Copyright (c) 1995 Association of Mizar Users
environ vocabulary CQC_LANG, QC_LANG1, QMAX_1, CQC_THE1, BOOLE, ZF_LANG, PRE_TOPC, FINSEQ_1, FUNCT_1, CAT_1, QC_LANG3, CQC_THE3; notation TARSKI, XBOOLE_0, SUBSET_1, XREAL_0, DOMAIN_1, NAT_1, FUNCT_1, FINSEQ_1, QC_LANG1, QC_LANG3, CQC_LANG, CQC_THE1; constructors DOMAIN_1, NAT_1, QC_LANG3, CQC_THE1, XREAL_0, MEMBERED, XBOOLE_0; clusters RELSET_1, CQC_LANG, ARYTM_3, MEMBERED, ZFMISC_1, XBOOLE_0; requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM; definitions TARSKI; theorems TARSKI, ZFMISC_1, SUBSET_1, CQC_THE1, CQC_THE2, LUKASI_1, NAT_1, RLVECT_1, QC_LANG1, QC_LANG2, QC_LANG3, PROCAL_1, CQC_LANG, XBOOLE_0, XBOOLE_1; schemes NAT_1, QC_LANG1; begin reserve p, q, r, s, p1, q1 for Element of CQC-WFF, X, Y, Z, X1, X2 for Subset of CQC-WFF, h for QC-formula, x, y for bound_QC-variable, n for Nat; theorem Th1: p in X implies X |- p proof assume A1: p in X; X c= Cn(X) by CQC_THE1:38; hence thesis by A1,CQC_THE1:def 9; end; theorem Th2: X c= Cn(Y) implies Cn(X) c= Cn(Y) proof assume A1: X c= Cn(Y); Cn(Y) is being_a_theory by CQC_THE1:36; hence thesis by A1,CQC_THE1:37; end; theorem Th3: X |- p & {p} |- q implies X |- q proof assume that A1: X |- p and A2: {p} |- q; p in Cn(X) by A1,CQC_THE1:def 9; then {p} c= Cn(X) by ZFMISC_1:37; then A3: Cn({p}) c= Cn(X) by Th2; q in Cn({p}) by A2,CQC_THE1:def 9; hence X |- q by A3,CQC_THE1:def 9; end; theorem Th4: X |- p & X c= Y implies Y |- p proof assume that A1: X |- p and A2: X c= Y; A3: Cn(X) c= Cn(Y) by A2,CQC_THE1:39; p in Cn(X) by A1,CQC_THE1:def 9; hence Y |- p by A3,CQC_THE1:def 9; end; definition let p, q be Element of CQC-WFF; pred p |- q means :Def1: {p} |- q; end; theorem Th5: p |- p proof {p} |- p by CQC_THE2:91; hence thesis by Def1; end; theorem Th6: p |- q & q |- r implies p |- r proof assume that A1: p |- q and A2: q |- r; A3: {p} |- q by A1,Def1; {q} |- r by A2,Def1; then {p} |- r by A3,Th3; hence thesis by Def1; end; definition let X, Y be Subset of CQC-WFF; pred X |- Y means :Def2: for p being Element of CQC-WFF st p in Y holds X |- p; end; theorem Th7: X |- Y iff Y c= Cn(X) proof A1: now assume A2: X |- Y; now let p be set; assume A3: p in Y; then reconsider p' = p as Element of CQC-WFF; X |- p' by A2,A3,Def2; hence p in Cn(X) by CQC_THE1:def 9; end; hence Y c= Cn(X) by TARSKI:def 3; end; now assume Y c= Cn(X); then for p st p in Y holds X |- p by CQC_THE1:def 9; hence X |- Y by Def2; end; hence thesis by A1; end; theorem X |- X proof for p st p in X holds X |- p by Th1; hence thesis by Def2; end; theorem Th9: X |- Y & Y |- Z implies X |- Z proof assume that A1: X |- Y and A2: Y |- Z; for p st p in Z holds X |- p proof let p; assume p in Z; then Y |- p by A2,Def2; then A3: p in Cn(Y) by CQC_THE1:def 9; Y c= Cn(X) by A1,Th7; then Cn(Y) c= Cn(X) by Th2; hence X |- p by A3,CQC_THE1:def 9; end; hence X |- Z by Def2; end; theorem Th10: X |- {p} iff X |- p proof A1: now assume A2: X |- {p}; p in {p} by TARSKI:def 1; hence X |- p by A2,Def2; end; now assume X |- p; then for q st q in {p} holds X |- q by TARSKI:def 1; hence X |- {p} by Def2; end; hence thesis by A1; end; theorem Th11: {p} |- {q} iff p |- q proof A1: now assume {p} |- {q}; then {p} |- q by Th10; hence p |- q by Def1; end; now assume p |- q; then {p} |- q by Def1; hence {p} |- {q} by Th10; end; hence thesis by A1; end; theorem X c= Y implies Y |- X proof assume X c= Y; then for p st p in X holds Y |- p by Th1; hence Y |- X by Def2; end; theorem Th13: X |- TAUT proof TAUT c= Cn(X) by CQC_THE1:75; hence thesis by Th7; end; theorem Th14: {}(CQC-WFF) |- TAUT by Th13; definition let X be Subset of CQC-WFF; pred |- X means :Def3: for p