Copyright (c) 1999 Association of Mizar Users
environ vocabulary FINSET_1, LATTICES, LATTICE3, BOOLE, FINSEQ_1, RELAT_1, FUNCT_1, ORDERS_1, FILTER_1, RELAT_2, BHSP_3, WELLORD1, ARYTM_3, WAYBEL_6, ZF_LANG, REALSET1, BINOP_1, TARSKI, LATTICE6; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XREAL_0, RELAT_1, FUNCT_1, NAT_1, STRUCT_0, BINOP_1, LATTICES, LATTICE3, FINSET_1, WELLORD1, WAYBEL_6, GROUP_1, ORDERS_1, FINSEQ_1, WELLFND1, YELLOW_0, LATTICE2, REALSET1; constructors LATTICE2, WAYBEL_1, WAYBEL_6, NAT_1, GROUP_1, WELLFND1, WELLORD1, BINOP_1, REALSET1, MEMBERED; clusters SUBSET_1, STRUCT_0, LATTICE2, LATTICE3, YELLOW_1, FINSET_1, ORDERS_1, INT_1, FINSEQ_1, FINSEQ_6, WAYBEL_0, KNASTER, LATTICES; requirements NUMERALS, BOOLE, SUBSET, ARITHM; definitions WELLFND1, WELLORD1; theorems AXIOMS, TARSKI, FUNCT_1, LATTICES, LATTICE3, STRUCT_0, ORDERS_1, VECTSP_8, FINSEQ_1, WAYBEL_6, FINSET_1, GROUP_1, NAT_1, INT_1, REAL_1, RELAT_1, YELLOW_0, WELLORD1, WELLFND1, REALSET1, XBOOLE_0; schemes NAT_1, WELLFND1; begin definition cluster finite Lattice; existence proof set L = BooleLatt {}; A1: the carrier of L = bool {} by LATTICE3:def 1; bool {} is finite by FINSET_1:24; then L is finite by A1,GROUP_1:def 13; hence thesis; end; end; Lm1:for L being finite Lattice for X being Subset of L holds X is empty or ex a being Element of LattPOSet L st a in X & for b being Element of LattPOSet L st b in X & b <> a holds not(b <= a) proof let L be finite Lattice; let X be Subset of L; defpred P[Nat] means for L being finite Lattice for X being Subset of LattPOSet L for p being FinSequence st rng p = X & len p = $1 holds X is empty or ex a being Element of LattPOSet L st a in X & for b being Element of LattPOSet L st b in X & b <> a holds not(b <= a); A1: P[0] proof let L be finite Lattice; let X be Subset of LattPOSet L; let p be FinSequence; assume A2: rng p = X & len p = 0; then Seg 0 = dom p by FINSEQ_1:def 3; hence thesis by A2,FINSEQ_1:4,RELAT_1:65; end; A3: for k being Nat st P[k] holds P[k + 1] proof let k be Nat; assume A4: P[k]; per cases; suppose A5: k = 0; for L being finite Lattice for X being Subset of LattPOSet L for p being FinSequence st rng p = X & len p = 1 holds ex a being Element of LattPOSet L st a in X & for b being Element of LattPOSet L st b in X & b <> a holds not(b <= a) proof let L be finite Lattice; let X be Subset of LattPOSet L; let p be FinSequence; assume A6: rng p = X & len p = 1; then A7: Seg 1 = dom p by FINSEQ_1:def 3; consider a being set such that A8: p.1 = a; 1 in dom p by A7,FINSEQ_1:4,TARSKI:def 1; then A9: a in rng p by A8,FUNCT_1:def 5; then reconsider a as Element of LattPOSet L by A6; rng p = { a } by A7,A8,FINSEQ_1:4,FUNCT_1:14; then for b being Element of LattPOSet L st b in X & b <> a holds not(b <= a) by A6,TARSKI:def 1; hence thesis by A6,A9; end; hence thesis by A5; suppose A10: k <> 0; for L being finite Lattice for X being Subset of LattPOSet L for p being FinSequence st rng p = X & len p = k + 1 holds ex a being Element of LattPOSet L st a in X & for b being Element of LattPOSet L st b in X & b <> a holds not(b <= a) proof let L be finite Lattice; let X be Subset of LattPOSet L; let p be FinSequence; assume A11: rng p = X & len p = k + 1; set q = p|(Seg k), X' = rng q; A12: k <= len p by A11,NAT_1:29; then A13: len q = k by FINSEQ_1:21; A14: rng q c= rng p by FUNCT_1:76; for x being set holds x in X' implies x in the carrier of LattPOSet L proof let x be set; assume x in X'; then x in rng p by A14; hence thesis by A11; end; then A15: X' is Subset of LattPOSet L by TARSKI:def 3; q <> {} by A10,A13,FINSEQ_1:25; then X' <> {} by FINSEQ_1:27; then consider a being Element of LattPOSet L such that A16: a in X' & for b being Element of LattPOSet L st b in X' & b <> a holds not(b <= a) by A4,A13,A15; consider b being set such that A17: p.(k+1) = b; Seg (k+1) = dom p by A11,FINSEQ_1:def 3; then k + 1 in dom p by FINSEQ_1:6; then A18: b in rng p by A17,FUNCT_1:def 5; then reconsider b as Element of LattPOSet L by A11; per cases; suppose ex c being Element of LattPOSet L st c in X & c <= b & c <> b; then consider c being Element of LattPOSet L such that A19: c in X & c <= b & c <> b; for u being Element of LattPOSet L st u in X & u <> a holds not(u <= a) proof let u be Element of LattPOSet L; assume A20: u in X & u <> a; per cases; suppose u in X'; hence thesis by A16,A20; suppose A21: not(u in X'); consider n being set such that A22: n in dom p & p.n = u by A11,A20,FUNCT_1:def 5; A23: Seg(k+1) = dom p by A11,FINSEQ_1:def 3; reconsider n as Nat by A22; A24: 1 <= n & n <= k+1 by A22,A23,FINSEQ_1:3; A25: n = k + 1 proof assume n <> k + 1; then n < k + 1 by A24,REAL_1:def 5; then n <= k by NAT_1:38; then n in Seg(k) by A24,FINSEQ_1:3; then A26: n in dom q by A12,FINSEQ_1:21; then q.n = u by A22,FUNCT_1:70; hence thesis by A21,A26,FUNCT_1:def 5; end; assume A27: u <= a; then A28: c <= a by A17,A19,A22,A25,ORDERS_1:26; A29: c in X' proof consider n being set such that A30: n in dom p & p.n = c by A11,A19,FUNCT_1:def 5; A31: Seg(k+1) = dom p by A11,FINSEQ_1:def 3; reconsider n as Nat by A30; A32: 1 <= n & n <= k+1 by A30,A31,FINSEQ_1:3; then n < k + 1 by A17,A19,A30,REAL_1:def 5; then n <= k by NAT_1:38; then n in Seg(k) by A32,FINSEQ_1:3; then A33: n in dom q by A12,FINSEQ_1:21; then q.n = c by A30,FUNCT_1:70; hence thesis by A33,FUNCT_1:def 5; end; c <> a by A17,A19,A22,A25,A27,ORDERS_1:25; hence thesis by A16,A28,A29; end; hence thesis by A11,A14,A16; suppose not(ex c being Element of LattPOSet L st c in X & c <= b & c <> b); then for u being Element of LattPOSet L st u in X & u <> b holds not(u <= b); hence thesis by A11,A18; end; hence thesis; end; A34: for k being Nat holds P[k] from Ind(A1,A3); the carrier of L is finite by GROUP_1:def 13; then reconsider X as finite Subset of L by FINSET_1:13; LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2; then reconsider X' = X as finite Subset of LattPOSet L; consider p being FinSequence such that A35: rng p = X' by FINSEQ_1:73; dom p = Seg len p by FINSEQ_1:def 3; hence thesis by A34,A35; end; Lm2:for L being finite Lattice for X being Subset of L holds X is empty or ex a being Element of LattPOSet L st a in X & for b being Element of LattPOSet L st b in X & b <> a holds not(a <= b) proof let L be finite Lattice; let X be Subset of L; defpred P[Nat] means for L being finite Lattice for X being Subset of LattPOSet L for p being FinSequence st rng p = X & len p = $1 holds X is empty or ex a being Element of LattPOSet L st a in X & for b being Element of LattPOSet L st b in X & b <> a holds not(a <= b); A1: P[0] proof let L be finite Lattice; let X be Subset of LattPOSet L; let p be FinSequence; assume A2: rng p = X & len p = 0; then Seg 0 = dom p by FINSEQ_1:def 3; hence thesis by A2,FINSEQ_1:4,RELAT_1:65; end; A3: for k being Nat st P[k] holds P[k + 1] proof let k be Nat; assume A4: P[k]; per cases; suppose A5: k = 0; for L being finite Lattice for X being Subset of LattPOSet L for p being FinSequence st rng p = X & len p = 1 holds ex a being Element of LattPOSet L st a in X & for b being Element of LattPOSet L st b in X & b <> a holds not(a <= b) proof let L be finite Lattice; let X be Subset of LattPOSet L; let p be FinSequence; assume A6: rng p = X & len p = 1; then A7: dom p = { 1 } by FINSEQ_1:4,def 3; consider a being set such that A8: p.1 = a; 1 in dom p by A7,TARSKI:def 1; then A9: a in rng p by A8,FUNCT_1:def 5; then reconsider a as Element of LattPOSet L by A6; rng p = { a } by A7,A8,FUNCT_1:14; then for b being Element of LattPOSet L st b in X & b <> a holds not(a <= b) by A6,TARSKI:def 1; hence thesis by A6,A9; end; hence thesis by A5; suppose A10: k <> 0; for L being finite Lattice for X being Subset of LattPOSet L for p being FinSequence st rng p = X & len p = k + 1 holds ex a being Element of LattPOSet L st a in X & for b being Element of LattPOSet L st b in X & b <> a holds not(a <= b) proof let L be finite Lattice; let X be Subset of LattPOSet L; let p be FinSequence; assume A11: rng p = X & len p = k + 1; set q = p|(Seg k), X' = rng q; A12: k <= len p by A11,NAT_1:29; then A13: len q = k by FINSEQ_1:21; A14: rng q c= rng p by FUNCT_1:76; for x being set holds x in X' implies x in the carrier of LattPOSet L proof let x be set; assume x in X'; then x in rng p by A14; hence thesis by A11; end; then A15: X' is Subset of LattPOSet L by TARSKI:def 3; q <> {} by A10,A13,FINSEQ_1:25; then X' <> {} by FINSEQ_1:27; then consider a being Element of LattPOSet L such that A16: a in X' & for b being Element of LattPOSet L st b in X' & b <> a holds not(a <= b) by A4,A13,A15; consider b being set such that A17: p.