### The Mizar article:

### Intuitionistic Propositional Calculus in the Extended Framework with Modal Operator. Part I

**by****Takao Inoue**

- Received April 3, 2003
Copyright (c) 2003 Association of Mizar Users

- MML identifier: INTPRO_1
- [ MML identifier index ]

environ vocabulary FINSEQ_1, RELAT_1, FUNCT_1, ZF_LANG, BOOLE, INTPRO_1, QC_LANG2; notation TARSKI, XBOOLE_0, SUBSET_1, NUMBERS, RELAT_1, FUNCT_1, NAT_1, FINSEQ_1; constructors NAT_1, CQC_LANG, MEMBERED; clusters RELSET_1, MEMBERED, NUMBERS, ORDINAL2; requirements NUMERALS, BOOLE, SUBSET; definitions TARSKI; theorems TARSKI, FINSEQ_1, XBOOLE_0, XBOOLE_1; schemes XBOOLE_0; begin :: Intuitionistic propositional calculus IPC in the extended :: language with modal operator definition let E be set; attr E is with_FALSUM means :Def1: <*0*>in E; end; definition let E be set; attr E is with_int_implication means :Def2: for p, q being FinSequence st p in E & q in E holds <*1*>^p^q in E; end; definition let E be set; attr E is with_int_conjunction means :Def3: for p, q being FinSequence st p in E & q in E holds <*2*>^p^q in E; end; definition let E be set; attr E is with_int_disjunction means :Def4: for p, q being FinSequence st p in E & q in E holds <*3*>^p^q in E; end; definition let E be set; attr E is with_int_propositional_variables means :Def5: for n being Nat holds <* 5+2*n *> in E; end; definition let E be set; attr E is with_modal_operator means :Def6: for p being FinSequence st p in E holds <*6*>^p in E; end; :: We reserve <*4*> for verum for a possible formulation. :: So do we <* 5+2*n+1 *> for every n >= 1 for introduction of a number of :: other logical connectives (e.g. for polymodal logics, :: hybrid logics, recent computer-oriented logics and so on). definition let E be set; attr E is MC-closed means :Def7: E c= NAT* & E is with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator; end; Lm1: for E be set st E is MC-closed holds E is non empty proof let E be set; assume E is MC-closed; then E is with_FALSUM by Def7; hence thesis by Def1; end; definition cluster MC-closed -> with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator non empty set; coherence by Def7,Lm1; cluster with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator -> MC-closed Subset of NAT*; coherence by Def7; end; definition func MC-wff -> set means :Def8: it is MC-closed & for E being set st E is MC-closed holds it c= E; existence proof defpred P[set] means for E being set st E is MC-closed holds $1 in E; consider E0 being set such that A1: for x being set holds x in E0 iff x in NAT* & P[x] from Separation; A2: <*0*> in NAT* by FINSEQ_1:def 11; A3: for E being set st E is MC-closed holds <*0*> in E proof let E be set; assume E is MC-closed; then E is with_FALSUM by Def7; hence thesis by Def1; end; then reconsider E0 as non empty set by A1,A2; take E0; A4: E0 c= NAT* proof let x be set; thus thesis by A1; end; <*0*> in E0 by A1,A2,A3; then A5:E0 is with_FALSUM by Def1; for n being Nat holds <* 5+2*n *> in E0 proof let n be Nat; set p = 5+2*n; reconsider h = <*p*> as FinSequence of NAT; A6: h in NAT* by FINSEQ_1:def 11; for E being set st E is MC-closed holds <*p*> in E proof let E be set; assume E is MC-closed; then E is with_int_propositional_variables by Def7; hence thesis by Def5; end; hence thesis by A1,A6; end; then A7: E0 is with_int_propositional_variables by Def5; for p, q being FinSequence st p in E0 & q in E0 holds <*1*>^p^q in E0 proof let p, q be FinSequence such that A8: p in E0 & q in E0; p in NAT* & q in NAT* by A1,A8; then reconsider p'=p, q'=q as FinSequence of NAT by FINSEQ_1:def 11 ; A9: <*1*>^p'^q' in NAT* by FINSEQ_1:def 11; for E being set st E is MC-closed holds <*1*>^p^q in E proof let E be set; assume A10: E is MC-closed; then A11: E is with_int_implication by Def7; p in E & q in E by A1,A8,A10; hence thesis by A11,Def2; end; hence <*1*>^p^q in E0 by A1,A9; end; then A12: E0 is with_int_implication by Def2; for p, q being FinSequence st p in E0 & q in E0 holds <*2*>^p^q in E0 proof let p, q be FinSequence such that A13: p in E0 & q in E0; p in NAT* & q in NAT* by A1,A13; then reconsider p'=p, q'=q as FinSequence of NAT by FINSEQ_1:def 11 ; A14: <*2*>^p'^q' in NAT* by FINSEQ_1:def 11; for E being set st E is MC-closed holds <*2*>^p^q in E proof let E be set; assume A15: E is MC-closed; then A16: E is with_int_conjunction by Def7; p in E & q in E by A1,A13,A15; hence thesis by A16,Def3; end; hence <*2*>^p^q in E0 by A1,A14; end; then A17: E0 is with_int_conjunction by Def3; for p, q being FinSequence st p in E0 & q in E0 holds <*3*>^p^q in E0 proof let p, q be FinSequence such that A18: p in E0 & q in E0; p in NAT* & q in NAT* by A1,A18; then reconsider p'=p, q'=q as FinSequence of NAT by FINSEQ_1:def 11 ; A19: <*3*>^p'^q' in NAT* by FINSEQ_1:def 11; for E being set st E is MC-closed holds <*3*>^p^q in E proof let E be set; assume A20: E is MC-closed; then A21: E is with_int_disjunction by Def7; p in E & q in E by A1,A18,A20; hence thesis by A21,Def4; end; hence <*3*>^p^q in E0 by A1,A19; end; then A22: E0 is with_int_disjunction by Def4; for p being FinSequence st p in E0 holds <*6*>^p in E0 proof let p be FinSequence such that A23: p in E0; p in NAT* by A1,A23; then reconsider p'=p as FinSequence of NAT by FINSEQ_1:def 11; A24: <*6*>^p' in NAT* by FINSEQ_1:def 11; for E being set st E is MC-closed holds <*6*>^p in E proof let E be set; assume A25: E is MC-closed; then A26: E is with_modal_operator by Def7; p in E by A1,A23,A25; hence thesis by A26,Def6; end; hence <*6*>^p in E0 by A1,A24; end; then E0 is with_modal_operator by Def6; hence E0 is MC-closed by A4,A5,A7,A12,A17,A22,Def7; let E be set such that A27: E is MC-closed; let x be set; assume x in E0; hence thesis by A1,A27; end; uniqueness proof let E1, E2 be set such that A28: E1 is MC-closed & for E being set st E is MC-closed holds E1 c= E and A29: E2 is MC-closed & for E being set st E is MC-closed holds E2 c= E; E1 c= E2 & E2 c= E1 by