being Element of CQC-WFF st p in X holds |- p; end; theorem Th15: |- X iff {}(CQC-WFF) |- X proof A1: now assume A2: |- X; now let p; assume p in X; then |- p by A2,Def3; hence {}(CQC-WFF) |- p by CQC_THE1:def 10; end; hence {}(CQC-WFF) |- X by Def2; end; now assume A3: {}(CQC-WFF) |- X; now let p; assume p in X; then {}(CQC-WFF) |- p by A3,Def2; hence |- p by CQC_THE1:def 10; end; hence |- X by Def3; end; hence thesis by A1; end; theorem |- TAUT by Th14,Th15; theorem |- X iff X c= TAUT proof A1: now assume A2: |- X; now let p; assume p in X; then |- p by A2,Def3; hence p in TAUT by CQC_THE1:def 11; end; hence X c= TAUT by SUBSET_1:7; end; now assume X c= TAUT; then for p st p in X holds |- p by CQC_THE1:def 11; hence |- X by Def3; end; hence thesis by A1; end; definition let X, Y; pred X |-| Y means :Def4: for p holds (X |- p iff Y |- p); reflexivity; symmetry; end; theorem Th18: X |-| Y iff (X |- Y & Y |- X) proof A1: now assume A2: X |-| Y; now let p; assume p in Y; then Y |- p by Th1; hence X |- p by A2,Def4; end; hence X |- Y by Def2; now let p; assume p in X; then X |- p by Th1; hence Y |- p by A2,Def4; end; hence Y |- X by Def2; end; now assume that A3: X |- Y and A4: Y |- X; now let p; A5: now assume X |- p; then X |- {p} by Th10; then Y |- {p} by A4,Th9; hence Y |- p by Th10; end; now assume Y |- p; then Y |- {p} by Th10; then X |- {p} by A3,Th9; hence X |- p by Th10; end; hence X |- p iff Y |- p by A5; end; hence X |-| Y by Def4; end; hence thesis by A1; end; theorem Th19: X |-| Y & Y |-| Z implies X |-| Z proof assume that A1: X |-| Y and A2: Y |-| Z; A3: X |- Y by A1,Th18; Y |- Z by A2,Th18; then A4: X |- Z by A3,Th9; A5: Y |- X by A1,Th18; Z |- Y by A2,Th18; then Z |- X by A5,Th9; hence X |-| Z by A4,Th18; end; theorem Th20: X |-| Y iff Cn(X) = Cn(Y) proof A1: now assume X |-| Y; then X |- Y & Y |- X by Th18; then Y c= Cn(X) & X c= Cn(Y) by Th7; then Cn(Y) c= Cn(X) & Cn(X) c= Cn(Y) by Th2; hence Cn(X) = Cn(Y) by XBOOLE_0:def 10; end; now assume A2: Cn(X) = Cn(Y); Y c= Cn(Y) & X c= Cn(X) by CQC_THE1:38; then X |- Y & Y |- X by A2,Th7; hence X |-| Y by Th18; end; hence thesis by A1; end; Lm1: X \/ Y c= Cn(X) \/ Cn(Y) proof X c= Cn(X) & Y c= Cn(Y) by CQC_THE1:38; hence thesis by XBOOLE_1:13; end; theorem Th21: Cn(X) \/ Cn(Y) c= Cn(X \/ Y) proof X c= X \/ Y & Y c= X \/ Y by XBOOLE_1:7; then Cn(X) c= Cn(X \/ Y) & Cn(Y) c= Cn(X \/ Y) by CQC_THE1:39; hence thesis by XBOOLE_1:8; end; theorem Th22: Cn(X \/ Y) = Cn(Cn(X) \/ Cn(Y)) proof X \/ Y c= Cn(X) \/ Cn(Y) by Lm1; then A1: Cn(X \/ Y) c= Cn(Cn(X) \/ Cn(Y)) by CQC_THE1:39; Cn(X) \/ Cn(Y) c= Cn(X \/ Y) by Th21; then Cn(Cn(X) \/ Cn(Y)) c= Cn(X \/ Y) by Th2; hence thesis by A1,XBOOLE_0:def 10; end; theorem X |-| Cn(X) proof Cn(X) = Cn(Cn(X)) by CQC_THE1:40; hence thesis by Th20; end; theorem X \/ Y |-| Cn(X) \/ Cn(Y) proof Cn(X \/ Y) = Cn(Cn(X) \/ Cn(Y)) by Th22; hence thesis by Th20; end; theorem Th25: X1 |-| X2 implies X1 \/ Y |-| X2 \/ Y proof assume X1 |-| X2; then Cn(X1) = Cn(X2) by Th20; then Cn(X1 \/ Y) = Cn(Cn(X2) \/ Cn(Y)) by Th22 .= Cn(X2 \/ Y) by Th22; hence X1 \/ Y |-| X2 \/ Y by Th20; end; theorem Th26: X1 |-| X2 & X1 \/ Y |- Z implies X2 \/ Y |- Z proof assume that A1: X1 |-| X2 and A2: X1 \/ Y |- Z; X1 \/ Y |-| X2 \/ Y by A1,Th25; then Cn(X1 \/ Y) = Cn(X2 \/ Y) by Th20; then Z c= Cn(X2 \/ Y) by A2,Th7; hence X2 \/ Y |- Z by Th7; end; theorem Th27: X1 |-| X2 & Y |- X1 implies Y |- X2 proof assume that A1: X1 |-| X2 and A2: Y |- X1; X1 |- X2 by A1,Th18; hence Y |- X2 by A2,Th9; end; definition let p, q be Element of CQC-WFF; pred p |-| q means :Def5: p |- q & q |- p; reflexivity by Th5; symmetry; end; theorem Th28: p |-| q & q |-| r implies p |-| r proof assume that A1: p |-| q and A2: q |-| r; A3: p |- q by A1,Def5; q |- r by A2,Def5; then A4: p |- r by A3,Th6; A5: q |- p by A1,Def5; r |- q by A2,Def5; then r |- p by A5,Th6; hence p |-| r by A4,Def5; end; theorem Th29: p |-| q iff {p} |-| {q} proof A1: now assume p |-| q; then p |- q & q |- p by Def5; then {p} |- {q} & {q} |- {p} by Th11; hence {p} |-| {q} by Th18; end; now assume {p} |-| {q}; then {p} |- {q} & {q} |- {p} by Th18; then p |- q & q |- p by Th11; hence p |-| q by Def5; end; hence thesis by A1; end; theorem p |-| q & X |- p implies X |- q proof assume that A1: p |-| q and A2: X |- p; A3: {p} |-| {q} by A1,Th29; X |- {p} by A2,Th10; then X |- {q} by A3,Th27; hence X |- q by Th10; end; theorem Th31: {p,q} |-| {p '&' q} proof for r holds {p,q} |- r iff {p '&' q} |- r by CQC_THE2:93; hence thesis by Def4; end; theorem p '&' q |-| q '&' p proof A1: {p '&' q} |-| {q,p} by Th31; {q,p} |-| {q '&' p} by Th31; then {p '&' q} |-| {q '&' p} by A1,Th19; hence p '&' q |-| q '&' p by Th29; end; Lm2: X |- p & X |- q implies X |- p '&' q proof assume that A1: X |- p and A2: X |- q; for r st r in {p,q} holds X |- r by A1,A2,TARSKI:def 2; then A3: X |- {p,q} by Def2; {p,q} |-| {p '&' q} by Th31; then X |- {p '&' q} by A3,Th27; hence X |- p '&' q by Th10; end; Lm3: X |- p '&' q implies (X |- p & X |- q) proof assume X |- p '&' q; then A1: X |- {p '&' q} by Th10; {p,q} |-| {p '&' q} by Th31; then A2: X |- {p,q} by A1,Th27; p in {p,q} by TARSKI:def 2; hence X |- p by A2,Def2; q in {p,q} by TARSKI:def 2; hence X |- q by A2,Def2; end; theorem X |- p '&' q iff (X |- p & X |- q) by Lm2,Lm3; Lm4: p |-| q & r |-| s implies p '&' r |- q '&' s proof assume that A1: p |-| q and A2: r |-| s; p |- q by A1,Def5; then A3: {p} |- q by Def1; {p} c= {p,r} by ZFMISC_1:12; then A4: {p,r} |- q by A3,Th4; r |- s by A2,Def5; then A5: {r} |- s by Def1; {r} c= {p,r} by ZFMISC_1:12; then {p,r} |- s by A5,Th4; then {p,r} |- q '&' s by A4,Lm2; then A6: {p,r} |- {q '&' s} by Th10; {p,r} |-| {p '&' r} by Th31; then {p '&' r} |- {p,r} by Th18; then {p '&' r} |- {q '&' s} by A6,Th9; hence p '&' r |- q '&' s by Th11; end; theorem p |-| q & r |-| s implies p '&' r |-| q '&' s proof assume that A1: p |-| q and A2: r |-| s; A3: p '&' r |- q '&' s by A1,A2,Lm4; q '&' s |- p '&' r by A1,A2,Lm4; hence p '&' r |-| q '&' s by A3,Def5; end; theorem Th35: X |- All(x,p) iff X |- p proof thus X |- All(x,p) implies X |- p proof assume A1: X |- All(x,p); X |- All(x,p) => p by CQC_THE1:93; hence X |- p by A1,CQC_THE1:92; end; thus X |- p implies X |- All(x,p) by CQC_THE2:94; end; theorem Th36: All(x,p) |-| p proof A1: {All(x,p)} |- All(x,p) => p by CQC_THE1:93; {All(x,p)} |- All(x,p) by CQC_THE2:91; then {All(x,p)} |- p by A1,CQC_THE1:92; hence All(x,p) |- p by Def1; {p} |- p by CQC_THE2:91; then {p} |- All(x,p) by Th35; hence p |- All(x,p) by Def1; end; theorem p |-| q implies All(x,p) |-| All(y,q) proof assume A1: p |-| q; All(x,p) |-| p by Th36; then A2: All(x,p) |-| q by A1,Th28; q |-| All(y,q) by Th36; hence All(x,p) |-| All(y,q) by A2,Th28; end; definition let p, q be Element of CQC-WFF; pred p is_an_universal_closure_of q means :Def6: p is closed & ex n being Nat st 1 <= n & ex L being FinSequence st len L = n & L.1 = q & L.n = p & for k being Nat st 1 <= k & k < n holds ex x being bound_QC-variable st ex r being Element of CQC-WFF st r = L.k & L.(k+1) = All(x,r); end; theorem Th38: p is_an_universal_closure_of q implies p |-| q proof assume p is_an_universal_closure_of q; then consider n being Nat such that A1: 1 <= n & ex L being FinSequence st len L = n & L.1 = q & L.n = p & for k being Nat st 1 <= k & k < n holds ex x, r st r = L.k & L.