NO proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be disproven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof [EQUIVALENT, 459 ms] (4) AND (5) IRSwT (6) IntTRSCompressionProof [EQUIVALENT, 0 ms] (7) IRSwT (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (9) IRSwT (10) TempFilterProof [SOUND, 41 ms] (11) IntTRS (12) PolynomialOrderProcessor [EQUIVALENT, 13 ms] (13) IntTRS (14) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (15) YES (16) IRSwT (17) IntTRSCompressionProof [EQUIVALENT, 0 ms] (18) IRSwT (19) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (20) IRSwT (21) FilterProof [EQUIVALENT, 0 ms] (22) IntTRS (23) IntTRSPeriodicNontermProof [COMPLETE, 0 ms] (24) NO ---------------------------------------- (0) Obligation: Rules: l0(___rho_1HAT0, ___rho_2HAT0, pHAT0, startHAT0, xHAT0) -> l1(___rho_1HATpost, ___rho_2HATpost, pHATpost, startHATpost, xHATpost) :|: xHAT0 = xHATpost && startHAT0 = startHATpost && ___rho_2HAT0 = ___rho_2HATpost && ___rho_1HAT0 = ___rho_1HATpost && pHATpost = 0 && ___rho_2HAT0 <= 0 l0(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x1 = x6 && x = x5 && x7 = 1 && 1 <= x1 l1(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x10 = x15 && x16 = x16 l2(x20, x21, x22, x23, x24) -> l3(x25, x26, x27, x28, x29) :|: x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && x29 = -2 + x24 && x20 <= 0 l2(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 && x39 = -1 + x34 && 1 <= x30 l3(x40, x41, x42, x43, x44) -> l1(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && x44 <= 0 l3(x50, x51, x52, x53, x54) -> l2(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && 1 <= x54 l4(x60, x61, x62, x63, x64) -> l3(x65, x66, x67, x68, x69) :|: x67 = 0 && x70 = 0 && x71 = 1 && x68 = 0 && x60 = x65 && x61 = x66 && x64 = x69 l5(x72, x73, x74, x75, x76) -> l4(x77, x78, x79, x80, x81) :|: x76 = x81 && x75 = x80 && x74 = x79 && x73 = x78 && x72 = x77 Start term: l5(___rho_1HAT0, ___rho_2HAT0, pHAT0, startHAT0, xHAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(___rho_1HAT0, ___rho_2HAT0, pHAT0, startHAT0, xHAT0) -> l1(___rho_1HATpost, ___rho_2HATpost, pHATpost, startHATpost, xHATpost) :|: xHAT0 = xHATpost && startHAT0 = startHATpost && ___rho_2HAT0 = ___rho_2HATpost && ___rho_1HAT0 = ___rho_1HATpost && pHATpost = 0 && ___rho_2HAT0 <= 0 l0(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x1 = x6 && x = x5 && x7 = 1 && 1 <= x1 l1(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x10 = x15 && x16 = x16 l2(x20, x21, x22, x23, x24) -> l3(x25, x26, x27, x28, x29) :|: x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && x29 = -2 + x24 && x20 <= 0 l2(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 && x39 = -1 + x34 && 1 <= x30 l3(x40, x41, x42, x43, x44) -> l1(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && x44 <= 0 l3(x50, x51, x52, x53, x54) -> l2(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && 1 <= x54 l4(x60, x61, x62, x63, x64) -> l3(x65, x66, x67, x68, x69) :|: x67 = 0 && x70 = 0 && x71 = 1 && x68 = 0 && x60 = x65 && x61 = x66 && x64 = x69 l5(x72, x73, x74, x75, x76) -> l4(x77, x78, x79, x80, x81) :|: x76 = x81 && x75 = x80 && x74 = x79 && x73 = x78 && x72 = x77 Start term: l5(___rho_1HAT0, ___rho_2HAT0, pHAT0, startHAT0, xHAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(___rho_1HAT0, ___rho_2HAT0, pHAT0, startHAT0, xHAT0) -> l1(___rho_1HATpost, ___rho_2HATpost, pHATpost, startHATpost, xHATpost) :|: xHAT0 = xHATpost && startHAT0 = startHATpost && ___rho_2HAT0 = ___rho_2HATpost && ___rho_1HAT0 = ___rho_1HATpost && pHATpost = 0 && ___rho_2HAT0 <= 0 (2) l0(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x1 = x6 && x = x5 && x7 = 1 && 1 <= x1 (3) l1(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x10 = x15 && x16 = x16 (4) l2(x20, x21, x22, x23, x24) -> l3(x25, x26, x27, x28, x29) :|: x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && x29 = -2 + x24 && x20 <= 0 (5) l2(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 && x39 = -1 + x34 && 1 <= x30 (6) l3(x40, x41, x42, x43, x44) -> l1(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && x44 <= 0 (7) l3(x50, x51, x52, x53, x54) -> l2(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && 1 <= x54 (8) l4(x60, x61, x62, x63, x64) -> l3(x65, x66, x67, x68, x69) :|: x67 = 0 && x70 = 0 && x71 = 1 && x68 = 0 && x60 = x65 && x61 = x66 && x64 = x69 (9) l5(x72, x73, x74, x75, x76) -> l4(x77, x78, x79, x80, x81) :|: x76 = x81 && x75 = x80 && x74 = x79 && x73 = x78 && x72 = x77 Arcs: (1) -> (3) (2) -> (3) (3) -> (1), (2) (4) -> (6), (7) (5) -> (6), (7) (6) -> (3) (7) -> (4), (5) (8) -> (6), (7) (9) -> (8) This digraph is fully evaluated! ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Termination digraph: Nodes: (1) l2(x20, x21, x22, x23, x24) -> l3(x25, x26, x27, x28, x29) :|: x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && x29 = -2 + x24 && x20 <= 0 (2) l3(x50, x51, x52, x53, x54) -> l2(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x55 = x55 && 1 <= x54 (3) l2(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 && x39 = -1 + x34 && 1 <= x30 Arcs: (1) -> (2) (2) -> (1), (3) (3) -> (2) This digraph is fully evaluated! ---------------------------------------- (6) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (7) Obligation: Rules: l2(x20:0, x21:0, x22:0, x23:0, x24:0) -> l2(x55:0, x21:0, x22:0, x23:0, -2 + x24:0) :|: x20:0 < 1 && x24:0 > 2 l2(x, x1, x2, x3, x4) -> l2(x5, x1, x2, x3, -1 + x4) :|: x > 0 && x4 > 1 ---------------------------------------- (8) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l2(x1, x2, x3, x4, x5) -> l2(x1, x5) ---------------------------------------- (9) Obligation: Rules: l2(x20:0, x24:0) -> l2(x55:0, -2 + x24:0) :|: x20:0 < 1 && x24:0 > 2 l2(x, x4) -> l2(x5, -1 + x4) :|: x > 0 && x4 > 1 ---------------------------------------- (10) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l2(VARIABLE, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (11) Obligation: Rules: l2(x20:0, x24:0) -> l2(x55:0, c) :|: c = -2 + x24:0 && (x20:0 < 1 && x24:0 > 2) l2(x, x4) -> l2(x5, c1) :|: c1 = -1 + x4 && (x > 0 && x4 > 1) ---------------------------------------- (12) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l2(x, x1)] = -3 + x1 The following rules are decreasing: l2(x20:0, x24:0) -> l2(x55:0, c) :|: c = -2 + x24:0 && (x20:0 < 1 && x24:0 > 2) l2(x, x4) -> l2(x5, c1) :|: c1 = -1 + x4 && (x > 0 && x4 > 1) The following rules are bounded: l2(x20:0, x24:0) -> l2(x55:0, c) :|: c = -2 + x24:0 && (x20:0 < 1 && x24:0 > 2) ---------------------------------------- (13) Obligation: Rules: l2(x, x4) -> l2(x5, c1) :|: c1 = -1 + x4 && (x > 0 && x4 > 1) ---------------------------------------- (14) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l2(x, x1)] = x1 The following rules are decreasing: l2(x, x4) -> l2(x5, c1) :|: c1 = -1 + x4 && (x > 0 && x4 > 1) The following rules are bounded: l2(x, x4) -> l2(x5, c1) :|: c1 = -1 + x4 && (x > 0 && x4 > 1) ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Termination digraph: Nodes: (1) l0(___rho_1HAT0, ___rho_2HAT0, pHAT0, startHAT0, xHAT0) -> l1(___rho_1HATpost, ___rho_2HATpost, pHATpost, startHATpost, xHATpost) :|: xHAT0 = xHATpost && startHAT0 = startHATpost && ___rho_2HAT0 = ___rho_2HATpost && ___rho_1HAT0 = ___rho_1HATpost && pHATpost = 0 && ___rho_2HAT0 <= 0 (2) l1(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x10 = x15 && x16 = x16 (3) l0(x, x1, x2, x3, x4) -> l1(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x1 = x6 && x = x5 && x7 = 1 && 1 <= x1 Arcs: (1) -> (2) (2) -> (1), (3) (3) -> (2) This digraph is fully evaluated! ---------------------------------------- (17) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (18) Obligation: Rules: l0(x15:0, x1:0, x2:0, x18:0, x19:0) -> l0(x15:0, x16:0, 1, x18:0, x19:0) :|: x1:0 > 0 l0(x, x1, x2, x3, x4) -> l0(x, x5, 0, x3, x4) :|: x1 < 1 ---------------------------------------- (19) 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(x2) ---------------------------------------- (20) Obligation: Rules: l0(x1:0) -> l0(x16:0) :|: x1:0 > 0 l0(x1) -> l0(x5) :|: x1 < 1 ---------------------------------------- (21) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: l0(VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (22) Obligation: Rules: l0(x1:0) -> l0(x16:0) :|: x1:0 > 0 l0(x1) -> l0(x5) :|: x1 < 1 ---------------------------------------- (23) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, x1:0) -> f(1, x16:0) :|: pc = 1 && x1:0 > 0 f(pc, x1) -> f(1, x5) :|: pc = 1 && x1 < 1 Witness term starting non-terminating reduction: f(1, -8) ---------------------------------------- (24) NO