YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 342 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 28 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) IRSwTChainingProof [EQUIVALENT, 0 ms] (10) IRSwT (11) IRSwTTerminationDigraphProof [EQUIVALENT, 25 ms] (12) AND (13) IRSwT (14) IntTRSCompressionProof [EQUIVALENT, 0 ms] (15) IRSwT (16) TempFilterProof [SOUND, 3 ms] (17) IntTRS (18) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (19) YES (20) IRSwT (21) IntTRSCompressionProof [EQUIVALENT, 0 ms] (22) IRSwT (23) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (24) IRSwT (25) TempFilterProof [SOUND, 9 ms] (26) IntTRS (27) RankingReductionPairProof [EQUIVALENT, 0 ms] (28) YES ---------------------------------------- (0) Obligation: Rules: l0(c0HAT0, deltaextHAT0, deltaext_newHAT0, wntHAT0) -> l2(c0HATpost, deltaextHATpost, deltaext_newHATpost, wntHATpost) :|: wntHAT0 = wntHATpost && deltaext_newHAT0 = deltaext_newHATpost && deltaextHAT0 = deltaextHATpost && c0HAT0 = c0HATpost && 1 + deltaextHAT0 <= deltaext_newHAT0 l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 && 1 + x2 <= x1 l2(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x8 = x12 && x13 = x10 l1(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x17 = x21 && x16 = x20 && x22 = x17 && x16 + x19 <= 2 * x17 && -1 + 2 * x17 <= x16 + x19 l1(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x24 = x28 && x30 = -1 + x25 && 1 + x24 + x27 <= -1 + 2 * x25 l1(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x33 = x37 && x32 = x36 && x38 = 1 + x33 && 1 + 2 * x33 <= x32 + x35 l3(x40, x41, x42, x43) -> l1(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x44 = 2 && x41 <= 3 && 0 <= x41 && x43 <= 3 && 0 <= x43 l4(x48, x49, x50, x51) -> l3(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 Start term: l4(c0HAT0, deltaextHAT0, deltaext_newHAT0, wntHAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(c0HAT0, deltaextHAT0, deltaext_newHAT0, wntHAT0) -> l2(c0HATpost, deltaextHATpost, deltaext_newHATpost, wntHATpost) :|: wntHAT0 = wntHATpost && deltaext_newHAT0 = deltaext_newHATpost && deltaextHAT0 = deltaextHATpost && c0HAT0 = c0HATpost && 1 + deltaextHAT0 <= deltaext_newHAT0 l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 && 1 + x2 <= x1 l2(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x8 = x12 && x13 = x10 l1(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x17 = x21 && x16 = x20 && x22 = x17 && x16 + x19 <= 2 * x17 && -1 + 2 * x17 <= x16 + x19 l1(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x24 = x28 && x30 = -1 + x25 && 1 + x24 + x27 <= -1 + 2 * x25 l1(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x33 = x37 && x32 = x36 && x38 = 1 + x33 && 1 + 2 * x33 <= x32 + x35 l3(x40, x41, x42, x43) -> l1(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x44 = 2 && x41 <= 3 && 0 <= x41 && x43 <= 3 && 0 <= x43 l4(x48, x49, x50, x51) -> l3(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 Start term: l4(c0HAT0, deltaextHAT0, deltaext_newHAT0, wntHAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(c0HAT0, deltaextHAT0, deltaext_newHAT0, wntHAT0) -> l2(c0HATpost, deltaextHATpost, deltaext_newHATpost, wntHATpost) :|: wntHAT0 = wntHATpost && deltaext_newHAT0 = deltaext_newHATpost && deltaextHAT0 = deltaextHATpost && c0HAT0 = c0HATpost && 1 + deltaextHAT0 <= deltaext_newHAT0 (2) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 && 1 + x2 <= x1 (3) l2(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x8 = x12 && x13 = x10 (4) l1(x16, x17, x18, x19) -> l0(x20, x21, x22, x23) :|: