NO proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be disproven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 274 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 0 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) IRSwTChainingProof [EQUIVALENT, 0 ms] (10) IRSwT (11) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (12) AND (13) IRSwT (14) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (15) IRSwT (16) FilterProof [EQUIVALENT, 0 ms] (17) IntTRS (18) IntTRSNonPeriodicNontermProof [COMPLETE, 0 ms] (19) NO (20) IRSwT (21) IntTRSCompressionProof [EQUIVALENT, 0 ms] (22) IRSwT ---------------------------------------- (0) Obligation: Rules: l0(Result_4HAT0, __const_19HAT0, __const_29HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, __const_19HATpost, __const_29HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && __const_29HAT0 = __const_29HATpost && __const_19HAT0 = __const_19HATpost && Result_4HATpost = Result_4HATpost && 1 + __const_19HAT0 - x_5HAT0 <= 0 l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 0 <= x1 - x3 l3(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 l2(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x22 = x27 && x21 = x26 && x20 = x25 && x28 = -1 + x23 && 1 + x22 - x24 <= 0 l2(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x34 = x39 && x32 = x37 && x31 = x36 && x30 = x35 && x38 = -1 + x33 && 0 <= x32 - x34 l4(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 l5(x50, x51, x52, x53, x54) -> l3(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 Start term: l5(Result_4HAT0, __const_19HAT0, __const_29HAT0, x_5HAT0, y_6HAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(Result_4HAT0, __const_19HAT0, __const_29HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, __const_19HATpost, __const_29HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && __const_29HAT0 = __const_29HATpost && __const_19HAT0 = __const_19HATpost && Result_4HATpost = Result_4HATpost && 1 + __const_19HAT0 - x_5HAT0 <= 0 l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 0 <= x1 - x3 l3(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 l2(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x22 = x27 && x21 = x26 && x20 = x25 && x28 = -1 + x23 && 1 + x22 - x24 <= 0 l2(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x34 = x39 && x32 = x37 && x31 = x36 && x30 = x35 && x38 = -1 + x33 && 0 <= x32 - x34 l4(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 l5(x50, x51, x52, x53, x54) -> l3(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 Start term: l5(Result_4HAT0, __const_19HAT0, __const_29HAT0, x_5HAT0, y_6HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(Result_4HAT0, __const_19HAT0, __const_29HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, __const_19HATpost, __const_29HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && __const_29HAT0 = __const_29HATpost && __const_19HAT0 = __const_19HATpost && Result_4HATpost = Result_4HATpost && 1 + __const_19HAT0 - x_5HAT0 <= 0 (2) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 0 <= x1 - x3 (3) l3(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 (4) l2(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x22 = x27 && x21 = x26 && x20 = x25 && x28 = -1 + x23 && 1 + x22 - x24 <= 0 (5) l2(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x34 = x39 && x32 = x37 && x31 = x36 && x30 = x35 && x38 = -1 + x33 && 0 <= x32 - x34 (6) l4(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 (7) l5(x50, x51, x52, x53, x54) -> l3(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 Arcs: (2) -> (4), (5) (3) -> (1), (2) (4) -> (1), (2) (5) -> (6) (6) -> (4), (5) (7) -> (3) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 0 <= x1 - x3 (2) l2(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x22 = x27 && x21 = x26 && x20 = x25 && x28 = -1 + x23 && 1 + x22 - x24 <= 0 (3) l4(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 (4) l2(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x34 = x39 && x32 = x37 && x31 = x36 && x30 = x35 && x38 = -1 + x33 && 0 <= x32 - x34 Arcs: (1) -> (2), (4) (2) -> (1) (3) -> (2), (4) (4) -> (3) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l2(x20:0, x21:0, x22:0, x23:0, x24:0) -> l2(x20:0, x21:0, x22:0, -1 + x23:0, x24:0) :|: 1 + x22:0 - x24:0 <= 0 && x21:0 - (-1 + x23:0) >= 0 l2(x30:0, x31:0, x32:0, x33:0, x34:0) -> l2(x30:0, x31:0, x32:0, -1 + x33:0, x34:0) :|: x32:0 - x34:0 >= 0 ---------------------------------------- (7) 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(x2, x3, x4, x5) ---------------------------------------- (8) Obligation: Rules: l2(x21:0, x22:0, x23:0, x24:0) -> l2(x21:0, x22:0, -1 + x23:0, x24:0) :|: 1 + x22:0 - x24:0 <= 0 && x21:0 - (-1 + x23:0) >= 0 l2(x31:0, x32:0, x33:0, x34:0) -> l2(x31:0, x32:0, -1 + x33:0, x34:0) :|: x32:0 - x34:0 >= 0 ---------------------------------------- (9) IRSwTChainingProof (EQUIVALENT) Chaining! ---------------------------------------- (10) Obligation: Rules: l2(x, x1, x2, x3) -> l2(x, x1, -2 + x2, x3) :|: TRUE && x1 + -1 * x3 <= -1 && x + -1 * x2 >= -1 l2(x31:0, x32:0, x33:0, x34:0) -> l2(x31:0, x32:0, -1 + x33:0, x34:0) :|: x32:0 - x34:0 >= 0 ---------------------------------------- (11) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l2(x, x1, x2, x3) -> l2(x, x1, -2 + x2, x3) :|: TRUE && x1 + -1 * x3 <= -1 && x + -1 * x2 >= -1 (2) l2(x31:0, x32:0, x33:0, x34:0) -> l2(x31:0, x32:0, -1 + x33:0, x34:0) :|: x32:0 - x34:0 >= 0 Arcs: (1) -> (1) (2) -> (2) This digraph is fully evaluated! ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Termination digraph: Nodes: (1) l2(x31:0, x32:0, x33:0, x34:0) -> l2(x31:0, x32:0, -1 + x33:0, x34:0) :|: x32:0 - x34:0 >= 0 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (14) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l2(x1, x2, x3, x4) -> l2(x2, x4) ---------------------------------------- (15) Obligation: Rules: l2(x32:0, x34:0) -> l2(x32:0, x34:0) :|: x32:0 - x34:0 >= 0 ---------------------------------------- (16) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: l2(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (17) Obligation: Rules: l2(x32:0, x34:0) -> l2(x32:0, x34:0) :|: x32:0 - x34:0 >= 0 ---------------------------------------- (18) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, x32:0, x34:0) -> f(1, x32:0, x34:0) :|: pc = 1 && x32:0 - x34:0 >= 0 Proved unsatisfiability of the following formula, indicating that the system is never left after entering: (((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1)) and run2_2 = ((run1_2 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and ((run1_1 * 1) + (run1_2 * -1)) >= 0)) and !(((run2_0 * 1)) = ((1 * 1)) and ((run2_1 * 1) + (run2_2 * -1)) >= 0)) Proved satisfiability of the following formula, indicating that the system is entered at least once: ((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1)) and run2_2 = ((run1_2 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and ((run1_1 * 1) + (run1_2 * -1)) >= 0)) ---------------------------------------- (19) NO ---------------------------------------- (20) Obligation: Termination digraph: Nodes: (1) l2(x, x1, x2, x3) -> l2(x, x1, -2 + x2, x3) :|: TRUE && x1 + -1 * x3 <= -1 && x + -1 * x2 >= -1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (21) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (22) Obligation: Rules: l2(x:0, x1:0, x2:0, x3:0) -> l2(x:0, x1:0, -2 + x2:0, x3:0) :|: x:0 + -1 * x2:0 >= -1 && x1:0 + -1 * x3:0 <= -1