YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 1290 ms] (4) AND (5) IRSwT (6) IntTRSCompressionProof [EQUIVALENT, 8 ms] (7) IRSwT (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (9) IRSwT (10) TempFilterProof [SOUND, 13 ms] (11) IntTRS (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (13) YES (14) IRSwT (15) IntTRSCompressionProof [EQUIVALENT, 12 ms] (16) IRSwT (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (18) IRSwT (19) TempFilterProof [SOUND, 4 ms] (20) IntTRS (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (22) YES (23) IRSwT (24) IntTRSCompressionProof [EQUIVALENT, 8 ms] (25) IRSwT (26) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (27) IRSwT (28) TempFilterProof [SOUND, 3 ms] (29) IntTRS (30) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (31) YES ---------------------------------------- (0) Obligation: Rules: l0(__const_999HAT0, i2HAT0, i34HAT0, i6HAT0, i8HAT0) -> l1(__const_999HATpost, i2HATpost, i34HATpost, i6HATpost, i8HATpost) :|: i8HAT0 = i8HATpost && i6HAT0 = i6HATpost && i34HAT0 = i34HATpost && i2HAT0 = i2HATpost && __const_999HAT0 = __const_999HATpost && 1 + i34HAT0 <= 0 l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x1 = x6 && x = x5 && x7 = -1 + x2 && 0 <= x2 l3(x10, x11, x12, x13, x14) -> l1(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && 1 <= x11 l3(x20, x21, x22, x23, x24) -> l1(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && 1 + x21 <= 0 l3(x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x31 = x36 && x30 = x35 && x37 = x30 && 0 <= x31 && x31 <= 0 l2(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 l4(x50, x51, x52, x53, x54) -> l5(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 l6(x60, x61, x62, x63, x64) -> l7(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && 1 + x64 <= 0 l6(x70, x71, x72, x73, x74) -> l8(x75, x76, x77, x78, x79) :|: x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && x79 = -1 + x74 && 0 <= x74 l8(x80, x81, x82, x83, x84) -> l6(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 l5(x90, x91, x92, x93, x94) -> l8(x95, x96, x97, x98, x99) :|: x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95 && x99 = x90 && 1 + x93 <= 0 l5(x100, x101, x102, x103, x104) -> l4(x105, x106, x107, x108, x109) :|: x104 = x109 && x102 = x107 && x101 = x106 && x100 = x105 && x108 = -1 + x103 && 0 <= x103 l1(x110, x111, x112, x113, x114) -> l4(x115, x116, x117, x118, x119) :|: x114 = x119 && x112 = x117 && x111 = x116 && x110 = x115 && x118 = x110 l9(x120, x121, x122, x123, x124) -> l3(x125, x126, x127, x128, x129) :|: x124 = x129 && x123 = x128 && x122 = x127 && x120 = x125 && x126 = 1 l10(x130, x131, x132, x133, x134) -> l9(x135, x136, x137, x138, x139) :|: x134 = x139 && x133 = x138 && x132 = x137 && x131 = x136 && x130 = x135 Start term: l10(__const_999HAT0, i2HAT0, i34HAT0, i6HAT0, i8HAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(__const_999HAT0, i2HAT0, i34HAT0, i6HAT0, i8HAT0) -> l1(__const_999HATpost, i2HATpost, i34HATpost, i6HATpost, i8HATpost) :|: i8HAT0 = i8HATpost && i6HAT0 = i6HATpost && i34HAT0 = i34HATpost && i2HAT0 = i2HATpost && __const_999HAT0 = __const_999HATpost && 1 + i34HAT0 <= 0 l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x1 = x6 && x = x5 && x7 = -1 + x2 && 0 <= x2 l3(x10, x11, x12, x13, x14) -> l1(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && 1 <= x11 l3(x20, x21, x22, x23, x24) -> l1(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && 1 + x21 <= 0 l3(x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x31 = x36 && x30 = x35 && x37 = x30 && 0 <= x31 && x31 <= 0 l2(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 