A28,A29; hence E1 = E2 by XBOOLE_0:def 10; end; end; definition cluster MC-wff -> MC-closed; coherence by Def8; end; definition cluster MC-closed non empty set; existence proof take MC-wff; thus thesis; end; end; definition cluster -> Relation-like Function-like Element of MC-wff; coherence proof let p be Element of MC-wff; MC-wff c= NAT* & p in MC-wff by Def7; hence thesis by FINSEQ_1:def 11; end; end; definition cluster -> FinSequence-like Element of MC-wff; coherence proof let p be Element of MC-wff; MC-wff c= NAT* & p in MC-wff by Def7; hence thesis by FINSEQ_1:def 11; end; end; definition mode MC-formula is Element of MC-wff; end; definition func FALSUM -> MC-formula equals <*0*>; correctness by Def1; let p, q be Element of MC-wff; func p => q -> MC-formula equals <*1*>^p^q; correctness by Def2; func p '&' q -> MC-formula equals <*2*>^p^q; correctness by Def3; func p 'or' q -> MC-formula equals <*3*>^p^q; correctness by Def4; end; definition let p be Element of MC-wff; func Nes p -> MC-formula equals <*6*>^p; correctness by Def6; end; reserve T, X, Y for Subset of MC-wff; reserve p, q, r, s for Element of MC-wff; definition let T be Subset of MC-wff; attr T is IPC_theory means :Def14: for p, q, r being Element of MC-wff holds p => (q => p) in T & (p => (q => r)) => ((p => q) => (p => r)) in T & (p '&' q) => p in T & (p '&' q) => q in T & p => (q => (p '&' q)) in T & p => (p 'or' q) in T & q => (p 'or' q) in T & (p => r) => ((q => r) => ((p 'or' q) => r)) in T & FALSUM => p in T & (p in T & p => q in T implies q in T); correctness; end; definition let X; func CnIPC X -> Subset of MC-wff means :Def15: r in it iff for T st T is IPC_theory & X c= T holds r in T; existence proof defpred P[set] means for T st T is IPC_theory & X c= T holds $1 in T; consider Y being set such that A1: for a being set holds a in Y iff a in MC-wff & P[a] from Separation; Y c= MC-wff proof let a be set; assume a in Y; hence a in MC-wff by A1; end; then reconsider Z = Y as Subset of MC-wff; take Z; thus r in Z iff for T st T is IPC_theory & X c= T holds r in T by A1; end; uniqueness proof let Y,Z be Subset of MC-wff such that A2: r in Y iff for T st T is IPC_theory & X c= T holds r in T and A3: r in Z iff for T st T is IPC_theory & X c= T holds r in T; for a being set holds a in Y iff a in Z proof let a be set; hereby assume A4: a in Y; then reconsider t=a as Element of MC-wff; for T st T is IPC_theory & X c= T holds t in T by A2,A4; hence a in Z by A3; end; assume A5: a in Z; then reconsider t=a as Element of MC-wff; for T st T is IPC_theory & X c= T holds t in T by A3,A5; hence a in Y by A2; end; hence thesis by TARSKI:2; end; end; definition func IPC-Taut -> Subset of MC-wff equals :Def16: CnIPC({}(MC-wff)); correctness; end; definition let p be Element of MC-wff; func neg p -> MC-formula equals (p => FALSUM); correctness; end; definition func IVERUM -> MC-formula equals :Def18: (FALSUM => FALSUM); correctness; end; theorem Th1: p => (q => p) in CnIPC (X) proof T is IPC_theory & X c= T implies p => (q => p) in T by Def14; hence thesis by Def15; end; theorem Th2: (p => (q => r)) => ((p => q) => (p => r)) in CnIPC (X) proof T is IPC_theory & X c= T implies (p => (q => r)) => ((p => q) => (p => r)) in T by Def14; hence thesis by Def15; end; theorem Th3: p '&' q => p in CnIPC(X) proof T is IPC_theory & X c= T implies p '&' q => p in T by Def14; hence thesis by Def15; end; theorem Th4: p '&' q => q in CnIPC(X) proof T is IPC_theory & X c= T implies p '&' q => q in T by Def14; hence thesis by Def15; end; theorem Th5: p => (q => (p '&' q)) in CnIPC (X) proof T is IPC_theory & X c= T implies p => (q => (p '&' q)) in T by Def14; hence thesis by Def15; end; theorem Th6: p => (p 'or' q) in CnIPC (X) proof T is IPC_theory & X c= T implies p => (p 'or' q) in T by Def14; hence thesis by Def15; end; theorem Th7: q => (p 'or' q) in CnIPC (X) proof T is IPC_theory & X c= T implies q => (p 'or' q) in T by Def14; hence thesis by Def15; end; theorem Th8: (p => r) => ((q => r) => ((p 'or' q) => r)) in CnIPC (X) proof T is IPC_theory & X c= T implies (p => r) => ((q => r) => ((p 'or' q) => r)) in T by Def14; hence thesis by Def15; end; theorem Th9: FALSUM => p in CnIPC (X) proof T is IPC_theory & X c= T implies FALSUM => p in T by Def14; hence thesis by Def15; end; theorem Th10: p in CnIPC(X) & p => q in CnIPC(X) implies q in CnIPC(X) proof assume A1: p in CnIPC(X) & p => q in CnIPC(X); T is IPC_theory & X c= T implies q in T proof assume A2: T is IPC_theory & X c= T; then p in T & p => q in T by A1,Def15; hence thesis by A2,Def14; end; hence thesis by Def15; end; theorem Th11: T is IPC_theory & X c= T implies CnIPC(X) c= T proof assume that A1:T is IPC_theory and A2: X c= T; let a be set; thus thesis by A1,A2,Def15; end; theorem Th12: X c= CnIPC(X) proof let a be set; assume A1: a in X; then reconsider t = a as Element of MC-wff; for T st T is IPC_theory & X c= T holds t in T by A1; hence a in CnIPC(X) by Def15; end; theorem Th13: X c= Y implies CnIPC(X) c= CnIPC(Y) proof assume A1: X c= Y; thus CnIPC(X) c= CnIPC(Y) proof let a be set; assume A2: a in CnIPC(X); then reconsider t = a as Element of MC-wff; for T st T is IPC_theory & Y c= T holds t in T proof let T such that A3: T is IPC_theory and A4: Y c= T; X c= T by A1,A4,XBOOLE_1:1; hence t in T by A2,A3,Def15; end; hence thesis by Def15; end; end; Lm2: CnIPC(CnIPC(X)) c= CnIPC(X) proof let a be set; assume A1: a in CnIPC(CnIPC(X)); then reconsider t = a as Element of MC-wff; for T st T is IPC_theory & X c= T holds t in T proof let T; assume that A2: T is IPC_theory and A3: X c= T; CnIPC(X) c= T by A2,A3,Th11; hence t in T by A1,A2,Def15; end; hence thesis by Def15; end; theorem :: Th14: CnIPC(CnIPC(X)) = CnIPC(X) proof CnIPC(CnIPC(X)) c= CnIPC(X) & CnIPC(X) c= CnIPC(CnIPC(X)) by Lm2,Th12; hence thesis by XBOOLE_0:def 10; end; Lm3: CnIPC(X) is IPC_theory proof let p, q, r; thus p => (q => p) in CnIPC(X) by Th1; thus (p => (q => r)) => ((p => q) => (p => r)) in CnIPC(X) by