(k+1) = b; Seg (k+1) = dom p by A11,FINSEQ_1:def 3; then k + 1 in dom p by FINSEQ_1:6; then A18: b in rng p by A17,FUNCT_1:def 5; then reconsider b as Element of LattPOSet L by A11; per cases; suppose ex c being Element of LattPOSet L st c in X & b <= c & c <> b; then consider c being Element of LattPOSet L such that A19: c in X & b <= c & c <> b; for u being Element of LattPOSet L st u in X & u <> a holds not(a <= u) proof let u be Element of LattPOSet L; assume A20: u in X & u <> a; now per cases; case u in X'; hence thesis by A16,A20; case A21: not(u in X'); consider n being set such that A22: n in dom p & p.n = u by A11,A20,FUNCT_1:def 5; A23: Seg(k+1) = dom p by A11,FINSEQ_1:def 3; reconsider n as Nat by A22; A24: 1 <= n & n <= k+1 by A22,A23,FINSEQ_1:3; A25: n = k + 1 proof assume n <> k + 1; then n < k + 1 by A24,REAL_1:def 5; then n <= k by NAT_1:38; then n in Seg(k) by A24,FINSEQ_1:3; then A26: n in dom q by A12,FINSEQ_1:21; then q.n = u by A22,FUNCT_1:70; hence thesis by A21,A26,FUNCT_1:def 5; end; assume A27: a <= u; then A28: a <= c by A17,A19,A22,A25,ORDERS_1:26; A29: c in X' proof consider n being set such that A30: n in dom p & p.n = c by A11,A19,FUNCT_1:def 5; A31: Seg(k+1) = dom p by A11,FINSEQ_1:def 3; reconsider n as Nat by A30; A32: 1 <= n & n <= k+1 by A30,A31,FINSEQ_1:3; then n < k + 1 by A17,A19,A30,REAL_1:def 5; then n <= k by NAT_1:38; then n in Seg(k) by A32,FINSEQ_1:3; then A33: n in dom q by A12,FINSEQ_1:21; then q.n = c by A30,FUNCT_1:70; hence thesis by A33,FUNCT_1:def 5; end; c <> a by A17,A19,A22,A25,A27,ORDERS_1:25; hence thesis by A16,A28,A29; end; :: cases hence thesis; end; hence thesis by A11,A14,A16; suppose not(ex c being Element of LattPOSet L st c in X & b <= c & c <> b); then for u being Element of LattPOSet L st u in X & u <> b holds not(b <= u); hence thesis by A11,A18; end; hence thesis; end; A34: for k being Nat holds P[k] from Ind(A1,A3); the carrier of L is finite by GROUP_1:def 13; then reconsider X as finite Subset of L by FINSET_1:13; LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2; then reconsider X' = X as finite Subset of LattPOSet L; consider p being FinSequence such that A35: rng p = X' by FINSEQ_1:73; dom p = Seg len p by FINSEQ_1:def 3; hence thesis by A34,A35; end; Lm3:for L being finite Lattice for X being Subset of L holds X is empty or ex a being Element of LattPOSet L st for b being Element of LattPOSet L st b in X holds b <= a proof let L be finite Lattice; let X be Subset of L; defpred P[Nat] means for L being finite Lattice for X being Subset of LattPOSet L for p being FinSequence st rng p = X & len p = $1 holds X is empty or ex a being Element of LattPOSet L st for b being Element of LattPOSet L st b in X holds b <= a; A1: P[0] proof let L be finite Lattice; let X be Subset of LattPOSet L; let p be FinSequence; assume A2: rng p = X & len p = 0; then dom p = {} by FINSEQ_1:4,def 3; hence thesis by A2,RELAT_1:65; end; A3: for k being Nat st P[k] holds P[k + 1] proof let k be Nat; assume A4: P[k]; per cases; suppose A5: k = 0; for L being finite Lattice for X being Subset of LattPOSet L for p being FinSequence st rng p = X & len p = 1 holds ex a being Element of LattPOSet L st for b being Element of LattPOSet L st b in X holds b <= a proof let L be finite Lattice; let X be Subset of LattPOSet L; let p be FinSequence; assume A6: rng p = X & len p = 1; then A7: Seg 1 = dom p by FINSEQ_1:def 3; consider a being set such that A8: p.1 = a; 1 in dom p by A7,FINSEQ_1:4,TARSKI:def 1; then a in rng p by A8,FUNCT_1:def 5; then reconsider a as Element of LattPOSet L by A6; rng p = { a } by A7,A8,FINSEQ_1:4,FUNCT_1:14; then for b being Element of LattPOSet L st b in X holds b <= a by A6,TARSKI:def 1; hence thesis; end; hence thesis by A5; suppose A9: k <> 0; for L being finite Lattice for X being Subset of LattPOSet L for p being FinSequence st rng p = X & len p = k + 1 holds ex a being Element of LattPOSet L st for b being Element of LattPOSet L st b in X holds b <= a proof let L be finite Lattice; let X be Subset of LattPOSet L; let p be FinSequence; assume A10: rng p = X & len p = k + 1; set q = p|(Seg k), X' = rng q; A11: k <= k + 1 by NAT_1:29; then A12: len q = k by A10,FINSEQ_1:21; for x being set holds x in X' implies x in the carrier of LattPOSet L proof let x be set; assume A13: x in X'; rng q c= rng p by FUNCT_1:76; then x in rng p by A13; hence thesis by A10; end; then A14: X' is Subset of LattPOSet L by TARSKI:def 3; q <> {} by A9,A12,FINSEQ_1:25; then X' <> {} by FINSEQ_1:27; then consider a being Element of LattPOSet L such that A15: for b being Element of LattPOSet L st b in X' holds b <= a by A4,A12,A14; consider b being set such that A16: p.(k+1) = b; Seg (k+1) = dom p by A10,FINSEQ_1:def 3; then k + 1 in dom p by FINSEQ_1:6; then b in rng p by A16,FUNCT_1:def 5; then reconsider b as Element of LattPOSet L by A10; set a2 = %a "\/" %b; A17: %a "\/" a2 = (%a "\/" %a) "\/" %b by LATTICES:def 5 .= %a "\/" %b by LATTICES:17; %b "\/" a2 = (%b "\/" %b) "\/" %a by LATTICES:def 5 .= %b "\/" %a by LATTICES:17; then A18: %a [= a2 & %b [= a2 by A17,LATTICES:def 3; A19: (%a)% = %a by LATTICE3:def 3 .= a by LATTICE3:def 4; A20: (%b)% = %b by LATTICE3:def 3 .= b by LATTICE3:def 4; for c being Element of LattPOSet L st c in X holds c <= a2% proof let c be Element of LattPOSet L; assume A21: c in X; per cases; suppose c in X'; then a <= a2% & c <= a by A15,A18,A19,LATTICE3:7; hence thesis by ORDERS_1:26; suppose A22: not(c in X'); consider n being set such that A23: n in dom p & c = p.n by A10,A21,FUNCT_1:def 5; reconsider n as Nat by A23; A24: n in Seg(k + 1) by A10,A23,FINSEQ_1:def 3; then A25: 1 <= n & n <= k + 1 by FINSEQ_1:3; n >= k + 1 proof assume not(n >= k + 1); then 1 <= n & n <= k by A24,FINSEQ_1:3,INT_1:20; then A26: n in Seg(k) by FINSEQ_1:3; then A27: n in dom q by A10,A11,FINSEQ_1:21; dom(p|(Seg k)) = dom p /\ Seg(k) by RELAT_1:90 .= Seg(k+1) /\ Seg(k) by A10,FINSEQ_1:def 3 .= Seg(k) by A11,FINSEQ_1:9; then q.n = c by A23,A26,FUNCT_1:70; hence thesis by A22,A27,FUNCT_1:def 5; end; then c = b by A16,A23,A25,AXIOMS:21; hence thesis by A18,A20,LATTICE3:7; end; hence thesis; end; hence thesis; end; A28: for k being Nat holds P[k] from Ind(A1,A3); the carrier of L is finite by GROUP_1:def 13; then reconsider X as finite Subset of L by FINSET_1:13; LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2; then reconsider X' = X as finite Subset of LattPOSet L; consider p being FinSequence such that A29: rng p = X' by FINSEQ_1:73; dom p = Seg len p by FINSEQ_1:def 3; hence thesis by A28,A29; end; Lm4:for L being finite Lattice holds ex a being Element of LattPOSet L st for b being Element of LattPOSet L holds b <= a proof let L be finite Lattice; the carrier of L c= the carrier of L; then reconsider L' = the carrier of L as Subset of L; consider a being Element of LattPOSet L such that A1: for b being Element of LattPOSet L st b in L' holds b <= a by Lm3; for b being Element of LattPOSet L holds b <= a proof let b be Element of LattPOSet L; LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2; hence thesis by A1; end; hence thesis; end; Lm5:for L being finite Lattice holds L is complete proof let L be finite Lattice; for X being Subset of L ex a being Element of L st a is_less_than X & for b being Element of L st b is_less_than X holds b [= a proof let X be Subset of L; defpred P[Nat] means for X' being finite Subset of LattPOSet L for p being FinSequence st rng p = X' & len p = $1 holds ex a being Element of LattPOSet L st (for x being Element of LattPOSet L st x in X' holds a <= x) & (for x' being Element of LattPOSet L st for x being Element of LattPOSet L st x in X' holds x' <= x holds x' <= a); A1: P[0] proof for X' being finite Subset of LattPOSet L for p being FinSequence st rng p = X' & len p = 0 holds ex a being Element of LattPOSet L st (for x being Element of LattPOSet L st x in X' holds a <= x) & (for x' being Element of LattPOSet L st for x being Element of LattPOSet L st x in X' holds x' <= x holds x' <= a) proof let X' be finite Subset of LattPOSet L; let p be FinSequence; assume A2: rng p = X' & len p = 0; then A3: dom p = {} by FINSEQ_1:4,def 3; ex a being Element of LattPOSet L st (for x being Element of LattPOSet L st x in X' holds a <= x) & (for x' being Element of LattPOSet L st for x being Element of LattPOSet L st x in X' holds x' <= x holds x' <= a) proof consider a being Element of LattPOSet L such that A4: for b being Element of LattPOSet L holds b <= a by Lm4; A5: for x being Element of LattPOSet L st x in X' holds a <= x by A2,A3,RELAT_1:65; for x' being Element of LattPOSet L st for x being Element of LattPOSet L st x in X' holds x' <= x holds x' <= a by A4; hence thesis by A5; end; hence thesis; end; hence thesis; end; A6: for k being Nat st P[k] holds P[k + 1] proof let k be Nat; assume A7: P[k]; for X' being finite Subset of LattPOSet L for p being FinSequence st rng p = X' & len p = k + 1 holds ex a being Element of LattPOSet L st (for x being Element of LattPOSet L st x in X' holds a <= x) & (for x' being Element of LattPOSet L st for x being Element of LattPOSet L st x in X' holds x' <= x holds x' <= a) proof let X be finite Subset of LattPOSet L; let p be FinSequence; assume A8: rng p = X & len p = k + 1; set q = p|(Seg k), X' = rng q; A9: k <= k + 1 by NAT_1:29; then A10: len q = k by A8,FINSEQ_1:21; for x being set holds x in X' implies x in the carrier of LattPOSet L proof let x be set; assume A11: x in X'; rng q c= rng p by FUNCT_1:76; then x in rng p by A11; hence thesis by A8; end; then X' is Subset of LattPOSet L by TARSKI:def 3; then consider a being Element of LattPOSet L such that A12: (for x being Element of LattPOSet L st x in X' holds a <= x) & (for x' being Element of LattPOSet L st for x being Element of LattPOSet L st x in X' holds x' <= x holds x' <= a) by A7,A10; consider b being set such that A13: p.(k+1) = b; Seg (k+1) = dom p by A8,FINSEQ_1:def 3; then k + 1 in dom p by FINSEQ_1:6; then A14: b in rng p by A13,FUNCT_1:def 5; then reconsider b as Element of LattPOSet L by A8; set a2 = %a "/\" %b; A15: a2 "\/" %a = %a by LATTICES:def 8; a2 "\/" %b = %b by LATTICES:def 8; then A16: a2 [= %a & a2 [= %b by A15,LATTICES:def 3; A17: (%a)% = %a by LATTICE3:def 3 .= a by LATTICE3:def 4; A18: (%b)% = %b by LATTICE3:def 3 .