YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 451 ms] (4) AND (5) IRSwT (6) IntTRSCompressionProof [EQUIVALENT, 17 ms] (7) IRSwT (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (9) IRSwT (10) TempFilterProof [SOUND, 8 ms] (11) IntTRS (12) RankingReductionPairProof [EQUIVALENT, 0 ms] (13) YES (14) IRSwT (15) IntTRSCompressionProof [EQUIVALENT, 0 ms] (16) IRSwT (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (18) IRSwT (19) TempFilterProof [SOUND, 16 ms] (20) IntTRS (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (22) YES ---------------------------------------- (0) Obligation: Rules: l0(__const_100HAT0, __const_200HAT0, iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0) -> l1(__const_100HATpost, __const_200HATpost, iHATpost, jHATpost, y4HATpost, y6HATpost, y8HATpost) :|: y8HAT0 = y8HATpost && y6HAT0 = y6HATpost && y4HAT0 = y4HATpost && iHAT0 = iHATpost && __const_200HAT0 = __const_200HATpost && __const_100HAT0 = __const_100HATpost && jHATpost = __const_100HAT0 && __const_100HAT0 <= iHAT0 l0(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x3 = x10 && x1 = x8 && x = x7 && x9 = 1 + x2 && x12 = x2 && x11 = x2 && 1 + x2 <= x l2(x14, x15, x16, x17, x18, x19, x20) -> l0(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21 l3(x28, x29, x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35 && x29 <= x31 l3(x42, x43, x44, x45, x46, x47, x48) -> l1(x49, x50, x51, x52, x53, x54, x55) :|: x47 = x54 && x46 = x53 && x44 = x51 && x43 = x50 && x42 = x49 && x52 = 1 + x45 && x55 = x45 && 1 + x45 <= x43 l1(x56, x57, x58, x59, x60, x61, x62) -> l3(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63 l5(x70, x71, x72, x73, x74, x75, x76) -> l2(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x71 = x78 && x70 = x77 && x79 = 0 l6(x84, x85, x86, x87, x88, x89, x90) -> l5(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 Start term: l6(__const_100HAT0, __const_200HAT0, iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(__const_100HAT0, __const_200HAT0, iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0) -> l1(__const_100HATpost, __const_200HATpost, iHATpost, jHATpost, y4HATpost, y6HATpost, y8HATpost) :|: y8HAT0 = y8HATpost && y6HAT0 = y6HATpost && y4HAT0 = y4HATpost && iHAT0 = iHATpost && __const_200HAT0 = __const_200HATpost && __const_100HAT0 = __const_100HATpost && jHATpost = __const_100HAT0 && __const_100HAT0 <= iHAT0 l0(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x3 = x10 && x1 = x8 && x = x7 && x9 = 1 + x2 && x12 = x2 && x11 = x2 && 1 + x2 <= x l2(x14, x15, x16, x17, x18, x19, x20) -> l0(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21 l3(x28, x29, x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35 && x29 <= x31 l3(x42, x43, x44, x45, x46, x47, x48) -> l1(x49, x50, x51, x52, x53, x54, x55) :|: x47 = x54 && x46 = x53 && x44 = x51 && x43 = x50 && x42 = x49 && x52 = 1 + x45 && x55 = x45 && 1 + x45 <= x43 l1(x56, x57, x58, x59, x60, x61, x62) -> l3(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63 l5(x70, x71, x72, x73, x74, x75, x76) -> l2(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x71 = x78 && x70 = x77 && x79 = 0 l6(x84, x85, x86, x87, x88, x89, x90) -> l5(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 Start term: l6(__const_100HAT0, __const_200HAT0, iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(__const_100HAT0, __const_200HAT0, iHAT0, jHAT0, y4HAT0, y6HAT0, y8HAT0) -> l1(__const_100HATpost, __const_200HATpost, iHATpost, jHATpost, y4HATpost, y6HATpost, y8HATpost) :|: y8HAT0 = y8HATpost && y6HAT0 = y6HATpost && y4HAT0 = y4HATpost && iHAT0 = iHATpost && __const_200HAT0 = __const_200HATpost && __const_100HAT0 = __const_100HATpost && jHATpost = __const_100HAT0 && __const_100HAT0 <= iHAT0 (2) l0(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x3 = x10 && x1 = x8 && x = x7 && x9 = 1 + x2 && x12 = x2 && x11 = x2 && 1 + x2 <= x (3) l2(x14, x15, x16, x17, x18, x19, x20) -> l0(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21 (4) l3(x28, x29, x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35 && x29 <= x31 (5) l3(x42, x43, x44, x45, x46, x47, x48) -> l1(x49, x50, x51, x52, x53, x54, x55) :|: x47 = x54 && x46 = x53 && x44 = x51 && x43 = x50 && x42 = x49 && x52 = 1 + x45 && x55 = x45 && 1 + x45 <= x43 (6) l1(x56, x57, x58, x59, x60, x61, x62) -> l3(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63 (7) l5(x70, x71, x72, x73, x74, x75, x76) -> l2(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x71 = x78 && x70 = x77 && x79 = 0 (8) l6(x84, x85, x86, x87, x88, x89, x90) -> l5(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91 Arcs: (1) -> (6) (2) -> (3) (3) -> (1), (2) (5) -> (6) (6) -> (4), (5) (7) -> (3) (8) -> (7) 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 && x3 = x10 && x1 = x8 && x = x7 && x9 = 1 + x2 && x12 = x2 && x11 = x2 && 1 + x2 <= x (2) l2(x14, x15, x16, x17, x18, x19, x20) -> l0(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (6) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (7) Obligation: Rules: l0(x21:0, x1:0, x11:0, x10:0, x4:0, x5:0, x13:0) -> l0(x21:0, x1:0, 1 + x11:0, x10:0, x11:0, x11:0, x13:0) :|: x21:0 >= 1 + x11:0 ---------------------------------------- (8) 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) -> l0(x1, x3) ---------------------------------------- (9) Obligation: Rules: l0(x21:0, x11:0) -> l0(x21:0, 1 + x11:0) :|: x21:0 >= 1 + x11:0 ---------------------------------------- (10) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l0(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (11) Obligation: Rules: l0(x21:0, x11:0) -> l0(x21:0, c) :|: c = 1 + x11:0 && x21:0 >= 1 + x11:0 ---------------------------------------- (12) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l0 ] = l0_1 + -1*l0_2 The following rules are decreasing: l0(x21:0, x11:0) -> l0(x21:0, c) :|: c = 1 + x11:0 && x21:0 >= 1 + x11:0 The following rules are bounded: l0(x21:0, x11:0) -> l0(x21:0, c) :|: c = 1 + x11:0 && x21:0 >= 1 + x11:0 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Termination digraph: Nodes: (1) l1(x56, x57, x58, x59, x60, x61, x62) -> l3(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63 (2) l3(x42, x43, x44, x45, x46, x47, x48) -> l1(x49, x50, x51, x52, x53, x54, x55) :|: x47 = x54 && x46 = x53 && x44 = x51 && x43 = x50 && x42 = x49 && x52 = 1 + x45 && x55 = x45 && 1 + x45 <= x43 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (15) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (16) Obligation: Rules: l1(x49:0, x50:0, x51:0, x55:0, x53:0, x54:0, x62:0) -> l1(x49:0, x50:0, x51:0, 1 + x55:0, x53:0, x54:0, x55:0) :|: x50:0 >= 1 + x55:0 ---------------------------------------- (17) 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(x2, x4) ---------------------------------------- (18) Obligation: Rules: l1(x50:0, x55:0) -> l1(x50:0, 1 + x55:0) :|: x50:0 >= 1 + x55:0 ---------------------------------------- (19) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l1(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (20) Obligation: Rules: l1(x50:0, x55:0) -> l1(x50:0, c) :|: c = 1 + x55:0 && x50:0 >= 1 + x55:0 ---------------------------------------- (21) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l1(x, x1)] = x - x1 The following rules are decreasing: l1(x50:0, x55:0) -> l1(x50:0, c) :|: c = 1 + x55:0 && x50:0 >= 1 + x55:0 The following rules are bounded: l1(x50:0, x55:0) -> l1(x50:0, c) :|: c = 1 + x55:0 && x50:0 >= 1 + x55:0 ---------------------------------------- (22) YES