Th2; thus (p '&' q) => p in CnIPC(X) by Th3; thus (p '&' q) => q in CnIPC(X) by Th4; thus p => (q => (p '&' q)) in CnIPC(X) by Th5; thus p => (p 'or' q) in CnIPC(X) by Th6; thus q => (p 'or' q) in CnIPC(X) by Th7; thus (p => r) => ((q => r) => ((p 'or' q) => r)) in CnIPC(X) by Th8; thus FALSUM => p in CnIPC(X) by Th9; thus (p in CnIPC(X) & p => q in CnIPC(X) implies q in CnIPC(X)) by Th10; end; definition let X be Subset of MC-wff; cluster CnIPC(X) -> IPC_theory; coherence by Lm3; end; theorem Th15: T is IPC_theory iff CnIPC(T) = T proof hereby assume A1: T is IPC_theory; A2: T c= CnIPC(T) by Th12; CnIPC(T) c= T proof let a be set; assume a in CnIPC(T); hence a in T by A1,Def15; end; hence CnIPC(T) = T by A2,XBOOLE_0:def 10; end; thus CnIPC(T) = T implies T is IPC_theory; end; theorem :: Th16: T is IPC_theory implies IPC-Taut c= T proof assume A1: T is IPC_theory; {}(MC-wff) c= T by XBOOLE_1:2; then IPC-Taut c= CnIPC(T) by Def16,Th13; hence thesis by A1,Th15; end; definition cluster IPC-Taut -> IPC_theory; coherence by Def16; end; begin :: Formulas provable in IPC : implication theorem Th17: :: Identity law p => p in IPC-Taut proof A1:p => (p => p) in IPC-Taut by Def14; A2:(p => ((p => p) => p)) => ((p => (p => p)) => (p => p)) in IPC-Taut by Def14; p => ((p => p) => p) in IPC-Taut by Def14; then (p => (p => p)) => (p => p) in IPC-Taut by A2,Def14; hence thesis by A1,Def14; end; theorem Th18: q in IPC-Taut implies p => q in IPC-Taut proof q => (p => q) in IPC-Taut by Def14; hence thesis by Def14; end; theorem :: Th19: IVERUM in IPC-Taut by Def14,Def18; theorem :: Th20: (p => q) => (p => p) in IPC-Taut proof (p => p) in IPC-Taut by Th17; hence thesis by Th18; end; theorem :: Th21: (q => p) => (p => p) in IPC-Taut proof (p => p) in IPC-Taut by Th17; hence thesis by Th18; end; theorem Th22: (q => r) => ((p => q) => (p => r)) in IPC-Taut proof A1:((p => (q => r)) => ((p => q) => (p => r))) => ( (q => r) => ((p => (q => r)) => ((p => q) => (p => r)))) in IPC-Taut by Def14; ((p => (q => r)) => ((p => q) => (p => r))) in IPC-Taut by Def14; then A2: ( (q => r) => ((p => (q => r)) => ((p => q) => (p => r)))) in IPC-Taut by A1,Def14; ( (q => r) => ((p => (q => r)) => ((p => q) => (p => r)))) => (((q => r) => (p => (q => r))) => ((q => r) => ((p => q) => (p => r)))) in IPC-Taut by Def14; then A3: ((q => r) => (p => (q => r))) => ((q => r) => ((p => q) => (p => r))) in IPC-Taut by A2,Def14; (q => r) => (p => (q => r)) in IPC-Taut by Def14; hence thesis by A3,Def14; end; theorem Th23: p => (q => r) in IPC-Taut implies q => (p => r) in IPC-Taut proof assume A1: p => (q => r) in IPC-Taut; (p => (q => r)) => ((p => q) => (p => r)) in IPC-Taut by Def14; then A2: ((p => q) => (p => r)) in IPC-Taut by A1,Def14; ((p => q) => (p => r)) => ((q => (p => q)) => (q => (p => r))) in IPC-Taut by Th22; then A3:((q => (p => q)) => (q => (p => r))) in IPC-Taut by A2,Def14; q => (p => q) in IPC-Taut by Def14; hence thesis by A3,Def14; end; theorem Th24: :: Hypothetical syllogism (p => q) => ((q => r) => (p => r)) in IPC-Taut proof (q => r) => ((p => q) => (p => r)) in IPC-Taut by Th22; hence thesis by Th23; end; theorem Th25: p => q in IPC-Taut implies (q => r) => (p => r) in IPC-Taut proof assume A1: p => q in IPC-Taut; (p => q) => ((q => r) => (p => r)) in IPC-Taut by Th24; hence (q => r) => (p => r) in IPC-Taut by A1,Def14; end; theorem Th26: p => q in IPC-Taut & q => r in IPC-Taut implies p => r in IPC-Taut proof assume that A1: p => q in IPC-Taut and A2: q => r in IPC-Taut; (p => q) => ((q => r) => (p => r)) in IPC-Taut by Th24; then (q => r) => (p => r) in IPC-Taut by A1,Def14; hence p => r in IPC-Taut by A2,Def14; end; Lm4: (((q => r) => (p => r)) => s) => ((p => q) => s) in IPC-Taut proof (p => q) => ((q => r) => (p => r)) in IPC-Taut by Th24; hence thesis by Th25; end; theorem Th27: (p => (q => r)) => ((s => q) => (p => (s => r))) in IPC-Taut proof A1: ((((q => r) => (s => r)) => (p => (s => r))) => ((s => q) => (p => (s => r)))) => ((p => (q => r)) => ((s => q) => (p => (s => r)))) in IPC-Taut by Lm4; (((q => r) => (s => r)) => (p => (s => r))) => ((s => q) => (p => (s => r))) in IPC-Taut by Lm4; hence thesis by A1,Def14; end; theorem :: Th28: ((p => q) => r) => (q => r) in IPC-Taut proof q => (p => q) in IPC-Taut & (q => (p => q)) => (((p => q) => r) => (q => r)) in IPC-Taut by Def14,Th24; hence thesis by Def14; end; theorem Th29: :: Contraposition (p => (q => r)) => (q => (p => r)) in IPC-Taut proof (p => (q => r)) => ((p => q) => (p => r)) in IPC-Taut by Def14; then A1:(p => q) => ((p => (q => r)) => (p => r)) in IPC-Taut by Th23; ((p => q) => ((p => (q => r)) => (p => r))) => ((q => (p => q)) => (q => ((p => (q => r)) => (p => r)))) in IPC-Taut by Th22; then A2:(q => (p => q)) => (q => ((p => (q => r)) => (p => r))) in IPC-Taut by A1,Def14; q => (p => q) in IPC-Taut by Def14; then (q => ((p => (q => r)) => (p => r))) in IPC-Taut by A2,Def14; hence thesis by Th23; end; theorem Th30: :: A Hilbert axiom (p => (p => q)) => (p => q) in IPC-Taut proof A1:p => p in IPC-Taut by Th17; (p => (p => q)) => ((p => p) => (p => q)) in IPC-Taut by Def14; then (p => p) => ((p => (p => q)) => (p => q)) in IPC-Taut by Th23; hence thesis by A1,Def14; end; theorem :: Th31: :: Modus ponendo ponens q => ((q => p) => p) in IPC-Taut proof A1:(q => p) => (q => p) in IPC-Taut by Th17; ((q => p) => (q => p)) => (q => ((q => p) => p)) in IPC-Taut by Th29; hence thesis by A1,Def14; end; theorem Th32: s => (q => p) in IPC-Taut & q in IPC-Taut implies s => p in IPC-Taut proof assume A1: s => (q => p) in IPC-Taut & q in IPC-Taut; (s => (q => p)) => (q => (s => p)) in IPC-Taut by Th29; then q => (s => p) in IPC-Taut by A1,Def14; hence thesis by A1,Def14; end; begin :: Formulas provable in IPC : conjunction theorem Th33: p => (p '&' p) in IPC-Taut proof A1:p => (p => (p '&' p)) in IPC-Taut by Def14; (p => (p => (p '&' p))) => (p => (p '&' p)) in IPC-Taut by Th30; hence thesis by A1,Def14; end; theorem Th34: (p '&' q) in IPC-Taut iff p