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|
% --------------------------------------------------------------------- %
% Copyright (c) Jim Grundy 1992 %
% All rights reserved %
% %
% Jim Grundy, hereafter referred to as `the Author', retains the %
% copyright and all other legal rights to the Software contained in %
% this file, hereafter referred to as `the Software'. %
% %
% The Software is made available free of charge on an `as is' basis. %
% No guarantee, either express or implied, of maintenance, reliability %
% or suitability for any purpose is made by the Author. %
% %
% The user is granted the right to make personal or internal use %
% of the Software provided that both: %
% 1. The Software is not used for commercial gain. %
% 2. The user shall not hold the Author liable for any consequences %
% arising from use of the Software. %
% %
% The user is granted the right to further distribute the Software %
% provided that both: %
% 1. The Software and this statement of rights is not modified. %
% 2. The Software does not form part or the whole of a system %
% distributed for commercial gain. %
% %
% The user is granted the right to modify the Software for personal or %
% internal use provided that all of the following conditions are %
% observed: %
% 1. The user does not distribute the modified software. %
% 2. The modified software is not used for commercial gain. %
% 3. The Author retains all rights to the modified software. %
% %
% Anyone seeking a licence to use this software for commercial purposes %
% is invited to contact the Author. %
% --------------------------------------------------------------------- %
% CONTENTS: conversions for moveing qantifiers about. %
% --------------------------------------------------------------------- %
%$Id: conv.ml,v 3.1 1993/12/07 14:42:10 jg Exp $%
begin_section convsec;;
% ------------------------------------------------------------------------- %
% NOT_FORALL_THM = |- !f. ~(!x. f x) = (?x. ~f x) %
% ------------------------------------------------------------------------- %
let NOT_FORALL_THM =
let f = "f:*->bool" in
let x = "x:*" in
let t = mk_comb(f,x) in
let all = mk_forall(x,t) and exists = mk_exists(x,mk_neg t) in
let nott = ASSUME (mk_neg t) in
%WW% let th1 = DISCH all (MP (NOT_ELIM nott) (SPEC x (ASSUME all))) in
let imp1 = DISCH exists (CHOOSE (x, ASSUME exists) (NOT_INTRO th1)) in
let th2 = CCONTR t
%WW% (MP (NOT_ELIM(ASSUME(mk_neg exists))) (EXISTS(exists,x)nott)) in
let th3 = CCONTR exists
%WW% (MP (NOT_ELIM(ASSUME (mk_neg all))) (GEN x th2)) in
let imp2 = DISCH (mk_neg all) th3 in
GEN f (IMP_ANTISYM_RULE imp2 imp1);;
% ------------------------------------------------------------------------- %
% NOT_EXISTS_THM = |- !f. ~(?x. f x) = (!x. ~f x) %
% ------------------------------------------------------------------------- %
let NOT_EXISTS_THM =
let f = "f:*->bool" in
let x = "x:*" in
let t = mk_comb(f,x) in
let tm = mk_neg(mk_exists(x,t)) in
let all = mk_forall(x,mk_neg t) in
let asm1 = ASSUME t in
%WW% let thm1 = MP (NOT_ELIM(ASSUME tm)) (EXISTS (rand tm, x) asm1) in
let imp1 = DISCH tm (GEN x (NOT_INTRO (DISCH t thm1))) in
let asm2 = ASSUME all and asm3 = ASSUME (rand tm) in
let thm2 = DISCH (rand tm) (CHOOSE (x,asm3)
%WW% (MP (NOT_ELIM(SPEC x asm2)) asm1)) in
