YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 449 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 28 ms] (6) IRSwT (7) TempFilterProof [SOUND, 123 ms] (8) IntTRS (9) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 8 ms] (12) IntTRS (13) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (14) IntTRS (15) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (16) YES ---------------------------------------- (0) Obligation: Rules: l0(cHAT0, oxHAT0, xHAT0, yHAT0) -> l1(cHATpost, oxHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && 1 + xHAT0 <= oxHAT0 l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 && x1 <= x2 l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x8 <= 0 l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x20 = 1 && x21 = x18 l4(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28 && x24 <= 0 l4(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 && 1 <= x32 l5(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x42 = x46 && x41 = x45 && x40 = x44 && x47 = -1 + x43 l5(x48, x49, x50, x51) -> l4(x52, x53, x54, x55) :|: x51 = x55 && x49 = x53 && x48 = x52 && x54 = -1 + x50 l1(x56, x57, x58, x59) -> l5(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 && 1 <= x59 && 1 <= x58 l6(x64, x65, x66, x67) -> l1(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68 && x64 <= 0 l7(x72, x73, x74, x75) -> l6(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 Start term: l7(cHAT0, oxHAT0, xHAT0, yHAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(cHAT0, oxHAT0, xHAT0, yHAT0) -> l1(cHATpost, oxHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && 1 + xHAT0 <= oxHAT0 l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 && x1 <= x2 l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x8 <= 0 l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x20 = 1 && x21 = x18 l4(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28 && x24 <= 0 l4(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 && 1 <= x32 l5(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x42 = x46 && x41 = x45 && x40 = x44 && x47 = -1 + x43 l5(x48, x49, x50, x51) -> l4(x52, x53, x54, x55) :|: x51 = x55 && x49 = x53 && x48 = x52 && x54 = -1 + x50 l1(x56, x57, x58, x59) -> l5(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 && 1 <= x59 && 1 <= x58 l6(x64, x65, x66, x67) -> l1(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68 && x64 <= 0 l7(x72, x73, x74, x75) -> l6(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 Start term: l7(cHAT0, oxHAT0, xHAT0, yHAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(cHAT0, oxHAT0, xHAT0, yHAT0) -> l1(cHATpost, oxHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && 1 + xHAT0 <= oxHAT0 (2) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 && x1 <= x2 (3) l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x8 <= 0 (4) l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x20 = 1 && x21 = x18 (5) l4(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28 && x24 <= 0 (6) l4(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 && 1 <= x32 (7) l5(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x42 = x46 && x41 = x45 && x40 = x44 && x47 = -1 + x43 (8) l5(x48, x49, x50, x51) -> l4(x52, x53, x54, x55) :|: x51 = x55 && x49 = x53 && x48 = x52 && x54 = -1 + x50 (9) l1(x56, x57, x58, x59) -> l5(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 && 1 <= x59 && 1 <= x58 (10) l6(x64, x65, x66, x67) -> l1(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68 && x64 <= 0 (11) l7(x72, x73, x74, x75) -> l6(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 Arcs: (1) -> (9) (3) -> (9) (4) -> (9) (5) -> (3), (4) (6) -> (1), (2) (7) -> (5), (6) (8) -> (5), (6) (9) -> (7), (8) (10) -> (9) (11) -> (10) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l0(cHAT0, oxHAT0, xHAT0, yHAT0) -> l1(cHATpost, oxHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && 1 + xHAT0 <= oxHAT0 (2) l4(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 && 1 <= x32 (3) l5(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x42 = x46 && x41 = x45 && x40 = x44 && x47 = -1 + x43 (4) l1(x56, x57, x58, x59) -> l5(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 && 1 <= x59 && 1 <= x58 (5) l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x20 = 1 && x21 = x18 (6) l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x8 <= 0 (7) l4(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28 && x24 <= 0 (8) l5(x48, x49, x50, x51) -> l4(x52, x53, x54, x55) :|: x51 = x55 && x49 = x53 && x48 = x52 && x54 = -1 + x50 Arcs: (1) -> (4) (2) -> (1) (3) -> (2), (7) (4) -> (3), (8) (5) -> (4) (6) -> (4) (7) -> (5), (6) (8) -> (2), (7) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l4(x12:0, x13:0, x14:0, x15:0) -> l5(x12:0, x13:0, x14:0, x15:0) :|: x12:0 < 1 && x15:0 > 0 && x14:0 > 0 l5(x48:0, x49:0, x50:0, x51:0) -> l4(x48:0, x49:0, -1 + x50:0, x51:0) :|: TRUE l5(x40:0, x41:0, x42:0, x43:0) -> l4(x40:0, x41:0, x42:0, -1 + x43:0) :|: TRUE l4(x24:0, x25:0, x21:0, x23:0) -> l5(1, x21:0, x21:0, x23:0) :|: x21:0 > 0 && x23:0 > 0 && x24:0 < 1 l4(cHATpost:0, oxHATpost:0, x34:0, x35:0) -> l5(cHATpost:0, oxHATpost:0, x34:0, x35:0) :|: oxHATpost:0 >= 1 + x34:0 && cHATpost:0 > 0 && x35:0 > 0 && x34:0 > 0 ---------------------------------------- (7) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l4(VARIABLE, VARIABLE, VARIABLE, VARIABLE) l5(VARIABLE, VARIABLE, VARIABLE, VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (8) Obligation: Rules: l4(x12:0, x13:0, x14:0, x15:0) -> l5(x12:0, x13:0, x14:0, x15:0) :|: x12:0 < 1 && x15:0 > 0 && x14:0 > 0 l5(x48:0, x49:0, x50:0, x51:0) -> l4(x48:0, x49:0, c, x51:0) :|: c = -1 + x50:0 && TRUE l5(x40:0, x41:0, x42:0, x43:0) -> l4(x40:0, x41:0, x42:0, c1) :|: c1 = -1 + x43:0 && TRUE l4(x24:0, x25:0, x21:0, x23:0) -> l5(c2, x21:0, x21:0, x23:0) :|: c2 = 1 && (x21:0 > 0 && x23:0 > 0 && x24:0 < 1) l4(cHATpost:0, oxHATpost:0, x34:0, x35:0) -> l5(cHATpost:0, oxHATpost:0, x34:0, x35:0) :|: oxHATpost:0 >= 1 + x34:0 && cHATpost:0 > 0 && x35:0 > 0 && x34:0 > 0 ---------------------------------------- (9) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l4(x, x1, x2, x3)] = -x [l5(x4, x5, x6, x7)] = -x4 The following rules are decreasing: l4(x24:0, x25:0, x21:0, x23:0) -> l5(c2, x21:0, x21:0, x23:0) :|: c2 = 1 && (x21:0 > 0 && x23:0 > 0 && x24:0 < 