YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 215 ms] (4) AND (5) IRSwT (6) IntTRSCompressionProof [EQUIVALENT, 8 ms] (7) IRSwT (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (9) IRSwT (10) TempFilterProof [SOUND, 32 ms] (11) IntTRS (12) RankingReductionPairProof [EQUIVALENT, 0 ms] (13) YES (14) IRSwT (15) IntTRSCompressionProof [EQUIVALENT, 29 ms] (16) IRSwT (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (18) IRSwT (19) TempFilterProof [SOUND, 20 ms] (20) IntTRS (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (22) YES ---------------------------------------- (0) Obligation: Rules: l0(__const_10HAT0, i4HAT0, i7HAT0, tmpHAT0) -> l1(__const_10HATpost, i4HATpost, i7HATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && i4HAT0 = i4HATpost && __const_10HAT0 = __const_10HATpost && i7HATpost = 0 && __const_10HAT0 <= i4HAT0 l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x = x4 && x5 = 1 + x1 && 1 + x1 <= x l2(x8, x9, x10, x11) -> l0(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 l1(x16, x17, x18, x19) -> l3(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 l3(x24, x25, x26, x27) -> l4(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28 && x24 <= x26 l3(x32, x33, x34, x35) -> l1(x36, x37, x38, x39) :|: x35 = x39 && x33 = x37 && x32 = x36 && x38 = 1 + x34 && 1 + x34 <= x32 l5(x40, x41, x42, x43) -> l2(x44, x45, x46, x47) :|: x42 = x46 && x40 = x44 && x45 = 0 && x47 = x47 l6(x48, x49, x50, x51) -> l5(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 Start term: l6(__const_10HAT0, i4HAT0, i7HAT0, tmpHAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(__const_10HAT0, i4HAT0, i7HAT0, tmpHAT0) -> l1(__const_10HATpost, i4HATpost, i7HATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && i4HAT0 = i4HATpost && __const_10HAT0 = __const_10HATpost && i7HATpost = 0 && __const_10HAT0 <= i4HAT0 l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x = x4 && x5 = 1 + x1 && 1 + x1 <= x l2(x8, x9, x10, x11) -> l0(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 l1(x16, x17, x18, x19) -> l3(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 l3(x24, x25, x26, x27) -> l4(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28 && x24 <= x26 l3(x32, x33, x34, x35) -> l1(x36, x37, x38, x39) :|: x35 = x39 && x33 = x37 && x32 = x36 && x38 = 1 + x34 && 1 + x34 <= x32 l5(x40, x41, x42, x43) -> l2(x44, x45, x46, x47) :|: x42 = x46 && x40 = x44 && x45 = 0 && x47 = x47 l6(x48, x49, x50, x51) -> l5(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 Start term: l6(__const_10HAT0, i4HAT0, i7HAT0, tmpHAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(__const_10HAT0, i4HAT0, i7HAT0, tmpHAT0) -> l1(__const_10HATpost, i4HATpost, i7HATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && i4HAT0 = i4HATpost && __const_10HAT0 = __const_10HATpost && i7HATpost = 0 && __const_10HAT0 <= i4HAT0 (2) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x = x4 && x5 = 1 + x1 && 1 + x1 <= x (3) l2(x8, x9, x10, x11) -> l0(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 (4) l1(x16, x17, x18, x19) -> l3(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 (5) l3(x24, x25, x26, x27) -> l4(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28 && x24 <= x26 (6) l3(x32, x33, x34, x35) -> l1(x36, x37, x38, x39) :|: x35 = x39 && x33 = x37 && x32 = x36 && x38 = 1 + x34 && 1 + x34 <= x32 (7) l5(x40, x41, x42, x43) -> l2(x44, x45, x46, x47) :|: x42 = x46 && x40 = x44 && x45 = 0 && x47 = x47 (8) l6(x48, x49, x50, x51) -> l5(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 Arcs: (1) -> (4) (2) -> (3) (3) -> (1), (2) (4) -> (5), (6) (6) -> (4) (7) -> (3) (8) -> (7) 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 && 1 + x1 <= x (2) l2(x8, x9, x10, x11) -> l0(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (6) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (7) Obligation: Rules: l0(x12:0, x1:0, x14:0, x15:0) -> l0(x12:0, 1 + x1:0, x14:0, x15:0) :|: x12:0 >= 1 + x1:0 ---------------------------------------- (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(x1, x2) ---------------------------------------- (9) Obligation: Rules: l0(x12:0, x1:0) -> l0(x12:0, 1 + x1:0) :|: x12:0 >= 1 + x1:0 ---------------------------------------- (10) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l0(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (11) Obligation: Rules: l0(x12:0, x1:0) -> l0(x12:0, c) :|: c = 1 + x1:0 && x12:0 >= 1 + x1:0 ---------------------------------------- (12) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l0 ] = l0_1 + -1*l0_2 The following rules are decreasing: l0(x12:0, x1:0) -> l0(x12:0, c) :|: c = 1 + x1:0 && x12:0 >= 1 + x1:0 The following rules are bounded: l0(x12:0, x1:0) -> l0(x12:0, c) :|: c = 1 + x1:0 && x12:0 >= 1 + x1:0 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Termination digraph: Nodes: (1) l1(x16, x17, x18, x19) -> l3(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 (2) l3(x32, x33, x34, x35) -> l1(x36, x37, x38, x39) :|: x35 = x39 && x33 = x37 && x32 = x36 && x38 = 1 + x34 && 1 + x34 <= x32 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (15) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (16) Obligation: Rules: l1(x16:0, x17:0, x18:0, x19:0) -> l1(x16:0, x17:0, 1 + x18:0, x19:0) :|: x16:0 >= 1 + x18:0 ---------------------------------------- (17) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l1(x1, x2, x3, x4) -> l1(x1, x3) ---------------------------------------- (18) Obligation: Rules: l1(x16:0, x18:0) -> l1(x16:0, 1 + x18:0) :|: x16:0 >= 1 + x18:0 ---------------------------------------- (19) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l1(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (20) Obligation: Rules: l1(x16:0, x18:0) -> l1(x16:0, c) :|: c = 1 + x18:0 && x16:0 >= 1 + x18:0 ---------------------------------------- (21) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l1(x, x1)] = x - x1 The following rules are decreasing: l1(x16:0, x18:0) -> l1(x16:0, c) :|: c = 1 + x18:0 && x16:0 >= 1 + x18:0 The following rules are bounded: l1(x16:0, x18:0) -> l1(x16:0, c) :|: c = 1 + x18:0 && x16:0 >= 1 + x18:0 ---------------------------------------- (22) YES