(k+1) = All(x,r) by Def6; consider L being FinSequence such that len L = n and A2: L.1 = q and A3: L.n = p and A4: for k being Nat st 1 <= k & k < n holds ex x, r st r = L.k & L.(k+1) = All(x,r) by A1; for k being Nat st 1 <= k & k <= n holds ex r st r = L.k & q |-| r proof defpred P[Nat] means 1 <= $1 & $1 <= n implies ex r st r = L.$1 & q |-| r; A5: P[0]; A6: for k being Nat st P[k] holds P[k+1] proof let k be Nat such that A7: P[k]; now assume that 1 <= k+1 and A8: k+1 <= n; per cases by A8,NAT_1:38,RLVECT_1:99; case A9: k = 0; take p=q; thus ex r st r = L.(k+1) & q |-| r by A2,A9; case A10: 1 <= k & k < n; then consider x, r such that A11: r = L.k and A12: L.(k+1) = All(x,r) by A4; consider s such that A13: s = All(x,r); s |-| r by A13,Th36; then q |-| s by A7,A10,A11,Th28; hence ex s st s = L.(k+1) & q |-| s by A12,A13; end; hence P[k+1]; end; thus for k being Nat holds P[k] from Ind(A5,A6); end; then ex r st r = L.n & q |-| r by A1; hence p |-| q by A3; end; theorem Th39: |- p => q implies p |- q proof assume |- p => q; then A1: {p} |- p => q by CQC_THE1:98; {p} |- p by CQC_THE2:91; then {p} |- q by A1,CQC_THE1:92; hence p |- q by Def1; end; theorem Th40: X |- p => q implies X \/ {p} |- q proof assume A1: X |- p => q; X c= X \/ {p} by XBOOLE_1:7; then A2: X \/ {p} |- p => q by A1,Th4; p in {p} by TARSKI:def 1; then p in X \/ {p} by XBOOLE_0:def 2; then X \/ {p} |- p by Th1; hence X \/ {p} |- q by A2,CQC_THE1:92; end; theorem Th41: p is closed & p |- q implies |- p => q proof assume that A1: p is closed and A2: p |- q; {}(CQC-WFF) \/ {p} |- q by A2,Def1; then {}(CQC-WFF) |- p => q by A1,CQC_THE2:96; hence |- p => q by CQC_THE1:def 10; end; theorem p1 is_an_universal_closure_of p implies (X \/ {p} |- q iff X |- p1 => q) proof assume A1: p1 is_an_universal_closure_of p; now assume A2: X \/ {p} |- q; p |-| p1 by A1,Th38; then A3: {p} |-| {p1} by Th29; X \/ {p} |- {q} by A2,Th10; then X \/ {p1} |- {q} by A3,Th26; then A4: X \/ {p1} |- q by Th10; p1 is closed by A1,Def6; hence X |- p1 => q by A4,CQC_THE2:96; end; hence X \/ {p} |- q implies X |- p1 => q; now assume X |- p1 => q; then X \/ {p1} |- q by Th40; then A5: X \/ {p1} |- {q} by Th10; p |-| p1 by A1,Th38; then {p} |-| {p1} by Th29; then X \/ {p} |- {q} by A5,Th26; hence X \/ {p} |- q by Th10; end; hence X |- p1 => q implies X \/ {p} |- q; end; theorem Th43: p is closed & p |- q implies 'not' q |- 'not' p proof assume that A1: p is closed and A2: p |- q; |- p => q by A1,A2,Th41; then |- 'not' q => 'not' p by LUKASI_1:63; hence 'not' q |- 'not' p by Th39; end; theorem p is closed & X \/ {p} |- q implies X \/ {'not' q} |- 'not' p proof assume that A1: p is closed and A2: X \/ {p} |- q; X |- p => q by A1,A2,CQC_THE2:96; then X |- 'not' q => 'not' p by LUKASI_1:86; hence X \/ {'not' q} |- 'not' p by Th40; end; theorem Th45: p is closed & 'not' p |- 'not' q implies q |- p proof assume that A1: p is closed and A2: 'not' p |- 'not' q; 'not' p is closed by A1,QC_LANG3:25; then |- 'not' p => 'not' q by A2,Th41; then |- q => p by LUKASI_1:63; hence q |- p by Th39; end; theorem p is closed & X \/ {'not' p} |- 'not' q implies X \/ {q} |- p proof assume that A1: p is closed and A2: X \/ {'not' p} |- 'not' q; 'not' p is closed by A1,QC_LANG3:25; then X |- 'not' p => 'not' q by A2,CQC_THE2:96; then X |- q => p by LUKASI_1:86; hence X \/ {q} |- p by Th40; end; theorem p is closed & q is closed implies (p |- q iff 'not' q |- 'not' p) by Th43,Th45; theorem Th48: p1 is_an_universal_closure_of p & q1 is_an_universal_closure_of q implies (p |- q iff 'not' q1 |- 'not' p1) proof assume that A1: p1 is_an_universal_closure_of p and