in IPC-Taut & q in IPC-Taut proof hereby assume A1: p '&' q in IPC-Taut; (p '&' q) => p in IPC-Taut & (p '&' q) => q in IPC-Taut by Def14; hence p in IPC-Taut & q in IPC-Taut by A1,Def14; end; assume A2: p in IPC-Taut & q in IPC-Taut; p => (q => (p '&' q)) in IPC-Taut by Def14; then q => (p '&' q) in IPC-Taut by A2,Def14; hence thesis by A2,Def14; end; theorem :: Th35: (p '&' q) in IPC-Taut iff (q '&' p) in IPC-Taut proof hereby assume p '&' q in IPC-Taut; then p in IPC-Taut & q in IPC-Taut by Th34; hence q '&' p in IPC-Taut by Th34; end; assume q '&' p in IPC-Taut; then q in IPC-Taut & p in IPC-Taut by Th34; hence p '&' q in IPC-Taut by Th34; end; theorem Th36: (( p '&' q ) => r ) => ( p => ( q => r )) in IPC-Taut proof ( q => ( p '&' q )) => ((( p '&' q ) => r ) => ( q => r )) in IPC-Taut by Th24; then A1: p => (( q => ( p '&' q )) => ((( p '&' q ) => r ) => ( q => r ))) in IPC-Taut by Th18; set qp = ( q => ( p '&' q )); set pr = (( p '&' q ) => r) => ( q => r ); ( p => ( qp => pr )) => ( ( p => qp ) => ( p => pr )) in IPC-Taut by Def14 ; then A2: ( ( p => qp ) => ( p => pr )) in IPC-Taut by A1,Def14; p => ( q => ( p '&' q )) in IPC-Taut by Def14; then A3: p => ((( p '&' q ) => r ) => ( q => r )) in IPC-Taut by A2,Def14; (p => ((( p '&' q ) => r ) => ( q => r ))) => ((( p '&' q ) => r ) => ( p => ( q => r ))) in IPC-Taut by Th29; hence thesis by A3,Def14; end; theorem Th37: ( p => ( q => r )) => (( p '&' q ) => r ) in IPC-Taut proof A1: ( p '&' q ) => q in IPC-Taut by Def14; (( p '&' q ) => q) => (( q => r ) => (( p '&' q ) => r )) in IPC-Taut by Th24; then ( q => r ) => (( p '&' q ) => r ) in IPC-Taut by A1,Def14; then A2: p => (( q => r ) => (( p '&' q ) => r )) in IPC-Taut by Th18; p => (( q => r ) => (( p '&' q ) => r )) => ((p => ( q => r )) => ( p => (( p '&' q ) => r ))) in IPC-Taut by Def14; then A3: (p => ( q => r )) => ( p => (( p '&' q ) => r )) in IPC-Taut by A2,Def14; ( p => (( p '&' q ) => r )) => ((p '&' q ) => ( p => r )) in IPC-Taut by Th29; then A4: (p => ( q => r )) => ((p '&' q ) => ( p => r )) in IPC-Taut by A3, Th26; A5: ((p '&' q ) => ( p => r )) => ((( p '&' q ) => p ) => (( p '&' q ) => r )) in IPC-Taut by Def14; ( p '&' q ) => p in IPC-Taut by Def14; then ((p '&' q ) => ( p => r )) => (( p '&' q ) => r ) in IPC-Taut by A5, Th32; hence thesis by A4,Th26; end; theorem Th38: ( r => p ) => (( r => q ) => ( r => ( p '&' q ))) in IPC-Taut proof p => ( q => ( p '&' q )) in IPC-Taut by Def14; then A1: r => ( p => ( q => ( p '&' q ))) in IPC-Taut by Th18; (r => ( p => ( q => ( p '&' q )))) => (( r => p ) => ( r => ( q => ( p '&' q )))) in IPC-Taut by Def14; then A2: ( r => p ) => ( r => ( q => ( p '&' q ))) in IPC-Taut by A1,Def14; ( r => ( q => ( p '&' q ))) => (( r => q ) => ( r => ( p '&' q ))) in IPC-Taut by Def14; hence thesis by A2,Th26; end; theorem Th39: ( (p => q) '&' p ) => q in IPC-Taut proof set P = p => q; A1:( P => P ) => (( P '&' p ) => q ) in IPC-Taut by Th37; P => P in IPC-Taut by Th17; hence thesis by A1,Def14; end; theorem :: Th40: (( (p => q) '&' p ) '&' s ) => q in IPC-Taut proof set P = (p => q) '&' p; A1:(P '&' s) => P in IPC-Taut by Def14; P => q in IPC-Taut by Th39; hence thesis by A1,Th26; end; theorem :: Th41: (q => s) => (( p '&' q ) => s) in IPC-Taut proof set P = p '&' q; A1:(P => q) => ((q => s) => (P => s)) in IPC-Taut by Th24; P => q in IPC-Taut by Def14; hence thesis by A1,Def14; end; theorem Th42: (q => s) => (( q '&' p ) => s) in IPC-Taut proof set P = q '&' p; A1:(P => q) => ((q => s) => (P => s)) in IPC-Taut by Th24; P => q in IPC-Taut by Def14; hence thesis by A1,Def14; end; theorem Th43: ( (p '&' s) => q ) => ((p '&' s) => (q '&' s)) in IPC-Taut proof set P = p '&' s; A1:P => s in IPC-Taut by Def14; ( P => q ) => (( P => s ) => ( P => ( q '&' s ))) in IPC-Taut by Th38; hence thesis by A1,Th32; end; theorem Th44: ( p => q ) => ((p '&' s) => (q '&' s)) in IPC-Taut proof A1:(p => q) => (( p '&' s ) => q) in IPC-Taut by Th42; ( (p '&' s) => q ) => ((p '&' s) => (q '&' s)) in IPC-Taut by Th43; hence thesis by A1,Th26; end; theorem Th45: (( p => q ) '&' ( p '&' s )) => ( q '&' s ) in IPC-Taut proof set P = p => q, Q = p '&' s, S = q '&' s; A1:( P => ( Q => S )) => (( P '&' Q ) => S ) in IPC-Taut by Th37; P => (Q => S) in IPC-Taut by Th44; hence thesis by A1,Def14; end; theorem Th46: ( p '&' q ) => ( q '&' p ) in IPC-Taut proof set P = p '&' q; A1:( P => q ) => (( P => p ) => ( P => ( q '&' p ))) in IPC-Taut by Th38; P => q in IPC-Taut by Def14; then A2:( P => p ) => ( P => ( q '&' p )) in IPC-Taut by A1,Def14; P => p in IPC-Taut by Def14; hence thesis by A2,Def14; end; theorem Th47: ( p => q ) '&' ( p '&' s ) => ( s '&' q ) in IPC-Taut proof A1:( p => q ) '&' ( p '&' s ) => ( q '&' s ) in IPC-Taut by Th45; ( q '&' s ) => ( s '&' q ) in IPC-Taut by Th46; hence thesis by A1,Th26; end; theorem Th48: ( p => q ) => (( p '&' s ) => ( s '&' q )) in IPC-Taut proof set P = p => q, Q = p '&' s, S = s '&' q; A1:(( P '&' Q ) => S ) => ( P => ( Q => S )) in IPC-Taut by Th36; P '&' Q => S in IPC-Taut by Th47; hence thesis by A1,Def14; end; theorem Th49: ( p => q ) => (( s '&' p ) => ( s '&' q )) in IPC-Taut proof set P = p => q, Q = p '&' s, S = s '&' q, T = s '&' p; A1:(P => (Q => S)) => ((T => Q) => (P => (T => S))) in IPC-Taut by Th27; P => (Q => S) in IPC-Taut by Th48; then A2:(T => Q) => (P => (T => S)) in IPC-Taut by A1,Def14; T => Q in IPC-Taut by Th46; hence thesis by A2,Def14; end; theorem :: Th50: ( p '&' (s '&' q) ) => ( p '&' (q '&' s) ) in IPC-Taut proof set P = s '&' q, Q = q '&' s; A1:( P => Q ) => (( p '&' P ) => ( p '&' Q )) in IPC-Taut by Th49; P => Q in IPC-Taut by Th46; hence thesis by A1,Def14; end; theorem :: Th51: ( ( p => q ) '&' (p => s) ) => ( p => (q '&' s) ) in IPC-Taut proof set P = p => q, Q = p => s, S = p => (q '&' s); A1:( P => ( Q => S )) => (( P '&' Q ) => S ) in IPC-Taut by Th37; P => (Q => S) in IPC-Taut by Th38; hence thesis by A1,Def14; end; Lm5: ( (p '&' q) '&' s ) => q in IPC-Taut proof A1:((p '&' q) '&' s) => (p '&' q) in IPC-Taut by Def14; (p '&' q) => q in IPC-Taut by Def14; hence thesis by A1,Th26; end; Lm6: ( (p '&' q) '&' s ) '&' ( (p '&' q) '&' s ) => ( (p '&' q) '&' s ) '&' q in IPC-Taut proof set P = (p '&' q) '&' s; A1:( P => q ) => (( P '&' P ) => ( P '&' q )) in IPC-Taut by Th49; P => q in IPC-Taut by Lm5; hence thesis by A1,Def14; end; Lm7: (p '&' q) '&' s => ((p '&' q) '&' s) '&' q in IPC-Taut proof A1:( (p '&' q) '&' s ) => ( (p '&' q) '&' s ) '&' ( (p '&' q) '&' s ) in IPC-Taut by Th33; ( (p '&' q) '&' s ) '&' ( (p '&' q) '&' s ) => ( (p '&' q) '&' s ) '&' q in IPC-Taut by Lm6; hence thesis by A1,Th26; end; Lm8: (p '&' q) '&' s => (p '&' s) in IPC-Taut proof set P = p '&' q; A1:( P => p ) => ((P '&' s) => (p '&' s)) in IPC-Taut by Th44; P => p in IPC-Taut by Def14; hence thesis by A1,Def14; end; Lm9: (p '&' q) '&' s '&' q => (p '&' s) '&' q in IPC-Taut proof set P = p '&' q '&' s, Q = p '&' s; A1:( P => Q ) => ((P '&' q) => (Q '&' q)) in IPC-Taut by Th44; P => Q in IPC-Taut by Lm8; hence thesis by A1,Def14; end; Lm10: (p '&' q) '&' s => (p '&' s) '&' q in IPC-Taut proof A1:(p '&' q) '&' s => ((p '&' q) '&' s) '&' q in IPC-Taut by Lm7; (p '&' q) '&' s '&' q => (p '&' s) '&' q in IPC-Taut by Lm9; hence thesis by A1,Th26; end; Lm11: (p '&' s) '&' q => (s '&' p) '&' q in IPC-Taut proof set P = p '&' s, Q = s '&' p; A1:( P => Q ) => ((P '&' q) => (Q '&' q)) in IPC-Taut by Th44; P => Q in IPC-Taut by Th46; hence thesis by A1,Def14; end; Lm12: (p '&' q) '&' s => (s '&' p) '&' q in IPC-Taut proof A1:(p '&' q) '&' s => (p '&' s) '&' q in IPC-Taut by Lm10; (p '&' s) '&' q => (s '&' p) '&' q in IPC-Taut by Lm11; hence thesis by A1,Th26; end; Lm13: (p '&' q) '&' s => (s '&' q) '&' p in IPC-Taut proof A1:(p '&' q) '&' s => (s '&' p) '&' q in IPC-Taut by Lm12; (s '&' p) '&' q => (s '&' q) '&' p in IPC-Taut by Lm10; hence thesis by A1,Th26; end; Lm14: (p '&' q) '&' s => p '&' (s '&' q) in IPC-Taut proof A1:(p '&' q) '&' s => (s '&' q) '&' p in IPC-Taut by Lm13; (s '&' q) '&' p => p '&' (s '&' q) in IPC-Taut by Th46; hence thesis by A1,Th26; end; Lm15: p '&' (s '&' q) => p '&' (q '&' s) in IPC-Taut proof set P = s '&' q, Q = q '&' s; A1:( P => Q ) => (( p '&' P ) => ( p '&' Q )) in IPC-Taut by Th49; s '&' q => q '&' s in IPC-Taut by Th46; hence thesis by A1,Def14; end; theorem :: Th52: (p '&' q) '&' s => p '&' (q '&' s) in IPC-Taut proof A1:(p '&' q) '&' s => p '&' (s '&' q) in IPC-Taut by Lm14; p '&' (s '&' q) => p '&' (q '&' s) in IPC-Taut by Lm15; hence thesis by A1,Th26; end; Lm16: p '&' (q '&' s) => (s '&' q) '&' p in IPC-Taut proof A1:p '&' (q '&' s) => p '&' (s '&' q) in IPC-Taut by Lm15; p '&' (s '&' q) => (s '&' q) '&' p in IPC-Taut by Th46; hence thesis by A1,Th26; end; Lm17: (s '&' q) '&' p => (q '&' s) '&' p in IPC-Taut proof set P = s '&' q, Q = q '&' s; A1:( P => Q ) => ((P '&' p) => (Q '&' p)) in IPC-Taut by Th44; s '&' q => q '&' s in IPC-Taut by Th46; hence thesis by A1,Def14; end; Lm18: p '&' (q '&' s) => (q '&' s) '&' p in IPC-Taut proof A1:p '&' (q '&' s) => (s '&' q) '&' p in IPC-Taut by Lm16; (s '&' q) '&' p => (q '&' s) '&' p in IPC-Taut by Lm17; hence thesis by A1,Th26; end; Lm19: p '&' (q '&' s) => (p '&' s) '&' q in IPC-Taut proof A1:p '&' (q '&' s) => (q '&' s) '&' p in IPC-Taut by Lm18; (q '&' s) '&' p => (p '&' s) '&' q in IPC-Taut by Lm13; hence thesis by A1,Th26; end; Lm20: p '&' (q '&' s) => p '&' (s '&' q) in IPC-Taut proof set P = q '&' s, Q = s '&' q; A1:( P => Q ) => (( p '&' P ) => ( p '&' Q )) in IPC-Taut by Th49; q '&' s => s '&' q in IPC-Taut by Th46; hence thesis by A1,Def14; end; theorem :: Th53: p '&' (q '&' s) => (p '&' q) '&' s in IPC-Taut proof A1:p '&' (q '&' s) => p '&' (s '&' q) in IPC-Taut by Lm20; p '&' (s '&' q) => (p '&' q) '&' s in IPC-Taut by Lm19; hence thesis by A1,Th26; end; begin :: Formulas provable in IPC: disjunction theorem Th54: (p 'or' p) => p in IPC-Taut proof A1:p => p in IPC-Taut by Th17; (p => p) => ((p => p) => ((p 'or' p) => p)) in IPC-Taut by Def14; then (p => p) => ((p 'or' p) => p) in IPC-Taut by A1,Def14; hence thesis by A1,Def14; end; theorem :: Th55: p in IPC-Taut or q in IPC-Taut implies (p 'or' q) in IPC-Taut proof assume A1: p in IPC-Taut or q in IPC-Taut; now per cases by A1; case A2: p in IPC-Taut; p => (p 'or' q) in IPC-Taut by Def14; hence thesis by A2,Def14; case A3: q in IPC-Taut; q => (p 'or' q) in IPC-Taut by Def14; hence thesis by A3,Def14; end; hence thesis; end; theorem Th56: (p 'or' q) => (q 'or' p) in IPC-Taut proof A1: (p => (q 'or' p)) => ((q => (q 'or' p)) => ((p 'or' q) => (q 'or' p))) in IPC-Taut by Def14; A2: p => (q 'or' p) in IPC-Taut by Def14; A3: q => (q 'or' p) in IPC-Taut by Def14; ((q => (q 'or' p)) => ((p 'or' q) => (q 'or' p))) in IPC-Taut by A1,A2,Def14; hence thesis by A3,Def14; end; theorem :: Th57: (p 'or' q) in IPC-Taut iff (q 'or' p) in IPC-Taut proof hereby assume A1: p 'or' q in IPC-Taut; (p 'or' q) => (q 'or' p) in IPC-Taut by Th56; hence q 'or' p in IPC-Taut by A1,Def14; end; assume A2: q 'or' p in IPC-Taut; (q 'or' p) => (p 'or' q) in IPC-Taut by Th56; hence p 'or' q in IPC-Taut by A2,Def14; end; theorem Th58: (p => q) => (p => (q 'or' s)) in IPC-Taut proof A1: q => (q 'or' s) in IPC-Taut by Def14; (q => (q 'or' s)) => ((p => q) => (p => (q 'or' s))) in IPC-Taut by Th22; hence thesis by A1,Def14; end; theorem Th59: (p => q) => (p => (s 'or' q)) in IPC-Taut proof A1: q => (s 'or' q) in IPC-Taut by Def14; (q => (s 'or' q)) => ((p => q) => (p => (s 'or' q))) in IPC-Taut by Th22; hence thesis by A1,Def14; end; theorem Th60: ( p => q ) => ((p 'or' s) => (q 'or' s)) in IPC-Taut proof set P = p 'or' s, Q = q 'or' s; (p => Q) => ((s => Q) => (P => Q)) in IPC-Taut by Def14; then A1: (s => Q) => ((p => Q) => (P => Q)) in IPC-Taut by Th23; s => Q in IPC-Taut by Def14; then A2: (p => Q) => (P => Q) in IPC-Taut by A1,Def14; (p => q) => (p => Q) in IPC-Taut by Th58; hence thesis by A2,Th26; end; theorem Th61: p => q in IPC-Taut implies (p 'or' s) => (q 'or' s) in IPC-Taut