YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 3564 ms] (4) AND (5) IRSwT (6) IntTRSCompressionProof [EQUIVALENT, 0 ms] (7) IRSwT (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (9) IRSwT (10) TempFilterProof [SOUND, 27 ms] (11) IntTRS (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (13) YES (14) IRSwT (15) IntTRSCompressionProof [EQUIVALENT, 0 ms] (16) IRSwT (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (18) IRSwT (19) TempFilterProof [SOUND, 11 ms] (20) IntTRS (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (22) YES ---------------------------------------- (0) Obligation: Rules: l0(ilHAT0, ret_foo5HAT0, ret_foo7HAT0, tmpHAT0, tmp___0HAT0, x4HAT0, x6HAT0) -> l1(ilHATpost, ret_foo5HATpost, ret_foo7HATpost, tmpHATpost, tmp___0HATpost, x4HATpost, x6HATpost) :|: x6HAT0 = x6HATpost && x4HAT0 = x4HATpost && tmp___0HAT0 = tmp___0HATpost && tmpHAT0 = tmpHATpost && ret_foo7HAT0 = ret_foo7HATpost && ret_foo5HAT0 = ret_foo5HATpost && ilHAT0 = ilHATpost && 0 <= tmpHAT0 && tmpHAT0 <= 0 l0(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x3 = x10 && x2 = x9 && x1 = x8 && x = x7 && 1 <= x3 l0(x14, x15, x16, x17, x18, x19, x20) -> l2(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21 && 1 + x17 <= 0 l3(x28, x29, x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x32 = x39 && x30 = x37 && x28 = x35 && x38 = x36 && x36 = x36 && x40 = x28 l4(x42, x43, x44, x45, x46, x47, x48) -> l5(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x46 = x53 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49 l6(x56, x57, x58, x59, x60, x61, x62) -> l7(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x63 = 1 + x56 l8(x70, x71, x72, x73, x74, x75, x76) -> l4(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77 && 0 <= x74 && x74 <= 0 l8(x84, x85, x86, x87, x88, x89, x90) -> l6(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 && 1 <= x88 l8(x98, x99, x100, x101, x102, x103, x104) -> l6(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105 && 1 + x102 <= 0 l7(x112, x113, x114, x115, x116, x117, x118) -> l9(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x114 = x121 && x113 = x120 && x112 = x119 l9(x126, x127, x128, x129, x130, x131, x132) -> l4(x133, x134, x135, x136, x137, x138, x139) :|: x132 = x139 && x131 = x138 && x130 = x137 && x129 = x136 && x128 = x135 && x127 = x134 && x126 = x133 && 10 <= x126 l9(x140, x141, x142, x143, x144, x145, x146) -> l8(x147, x148, x149, x150, x151, x152, x153) :|: x145 = x152 && x143 = x150 && x141 = x148 && x140 = x147 && x151 = x149 && x149 = x149 && x153 = x140 && 1 + x140 <= 10 l1(x154, x155, x156, x157, x158, x159, x160) -> l7(x161, x162, x163, x164, x165, x166, x167) :|: x160 = x167 && x159 = x166 && x158 = x165 && x157 = x164 && x156 = x163 && x155 = x162 && x161 = 0 l2(x168, x169, x170, x171, x172, x173, x174) -> l1(x175, x176, x177, x178, x179, x180, x181) :|: x174 = x181 && x173 = x180 && x172 = x179 && x171 = x178 && x170 = x177 && x169 = x176 && x168 = x175 && 10 <= x168 l2(x182, x183, x184, x185, x186, x187, x188) -> l3(x189, x190, x191, x192, x193, x194, x195) :|: x188 = x195 && x187 = x194 && x186 = x193 && x185 = x192 && x184 = x191 && x183 = x190 && x189 = 1 + x182 && 1 + x182 <= 10 l10(x196, x197, x198, x199, x200, x201, x202) -> l3(x203, x204, x205, x206, x207, x208, x209) :|: x202 = x209 && x201 = x208 && x200 = x207 && x199 = x206 && x198 = x205 && x197 = x204 && x203 = 0 l11(x210, x211, x212, x213, x214, x215, x216) -> l10(x217, x218, x219, x220, x221, x222, x223) :|: x216 = x223 && x215 = x222 && x214 = x221 && x213 = x220 && x212 = x219 && x211 = x218 && x210 = x217 Start term: l11(ilHAT0, ret_foo5HAT0, ret_foo7HAT0, tmpHAT0, tmp___0HAT0, x4HAT0, x6HAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(ilHAT0, ret_foo5HAT0, ret_foo7HAT0, tmpHAT0, tmp___0HAT0, x4HAT0, x6HAT0) -> l1(ilHATpost, ret_foo5HATpost, ret_foo7HATpost, tmpHATpost, tmp___0HATpost, x4HATpost, x6HATpost) :|: x6HAT0 = x6HATpost && x4HAT0 = x4HATpost && tmp___0HAT0 = tmp___0HATpost && tmpHAT0 = tmpHATpost && ret_foo7HAT0 = ret_foo7HATpost && ret_foo5HAT0 = ret_foo5HATpost && ilHAT0 = ilHATpost && 0 <= tmpHAT0 && tmpHAT0 <= 0 l0(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x3 = x10 && x2 = x9 && x1 = x8 && x = x7 && 1 <= x3 l0(x14, x15, x16, x17, x18, x19, x20) -> l2(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21 && 1 + x17 <= 0 l3(x28, x29, x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x32 = x39 && x30 = x37 && x28 = x35 && x38 = x36 && x36 = x36 && x40 = x28 l4(x42, x43, x44, x45, x46, x47, x48) -> l5(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x46 = x53 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49 l6(x56, x57, x58, x59, x60, x61, x62) -> l7(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x63 = 1 + x56 l8(x70, x71, x72, x73, x74, x75, x76) -> l4(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77 && 0 <= x74 && x74 <= 0 l8(x84, x85, x86, x87, x88, x89, x90) -> l6(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 && 1 <= x88 l8(x98, x99, x100, x101, x102, x103, x104) -> l6(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105 && 1 + x102 <= 0 l7(x112, x113, x114, x115, x116, x117, x118) -> l9(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x114 = x121 && x113 = x120 && x112 = x119 l9(x126, x127, x128, x129, x130, x131, x132) -> l4(x133, x134, x135, x136, x137, x138, x139) :|: x132 = x139 && x131 = x138 && x130 = x137 && x129 = x136 && x128 = x135 && x127 = x134 && x126 = x133 && 10 <= x126 l9(x140, x141, x142, x143, x144, x145, x146) -> l8(x147, x148, x149, x150, x151, x152, x153) :|: x145 = x152 && x143 = x150 && x141 = x148 && x140 = x147 && x151 = x149 && x149 = x149 && x153 = x140 && 1 + x140 <= 10 l1(x154, x155, x156, x157, x158, x159, x160) -> l7(x161, x162, x163, x164, x165, x166, x167) :|: x160 = x167 && x159 = x166 && x158 = x165 && x157 = x164 && x156 = x163 && x155 = x162 && x161 = 0 l2(x168, x169, x170, x171, x172, x173, x174) -> l1(x175, x176, x177, x178, x179, x180, x181) :|: x174 = x181 && x173 = x180 && x172 = x179 && x171 = x178 && x170 = x177 && x169 = x176 && x168 = x175 && 10 <= x168 l2(x182, x183, x184, x185, x186, x187, x188) -> l3(x189, x190, x191, x192, x193, x194, x195) :|: x188 = x195 && x187 = x194 && x186 = x193 && x185 = x192 && x184 = x191 && x183 = x190 && x189 = 1 + x182 && 1 + x182 <= 10 l10(x196, x197, x198, x199, x200, x201, x202) -> l3(x203, x204, x205, x206, x207, x208, x209) :|: x202 = x209 && x201 = x208 && x200 = x207 && x199 = x206 && x198 = x205 && x197 = x204 && x203 = 0 l11(x210, x211, x212, x213, x214, x215, x216) -> l10(x217, x218, x219, x220, x221, x222, x223) :|: x216 = x223 && x215 = x222 && x214 = x221 && x213 = x220 && x212 = x219 && x211 = x218 && x210 = x217 Start term: l11(ilHAT0, ret_foo5HAT0, ret_foo7HAT0, tmpHAT0, tmp___0HAT0, x4HAT0, x6HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(ilHAT0, ret_foo5HAT0, ret_foo7HAT0, tmpHAT0, tmp___0HAT0, x4HAT0, x6HAT0) -> l1(ilHATpost, ret_foo5HATpost, ret_foo7HATpost, tmpHATpost, tmp___0HATpost, x4HATpost, x6HATpost) :|: x6HAT0 = x6HATpost && x4HAT0 = x4HATpost && tmp___0HAT0 = tmp___0HATpost && tmpHAT0 = tmpHATpost && ret_foo7HAT0 = ret_foo7HATpost && ret_foo5HAT0 = ret_foo5HATpost && ilHAT0 = ilHATpost && 0 <= tmpHAT0 && tmpHAT0 <= 0 (2) l0(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x3 = x10 && x2 = x9 && x1 = x8 && x = x7 && 1 <= x3 (3) l0(x14, x15, x16, x17, x18, x19, x20) -> l2(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21 && 1 + x17 <= 0 (4) l3(x28, x29, x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x32 = x39 && x30 = x37 && x28 = x35 && x38 = x36 && x36 = x36 && x40 = x28 (5) l4(x42, x43, x44, x45, x46, x47, x48) -> l5(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x46 = x53 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49 (6) l6(x56, x57, x58, x59, x60, x61, x62) -> l7(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x63 = 1 + x56 (7) l8(x70, x71, x72, x73, x74, x75, x76) -> l4(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77 && 0 <= x74 && x74 <= 0 (8) l8(x84, x85, x86, x87, x88, x89, x90) -> l6(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 && 1 <= x88 (9) l8(x98, x99, x100, x101, x102, x103, x104) -> l6(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105 && 1 + x102 <= 0 (10) l7(x112, x113, x114, x115, x116, x117, x118) -> l9(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x114 = x121 && x113 = x120 && x112 = x119 (11) l9(x126, x127, x128, x129, x130, x131, x132) -> l4(x133, x134, x135, x136, x137, x138, x139) :|: x132 = x139 && x131 = x138 && x130 = x137 && x129 = x136 && x128 = x135 && x127 = x134 && x126 = x133 && 10 <= x126 (12) l9(x140, x141, x142, x143, x144, x145, x146) -> l8(x147, x148, x149, x150, x151, x152, x153) :|: x145 = x152 && x143 = x150 && x141 = x148 && x140 = x147 && x151 = x149 && x149 = x149 && x153 = x140 && 1 + x140 <= 10 (13) l1(x154, x155, x156, x157, x158, x159, x160) -> l7(x161, x162, x163, x164, x165, x166, x167) :|: x160 = x167 && x159 = x166 && x158 = x165 && x157 = x164 && x156 = x163 && x155 = x162 && x161 = 0 (14) l2(x168, x169, x170, x171, x172, x173, x174) -> l1(x175, x176, x177, x178, x179, x180, x181) :|: x174 = x181 && x173 = x180 && x172 = x179 && x171 = x178 && x170 = x177 && x169 = x176 && x168 = x175 && 10 <= x168 (15) l2(x182, x183, x184, x185, x186, x187, x188) -> l3(x189, x190, x191, x192, x193, x194, x195) :|: x188 = x195 && x187 = x194 && x186 = x193 && x185 = x192 && x184 = x191 && x183 = x190 && x189 = 1 + x182 && 1 + x182 <= 10 (16) l10(x196, x197, x198, x199, x200, x201, x202) -> l3(x203, x204, x205, x206, x207, x208, x209) :|: x202 = x209 && x201 = x208 && x200 = x207 && x199 = x206 && x198 = x205 && x197 = x204 && x203 = 0 (17) l11(x210, x211, x212, x213, x214, x215, x216) -> l10(x217, x218, x219, x220, x221, x222, x223) :|: x216 = x223 && x215 = x222 && x214 = x221 && x213 = x220 && x212 = x219 && x211 = x218 && x210 = x217 Arcs: (1) -> (13) (2) -> (14), (15) (3) -> (14), (15) (4) -> (1), (2), (3) (6) -> (10) (7) -> (5) (8) -> (6) (9) -> (6) (10) -> (11), (12) (11) -> (5) (12) -> (7), (8), (9) (13) -> (10) (14) -> (13) (15) -> (4) (16) -> (4) (17) -> (16) This digraph is fully evaluated! ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Termination digraph: Nodes: (1) l0(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x3 = x10 && x2 = x9 && x1 = x8 && x = x7 && 1 <= x3 (2) l3(x28, x29, x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x32 = x39 && x30 = x37 && x28 = x35 && x38 = x36 && x36 = x36 && x40 = x28 (3) l2(x182, x183, x184, x185, x186, x187, x188) -> l3(x189, x190, x191, x192, x193, x194, x195) :|: x188 = x195 && x187 = x194 && x186 = x193 && x185 = x192 && x184 = x191 && x183 = x190 && x189 = 1 + x182 && 1 + x182 <= 10 (4) l0(x14, x15, x16, x17, x18, x19, x20) -> l2(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21 && 1 + x17 <= 0 Arcs: (1) -> (3) (2) -> (1), (4) (3) -> (2) (4) -> (3) This digraph is fully evaluated! ---------------------------------------- (6) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (7) Obligation: Rules: l3(x194:0, x29:0, x191:0, x31:0, x193:0, x33:0, x195:0) -> l3(1 + x194:0, x190:0, x191:0, x190:0, x193:0, x194:0, x195:0) :|: x190:0 < 0 && x194:0 < 10 l3(x, x1, x2, x3, x4, x5, x6) -> l3(1 + x, x7, x2, x7, x4, x, x6) :|: x7 > 0 && x < 10 ---------------------------------------- (8) 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, x7) -> l3(x1) ---------------------------------------- (9) Obligation: Rules: l3(x194:0) -> l3(1 + x194:0) :|: x190:0 < 0 && x194:0 < 10 l3(x) -> l3(1 + x) :|: x7 > 0 && x < 10 ---------------------------------------- (10) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l3(INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (11) Obligation: Rules: l3(x194:0) -> l3(c) :|: c = 1 + x194:0 && (x190:0 < 0 && x194:0 < 10) l3(x) -> l3(c1) :|: c1 = 1 + x && (x7 > 0 && x < 10) ---------------------------------------- (12) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l3(x)] = 9 - x The following rules are decreasing: l3(x194:0) -> l3(c) :|: c = 1 + x194:0 && (x190:0 < 0 && x194:0 < 10) l3(x) -> l3(c1) :|: c1 = 1 + x && (x7 > 0 && x < 10) The following rules are bounded: l3(x194:0) -> l3(c) :|: c = 1 + x194:0 && (x190:0 < 0 && x194:0 < 10) l3(x) -> l3(c1) :|: c1 = 1 + x && (x7 > 0 && x < 10) ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Termination digraph: Nodes: (1) l7(x112, x113, x114, x115, x116, x117, x118) -> l9(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x114 = x121 && x113 = x120 && x112 = x119 (2) l6(x56, x57, x58, x59, x60, x61, x62) -> l7(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x63 = 1 + x56 (3) l8(x98, x99, x100, x101, x102, x103, x104) -> l6(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105 && 1 + x102 <= 0 (4) l8(x84, x85, x86, x87, x88, x89, x90) -> l6(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 && 1 <= x88 (5) l9(x140, x141, x142, x143, x144, x145, x146) -> l8(x147, x148, x149, x150, x151, x152, x153) :|: x145 = x152 && x143 = x150 && x141 = x148 && x140 = x147 && x151 = x149 && x149 = x149 && x153 = x140 && 1 + x140 <= 10 Arcs: (1) -> (5) (2) -> (1) (3) -> (2) (4) -> (2) (5) -> (3), (4) This digraph is fully evaluated! ---------------------------------------- (15) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (16) Obligation: Rules: l8(x84:0, x120:0, x121:0, x122:0, x123:0, x124:0, x125:0) -> l8(1 + x84:0, x120:0, x149:0, x122:0, x149:0, x124:0, 1 + x84:0) :|: x123:0 > 0 && x84:0 < 9 l8(x, x1, x2, x3, x4, x5, x6) -> l8(1 + x, x1, x7, x3, x7, x5, 1 + x) :|: x4 < 0 && x < 9 ---------------------------------------- (17) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l8(x1, x2, x3, x4, x5, x6, x7) -> l8(x1, x5) ---------------------------------------- (18) Obligation: Rules: l8(x84:0, x123:0) -> l8(1 + x84:0, x149:0) :|: x123:0 > 0 && x84:0 < 9 l8(x, x4) -> l8(1 + x, x7) :|: x4 < 0 && x < 9 ---------------------------------------- (19) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l8(INTEGER, VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (20) Obligation: Rules: l8(x84:0, x123:0) -> l8(c, x149:0) :|: c = 1 + x84:0 && (x123:0 > 0 && x84:0 < 9) l8(x, x4) -> l8(c1, x7) :|: c1 = 1 + x && (x4 < 0 && x < 9) ---------------------------------------- (21) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l8(x, x1)] = 8 - x The following rules are decreasing: l8(x84:0, x123:0) -> l8(c, x149:0) :|: c = 1 + x84:0 && (x123:0 > 0 && x84:0 < 9) l8(x, x4) -> l8(c1, x7) :|: c1 = 1 + x && (x4 < 0 && x < 9) The following rules are bounded: l8(x84:0, x123:0) -> l8(c, x149:0) :|: c = 1 + x84:0 && (x123:0 > 0 && x84:0 < 9) l8(x, x4) -> l8(c1, x7) :|: c1 = 1 + x && (x4 < 0 && x < 9) ---------------------------------------- (22) YES