let imp2 = DISCH all (NOT_INTRO thm2) in
GEN f (IMP_ANTISYM_RULE imp1 imp2);;
% ------------------------------------------------------------------------- %
% NOT_PFORALL_CONV "~!p.t" = |- (~!p.t) = (?p.~t) %
% ------------------------------------------------------------------------- %
let NOT_PFORALL_CONV tm =
(let (p,_) = dest_pforall (dest_neg tm) in
let f = rand (rand tm) in
let th = ISPEC f NOT_FORALL_THM in
let th1 = CONV_RULE (RATOR_CONV (RAND_CONV (RAND_CONV (RAND_CONV
ETA_CONV)))) th in
let th2 = CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV p))) th1 in
CONV_RULE
(RAND_CONV (RAND_CONV (PABS_CONV (RAND_CONV PBETA_CONV))))
th2
) ? failwith `NOT_PFORALL_CONV: argyment must have the form "~!p.tm"`;;
% ------------------------------------------------------------------------- %
% NOT_PEXISTS_CONV "~?p.t" = |- (~?p.t) = (!p.~t) %
% ------------------------------------------------------------------------- %
let NOT_PEXISTS_CONV tm =
(let (p,_) = dest_pexists (dest_neg tm) in
let f = rand (rand tm) in
let th = ISPEC f NOT_EXISTS_THM in
let th1 = CONV_RULE (RATOR_CONV (RAND_CONV (RAND_CONV (RAND_CONV
ETA_CONV)))) th in
let th2 = CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV p))) th1 in
CONV_RULE
(RAND_CONV (RAND_CONV (PABS_CONV (RAND_CONV PBETA_CONV))))
th2
) ? failwith `NOT_PEXISTS_CONV: argyment must have the form "~!p.tm"`;;
% ------------------------------------------------------------------------- %
% PEXISTS_NOT_CONV "?p.~t" = |- (?p.~t) = (~!p.t) %
% ------------------------------------------------------------------------- %
let PEXISTS_NOT_CONV tm =
(let xtm = mk_pforall ((I # dest_neg) (dest_pexists tm)) in
SYM (NOT_PFORALL_CONV (mk_neg xtm))
) ? failwith `PEXISTS_NOT_CONV: argument must have the form "?x.~tm"`;;
% ------------------------------------------------------------------------- %
% PFORALL_NOT_CONV "!p.~t" = |- (!p.~t) = (~?p.t) %
% ------------------------------------------------------------------------- %
let PFORALL_NOT_CONV tm =
(let xtm = mk_pexists ((I # dest_neg) (dest_pforall tm)) in
SYM (NOT_PEXISTS_CONV (mk_neg xtm))
) ? failwith `PFORALL_NOT_CONV: argument must have the form "!x.~tm"`;;
% ------------------------------------------------------------------------- %
% FORALL_AND_THM |- !f g. (!x. f x /\ g x) = ((!x. f x) /\ (!x. g x)) %
% ------------------------------------------------------------------------- %
let FORALL_AND_THM =
let f = "f:*->bool" in
let g = "g:*->bool" in
let x = "x:*" in
let th1 = ASSUME "!x:*. (f x) /\ (g x)" in
let imp1 =
(uncurry CONJ) (((GEN x) # (GEN x)) (CONJ_PAIR (SPEC x th1))) in
let th2 = ASSUME "(!x:*. f x) /\ (!x:*. g x)" in
let imp2 =
GEN x ((uncurry CONJ) (((SPEC x) # (SPEC x)) (CONJ_PAIR th2)))
in
GENL [f;g] (IMP_ANTISYM_RULE (DISCH_ALL imp1) (DISCH_ALL imp2));;
% ------------------------------------------------------------------------- %
% PFORALL_AND_CONV "!x. P /\ Q" = |- (!x. P /\ Q) = (!x.P) /\ (!x.Q) %
% ------------------------------------------------------------------------- %
let PFORALL_AND_CONV tm =
(let (x,(P,Q)) = (I # dest_conj) (dest_pforall tm) in
let f = mk_pabs(x,P) in
let g = mk_pabs(x,Q) in
let th = ISPECL [f;g] FORALL_AND_THM in
let th1 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV
(PALPHA_CONV x))))
