YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 851 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 0 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 39 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 12 ms] (12) IntTRS (13) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Rules: l0(__disjvr_0HAT0, nd_12HAT0, rt_11HAT0, rv_15HAT0, st_14HAT0, st_16HAT0, x_13HAT0, y_17HAT0) -> l1(__disjvr_0HATpost, nd_12HATpost, rt_11HATpost, rv_15HATpost, st_14HATpost, st_16HATpost, x_13HATpost, y_17HATpost) :|: y_17HAT0 = y_17HATpost && x_13HAT0 = x_13HATpost && st_16HAT0 = st_16HATpost && st_14HAT0 = st_14HATpost && rv_15HAT0 = rv_15HATpost && rt_11HAT0 = rt_11HATpost && nd_12HAT0 = nd_12HATpost && __disjvr_0HAT0 = __disjvr_0HATpost l1(x, x1, x2, x3, x4, x5, x6, x7) -> l2(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x5 = x13 && x4 = x12 && x3 = x11 && x1 = x9 && x = x8 && x10 = x4 && x6 <= 0 l1(x16, x17, x18, x19, x20, x21, x22, x23) -> l3(x24, x25, x26, x27, x28, x29, x30, x31) :|: 1 <= x22 && x32 = x32 && x27 = x32 && x25 = x25 && 0 <= x27 && x27 <= 0 && x31 = -1 + x23 && x29 = x29 && 2 <= x31 && x16 = x24 && x18 = x26 && x20 = x28 && x22 = x30 l3(x33, x34, x35, x36, x37, x38, x39, x40) -> l1(x41, x42, x43, x44, x45, x46, x47, x48) :|: x40 = x48 && x39 = x47 && x38 = x46 && x37 = x45 && x36 = x44 && x35 = x43 && x34 = x42 && x33 = x41 l1(x49, x50, x51, x52, x53, x54, x55, x56) -> l5(x57, x58, x59, x60, x61, x62, x63, x64) :|: 1 <= x55 && x65 = x65 && x60 = x65 && x58 = x58 && x49 = x57 && x51 = x59 && x53 = x61 && x54 = x62 && x55 = x63 && x56 = x64 l5(x66, x67, x68, x69, x70, x71, x72, x73) -> l6(x74, x75, x76, x77, x78, x79, x80, x81) :|: x73 = x81 && x72 = x80 && x71 = x79 && x70 = x78 && x69 = x77 && x68 = x76 && x67 = x75 && x66 = x74 && x74 = x66 l6(x82, x83, x84, x85, x86, x87, x88, x89) -> l4(x90, x91, x92, x93, x94, x95, x96, x97) :|: x96 = -1 + x88 && x98 = x98 && x97 = x98 && x91 = x91 && x82 = x90 && x84 = x92 && x85 = x93 && x86 = x94 && x87 = x95 l4(x99, x100, x101, x102, x103, x104, x105, x106) -> l1(x107, x108, x109, x110, x111, x112, x113, x114) :|: x106 = x114 && x105 = x113 && x104 = x112 && x103 = x111 && x102 = x110 && x101 = x109 && x100 = x108 && x99 = x107 l7(x115, x116, x117, x118, x119, x120, x121, x122) -> l0(x123, x124, x125, x126, x127, x128, x129, x130) :|: x122 = x130 && x121 = x129 && x120 = x128 && x119 = x127 && x118 = x126 && x117 = x125 && x116 = x124 && x115 = x123 Start term: l7(__disjvr_0HAT0, nd_12HAT0, rt_11HAT0, rv_15HAT0, st_14HAT0, st_16HAT0, x_13HAT0, y_17HAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(__disjvr_0HAT0, nd_12HAT0, rt_11HAT0, rv_15HAT0, st_14HAT0, st_16HAT0, x_13HAT0, y_17HAT0) -> l1(__disjvr_0HATpost, nd_12HATpost, rt_11HATpost, rv_15HATpost, st_14HATpost, st_16HATpost, x_13HATpost, y_17HATpost) :|: y_17HAT0 = y_17HATpost && x_13HAT0 = x_13HATpost && st_16HAT0 = st_16HATpost && st_14HAT0 = st_14HATpost && rv_15HAT0 = rv_15HATpost && rt_11HAT0 = rt_11HATpost && nd_12HAT0 = nd_12HATpost && __disjvr_0HAT0 = __disjvr_0HATpost l1(x, x1, x2, x3, x4, x5, x6, x7) -> l2(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x5 = x13 && x4 = x12 && x3 = x11 && x1 = x9 && x = x8 && x10 = x4 && x6 <= 0 l1(x16, x17, x18, x19, x20, x21, x22, x23) -> l3(x24, x25, x26, x27, x28, x29, x30, x31) :|: 1 <= x22 && x32 = x32 && x27 = x32 && x25 = x25 && 0 <= x27 && x27 <= 0 && x31 = -1 + x23 && x29 = x29 && 2 <= x31 && x16 = x24 && x18 = x26 && x20 = x28 && x22 = x30 l3(x33, x34, x35, x36, x37, x38, x39, x40) -> l1(x41, x42, x43, x44, x45, x46, x47, x48) :|: x40 = x48 && x39 = x47 && x38 = x46 && x37 = x45 && x36 = x44 && x35 = x43 && x34 = x42 && x33 = x41 l1(x49, x50, x51, x52, x53, x54, x55, x56) -> l5(x57, x58, x59, x60, x61, x62, x63, x64) :|: 1 <= x55 && x65 = x65 && x60 = x65 && x58 = x58 && x49 = x57 && x51 = x59 && x53 = x61 && x54 = x62 && x55 = x63 && x56 = x64 l5(x66, x67, x68, x69, x70, x71, x72, x73) -> l6(x74, x75, x76, x77, x78, x79, x80, x81) :|: x73 = x81 && x72 = x80 && x71 = x79 && x70 = x78 && x69 = x77 && x68 = x76 && x67 = x75 && x66 = x74 && x74 = x66 l6(x82, x83, x84, x85, x86, x87, x88, x89) -> l4(x90, x91, x92, x93, x94, x95, x96, x97) :|: x96 = -1 + x88 && x98 = x98 && x97 = x98 && x91 = x91 && x82 = x90 && x84 = x92 && x85 = x93 && x86 = x94 && x87 = x95 l4(x99, x100, x101, x102, x103, x104, x105, x106) -> l1(x107, x108, x109, x110, x111, x112, x113, x114) :|: x106 = x114 && x105 = x113 && x104 = x112 && x103 = x111 && x102 = x110 && x101 = x109 && x100 = x108 && x99 = x107 l7(x115, x116, x117, x118, x119, x120, x121, x122) -> l0(x123, x124, x125, x126, x127, x128, x129, x130) :|: x122 = x130 && x121 = x129 && x120 = x128 && x119 = x127 && x118 = x126 && x117 = x125 && x116 = x124 && x115 = x123 Start term: l7(__disjvr_0HAT0, nd_12HAT0, rt_11HAT0, rv_15HAT0, st_14HAT0, st_16HAT0, x_13HAT0, y_17HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(__disjvr_0HAT0, nd_12HAT0, rt_11HAT0, rv_15HAT0, st_14HAT0, st_16HAT0, x_13HAT0, y_17HAT0) -> l1(__disjvr_0HATpost, nd_12HATpost, rt_11HATpost, rv_15HATpost, st_14HATpost, st_16HATpost, x_13HATpost, y_17HATpost) :|: y_17HAT0 = y_17HATpost && x_13HAT0 = x_13HATpost && st_16HAT0 = st_16HATpost && st_14HAT0 = st_14HATpost && rv_15HAT0 = rv_15HATpost && rt_11HAT0 = rt_11HATpost && nd_12HAT0 = nd_12HATpost && __disjvr_0HAT0 = __disjvr_0HATpost (2) l1(x, x1, x2, x3, x4, x5, x6, x7) -> l2(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x5 = x13 && x4 = x12 && x3 = x11 && x1 = x9 && x = x8 && x10 = x4 && x6 <= 0 (3) l1(x16, x17, x18, x19, x20, x21, x22, x23) -> l3(x24, x25, x26, x27, x28, x29, x30, x31) :|: 1 <= x22 && x32 = x32 && x27 = x32 && x25 = x25 && 0 <= x27 && x27 <= 0 && x31 = -1 + x23 && x29 = x29 && 2 <= x31 && x16 = x24 && x18 = x26 && x20 = x28 && x22 = x30 (4) l3(x33, x34, x35, x36, x37, x38, x39, x40) -> l1(x41, x42, x43, x44, x45, x46, x47, x48) :|: x40 = x48 && x39 = x47 && x38 = x46 && x37 = x45 && x36 = x44 && x35 = x43 && x34 = x42 && x33 = x41 (5) l1(x49, x50, x51, x52, x53, x54, x55, x56) -> l5(x57, x58, x59, x60, x61, x62, x63, x64) :|: 1 <= x55 && x65 = x65 && x60 = x65 && x58 = x58 && x49 = x57 && x51 = x59 && x53 = x61 && x54 = x62 && x55 = x63 && x56 = x64 (6) l5(x66, x67, x68, x69, x70, x71, x72, x73) -> l6(x74, x75, x76, x77, x78, x79, x80, x81) :|: x73 = x81 && x72 = x80 && x71 = x79 && x70 = x78 && x69 = x77 && x68 = x76 && x67 = x75 && x66 = x74 && x74 = x66 (7) l6(x82, x83, x84, x85, x86, x87, x88, x89) -> l4(x90, x91, x92, x93, x94, x95, x96, x97) :|: x96 = -1 + x88 && x98 = x98 && x97 = x98 && x91 = x91 && x82 = x90 && x84 = x92 && x85 = x93 && x86 = x94 && x87 = x95 (8) l4(x99, x100, x101, x102, x103, x104, x105, x106) -> l1(x107, x108, x109, x110, x111, x112, x113, x114) :|: x106 = x114 && x105 = x113 && x104 = x112 && x103 = x111 && x102 = x110 && x101 = x109 && x100 = x108 && x99 = x107 (9) l7(x115, x116, x117, x118, x119, x120, x121, x122) -> l0(x123, x124, x125, x126, x127, x128, x129, x130) :|: x122 = x130 && x121 = x129 && x120 = x128 && x119 = x127 && x118 = x126 && x117 = x125 && x116 = x124 && x115 = x123 Arcs: (1) -> (2), (3), (5) (3) -> (4) (4) -> (2), (3), (5) (5) -> (6) (6) -> (7) (7) -> (8) (8) -> (2), (3), (5) (9) -> (1) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l1(x16, x17, x18, x19, x20, x21, x22, x23) -> l3(x24, x25, x26, x27, x28, x29, x30, x31) :|: 1 <= x22 && x32 = x32 && x27 = x32 && x25 = x25 && 0 <= x27 && x27 <= 0 && x31 = -1 + x23 && x29 = x29 && 2 <= x31 && x16 = x24 && x18 = x26 && x20 = x28 && x22 = x30 (2) l4(x99, x100, x101, x102, x103, x104, x105, x106) -> l1(x107, x108, x109, x110, x111, x112, x113, x114) :|: x106 = x114 && x105 = x113 && x104 = x112 && x103 = x111 && x102 = x110 && x101 = x109 && x100 = x108 && x99 = x107 (3) l6(x82, x83, x84, x85, x86, x87, x88, x89) -> l4(x90, x91, x92, x93, x94, x95, x96, x97) :|: x96 = -1 + x88 && x98 = x98 && x97 = x98 && x91 = x91 && x82 = x90 && x84 = x92 && x85 = x93 && x86 = x94 && x87 = x95 (4) l5(x66, x67, x68, x69, x70, x71, x72, x73) -> l6(x74, x75, x76, x77, x78, x79, x80, x81) :|: x73 = x81 && x72 = x80 && x71 = x79 && x70 = x78 && x69 = x77 && x68 = x76 && x67 = x75 && x66 = x74 && x74 = x66 (5) l1(x49, x50, x51, x52, x53, x54, x55, x56) -> l5(x57, x58, x59, x60, x61, x62, x63, x64) :|: 1 <= x55 && x65 = x65 && x60 = x65 && x58 = x58 && x49 = x57 && x51 = x59 && x53 = x61 && x54 = x62 && x55 = x63 && x56 = x64 (6) l3(x33, x34, x35, x36, x37, x38, x39, x40) -> l1(x41, x42, x43, x44, x45, x46, x47, x48) :|: x40 = x48 && x39 = x47 && x38 = x46 && x37 = x45 && x36 = x44 && x35 = x43 && x34 = x42 && x33 = x41 Arcs: (1) -> (6) (2) -> (1), (5) (3) -> (2) (4) -> (3) (5) -> (4) (6) -> (1), (5) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l1(x107:0, x50:0, x109:0, x52:0, x111:0, x112:0, x55:0, x56:0) -> l1(x107:0, x108:0, x109:0, x110:0, x111:0, x112:0, -1 + x55:0, x114:0) :|: x55:0 > 0 l1(x16:0, x17:0, x18:0, x19:0, x20:0, x21:0, x22:0, x23:0) -> l1(x16:0, x25:0, x18:0, x27:0, x20:0, x29:0, x22:0, -1 + x23:0) :|: x27:0 < 1 && x23:0 > 2 && x22:0 > 0 && x27:0 > -1 ---------------------------------------- (7) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l1(x1, x2, x3, x4, x5, x6, x7, x8) -> l1(x7, x8) ---------------------------------------- (8) Obligation: Rules: l1(x55:0, x56:0) -> l1(-1 + x55:0, x114:0) :|: x55:0 > 0 l1(x22:0, x23:0) -> l1(x22:0, -1 + x23:0) :|: x27:0 < 1 && x23:0 > 2 && x22:0 > 0 && x27:0 > -1 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l1(INTEGER, VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l1(x55:0, x56:0) -> l1(c, x114:0) :|: c = -1 + x55:0 && x55:0 > 0 l1(x22:0, x23:0) -> l1(x22:0, c1) :|: c1 = -1 + x23:0 && (x27:0 < 1 && x23:0 > 2 && x22:0 > 0 && x27:0 > -1) ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l1(x, x1)] = -1 + x The following rules are decreasing: l1(x55:0, x56:0) -> l1(c, x114:0) :|: c = -1 + x55:0 && x55:0 > 0 The following rules are bounded: l1(x55:0, x56:0) -> l1(c, x114:0) :|: c = -1 + x55:0 && x55:0 > 0 l1(x22:0, x23:0) -> l1(x22:0, c1) :|: c1 = -1 + x23:0 && (x27:0 < 1 && x23:0 > 2 && x22:0 > 0 && x27:0 > -1) ---------------------------------------- (12) Obligation: Rules: l1(x22:0, x23:0) -> l1(x22:0, c1) :|: c1 = -1 + x23:0 && (x27:0 < 1 && x23:0 > 2 && x22:0 > 0 && x27:0 > -1) ---------------------------------------- (13) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l1(x, x1)] = x1 The following rules are decreasing: l1(x22:0, x23:0) -> l1(x22:0, c1) :|: c1 = -1 + x23:0 && (x27:0 < 1 && x23:0 > 2 && x22:0 > 0 && x27:0 > -1) The following rules are bounded: l1(x22:0, x23:0) -> l1(x22:0, c1) :|: c1 = -1 + x23:0 && (x27:0 < 1 && x23:0 > 2 && x22:0 > 0 && x27:0 > -1) ---------------------------------------- (14) YES