proof assume A1: p => q in IPC-Taut; ( p => q ) => ((p 'or' s) => (q 'or' s)) in IPC-Taut by Th60; hence thesis by A1,Def14; end; theorem Th62: ( p => q ) => (( s 'or' p ) => ( s 'or' q )) in IPC-Taut proof set P = s 'or' p, Q = s 'or' q; A1: (s => Q) => ((p => Q) => (P => Q)) in IPC-Taut by Def14; s => Q in IPC-Taut by Def14; then A2: (p => Q) => (P => Q) in IPC-Taut by A1,Def14; (p => q) => (p => Q) in IPC-Taut by Th59; hence thesis by A2,Th26; end; theorem Th63: p => q in IPC-Taut implies ( s 'or' p ) => ( s 'or' q ) in IPC-Taut proof assume A1: p => q in IPC-Taut; ( p => q ) => (( s 'or' p ) => ( s 'or' q )) in IPC-Taut by Th62; hence thesis by A1,Def14; end; theorem Th64: ( p 'or' (q 'or' s) ) => ( q 'or' (p 'or' s) ) in IPC-Taut proof set P = p 'or' s, Q = q 'or' s; s => P in IPC-Taut by Def14; then (q 'or' s) => (q 'or' P) in IPC-Taut by Th63; :: Q => (q 'or' P) in IPC-Taut; then then A1: (p 'or' Q) => (p 'or' (q 'or' P)) in IPC-Taut by Th63; (p 'or' (q 'or' P)) => ((q 'or' P) 'or' p) in IPC-Taut by Th56; then A2: (p 'or' Q) => ((q 'or' P) 'or' p) in IPC-Taut by A1,Th26; A3: p => P in IPC-Taut by Def14; P => (q 'or' P) in IPC-Taut by Def14; then p => (q 'or' P) in IPC-Taut by A3,Th26; then A4: ((q 'or' P) 'or' p) => ((q 'or' P) 'or' (q 'or' P)) in IPC-Taut by Th63; ((q 'or' P) 'or' (q 'or' P)) => (q 'or' P) in IPC-Taut by Th54; then ((q 'or' P) 'or' p) => (q 'or' P) in IPC-Taut by A4,Th26; hence thesis by A2,Th26; end; theorem :: Th65: ( p 'or' (q 'or' s) ) => ( (p 'or' q) 'or' s ) in IPC-Taut proof (q 'or' s) => (s 'or' q) in IPC-Taut by Th56; then A1: (p 'or' (q 'or' s)) => (p 'or' (s 'or' q)) in IPC-Taut by Th63; (p 'or' (s 'or' q)) => (s 'or' (p 'or' q)) in IPC-Taut by Th64; then A2: (p 'or' (q 'or' s)) => (s 'or' (p 'or' q)) in IPC-Taut by A1,Th26; (s 'or' (p 'or' q)) => ((p 'or' q) 'or' s) in IPC-Taut by Th56; hence thesis by A2,Th26; end; theorem :: Th66: ( (p 'or' q) 'or' s ) => ( p 'or' (q 'or' s) ) in IPC-Taut proof (p 'or' q) => (q 'or' p) in IPC-Taut by Th56; then A1: ((p 'or' q) 'or' s) => ((q 'or' p) 'or' s) in IPC-Taut by Th61; ((q 'or' p) 'or' s) => (s 'or' (q 'or' p)) in IPC-Taut by Th56; then A2: ((p 'or' q) 'or' s) => (s 'or' (q 'or' p)) in IPC-Taut by A1,Th26; (q 'or' p) => (p 'or' q) in IPC-Taut by Th56; then (s 'or' (q 'or' p)) => (s 'or' (p 'or' q)) in IPC-Taut by Th63; then A3: ((p 'or' q) 'or' s) => (s 'or' (p 'or' q)) in IPC-Taut by A2,Th26; (s 'or' (p 'or' q)) => (p 'or' (s 'or' q)) in IPC-Taut by Th64; then A4: ((p 'or' q) 'or' s) => (p 'or' (s 'or' q)) in IPC-Taut by A3,Th26; (s 'or' q) => (q 'or' s) in IPC-Taut by Th56; then (p 'or' (s 'or' q)) => (p 'or' (q 'or' s)) in IPC-Taut by Th63; hence thesis by A4,Th26; end; begin :: Classical propositional calculus CPC reserve T, X, Y for Subset of MC-wff; reserve p, q, r for Element of MC-wff; definition let T be Subset of MC-wff; attr T is CPC_theory means :Def19: for p, q, r being Element of MC-wff holds p => (q => p) in T & (p => (q => r)) => ((p => q) => (p => r)) in T & (p '&' q) => p in T & (p '&' q) => q in T & p => (q => (p '&' q)) in T & p => (p 'or' q) in T & q => (p 'or' q) in T & (p => r) => ((q => r) => ((p 'or' q) => r)) in T & FALSUM => p in T & p 'or' (p => FALSUM) in T & (p in T & p => q in T implies q in T); correctness; end; theorem Th67: T is CPC_theory implies T is IPC_theory proof assume A1: T is CPC_theory; let p, q, r be Element of MC-wff; thus thesis by A1,Def19; end; definition let X; func CnCPC X -> Subset of MC-wff means :Def20: r in it iff for T st T is CPC_theory & X c= T holds r in T; existence proof defpred P[set] means for T st T is CPC_theory & X c= T holds $1 in T; consider Y being set such that A1: for a being set holds a in Y iff a in MC-wff & P[a] from Separation; Y c= MC-wff proof let a be set; assume a in Y; hence a in MC-wff by A1; end; then reconsider Z = Y as Subset of MC-wff; take Z; thus r in Z iff for T st T is CPC_theory & X c= T holds r in T by A1; end; uniqueness proof let Y,Z be Subset of MC-wff such that A2: r in Y iff for T st T is CPC_theory & X c= T holds r in T and A3: r in Z iff for T st T is CPC_theory & X c= T holds r in T; for a being set holds a in Y iff a in Z proof let a be set; hereby assume A4: a in Y; then reconsider t=a as Element of MC-wff; for T st T is CPC_theory & X c= T holds t in T by A2,A4; hence a in Z by A3; end; assume A5: a in Z; then reconsider t=a as Element of MC-wff; for T st T is CPC_theory & X c= T holds t in T by A3,A5; hence a in Y by A2; end; hence thesis by TARSKI:2; end; end; definition func CPC-Taut -> Subset of MC-wff equals :Def21: CnCPC({}(MC-wff)); correctness; end; theorem Th68: CnIPC (X) c= CnCPC (X) proof let a be set; assume A1: a in CnIPC (X); then reconsider r = a as Element of MC-wff; for T st T is CPC_theory & X c= T holds r in T proof let T such that A2: T is CPC_theory and A3: X c= T; T is IPC_theory by A2,Th67; hence r in T by A1,A3,Def15; end; hence thesis by Def20; end; theorem Th69: p => (q => p) in CnCPC (X) & (p => (q => r)) => ((p => q) => (p => r)) in CnCPC (X) & p '&' q => p in CnCPC (X) & p '&' q => q in CnCPC (X) & p => (q => (p '&' q)) in CnCPC (X) & p => (p 'or' q) in CnCPC (X) & q => (p 'or' q) in CnCPC (X) & (p => r) => ((q => r) => ((p 'or' q) => r)) in CnCPC (X) & FALSUM => p in CnCPC (X) & p 'or' (p => FALSUM) in CnCPC (X) proof thus p => (q => p) in CnCPC (X) proof A1: p => (q => p) in CnIPC (X) by Th1; CnIPC(X) c= CnCPC(X) by Th68; hence thesis by A1; end; thus (p => (q => r)) => ((p => q) => (p => r)) in CnCPC (X) proof A2: (p => (q => r)) => ((p => q) => (p => r)) in CnIPC (X) by Th2; CnIPC(X) c= CnCPC(X) by Th68; hence thesis by A2; end; thus p '&' q => p in CnCPC (X) proof A3: p '&' q => p in CnIPC (X) by Th3; CnIPC(X) c= CnCPC(X) by Th68; hence thesis by A3; end; thus p '&' q => q in CnCPC (X) proof A4: p '&' q => q in CnIPC (X) by Th4; CnIPC(X) c= CnCPC(X) by Th68; hence thesis by A4; end; thus p => (q => (p '&' q)) in CnCPC (X) proof A5: p => (q => (p '&' q)) in CnIPC (X) by