= b by LATTICE3:def 4; A19: for x being Element of LattPOSet L st x in X holds a2% <= x proof let c be Element of LattPOSet L; assume A20: c in X; per cases; suppose c in X'; then a2% <= a & a <= c by A12,A16,A17,LATTICE3:7; hence thesis by ORDERS_1:26; suppose A21: not c in X'; consider n being set such that A22: n in dom p & c = p.n by A8,A20,FUNCT_1:def 5; reconsider n as Nat by A22; A23: n in Seg(k + 1) by A8,A22,FINSEQ_1:def 3; then A24: 1 <= n & n <= k + 1 by FINSEQ_1:3; n >= k + 1 proof assume not(n >= k + 1); then 1 <= n & n <= k by A23,FINSEQ_1:3,INT_1:20; then A25: n in Seg(k) by FINSEQ_1:3; then A26: n in dom q by A8,A9,FINSEQ_1:21; dom(p|(Seg k)) = dom p /\ Seg(k) by RELAT_1:90 .= Seg(k+1) /\ Seg(k) by A8,FINSEQ_1:def 3 .= Seg(k) by A9,FINSEQ_1:9; then q.n = c by A22,A25,FUNCT_1:70; hence thesis by A21,A26,FUNCT_1:def 5; end; then c = b by A13,A22,A24,AXIOMS:21; hence thesis by A16,A18,LATTICE3:7; end; for x' being Element of LattPOSet L st for x being Element of LattPOSet L st x in X holds x' <= x holds x' <= a2% proof let x' be Element of LattPOSet L; assume A27: for x being Element of LattPOSet L st x in X holds x' <= x; for x being Element of LattPOSet L st x in X' holds x' <= x proof let x be Element of LattPOSet L; rng q c= rng p by FUNCT_1:76; hence thesis by A8,A27; end; then A28: x' <= a by A12; A29: x' <= b by A8,A14,A27; A30: (%a)% = %a by LATTICE3:def 3 .= a by LATTICE3:def 4; A31: (%b)% = %b by LATTICE3:def 3 .= b by LATTICE3:def 4; A32: (%(x'))% = %(x') by LATTICE3:def 3 .= x' by LATTICE3:def 4; then A33: %(x') [= %a & %(x') [= %b by A28,A29,A30,A31,LATTICE3:7; %(x') "/\" a2 = (%(x') "/\" %a) "/\" %b by LATTICES:def 7 .= %(x') "/\" %b by A33,LATTICES:21 .= %(x') by A33,LATTICES:21; then %(x') [= a2 by LATTICES:21; hence thesis by A32,LATTICE3:7; end; hence thesis by A19; end; hence thesis; end; A34: for k being Nat holds P[k] from Ind(A1,A6); the carrier of L is finite by GROUP_1:def 13; then reconsider X as finite Subset of L by FINSET_1:13; LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2; then reconsider X' = X as finite Subset of LattPOSet L; consider p being FinSequence such that A35: rng p = X' by FINSEQ_1:73; dom p = Seg len p by FINSEQ_1:def 3; then consider a being Element of LattPOSet L such that A36: (for x being Element of LattPOSet L st x in X' holds a <= x) & (for x' being Element of LattPOSet L st for x being Element of LattPOSet L st x in X' holds x' <= x holds x' <= a) by A34,A35; for x being Element of L st x in X holds %a [= x proof let x be Element of L; assume x in X; then consider x' being Element of LattPOSet L such that A37: x' = x & x' in X'; A38: a <= x' by A36,A37; A39: (%a)% = %a by LATTICE3:def 3 .= a by LATTICE3:def 4; x' = x% by A37,LATTICE3:def 3; hence thesis by A38,A39,LATTICE3:7; end; then A40: %a is_less_than X by LATTICE3:def 16; for b being Element of L st b is_less_than X holds b [= %a proof let b be Element of L; assume A41: b is_less_than X; for x being Element of LattPOSet L st x in X' holds b% <= x proof let x be Element of LattPOSet L; assume x in X'; then consider x' being Element of L such that A42: x' = x & x' in X; b [= x' by A41,A42,LATTICE3:def 16; then b% <= (x')% by LATTICE3:7; hence thesis by A42,LATTICE3:def 3; end; then A43: b% <= a by A36; (%a)% = %a by LATTICE3:def 3 .= a by LATTICE3:def 4; hence thesis by A43,LATTICE3:7; end; hence thesis by A40; end; hence thesis by VECTSP_8:def 6; end; definition cluster finite -> complete Lattice; coherence by Lm5; end; definition let L be Lattice; let D be Subset of L; func D% -> Subset of LattPOSet L equals :Def1: {d% where d is Element of L : d in D}; coherence proof set M' = {d% where d is Element of L : d in D}; for x being set holds x in M' implies x in the carrier of LattPOSet L proof let x be set; assume x in M'; then ex x' being Element of L st x = x'% & x' in D; hence thesis; end; then M' c= the carrier of (LattPOSet L) by TARSKI:def 3; hence thesis; end; end; definition let L be Lattice; let D be Subset of LattPOSet L; func %D -> Subset of L equals :Def2: {%d where d is Element of LattPOSet L : d in D}; coherence proof set M' = {%d where d is Element of LattPOSet L : d in D}; for x being set holds x in M' implies x in the carrier of L proof let x be set; assume x in M'; then ex x' being Element of LattPOSet L st x = %x' & x' in D; hence thesis; end; then M' c= the carrier of L by TARSKI:def 3; hence thesis; end; end; definition let L be finite Lattice; cluster LattPOSet L -> well_founded; coherence proof set R = LattPOSet L; let Y be set; assume A1: Y c= the carrier of R & Y <> {}; LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2; then reconsider Y as Subset of L by A1; consider a being Element of LattPOSet L such that A2: a in Y & for b being Element of LattPOSet L st b in Y & b <> a holds not(b <= a) by A1,Lm1; A3: not(ex x being Element of R st x <> a & [x,a] in the InternalRel of R & x in Y) proof given x being Element of R such that A4: x <> a & [x,a] in the InternalRel of R & x in Y; x <= a by A4,ORDERS_1:def 9; hence thesis by A2,A4; end; (the InternalRel of R)-Seg(a) /\ Y = {} proof assume A5: (the InternalRel of R)-Seg(a) /\ Y <> {}; consider z being Element of (the InternalRel of R)-Seg(a) /\ Y; A6: z in (the InternalRel of R)-Seg(a) & z in Y by A5,XBOOLE_0:def 3; then A7: z <> a & [z,a] in (the InternalRel of R) by WELLORD1:def 1; z is Element of R by A1,A6; hence thesis by A3,A6,A7; end; then (the InternalRel of R)-Seg(a) misses Y by XBOOLE_0:def 7; hence thesis by A2; end; end; Lm6:for L being finite Lattice holds (LattPOSet L)~ is well_founded proof let L be finite Lattice; set R = LattPOSet L; A1:(LattPOSet L)~ = RelStr(#the carrier of R, (the InternalRel of R)~#) by LATTICE3:def 5; for Y being set st Y c= the carrier of R~ & Y <> {} ex a being set st a in Y & (the InternalRel of R~)-Seg(a) misses Y proof let Y be set; assume A2: Y c= the carrier of R~ & Y <> {}; LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2; then reconsider Y as Subset of L by A1,A2; consider a being Element of R such that A3: a in Y & for b being Element of LattPOSet L st b in Y & b <> a holds not(a <= b) by A2,Lm2; reconsider a as Element of R; reconsider a' = a as Element of R~ by A1; A4: for b being Element of (LattPOSet L)~ st b in Y & b <> a holds not(b <= a') proof let b be Element of (LattPOSet L)~; assume A5: b in Y & b <> a; reconsider b' = b as Element of R by A1; A6: not(a <= b') by A3,A5; a~ = a' & (b')~ = b by LATTICE3:def 6; hence thesis by A6,LATTICE3:9; end; A7: not(ex x being Element of R~ st x <> a & [x,a] in the InternalRel of R~ & x in Y) proof given x being Element of R~ such that A8: x <> a & [x,a] in the InternalRel of R~ & x in Y; x <= a' by A8,ORDERS_1:def 9; hence thesis by A4,A8; end; (the InternalRel of R~)-Seg(a) /\ Y = {} proof assume A9: (the InternalRel of R~)-Seg(a) /\ Y <> {}; consider z being Element of (the InternalRel of R~)-Seg(a) /\ Y; A10: z in (the InternalRel of R~)-Seg(a) & z in Y by A9,XBOOLE_0:def 3; then A11: z <> a & [z,a] in (the InternalRel of R~) by WELLORD1:def 1; z is Element of R~ by A2,A10; hence thesis by A7,A10,A11; end; then (the InternalRel of R~)-Seg(a) misses Y by XBOOLE_0:def 7; hence thesis by A3; end; then the InternalRel of R~ is_well_founded_in the carrier of R~ by WELLORD1:def 3; hence thesis by WELLFND1:def 2; end; definition let L be Lattice; attr L is noetherian means :Def3: LattPOSet L is well_founded; attr L is co-noetherian means :Def4: (LattPOSet L)~ is well_founded; end; definition cluster noetherian upper-bounded lower-bounded complete Lattice; existence proof consider L being finite Lattice; take L; LattPOSet L is well_founded; hence thesis by Def3; end; end; definition cluster co-noetherian upper-bounded lower-bounded complete Lattice; existence proof consider L being finite Lattice; take L; (LattPOSet L)~ is well_founded by Lm6; hence thesis by Def4; end; end; theorem for L being Lattice holds L is noetherian iff L.: is co-noetherian proof let L be Lattice; set R = LattPOSet L; set Ri = (LattPOSet L)~; A1: now assume L is noetherian; then R is well_founded by Def3; then A2: (Ri)~ is well_founded by LATTICE3:8; (LattPOSet L)~ = LattPOSet (L.:) by LATTICE3:20; hence L.: is co-noetherian by A2,Def4; end; now assume A3: L.: is co-noetherian; (LattPOSet L)~ = LattPOSet (L.:) by LATTICE3:20; then (Ri)~ is well_founded by A3,Def4; then R is well_founded by LATTICE3:8; hence L is noetherian by Def3; end; hence thesis by A1; end; Lm7:for L being finite Lattice holds L is noetherian & L is co-noetherian proof let L be finite Lattice; LattPOSet L is well_founded & (LattPOSet L)~ is well_founded by Lm6; hence thesis by Def3,Def4; end; definition cluster finite -> noetherian Lattice; coherence by Lm7; cluster finite -> co-noetherian Lattice; coherence by Lm7; end; definition let L be Lattice; let a,b be Element of L; pred a is-upper-neighbour-of b means :Def5: a <> b & b [= a & for c being Element of L holds (b [= c & c [= a) implies (c = a or c = b); synonym b is-lower-neighbour-of a; end; theorem Th2:for L being Lattice for a being Element of L for b,c being Element of L st b <> c holds ((b is-upper-neighbour-of a & c is-upper-neighbour-of a) implies a = c "/\" b) & ((b is-lower-neighbour-of a & c is-lower-neighbour-of a) implies a = c "\/" b) proof let L be Lattice; let a be Element of L; let b,c be Element of L; assume A1: b <> c; A2: now assume A3: b is-upper-neighbour-of a & c is-upper-neighbour-of a; then A4: a [= b & a [= c by Def5; a = a "/\" a by LATTICES:18 .= (b "/\" a) "/\" a by A4,LATTICES:21 .