x118 = x115 && x117 = x114 (9) l6(x126, x127, x128, x129, x130, x131, x132, x133, x134) -> l7(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) l6(x144, x145, x146, x147, x148, x149, x150, x151, x152) -> l1(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) l6(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) l6(x180, x181, x182, x183, x184, x185, x186, x187, x188) -> l2(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) l6(x198, x199, x200, x201, x202, x203, x204, x205, x206) -> l3(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) l6(x216, x217, x218, x219, x220, x221, x222, x223, x224) -> l5(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) -> l4(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 (16) l8(x252, x253, x254, x255, x256, x257, x258, x259, x260) -> l6(x261, x262, x263, x264, x265, x266, x267, x268, x269) :|: x260 = x269 && x259 = x268 && x258 = x267 && x257 = x266 && x256 = x265 && x255 = x264 && x254 = x263 && x253 = x262 && x252 = x261 Arcs: (2) -> (4), (5) (3) -> (7) (4) -> (1) (5) -> (2), (3) (6) -> (4), (5) (7) -> (6) (11) -> (1) (12) -> (2), (3) (13) -> (4), (5) (14) -> (6) (15) -> (7) (16) -> (8), (9), (10), (11), (12), (13), (14), (15) This digraph is fully evaluated! ---------------------------------------- (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 && x17 = x12 && x16 = x11 && x15 = x9 && x12 = x12 && x11 = x8 && x10 = x7 && x9 = x6 (2) l3(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l2(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x59 = x68 && x58 = x67 && x57 = x66 && x71 = -1 + x64 && x70 = x64 && x69 = x63 && 1 <= -1 + x64 && x65 = x62 && x64 = x61 && x63 = x60 (3) l5(x72, x73, x74, x75, x76, x77, x78, x79, x80) -> l3(x81, x82, x83, x84, x85, x86, x87, x88, x89) :|: x77 = x86 && x76 = x85 && x89 = x84 && x88 = x81 && x87 = x81 && x84 = x84 && x83 = x80 && x82 = x79 && x81 = x78 (4) l4(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l5(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x95 = x104 && x107 = x103 && x106 = x102 && x105 = x99 && x103 = x103 && x102 = x102 && x101 = x98 && x100 = x97 && x99 = x96 (5) l2(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l4(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x23 = x32 && x35 = x31 && x34 = x30 && x33 = x29 && x31 = x31 && x30 = x30 && x29 = x26 && x28 = x25 && x27 = x24 Arcs: (1) -> (2) (2) -> (1), (5) (3) -> (2) (4) -> (3) (5) -> (4) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l2(x:0, x1:0, x2:0, x3:0, x13:0, x14:0, x15:0, x10:0, x11:0) -> l2(x15:0, x11:0, x12:0, x12:0, x13:0, x14:0, x15:0, x11:0, -1 + x11:0) :|: x11:0 > 1 l2(x18:0, x19:0, x20:0, x21:0, x22:0, x104:0, x24:0, x25:0, x105:0) -> l2(x105:0, x105:0, x65:0, x65:0, x103:0, x104:0, x105:0, x105:0, -1 + x105:0) :|: x105:0 > 1 ---------------------------------------- (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(x9) ---------------------------------------- (8) Obligation: Rules: l2(x11:0) -> l2(-1 + x11:0) :|: x11:0 > 1 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l2(INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l2(x11:0) -> l2(c) :|: c = -1 + x11:0 && x11:0 > 1 ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l2(x)] = x The following rules are decreasing: l2(x11:0) -> l2(c) :|: c = -1 + x11:0 && x11:0 > 1 The following rules are bounded: l2(x11:0) -> l2(c) :|: c = -1 + x11:0 && x11:0 > 1 ---------------------------------------- (12) YES