Th5; CnIPC(X) c= CnCPC(X) by Th68; hence thesis by A5; end; thus p => (p 'or' q) in CnCPC (X) proof A6: p => (p 'or' q) in CnIPC (X) by Th6; CnIPC(X) c= CnCPC(X) by Th68; hence thesis by A6; end; thus q => (p 'or' q) in CnCPC (X) proof A7: q => (p 'or' q) in CnIPC (X) by Th7; CnIPC(X) c= CnCPC(X) by Th68; hence thesis by A7; end; thus (p => r) => ((q => r) => ((p 'or' q) => r)) in CnCPC (X) proof A8: (p => r) => ((q => r) => ((p 'or' q) => r)) in CnIPC (X) by Th8; CnIPC(X) c= CnCPC(X) by Th68; hence thesis by A8; end; thus FALSUM => p in CnCPC (X) proof A9: FALSUM => p in CnIPC (X) by Th9; CnIPC(X) c= CnCPC(X) by Th68; hence thesis by A9; end; thus p 'or' (p => FALSUM) in CnCPC (X) proof T is CPC_theory & X c= T implies p 'or' (p => FALSUM) in T by Def19; hence thesis by Def20; end; end; theorem Th70: p in CnCPC(X) & p => q in CnCPC(X) implies q in CnCPC(X) proof assume A1: p in CnCPC(X) & p => q in CnCPC(X); T is CPC_theory & X c= T implies q in T proof assume A2: T is CPC_theory & X c= T; then p in T & p => q in T by A1,Def20; hence thesis by A2,Def19; end; hence thesis by Def20; end; theorem Th71: T is CPC_theory & X c= T implies CnCPC(X) c= T proof assume that A1:T is CPC_theory and A2: X c= T; let a be set; thus thesis by A1,A2,Def20; end; theorem Th72: X c= CnCPC(X) proof let a be set; assume A1: a in X; then reconsider t = a as Element of MC-wff; for T st T is CPC_theory & X c= T holds t in T by A1; hence a in CnCPC(X) by Def20; end; theorem Th73: X c= Y implies CnCPC(X) c= CnCPC(Y) proof assume A1: X c= Y; thus CnCPC(X) c= CnCPC(Y) proof let a be set; assume A2: a in CnCPC(X); then reconsider t = a as Element of MC-wff; for T st T is CPC_theory & Y c= T holds t in T proof let T such that A3: T is CPC_theory and A4: Y c= T; X c= T by A1,A4,XBOOLE_1:1; hence t in T by A2,A3,Def20; end; hence thesis by Def20; end; end; Lm21: CnCPC(CnCPC(X)) c= CnCPC(X) proof let a be set; assume A1: a in CnCPC(CnCPC(X)); then reconsider t = a as Element of MC-wff; for T st T is CPC_theory & X c= T holds t in T proof let T; assume that A2: T is CPC_theory and A3: X c= T; CnCPC(X) c= T by A2,A3,Th71; hence t in T by A1,A2,Def20; end; hence thesis by Def20; end; theorem :: Th6C: CnCPC(CnCPC(X)) = CnCPC(X) proof CnCPC(CnCPC(X)) c= CnCPC(X) & CnCPC(X) c= CnCPC(CnCPC(X)) by Lm21,Th72; hence thesis by XBOOLE_0:def 10; end; Lm22: CnCPC(X) is CPC_theory proof let p, q, r; thus p => (q => p) in CnCPC(X) by Th69; thus (p => (q => r)) => ((p => q) => (p => r)) in CnCPC(X) by Th69; thus (p '&' q) => p in CnCPC(X) by Th69; thus (p '&' q) => q in CnCPC(X) by Th69; thus p => (q => (p '&' q)) in CnCPC(X) by Th69; thus p => (p 'or' q) in CnCPC(X) by Th69; thus q => (p 'or' q) in CnCPC(X) by Th69; thus (p => r) => ((q => r) => ((p 'or' q) => r)) in CnCPC(X) by Th69; thus FALSUM => p in CnCPC(X) by Th69; thus p 'or' (p => FALSUM) in CnCPC (X) by Th69; thus (p in CnCPC(X) & p => q in CnCPC(X) implies q in CnCPC(X)) by Th70; end; definition let X be Subset of MC-wff; cluster CnCPC(X) -> CPC_theory; coherence by Lm22; end; theorem Th75: T is CPC_theory iff CnCPC(T) = T proof hereby assume A1: T is CPC_theory; A2: T c= CnCPC(T) by Th72; CnCPC(T) c= T proof let a be set; assume a in CnCPC(T); hence a in T by A1,Def20; end; hence CnCPC(T) = T by A2,XBOOLE_0:def 10; end; thus CnCPC(T) = T implies T is CPC_theory; end; theorem :: Th8C: T is CPC_theory implies CPC-Taut c= T proof assume A1: T is CPC_theory; {}(MC-wff) c= T by XBOOLE_1:2; then CPC-Taut c= CnCPC(T) by Def21,Th73; hence thesis by A1,Th75; end; definition cluster CPC-Taut -> CPC_theory; coherence by Def21; end; theorem :: CPCincIPC: IPC-Taut c= CPC-Taut by Def16,Def21,Th68; begin :: Modal calculus S4 reserve T, X, Y for Subset of MC-wff; reserve p, q, r for Element of MC-wff; definition let T be Subset of MC-wff; attr T is S4_theory means :Def22: for p, q, r being Element of MC-wff holds p => (q => p) in T & (p => (q => r)) => ((p => q) => (p => r)) in T & (p '&' q) => p in T & (p '&' q) => q in T & p => (q => (p '&' q)) in T & p => (p 'or' q) in T & q => (p 'or' q) in T & (p => r) => ((q => r) => ((p 'or' q) => r)) in T & FALSUM => p in T & p 'or' (p => FALSUM) in T & (Nes (p => q)) => ((Nes p) => (Nes q)) in T & (Nes p) => p in T & (Nes p) => Nes (Nes p) in T & (p in T & p => q in T implies q in T) & (p in T implies Nes p in T); correctness; end; theorem Th78: T is S4_theory implies T is CPC_theory proof assume A1: T is S4_theory; let p, q, r be Element of MC-wff; thus thesis by A1,Def22; end; theorem :: S4TincIPCT: T is S4_theory implies T is IPC_theory proof assume A1: T is S4_theory; let p, q, r be Element of MC-wff; thus thesis by A1,Def22; end; definition let X; func CnS4 X -> Subset of MC-wff means :Def23: r in it iff for T st T is S4_theory & X c= T holds r in T; existence proof defpred P[set] means for T st T is S4_theory & X c= T holds $1 in T; consider Y being set such that A1: for a being set holds a in Y iff a in MC-wff & P[a] from Separation; Y c= MC-wff proof let a be set; assume a in Y; hence a in MC-wff by A1; end; then reconsider Z = Y as Subset of MC-wff; take Z; thus r in Z iff for T st T is S4_theory & X c= T holds r in T by A1; end; uniqueness proof let Y,Z be Subset of MC-wff such that A2: r in Y iff for T st T is S4_theory & X c= T holds r in T and A3: r in Z iff for T st T is S4_theory & X c= T holds r in T; for a being set holds a in Y iff a in Z proof let a be set; hereby assume A4: a in Y; then reconsider t=a as Element of MC-wff; for T st T is S4_theory & X c= T holds t in T by A2,A4; hence a in Z by A3; end; assume A5: a in Z; then reconsider t=a as Element of MC-wff; for T st T is S4_theory & X c= T holds t in T by A3,A5; hence a in Y by A2; end; hence thesis by TARSKI:2; end; end; definition func S4-Taut -> Subset of MC-wff equals :Def24: CnS4({}(MC-wff)); correctness; end; theorem Th80: CnCPC (X) c= CnS4 (X) proof let a be set; assume A1: a in CnCPC (X); then reconsider r = a as Element of MC-wff; for T st T is S4_theory & X c= T holds r in T proof let T such that A2: T is S4_theory and A3: X c= T; T is CPC_theory by A2,Th78; hence thesis by A1,A3,Def20; end; hence thesis by Def23; end; theorem Th81: CnIPC (X) c= CnS4 (X) proof A1: CnIPC (X) c= CnCPC (X) by Th68; CnCPC (X) c= CnS4 (X) by Th80; hence thesis by A1,XBOOLE_1:1; end; theorem Th82: p => (q => p) in CnS4 (X) & (p => (q => r)) => ((p => q) => (p => r)) in CnS4 (X) & p '&' q => p in CnS4 (X) & p '&' q => q in CnS4 (X) & p => (q => (p '&' q)) in CnS4 (X) & p => (p 'or' q) in CnS4 (X) & q => (p 'or' q) in CnS4 (X) & (p => r) => ((q => r) => ((p 'or' q) => r)) in CnS4 (X) & FALSUM => p in CnS4 (X) & p 'or' (p => FALSUM) in CnS4 (X) proof thus p => (q => p) in CnS4 (X) proof A1: p => (q => p) in CnIPC (X) by Th1; CnIPC (X) c= CnS4 (X) by Th81; hence thesis by A1; end; thus (p => (q => r)) => ((p => q) => (p => r)) in CnS4 (X) proof A2: (p => (q => r)) => ((p => q) => (p => r)) in CnIPC (X) by Th2; CnIPC (X) c= CnS4 (X) by Th81; hence thesis by A2; end; thus p '&' q => p in CnS4 (X) proof A3: p '&' q => p in CnIPC (X) by Th3; CnIPC (X) c= CnS4 (X) by Th81; hence thesis by A3; end; thus p '&' q => q in CnS4 (X) proof A4: p '&' q => q in CnIPC (X) by Th4; CnIPC (X) c= CnS4 (X) by Th81; hence thesis by A4; end; thus p => (q => (p '&' q)) in CnS4 (X) proof A5: p => (q => (p '&' q)) in CnIPC (X) by Th5; CnIPC (X) c= CnS4 (X) by Th81; hence thesis by A5; end; thus p => (p 'or' q) in CnS4 (X) proof A6: p => (p 'or' q) in CnIPC (X) by Th6; CnIPC (X) c= CnS4 (X) by Th81; hence thesis by A6; end; thus q => (p 'or' q) in CnS4 (X) proof A7: q => (p 'or' q) in CnIPC (X) by Th7; CnIPC (X) c= CnS4 (X) by Th81; hence thesis by A7; end; thus (p => r) => ((q => r) => ((p 'or' q) => r)) in CnS4 (X) proof A8: (p => r) => ((q => r) => ((p 'or' q) => r)) in CnIPC (X) by Th8; CnIPC (X) c= CnS4 (X) by Th81; hence thesis by A8; end; thus FALSUM => p in CnS4 (X) proof A9: FALSUM => p in CnIPC (X) by Th9; CnIPC (X) c= CnS4 (X) by Th81; hence thesis by A9; end; thus p 'or' (p => FALSUM) in CnS4 (X) proof T is S4_theory & X c= T implies p 'or' (p => FALSUM) in T by Def22; hence thesis by Def23; end; end; theorem Th83: p in CnS4 (X) & p => q in CnS4 (X) implies q in CnS4 (X) proof assume A1: p in CnS4 (X) & p => q in CnS4 (X); T is S4_theory & X c= T implies q in T proof assume A2: T is S4_theory & X c= T; then p in T & p => q in T by A1,Def23; hence thesis by A2,Def22; end; hence thesis by Def23; end; theorem Th84: (Nes (p => q)) => ((Nes p) => (Nes q)) in CnS4 (X) proof T is S4_theory & X c= T implies (Nes (p => q)) => ((Nes p) => (Nes q)) in T by Def22; hence thesis by Def23; end; theorem Th85: (Nes p) => p in CnS4 (X) proof T is S4_theory & X c= T implies (Nes p) => p in T by Def22; hence thesis by Def23; end; theorem Th86: (Nes p) => Nes (Nes p) in CnS4 (X) proof T is S4_theory & X c= T implies (Nes p) => Nes (Nes p) in T by Def22; hence thesis by Def23; end; theorem Th87: p in CnS4 (X) implies Nes p in CnS4 (X) proof assume A1: p in CnS4 (X); T is S4_theory & X c= T implies Nes p in T proof assume A2: T is S4_theory & X c= T; then p in T by A1,Def23; hence thesis by A2,Def22; end; hence thesis by Def23; end; theorem Th88: T is S4_theory & X c= T implies CnS4(X) c= T proof assume that A1:T is S4_theory and A2: X c= T; let a be set; thus thesis by A1,A2,Def23; end; theorem Th89: X c= CnS4(X) proof let a be set; assume A1: a in X; then reconsider t = a as Element of MC-wff; for T st T is S4_theory & X c= T holds t in T by A1; hence a in CnS4(X) by Def23; end; theorem Th90: X c= Y implies CnS4(X) c= CnS4(Y) proof assume A1: X c= Y; thus CnS4(X) c= CnS4(Y) proof let a be set; assume A2: a in CnS4(X); then reconsider t = a as Element of MC-wff; for T st T is S4_theory & Y c= T holds t in T proof let T such that A3: T is S4_theory and A4: Y c= T; X c= T by A1,A4,XBOOLE_1:1; hence t in T by A2,A3,Def23; end; hence thesis by Def23; end; end; Lm23: CnS4(CnS4(X)) c= CnS4(X) proof let a be set; assume A1: a in CnS4(CnS4(X)); then reconsider t = a as Element of MC-wff; for T st T is S4_theory & X c= T holds t in T proof let T; assume that A2: T is S4_theory and A3: X c= T; CnS4(X) c= T by A2,A3,Th88; hence t in T by A1,A2,Def23; end; hence thesis by Def23; end; theorem :: Th10M: CnS4(CnS4(X)) = CnS4(X) proof CnS4(CnS4(X)) c= CnS4(X) & CnS4(X) c= CnS4(CnS4(X)) by Lm23,Th89; hence thesis by XBOOLE_0:def 10; end; Lm24: CnS4(X) is S4_theory proof let p, q, r; thus p => (q => p) in CnS4(X) by Th82; thus (p => (q => r)) => ((p => q) => (p => r)) in CnS4(X) by Th82; thus (p '&' q) => p in CnS4(X) by Th82; thus (p '&' q) => q in CnS4(X) by Th82; thus p => (q => (p '&' q)) in CnS4(X) by Th82; thus p => (p 'or' q) in CnS4(X) by Th82; thus q => (p 'or' q) in CnS4(X) by Th82; thus (p => r) => ((q => r) => ((p 'or' q) => r)) in CnS4(X) by Th82; thus FALSUM => p in CnS4(X) by Th82; thus p 'or' (p => FALSUM) in CnS4 (X) by Th82; thus (Nes (p => q)) => ((Nes p) => (Nes q)) in CnS4(X) by Th84; thus (Nes p) => p in CnS4(X) by Th85; thus (Nes p) => Nes (Nes p) in CnS4(X) by Th86; thus p in CnS4(X) & p => q in CnS4(X) implies q in CnS4(X) by Th83; thus (p in CnS4(X) implies Nes p in CnS4(X)) by Th87; end; definition let X be Subset of MC-wff; cluster CnS4(X) -> S4_theory; coherence by Lm24; end; theorem Th92: T is S4_theory iff CnS4(T) = T proof hereby assume A1: T is S4_theory; A2: T c= CnS4(T) by Th89; CnS4(T) c= T proof let a be set; assume a in CnS4(T); hence a in T by A1,Def23; end; hence CnS4(T) = T by A2,XBOOLE_0:def 10; end; thus CnS4(T) = T implies T is S4_theory; end; theorem :: Th12M: T is S4_theory implies S4-Taut c= T proof assume A1: T is S4_theory; {}(MC-wff) c= T by XBOOLE_1:2; then S4-Taut c= CnS4(T) by Def24,Th90; hence thesis by A1,Th92; end; definition cluster S4-Taut -> S4_theory; coherence by Def24; end; theorem :: S4incCPC: CPC-Taut c= S4-Taut by Def21,Def24,Th80; theorem :: S4incIPC: IPC-Taut c= S4-Taut by Def16,Def24,Th81;

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