= (b "/\" a) "/\" (c "/\" a) by A4,LATTICES:21 .= b "/\" (a "/\" (c "/\" a)) by LATTICES:def 7 .= b "/\" ((a "/\" a) "/\" c) by LATTICES:def 7 .= b "/\" (a "/\" c) by LATTICES:18 .= (b "/\" c) "/\" a by LATTICES:def 7; then A5: a [= b "/\" c by LATTICES:21; A6: b "/\" c [= c by LATTICES:23; not(b "/\" c = c) proof assume b "/\" c = c; then c [= b by LATTICES:21; then c = a or c = b by A3,A4,Def5; hence contradiction by A1,A3,Def5; end; hence b "/\" c = a by A3,A5,A6,Def5; end; now assume A7: b is-lower-neighbour-of a & c is-lower-neighbour-of a; then A8: b [= a & c [= a by Def5; a = a "\/" a by LATTICES:17 .= (b "\/" a) "\/" a by A8,LATTICES:def 3 .= (b "\/" a) "\/" (c "\/" a) by A8,LATTICES:def 3 .= b "\/" (a "\/" (a "\/" c)) by LATTICES:def 5 .= b "\/" ((a "\/" a) "\/" c) by LATTICES:def 5 .= b "\/" (a "\/" c) by LATTICES:17 .= (b "\/" c) "\/" a by LATTICES:def 5; then A9: b "\/" c [= a by LATTICES:def 3; A10: c [= b "\/" c by LATTICES:22; not(b "\/" c = c) proof assume b "\/" c = c; then b [= c by LATTICES:def 3; then c = a or c = b by A7,A8,Def5; hence contradiction by A1,A7,Def5; end; hence b "\/" c = a by A7,A9,A10,Def5; end; hence thesis by A2; end; theorem Th3:for L being noetherian Lattice for a being Element of L for d being Element of L st a [= d & a <> d holds ex c being Element of L st c [= d & c is-upper-neighbour-of a proof let L be noetherian Lattice; let a be Element of L; let d be Element of L; assume A1: a [= d & a <> d; defpred P[Element of LattPOSet L] means a [= %($1) & a <> %($1) implies ex c being Element of L st c [= %($1) & c is-upper-neighbour-of a; A2: LattPOSet L is well_founded by Def3; A3: for x being Element of LattPOSet L st for y being Element of LattPOSet L st y <> x & [y,x] in the InternalRel of LattPOSet L holds P[y] holds P[x] proof let x be Element of LattPOSet L; assume A4: for y being Element of LattPOSet L st y <> x & [y,x] in the InternalRel of LattPOSet L holds P[y]; a [= %x & a <> %x implies ex c being Element of L st c [= %x & c is-upper-neighbour-of a proof assume A5: a [= %x & a <> %x; A6: (%x)% = %x by LATTICE3:def 3 .= x by LATTICE3:def 4; per cases; suppose %x is-upper-neighbour-of a; hence thesis; suppose not(%x is-upper-neighbour-of a); then consider c being Element of L such that A7: a [= c & c [= %x & not(c = %x or c = a) by A5,Def5; A8: %(c%) = c% by LATTICE3:def 4 .= c by LATTICE3:def 3; c% <= x by A6,A7,LATTICE3:7; then [c%,x] in the InternalRel of LattPOSet L by ORDERS_1:def 9; then consider c' being Element of L such that A9: c' [= c & c' is-upper-neighbour-of a by A4,A7,A8; c' [= %x by A7,A9,LATTICES:25; hence thesis by A9; end; hence thesis; end; A10: for x being Element of LattPOSet L holds P[x] from WFInduction(A3,A2); %(d%) = d% by LATTICE3:def 4 .= d by LATTICE3:def 3; hence thesis by A1,A10; end; theorem Th4:for L being co-noetherian Lattice for a being Element of L for d being Element of L st d [= a & a <> d holds ex c being Element of L st d [= c & c is-lower-neighbour-of a proof let L be co-noetherian Lattice; let a be Element of L; let d be Element of L; assume A1: d [= a & a <> d; defpred P[Element of (LattPOSet L)~] means %(~($1)) [= a & a <> %(~($1)) implies ex c being Element of L st %(~($1)) [= c & c is-lower-neighbour-of a; A2: (LattPOSet L)~ is well_founded by Def4; A3: for x being Element of (LattPOSet L)~ st for y being Element of (LattPOSet L)~ st y <> x & [y,x] in the InternalRel of (LattPOSet L)~ holds P[y] holds P[x] proof let x be Element of (LattPOSet L)~; assume A4: for y being Element of (LattPOSet L)~ st y <> x & [y,x] in the InternalRel of (LattPOSet L)~ holds P[y]; %(~x) [= a & a <> %(~x) implies ex c being Element of L st %(~x) [= c & c is-lower-neighbour-of a proof assume A5: %(~x) [= a & a <> %(~x); A6: (%(~x))% = %(~x) by LATTICE3:def 3 .= ~x by LATTICE3:def 4; A7: (~x)~ = ~x by LATTICE3:def 6 .= x by LATTICE3:def 7; per cases; suppose %(~x) is-lower-neighbour-of a; hence thesis; suppose not(%(~x) is-lower-neighbour-of a); then consider c being Element of L such that A8: %(~x) [= c & c [= a & not(c = a or c = %(~x)) by A5,Def5; A9: %(~((c%)~)) = ~((c%)~) by LATTICE3:def 4 .= (c%)~ by LATTICE3:def 7 .= c% by LATTICE3:def 6 .= c by LATTICE3:def 3; ~x <= c% by A6,A8,LATTICE3:7; then (c%)~ <= x by A7,LATTICE3:9; then [(c%)~,x] in the InternalRel of (LattPOSet L)~ by ORDERS_1:def 9; then consider c' being Element of L such that A10: %(~((c%)~)) [= c' & c' is-lower-neighbour-of a by A4,A8,A9; %(~x) [= c' by A8,A9,A10,LATTICES:25; hence thesis by A10; end; hence thesis; end; A11: for x being Element of (LattPOSet L)~ holds P[x] from WFInduction(A3,A2); %(~((d%)~)) = ~((d%)~) by LATTICE3:def 4 .= (d%)~ by LATTICE3:def 7 .= d% by LATTICE3:def 6 .= d by LATTICE3:def 3; hence thesis by A1,A11; end; theorem Th5:for L being upper-bounded Lattice holds not(ex b being Element of L st b is-upper-neighbour-of Top L) proof let L be upper-bounded Lattice; given b being Element of L such that A1: b is-upper-neighbour-of Top L; A2: Top L [= b & Top L <> b by A1,Def5; b [= Top L by LATTICES:45; hence thesis by A2,LATTICES:26; end; theorem Th6:for L being noetherian upper-bounded Lattice for a being Element of L holds a = Top L iff not(ex b being Element of L st b is-upper-neighbour-of a) proof let L be noetherian upper-bounded Lattice; let a be Element of L; now assume A1: not(ex b being Element of L st b is-upper-neighbour-of a); for b being Element of L holds a "\/" b = a & b "\/" a = a proof let b be Element of L; consider c being Element of L such that A2: c = a "\/" b; A3: a [= c & b [= c by A2,LATTICES:22; per cases; suppose a <> c; then ex d being Element of L st d [= c & d is-upper-neighbour-of a by A3,Th3; hence thesis by A1; suppose a = c; hence thesis by A2; end; hence a = Top L by LATTICES:def 17; end; hence thesis by Th5; end; theorem Th7:for L being lower-bounded Lattice holds not(ex b being Element of L st b is-lower-neighbour-of Bottom L) proof let L be lower-bounded Lattice; given b being Element of L such that A1: b is-lower-neighbour-of Bottom L; A2: b [= Bottom L & b <> Bottom L by A1,Def5; Bottom L [= b by LATTICES:41; hence thesis by A2,LATTICES:26; end; theorem Th8:for L being co-noetherian lower-bounded Lattice for a being Element of L holds a = Bottom L iff not(ex b being Element of L st b is-lower-neighbour-of a) proof let L be co-noetherian lower-bounded Lattice; let a be Element of L; now assume A1: not(ex b being Element of L st b is-lower-neighbour-of a); for b being Element of L holds a "/\" b = a & b "/\" a = a proof let b be Element of L; consider c being Element of L such that A2: c = a "/\" b; A3: c [= a & c [= b by A2,LATTICES:23; per cases; suppose a <> c; then ex d being Element of L st c [= d & d is-lower-neighbour-of a by A3,Th4; hence thesis by A1; suppose a = c; hence thesis by A2; end; hence a = Bottom L by LATTICES:def 16; end; hence thesis by Th7; end; definition let L be complete Lattice; let a be Element of L; func a*' -> Element of L equals :Def6: "/\"({d where d is Element of L : a [= d & d <> a},L); correctness; func *'a -> Element of L equals :Def7: "\/"({d where d is Element of L : d [= a & d <> a},L); correctness; end; definition let L be complete Lattice; let a be Element of L; attr a is completely-meet-irreducible means :Def8: a*' <> a; attr a is completely-join-irreducible means :Def9: *'a <> a; end; theorem Th9:for L being complete Lattice for a being Element of L holds a [= a*' & *'a [= a proof let L be complete Lattice; let a be Element of L; set X = {d where d is Element of L : a [= d & d <> a}; A1: a is_less_than X proof for q being Element of L st q in X holds a [= q proof let q be Element of L; assume q in X; then ex q' being Element of L st q' = q & a [= q' & q' <> a; hence thesis; end; hence thesis by LATTICE3:def 16; end; A2: a*' = "/\"(X,L) by Def6; set X = {d where d is Element of L : d [= a & d <> a}; A3: X is_less_than a proof for q being Element of L st q in X holds q [= a proof let q be Element of L; assume q in X; then ex q' being Element of L st q' = q & q' [= a & q' <> a; hence thesis; end; hence thesis by LATTICE3:def 17; end; *'a = "\/"(X,L) by Def7; hence thesis by A1,A2,A3,LATTICE3:34,def 21; end; theorem for L being complete Lattice holds Top L*' = Top L & (Top L)% is meet-irreducible proof let L be complete Lattice; set X = {d where d is Element of L : Top L [= d & d <> Top L}; A1: X = {} proof assume X <> {}; then reconsider X as non empty set; consider x being Element of X; x in X; then consider x' being Element of L such that A2: x' = x & Top L [= x' & x' <> Top L; x' [= Top L by LATTICES:45; hence thesis by A2,LATTICES:26; end; A3: Top L is_less_than {} proof for q being Element of L st q in {} holds Top L [= q; hence thesis by LATTICE3:def 16; end; A4: for b being Element of L st b is_less_than {} holds b [= Top L by LATTICES:45; then A5: "/\"({},L) = Top L by A3,LATTICE3:34; (Top L)% = Top (LattPOSet L) proof Top (LattPOSet L) = "/\"({},LattPOSet L) by YELLOW_0:def 12 .= "/\"({},L) by YELLOW_0:29 .= Top L by A3,A4,LATTICE3:34; hence thesis by LATTICE3:def 3; end; hence thesis by A1,A5,Def6,WAYBEL_6:10; end; theorem for L being complete Lattice holds *'Bottom L = Bottom L & (Bottom L)% is join-irreducible proof let L be complete Lattice; set X = {d where d is Element of L : d [= Bottom L & d <> Bottom L}; A1: X = {} proof assume X <> {}; then reconsider X as non empty set; consider x being Element of X; x in X; then consider x' being Element of L such that A2: x' = x & x' [= Bottom L & x' <> Bottom L; Bottom L [= x' by LATTICES:41; hence thesis by A2,LATTICES:26; end; A3: Bottom L is_great_than {} proof for q being Element of L st q in {} holds q [= Bottom L; hence thesis by LATTICE3:def 17; end; A4: for b being Element of L st b is_great_than {} holds Bottom L [= b by LATTICES:41; then A5: "\/"({},L) = Bottom L by A3,LATTICE3:def 21; A6: (Bottom L)% = Bottom (LattPOSet L) proof Bottom (LattPOSet L) = "\/"({},LattPOSet L) by YELLOW_0:def 11 .