th in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PABS_CONV
(RATOR_CONV (RAND_CONV PBETA_CONV))))))
th1 in
let th3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PABS_CONV
(RAND_CONV PBETA_CONV)))))
th2 in
let th4 =
CONV_RULE
(RAND_CONV (RATOR_CONV (RAND_CONV (RAND_CONV ETA_CONV))))
th3 in
(CONV_RULE (RAND_CONV (RAND_CONV (RAND_CONV ETA_CONV)))) th4
) ? failwith
`PFORALL_AND_CONV: argument must have the form "!p. P /\\ Q"`;;
% ------------------------------------------------------------------------- %
% EXISTS_OR_THM |- !f g. (?x. f x \/ g x) = ((?x. f x) \/ (?x. g x)) %
% ------------------------------------------------------------------------- %
let EXISTS_OR_THM =
let f = "f:*->bool" in
let g = "g:*->bool" in
let x = "x:*" in
let P = mk_comb(f,x) in
let Q = mk_comb(g,x) in
let tm = mk_pexists (x,mk_disj(P,Q)) in
let ep = mk_exists(x,P) and eq = mk_exists(x,Q) in
let Pth = EXISTS(ep,x)(ASSUME P) and Qth = EXISTS(eq,x)(ASSUME Q) in
let thm1 = DISJ_CASES_UNION (ASSUME(mk_disj(P,Q))) Pth Qth in
let imp1 = DISCH tm (CHOOSE (x,ASSUME tm) thm1) in
let t1 = DISJ1 (ASSUME P) Q and t2 = DISJ2 P (ASSUME Q) in
let th1 = EXISTS(tm,x) t1 and th2 = EXISTS(tm,x) t2 in
let e1 = CHOOSE (x,ASSUME ep) th1 and e2 = CHOOSE (x,ASSUME eq) th2 in
let thm2 = DISJ_CASES (ASSUME(mk_disj(ep,eq))) e1 e2 in
let imp2 = DISCH (mk_disj(ep,eq)) thm2 in
GENL [f;g] (IMP_ANTISYM_RULE imp1 imp2);;
% ------------------------------------------------------------------------- %
% PEXISTS_OR_CONV "?x. P \/ Q" = |- (?x. P \/ Q) = (?x.P) \/ (?x.Q) %
% ------------------------------------------------------------------------- %
let PEXISTS_OR_CONV tm =
(let (x,(P,Q)) = (I # dest_disj) (dest_pexists tm) in
let f = mk_pabs(x,P) in
let g = mk_pabs(x,Q) in
let th = ISPECL [f;g] EXISTS_OR_THM in
let th1 = (CONV_RULE (RATOR_CONV (RAND_CONV (RAND_CONV
(PALPHA_CONV x))))) th in
let th2 = (CONV_RULE (RATOR_CONV (RAND_CONV (RAND_CONV (PABS_CONV
(RATOR_CONV (RAND_CONV PBETA_CONV))))))) th1 in
let th3 = (CONV_RULE (RATOR_CONV (RAND_CONV (RAND_CONV (PABS_CONV
(RAND_CONV PBETA_CONV)))))) th2 in
let th4 = (CONV_RULE (RAND_CONV (RATOR_CONV (RAND_CONV (RAND_CONV
ETA_CONV))))) th3 in
(CONV_RULE (RAND_CONV (RAND_CONV (RAND_CONV ETA_CONV)))) th4
) ? failwith
`PEXISTS_OR_CONV: argument must have the form "?p. P \\/ Q"`;;
% ------------------------------------------------------------------------- %
% AND_PFORALL_CONV "(!x. P) /\ (!x. Q)" = |- (!x.P)/\(!x.Q) = (!x. P /\ Q) %
% ------------------------------------------------------------------------- %
let AND_PFORALL_CONV tm =
(let (x,P),(y,Q) = (dest_pforall # dest_pforall) (dest_conj tm) in
if (not (x=y)) then fail else
let f = mk_pabs (x,P) in
let g = mk_pabs (x,Q) in
let th = SYM (ISPECL [f;g] FORALL_AND_THM) in
let th1 = (CONV_RULE (RATOR_CONV (RAND_CONV (RATOR_CONV (RAND_CONV
(RAND_CONV ETA_CONV)))))) th in
let th2 = (CONV_RULE (RATOR_CONV (RAND_CONV (RAND_CONV
(RAND_CONV ETA_CONV))))) th1 in
let th3 = (CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x)))) th2 in
let th4 = (CONV_RULE (RAND_CONV (RAND_CONV (PABS_CONV
(RATOR_CONV (RAND_CONV PBETA_CONV)))))) th3
in
(CONV_RULE (RAND_CONV (RAND_CONV (PABS_CONV (RAND_CONV
PBETA_CONV))))) th4
) ? failwith
`AND_PFORALL_CONV: arguments must have form "(!p.P)/\\(!p.Q)"`;;
