YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 831 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 65 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 48 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (12) IntTRS (13) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Rules: l0(nd_12HAT0, rt_11HAT0, rv_15HAT0, st_14HAT0, st_16HAT0, x_13HAT0, y_17HAT0) -> l1(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 l1(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 && x = x7 && x8 = x3 && x5 <= 0 l1(x14, x15, x16, x17, x18, x19, x20) -> l3(x21, x22, x23, x24, x25, x26, x27) :|: 1 <= x19 && x28 = x28 && x23 = x28 && x21 = x21 && 0 <= x23 && x23 <= 0 && x27 = -1 + x20 && x25 = x25 && 2 <= x27 && x15 = x22 && x17 = x24 && x19 = x26 l3(x29, x30, x31, x32, x33, x34, x35) -> l1(x36, x37, x38, x39, x40, x41, x42) :|: x35 = x42 && x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 l1(x43, x44, x45, x46, x47, x48, x49) -> l5(x50, x51, x52, x53, x54, x55, x56) :|: 1 <= x48 && x57 = x57 && x52 = x57 && x50 = x50 && x44 = x51 && x46 = x53 && x47 = x54 && x48 = x55 && x49 = x56 l5(x58, x59, x60, x61, x62, x63, x64) -> l6(x65, x66, x67, x68, x69, x70, x71) :|: x64 = x71 && x63 = x70 && x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && 1 <= x60 l5(x72, x73, x74, x75, x76, x77, x78) -> l6(x79, x80, x81, x82, x83, x84, x85) :|: x78 = x85 && x77 = x84 && x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && 1 + x74 <= 0 l6(x86, x87, x88, x89, x90, x91, x92) -> l4(x93, x94, x95, x96, x97, x98, x99) :|: x98 = -1 + x91 && x100 = x100 && x99 = x100 && x93 = x93 && x87 = x94 && x88 = x95 && x89 = x96 && x90 = x97 l4(x101, x102, x103, x104, x105, x106, x107) -> l1(x108, x109, x110, x111, x112, x113, x114) :|: x107 = x114 && x106 = x113 && x105 = x112 && x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 l7(x115, x116, x117, x118, x119, x120, x121) -> l0(x122, x123, x124, x125, x126, x127, x128) :|: x121 = x128 && x120 = x127 && x119 = x126 && x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 Start term: l7(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(nd_12HAT0, rt_11HAT0, rv_15HAT0, st_14HAT0, st_16HAT0, x_13HAT0, y_17HAT0) -> l1(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 l1(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 && x = x7 && x8 = x3 && x5 <= 0 l1(x14, x15, x16, x17, x18, x19, x20) -> l3(x21, x22, x23, x24, x25, x26, x27) :|: 1 <= x19 && x28 = x28 && x23 = x28 && x21 = x21 && 0 <= x23 && x23 <= 0 && x27 = -1 + x20 && x25 = x25 && 2 <= x27 && x15 = x22 && x17 = x24 && x19 = x26 l3(x29, x30, x31, x32, x33, x34, x35) -> l1(x36, x37, x38, x39, x40, x41, x42) :|: x35 = x42 && x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 l1(x43, x44, x45, x46, x47, x48, x49) -> l5(x50, x51, x52, x53, x54, x55, x56) :|: 1 <= x48 && x57 = x57 && x52 = x57 && x50 = x50 && x44 = x51 && x46 = x53 && x47 = x54 && x48 = x55 && x49 = x56 l5(x58, x59, x60, x61, x62, x63, x64) -> l6(x65, x66, x67, x68, x69, x70, x71) :|: x64 = x71 && x63 = x70 && x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && 1 <= x60 l5(x72, x73, x74, x75, x76, x77, x78) -> l6(x79, x80, x81, x82, x83, x84, x85) :|: x78 = x85 && x77 = x84 && x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && 1 + x74 <= 0 l6(x86, x87, x88, x89, x90, x91, x92) -> l4(x93, x94, x95, x96, x97, x98, x99) :|: x98 = -1 + x91 && x100 = x100 && x99 = x100 && x93 = x93 && x87 = x94 && x88 = x95 && x89 = x96 && x90 = x97 l4(x101, x102, x103, x104, x105, x106, x107) -> l1(x108, x109, x110, x111, x112, x113, x114) :|: x107 = x114 && x106 = x113 && x105 = x112 && x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 l7(x115, x116, x117, x118, x119, x120, x121) -> l0(x122, x123, x124, x125, x126, x127, x128) :|: x121 = x128 && x120 = x127 && x119 = x126 && x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 Start term: l7(nd_12HAT0, rt_11HAT0, rv_15HAT0, st_14HAT0, st_16HAT0, x_13HAT0, y_17HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(nd_12HAT0, rt_11HAT0, rv_15HAT0, st_14HAT0, st_16HAT0, x_13HAT0, y_17HAT0) -> l1(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 (2) l1(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 && x = x7 && x8 = x3 && x5 <= 0 (3) l1(x14, x15, x16, x17, x18, x19, x20) -> l3(x21, x22, x23, x24, x25, x26, x27) :|: 1 <= x19 && x28 = x28 && x23 = x28 && x21 = x21 && 0 <= x23 && x23 <= 0 && x27 = -1 + x20 && x25 = x25 && 2 <= x27 && x15 = x22 && x17 = x24 && x19 = x26 (4) l3(x29, x30, x31, x32, x33, x34, x35) -> l1(x36, x37, x38, x39, x40, x41, x42) :|: x35 = x42 && x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 (5) l1(x43, x44, x45, x46, x47, x48, x49) -> l5(x50, x51, x52, x53, x54, x55, x56) :|: 1 <= x48 && x57 = x57 && x52 = x57 && x50 = x50 && x44 = x51 && x46 = x53 && x47 = x54 && x48 = x55 && x49 = x56 (6) l5(x58, x59, x60, x61, x62, x63, x64) -> l6(x65, x66, x67, x68, x69, x70, x71) :|: x64 = x71 && x63 = x70 && x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && 1 <= x60 (7) l5(x72, x73, x74, x75, x76, x77, x78) -> l6(x79, x80, x81, x82, x83, x84, x85) :|: x78 = x85 && x77 = x84 && x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && 1 + x74 <= 0 (8) l6(x86, x87, x88, x89, x90, x91, x92) -> l4(x93, x94, x95, x96, x97, x98, x99) :|: x98 = -1 + x91 && x100 = x100 && x99 = x100 && x93 = x93 && x87 = x94 && x88 = x95 && x89 = x96 && x90 = x97 (9) l4(x101, x102, x103, x104, x105, x106, x107) -> l1(x108, x109, x110, x111, x112, x113, x114) :|: x107 = x114 && x106 = x113 && x105 = x112 && x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 (10) l7(x115, x116, x117, x118, x119, x120, x121) -> l0(x122, x123, x124, x125, x126, x127, x128) :|: x121 = x128 && x120 = x127 && x119 = x126 && x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 Arcs: (1) -> (2), (3), (5) (3) -> (4) (4) -> (2), (3), (5) (5) -> (6), (7) (6) -> (8) (7) -> (8) (8) -> (9) (9) -> (2), (3), (5) (10) -> (1) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l1(x14, x15, x16, x17, x18, x19, x20) -> l3(x21, x22, x23, x24, x25, x26, x27) :|: 1 <= x19 && x28 = x28 && x23 = x28 && x21 = x21 && 0 <= x23 && x23 <= 0 && x27 = -1 + x20 && x25 = x25 && 2 <= x27 && x15 = x22 && x17 = x24 && x19 = x26 (2) l4(x101, x102, x103, x104, x105, x106, x107) -> l1(x108, x109, x110, x111, x112, x113, x114) :|: x107 = x114 && x106 = x113 && x105 = x112 && x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 (3) l6(x86, x87, x88, x89, x90, x91, x92) -> l4(x93, x94, x95, x96, x97, x98, x99) :|: x98 = -1 + x91 && x100 = x100 && x99 = x100 && x93 = x93 && x87 = x94 && x88 = x95 && x89 = x96 && x90 = x97 (4) l5(x72, x73, x74, x75, x76, x77, x78) -> l6(x79, x80, x81, x82, x83, x84, x85) :|: x78 = x85 && x77 = x84 && x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && 1 + x74 <= 0 (5) l5(x58, x59, x60, x61, x62, x63, x64) -> l6(x65, x66, x67, x68, x69, x70, x71) :|: x64 = x71 && x63 = x70 && x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && 1 <= x60 (6) l1(x43, x44, x45, x46, x47, x48, x49) -> l5(x50, x51, x52, x53, x54, x55, x56) :|: 1 <= x48 && x57 = x57 && x52 = x57 && x50 = x50 && x44 = x51 && x46 = x53 && x47 = x54 && x48 = x55 && x49 = x56 (7) l3(x29, x30, x31, x32, x33, x34, x35) -> l1(x36, x37, x38, x39, x40, x41, x42) :|: x35 = x42 && x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 Arcs: (1) -> (7) (2) -> (1), (6) (3) -> (2) (4) -> (3) (5) -> (3) (6) -> (4), (5) (7) -> (1), (6) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l1(x14:0, x15:0, x16:0, x17:0, x18:0, x19:0, x20:0) -> l1(x21:0, x15:0, x23:0, x17:0, x25:0, x19:0, -1 + x20:0) :|: x23:0 < 1 && x20:0 > 2 && x19:0 > 0 && x23:0 > -1 l1(x43:0, x109:0, x45:0, x111:0, x112:0, x48:0, x49:0) -> l1(x108:0, x109:0, x110:0, x111:0, x112:0, -1 + x48:0, x100:0) :|: x110:0 < 0 && x48:0 > 0 l1(x, x1, x2, x3, x4, x5, x6) -> l1(x7, x1, x8, x3, x4, -1 + x5, x9) :|: x8 > 0 && x5 > 0 ---------------------------------------- (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) -> l1(x6, x7) ---------------------------------------- (8) Obligation: Rules: l1(x19:0, x20:0) -> l1(x19:0, -1 + x20:0) :|: x23:0 < 1 && x20:0 > 2 && x19:0 > 0 && x23:0 > -1 l1(x48:0, x49:0) -> l1(-1 + x48:0, x100:0) :|: x110:0 < 0 && x48:0 > 0 l1(x5, x6) -> l1(-1 + x5, x9) :|: x8 > 0 && x5 > 0 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l1(INTEGER, VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l1(x19:0, x20:0) -> l1(x19:0, c) :|: c = -1 + x20:0 && (x23:0 < 1 && x20:0 > 2 && x19:0 > 0 && x23:0 > -1) l1(x48:0, x49:0) -> l1(c1, x100:0) :|: c1 = -1 + x48:0 && (x110:0 < 0 && x48:0 > 0) l1(x5, x6) -> l1(c2, x9) :|: c2 = -1 + x5 && (x8 > 0 && x5 > 0) ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l1(x, x1)] = -1 + x The following rules are decreasing: l1(x48:0, x49:0) -> l1(c1, x100:0) :|: c1 = -1 + x48:0 && (x110:0 < 0 && x48:0 > 0) l1(x5, x6) -> l1(c2, x9) :|: c2 = -1 + x5 && (x8 > 0 && x5 > 0) The following rules are bounded: l1(x19:0, x20:0) -> l1(x19:0, c) :|: c = -1 + x20:0 && (x23:0 < 1 && x20:0 > 2 && x19:0 > 0 && x23:0 > -1) l1(x48:0, x49:0) -> l1(c1, x100:0) :|: c1 = -1 + x48:0 && (x110:0 < 0 && x48:0 > 0) l1(x5, x6) -> l1(c2, x9) :|: c2 = -1 + x5 && (x8 > 0 && x5 > 0) ---------------------------------------- (12) Obligation: Rules: l1(x19:0, x20:0) -> l1(x19:0, c) :|: c = -1 + x20:0 && (x23:0 < 1 && x20:0 > 2 && x19:0 > 0 && x23:0 > -1) ---------------------------------------- (13) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l1(x, x1)] = x1 The following rules are decreasing: l1(x19:0, x20:0) -> l1(x19:0, c) :|: c = -1 + x20:0 && (x23:0 < 1 && x20:0 > 2 && x19:0 > 0 && x23:0 > -1) The following rules are bounded: l1(x19:0, x20:0) -> l1(x19:0, c) :|: c = -1 + x20:0 && (x23:0 < 1 && x20:0 > 2 && x19:0 > 0 && x23:0 > -1) ---------------------------------------- (14) YES