= "\/"({},L) by YELLOW_0:29 .= Bottom L by A3,A4,LATTICE3:def 21; hence thesis by LATTICE3:def 3; end; Bottom (LattPOSet L) is join-irreducible proof for x,y being Element of LattPOSet L st Bottom (LattPOSet L) = x "\/" y holds x = Bottom (LattPOSet L) or y = Bottom (LattPOSet L) proof let x,y be Element of LattPOSet L; assume Bottom (LattPOSet L) = x "\/" y; then A7: Bottom (LattPOSet L) >= x & Bottom (LattPOSet L) >= y by YELLOW_0:22; reconsider L' = LattPOSet L as lower-bounded antisymmetric non empty RelStr; reconsider x' = x as Element of L'; reconsider y' = y as Element of L'; x' >= Bottom (L') or y' >= Bottom (L') by YELLOW_0:44; hence x = Bottom (LattPOSet L) or y = Bottom (LattPOSet L) by A7,ORDERS_1:25; end; hence thesis by WAYBEL_6:def 3; end; hence thesis by A1,A5,A6,Def7; end; theorem Th12:for L being complete Lattice for a being Element of L st a is completely-meet-irreducible holds a*' is-upper-neighbour-of a & for c being Element of L holds c is-upper-neighbour-of a implies c = a*' proof let L be complete Lattice; let a be Element of L; assume a is completely-meet-irreducible; then A1: a*' <> a by Def8; set X = { x where x is Element of L : a [= x & x <> a}; A2: a*' = "/\"(X,L) by Def6; A3: a [= a*' by Th9; for c being Element of L st a [= c & c [= a*' holds c = a or c = a*' proof let c be Element of L; assume A4: a [= c & c [= a*'; assume c <> a; then c in X by A4; then a*' [= c by A2,LATTICE3:38; hence thesis by A4,LATTICES:26; end; then A5: for c being Element of L holds (a [= c & c [= a*') implies (c = a*' or c = a); for c being Element of L holds c is-upper-neighbour-of a implies c = a*' proof let c be Element of L; assume A6: c is-upper-neighbour-of a; then a <> c & a [= c & for d being Element of L holds (a [= d & d [= c) implies (d = c or d = a) by Def5; then c in X; then a [= a*' & a*' [= c by A2,Th9,LATTICE3:38; hence thesis by A1,A6,Def5; end; hence thesis by A1,A3,A5,Def5; end; theorem Th13:for L being complete Lattice for a being Element of L st a is completely-join-irreducible holds *'a is-lower-neighbour-of a & for c being Element of L holds c is-lower-neighbour-of a implies c = *'a proof let L be complete Lattice; let a be Element of L; assume a is completely-join-irreducible; then A1: *'a <> a by Def9; set X = { x where x is Element of L : x [= a & x <> a}; A2: *'a = "\/"(X,L) by Def7; A3: *'a [= a by Th9; A4: for c being Element of L st *'a [= c & c [= a holds c = a or c = *'a proof let c be Element of L; assume A5: *'a [= c & c [= a; assume c <> a; then c in X by A5; then c [= *'a by A2,LATTICE3:38; hence thesis by A5,LATTICES:26; end; for c being Element of L holds c is-lower-neighbour-of a implies c = *'a proof let c be Element of L; assume A6: c is-lower-neighbour-of a; then a <> c & c [= a & for d being Element of L holds (c [= d & d [= a) implies (d = c or d = a) by Def5; then c in X; then *'a [= a & c [= *'a by A2,Th9,LATTICE3:38; hence thesis by A1,A6,Def5; end; hence thesis by A1,A3,A4,Def5; end; theorem Th14:for L being noetherian complete Lattice for a being Element of L holds a is completely-meet-irreducible iff ex b being Element of L st b is-upper-neighbour-of a & for c being Element of L holds c is-upper-neighbour-of a implies c = b proof let L be noetherian complete Lattice; let a be Element of L; set X = { x where x is Element of L : a [= x & x <> a}; hereby assume a is completely-meet-irreducible; then a*' is-upper-neighbour-of a & for c being Element of L holds c is-upper-neighbour-of a implies c = a*' by Th12; hence ex b being Element of L st b is-upper-neighbour-of a & for c being Element of L holds c is-upper-neighbour-of a implies c = b; end; given b being Element of L such that A1: b is-upper-neighbour-of a & for c being Element of L holds c is-upper-neighbour-of a implies c = b; A2: a [= b & a <> b by A1,Def5; then A3: b in X; for q being Element of L st q in X holds b [= q proof let q be Element of L; assume q in X; then ex q' being Element of L st q' = q & a [= q' & q' <> a; then ex c being Element of L st c [= q & c is-upper-neighbour-of a by Th3; hence thesis by A1; end; then b is_less_than X by LATTICE3:def 16; then b = "/\"(X,L) by A3,LATTICE3:42; hence a <> a*' by A2,Def6; end; theorem Th15:for L being co-noetherian complete Lattice for a being Element of L holds a is completely-join-irreducible iff ex b being Element of L st b is-lower-neighbour-of a & for c being Element of L holds c is-lower-neighbour-of a implies c = b proof let L be co-noetherian complete Lattice; let a be Element of L; set X = { x where x is Element of L : x [= a & x <> a}; A1: now assume a is completely-join-irreducible; then *'a is-lower-neighbour-of a & for c being Element of L holds c is-lower-neighbour-of a implies c = *'a by Th13; hence ex b being Element of L st b is-lower-neighbour-of a & for c being Element of L holds c is-lower-neighbour-of a implies c = b; end; now given b being Element of L such that A2: b is-lower-neighbour-of a & for c being Element of L holds c is-lower-neighbour-of a implies c = b; A3: b [= a & a <> b by A2,Def5; then A4: b in X; for q being Element of L st q in X holds q [= b proof let q be Element of L; assume q in X; then ex q' being Element of L st q' = q & q' [= a & q' <> a; then ex c being Element of L st q [= c & c is-lower-neighbour-of a by Th4; hence thesis by A2; end; then b is_great_than X by LATTICE3:def 17; then b = "\/"(X,L) by A4,LATTICE3:41; then a <> *'a by A3,Def7; hence a is completely-join-irreducible by Def9; end; hence thesis by A1; end; theorem Th16:for L being complete Lattice for a being Element of L holds a is completely-meet-irreducible implies a% is meet-irreducible proof let L be complete Lattice; let a be Element of L; set X = {d where d is Element of L : a [= d & d <> a}; assume a is completely-meet-irreducible; then A1: a <> a*' by Def8; for x,y being Element of LattPOSet L st a% = x "/\" y holds x = a% or y = a% proof let x,y be Element of LattPOSet L; assume A2: a% = x "/\" y; then A3: a% <= x & a% <= y & for d being Element of LattPOSet L st d <= x & d <= y holds a% >= d by YELLOW_0:23; A4: a = a% by LATTICE3:def 3 .= %(a%) by LATTICE3:def 4; A5: y = %y by LATTICE3:def 4 .= (%y)% by LATTICE3:def 3; A6: x = %x by LATTICE3:def 4 .= (%x)% by LATTICE3:def 3; then A7: a [= %x & a [= %y by A3,A5,LATTICE3:7; A8: x = %x & y = %y & %(a%) = a% by LATTICE3:def 4; A9: a [= a*' by Th9; assume x <> a% & y <> a%; then A10: %x in X & %y in X by A4,A7,A8; a*' = "/\"(X,L) by Def6; then a*' is_less_than X & for b being Element of L st b is_less_than X holds b [= a*' by LATTICE3:34; then a*' [= %x & a*' [= %y by A10,LATTICE3:def 16; then (a*')% <= (%x)% & (a*')% <= (%y)% by LATTICE3:7; then A11: (a*')% <= a% by A2,A5,A6,YELLOW_0:23; A12: a% <= (a*')% by A9,LATTICE3:7; a = a% & a*' = (a*')% by LATTICE3:def 3; hence contradiction by A1,A11,A12,ORDERS_1:25; end; hence thesis by WAYBEL_6:def 2; end; Lm8:for L being Lattice for a,b being Element of L holds a "/\" b = a% "/\" b% & a "\/" b = a% "\/" b% proof let L be Lattice; let a,b be Element of L; consider c being Element of L such that A1: c = a "/\" b; c [= a & c [= b by A1,LATTICES:23; then A2: c% <= a% & c% <= b% by LATTICE3:7; for d being Element of LattPOSet L st d <= a% & d <= b% holds c% >= d proof let d be Element of LattPOSet L; assume A3: d <= a% & d <= b%; A4: (%d)% = %d by LATTICE3:def 3 .= d by LATTICE3:def 4; then A5: %d [= a & %d [= b by A3,LATTICE3:7; %d "/\" c = (%d "/\" a) "/\" b by A1,LATTICES:def 7 .= %d "/\" b by A5,LATTICES:21 .= %d by A5,LATTICES:21; then %d [= c by LATTICES:21; hence thesis by A4,LATTICE3:7; end; then A6: c% = a% "/\" b% by A2,YELLOW_0:23; consider c being Element of L such that A7: c = a "\/" b; a [= c & b [= c by A7,LATTICES:22; then A8: a% <= c% & b% <= c% by LATTICE3:7; for d being Element of LattPOSet L st a% <= d & b% <= d holds d >= c% proof let d be Element of LattPOSet L; assume A9: a% <= d & b% <= d; A10: (%d)% = %d by LATTICE3:def 3 .= d by LATTICE3:def 4; then A11: a [= %d & b [= %d by A9,LATTICE3:7; %d "\/" c = (%d "\/" a) "\/" b by A7,LATTICES:def 5 .= %d "\/" b by A11,LATTICES:def 3 .= %d by A11,LATTICES:def 3; then c [= %d by LATTICES:def 3; hence thesis by A10,LATTICE3:7; end; then c% = a% "\/" b% by A8,YELLOW_0:22; hence thesis by A1,A6,A7,LATTICE3:def 3; end; theorem Th17:for L being complete noetherian Lattice for a being Element of L st a <> Top L holds a is completely-meet-irreducible iff a% is meet-irreducible proof let L be complete noetherian Lattice; let a be Element of L; assume a <> Top L; then consider b being Element of L such that A1: b is-upper-neighbour-of a by Th6; A2: b <> a & a [= b & for c being Element of L holds (a [= c & c [= b) implies (c = b or c = a) by A1,Def5; now assume A3: a% is meet-irreducible; for d being Element of L holds d is-upper-neighbour-of a implies d = b proof let d be Element of L; assume A4: d is-upper-neighbour-of a; then A5: d <> a & a [= d & for c being Element of L holds (a [= c & c [= d) implies (c = d or c = a) by Def5; assume d <> b; then A6: a = d "/\" b by A1,A4,Th2; A7: a% = a & d% = d & b% = b by LATTICE3:def 3; then a% = d% "/\" b% by A6,Lm8; hence thesis by A2,A3,A5,A7,WAYBEL_6:def 2; end; hence a is completely-meet-irreducible by A1,Th14; end; hence thesis by Th16; end; theorem Th18:for L being complete Lattice for a being Element of L holds a is completely-join-irreducible implies a% is join-irreducible proof let L be complete Lattice; let a be Element of L; set X = {d where d is Element of L : d [= a & d <> a}; assume a is completely-join-irreducible; then A1: a <> *'a by Def9; for x,y being Element of LattPOSet L st a% = x "\/" y holds x = a% or y = a% proof let x,y be Element of LattPOSet L; assume A2: a% = x "\/" y; then A3: x <= a% & y <= a% & for d being Element of LattPOSet L st d >= x & d >= y holds a% <= d by YELLOW_0:22; A4: a = a% by LATTICE3:def 3 .