A2: q1 is_an_universal_closure_of q; now assume A3: p |- q; A4: p1 is closed by A1,Def6; p1 |-| p by A1,Th38; then p1 |- p by Def5; then A5: p1 |- q by A3,Th6; q |-| q1 by A2,Th38; then q |- q1 by Def5; then p1 |- q1 by A5,Th6; hence 'not' q1 |- 'not' p1 by A4,Th43; end; hence p |- q implies 'not' q1 |- 'not' p1; now assume A6: 'not' q1 |- 'not' p1; q1 is closed by A2,Def6; then A7: p1 |- q1 by A6,Th45; p1 |-| p by A1,Th38; then p |- p1 by Def5; then A8: p |- q1 by A7,Th6; q1 |-| q by A2,Th38; then q1 |- q by Def5; hence p |- q by A8,Th6; end; hence 'not' q1 |- 'not' p1 implies p |- q; end; theorem p1 is_an_universal_closure_of p & q1 is_an_universal_closure_of q implies (p |-| q iff 'not' p1 |-| 'not' q1) proof assume that A1: p1 is_an_universal_closure_of p and A2: q1 is_an_universal_closure_of q; now assume A3: p |-| q; then q |- p by Def5; then A4: 'not' p1 |- 'not' q1 by A1,A2,Th48; p |- q by A3,Def5; then 'not' q1 |- 'not' p1 by A1,A2,Th48; hence 'not' p1 |-| 'not' q1 by A4,Def5; end; hence p |-| q implies 'not' p1 |-| 'not' q1; now assume A5: 'not' p1 |-| 'not' q1; then 'not' p1 |- 'not' q1 by Def5; then A6: q |- p by A1,A2,Th48; 'not' q1 |- 'not' p1 by A5,Def5; then p |- q by A1,A2,Th48; hence p |-| q by A6,Def5; end; hence 'not' p1 |-| 'not' q1 implies p |-| q; end; definition let p, q be Element of CQC-WFF; pred p <==> q means :Def7: |- p <=> q; reflexivity proof let p; |- p => p by LUKASI_1:44; then {}(CQC-WFF) |- p => p by CQC_THE1:def 10 ; then {}(CQC-WFF) |- (p => p) '&' (p => p) by Lm2; then |- (p => p) '&' (p => p) by CQC_THE1:def 10; hence |- p <=> p by QC_LANG2:def 4; end; symmetry proof let p, q; assume |- p <=> q; then |- (p => q) '&' (q => p) by QC_LANG2:def 4; then {}(CQC-WFF) |- (p => q) '&' (q => p) by CQC_THE1:def 10; then {}(CQC-WFF) |- p => q & {}(CQC-WFF) |- q => p by Lm3; then {}(CQC-WFF) |- (q => p) '&' (p => q) by Lm2; then |- (q => p) '&' (p => q) by CQC_THE1:def 10; hence |- q <=> p by QC_LANG2:def 4; end; end; theorem Th50: p <==> q iff (|- p => q & |- q => p) proof A1: now assume p <==> q; then |- p <=> q by Def7; then |- (p => q) '&' (q => p) by QC_LANG2:def 4; then {}(CQC-WFF) |- (p => q) '&' (q => p) by CQC_THE1:def 10; then {}(CQC-WFF) |- p => q & {}(CQC-WFF) |- q => p by Lm3; hence |- p => q & |- q => p by CQC_THE1:def 10; end; now assume |- p => q & |- q => p; then {}(CQC-WFF) |- p => q & {}(CQC-WFF) |- q => p by CQC_THE1:def 10; then {}(CQC-WFF) |- (p => q) '&' (q => p) by Lm2; then |- (p => q) '&' (q => p) by CQC_THE1:def 10; then |- p <=> q by QC_LANG2:def 4; hence p <==> q by Def7; end; hence thesis by A1; end; theorem p <==> q & q <==> r implies p <==> r proof assume that A1: p <==> q and A2: q <==> r; A3: |- p => q by A1,Th50; A4: |- q => p by A1,Th50; A5: |- q => r by A2,Th50; A6: |- r => q by A2,Th50; A7: |- p => r by A3,A5,LUKASI_1:43; |- r => p by A4,A6,LUKASI_1:43; hence p <==> r by A7,Th50; end; theorem p <==> q implies p |-| q proof assume p <==> q; then |- p <=> q by Def7; then |- (p => q) '&' (q => p) by QC_LANG2:def 4; then {}(CQC-WFF) |- (p => q) '&' (q => p) by CQC_THE1:def 10; then {}(CQC-WFF) |- p => q & {}(CQC-WFF) |- q => p by Lm3; then |- p => q & |- q => p by CQC_THE1:def 10; hence p |- q & q |- p by Th39; end; Lm5: p <==> q implies 'not' p <==> 'not' q proof assume p <==> q; then |- p => q & |- q => p by Th50; then |- 'not' p => 'not' q & |- 'not' q => 'not' p by LUKASI_1:63; hence 'not' p <==> 'not' q by Th50; end; Lm6: 'not' p <==> 'not' q implies p <==> q proof assume 'not' p <==> 'not' q; then |- 'not' p => 'not' q & |- 'not' q => 'not' p by Th50; then |- p => q & |- q => p by LUKASI_1:63; hence p <==> q by Th50; end; theorem