1) The following rules are bounded: l4(x12:0, x13:0, x14:0, x15:0) -> l5(x12:0, x13:0, x14:0, x15:0) :|: x12:0 < 1 && x15:0 > 0 && x14:0 > 0 l4(x24:0, x25:0, x21:0, x23:0) -> l5(c2, x21:0, x21:0, x23:0) :|: c2 = 1 && (x21:0 > 0 && x23:0 > 0 && x24:0 < 1) ---------------------------------------- (10) Obligation: Rules: l4(x12:0, x13:0, x14:0, x15:0) -> l5(x12:0, x13:0, x14:0, x15:0) :|: x12:0 < 1 && x15:0 > 0 && x14:0 > 0 l5(x48:0, x49:0, x50:0, x51:0) -> l4(x48:0, x49:0, c, x51:0) :|: c = -1 + x50:0 && TRUE l5(x40:0, x41:0, x42:0, x43:0) -> l4(x40:0, x41:0, x42:0, c1) :|: c1 = -1 + x43:0 && TRUE l4(cHATpost:0, oxHATpost:0, x34:0, x35:0) -> l5(cHATpost:0, oxHATpost:0, x34:0, x35:0) :|: oxHATpost:0 >= 1 + x34:0 && cHATpost:0 > 0 && x35:0 > 0 && x34:0 > 0 ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l4(x, x1, x2, x3)] = -2 - x + x2 + x3 [l5(x4, x5, x6, x7)] = -3 - x4 + x6 + x7 The following rules are decreasing: l4(x12:0, x13:0, x14:0, x15:0) -> l5(x12:0, x13:0, x14:0, x15:0) :|: x12:0 < 1 && x15:0 > 0 && x14:0 > 0 l4(cHATpost:0, oxHATpost:0, x34:0, x35:0) -> l5(cHATpost:0, oxHATpost:0, x34:0, x35:0) :|: oxHATpost:0 >= 1 + x34:0 && cHATpost:0 > 0 && x35:0 > 0 && x34:0 > 0 The following rules are bounded: l4(x12:0, x13:0, x14:0, x15:0) -> l5(x12:0, x13:0, x14:0, x15:0) :|: x12:0 < 1 && x15:0 > 0 && x14:0 > 0 ---------------------------------------- (12) Obligation: Rules: l5(x48:0, x49:0, x50:0, x51:0) -> l4(x48:0, x49:0, c, x51:0) :|: c = -1 + x50:0 && TRUE l5(x40:0, x41:0, x42:0, x43:0) -> l4(x40:0, x41:0, x42:0, c1) :|: c1 = -1 + x43:0 && TRUE l4(cHATpost:0, oxHATpost:0, x34:0, x35:0) -> l5(cHATpost:0, oxHATpost:0, x34:0, x35:0) :|: oxHATpost:0 >= 1 + x34:0 && cHATpost:0 > 0 && x35:0 > 0 && x34:0 > 0 ---------------------------------------- (13) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l5(x, x1, x2, x3)] = -1 + x + x1 + x2 + x3 [l4(x4, x5, x6, x7)] = x4 + x5 + x6 + x7 The following rules are decreasing: l4(cHATpost:0, oxHATpost:0, x34:0, x35:0) -> l5(cHATpost:0, oxHATpost:0, x34:0, x35:0) :|: oxHATpost:0 >= 1 + x34:0 && cHATpost:0 > 0 && x35:0 > 0 && x34:0 > 0 The following rules are bounded: l4(cHATpost:0, oxHATpost:0, x34:0, x35:0) -> l5(cHATpost:0, oxHATpost:0, x34:0, x35:0) :|: oxHATpost:0 >= 1 + x34:0 && cHATpost:0 > 0 && x35:0 > 0 && x34:0 > 0 ---------------------------------------- (14) Obligation: Rules: l5(x48:0, x49:0, x50:0, x51:0) -> l4(x48:0, x49:0, c, x51:0) :|: c = -1 + x50:0 && TRUE l5(x40:0, x41:0, x42:0, x43:0) -> l4(x40:0, x41:0, x42:0, c1) :|: c1 = -1 + x43:0 && TRUE ---------------------------------------- (15) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l5(x, x1, x2, x3)] = 0 [l4(x4, x5, x6, x7)] = -1 The following rules are decreasing: l5(x48:0, x49:0, x50:0, x51:0) -> l4(x48:0, x49:0, c, x51:0) :|: c = -1 + x50:0 && TRUE l5(x40:0, x41:0, x42:0, x43:0) -> l4(x40:0, x41:0, x42:0, c1) :|: c1 = -1 + x43:0 && TRUE The following rules are bounded: l5(x48:0, x49:0, x50:0, x51:0) -> l4(x48:0, x49:0, c, x51:0) :|: c = -1 + x50:0 && TRUE l5(x40:0, x41:0, x42:0, x43:0) -> l4(x40:0, x41:0, x42:0, c1) :|: c1 = -1 + x43:0 && TRUE ---------------------------------------- (16) YES