YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 409 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 50 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 58 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (12) IntTRS (13) RankingReductionPairProof [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Rules: l0(ResultHAT0, __const_99HAT0, __disjvr_0HAT0, xHAT0, yHAT0) -> l1(ResultHATpost, __const_99HATpost, __disjvr_0HATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && __disjvr_0HAT0 = __disjvr_0HATpost && __const_99HAT0 = __const_99HATpost && ResultHATpost = ResultHATpost && -1 * xHAT0 <= 0 l0(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 0 <= -1 - x3 l3(x10, x11, x12, x13, x14) -> l4(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && x17 = x12 l4(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x22 = x27 && x21 = x26 && x20 = x25 && x29 = 1 + x24 && x28 = 1 + x23 l2(x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 l0(x40, x41, x42, x43, x44) -> l5(x45, x46, x47, x48, x49) :|: x42 = x47 && x41 = x46 && x40 = x45 && x48 = -1 * x41 + x43 && x49 = 1 + x44 && -1 <= x44 && x44 <= -1 && 0 <= -1 - x43 l5(x50, x51, x52, x53, x54) -> l0(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 l6(x60, x61, x62, x63, x64) -> l0(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 l7(x70, x71, x72, x73, x74) -> l6(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 Start term: l7(ResultHAT0, __const_99HAT0, __disjvr_0HAT0, xHAT0, yHAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(ResultHAT0, __const_99HAT0, __disjvr_0HAT0, xHAT0, yHAT0) -> l1(ResultHATpost, __const_99HATpost, __disjvr_0HATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && __disjvr_0HAT0 = __disjvr_0HATpost && __const_99HAT0 = __const_99HATpost && ResultHATpost = ResultHATpost && -1 * xHAT0 <= 0 l0(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 0 <= -1 - x3 l3(x10, x11, x12, x13, x14) -> l4(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && x17 = x12 l4(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x22 = x27 && x21 = x26 && x20 = x25 && x29 = 1 + x24 && x28 = 1 + x23 l2(x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 l0(x40, x41, x42, x43, x44) -> l5(x45, x46, x47, x48, x49) :|: x42 = x47 && x41 = x46 && x40 = x45 && x48 = -1 * x41 + x43 && x49 = 1 + x44 && -1 <= x44 && x44 <= -1 && 0 <= -1 - x43 l5(x50, x51, x52, x53, x54) -> l0(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 l6(x60, x61, x62, x63, x64) -> l0(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 l7(x70, x71, x72, x73, x74) -> l6(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 Start term: l7(ResultHAT0, __const_99HAT0, __disjvr_0HAT0, xHAT0, yHAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(ResultHAT0, __const_99HAT0, __disjvr_0HAT0, xHAT0, yHAT0) -> l1(ResultHATpost, __const_99HATpost, __disjvr_0HATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && __disjvr_0HAT0 = __disjvr_0HATpost && __const_99HAT0 = __const_99HATpost && ResultHATpost = ResultHATpost && -1 * xHAT0 <= 0 (2) l0(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 0 <= -1 - x3 (3) l3(x10, x11, x12, x13, x14) -> l4(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && x17 = x12 (4) l4(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x22 = x27 && x21 = x26 && x20 = x25 && x29 = 1 + x24 && x28 = 1 + x23 (5) l2(x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 (6) l0(x40, x41, x42, x43, x44) -> l5(x45, x46, x47, x48, x49) :|: x42 = x47 && x41 = x46 && x40 = x45 && x48 = -1 * x41 + x43 && x49 = 1 + x44 && -1 <= x44 && x44 <= -1 && 0 <= -1 - x43 (7) l5(x50, x51, x52, x53, x54) -> l0(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 (8) l6(x60, x61, x62, x63, x64) -> l0(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 (9) l7(x70, x71, x72, x73, x74) -> l6(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 Arcs: (2) -> (3) (3) -> (4) (4) -> (5) (5) -> (1), (2), (6) (6) -> (7) (7) -> (1), (2), (6) (8) -> (1), (2), (6) (9) -> (8) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l0(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 0 <= -1 - x3 (2) l5(x50, x51, x52, x53, x54) -> l0(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 (3) l0(x40, x41, x42, x43, x44) -> l5(x45, x46, x47, x48, x49) :|: x42 = x47 && x41 = x46 && x40 = x45 && x48 = -1 * x41 + x43 && x49 = 1 + x44 && -1 <= x44 && x44 <= -1 && 0 <= -1 - x43 (4) l2(x30, x31, x32, x33, x34) -> l0(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 (5) l4(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x22 = x27 && x21 = x26 && x20 = x25 && x29 = 1 + x24 && x28 = 1 + x23 (6) l3(x10, x11, x12, x13, x14) -> l4(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && x17 = x12 Arcs: (1) -> (6) (2) -> (1), (3) (3) -> (2) (4) -> (1), (3) (5) -> (4) (6) -> (5) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l0(x15:0, x16:0, x17:0, x18:0, x19:0) -> l0(x15:0, x16:0, x17:0, 1 + x18:0, 1 + x19:0) :|: x18:0 < 0 l0(x40:0, x41:0, x42:0, x43:0, x44:0) -> l0(x40:0, x41:0, x42:0, -1 * x41:0 + x43:0, 1 + x44:0) :|: x44:0 < 0 && x44:0 > -2 && x43:0 < 0 ---------------------------------------- (7) 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) -> l0(x2, x4, x5) ---------------------------------------- (8) Obligation: Rules: l0(x16:0, x18:0, x19:0) -> l0(x16:0, 1 + x18:0, 1 + x19:0) :|: x18:0 < 0 l0(x41:0, x43:0, x44:0) -> l0(x41:0, -1 * x41:0 + x43:0, 1 + x44:0) :|: x44:0 < 0 && x44:0 > -2 && x43:0 < 0 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l0(VARIABLE, INTEGER, VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l0(x16:0, x18:0, x19:0) -> l0(x16:0, c, c1) :|: c1 = 1 + x19:0 && c = 1 + x18:0 && x18:0 < 0 l0(x41:0, x43:0, x44:0) -> l0(x41:0, c2, c3) :|: c3 = 1 + x44:0 && c2 = -1 * x41:0 + x43:0 && (x44:0 < 0 && x44:0 > -2 && x43:0 < 0) ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l0(x, x1, x2)] = -1 - x2 The following rules are decreasing: l0(x16:0, x18:0, x19:0) -> l0(x16:0, c, c1) :|: c1 = 1 + x19:0 && c = 1 + x18:0 && x18:0 < 0 l0(x41:0, x43:0, x44:0) -> l0(x41:0, c2, c3) :|: c3 = 1 + x44:0 && c2 = -1 * x41:0 + x43:0 && (x44:0 < 0 && x44:0 > -2 && x43:0 < 0) The following rules are bounded: l0(x41:0, x43:0, x44:0) -> l0(x41:0, c2, c3) :|: c3 = 1 + x44:0 && c2 = -1 * x41:0 + x43:0 && (x44:0 < 0 && x44:0 > -2 && x43:0 < 0) ---------------------------------------- (12) Obligation: Rules: l0(x16:0, x18:0, x19:0) -> l0(x16:0, c, c1) :|: c1 = 1 + x19:0 && c = 1 + x18:0 && x18:0 < 0 ---------------------------------------- (13) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l0 ] = -1*l0_2 The following rules are decreasing: l0(x16:0, x18:0, x19:0) -> l0(x16:0, c, c1) :|: c1 = 1 + x19:0 && c = 1 + x18:0 && x18:0 < 0 The following rules are bounded: l0(x16:0, x18:0, x19:0) -> l0(x16:0, c, c1) :|: c1 = 1 + x19:0 && c = 1 + x18:0 && x18:0 < 0 ---------------------------------------- (14) YES