p <==> q iff 'not' p <==> 'not' q by Lm5,Lm6; theorem Th54: p <==> q & r <==> s implies p '&' r <==> q '&' s proof assume that A1: p <==> q and A2: r <==> s; |- p => q & |- q => p by A1,Th50; then A3: p => q in TAUT & q => p in TAUT by CQC_THE1:def 11; |- r => s & |- s => r by A2,Th50; then r => s in TAUT & s => r in TAUT by CQC_THE1:def 11; then p '&' r => q '&' s in TAUT & q '&' s => p '&' r in TAUT by A3,PROCAL_1:56; then |- p '&' r => q '&' s & |- q '&' s => p '&' r by CQC_THE1:def 11; hence p '&' r <==> q '&' s by Th50; end; theorem Th55: p <==> q & r <==> s implies p => r <==> q => s proof assume that A1: p <==> q and A2: r <==> s; 'not' r <==> 'not' s by A2,Lm5; then p '&' 'not' r <==> q '&' 'not' s by A1,Th54; then A3: 'not' (p '&' 'not' r) <==> 'not' (q '&' 'not' s) by Lm5; p => r = 'not' (p '&' 'not' r) by QC_LANG2:def 2; hence p => r <==> q => s by A3,QC_LANG2:def 2; end; theorem p <==> q & r <==> s implies p 'or' r <==> q 'or' s proof assume that A1: p <==> q and A2: r <==> s; |- p => q & |- q => p by A1,Th50; then A3: p => q in TAUT & q => p in TAUT by CQC_THE1:def 11; |- r => s & |- s => r by A2,Th50; then r => s in TAUT & s => r in TAUT by CQC_THE1:def 11; then p 'or' r => q 'or' s in TAUT & q 'or' s => p 'or' r in TAUT by A3,PROCAL_1:57; then |- p 'or' r => q 'or' s & |- q 'or' s => p 'or' r by CQC_THE1:def 11; hence p 'or' r <==> q 'or' s by Th50; end; theorem p <==> q & r <==> s implies p <=> r <==> q <=> s proof assume that A1: p <==> q and A2: r <==> s; A3: p => r <==> q => s by A1,A2,Th55; r => p <==> s => q by A1,A2,Th55; then A4: (p => r) '&' (r => p) <==> (q => s) '&' (s => q) by A3,Th54; p <=> r = (p => r) '&' (r => p) by QC_LANG2:def 4; hence p <=> r <==> q <=> s by A4,QC_LANG2:def 4; end; theorem Th58: p <==> q implies All(x,p) <==> All(x,q) proof assume p <==> q; then |- p => q & |- q => p by Th50; then |- All(x,p => q) & |- All(x,q => p) by CQC_THE2:26; then |- All(x,p) => All(x,q) & |- All(x,q) => All(x,p) by CQC_THE2:35; hence All(x,p) <==> All(x,q) by Th50; end; theorem p <==> q implies Ex(x,p) <==> Ex(x,q) proof assume p <==> q; then 'not' p <==> 'not' q by Lm5; then All(x,'not' p) <==> All(x,'not' q) by Th58; then A1: 'not' All(x,'not' p) <==> 'not' All(x,'not' q) by Lm5; Ex(x,p) = 'not' All(x,'not' p) by QC_LANG2:def 5; hence Ex(x,p) <==> Ex(x,q) by A1,QC_LANG2:def 5; end; canceled; theorem Th61: for k being Nat, l being QC-variable_list of k, a being free_QC-variable, x being bound_QC-variable holds still_not-bound_in l c= still_not-bound_in Subst(l,a .--> x) proof let k be Nat, l be QC-variable_list of k, a be free_QC-variable, x be bound_QC-variable; now let y be set; A1: still_not-bound_in l = { l.n : 1 <= n & n <= len l & l.n in bound_QC-variables } by QC_LANG1:def 28; A2: still_not-bound_in Subst(l,a .--> x) = { Subst(l,a .--> x).n : 1 <= n & n <= len Subst(l,a .--> x) & Subst(l,a .--> x).n in bound_QC-variables } by QC_LANG1:def 28; assume A3: y in still_not-bound_in l; then reconsider y' = y as Element of still_not-bound_in l; consider n such that A4: y' = l.n and A5: 1 <= n and A6: n <= len l and A7: l.n in bound_QC-variables by A1,A3; l.n <> a by A7,QC_LANG3:42; then A8: l.n = Subst(l,a .--> x).n by A5,A6,CQC_LANG:11; n <= len Subst(l,a .--> x) by A6,CQC_LANG:def 3; hence y in still_not-bound_in Subst(l,a .--> x) by A2,A4,A5,A7,A8; end; hence still_not-bound_in l c= still_not-bound_in Subst(l,a .--> x) by TARSKI:def 3; end; theorem Th62: for k being Nat, l being QC-variable_list of k, a being free_QC-variable, x being bound_QC-variable holds still_not-bound_in Subst(l,a .