% ------------------------------------------------------------------------- %
% LEFT_AND_FORALL_THM = |- !f Q. (!x. f x) /\ Q = (!x. f x /\ Q) %
% ------------------------------------------------------------------------- %
let LEFT_AND_FORALL_THM =
let x = "x:*" in
let f = "f:*->bool" in
let Q = "Q:bool" in
let th1 = ASSUME "(!x:*. f x) /\ Q" in
let imp1 = GEN x ((uncurry CONJ) ((SPEC x # I) (CONJ_PAIR th1))) in
let th2 = ASSUME "!x:*. f x /\ Q" in
let imp2 = (uncurry CONJ) ((GEN x # I) (CONJ_PAIR (SPEC x th2))) in
GENL [Q;f] (IMP_ANTISYM_RULE (DISCH_ALL imp1) (DISCH_ALL imp2));;
% ------------------------------------------------------------------------- %
% LEFT_AND_PFORALL_CONV "(!x.P) /\ Q" = %
% |- (!x.P) /\ Q = (!x'. P[x'/x] /\ Q) %
% ------------------------------------------------------------------------- %
let LEFT_AND_PFORALL_CONV tm =
(let (x,P),Q = (dest_pforall # I) (dest_conj tm) in
let f = mk_pabs(x,P) in
let th = ISPECL [Q;f] LEFT_AND_FORALL_THM in
let th1 = (CONV_RULE (RATOR_CONV (RAND_CONV (RATOR_CONV (RAND_CONV
(RAND_CONV ETA_CONV)))))) th in
let th2 = (CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x)))) th1
in
(CONV_RULE (RAND_CONV (RAND_CONV (PABS_CONV (RATOR_CONV (RAND_CONV
PBETA_CONV)))))) th2
) ? failwith `LEFT_AND_PFORALL_CONV: expecting "(!p.P) /\\ Q"`;;
% ------------------------------------------------------------------------- %
% RIGHT_AND_FORALL_THM = |- !P g. P /\ (!x. g x) = (!x. P /\ g x) %
% ------------------------------------------------------------------------- %
let RIGHT_AND_FORALL_THM =
let x = "x:*" in
let P = "P:bool" in
let g = "g:*->bool" in
let th1 = ASSUME "P /\ (!x:*. g x)" in
let imp1 = GEN x ((uncurry CONJ) ((I # SPEC x) (CONJ_PAIR th1))) in
let th2 = ASSUME "!x:*. P /\ g x" in
let imp2 = (uncurry CONJ) ((I # GEN x) (CONJ_PAIR (SPEC x th2))) in
GENL [P;g] (IMP_ANTISYM_RULE (DISCH_ALL imp1) (DISCH_ALL imp2));;
% ------------------------------------------------------------------------- %
% RIGHT_AND_PFORALL_CONV "P /\ (!x.Q)" = %
% |- P /\ (!x.Q) = (!x'. P /\ Q[x'/x]) %
% ------------------------------------------------------------------------- %
let RIGHT_AND_PFORALL_CONV tm =
(let P,(x,Q) = (I # dest_pforall) (dest_conj tm) in
let g = mk_pabs(x,Q) in
let th = (ISPECL [P; g] RIGHT_AND_FORALL_THM) in
let th1 = (CONV_RULE (RATOR_CONV (RAND_CONV (RAND_CONV (RAND_CONV
ETA_CONV))))) th in
let th2 = (CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x)))) th1 in
CONV_RULE
(RAND_CONV (RAND_CONV (PABS_CONV (RAND_CONV PBETA_CONV))))
th2
) ? failwith `RIGHT_AND_PFORALL_CONV: expecting "P /\\ (!p.Q)"`;;
% ------------------------------------------------------------------------- %
% OR_PEXISTS_CONV "(?x. P) \/ (?x. Q)" = |- (?x.P) \/ (?x.Q) = (?x. P \/ Q) %
% ------------------------------------------------------------------------- %
let OR_PEXISTS_CONV tm =
(let (x,P),(y,Q) = (dest_pexists # dest_pexists) (dest_disj tm) in
if (not (x=y)) then fail else
let f = mk_pabs (x,P) in
let g = mk_pabs (x,Q) in
let th = SYM (ISPECL [f;g] EXISTS_OR_THM) in
let th1 = (CONV_RULE (RATOR_CONV (RAND_CONV (RATOR_CONV (RAND_CONV
(RAND_CONV ETA_CONV)))))) th in
let th2 = (CONV_RULE (RATOR_CONV (RAND_CONV (RAND_CONV
(RAND_CONV ETA_CONV))))) th1 in