= %(a%) by LATTICE3:def 4; A5: y = %y by LATTICE3:def 4 .= (%y)% by LATTICE3:def 3; A6: x = %x by LATTICE3:def 4 .= (%x)% by LATTICE3:def 3; then A7: %x [= a & %y [= a by A3,A5,LATTICE3:7; A8: x = %x & y = %y & %(a%) = a% by LATTICE3:def 4; A9: *'a [= a by Th9; assume x <> a% & y <> a%; then A10: %x in X & %y in X by A4,A7,A8; *'a = "\/"(X,L) by Def7; then X is_less_than *'a & for b being Element of L st X is_less_than b holds *'a [= b by LATTICE3:def 21; then %x [= *'a & %y [= *'a by A10,LATTICE3:def 17; then (*'a)% >= x & (*'a)% >= y by A5,A6,LATTICE3:7; then A11: (*'a)% >= a% by A2,YELLOW_0:22; A12: a% >= (*'a)% by A9,LATTICE3:7; a = a% & *'a = (*'a)% by LATTICE3:def 3; hence contradiction by A1,A11,A12,ORDERS_1:25; end; hence thesis by WAYBEL_6:def 3; end; theorem Th19:for L being complete co-noetherian Lattice for a being Element of L st a <> Bottom L holds a is completely-join-irreducible iff a% is join-irreducible proof let L be complete co-noetherian Lattice; let a be Element of L; assume a <> Bottom L; then consider b being Element of L such that A1: b is-lower-neighbour-of a by Th8; A2: a <> b & b [= a & for c being Element of L holds (b [= c & c [= a) implies (c = a or c = b) by A1,Def5; now assume A3: a% is join-irreducible; for d being Element of L holds d is-lower-neighbour-of a implies d = b proof let d be Element of L; assume A4: d is-lower-neighbour-of a; then A5: a <> d & d [= a & for c being Element of L holds (d [= c & c [= a) implies (c = a or c = d) by Def5; assume d <> b; then A6: a = d "\/" b by A1,A4,Th2; A7: a% = a & d% = d & b% = b by LATTICE3:def 3; then a% = d% "\/" b% by A6,Lm8; hence thesis by A2,A3,A5,A7,WAYBEL_6:def 3; end; hence a is completely-join-irreducible by A1,Th15; end; hence thesis by Th18; end; theorem for L being finite Lattice for a being Element of L st a <> Bottom L & a <> Top L holds (a is completely-meet-irreducible iff a% is meet-irreducible) & (a is completely-join-irreducible iff a% is join-irreducible) by Th17,Th19; definition let L be Lattice; let a be Element of L; attr a is atomic means :Def10: a is-upper-neighbour-of Bottom L; attr a is co-atomic means :Def11: a is-lower-neighbour-of Top L; end; theorem for L being complete Lattice for a being Element of L st a is atomic holds a is completely-join-irreducible proof let L be complete Lattice; let a be Element of L; assume a is atomic; then A1: a is-upper-neighbour-of Bottom L by Def10; then A2: a <> Bottom L & Bottom L [= a & for c being Element of L holds (Bottom L [= c & c [= a) implies (c = a or c = Bottom L) by Def5; set X = { x where x is Element of L : x [= a & x <> a}; A3: for x being set holds x in X implies x in {Bottom L} proof let x be set; assume x in X; then A4: ex x' being Element of L st x' = x & x' [= a & x' <> a; then reconsider x as Element of L; Bottom L [= x by LATTICES:41; then x = Bottom L by A1,A4,Def5; hence thesis by TARSKI:def 1; end; A5: for x being set holds x in {Bottom L} implies x in X proof let x be set; assume x in {Bottom L}; then x = Bottom L by TARSKI:def 1; hence thesis by A2; end; Bottom L = "\/"({Bottom L},L) by LATTICE3:43 .= "\/"(X,L) by A3,A5,TARSKI:2 .= *'a by Def7; hence thesis by A2,Def9; end; theorem for L being complete Lattice for a being Element of L st a is co-atomic holds a is completely-meet-irreducible proof let L be complete Lattice; let a be Element of L; assume a is co-atomic; then A1: a is-lower-neighbour-of Top L by Def11; then A2: a <> Top L & a [= Top L & for c being Element of L holds (a [= c & c [= Top L) implies (c = Top L or c = a) by Def5; set X = { x where x is Element of L : a [= x & x <> a}; A3: for x being set holds x in X implies x in {Top L} proof let x be set; assume x in X; then A4: ex x' being Element of L st x' = x & a [= x' & x' <> a; then reconsider x as Element of L; x [= Top L by LATTICES:45; then x = Top L by A1,A4,Def5; hence thesis by TARSKI:def 1; end; A5: for x being set holds x in {Top L} implies x in X proof let x be set; assume x in {Top L}; then x = Top L by TARSKI:def 1; hence thesis by A2; end; Top L = "/\"({Top L},L) by LATTICE3:43 .= "/\"(X,L) by A3,A5,TARSKI:2 .= a*' by Def6; hence thesis by A2,Def8; end; definition let L be Lattice; attr L is atomic means :Def12: for a being Element of L holds ex X being Subset of L st (for x being Element of L st x in X holds x is atomic) & a = "\/"(X,L); end; definition cluster strict non empty trivial Lattice; existence proof consider x being set; set X = {x}; consider XJ,XM being BinOp of X; reconsider L = LattStr(#X,XJ,XM#) as non empty LattStr by STRUCT_0:def 1; A1: the carrier of L is trivial by REALSET1:def 12; then A2: L is trivial by REALSET1:def 13; L is Lattice-like proof A3: for a,b being Element of L holds a"\/"b = b"\/"a by A2,REALSET1:def 20; A4: for a,b,c being Element of L holds a"\/"(b"\/"c) = (a"\/"b)"\/"c by A2,REALSET1:def 20; A5: for a,b being Element of L holds (a"/\"b)"\/"b = b by A2,REALSET1:def 20; A6: for a,b being Element of L holds a"/\"b = b"/\"a by A2,REALSET1:def 20; A7: for a,b,c being Element of L holds a"/\"(b"/\"c) = (a"/\"b)"/\"c by A2,REALSET1:def 20; for a,b being Element of L holds a"/\"(a"\/"b)=a by A2,REALSET1:def 20; then L is join-commutative join-associative meet-absorbing meet-commutative meet-associative join-absorbing by A3,A4,A5,A6,A7,LATTICES:def 4,def 5,def 6,def 7,def 8,def 9; hence thesis by LATTICES:def 10; end; then reconsider L as Lattice; take L; thus thesis by A1,REALSET1:def 13; end; end; Lm9:ex L being Lattice st L is atomic & L is complete proof consider L being strict non empty trivial Lattice; the carrier of L is trivial non empty by REALSET1:def 13; then consider x being set such that A1: the carrier of L = {x} by REALSET1:def 12; reconsider x as Element of L by A1,TARSKI:def 1; for Y being set ex p being Element of L st Y is_less_than p & for r being Element of L st Y is_less_than r holds p [= r proof let Y be set; for q being Element of L st q in Y holds q [= x by A1,TARSKI:def 1; then A2: Y is_less_than x by LATTICE3:def 17; for r being Element of L st Y is_less_than r holds x [= r by A1,TARSKI:def 1; hence thesis by A2; end; then reconsider L as complete Lattice by LATTICE3:def 18; for a being Element of L holds ex X being Subset of L st (for u being Element of L st u in X holds u is atomic) & a = "\/"(X,L) proof let a be Element of L; A3: a = x by A1,TARSKI:def 1; A4: x = "\/"({},L) by A1,TARSKI:def 1; A5: for u being Element of L st u in {} holds u is atomic; for u being set holds u in {} implies u in the carrier of L; then {} is Subset of L by TARSKI:def 3; hence thesis by A3,A4,A5; end; then L is atomic by Def12; hence thesis; end; definition cluster atomic complete Lattice; existence by Lm9; end; definition let L be complete Lattice; let D be Subset of L; attr D is supremum-dense means :Def13: for a being Element of L holds ex D' being Subset of D st a = "\/"(D',L); attr D is infimum-dense means :Def14: for a being Element of L holds ex D' being Subset of D st a = "/\"(D',L); end; theorem Th23:for L being complete Lattice for D being Subset of L holds D is supremum-dense iff for a being Element of L holds a = "\/"({d where d is Element of L: d in D & d [= a},L) proof let L be complete Lattice; let D be Subset of L; A1: now assume A2: D is supremum-dense; thus for a being Element of L holds a = "\/"({d where d is Element of L: d in D & d [= a},L) proof let a be Element of L; set X = {d where d is Element of L: d in D & d [= a}; consider D' being Subset of D such that A3: a = "\/"(D',L) by A2,Def13; for x being set holds x in D' implies x in X proof let x be set; assume A4: x in D'; then x in D; then reconsider x as Element of L; D' is_less_than a by A3,LATTICE3:def 21; then x [= a by A4,LATTICE3:def 17; hence thesis by A4; end; then D' c= X by TARSKI:def 3; then A5: a [= "\/"(X,L) by A3,LATTICE3:46; for q being Element of L st q in X holds q [= a proof let q be Element of L; assume q in X; then ex q' being Element of L st q' = q & q' in D & q' [= a; hence thesis; end; then X is_less_than a by LATTICE3:def 17; then "\/"(X,L) [= a by LATTICE3:def 21; hence thesis by A5,LATTICES:26; end; end; now assume A6: for a being Element of L holds a = "\/"({d where d is Element of L: d in D & d [= a},L); for a being Element of L holds ex D' being Subset of D st a = "\/"(D',L) proof let a be Element of L; set X = {d where d is Element of L: d in D & d [= a}; for x being set holds x in X implies x in D proof let x be set; assume x in X; then ex x' being Element of L st x' = x & x' in D & x' [= a; hence thesis; end; then A7: X is Subset of D by TARSKI:def 3; a = "\/"(X,L) by A6; hence thesis by A7; end; hence D is