--> x) c= still_not-bound_in l \/ {x} proof let k be Nat, l be QC-variable_list of k, a be free_QC-variable, x be bound_QC-variable; let y be set; A1: still_not-bound_in l = { l.n : 1 <= n & n <= len l & l.n in bound_QC-variables } by QC_LANG1:def 28; A2: still_not-bound_in Subst(l,a .--> x) = { Subst(l,a .--> x).n : 1 <= n & n <= len Subst(l,a .--> x) & Subst(l,a .--> x).n in bound_QC-variables } by QC_LANG1:def 28; assume A3: y in still_not-bound_in Subst(l,a .--> x); then reconsider y' = y as Element of still_not-bound_in Subst(l,a .--> x); consider n such that A4: y' = Subst(l,a .--> x).n and A5: 1 <= n and A6: n <= len Subst(l,a .--> x) and A7: Subst(l,a .--> x).n in bound_QC-variables by A2,A3; A8: n <= len l by A6,CQC_LANG:def 3; per cases; suppose l.n = a; then y' = x by A4,A5,A8,CQC_LANG:11; then y' in {x} by TARSKI:def 1; hence y in still_not-bound_in l \/ {x} by XBOOLE_0:def 2; suppose l.n <> a; then l.n = Subst(l,a .--> x).n by A5,A8,CQC_LANG:11; then y' in still_not-bound_in l by A1,A4,A5,A7,A8; hence y in still_not-bound_in l \/ {x} by XBOOLE_0:def 2; end; theorem Th63: for h holds still_not-bound_in h c= still_not-bound_in (h.x) proof defpred P[QC-formula] means still_not-bound_in $1 c= still_not-bound_in ($1.x); A1: for k being Nat, P being (QC-pred_symbol of k), ll being QC-variable_list of k holds P[P!ll] proof let k be Nat, P be (QC-pred_symbol of k), ll be QC-variable_list of k; A2: still_not-bound_in ll c= still_not-bound_in Subst(ll,a.0.-->x) by Th61; still_not-bound_in ((P!ll).x) = still_not-bound_in (P!Subst(ll,a.0.-->x)) by CQC_LANG:30 .= still_not-bound_in Subst(ll,a.0.-->x) by QC_LANG3:9; hence still_not-bound_in (P!ll) c= still_not-bound_in ((P!ll).x) by A2,QC_LANG3:9; end; A3: P[VERUM] by CQC_LANG:28; A4: for p being Element of QC-WFF st P[p] holds P['not' p] proof let p be Element of QC-WFF; still_not-bound_in (('not' p).x) = still_not-bound_in 'not' (p.x) by CQC_LANG:32 .= still_not-bound_in(p.x) by QC_LANG3:11; hence thesis by QC_LANG3:11; end; A5: for p, q being Element of QC-WFF st P[p] & P[q] holds P[p '&' q] proof let p, q be Element of QC-WFF such that A6: P[p] and A7: P[q]; A8: still_not-bound_in (p '&' q) = still_not-bound_in p \/ still_not-bound_in q by QC_LANG3:14; still_not-bound_in ((p '&' q).x) = still_not-bound_in ((p.x) '&' (q.x)) by CQC_LANG:34 .= still_not-bound_in (p.x) \/ still_not-bound_in (q.x) by QC_LANG3:14; hence still_not-bound_in (p '&' q) c= still_not-bound_in ((p '&' q).x) by A6,A7,A8,XBOOLE_1:13; end; A9: for x being bound_QC-variable, p being Element of QC-WFF st P[p] holds P[All(x, p)] proof let y be bound_QC-variable, p be Element of QC-WFF such that A10: P[p]; per cases; suppose y = x; hence still_not-bound_in All(y,p) c= still_not-bound_in (All(y,p).x) by CQC_LANG:37; suppose y <> x; then A11: still_not-bound_in (All(y,p).x) = still_not-bound_in All(y,p.x) by CQC_LANG:38 .= still_not-bound_in (p.x) \ {y} by QC_LANG3:16; still_not-bound_in All(y,p) = still_not-bound_in p \ {y} by QC_LANG3:16; hence still_not-bound_in All(y,p) c= still_not-bound_in (All(y,p).x) by A10,A11,XBOOLE_1:33; end; thus for h holds P[h] from QC_Ind(A1,A3,A4,A5,A9); end; theorem Th64: for h holds still_not-bound_in (h.x) c= still_not-bound_in h \/ {x} proof defpred P[QC-formula] means still_not-bound_in ($1.x) c= still_not-bound_in $1 \/ {x}; A1: for k being Nat, P being (QC-pred_symbol of k), ll being QC-variable_list of k holds P[P!ll] proof let k be Nat, P be (QC-pred_symbol of k), ll be QC-variable_list of k; A2: still_not-bound_in ((P!ll).x) = still_not-bound_in (P!Subst(ll,a.0.-->x)) by CQC_LANG:30 .= still_not-bound_in Subst(ll,a.0.-->x) by QC_LANG3:9; still_not-bound_in Subst(ll,a.0.