let th3 = (CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x)))) th2 in
let th4 = (CONV_RULE (RAND_CONV (RAND_CONV (PABS_CONV
(RATOR_CONV (RAND_CONV PBETA_CONV)))))) th3
in
(CONV_RULE (RAND_CONV (RAND_CONV (PABS_CONV (RAND_CONV
PBETA_CONV))))) th4
) ? failwith
`OR_PEXISTS_CONV: arguments must have form "(!p.P) \\/ (!p.Q)"`;;
% ------------------------------------------------------------------------- %
% LEFT_OR_EXISTS_THM = |- (?x. f x) \/ Q = (?x. f x \/ Q) %
% ------------------------------------------------------------------------- %
let LEFT_OR_EXISTS_THM =
let x = "x:*" in
let Q = "Q:bool" in
let f = "f:*->bool" in
let P = mk_comb (f,x) in
let ep = mk_exists(x,P) in
let tm = mk_disj (ep,Q) in
let otm = mk_exists (x,(mk_disj(P,Q))) in
let t1 = DISJ1 (ASSUME P) Q and t2 = DISJ2 P (ASSUME Q) in
let th1 = EXISTS(otm,x) t1 and th2 = EXISTS(otm,x) t2 in
let thm1 = DISJ_CASES (ASSUME tm) (CHOOSE(x,ASSUME ep)th1) th2 in
let imp1 = DISCH tm thm1 in
let Pth = EXISTS(ep,x)(ASSUME P) and Qth = ASSUME Q in
let thm2 = DISJ_CASES_UNION (ASSUME(mk_disj(P,Q))) Pth Qth in
let imp2 = DISCH otm (CHOOSE (x,ASSUME otm) thm2) in
GENL [Q;f] (IMP_ANTISYM_RULE imp1 imp2);;
% ------------------------------------------------------------------------- %
% LEFT_OR_PEXISTS_CONV "(?x.P) \/ Q" = %
% |- (?x.P) \/ Q = (?x'. P[x'/x] \/ Q) %
% ------------------------------------------------------------------------- %
let LEFT_OR_PEXISTS_CONV tm =
(let (x,P),Q = (dest_pexists # I) (dest_disj tm) in
let f = mk_pabs(x,P) in
let th = ISPECL [Q;f] LEFT_OR_EXISTS_THM in
let th1 = (CONV_RULE (RATOR_CONV (RAND_CONV (RATOR_CONV (RAND_CONV
(RAND_CONV ETA_CONV)))))) th in
let th2 = (CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x)))) th1
in
CONV_RULE
(RAND_CONV (RAND_CONV (PABS_CONV (RATOR_CONV (RAND_CONV
PBETA_CONV)))))
th2
) ? failwith `LEFT_OR_PEXISTS_CONV: expecting "(?p.P) \\/ Q"`;;
% ------------------------------------------------------------------------- %
% RIGHT_OR_EXISTS_THM = |- P \/ (?x. g x) = (?x. P \/ g x) %
% ------------------------------------------------------------------------- %
let RIGHT_OR_EXISTS_THM =
let x = "x:*" in
let P = "P:bool" in
let g = "g:*->bool" in
let Q = mk_comb (g,x) in
let eq = mk_exists(x,Q) in
let tm = mk_disj (P,eq) in
let otm = mk_exists (x,(mk_disj(P,Q))) in
let t1 = DISJ1 (ASSUME P) Q and t2 = DISJ2 P (ASSUME Q) in
let th1 = EXISTS(otm,x) t1 and th2 = EXISTS(otm,x) t2 in
let thm1 = DISJ_CASES (ASSUME tm) th1 (CHOOSE(x,ASSUME eq)th2) in
let imp1 = DISCH tm thm1 in
let Qth = EXISTS(eq,x)(ASSUME Q) and Pth = ASSUME P in
let thm2 = DISJ_CASES_UNION (ASSUME(mk_disj(P,Q))) Pth Qth in
let imp2 = DISCH otm (CHOOSE (x,ASSUME otm) thm2) in
GENL [P;g] (IMP_ANTISYM_RULE imp1 imp2);;
% ------------------------------------------------------------------------- %
% RIGHT_OR_PEXISTS_CONV "P \/ (?x.Q)" = %
% |- P \/ (?x.Q) = (?x'. P \/ Q[x'/x]) %
% ------------------------------------------------------------------------- %
let RIGHT_OR_PEXISTS_CONV tm =
(let P,(x,Q) = (I # dest_pexists) (dest_disj tm) in
let g = mk_pabs(x,Q) in
let th = (ISPECL [P; g] RIGHT_OR_EXISTS_THM) in
let th1 = (CONV_RULE (RATOR_CONV (RAND_CONV (RAND_CONV (RAND_CONV
ETA_CONV))))) th in
let th2 = (CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x)))) th1 in
CONV_RULE