supremum-dense by Def13; end; hence thesis by A1; end; theorem Th24:for L being complete Lattice for D being Subset of L holds D is infimum-dense iff for a being Element of L holds a = "/\"({d where d is Element of L : d in D & a [= d},L) proof let L be complete Lattice; let D be Subset of L; A1: now assume A2: D is infimum-dense; thus for a being Element of L holds a = "/\"({d where d is Element of L: d in D & a [= d},L) proof let a be Element of L; set X = {d where d is Element of L: d in D & a [= d}; consider D' being Subset of D such that A3: a = "/\"(D',L) by A2,Def14; for x being set holds x in D' implies x in X proof let x be set; assume A4: x in D'; then x in D; then reconsider x as Element of L; a is_less_than D' by A3,LATTICE3:34; then a [= x by A4,LATTICE3:def 16; hence thesis by A4; end; then D' c= X by TARSKI:def 3; then A5: "/\"(X,L) [= a by A3,LATTICE3:46; for q being Element of L st q in X holds a [= q proof let q be Element of L; assume q in X; then ex q' being Element of L st q' = q & q' in D & a [= q'; hence thesis; end; then a is_less_than X by LATTICE3:def 16; then a [= "/\"(X,L) by LATTICE3:40; hence thesis by A5,LATTICES:26; end; end; now assume A6: for a being Element of L holds a = "/\"({d where d is Element of L: d in D & a [= d},L); for a being Element of L holds ex D' being Subset of D st a = "/\"(D',L) proof let a be Element of L; set X = {d where d is Element of L: d in D & a [= d}; for x being set holds x in X implies x in D proof let x be set; assume x in X; then ex x' being Element of L st x' = x & x' in D & a [= x'; hence thesis; end; then A7: X is Subset of D by TARSKI:def 3; a = "/\"(X,L) by A6; hence thesis by A7; end; hence D is infimum-dense by Def14; end; hence thesis by A1; end; theorem for L being complete Lattice for D being Subset of L holds D is infimum-dense iff D% is order-generating proof let L be complete Lattice; let D be Subset of L; A1: now assume A2: D is infimum-dense; for a being Element of LattPOSet L ex Y being Subset of D% st a = "/\"(Y,LattPOSet L) proof let a be Element of LattPOSet L; consider D' being Subset of D such that A3: %a = "/\"(D',L) by A2,Def14; A4: %a is_less_than D' & for b being Element of L st b is_less_than D' holds b [= %a by A3,LATTICE3:34; for x being set holds x in D' implies x in the carrier of L proof let x be set; assume x in D'; then x in D; hence thesis; end; then reconsider D' as Subset of L by TARSKI:def 3; for u being Element of LattPOSet L st u in (D')% holds a <= u proof let u be Element of LattPOSet L; assume u in (D')%; then u in {d% where d is Element of L : d in D'} by Def1 ; then consider u' being Element of L such that A5: u = (u')% & u' in D'; A6: %a [= u' by A4,A5,LATTICE3:def 16; (%a)% = %a by LATTICE3:def 3 .= a by LATTICE3:def 4; hence thesis by A5,A6,LATTICE3:7; end; then A7: (D')% is_>=_than a by LATTICE3:def 8; for b being Element of LattPOSet L st (D')% is_>=_than b holds b <= a proof let b be Element of LattPOSet L; assume A8: (D')% is_>=_than b; A9: b = %b by LATTICE3:def 4 .= (%b)% by LATTICE3:def 3; A10: a = %a by LATTICE3:def 4 .= (%a)% by LATTICE3:def 3; for u being Element of L st u in D' holds %b [= u proof let u be Element of L; assume u in D'; then u% in {d% where d is Element of L : d in D'}; then u% in (D')% by Def1; then b <= u% by A8,LATTICE3:def 8; hence thesis by A9,LATTICE3:7; end; then %b is_less_than D' by LATTICE3:def 16; then %b [= %a by A3,LATTICE3:34; hence thesis by A9,A10,LATTICE3:7; end; then A11: a = "/\"((D')%,LattPOSet L)by A7,YELLOW_0:33; for x being set holds x in (D')% implies x in D% proof let x be set; assume x in (D')%; then x in {d% where d is Element of L : d in D'} by Def1 ; then ex x' being Element of L st x = (x')% & x' in D'; then x in {d% where d is Element of L : d in D}; hence thesis by Def1; end; then (D')% is Subset of D% by TARSKI:def 3; hence thesis by A11; end; hence D% is order-generating by WAYBEL_6:15; end; now assume A12: D% is order-generating; for a being Element of L holds ex D' being Subset of D st a = "/\"(D',L) proof let a be Element of L; consider Y being Subset of D% such that A13: a% = "/\"(Y,LattPOSet L) by A12,WAYBEL_6:15; A14: a% is_<=_than Y & for b being Element of LattPOSet L st b is_<=_than Y holds a% >= b by A13,YELLOW_0:33; for x being set holds x in Y implies x in the carrier of LattPOSet L proof let x be set; assume x in Y; then x in D%; hence thesis; end; then reconsider Y as Subset of LattPOSet L by TARSKI:def 3; for q being Element of L st q in %Y holds a [= q proof let q be Element of L; assume q in %Y; then q in {%d where d is Element of LattPOSet L : d in Y} by Def2; then consider q' being Element of LattPOSet L such that A15: q = %(q') & q' in Y; A16: a% <= q' by A14,A15,LATTICE3:def 8; q' = %(q') by LATTICE3:def 4 .= (%(q'))% by LATTICE3:def 3; hence thesis by A15,A16,LATTICE3:7; end; then A17: a is_less_than %Y by LATTICE3:def 16; for b being Element of L st b is_less_than %Y holds b [= a proof let b be Element of L; assume A18: b is_less_than %Y; for u being Element of LattPOSet L st u in Y holds b% <= u proof let u be Element of LattPOSet L; assume u in Y; then %u in {%d where d is Element of LattPOSet L : d in Y}; then %u in %Y by Def2; then A19: b [= %u by A18,LATTICE3:def 16; (%u)% = %u by LATTICE3:def 3 .= u by LATTICE3:def 4; hence thesis by A19,LATTICE3:7; end; then b% is_<=_than Y by LATTICE3:def 8; then b% <= a% by A13,YELLOW_0:33; hence thesis by LATTICE3:7; end; then A20: a = "/\"(%Y,L) by A17,LATTICE3:34; for x being set holds x in %Y implies x in D proof let x be set; assume x in %Y; then x in {%d where d is Element of LattPOSet L : d in Y} by Def2; then consider x' being Element of LattPOSet L such that A21: x = %(x') & x' in Y; x' in D% by A21; then x' in {d% where d is Element of L : d in D} by Def1; then consider y being Element of L such that A22: x' = y% & y in D; x = y% by A21,A22,LATTICE3:def 4 .= y by LATTICE3:def 3; hence thesis by A22; end; then %Y is Subset of D by TARSKI:def 3; hence thesis by A20; end; hence D is infimum-dense by Def14; end; hence thesis by A1; end; definition let L be complete Lattice; func MIRRS(L) -> Subset of L equals :Def15: {a where a is Element of L : a is completely-meet-irreducible}; correctness proof set X = {a where a is Element of L : a is completely-meet-irreducible}; for x being set holds x in X implies x in the carrier of L proof let x be set; assume x in X; then ex x' being Element of L st x' = x & x' is completely-meet-irreducible; hence thesis; end; then X is Subset of L by TARSKI:def 3; hence thesis; end; func JIRRS(L) -> Subset of L equals :Def16: {a where a is Element of L : a is completely-join-irreducible }; correctness proof set X = {a where a is Element of L : a is completely-join-irreducible }; for x being set holds x in X implies x in the carrier of L proof let x be set; assume x in X; then ex x' being Element of L st x' = x & x' is completely-join-irreducible; hence thesis; end; then X is Subset of L by TARSKI:def 3; hence thesis; end; end; theorem for L being complete Lattice for D being Subset of L st D is supremum-dense holds JIRRS(L) c= D proof let L be complete Lattice; let D be Subset of L; assume A1: D is supremum-dense; for x being set holds x in JIRRS(L) implies x in D proof let x be set; assume A2: x in JIRRS(L); assume A3: not(x in D); JIRRS(L) = {a where a is Element of L : a is completely-join-irreducible} by Def16; then consider x' being Element of L such that A4: x' = x & x' is completely-join-irreducible by A2; A5: x' = x & *'(x') <> x' by A4,Def9; reconsider x as Element of L by A4; set D' = {d where d is Element of L: d in D & d [= x}; set X = {d where d is Element of L: d [= x & d <> x}; A6: not(x in D') proof assume x in D'; then ex x' being Element of L st x' = x & x' in D & x' [= x; hence thesis by A3; end; for u being set holds u in D' implies u in X proof let u be set; assume u in D'; then consider u' being Element of L such that A7: u' = u & u' in D & u' [= x; reconsider u as Element of L by A7; u <> x by A6,A7; hence thesis by A7; end; then D' c= X by TARSKI:def 3; then "\/"(D',L) [= "\/"(X,L) by LATTICE3:46; then A8: x [= "\/"(X,L) by A1,Th23; for q being Element of L st q in X holds q [= x proof let q be Element of L; assume q in X; then ex q' being Element of L st q' = q & q' [= x & q' <> x; hence thesis; end; then X is_less_than x by LATTICE3:def 17; then "\/"(X,L) [= x by LATTICE3:def 21; then x = "\/"(X,L) by A8,LATTICES:26; hence thesis by A5,Def7; end; hence thesis by TARSKI:def 3; end; theorem for L being complete Lattice for D being Subset of L st D is infimum-dense holds MIRRS(L) c= D proof let L be complete Lattice; let D be Subset of L; assume A1: D is infimum-dense; for x being set holds x in MIRRS(L) implies x in D proof let x be set; assume A2: x in MIRRS(L); assume A3: not(x in D); MIRRS(L) = {a where a is Element of L : a is completely-meet-irreducible} by Def15; then consider x' being Element of L such that A4: x' = x & x' is completely-meet-irreducible by A2; A5: x' = x & (x')*' <> x' by A4,Def8; reconsider x as Element of L by A4; set D' = {d where d is Element of L: d in D & x [= d}; set X = {d where d is Element of L: x [= d & d <> x}; A6: not(x in D') proof assume x in D'; then ex x' being Element of L st x' = x & x' in D & x [= x'; hence thesis by A3; end; for u being set holds u in D' implies u in X proof let u be set; assume u in D'; then consider u' being Element of L such that A7: u' = u & u' in D & x [= u'; reconsider u as Element