YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 291 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 3 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 41 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 16 ms] (12) YES ---------------------------------------- (0) Obligation: Rules: l0(Result_4HAT0, __disjvr_0HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, __disjvr_0HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && __disjvr_0HAT0 = __disjvr_0HATpost && Result_4HAT0 = Result_4HATpost l1(x, x1, x2, x3) -> l3(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 && -1 * x2 + x3 <= 0 && -1 * x2 + x3 <= 0 l3(x8, x9, x10, x11) -> l4(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x13 = x9 l4(x16, x17, x18, x19) -> l2(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x20 = x20 l1(x24, x25, x26, x27) -> l5(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x24 = x28 && x30 = 1 + x26 && x27 <= x26 && x26 <= x27 && -1 * x26 + x27 <= 0 && -1 * x26 + x27 <= 0 l5(x32, x33, x34, x35) -> l1(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 l1(x40, x41, x42, x43) -> l6(x44, x45, x46, x47) :|: x43 = x47 && x41 = x45 && x40 = x44 && x46 = 1 + x42 && 0 <= -1 - x42 + x43 l6(x48, x49, x50, x51) -> l1(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 l7(x56, x57, x58, x59) -> l0(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 Start term: l7(Result_4HAT0, __disjvr_0HAT0, 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, __disjvr_0HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, __disjvr_0HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && __disjvr_0HAT0 = __disjvr_0HATpost && Result_4HAT0 = Result_4HATpost l1(x, x1, x2, x3) -> l3(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 && -1 * x2 + x3 <= 0 && -1 * x2 + x3 <= 0 l3(x8, x9, x10, x11) -> l4(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x13 = x9 l4(x16, x17, x18, x19) -> l2(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x20 = x20 l1(x24, x25, x26, x27) -> l5(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x24 = x28 && x30 = 1 + x26 && x27 <= x26 && x26 <= x27 && -1 * x26 + x27 <= 0 && -1 * x26 + x27 <= 0 l5(x32, x33, x34, x35) -> l1(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 l1(x40, x41, x42, x43) -> l6(x44, x45, x46, x47) :|: x43 = x47 && x41 = x45 && x40 = x44 && x46 = 1 + x42 && 0 <= -1 - x42 + x43 l6(x48, x49, x50, x51) -> l1(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 l7(x56, x57, x58, x59) -> l0(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 Start term: l7(Result_4HAT0, __disjvr_0HAT0, x_5HAT0, y_6HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(Result_4HAT0, __disjvr_0HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, __disjvr_0HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && __disjvr_0HAT0 = __disjvr_0HATpost && Result_4HAT0 = Result_4HATpost (2) l1(x, x1, x2, x3) -> l3(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 && -1 * x2 + x3 <= 0 && -1 * x2 + x3 <= 0 (3) l3(x8, x9, x10, x11) -> l4(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 && x13 = x9 (4) l4(x16, x17, x18, x19) -> l2(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x20 = x20 (5) l1(x24, x25, x26, x27) -> l5(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x24 = x28 && x30 = 1 + x26 && x27 <= x26 && x26 <= x27 && -1 * x26 + x27 <= 0 && -1 * x26 + x27 <= 0 (6) l5(x32, x33, x34, x35) -> l1(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 (7) l1(x40, x41, x42, x43) -> l6(x44, x45, x46, x47) :|: x43 = x47 && x41 = x45 && x40 = x44 && x46 = 1 + x42 && 0 <= -1 - x42 + x43 (8) l6(x48, x49, x50, x51) -> l1(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 (9) l7(x56, x57, x58, x59) -> l0(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 Arcs: (1) -> (2), (5), (7) (2) -> (3) (3) -> (4) (5) -> (6) (6) -> (2), (5), (7) (7) -> (8) (8) -> (2), (5), (7) (9) -> (1) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l1(x24, x25, x26, x27) -> l5(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x24 = x28 && x30 = 1 + x26 && x27 <= x26 && x26 <= x27 && -1 * x26 + x27 <= 0 && -1 * x26 + x27 <= 0 (2) l6(x48, x49, x50, x51) -> l1(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 (3) l1(x40, x41, x42, x43) -> l6(x44, x45, x46, x47) :|: x43 = x47 && x41 = x45 && x40 = x44 && x46 = 1 + x42 && 0 <= -1 - x42 + x43 (4) l5(x32, x33, x34, x35) -> l1(x36, x37, x38, x39) :|: x35 = x39 && x34 = x38 && x33 = x37 && x32 = x36 Arcs: (1) -> (4) (2) -> (1), (3) (3) -> (2) (4) -> (1), (3) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l1(x24:0, x25:0, x26:0, x26:0) -> l1(x24:0, x25:0, 1 + x26:0, x26:0) :|: 0 >= -1 * x26:0 + x26:0 l1(x40:0, x41:0, x42:0, x43:0) -> l1(x40:0, x41:0, 1 + x42:0, x43:0) :|: 0 <= -1 - x42:0 + x43:0 ---------------------------------------- (7) 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(x3, x4) ---------------------------------------- (8) Obligation: Rules: l1(x26:0, x26:0) -> l1(1 + x26:0, x26:0) :|: 0 >= -1 * x26:0 + x26:0 l1(x42:0, x43:0) -> l1(1 + x42:0, x43:0) :|: 0 <= -1 - x42:0 + x43:0 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l1(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l1(x26:0, x26:0) -> l1(c, x26:0) :|: c = 1 + x26:0 && 0 >= -1 * x26:0 + x26:0 l1(x42:0, x43:0) -> l1(c1, x43:0) :|: c1 = 1 + x42:0 && 0 <= -1 - x42:0 + x43:0 ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l1(x, x1)] = -x + x1 The following rules are decreasing: l1(x26:0, x26:0) -> l1(c, x26:0) :|: c = 1 + x26:0 && 0 >= -1 * x26:0 + x26:0 l1(x42:0, x43:0) -> l1(c1, x43:0) :|: c1 = 1 + x42:0 && 0 <= -1 - x42:0 + x43:0 The following rules are bounded: l1(x26:0, x26:0) -> l1(c, x26:0) :|: c = 1 + x26:0 && 0 >= -1 * x26:0 + x26:0 l1(x42:0, x43:0) -> l1(c1, x43:0) :|: c1 = 1 + x42:0 && 0 <= -1 - x42:0 + x43:0 ---------------------------------------- (12) YES