(RAND_CONV (RAND_CONV (PABS_CONV (RAND_CONV PBETA_CONV))))
th2
) ? failwith `RIGHT_OR_PEXISTS_CONV: expecting "P \\/ (?p.Q)"`;;
% ------------------------------------------------------------------------- %
% BOTH_EXISTS_AND_THM = |- !P Q. (?x. P /\ Q) = (?x. P) /\ (?x. Q) %
% ------------------------------------------------------------------------- %
let BOTH_EXISTS_AND_THM =
let x = "x:*" in
let P = "P:bool" in
let Q = "Q:bool" in
let t = mk_conj(P,Q) in
let exi = mk_exists(x,t) in
let (t1,t2) = CONJ_PAIR (ASSUME t) in
let t11 = EXISTS ((mk_exists(x,P)),x) t1 in
let t21 = EXISTS ((mk_exists(x,Q)),x) t2 in
let imp1 =
DISCH_ALL
(CHOOSE (x, ASSUME (mk_exists(x,mk_conj(P,Q)))) (CONJ t11 t21))
in
let th21 = EXISTS (exi,x) (CONJ (ASSUME P) (ASSUME Q)) in
let th22 = CHOOSE(x,ASSUME(mk_exists(x,P))) th21 in
let th23 = CHOOSE(x,ASSUME(mk_exists(x,Q))) th22 in
let (u1,u2) =
CONJ_PAIR (ASSUME (mk_conj(mk_exists(x,P),mk_exists(x,Q)))) in
let th24 = PROVE_HYP u1 (PROVE_HYP u2 th23) in
let imp2 = DISCH_ALL th24 in
GENL [P;Q] (IMP_ANTISYM_RULE imp1 imp2) ;;
% ------------------------------------------------------------------------- %
% LEFT_EXISTS_AND_THM = |- !Q f. (?x. f x /\ Q) = (?x. f x) /\ Q %
% ------------------------------------------------------------------------- %
let LEFT_EXISTS_AND_THM =
let x = "x:*" in
let f = "f:*->bool" in
let P = mk_comb (f,x) in
let Q = "Q:bool" in
let t = mk_conj(P,Q) in
let exi = mk_exists(x,t) in
let (t1,t2) = CONJ_PAIR (ASSUME t) in
let t11 = EXISTS ((mk_exists(x,P)),x) t1 in
let imp1 =
DISCH_ALL
(CHOOSE
(x, ASSUME (mk_exists(x,mk_conj(P,Q))))
(CONJ t11 t2)) in
let th21 = EXISTS (exi,x) (CONJ (ASSUME P) (ASSUME Q)) in
let th22 = CHOOSE(x,ASSUME(mk_exists(x,P))) th21 in
let (u1,u2) = CONJ_PAIR (ASSUME (mk_conj(mk_exists(x,P),Q))) in
let th23 = PROVE_HYP u1 (PROVE_HYP u2 th22) in
let imp2 = DISCH_ALL th23 in
GENL [Q;f] (IMP_ANTISYM_RULE imp1 imp2) ;;
% ------------------------------------------------------------------------- %
% RIGHT_EXISTS_AND_THM = |- !P g. (?x. P /\ g x) = P /\ (?x. g x) %
% ------------------------------------------------------------------------- %
let RIGHT_EXISTS_AND_THM =
let x = "x:*" in
let P = "P:bool" in
let g = "g:*->bool" in
let Q = mk_comb(g,x) in
let t = mk_conj(P,Q) in
let exi = mk_exists(x,t) in
let (t1,t2) = CONJ_PAIR (ASSUME t) in
let t21 = EXISTS ((mk_exists(x,Q)),x) t2 in
let imp1 =
DISCH_ALL
(CHOOSE
(x, ASSUME (mk_exists(x,mk_conj(P,Q))))
(CONJ t1 t21)) in
let th21 = EXISTS (exi,x) (CONJ (ASSUME P) (ASSUME Q)) in
let th22 = CHOOSE(x,ASSUME(mk_exists(x,Q))) th21 in
let (u1,u2) = CONJ_PAIR (ASSUME (mk_conj(P,mk_exists(x,Q)))) in
let th23 = PROVE_HYP u1 (PROVE_HYP u2 th22) in
let imp2 = DISCH_ALL th23 in
GENL [P;g] (IMP_ANTISYM_RULE imp1 imp2) ;;
% ------------------------------------------------------------------------- %
% PEXISTS_AND_CONV : move existential quantifier into conjunction. %
% %
% A call to PEXISTS_AND_CONV "?x. P /\ Q" returns: %
% %
% |- (?x. P /\ Q) = (?x.P) /\ Q [x not free in Q] %
% |- (?x. P /\ Q) = P /\ (?x.Q) [x not free in P] %
% |- (?x. P /\ Q) = (?x.P) /\ (?x.Q) [x not free in P /\ Q] %
% ------------------------------------------------------------------------- %
let PEXISTS_AND_CONV tm =
(let (x,(P,Q)) = (I # dest_conj) (dest_pexists tm) ?
