YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 631 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 0 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 69 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 6 ms] (12) IntTRS (13) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Rules: l0(cHAT0, oxHAT0, oyHAT0, xHAT0, yHAT0) -> l1(cHATpost, oxHATpost, oyHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && oyHAT0 = oyHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && oyHAT0 <= yHAT0 && oxHAT0 <= xHAT0 l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x2 = x7 && x1 = x6 && x = x5 && x8 = x3 l3(x10, x11, x12, x13, x14) -> l2(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && x10 <= 0 l3(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x25 = 1 && x27 = x24 && x26 = x23 l4(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 && x30 <= 0 l4(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && 1 <= x40 l5(x50, x51, x52, x53, x54) -> l4(x55, x56, x57, x58, x59) :|: x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 && x59 = -1 + x54 l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x62 = x67 && x61 = x66 && x60 = x65 && x68 = -1 + x63 l2(x70, x71, x72, x73, x74) -> l5(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 1 <= x74 && 1 <= x73 l6(x80, x81, x82, x83, x84) -> l2(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && x80 <= 0 l7(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95 Start term: l7(cHAT0, oxHAT0, oyHAT0, xHAT0, yHAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(cHAT0, oxHAT0, oyHAT0, xHAT0, yHAT0) -> l1(cHATpost, oxHATpost, oyHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && oyHAT0 = oyHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && oyHAT0 <= yHAT0 && oxHAT0 <= xHAT0 l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x2 = x7 && x1 = x6 && x = x5 && x8 = x3 l3(x10, x11, x12, x13, x14) -> l2(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && x10 <= 0 l3(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x25 = 1 && x27 = x24 && x26 = x23 l4(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 && x30 <= 0 l4(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && 1 <= x40 l5(x50, x51, x52, x53, x54) -> l4(x55, x56, x57, x58, x59) :|: x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 && x59 = -1 + x54 l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x62 = x67 && x61 = x66 && x60 = x65 && x68 = -1 + x63 l2(x70, x71, x72, x73, x74) -> l5(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 1 <= x74 && 1 <= x73 l6(x80, x81, x82, x83, x84) -> l2(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && x80 <= 0 l7(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95 Start term: l7(cHAT0, oxHAT0, oyHAT0, xHAT0, yHAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(cHAT0, oxHAT0, oyHAT0, xHAT0, yHAT0) -> l1(cHATpost, oxHATpost, oyHATpost, xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHAT0 = xHATpost && oyHAT0 = oyHATpost && oxHAT0 = oxHATpost && cHAT0 = cHATpost && oyHAT0 <= yHAT0 && oxHAT0 <= xHAT0 (2) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x2 = x7 && x1 = x6 && x = x5 && x8 = x3 (3) l3(x10, x11, x12, x13, x14) -> l2(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && x10 <= 0 (4) l3(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x25 = 1 && x27 = x24 && x26 = x23 (5) l4(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 && x30 <= 0 (6) l4(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && 1 <= x40 (7) l5(x50, x51, x52, x53, x54) -> l4(x55, x56, x57, x58, x59) :|: x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 && x59 = -1 + x54 (8) l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x62 = x67 && x61 = x66 && x60 = x65 && x68 = -1 + x63 (9) l2(x70, x71, x72, x73, x74) -> l5(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 1 <= x74 && 1 <= x73 (10) l6(x80, x81, x82, x83, x84) -> l2(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && x80 <= 0 (11) l7(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95 Arcs: (2) -> (9) (3) -> (9) (4) -> (9) (5) -> (3), (4) (6) -> (1), (2) (7) -> (5), (6) (8) -> (5), (6) (9) -> (7), (8) (10) -> (9) (11) -> (10) 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 && x2 = x7 && x1 = x6 && x = x5 && x8 = x3 (2) l4(x40, x41, x42, x43, x44) -> l0(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && 1 <= x40 (3) l5(x50, x51, x52, x53, x54) -> l4(x55, x56, x57, x58, x59) :|: x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55 && x59 = -1 + x54 (4) l2(x70, x71, x72, x73, x74) -> l5(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 1 <= x74 && 1 <= x73 (5) l3(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x25 = 1 && x27 = x24 && x26 = x23 (6) l3(x10, x11, x12, x13, x14) -> l2(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && x10 <= 0 (7) l4(x30, x31, x32, x33, x34) -> l3(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 && x30 <= 0 (8) l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x62 = x67 && x61 = x66 && x60 = x65 && x68 = -1 + x63 Arcs: (1) -> (4) (2) -> (1) (3) -> (2), (7) (4) -> (3), (8) (5) -> (4) (6) -> (4) (7) -> (5), (6) (8) -> (2), (7) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l5(x50:0, x51:0, x52:0, x53:0, x54:0) -> l4(x50:0, x51:0, x52:0, x53:0, -1 + x54:0) :|: TRUE l4(x40:0, x41:0, x42:0, x43:0, x44:0) -> l5(x40:0, x41:0, x42:0, x43:0, x44:0) :|: x43:0 > 0 && x44:0 > 0 && x40:0 > 0 l5(x60:0, x61:0, x62:0, x63:0, x64:0) -> l4(x60:0, x61:0, x62:0, -1 + x63:0, x64:0) :|: TRUE l4(x30:0, x31:0, x32:0, x26:0, x27:0) -> l5(1, x26:0, x27:0, x26:0, x27:0) :|: x26:0 > 0 && x27:0 > 0 && x30:0 < 1 l4(x15:0, x16:0, x17:0, x18:0, x19:0) -> l5(x15:0, x16:0, x17:0, x18:0, x19:0) :|: x15:0 < 1 && x19:0 > 0 && x18:0 > 0 ---------------------------------------- (7) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l4(x1, x2, x3, x4, x5) -> l4(x1, x4, x5) l5(x1, x2, x3, x4, x5) -> l5(x1, x4, x5) ---------------------------------------- (8) Obligation: Rules: l5(x50:0, x53:0, x54:0) -> l4(x50:0, x53:0, -1 + x54:0) :|: TRUE l4(x40:0, x43:0, x44:0) -> l5(x40:0, x43:0, x44:0) :|: x43:0 > 0 && x44:0 > 0 && x40:0 > 0 l5(x60:0, x63:0, x64:0) -> l4(x60:0, -1 + x63:0, x64:0) :|: TRUE l4(x30:0, x26:0, x27:0) -> l5(1, x26:0, x27:0) :|: x26:0 > 0 && x27:0 > 0 && x30:0 < 1 l4(x15:0, x18:0, x19:0) -> l5(x15:0, x18:0, x19:0) :|: x15:0 < 1 && x19:0 > 0 && x18:0 > 0 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l5(VARIABLE, VARIABLE, VARIABLE) l4(VARIABLE, VARIABLE, VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l5(x50:0, x53:0, x54:0) -> l4(x50:0, x53:0, c) :|: c = -1 + x54:0 && TRUE l4(x40:0, x43:0, x44:0) -> l5(x40:0, x43:0, x44:0) :|: x43:0 > 0 && x44:0 > 0 && x40:0 > 0 l5(x60:0, x63:0, x64:0) -> l4(x60:0, c1, x64:0) :|: c1 = -1 + x63:0 && TRUE l4(x30:0, x26:0, x27:0) -> l5(c2, x26:0, x27:0) :|: c2 = 1 && (x26:0 > 0 && x27:0 > 0 && x30:0 < 1) l4(x15:0, x18:0, x19:0) -> l5(x15:0, x18:0, x19:0) :|: x15:0 < 1 && x19:0 > 0 && x18:0 > 0 ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l5(x, x1, x2)] = -1 + x1 + x2 [l4(x3, x4, x5)] = x4 + x5 The following rules are decreasing: l4(x40:0, x43:0, x44:0) -> l5(x40:0, x43:0, x44:0) :|: x43:0 > 0 && x44:0 > 0 && x40:0 > 0 l4(x30:0, x26:0, x27:0) -> l5(c2, x26:0, x27:0) :|: c2 = 1 && (x26:0 > 0 && x27:0 > 0 && x30:0 < 1) l4(x15:0, x18:0, x19:0) -> l5(x15:0, x18:0, x19:0) :|: x15:0 < 1 && x19:0 > 0 && x18:0 > 0 The following rules are bounded: l4(x40:0, x43:0, x44:0) -> l5(x40:0, x43:0, x44:0) :|: x43:0 > 0 && x44:0 > 0 && x40:0 > 0 l4(x30:0, x26:0, x27:0) -> l5(c2, x26:0, x27:0) :|: c2 = 1 && (x26:0 > 0 && x27:0 > 0 && x30:0 < 1) l4(x15:0, x18:0, x19:0) -> l5(x15:0, x18:0, x19:0) :|: x15:0 < 1 && x19:0 > 0 && x18:0 > 0 ---------------------------------------- (12) Obligation: Rules: l5(x50:0, x53:0, x54:0) -> l4(x50:0, x53:0, c) :|: c = -1 + x54:0 && TRUE l5(x60:0, x63:0, x64:0) -> l4(x60:0, c1, x64:0) :|: c1 = -1 + x63:0 && TRUE ---------------------------------------- (13) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l5(x, x1, x2)] = 0 [l4(x3, x4, x5)] = -1 The following rules are decreasing: l5(x50:0, x53:0, x54:0) -> l4(x50:0, x53:0, c) :|: c = -1 + x54:0 && TRUE l5(x60:0, x63:0, x64:0) -> l4(x60:0, c1, x64:0) :|: c1 = -1 + x63:0 && TRUE The following rules are bounded: l5(x50:0, x53:0, x54:0) -> l4(x50:0, x53:0, c) :|: c = -1 + x54:0 && TRUE l5(x60:0, x63:0, x64:0) -> l4(x60:0, c1, x64:0) :|: c1 = -1 + x63:0 && TRUE ---------------------------------------- (14) YES