YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 636 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 66 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 94 ms] (10) IntTRS (11) RankingReductionPairProof [EQUIVALENT, 0 ms] (12) IntTRS (13) RankingReductionPairProof [EQUIVALENT, 7 ms] (14) YES ---------------------------------------- (0) Obligation: Rules: l0(guardHAT0, iHAT0, jHAT0, nHAT0) -> l1(guardHATpost, iHATpost, jHATpost, nHATpost) :|: nHAT0 = nHATpost && jHAT0 = jHATpost && guardHAT0 = guardHATpost && iHATpost = 1 + iHAT0 && nHAT0 <= jHAT0 l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x4 = x4 && 1 + x2 <= x3 l3(x8, x9, x10, x11) -> l4(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x11 <= x9 l3(x16, x17, x18, x19) -> l5(x20, x21, x22, x23) :|: x19 = x23 && x17 = x21 && x16 = x20 && x22 = 1 + x17 && 1 + x17 <= x19 l1(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28 l5(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 l6(x40, x41, x42, x43) -> l5(x44, x45, x46, x47) :|: x43 = x47 && x41 = x45 && x40 = x44 && x46 = 1 + x42 l7(x48, x49, x50, x51) -> l6(x52, x53, x54, x55) :|: x49 = x53 && x48 = x52 && x55 = -1 + x51 && x54 = -1 + x50 l2(x56, x57, x58, x59) -> l6(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 && 0 <= x56 && x56 <= 0 l2(x64, x65, x66, x67) -> l7(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68 && 1 <= x64 l2(x72, x73, x74, x75) -> l7(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 && 1 + x72 <= 0 l8(x80, x81, x82, x83) -> l1(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x80 = x84 && x85 = 0 l9(x88, x89, x90, x91) -> l8(x92, x93, x94, x95) :|: x91 = x95 && x90 = x94 && x89 = x93 && x88 = x92 Start term: l9(guardHAT0, iHAT0, jHAT0, nHAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(guardHAT0, iHAT0, jHAT0, nHAT0) -> l1(guardHATpost, iHATpost, jHATpost, nHATpost) :|: nHAT0 = nHATpost && jHAT0 = jHATpost && guardHAT0 = guardHATpost && iHATpost = 1 + iHAT0 && nHAT0 <= jHAT0 l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x4 = x4 && 1 + x2 <= x3 l3(x8, x9, x10, x11) -> l4(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x11 <= x9 l3(x16, x17, x18, x19) -> l5(x20, x21, x22, x23) :|: x19 = x23 && x17 = x21 && x16 = x20 && x22 = 1 + x17 && 1 + x17 <= x19 l1(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28 l5(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 l6(x40, x41, x42, x43) -> l5(x44, x45, x46, x47) :|: x43 = x47 && x41 = x45 && x40 = x44 && x46 = 1 + x42 l7(x48, x49, x50, x51) -> l6(x52, x53, x54, x55) :|: x49 = x53 && x48 = x52 && x55 = -1 + x51 && x54 = -1 + x50 l2(x56, x57, x58, x59) -> l6(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 && 0 <= x56 && x56 <= 0 l2(x64, x65, x66, x67) -> l7(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68 && 1 <= x64 l2(x72, x73, x74, x75) -> l7(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 && 1 + x72 <= 0 l8(x80, x81, x82, x83) -> l1(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x80 = x84 && x85 = 0 l9(x88, x89, x90, x91) -> l8(x92, x93, x94, x95) :|: x91 = x95 && x90 = x94 && x89 = x93 && x88 = x92 Start term: l9(guardHAT0, iHAT0, jHAT0, nHAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(guardHAT0, iHAT0, jHAT0, nHAT0) -> l1(guardHATpost, iHATpost, jHATpost, nHATpost) :|: nHAT0 = nHATpost && jHAT0 = jHATpost && guardHAT0 = guardHATpost && iHATpost = 1 + iHAT0 && nHAT0 <= jHAT0 (2) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x4 = x4 && 1 + x2 <= x3 (3) l3(x8, x9, x10, x11) -> l4(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x11 <= x9 (4) l3(x16, x17, x18, x19) -> l5(x20, x21, x22, x23) :|: x19 = x23 && x17 = x21 && x16 = x20 && x22 = 1 + x17 && 1 + x17 <= x19 (5) l1(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28 (6) l5(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 (7) l6(x40, x41, x42, x43) -> l5(x44, x45, x46, x47) :|: x43 = x47 && x41 = x45 && x40 = x44 && x46 = 1 + x42 (8) l7(x48, x49, x50, x51) -> l6(x52, x53, x54, x55) :|: x49 = x53 && x48 = x52 && x55 = -1 + x51 && x54 = -1 + x50 (9) l2(x56, x57, x58, x59) -> l6(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 && 0 <= x56 && x56 <= 0 (10) l2(x64, x65, x66, x67) -> l7(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68 && 1 <= x64 (11) l2(x72, x73, x74, x75) -> l7(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 && 1 + x72 <= 0 (12) l8(x80, x81, x82, x83) -> l1(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x80 = x84 && x85 = 0 (13) l9(x88, x89, x90, x91) -> l8(x92, x93, x94, x95) :|: x91 = x95 && x90 = x94 && x89 = x93 && x88 = x92 Arcs: (1) -> (5) (2) -> (9), (10), (11) (4) -> (6) (5) -> (3), (4) (6) -> (1), (2) (7) -> (6) (8) -> (7) (9) -> (7) (10) -> (8) (11) -> (8) (12) -> (5) (13) -> (12) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l0(guardHAT0, iHAT0, jHAT0, nHAT0) -> l1(guardHATpost, iHATpost, jHATpost, nHATpost) :|: nHAT0 = nHATpost && jHAT0 = jHATpost && guardHAT0 = guardHATpost && iHATpost = 1 + iHAT0 && nHAT0 <= jHAT0 (2) l5(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 (3) l6(x40, x41, x42, x43) -> l5(x44, x45, x46, x47) :|: x43 = x47 && x41 = x45 && x40 = x44 && x46 = 1 + x42 (4) l2(x56, x57, x58, x59) -> l6(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 && 0 <= x56 && x56 <= 0 (5) l7(x48, x49, x50, x51) -> l6(x52, x53, x54, x55) :|: x49 = x53 && x48 = x52 && x55 = -1 + x51 && x54 = -1 + x50 (6) l2(x72, x73, x74, x75) -> l7(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 && 1 + x72 <= 0 (7) l2(x64, x65, x66, x67) -> l7(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68 && 1 <= x64 (8) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x4 = x4 && 1 + x2 <= x3 (9) l3(x16, x17, x18, x19) -> l5(x20, x21, x22, x23) :|: x19 = x23 && x17 = x21 && x16 = x20 && x22 = 1 + x17 && 1 + x17 <= x19 (10) l1(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28 Arcs: (1) -> (10) (2) -> (1), (8) (3) -> (2) (4) -> (3) (5) -> (3) (6) -> (5) (7) -> (5) (8) -> (4), (6), (7) (9) -> (2) (10) -> (9) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l5(guardHATpost:0, x33:0, jHATpost:0, nHATpost:0) -> l5(guardHATpost:0, 1 + x33:0, 1 + (1 + x33:0), nHATpost:0) :|: nHATpost:0 >= 1 + (1 + x33:0) && nHATpost:0 <= jHATpost:0 l5(x, x1, x2, x3) -> l5(x4, x1, 1 + (-1 + x2), -1 + x3) :|: x3 >= 1 + x2 && x4 < 0 l5(x5, x6, x7, x8) -> l5(x9, x6, 1 + (-1 + x7), -1 + x8) :|: x8 >= 1 + x7 && x9 > 0 l5(x10, x11, x12, x13) -> l5(x14, x11, 1 + x12, x13) :|: x14 < 1 && x14 > -1 && x13 >= 1 + x12 ---------------------------------------- (7) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l5(x1, x2, x3, x4) -> l5(x2, x3, x4) ---------------------------------------- (8) Obligation: Rules: l5(x33:0, jHATpost:0, nHATpost:0) -> l5(1 + x33:0, 1 + (1 + x33:0), nHATpost:0) :|: nHATpost:0 >= 1 + (1 + x33:0) && nHATpost:0 <= jHATpost:0 l5(x1, x2, x3) -> l5(x1, 1 + (-1 + x2), -1 + x3) :|: x3 >= 1 + x2 && x4 < 0 l5(x6, x7, x8) -> l5(x6, 1 + (-1 + x7), -1 + x8) :|: x8 >= 1 + x7 && x9 > 0 l5(x11, x12, x13) -> l5(x11, 1 + x12, x13) :|: x14 < 1 && x14 > -1 && x13 >= 1 + x12 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l5(VARIABLE, INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l5(x33:0, jHATpost:0, nHATpost:0) -> l5(c, c1, nHATpost:0) :|: c1 = 1 + (1 + x33:0) && c = 1 + x33:0 && (nHATpost:0 >= 1 + (1 + x33:0) && nHATpost:0 <= jHATpost:0) l5(x1, x2, x3) -> l5(x1, c2, c3) :|: c3 = -1 + x3 && c2 = 1 + (-1 + x2) && (x3 >= 1 + x2 && x4 < 0) l5(x6, x7, x8) -> l5(x6, c4, c5) :|: c5 = -1 + x8 && c4 = 1 + (-1 + x7) && (x8 >= 1 + x7 && x9 > 0) l5(x11, x12, x13) -> l5(x11, c6, x13) :|: c6 = 1 + x12 && (x14 < 1 && x14 > -1 && x13 >= 1 + x12) ---------------------------------------- (11) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l5 ] = 2*l5_3 + -2*l5_1 The following rules are decreasing: l5(x33:0, jHATpost:0, nHATpost:0) -> l5(c, c1, nHATpost:0) :|: c1 = 1 + (1 + x33:0) && c = 1 + x33:0 && (nHATpost:0 >= 1 + (1 + x33:0) && nHATpost:0 <= jHATpost:0) l5(x1, x2, x3) -> l5(x1, c2, c3) :|: c3 = -1 + x3 && c2 = 1 + (-1 + x2) && (x3 >= 1 + x2 && x4 < 0) l5(x6, x7, x8) -> l5(x6, c4, c5) :|: c5 = -1 + x8 && c4 = 1 + (-1 + x7) && (x8 >= 1 + x7 && x9 > 0) The following rules are bounded: l5(x33:0, jHATpost:0, nHATpost:0) -> l5(c, c1, nHATpost:0) :|: c1 = 1 + (1 + x33:0) && c = 1 + x33:0 && (nHATpost:0 >= 1 + (1 + x33:0) && nHATpost:0 <= jHATpost:0) ---------------------------------------- (12) Obligation: Rules: l5(x1, x2, x3) -> l5(x1, c2, c3) :|: c3 = -1 + x3 && c2 = 1 + (-1 + x2) && (x3 >= 1 + x2 && x4 < 0) l5(x6, x7, x8) -> l5(x6, c4, c5) :|: c5 = -1 + x8 && c4 = 1 + (-1 + x7) && (x8 >= 1 + x7 && x9 > 0) l5(x11, x12, x13) -> l5(x11, c6, x13) :|: c6 = 1 + x12 && (x14 < 1 && x14 > -1 && x13 >= 1 + x12) ---------------------------------------- (13) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l5 ] = l5_3 + -1*l5_2 The following rules are decreasing: l5(x1, x2, x3) -> l5(x1, c2, c3) :|: c3 = -1 + x3 && c2 = 1 + (-1 + x2) && (x3 >= 1 + x2 && x4 < 0) l5(x6, x7, x8) -> l5(x6, c4, c5) :|: c5 = -1 + x8 && c4 = 1 + (-1 + x7) && (x8 >= 1 + x7 && x9 > 0) l5(x11, x12, x13) -> l5(x11, c6, x13) :|: c6 = 1 + x12 && (x14 < 1 && x14 > -1 && x13 >= 1 + x12) The following rules are bounded: l5(x1, x2, x3) -> l5(x1, c2, c3) :|: c3 = -1 + x3 && c2 = 1 + (-1 + x2) && (x3 >= 1 + x2 && x4 < 0) l5(x6, x7, x8) -> l5(x6, c4, c5) :|: c5 = -1 + x8 && c4 = 1 + (-1 + x7) && (x8 >= 1 + x7 && x9 > 0) l5(x11, x12, x13) -> l5(x11, c6, x13) :|: c6 = 1 + x12 && (x14 < 1 && x14 > -1 && x13 >= 1 + x12) ---------------------------------------- (14) YES