-->x) c= still_not-bound_in ll \/ {x} by Th62; hence still_not-bound_in ((P!ll).x) c= still_not-bound_in (P!ll) \/ {x} by A2,QC_LANG3:9; end; VERUM.x = VERUM by CQC_LANG:28; then A3: P[VERUM] by QC_LANG3:7,XBOOLE_1:2; A4: for p being Element of QC-WFF st P[p] holds P['not' p] proof let p be Element of QC-WFF; still_not-bound_in (('not' p).x) = still_not-bound_in 'not' (p.x) by CQC_LANG:32 .= still_not-bound_in(p.x) by QC_LANG3:11; hence thesis by QC_LANG3:11; end; A5: for p, q being Element of QC-WFF st P[p] & P[q] holds P[p '&' q] proof let p, q be Element of QC-WFF such that A6: P[p] and A7: P[q]; A8: still_not-bound_in ((p '&' q).x) = still_not-bound_in ((p.x) '&' (q.x)) by CQC_LANG:34 .= still_not-bound_in (p.x) \/ still_not-bound_in (q.x) by QC_LANG3:14; A9: still_not-bound_in (p.x) \/ still_not-bound_in (q.x) c= still_not-bound_in p \/ {x} \/ (still_not-bound_in q \/ {x}) by A6,A7,XBOOLE_1:13; still_not-bound_in p \/ {x} \/ (still_not-bound_in q \/ {x}) = still_not-bound_in p \/ ({x} \/ still_not-bound_in q) \/ {x} by XBOOLE_1:4 .= still_not-bound_in p \/ still_not-bound_in q \/ {x} \/ {x} by XBOOLE_1:4 .= still_not-bound_in p \/ still_not-bound_in q \/ ({x} \/ {x}) by XBOOLE_1:4 .= still_not-bound_in p \/ still_not-bound_in q \/ {x}; hence still_not-bound_in ((p '&' q).x) c= still_not-bound_in (p '&' q) \/ {x} by A8,A9,QC_LANG3:14; end; A10: for x being bound_QC-variable, p being Element of QC-WFF st P[p] holds P[All(x, p)] proof let y be bound_QC-variable, p be Element of QC-WFF such that A11: P[p]; per cases; suppose A12: y = x; A13: still_not-bound_in All(x,p) = (still_not-bound_in p) \ {x} by QC_LANG3:16; A14: (still_not-bound_in p) \ {x} c= still_not-bound_in p by XBOOLE_1:36; still_not-bound_in p c= still_not-bound_in (p.x) by Th63; then still_not-bound_in All(x,p) c= still_not-bound_in (p.x) by A13,A14,XBOOLE_1:1; then A15: still_not-bound_in All(x,p) c= still_not-bound_in p \/ {x} by A11,XBOOLE_1:1; still_not-bound_in All(y,p) \/ {x} = ((still_not-bound_in p) \ {x}) \/ {x} by A12,QC_LANG3:16 .= still_not-bound_in p \/ {x} by XBOOLE_1:39; hence still_not-bound_in (All(y,p).x) c= still_not-bound_in All(y,p) \/ {x} by A12,A15,CQC_LANG:37; suppose A16: y <> x; then A17: still_not-bound_in (All(y,p).x) = still_not-bound_in All(y,p.x) by CQC_LANG:38 .= still_not-bound_in (p.x) \ {y} by QC_LANG3:16; A18: {x} misses {y} by A16,ZFMISC_1:17; still_not-bound_in All(y,p) \/ {x} = (still_not-bound_in p \ {y}) \/ {x} by QC_LANG3:16 .= (still_not-bound_in p \/ {x}) \ {y} by A18,XBOOLE_1:87; hence still_not-bound_in (All(y,p).x) c= still_not-bound_in All(y,p) \/ {x} by A11,A17,XBOOLE_1:33; end; thus for h holds P[h] from QC_Ind(A1,A3,A4,A5,A10); end; theorem Th65: p = h.x & x <> y & not y in still_not-bound_in h implies not y in still_not-bound_in p proof assume that A1: p = h.x and A2: x <> y and A3: not y in still_not-bound_in h and A4: y in still_not-bound_in p; A5: still_not-bound_in p c= still_not-bound_in h \/ {x} by A1,Th64; per cases by A4,A5,XBOOLE_0:def 2; suppose y in still_not-bound_in h; hence contradiction by A3; suppose y in {x}; hence contradiction by A2,TARSKI:def 1; end; theorem p = h.x & q = h.y & not x in still_not-bound_in h & not y in still_not-bound_in h implies All(x,p) <==> All(y,q) proof assume that A1: p = h.x and A2: q = h.y and A3: not x in still_not-bound_in h and A4: not y in still_not-bound_in h; per cases; suppose x = y; hence All(x,p) <==> All(y,q) by A1,A2; suppose A5: x <> y; then A6: not y in still_not-bound_in p by A1,A4,Th65; not x in still_not-bound_in q by A2,A3,A5,Th65; then |- All(x,p) => All(y,q) & |- All(y,q) => All(x,p) by A1,A2,A3,A4,A6,CQC_THE2:30; hence All(x,p) <==> All(y,q) by Th50; end;