failwith `expecting "?x. P /\\ Q"` in
let oP = occs_in x P and oQ = occs_in x Q in
if (oP & oQ) then
failwith `bound pair occurs in both conjuncts`
else if ((not oP) & (not oQ)) then
let th1 =
INST_TYPE
[(type_of x, mk_vartype `*`)]
BOTH_EXISTS_AND_THM in
let th2 = SPECL [P;Q] th1 in
let th3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th2 in
let th4 =
CONV_RULE
(RAND_CONV (RATOR_CONV (RAND_CONV (RAND_CONV
(PALPHA_CONV x)))))
th3 in
let th5 =
CONV_RULE
(RAND_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th4
in
th5
else if oP then % not free in Q %
let f = mk_pabs(x,P) in
let th1 = ISPECL [Q;f] LEFT_EXISTS_AND_THM in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th1 in
let th3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV
(PABS_CONV (RATOR_CONV (RAND_CONV PBETA_CONV))))))
th2 in
let th4 =
CONV_RULE
(RAND_CONV
(RATOR_CONV (RAND_CONV (RAND_CONV ETA_CONV))))
th3
in
th4
else % not free in P%
let g = mk_pabs(x,Q) in
let th1 = ISPECL [P;g] RIGHT_EXISTS_AND_THM in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th1 in
let th3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV
(PABS_CONV (RAND_CONV PBETA_CONV)))))
th2 in
let th4 =
CONV_RULE (RAND_CONV (RAND_CONV (RAND_CONV ETA_CONV))) th3
in
th4
) ?\st failwith `PEXISTS_AND_CONV: `^st;;
% ------------------------------------------------------------------------- %
% AND_PEXISTS_CONV "(?x.P) /\ (?x.Q)" = |- (?x.P) /\ (?x.Q) = (?x. P /\ Q) %
% ------------------------------------------------------------------------- %
let AND_PEXISTS_CONV tm =
(let ((x,P),(y,Q)) = (dest_pexists # dest_pexists) (dest_conj tm)
? failwith `expecting "(?x.P) /\\ (?x.Q)"`
in
if not (x=y) then
failwith `expecting "(?x.P) /\\ (?x.Q)"`
else if (occs_in x P) or (occs_in x Q) then
failwith `"` ^ (fst(dest_var x)) ^ `" free in conjunct(s)`
else
SYM (PEXISTS_AND_CONV (mk_pexists (x,mk_conj(P,Q))))
) ?\st failwith `AND_EXISTS_CONV: ` ^ st;;
% ------------------------------------------------------------------------- %
% LEFT_AND_PEXISTS_CONV "(?x.P) /\ Q" = %
% |- (?x.P) /\ Q = (?x'. P[x'/x] /\ Q) %
% ------------------------------------------------------------------------- %
let LEFT_AND_PEXISTS_CONV tm =
(let ((x,P),Q) = (dest_pexists # I) (dest_conj tm) in
let f = mk_pabs(x,P) in
let th1 = SYM (ISPECL [Q;f] LEFT_EXISTS_AND_THM) in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RATOR_CONV (RAND_CONV (RAND_CONV
ETA_CONV)))))
th1 in
let th3 = (CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x)))) th2 in
let th4 =
CONV_RULE
(RAND_CONV (RAND_CONV (PABS_CONV (RATOR_CONV (RAND_CONV
PBETA_CONV)))))
th3
in
th4
) ? failwith `LEFT_AND_PEXISTS_CONV: expecting "(?x.P) /\\ Q"`;;
% ------------------------------------------------------------------------- %
% RIGHT_AND_PEXISTS_CONV "P /\ (?x.Q)" = %
% |- P /\ (?x.Q) = (?x'. P /\ (Q[x'/x]) %
% ------------------------------------------------------------------------- %
let RIGHT_AND_PEXISTS_CONV tm =
(let (P,(x,Q)) = (I # dest_pexists) (dest_conj tm) in
let g = mk_pabs(x,Q) in
let th1 = SYM (ISPECL [P;g] RIGHT_EXISTS_AND_THM) in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (RAND_CONV ETA_CONV))))
th1 in