of L by A7; u <> x by A6,A7; hence thesis by A7; end; then D' c= X by TARSKI:def 3; then "/\"(X,L) [= "/\"(D',L) by LATTICE3:46; then A8: "/\"(X,L) [= x by A1,Th24; for q being Element of L st q in X holds x [= q proof let q be Element of L; assume q in X; then ex q' being Element of L st q' = q & x [= q' & q' <> x; hence thesis; end; then x is_less_than X by LATTICE3:def 16; then x [= "/\"(X,L) by LATTICE3:40; then x = "/\"(X,L) by A8,LATTICES:26; hence thesis by A5,Def6; end; hence thesis by TARSKI:def 3; end; Lm10:for L being co-noetherian complete Lattice holds MIRRS(L) is infimum-dense proof let L be co-noetherian complete Lattice; defpred P[Element of (LattPOSet L)~] means ex D' being Subset of MIRRS(L) st $1 = ("/\"(D',L))%; A1: (LattPOSet L)~ is well_founded by Def4; A2: for x being Element of (LattPOSet L)~ st for y being Element of (LattPOSet L)~ st y <> x & [y,x] in the InternalRel of (LattPOSet L)~ holds P[y] holds P[x] proof let x be Element of (LattPOSet L)~; assume A3: for y being Element of (LattPOSet L)~ st y <> x & [y,x] in the InternalRel of (LattPOSet L)~ holds P[y]; set X = {d where d is Element of L: %(~x) [= d & d <> %(~x)}; A4: for u being Element of L st u in X holds (u%)~ <> x & [(u%)~,x] in the InternalRel of (LattPOSet L)~ proof let u be Element of L; assume u in X; then consider u' being Element of L such that A5: u' = u & %(~x) [= u' & %(~x) <> u'; A6: (%(~x))% <= u% by A5,LATTICE3:7; ((%(~x))%)~ = (%(~x))% by LATTICE3:def 6 .= %(~x) by LATTICE3:def 3 .= ~x by LATTICE3:def 4 .= x by LATTICE3:def 7; then A7: (u%)~ <= x by A6,LATTICE3:9; A8: %(~x) = ~x by LATTICE3:def 4 .= x by LATTICE3:def 7; (u%)~ = u% by LATTICE3:def 6 .= u by LATTICE3:def 3; hence thesis by A5,A7,A8,ORDERS_1:def 9; end; A9: for u being Element of L st u in X holds P[(u%)~] proof let u be Element of L; assume u in X; then (u%)~ <> x & [(u%)~,x] in the InternalRel of (LattPOSet L)~ by A4; hence thesis by A3; end; per cases; suppose %(~x) is completely-meet-irreducible; then %(~x) in {a where a is Element of L : a is completely-meet-irreducible }; then %(~x) in MIRRS(L) by Def15; then for y being set holds y in {%(~x)} implies y in MIRRS(L) by TARSKI:def 1; then A10: {%(~x)} is Subset of MIRRS(L) by TARSKI:def 3; ("/\"({%(~x)},L))% = "/\"({%(~x)},L) by LATTICE3:def 3 .= %(~x) by LATTICE3:43 .= ~x by LATTICE3:def 4 .= x by LATTICE3:def 7; hence thesis by A10; suppose not( %(~x) is completely-meet-irreducible ); then (%(~x))*' = %(~x) by Def8; then A11: %(~x) = "/\"(X,L) by Def6; set G = {H where H is Subset of MIRRS(L) : ex u being Element of LattPOSet L st %u in X & %u = "/\" (H,L)}; set D' = union G; for v being set holds v in D' implies v in MIRRS(L) proof let v be set; assume v in D'; then consider v' being set such that A12: v in v' & v' in G by TARSKI:def 4; consider H being Subset of MIRRS(L) such that A13: H = v' & ex u being Element of LattPOSet L st %u in X & %u = "/\" (H,L) by A12; thus thesis by A12,A13; end; then A14: D' is Subset of MIRRS(L) by TARSKI:def 3; for q being Element of L st q in D' holds %(~x) [= q proof let q be Element of L; assume q in D'; then consider h being set such that A15: q in h & h in G by TARSKI:def 4; consider h' being Subset of MIRRS(L) such that A16: h' = h & ex u being Element of LattPOSet L st %u in X & %u = "/\"(h',L) by A15; consider u being Element of LattPOSet L such that A17: %u in X & %u = "/\"(h,L) by A16; %u is_less_than h by A17,LATTICE3:34; then A18: %u [= q by A15,LATTICE3:def 16; %(~x) is_less_than X by A11,LATTICE3:34; then %(~x) [= %u by A17,LATTICE3:def 16; hence thesis by A18,LATTICES:25; end; then A19: %(~x) is_less_than D' by LATTICE3:def 16; for r being Element of L st r is_less_than D' holds r [= %(~x) proof let r be Element of L; assume A20: r is_less_than D'; for q being Element of L st q in X holds r [= q proof let q be Element of L; assume A21: q in X; A22: q% = q by LATTICE3:def 3; consider D being Subset of MIRRS(L) such that A23: (q%)~ = ("/\"(D,L))% by A9,A21; A24: q = (q%)~ by A22,LATTICE3:def 6 .= "/\"(D,L) by A23,LATTICE3:def 3; for v being set holds v in D implies v in D' proof let v be set; assume A25: v in D; A26: %(q%) in X by A21,A22,LATTICE3:def 4; (q%)~ = q% by LATTICE3:def 6.= %(q%) by LATTICE3:def 4; then %(q%) = "/\"(D,L) by A23,LATTICE3:def 3; then D in G by A26; hence thesis by A25,TARSKI:def 4; end; then D c= D' by TARSKI:def 3; then for p being Element of L st p in D holds r [= p by A20,LATTICE3:def 16; then r is_less_than D by LATTICE3:def 16; hence thesis by A24,LATTICE3:34; end; then r is_less_than X by LATTICE3:def 16; hence thesis by A11,LATTICE3:34; end; then A27: %(~x) = "/\"(D',L) by A19,LATTICE3:34; A28: %(~x) = ~x by LATTICE3:def 4 .= x by LATTICE3:def 7; ("/\"(D',L))% = "/\"(D',L) by LATTICE3:def 3; hence thesis by A14,A27,A28; end; A29: for x being Element of (LattPOSet L)~ holds P[x] from WFInduction(A2,A1); for a being Element of L holds ex D' being Subset of MIRRS(L) st a = "/\"(D',L) proof let a be Element of L; (LattPOSet L)~ = RelStr(#the carrier of LattPOSet L, (the InternalRel of LattPOSet L)~#) by LATTICE3:def 5; then reconsider a' = a% as Element of (LattPOSet L)~; consider D' being Subset of MIRRS(L) such that A30: a' = ("/\" (D',L))% by A29; A31: ("/\"(D',L))% = "/\"(D',L) by LATTICE3:def 3; a% = a by LATTICE3:def 3; hence thesis by A30,A31; end; hence thesis by Def14; end; definition let L be co-noetherian complete Lattice; cluster MIRRS(L) -> infimum-dense; coherence by Lm10; end; Lm11:for L being noetherian complete Lattice holds JIRRS(L) is supremum-dense proof let L be noetherian complete Lattice; defpred P[Element of LattPOSet L] means ex D' being Subset of JIRRS(L) st %($1) = "\/"(D',L); A1: LattPOSet L is well_founded by Def3; A2: for x being Element of LattPOSet L st for y being Element of LattPOSet L st y <> x & [y,x] in the InternalRel of LattPOSet L holds P[y] holds P[x] proof let x be Element of LattPOSet L; assume A3: for y being Element of LattPOSet L st y <> x & [y,x] in the InternalRel of LattPOSet L holds P[y]; set X = {d where d is Element of L: d [= %x & d <> %x}; A4: for u being Element of L st u in X holds u% <> x & [u%,x] in the InternalRel of LattPOSet L proof let u be Element of L; assume u in X; then consider u' being Element of L such that A5: u' = u & u' [= %x & %x <> u'; A6: %x = x by LATTICE3:def 4; then (%x)% = x by LATTICE3:def 3; then u% <= x by A5,LATTICE3:7; hence thesis by A5,A6,LATTICE3:def 3,ORDERS_1:def 9; end; A7: for u being Element of L st u in X holds P[u%] proof let u be Element of L; assume u in X; then u% <> x & [u%,x] in the InternalRel of LattPOSet L by A4; hence thesis by A3; end; per cases; suppose %x is completely-join-irreducible; then %x in {a where a is Element of L : a is completely-join-irreducible}; then %x in JIRRS(L) by Def16; then for y being set holds y in {%x} implies y in JIRRS(L) by TARSKI:def 1; then A8: {%x} is Subset of JIRRS(L) by TARSKI:def 3; "\/"({%x},L) = %x by LATTICE3:43; hence thesis by A8; suppose not( %x is completely-join-irreducible ); then *'(%x) = %x by Def9; then A9: %x = "\/"(X,L) by Def7; set G = {H where H is Subset of JIRRS(L) : ex u being Element of LattPOSet L st %u in X & %u = "\/" (H,L)}; set D' = union G; for v being set holds v in D' implies v in JIRRS(L) proof let v be set; assume v in D'; then consider v' being set such that A10: v in v' & v' in G by TARSKI:def 4; consider H being Subset of JIRRS(L) such that A11: H = v' & ex u being Element of LattPOSet L st %u in X & %u = "\/" (H,L) by A10; thus thesis by A10,A11; end; then A12: D' is Subset of JIRRS(L) by TARSKI:def 3; for q being Element of L st q in D' holds q [= %x proof let q be Element of L; assume q in D'; then consider h being set such that A13: q in h & h in G by TARSKI:def 4; consider h' being Subset of JIRRS(L) such that A14: h' = h & ex u being Element of LattPOSet L st %u in X & %u = "\/"(h',L) by A13; consider u being Element of LattPOSet L such that A15: %u in X & %u = "\/"(h,L) by A14; h is_less_than %u by A15,LATTICE3:def 21; then A16: q [= %u by A13,LATTICE3:def 17; X is_less_than %x by A9,LATTICE3:def 21; then %u [= %x by A15,LATTICE3:def 17; hence thesis by A16,LATTICES:25; end; then A17: D' is_less_than %x by LATTICE3:def 17; for r being Element of L st D' is_less_than r holds %x [= r proof let r be Element of L; assume A18: D' is_less_than r; for q being Element of L st q in X holds q [= r proof let q be Element of L; assume A19: q in X; A20: q% = q by LATTICE3:def 3; consider D being Subset of JIRRS(L) such that A21: %(q%) = "\/"(D,L) by A7,A19; A22: q = "\/"(D,L) by A20,A21,LATTICE3:def 4; for v being set holds v in D implies v in D' proof let v be set; assume A23: v in D; D in G by A19,A21,A22; hence thesis by A23,TARSKI:def 4; end; then D c= D' by TARSKI:def 3; then for p being Element of L st p in D holds p [= r by A18,LATTICE3:def 17; then D is_less_than r by LATTICE3:def 17; hence thesis by A22,LATTICE3:def 21; end; then X is_less_than r by LATTICE3:def 17; hence thesis by A9,LATTICE3:def 21; end; then %x = "\/"(D',L) by A17,LATTICE3:def 21; hence thesis by A12; end; A24: for x being Element of LattPOSet L holds P[x] from WFInduction(A2,A1); for a being Element of L holds ex D' being Subset of JIRRS(L) st a = "\/"(D',L) proof let a be Element of L; %(a%) = a% by LATTICE3:def 4 .= a by LATTICE3:def 3; hence thesis by A24; end; hence thesis by Def13; end; definition let L be noetherian complete Lattice; cluster JIRRS(L) -> supremum-dense; coherence by Lm11; end;