YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 897 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 0 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 46 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 7 ms] (12) IntTRS (13) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Rules: l0(iHAT0, jHAT0, tmp5HAT0, x3HAT0, y4HAT0) -> l1(iHATpost, jHATpost, tmp5HATpost, x3HATpost, y4HATpost) :|: y4HAT0 = y4HATpost && x3HAT0 = x3HATpost && tmp5HAT0 = tmp5HATpost && jHAT0 = jHATpost && iHAT0 = iHATpost l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 l4(x10, x11, x12, x13, x14) -> l2(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 l5(x20, x21, x22, x23, x24) -> l6(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x20 = x25 && x26 = 1 + x21 l6(x30, x31, x32, x33, x34) -> l7(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 l8(x40, x41, x42, x43, x44) -> l5(x45, x46, x47, x48, x49) :|: x41 = x46 && x40 = x45 && x47 = x47 && x49 = 1 + x41 && x48 = x41 l8(x50, x51, x52, x53, x54) -> l5(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 l7(x60, x61, x62, x63, x64) -> l0(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x65 = -1 + x60 && x60 <= x61 l7(x70, x71, x72, x73, x74) -> l8(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 1 + x71 <= x70 l1(x80, x81, x82, x83, x84) -> l4(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && 1 + x80 <= 0 l1(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95 && 0 <= x90 l9(x100, x101, x102, x103, x104) -> l0(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x105 = 4 && x106 = 0 l10(x110, x111, x112, x113, x114) -> l9(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x110 = x115 Start term: l10(iHAT0, jHAT0, tmp5HAT0, x3HAT0, y4HAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(iHAT0, jHAT0, tmp5HAT0, x3HAT0, y4HAT0) -> l1(iHATpost, jHATpost, tmp5HATpost, x3HATpost, y4HATpost) :|: y4HAT0 = y4HATpost && x3HAT0 = x3HATpost && tmp5HAT0 = tmp5HATpost && jHAT0 = jHATpost && iHAT0 = iHATpost l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 l4(x10, x11, x12, x13, x14) -> l2(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 l5(x20, x21, x22, x23, x24) -> l6(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x20 = x25 && x26 = 1 + x21 l6(x30, x31, x32, x33, x34) -> l7(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 l8(x40, x41, x42, x43, x44) -> l5(x45, x46, x47, x48, x49) :|: x41 = x46 && x40 = x45 && x47 = x47 && x49 = 1 + x41 && x48 = x41 l8(x50, x51, x52, x53, x54) -> l5(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 l7(x60, x61, x62, x63, x64) -> l0(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x65 = -1 + x60 && x60 <= x61 l7(x70, x71, x72, x73, x74) -> l8(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 1 + x71 <= x70 l1(x80, x81, x82, x83, x84) -> l4(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && 1 + x80 <= 0 l1(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95 && 0 <= x90 l9(x100, x101, x102, x103, x104) -> l0(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x105 = 4 && x106 = 0 l10(x110, x111, x112, x113, x114) -> l9(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x110 = x115 Start term: l10(iHAT0, jHAT0, tmp5HAT0, x3HAT0, y4HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(iHAT0, jHAT0, tmp5HAT0, x3HAT0, y4HAT0) -> l1(iHATpost, jHATpost, tmp5HATpost, x3HATpost, y4HATpost) :|: y4HAT0 = y4HATpost && x3HAT0 = x3HATpost && tmp5HAT0 = tmp5HATpost && jHAT0 = jHATpost && iHAT0 = iHATpost (2) l2(x, x1, x2, x3, x4) -> l3(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 (3) l4(x10, x11, x12, x13, x14) -> l2(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 (4) l5(x20, x21, x22, x23, x24) -> l6(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x20 = x25 && x26 = 1 + x21 (5) l6(x30, x31, x32, x33, x34) -> l7(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 (6) l8(x40, x41, x42, x43, x44) -> l5(x45, x46, x47, x48, x49) :|: x41 = x46 && x40 = x45 && x47 = x47 && x49 = 1 + x41 && x48 = x41 (7) l8(x50, x51, x52, x53, x54) -> l5(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 (8) l7(x60, x61, x62, x63, x64) -> l0(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x65 = -1 + x60 && x60 <= x61 (9) l7(x70, x71, x72, x73, x74) -> l8(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 1 + x71 <= x70 (10) l1(x80, x81, x82, x83, x84) -> l4(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && 1 + x80 <= 0 (11) l1(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95 && 0 <= x90 (12) l9(x100, x101, x102, x103, x104) -> l0(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x105 = 4 && x106 = 0 (13) l10(x110, x111, x112, x113, x114) -> l9(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x110 = x115 Arcs: (1) -> (10), (11) (3) -> (2) (4) -> (5) (5) -> (8), (9) (6) -> (4) (7) -> (4) (8) -> (1) (9) -> (6), (7) (10) -> (3) (11) -> (5) (12) -> (1) (13) -> (12) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l0(iHAT0, jHAT0, tmp5HAT0, x3HAT0, y4HAT0) -> l1(iHATpost, jHATpost, tmp5HATpost, x3HATpost, y4HATpost) :|: y4HAT0 = y4HATpost && x3HAT0 = x3HATpost && tmp5HAT0 = tmp5HATpost && jHAT0 = jHATpost && iHAT0 = iHATpost (2) l7(x60, x61, x62, x63, x64) -> l0(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x65 = -1 + x60 && x60 <= x61 (3) l6(x30, x31, x32, x33, x34) -> l7(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 (4) l1(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95 && 0 <= x90 (5) l5(x20, x21, x22, x23, x24) -> l6(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x20 = x25 && x26 = 1 + x21 (6) l8(x50, x51, x52, x53, x54) -> l5(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 (7) l8(x40, x41, x42, x43, x44) -> l5(x45, x46, x47, x48, x49) :|: x41 = x46 && x40 = x45 && x47 = x47 && x49 = 1 + x41 && x48 = x41 (8) l7(x70, x71, x72, x73, x74) -> l8(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 1 + x71 <= x70 Arcs: (1) -> (4) (2) -> (1) (3) -> (2), (8) (4) -> (3) (5) -> (3) (6) -> (5) (7) -> (5) (8) -> (6), (7) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l6(x30:0, jHATpost:0, tmp5HATpost:0, x33:0, x34:0) -> l6(-1 + x30:0, jHATpost:0, tmp5HATpost:0, x33:0, x34:0) :|: x30:0 <= jHATpost:0 && x30:0 > 0 l6(x, x1, x2, x3, x4) -> l6(x, 1 + x1, x5, x1, 1 + x1) :|: x >= 1 + x1 l6(x25:0, x31:0, x27:0, x28:0, x29:0) -> l6(x25:0, 1 + x31:0, x27:0, x28:0, x29:0) :|: x25:0 >= 1 + x31:0 ---------------------------------------- (7) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l6(x1, x2, x3, x4, x5) -> l6(x1, x2) ---------------------------------------- (8) Obligation: Rules: l6(x30:0, jHATpost:0) -> l6(-1 + x30:0, jHATpost:0) :|: x30:0 <= jHATpost:0 && x30:0 > 0 l6(x, x1) -> l6(x, 1 + x1) :|: x >= 1 + x1 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l6(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l6(x30:0, jHATpost:0) -> l6(c, jHATpost:0) :|: c = -1 + x30:0 && (x30:0 <= jHATpost:0 && x30:0 > 0) l6(x, x1) -> l6(x, c1) :|: c1 = 1 + x1 && x >= 1 + x1 ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l6(x, x1)] = -1 + x - x1 The following rules are decreasing: l6(x30:0, jHATpost:0) -> l6(c, jHATpost:0) :|: c = -1 + x30:0 && (x30:0 <= jHATpost:0 && x30:0 > 0) l6(x, x1) -> l6(x, c1) :|: c1 = 1 + x1 && x >= 1 + x1 The following rules are bounded: l6(x, x1) -> l6(x, c1) :|: c1 = 1 + x1 && x >= 1 + x1 ---------------------------------------- (12) Obligation: Rules: l6(x30:0, jHATpost:0) -> l6(c, jHATpost:0) :|: c = -1 + x30:0 && (x30:0 <= jHATpost:0 && x30:0 > 0) ---------------------------------------- (13) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l6(x, x1)] = x The following rules are decreasing: l6(x30:0, jHATpost:0) -> l6(c, jHATpost:0) :|: c = -1 + x30:0 && (x30:0 <= jHATpost:0 && x30:0 > 0) The following rules are bounded: l6(x30:0, jHATpost:0) -> l6(c, jHATpost:0) :|: c = -1 + x30:0 && (x30:0 <= jHATpost:0 && x30:0 > 0) ---------------------------------------- (14) YES