x19 = x23 && x17 = x21 && x16 = x20 && x22 = x17 && x16 + x19 <= 2 * x17 && -1 + 2 * x17 <= x16 + x19 (5) l1(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x24 = x28 && x30 = -1 + x25 && 1 + x24 + x27 <= -1 + 2 * x25 (6) l1(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x33 = x37 && x32 = x36 && x38 = 1 + x33 && 1 + 2 * x33 <= x32 + x35 (7) l3(x40, x41, x42, x43) -> l1(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x41 = x45 && x44 = 2 && x41 <= 3 && 0 <= x41 && x43 <= 3 && 0 <= x43 (8) l4(x48, x49, x50, x51) -> l3(x52, x53, x54, x55) :|: x51 = x55 && x50 = x54 && x49 = x53 && x48 = x52 Arcs: (1) -> (3) (2) -> (3) (3) -> (4), (5), (6) (5) -> (2) (6) -> (1) (7) -> (4), (5), (6) (8) -> (7) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l0(c0HAT0, deltaextHAT0, deltaext_newHAT0, wntHAT0) -> l2(c0HATpost, deltaextHATpost, deltaext_newHATpost, wntHATpost) :|: wntHAT0 = wntHATpost && deltaext_newHAT0 = deltaext_newHATpost && deltaextHAT0 = deltaextHATpost && c0HAT0 = c0HATpost && 1 + deltaextHAT0 <= deltaext_newHAT0 (2) l1(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x33 = x37 && x32 = x36 && x38 = 1 + x33 && 1 + 2 * x33 <= x32 + x35 (3) l2(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x8 = x12 && x13 = x10 (4) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 && 1 + x2 <= x1 (5) l1(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x24 = x28 && x30 = -1 + x25 && 1 + x24 + x27 <= -1 + 2 * x25 Arcs: (1) -> (3) (2) -> (1) (3) -> (2), (5) (4) -> (3) (5) -> (4) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l1(x32:0, x33:0, x34:0, x35:0) -> l0(x32:0, x33:0, 1 + x33:0, x35:0) :|: x32:0 + x35:0 >= 1 + 2 * x33:0 l0(x12:0, x1:0, x13:0, x15:0) -> l1(x12:0, x13:0, x13:0, x15:0) :|: x1:0 >= 1 + x13:0 l0(c0HAT0:0, deltaextHAT0:0, deltaext_newHAT0:0, wntHAT0:0) -> l1(c0HAT0:0, deltaext_newHAT0:0, deltaext_newHAT0:0, wntHAT0:0) :|: deltaext_newHAT0:0 >= 1 + deltaextHAT0:0 l1(x24:0, x25:0, x26:0, x27:0) -> l0(x24:0, x25:0, -1 + x25:0, x27:0) :|: 1 + x24:0 + x27:0 <= -1 + 2 * x25:0 ---------------------------------------- (7) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: none ---------------------------------------- (8) Obligation: Rules: l1(x32:0, x33:0, x34:0, x35:0) -> l0(x32:0, x33:0, 1 + x33:0, x35:0) :|: x32:0 + x35:0 >= 1 + 2 * x33:0 l0(x12:0, x1:0, x13:0, x15:0) -> l1(x12:0, x13:0, x13:0, x15:0) :|: x1:0 >= 1 + x13:0 l0(c0HAT0:0, deltaextHAT0:0, deltaext_newHAT0:0, wntHAT0:0) -> l1(c0HAT0:0, deltaext_newHAT0:0, deltaext_newHAT0:0, wntHAT0:0) :|: deltaext_newHAT0:0 >= 1 + deltaextHAT0:0 l1(x24:0, x25:0, x26:0, x27:0) -> l0(x24:0, x25:0, -1 + x25:0, x27:0) :|: 1 + x24:0 + x27:0 <= -1 + 2 * x25:0 ---------------------------------------- (9) IRSwTChainingProof (EQUIVALENT) Chaining! ---------------------------------------- (10) Obligation: Rules: l0(x12:0, x1:0, x13:0, x15:0) -> l1(x12:0, x13:0, x13:0, x15:0) :|: x1:0 >= 1 + x13:0 l1(x8, x9, x10, x11) -> l1(x8, 1 + x9, 1 + x9, x11) :|: TRUE && x8 + x11 + -2 * x9 >= 1 && 0 >= 2 l0(c0HAT0:0, deltaextHAT0:0, deltaext_newHAT0:0, wntHAT0:0) -> l1(c0HAT0:0, deltaext_newHAT0:0, deltaext_newHAT0:0, wntHAT0:0) :|: deltaext_newHAT0:0 >= 1 + deltaextHAT0:0 l1(x16, x17, x18, x19) -> l1(x16, 1 + x17, 1 + x17, x19) :|: TRUE && x16 + x19 + -2 * x17 >= 1 l1(x24:0, x25:0, x26:0, x27:0) -> l0(x24:0, x25:0, -1 + x25:0, x27:0) :|: 1 + x24:0 + x27:0 <= -1 + 2 * x25:0 ---------------------------------------- (11) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(x12:0, x1:0, x13:0, x15:0) -> l1(x12:0, x13:0, x13:0, x15:0) :|: x1:0 >= 1 + x13:0 (2) l1(x8, x9, x10, x11) -> l1(x8, 1 + x9, 1 + x9, x11) :|: TRUE && x8 + x11 + -2 * x9 >= 1 && 0 >= 2 (3) l0(c0HAT0:0, deltaextHAT0:0, deltaext_newHAT0:0, wntHAT0:0) -> l1(c0HAT0:0, deltaext_newHAT0:0, deltaext_newHAT0:0, wntHAT0:0) :|: deltaext_newHAT0:0 >= 1 + deltaextHAT0:0 (4) l1(x16, x17, x18, x19) -> l1(x16, 1 + x17, 1 + x17, x19) :|: TRUE && x16 + x19 + -2 * x17 >= 1 (5) l1(x24:0, x25:0, x26:0, x27:0) -> l0(x24:0, x25:0, -1 + x25:0, x27:0) :|: 1 + x24:0 + x27:0 <= -1 + 2 * x25:0 Arcs: (1) -> (4), (5) (3) -> (4), (5) (4) -> (4) (5) -> (1) This digraph is fully evaluated! ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Termination digraph: Nodes: (1) l0(x12:0, x1:0, x13:0, x15:0) -> l1(x12:0, x13:0, x13:0, x15:0) :|: x1:0 >= 1 + x13:0 (2) l1(x24:0, x25:0, x26:0, x27:0) -> l0(x24:0, x25:0, -1 + x25:0, x27:0) :|: 1 + x24:0 + x27:0 <= -1 + 2 * x25:0 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (14) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (15) Obligation: Rules: l0(x12:0:0, x1:0:0, x13:0:0, x15:0:0) -> l0(x12:0:0, x13:0:0, -1 + x13:0:0, x15:0:0) :|: x1:0:0 >= 1 + x13:0:0 && 1 + x12:0:0 + x15:0:0 <= -1 + 2 * x13:0:0 ---------------------------------------- (16) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l0(INTEGER, INTEGER, INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (17) Obligation: Rules: l0(x12:0:0, x1:0:0, x13:0:0, x15:0:0) -> l0(x12:0:0, x13:0:0, c, x15:0:0) :|: c = -1 + x13:0:0 && (x1:0:0 >= 1 + x13:0:0 && 1 + x12:0:0 + x15:0:0 <= -1 + 2 * x13:0:0) ---------------------------------------- (18) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l0(x, x1, x2, x3)] = -1 - x + 2*x2 - x3 The following rules are decreasing: l0(x12:0:0, x1:0:0, x13:0:0, x15:0:0) -> l0(x12:0:0, x13:0:0, c, x15:0:0) :|: c = -1 + x13:0:0 && (x1:0:0 >= 1 + x13:0:0 && 1 + x12:0:0 + x15:0:0 <= -1 + 2 * x13:0:0) The following rules are bounded: l0(x12:0:0, x1:0:0, x13:0:0, x15:0:0) -> l0(x12:0:0, x13:0:0, c, x15:0:0) :|: c = -1 + x13:0:0 && (x1:0:0 >= 1 + x13:0:0 && 1 + x12:0:0 + x15:0:0 <= -1 + 2 * x13:0:0) ---------------------------------------- (19) YES ---------------------------------------- (20) Obligation: Termination digraph: Nodes: (1) l1(x16, x17, x18, x19) -> l1(x16, 1 + x17, 1 + x17, x19) :|: TRUE && x16 + x19 + -2 * x17 >= 1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (21) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (22) Obligation: Rules: l1(x16:0, x17:0, x18:0, x19:0) -> l1(x16:0, 1 + x17:0, 1 + x17:0, x19:0) :|: x16:0 + x19:0 + -2 * x17:0 >= 1 ---------------------------------------- (23) 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, x2, x4) ---------------------------------------- (24) Obligation: Rules: l1(x16:0, x17:0, x19:0) -> l1(x16:0, 1 + x17:0, x19:0) :|: x16:0 + x19:0 + -2 * x17:0 >= 1 ---------------------------------------- (25) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l1(INTEGER, INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (26) Obligation: Rules: l1(x16:0, x17:0, x19:0) -> l1(x16:0, c, x19:0) :|: c = 1 + x17:0 && x16:0 + x19:0 + -2 * x17:0 >= 1 ---------------------------------------- (27) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l1 ] = 1/2*l1_1 + 1/2*l1_3 + -1*l1_2 The following rules are decreasing: l1(x16:0, x17:0, x19:0) -> l1(x16:0, c, x19:0) :|: c = 1 + x17:0 && x16:0 + x19:0 + -2 * x17:0 >= 1 The following rules are bounded: l1(x16:0, x17:0, x19:0) -> l1(x16:0, c, x19:0) :|: c = 1 + x17:0 && x16:0 + x19:0 + -2 * x17:0 >= 1 ---------------------------------------- (28) YES