YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 960 ms] (4) AND (5) IRSwT (6) IntTRSCompressionProof [EQUIVALENT, 0 ms] (7) IRSwT (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (9) IRSwT (10) TempFilterProof [SOUND, 8 ms] (11) IntTRS (12) RankingReductionPairProof [EQUIVALENT, 3 ms] (13) YES (14) IRSwT (15) IntTRSCompressionProof [EQUIVALENT, 0 ms] (16) IRSwT (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (18) IRSwT (19) TempFilterProof [SOUND, 28 ms] (20) IntTRS (21) PolynomialOrderProcessor [EQUIVALENT, 12 ms] (22) YES (23) IRSwT (24) IntTRSCompressionProof [EQUIVALENT, 0 ms] (25) IRSwT (26) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (27) IRSwT (28) TempFilterProof [SOUND, 10 ms] (29) IntTRS (30) RankingReductionPairProof [EQUIVALENT, 4 ms] (31) YES ---------------------------------------- (0) Obligation: Rules: l0(i2HAT0, i34HAT0, i6HAT0, i8HAT0) -> l1(i2HATpost, i34HATpost, i6HATpost, i8HATpost) :|: i8HAT0 = i8HATpost && i6HAT0 = i6HATpost && i34HAT0 = i34HATpost && i2HAT0 = i2HATpost && 1 + i34HAT0 <= 0 l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x = x4 && x5 = -1 + x1 && 0 <= x1 l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && 1 <= x8 l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 && 1 + x16 <= 0 l3(x24, x25, x26, x27) -> l2(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x24 = x28 && x29 = 999 && 0 <= x24 && x24 <= 0 l2(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 l4(x40, x41, x42, x43) -> l5(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44 l6(x48, x49, x50, x51) -> l7(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 && 1 + x51 <= 0 l6(x56, x57, x58, x59) -> l8(x60, x61, x62, x63) :|: x58 = x62 && x57 = x61 && x56 = x60 && x63 = -1 + x59 && 0 <= x59 l8(x64, x65, x66, x67) -> l6(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68 l5(x72, x73, x74, x75) -> l8(x76, x77, x78, x79) :|: x74 = x78 && x73 = x77 && x72 = x76 && x79 = 999 && 1 + x74 <= 0 l5(x80, x81, x82, x83) -> l4(x84, x85, x86, x87) :|: x83 = x87 && x81 = x85 && x80 = x84 && x86 = -1 + x82 && 0 <= x82 l1(x88, x89, x90, x91) -> l4(x92, x93, x94, x95) :|: x91 = x95 && x89 = x93 && x88 = x92 && x94 = 999 l9(x96, x97, x98, x99) -> l3(x100, x101, x102, x103) :|: x99 = x103 && x98 = x102 && x97 = x101 && x100 = 1 l10(x104, x105, x106, x107) -> l9(x108, x109, x110, x111) :|: x107 = x111 && x106 = x110 && x105 = x109 && x104 = x108 Start term: l10(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(i2HAT0, i34HAT0, i6HAT0, i8HAT0) -> l1(i2HATpost, i34HATpost, i6HATpost, i8HATpost) :|: i8HAT0 = i8HATpost && i6HAT0 = i6HATpost && i34HAT0 = i34HATpost && i2HAT0 = i2HATpost && 1 + i34HAT0 <= 0 l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x = x4 && x5 = -1 + x1 && 0 <= x1 l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && 1 <= x8 l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 && 1 + x16 <= 0 l3(x24, x25, x26, x27) -> l2(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x24 = x28 && x29 = 999 && 0 <= x24 && x24 <= 0 l2(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 l4(x40, x41, x42, x43) -> l5(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44 l6(x48, x49, x50, x51) -> l7(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 && 1 + x51 <= 0 l6(x56, x57, x58, x59) -> l8(x60, x61, x62, x63) :|: x58 = x62 && x57 = x61 && x56 = x60 && x63 = -1 + x59 && 0 <= x59 l8(x64, x65, x66, x67) -> l6(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68 l5(x72, x73, x74, x75) -> l8(x76, x77, x78, x79) :|: x74 = x78 && x73 = x77 && x72 = x76 && x79 = 999 && 1 + x74 <= 0 l5(x80, x81, x82, x83) -> l4(x84, x85, x86, x87) :|: x83 = x87 && x81 = x85 && x80 = x84 && x86 = -1 + x82 && 0 <= x82 l1(x88, x89, x90, x91) -> l4(x92, x93, x94, x95) :|: x91 = x95 && x89 = x93 && x88 = x92 && x94 = 999 l9(x96, x97, x98, x99) -> l3(x100, x101, x102, x103) :|: x99 = x103 && x98 = x102 && x97 = x101 && x100 = 1 l10(x104, x105, x106, x107) -> l9(x108, x109, x110, x111) :|: x107 = x111 && x106 = x110 && x105 = x109 && x104 = x108 Start term: l10(i2HAT0, i34HAT0, i6HAT0, i8HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(i2HAT0, i34HAT0, i6HAT0, i8HAT0) -> l1(i2HATpost, i34HATpost, i6HATpost, i8HATpost) :|: i8HAT0 = i8HATpost && i6HAT0 = i6HATpost && i34HAT0 = i34HATpost && i2HAT0 = i2HATpost && 1 + i34HAT0 <= 0 (2) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x = x4 && x5 = -1 + x1 && 0 <= x1 (3) l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && 1 <= x8 (4) l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 && 1 + x16 <= 0 (5) l3(x24, x25, x26, x27) -> l2(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x24 = x28 && x29 = 999 && 0 <= x24 && x24 <= 0 (6) l2(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 (7) l4(x40, x41, x42, x43) -> l5(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44 (8) l6(x48, x49, x50, x51) -> l7(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 && 1 + x51 <= 0 (9) l6(x56, x57, x58, x59) -> l8(x60, x61, x62, x63) :|: x58 = x62 && x57 = x61 && x56 = x60 && x63 = -1 + x59 && 0 <= x59 (10) l8(x64, x65, x66, x67) -> l6(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68 (11) l5(x72, x73, x74, x75) -> l8(x76, x77, x78, x79) :|: x74 = x78 && x73 = x77 && x72 = x76 && x79 = 999 && 1 + x74 <= 0 (12) l5(x80, x81, x82, x83) -> l4(x84, x85, x86, x87) :|: x83 = x87 && x81 = x85 && x80 = x84 && x86 = -1 + x82 && 0 <= x82 (13) l1(x88, x89, x90, x91) -> l4(x92, x93, x94, x95) :|: x91 = x95 && x89 = x93 && x88 = x92 && x94 = 999 (14) l9(x96, x97, x98, x99) -> l3(x100, x101, x102, x103) :|: x99 = x103 && x98 = x102 && x97 = x101 && x100 = 1 (15) l10(x104, x105, x106, x107) -> l9(x108, x109, x110, x111) :|: x107 = x111 && x106 = x110 && x105 = x109 && x104 = x108 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) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x = x4 && x5 = -1 + x1 && 0 <= x1 (2) l2(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (6) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (7) Obligation: Rules: l0(x36:0, x1:0, x2:0, x39:0) -> l0(x36:0, -1 + x1:0, x2:0, x39:0) :|: x1:0 > -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) -> l0(x2) ---------------------------------------- (9) Obligation: Rules: l0(x1:0) -> l0(-1 + x1:0) :|: x1:0 > -1 ---------------------------------------- (10) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l0(INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (11) Obligation: Rules: l0(x1:0) -> l0(c) :|: c = -1 + x1:0 && x1:0 > -1 ---------------------------------------- (12) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l0 ] = l0_1 The following rules are decreasing: l0(x1:0) -> l0(c) :|: c = -1 + x1:0 && x1:0 > -1 The following rules are bounded: l0(x1:0) -> l0(c) :|: c = -1 + x1:0 && x1:0 > -1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Termination digraph: Nodes: (1) l4(x40, x41, x42, x43) -> l5(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x40 = x44 (2) l5(x80, x81, x82, x83) -> l4(x84, x85, x86, x87) :|: x83 = x87 && x81 = x85 && x80 = x84 && x86 = -1 + x82 && 0 <= x82 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (15) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (16) Obligation: Rules: l4(x40:0, x41:0, x42:0, x43:0) -> l4(x40:0, x41:0, -1 + x42:0, x43:0) :|: x42:0 > -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) -> l4(x3) ---------------------------------------- (18) Obligation: Rules: l4(x42:0) -> l4(-1 + x42:0) :|: x42:0 > -1 ---------------------------------------- (19) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l4(INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (20) Obligation: Rules: l4(x42:0) -> l4(c) :|: c = -1 + x42:0 && x42:0 > -1 ---------------------------------------- (21) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l4(x)] = x The following rules are decreasing: l4(x42:0) -> l4(c) :|: c = -1 + x42:0 && x42:0 > -1 The following rules are bounded: l4(x42:0) -> l4(c) :|: c = -1 + x42:0 && x42:0 > -1 ---------------------------------------- (22) YES ---------------------------------------- (23) Obligation: Termination digraph: Nodes: (1) l8(x64, x65, x66, x67) -> l6(x68, x69, x70, x71) :|: x67 = x71 && x66 = x70 && x65 = x69 && x64 = x68 (2) l6(x56, x57, x58, x59) -> l8(x60, x61, x62, x63) :|: x58 = x62 && x57 = x61 && x56 = x60 && x63 = -1 + x59 && 0 <= x59 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (24) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (25) Obligation: Rules: l8(x60:0, x61:0, x62:0, x67:0) -> l8(x60:0, x61:0, x62:0, -1 + x67:0) :|: x67:0 > -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) -> l8(x4) ---------------------------------------- (27) Obligation: Rules: l8(x67:0) -> l8(-1 + x67:0) :|: x67:0 > -1 ---------------------------------------- (28) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l8(INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (29) Obligation: Rules: l8(x67:0) -> l8(c) :|: c = -1 + x67:0 && x67:0 > -1 ---------------------------------------- (30) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l8 ] = l8_1 The following rules are decreasing: l8(x67:0) -> l8(c) :|: c = -1 + x67:0 && x67:0 > -1 The following rules are bounded: l8(x67:0) -> l8(c) :|: c = -1 + x67:0 && x67:0 > -1 ---------------------------------------- (31) YES