l4(x50, x51, x52, x53, x54) -> l5(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 l6(x60, x61, x62, x63, x64) -> l7(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && 1 + x64 <= 0 l6(x70, x71, x72, x73, x74) -> l8(x75, x76, x77, x78, x79) :|: x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && x79 = -1 + x74 && 0 <= x74 l8(x80, x81, x82, x83, x84) -> l6(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 l5(x90, x91, x92, x93, x94) -> l8(x95, x96, x97, x98, x99) :|: x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95 && x99 = x90 && 1 + x93 <= 0 l5(x100, x101, x102, x103, x104) -> l4(x105, x106, x107, x108, x109) :|: x104 = x109 && x102 = x107 && x101 = x106 && x100 = x105 && x108 = -1 + x103 && 0 <= x103 l1(x110, x111, x112, x113, x114) -> l4(x115, x116, x117, x118, x119) :|: x114 = x119 && x112 = x117 && x111 = x116 && x110 = x115 && x118 = x110 l9(x120, x121, x122, x123, x124) -> l3(x125, x126, x127, x128, x129) :|: x124 = x129 && x123 = x128 && x122 = x127 && x120 = x125 && x126 = 1 l10(x130, x131, x132, x133, x134) -> l9(x135, x136, x137, x138, x139) :|: x134 = x139 && x133 = x138 && x132 = x137 && x131 = x136 && x130 = x135 Start term: l10(__const_999HAT0, i2HAT0, i34HAT0, i6HAT0, i8HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(__const_999HAT0, i2HAT0, i34HAT0, i6HAT0, i8HAT0) -> l1(__const_999HATpost, i2HATpost, i34HATpost, i6HATpost, i8HATpost) :|: i8HAT0 = i8HATpost && i6HAT0 = i6HATpost && i34HAT0 = i34HATpost && i2HAT0 = i2HATpost && __const_999HAT0 = __const_999HATpost && 1 + i34HAT0 <= 0 (2) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x1 = x6 && x = x5 && x7 = -1 + x2 && 0 <= x2 (3) l3(x10, x11, x12, x13, x14) -> l1(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && 1 <= x11 (4) l3(x20, x21, x22, x23, x24) -> l1(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && 1 + x21 <= 0 (5) l3(x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x31 = x36 && x30 = x35 && x37 = x30 && 0 <= x31 && x31 <= 0 (6) l2(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 (7) l4(x50, x51, x52, x53, x54) -> l5(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 (8) l6(x60, x61, x62, x63, x64) -> l7(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 && 1 + x64 <= 0 (9) l6(x70, x71, x72, x73, x74) -> l8(x75, x76, x77, x78, x79) :|: x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && x79 = -1 + x74 && 0 <= x74 (10) l8(x80, x81, x82, x83, x84) -> l6(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 (11) l5(x90, x91, x92, x93, x94) -> l8(x95, x96, x97, x98, x99) :|: x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95 && x99 = x90 && 1 + x93 <= 0 (12) l5(x100, x101, x102, x103, x104) -> l4(x105, x106, x107, x108, x109) :|: x104 = x109 && x102 = x107 && x101 = x106 && x100 = x105 && x108 = -1 + x103 && 0 <= x103 (13) l1(x110, x111, x112, x113, x114) -> l4(x115, x116, x117, x118, x119) :|: x114 = x119 && x112 = x117 && x111 = x116 && x110 = x115 && x118 = x110 (14) l9(x120, x121, x122, x123, x124) -> l3(x125, x126, x127, x128, x129) :|: x124 = x129 && x123 = x128 && x122 = x127 && x120 = x125 && x126 = 1 (15) l10(x130, x131, x132, x133, x134) -> l9(x135, x136, x137, x138, x139) :|: x134 = x139 && x133 = x138 && x132 = x137 && x131 = x136 && x130 = x135 Arcs: (1) -> (13) (2) -> (6) (3) -> (13) (4) -> (13) (5) -> (6) (6) -> (1), (2) (7) -> (11), (12) (9) -> (10) (10) -> (8), (9) (11) -> (10) (12) -> (7) (13) -> (7) (14) -> (3) (15) -> (14) This digraph is fully evaluated! ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Termination digraph: Nodes: (1) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x1 = x6 && x = x5 && x7 = -1 + x2 && 0 <= x2 (2) l2(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (6) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (7) Obligation: Rules: l0(x45:0, x1:0, x2:0, x3:0, x49:0) -> l0(x45:0, x1:0, -1 + x2:0, x3:0, x49:0) :|: x2:0 > -1 ---------------------------------------- (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) -> l0(x3) ---------------------------------------- (9) Obligation: Rules: l0(x2:0) -> l0(-1 + x2:0) :|: x2:0 > -1 ---------------------------------------- (10) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l0(INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (11) Obligation: Rules: l0(x2:0) -> l0(c) :|: c = -1 + x2:0 && x2:0 > -1 ---------------------------------------- (12) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l0(x)] = x The following rules are decreasing: l0(x2:0) -> l0(c) :|: c = -1 + x2:0 && x2:0 > -1 The following rules are bounded: l0(x2:0) -> l0(c) :|: c = -1 + x2:0 && x2:0 > -1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Termination digraph: Nodes: (1) l4(x50, x51, x52, x53, x54) -> l5(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 (2) l5(x100, x101, x102, x103, x104) -> l4(x105, x106, x107, x108, x109) :|: x104 = x109 && x102 = x107 && x101 = x106 && x100 = x105 && x108 = -1 + x103 && 0 <= x103 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (15) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (16) Obligation: Rules: l4(x105:0, x106:0, x107:0, x53:0, x109:0) -> l4(x105:0, x106:0, x107:0, -1 + x53:0, x109:0) :|: x53:0 > -1 ---------------------------------------- (17) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l4(x1, x2, x3, x4, x5) -> l4(x4) ---------------------------------------- (18) Obligation: Rules: l4(x53:0) -> l4(-1 + x53:0) :|: x53:0 > -1 ---------------------------------------- (19) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l4(INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (20) Obligation: Rules: l4(x53:0) -> l4(c) :|: c = -1 + x53:0 && x53:0 > -1 ---------------------------------------- (21) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l4(x)] = x The following rules are decreasing: l4(x53:0) -> l4(c) :|: c = -1 + x53:0 && x53:0 > -1 The following rules are bounded: l4(x53:0) -> l4(c) :|: c = -1 + x53:0 && x53:0 > -1 ---------------------------------------- (22) YES ---------------------------------------- (23) Obligation: Termination digraph: Nodes: (1) l8(x80, x81, x82, x83, x84) -> l6(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 (2) l6(x70, x71, x72, x73, x74) -> l8(x75, x76, x77, x78, x79) :|: x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && x79 = -1 + x74 && 0 <= x74 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (24) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (25) Obligation: Rules: l8(x75:0, x76:0, x77:0, x78:0, x84:0) -> l8(x75:0, x76:0, x77:0, x78:0, -1 + x84:0) :|: x84:0 > -1 ---------------------------------------- (26) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l8(x1, x2, x3, x4, x5) -> l8(x5) ---------------------------------------- (27) Obligation: Rules: l8(x84:0) -> l8(-1 + x84:0) :|: x84:0 > -1 ---------------------------------------- (28) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l8(INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (29) Obligation: Rules: l8(x84:0) -> l8(c) :|: c = -1 + x84:0 && x84:0 > -1 ---------------------------------------- (30) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l8(x)] = x The following rules are decreasing: l8(x84:0) -> l8(c) :|: c = -1 + x84:0 && x84:0 > -1 The following rules are bounded: l8(x84:0) -> l8(c) :|: c = -1 + x84:0 && x84:0 > -1 ---------------------------------------- (31) YES