YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 1362 ms] (4) AND (5) IRSwT (6) IntTRSCompressionProof [EQUIVALENT, 0 ms] (7) IRSwT (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (9) IRSwT (10) TempFilterProof [SOUND, 44 ms] (11) IntTRS (12) PolynomialOrderProcessor [EQUIVALENT, 16 ms] (13) IntTRS (14) RankingReductionPairProof [EQUIVALENT, 0 ms] (15) YES (16) IRSwT (17) IntTRSCompressionProof [EQUIVALENT, 0 ms] (18) IRSwT (19) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (20) IRSwT (21) TempFilterProof [SOUND, 88 ms] (22) IntTRS (23) RankingReductionPairProof [EQUIVALENT, 18 ms] (24) IntTRS (25) RankingReductionPairProof [EQUIVALENT, 0 ms] (26) IntTRS (27) RankingReductionPairProof [EQUIVALENT, 0 ms] (28) IntTRS (29) RankingReductionPairProof [EQUIVALENT, 0 ms] (30) YES ---------------------------------------- (0) Obligation: Rules: l0(__const_5HAT0, i4HAT0, j5HAT0, k6HAT0) -> l1(__const_5HATpost, i4HATpost, j5HATpost, k6HATpost) :|: k6HAT0 = k6HATpost && j5HAT0 = j5HATpost && i4HAT0 = i4HATpost && __const_5HAT0 = __const_5HATpost l2(x, x1, x2, x3) -> l3(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 l4(x8, x9, x10, x11) -> l5(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 l6(x16, x17, x18, x19) -> l7(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 l8(x24, x25, x26, x27) -> l6(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x24 = x28 && x30 = 1 + x26 && 1 + x24 <= x27 l8(x32, x33, x34, x35) -> l9(x36, x37, x38, x39) :|: x34 = x38 && x33 = x37 && x32 = x36 && x39 = 1 + x35 && x35 <= x32 l7(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x40 = x44 && x45 = 1 + x41 && 1 + x40 <= x42 l7(x48, x49, x50, x51) -> l9(x52, x53, x54, x55) :|: x50 = x54 && x49 = x53 && x48 = x52 && x55 = 1 && x50 <= x48 l5(x56, x57, x58, x59) -> l10(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 && 1 + x56 <= x57 l5(x64, x65, x66, x67) -> l6(x68, x69, x70, x71) :|: x67 = x71 && x65 = x69 && x64 = x68 && x70 = 1 && x65 <= x64 l9(x72, x73, x74, x75) -> l8(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 l3(x80, x81, x82, x83) -> l0(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x80 = x84 && x85 = 1 + x81 && 1 + x80 <= x82 l3(x88, x89, x90, x91) -> l2(x92, x93, x94, x95) :|: x91 = x95 && x89 = x93 && x88 = x92 && x94 = 1 + x90 && x90 <= x88 l1(x96, x97, x98, x99) -> l4(x100, x101, x102, x103) :|: x99 = x103 && x98 = x102 && x96 = x100 && x101 = 1 && 1 + x96 <= x97 l1(x104, x105, x106, x107) -> l2(x108, x109, x110, x111) :|: x107 = x111 && x105 = x109 && x104 = x108 && x110 = 1 && x105 <= x104 l11(x112, x113, x114, x115) -> l0(x116, x117, x118, x119) :|: x115 = x119 && x114 = x118 && x112 = x116 && x117 = 1 l12(x120, x121, x122, x123) -> l11(x124, x125, x126, x127) :|: x123 = x127 && x122 = x126 && x121 = x125 && x120 = x124 Start term: l12(__const_5HAT0, i4HAT0, j5HAT0, k6HAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(__const_5HAT0, i4HAT0, j5HAT0, k6HAT0) -> l1(__const_5HATpost, i4HATpost, j5HATpost, k6HATpost) :|: k6HAT0 = k6HATpost && j5HAT0 = j5HATpost && i4HAT0 = i4HATpost && __const_5HAT0 = __const_5HATpost l2(x, x1, x2, x3) -> l3(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 l4(x8, x9, x10, x11) -> l5(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 l6(x16, x17, x18, x19) -> l7(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 l8(x24, x25, x26, x27) -> l6(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x24 = x28 && x30 = 1 + x26 && 1 + x24 <= x27 l8(x32, x33, x34, x35) -> l9(x36, x37, x38, x39) :|: x34 = x38 && x33 = x37 && x32 = x36 && x39 = 1 + x35 && x35 <= x32 l7(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x40 = x44 && x45 = 1 + x41 && 1 + x40 <= x42 l7(x48, x49, x50, x51) -> l9(x52, x53, x54, x55) :|: x50 = x54 && x49 = x53 && x48 = x52 && x55 = 1 && x50 <= x48 l5(x56, x57, x58, x59) -> l10(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 && 1 + x56 <= x57 l5(x64, x65, x66, x67) -> l6(x68, x69, x70, x71) :|: x67 = x71 && x65 = x69 && x64 = x68 && x70 = 1 && x65 <= x64 l9(x72, x73, x74, x75) -> l8(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 l3(x80, x81, x82, x83) -> l0(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x80 = x84 && x85 = 1 + x81 && 1 + x80 <= x82 l3(x88, x89, x90, x91) -> l2(x92, x93, x94, x95) :|: x91 = x95 && x89 = x93 && x88 = x92 && x94 = 1 + x90 && x90 <= x88 l1(x96, x97, x98, x99) -> l4(x100, x101, x102, x103) :|: x99 = x103 && x98 = x102 && x96 = x100 && x101 = 1 && 1 + x96 <= x97 l1(x104, x105, x106, x107) -> l2(x108, x109, x110, x111) :|: x107 = x111 && x105 = x109 && x104 = x108 && x110 = 1 && x105 <= x104 l11(x112, x113, x114, x115) -> l0(x116, x117, x118, x119) :|: x115 = x119 && x114 = x118 && x112 = x116 && x117 = 1 l12(x120, x121, x122, x123) -> l11(x124, x125, x126, x127) :|: x123 = x127 && x122 = x126 && x121 = x125 && x120 = x124 Start term: l12(__const_5HAT0, i4HAT0, j5HAT0, k6HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(__const_5HAT0, i4HAT0, j5HAT0, k6HAT0) -> l1(__const_5HATpost, i4HATpost, j5HATpost, k6HATpost) :|: k6HAT0 = k6HATpost && j5HAT0 = j5HATpost && i4HAT0 = i4HATpost && __const_5HAT0 = __const_5HATpost (2) l2(x, x1, x2, x3) -> l3(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 (3) l4(x8, x9, x10, x11) -> l5(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 (4) l6(x16, x17, x18, x19) -> l7(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 (5) l8(x24, x25, x26, x27) -> l6(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x24 = x28 && x30 = 1 + x26 && 1 + x24 <= x27 (6) l8(x32, x33, x34, x35) -> l9(x36, x37, x38, x39) :|: x34 = x38 && x33 = x37 && x32 = x36 && x39 = 1 + x35 && x35 <= x32 (7) l7(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x40 = x44 && x45 = 1 + x41 && 1 + x40 <= x42 (8) l7(x48, x49, x50, x51) -> l9(x52, x53, x54, x55) :|: x50 = x54 && x49 = x53 && x48 = x52 && x55 = 1 && x50 <= x48 (9) l5(x56, x57, x58, x59) -> l10(x60, x61, x62, x63) :|: x59 = x63 && x58 = x62 && x57 = x61 && x56 = x60 && 1 + x56 <= x57 (10) l5(x64, x65, x66, x67) -> l6(x68, x69, x70, x71) :|: x67 = x71 && x65 = x69 && x64 = x68 && x70 = 1 && x65 <= x64 (11) l9(x72, x73, x74, x75) -> l8(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 (12) l3(x80, x81, x82, x83) -> l0(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x80 = x84 && x85 = 1 + x81 && 1 + x80 <= x82 (13) l3(x88, x89, x90, x91) -> l2(x92, x93, x94, x95) :|: x91 = x95 && x89 = x93 && x88 = x92 && x94 = 1 + x90 && x90 <= x88 (14) l1(x96, x97, x98, x99) -> l4(x100, x101, x102, x103) :|: x99 = x103 && x98 = x102 && x96 = x100 && x101 = 1 && 1 + x96 <= x97 (15) l1(x104, x105, x106, x107) -> l2(x108, x109, x110, x111) :|: x107 = x111 && x105 = x109 && x104 = x108 && x110 = 1 && x105 <= x104 (16) l11(x112, x113, x114, x115) -> l0(x116, x117, x118, x119) :|: x115 = x119 && x114 = x118 && x112 = x116 && x117 = 1 (17) l12(x120, x121, x122, x123) -> l11(x124, x125, x126, x127) :|: x123 = x127 && x122 = x126 && x121 = x125 && x120 = x124 Arcs: (1) -> (14), (15) (2) -> (12), (13) (3) -> (9), (10) (4) -> (7), (8) (5) -> (4) (6) -> (11) (7) -> (3) (8) -> (11) (10) -> (4) (11) -> (5), (6) (12) -> (1) (13) -> (2) (14) -> (3) (15) -> (2) (16) -> (1) (17) -> (16) This digraph is fully evaluated! ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Termination digraph: Nodes: (1) l0(__const_5HAT0, i4HAT0, j5HAT0, k6HAT0) -> l1(__const_5HATpost, i4HATpost, j5HATpost, k6HATpost) :|: k6HAT0 = k6HATpost && j5HAT0 = j5HATpost && i4HAT0 = i4HATpost && __const_5HAT0 = __const_5HATpost (2) l3(x80, x81, x82, x83) -> l0(x84, x85, x86, x87) :|: x83 = x87 && x82 = x86 && x80 = x84 && x85 = 1 + x81 && 1 + x80 <= x82 (3) l2(x, x1, x2, x3) -> l3(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x1 = x5 && x = x4 (4) l1(x104, x105, x106, x107) -> l2(x108, x109, x110, x111) :|: x107 = x111 && x105 = x109 && x104 = x108 && x110 = 1 && x105 <= x104 (5) l3(x88, x89, x90, x91) -> l2(x92, x93, x94, x95) :|: x91 = x95 && x89 = x93 && x88 = x92 && x94 = 1 + x90 && x90 <= x88 Arcs: (1) -> (4) (2) -> (1) (3) -> (2), (5) (4) -> (3) (5) -> (3) This digraph is fully evaluated! ---------------------------------------- (6) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (7) Obligation: Rules: l2(x4:0, x1:0, x2:0, x3:0) -> l2(x4:0, x1:0, 1 + x2:0, x3:0) :|: x4:0 >= x2:0 l2(x, x1, x2, x3) -> l2(x, 1 + x1, 1, x3) :|: x2 >= 1 + x && x >= 1 + x1 ---------------------------------------- (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) -> l2(x1, x2, x3) ---------------------------------------- (9) Obligation: Rules: l2(x4:0, x1:0, x2:0) -> l2(x4:0, x1:0, 1 + x2:0) :|: x4:0 >= x2:0 l2(x, x1, x2) -> l2(x, 1 + x1, 1) :|: x2 >= 1 + x && x >= 1 + x1 ---------------------------------------- (10) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l2(INTEGER, VARIABLE, VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (11) Obligation: Rules: l2(x4:0, x1:0, x2:0) -> l2(x4:0, x1:0, c) :|: c = 1 + x2:0 && x4:0 >= x2:0 l2(x, x1, x2) -> l2(x, c1, c2) :|: c2 = 1 && c1 = 1 + x1 && (x2 >= 1 + x && x >= 1 + x1) ---------------------------------------- (12) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l2(x, x1, x2)] = -1 + x - x1 The following rules are decreasing: l2(x, x1, x2) -> l2(x, c1, c2) :|: c2 = 1 && c1 = 1 + x1 && (x2 >= 1 + x && x >= 1 + x1) The following rules are bounded: l2(x, x1, x2) -> l2(x, c1, c2) :|: c2 = 1 && c1 = 1 + x1 && (x2 >= 1 + x && x >= 1 + x1) ---------------------------------------- (13) Obligation: Rules: l2(x4:0, x1:0, x2:0) -> l2(x4:0, x1:0, c) :|: c = 1 + x2:0 && x4:0 >= x2:0 ---------------------------------------- (14) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l2 ] = l2_1 + -1*l2_3 The following rules are decreasing: l2(x4:0, x1:0, x2:0) -> l2(x4:0, x1:0, c) :|: c = 1 + x2:0 && x4:0 >= x2:0 The following rules are bounded: l2(x4:0, x1:0, x2:0) -> l2(x4:0, x1:0, c) :|: c = 1 + x2:0 && x4:0 >= x2:0 ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Termination digraph: Nodes: (1) l4(x8, x9, x10, x11) -> l5(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && x8 = x12 (2) l7(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x42 = x46 && x40 = x44 && x45 = 1 + x41 && 1 + x40 <= x42 (3) l6(x16, x17, x18, x19) -> l7(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20 (4) l5(x64, x65, x66, x67) -> l6(x68, x69, x70, x71) :|: x67 = x71 && x65 = x69 && x64 = x68 && x70 = 1 && x65 <= x64 (5) l8(x24, x25, x26, x27) -> l6(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x24 = x28 && x30 = 1 + x26 && 1 + x24 <= x27 (6) l9(x72, x73, x74, x75) -> l8(x76, x77, x78, x79) :|: x75 = x79 && x74 = x78 && x73 = x77 && x72 = x76 (7) l7(x48, x49, x50, x51) -> l9(x52, x53, x54, x55) :|: x50 = x54 && x49 = x53 && x48 = x52 && x55 = 1 && x50 <= x48 (8) l8(x32, x33, x34, x35) -> l9(x36, x37, x38, x39) :|: x34 = x38 && x33 = x37 && x32 = x36 && x39 = 1 + x35 && x35 <= x32 Arcs: (1) -> (4) (2) -> (1) (3) -> (2), (7) (4) -> (3) (5) -> (3) (6) -> (5), (8) (7) -> (6) (8) -> (6) This digraph is fully evaluated! ---------------------------------------- (17) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (18) Obligation: Rules: l9(x28:0, x29:0, x74:0, x31:0) -> l6(x28:0, x29:0, 1 + x74:0, x31:0) :|: x31:0 >= 1 + x28:0 l6(x16:0, x17:0, x18:0, x19:0) -> l9(x16:0, x17:0, x18:0, 1) :|: x18:0 <= x16:0 l6(x, x1, x2, x3) -> l6(x, 1 + x1, 1, x3) :|: x2 >= 1 + x && x >= 1 + x1 l9(x36:0, x37:0, x38:0, x75:0) -> l9(x36:0, x37:0, x38:0, 1 + x75:0) :|: x75:0 <= x36:0 ---------------------------------------- (19) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: none ---------------------------------------- (20) Obligation: Rules: l9(x28:0, x29:0, x74:0, x31:0) -> l6(x28:0, x29:0, 1 + x74:0, x31:0) :|: x31:0 >= 1 + x28:0 l6(x16:0, x17:0, x18:0, x19:0) -> l9(x16:0, x17:0, x18:0, 1) :|: x18:0 <= x16:0 l6(x, x1, x2, x3) -> l6(x, 1 + x1, 1, x3) :|: x2 >= 1 + x && x >= 1 + x1 l9(x36:0, x37:0, x38:0, x75:0) -> l9(x36:0, x37:0, x38:0, 1 + x75:0) :|: x75:0 <= x36:0 ---------------------------------------- (21) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l9(INTEGER, VARIABLE, VARIABLE, VARIABLE) l6(INTEGER, VARIABLE, VARIABLE, VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (22) Obligation: Rules: l9(x28:0, x29:0, x74:0, x31:0) -> l6(x28:0, x29:0, c, x31:0) :|: c = 1 + x74:0 && x31:0 >= 1 + x28:0 l6(x16:0, x17:0, x18:0, x19:0) -> l9(x16:0, x17:0, x18:0, c1) :|: c1 = 1 && x18:0 <= x16:0 l6(x, x1, x2, x3) -> l6(x, c2, c3, x3) :|: c3 = 1 && c2 = 1 + x1 && (x2 >= 1 + x && x >= 1 + x1) l9(x36:0, x37:0, x38:0, x75:0) -> l9(x36:0, x37:0, x38:0, c4) :|: c4 = 1 + x75:0 && x75:0 <= x36:0 ---------------------------------------- (23) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l9 ] = 4*l9_1 + -4*l9_2 [ l6 ] = 4*l6_1 + -4*l6_2 The following rules are decreasing: l6(x, x1, x2, x3) -> l6(x, c2, c3, x3) :|: c3 = 1 && c2 = 1 + x1 && (x2 >= 1 + x && x >= 1 + x1) The following rules are bounded: l6(x, x1, x2, x3) -> l6(x, c2, c3, x3) :|: c3 = 1 && c2 = 1 + x1 && (x2 >= 1 + x && x >= 1 + x1) ---------------------------------------- (24) Obligation: Rules: l9(x28:0, x29:0, x74:0, x31:0) -> l6(x28:0, x29:0, c, x31:0) :|: c = 1 + x74:0 && x31:0 >= 1 + x28:0 l6(x16:0, x17:0, x18:0, x19:0) -> l9(x16:0, x17:0, x18:0, c1) :|: c1 = 1 && x18:0 <= x16:0 l9(x36:0, x37:0, x38:0, x75:0) -> l9(x36:0, x37:0, x38:0, c4) :|: c4 = 1 + x75:0 && x75:0 <= x36:0 ---------------------------------------- (25) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l9 ] = 4*l9_1 + -4*l9_3 + -1 [ l6 ] = 4*l6_1 + -4*l6_3 The following rules are decreasing: l9(x28:0, x29:0, x74:0, x31:0) -> l6(x28:0, x29:0, c, x31:0) :|: c = 1 + x74:0 && x31:0 >= 1 + x28:0 l6(x16:0, x17:0, x18:0, x19:0) -> l9(x16:0, x17:0, x18:0, c1) :|: c1 = 1 && x18:0 <= x16:0 The following rules are bounded: l6(x16:0, x17:0, x18:0, x19:0) -> l9(x16:0, x17:0, x18:0, c1) :|: c1 = 1 && x18:0 <= x16:0 ---------------------------------------- (26) Obligation: Rules: l9(x28:0, x29:0, x74:0, x31:0) -> l6(x28:0, x29:0, c, x31:0) :|: c = 1 + x74:0 && x31:0 >= 1 + x28:0 l9(x36:0, x37:0, x38:0, x75:0) -> l9(x36:0, x37:0, x38:0, c4) :|: c4 = 1 + x75:0 && x75:0 <= x36:0 ---------------------------------------- (27) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l9 ] = 0 [ l6 ] = -1 The following rules are decreasing: l9(x28:0, x29:0, x74:0, x31:0) -> l6(x28:0, x29:0, c, x31:0) :|: c = 1 + x74:0 && x31:0 >= 1 + x28:0 The following rules are bounded: l9(x28:0, x29:0, x74:0, x31:0) -> l6(x28:0, x29:0, c, x31:0) :|: c = 1 + x74:0 && x31:0 >= 1 + x28:0 l9(x36:0, x37:0, x38:0, x75:0) -> l9(x36:0, x37:0, x38:0, c4) :|: c4 = 1 + x75:0 && x75:0 <= x36:0 ---------------------------------------- (28) Obligation: Rules: l9(x36:0, x37:0, x38:0, x75:0) -> l9(x36:0, x37:0, x38:0, c4) :|: c4 = 1 + x75:0 && x75:0 <= x36:0 ---------------------------------------- (29) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l9 ] = -1*l9_4 + l9_1 The following rules are decreasing: l9(x36:0, x37:0, x38:0, x75:0) -> l9(x36:0, x37:0, x38:0, c4) :|: c4 = 1 + x75:0 && x75:0 <= x36:0 The following rules are bounded: l9(x36:0, x37:0, x38:0, x75:0) -> l9(x36:0, x37:0, x38:0, c4) :|: c4 = 1 + x75:0 && x75:0 <= x36:0 ---------------------------------------- (30) YES