let th3 = CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x))) th2 in
let th4 =
CONV_RULE
(RAND_CONV (RAND_CONV (PABS_CONV (RAND_CONV PBETA_CONV))))
th3
in
th4
) ? failwith `RIGHT_AND_EXISTS_CONV: expecting "P /\\ (?x.Q)"`;;
% ------------------------------------------------------------------------- %
% BOTH_FORALL_OR_THM = |- !P Q. (!x. P \/ Q) = (!x. P) \/ (!x. Q) %
% ------------------------------------------------------------------------- %
let BOTH_FORALL_OR_THM =
let x = "x:*" in
let P = "P:bool" in
let Q = "Q:bool" in
let imp11 = DISCH_ALL (SPEC x (ASSUME (mk_forall(x,P)))) in
let imp12 = DISCH_ALL (GEN x (ASSUME P)) in
let fath = IMP_ANTISYM_RULE imp11 imp12 in
let th1 = REFL (mk_forall(x, mk_disj (P,Q))) in
let th2 =
CONV_RULE (RAND_CONV (K (INST [(mk_disj(P,Q),P)] fath))) th1 in
let th3 =
CONV_RULE (RAND_CONV (RATOR_CONV (RAND_CONV (K (SYM fath))))) th2 in
let th4 =
CONV_RULE (RAND_CONV (RAND_CONV (K (SYM (INST [(Q,P)] fath))))) th3
in
GENL [P;Q] th4 ;;
% ------------------------------------------------------------------------- %
% LEFT_FORALL_OR_THM = |- !Q f. (!x. f x \/ Q) = (!x. f x) \/ Q %
% ------------------------------------------------------------------------- %
let LEFT_FORALL_OR_THM =
let x = "x:*" in
let f = "f:*->bool" in
let P = mk_comb (f,x) in
let Q = "Q:bool" in
let tm = (mk_forall(x,mk_disj(P,Q))) in
let thm1 = SPEC x (ASSUME tm) in
%WW% let thm2 = CONTR P (MP (NOT_ELIM(ASSUME (mk_neg Q))) (ASSUME Q)) in
let thm3 = DISJ1 (GEN x (DISJ_CASES thm1 (ASSUME P) thm2)) Q in
let thm4 = DISJ2 (mk_forall(x,P)) (ASSUME Q) in
let imp1 = DISCH tm (DISJ_CASES (SPEC Q EXCLUDED_MIDDLE) thm4 thm3) in
let thm5 = SPEC x (ASSUME (mk_forall(x,P))) in
let thm6 = ASSUME Q in
let imp2 =
(DISCH_ALL (GEN x (DISJ_CASES_UNION
(ASSUME (mk_disj(mk_forall(x,P),Q)))
thm5
thm6)))
in
GENL [Q;f] (IMP_ANTISYM_RULE imp1 imp2);;
% ------------------------------------------------------------------------- %
% RIGHT_FORALL_OR_THM = |- !P g. (!x. P \/ g x) = P \/ (!x. g x) %
% ------------------------------------------------------------------------- %
let RIGHT_FORALL_OR_THM =
let x = "x:*" in
let P = "P:bool" in
let g = "g:*->bool" in
let Q = mk_comb(g,x) in
let tm = (mk_forall(x,mk_disj(P,Q))) in
let thm1 = SPEC x (ASSUME tm) in
%WW% let thm2 = CONTR Q (MP (NOT_ELIM (ASSUME (mk_neg P))) (ASSUME P)) in
let thm3 = DISJ2 P (GEN x (DISJ_CASES thm1 thm2 (ASSUME Q))) in
let thm4 = DISJ1 (ASSUME P) (mk_forall(x,Q)) in
let imp1 = DISCH tm (DISJ_CASES (SPEC P EXCLUDED_MIDDLE) thm4 thm3) in
let thm5 = ASSUME P in
let thm6 = SPEC x (ASSUME (mk_forall(x,Q))) in
let imp2 =
(DISCH_ALL (GEN x (DISJ_CASES_UNION
(ASSUME (mk_disj(P,mk_forall(x,Q))))
thm5
thm6)))
in
GENL [P;g] (IMP_ANTISYM_RULE imp1 imp2);;
% ------------------------------------------------------------------------- %
% PFORALL_OR_CONV : move universal quantifier into disjunction. %
% %
% A call to PFORALL_OR_CONV "!x. P \/ Q" returns: %
% %
% |- (!x. P \/ Q) = (!x.P) \/ Q [if x not free in Q] %
% |- (!x. P \/ Q) = P \/ (!x.Q) [if x not free in P] %
% |- (!x. P \/ Q) = (!x.P) \/ (!x.Q) [if x free in neither P nor Q] %
% ------------------------------------------------------------------------- %
let PFORALL_OR_CONV tm =
(let (x,(P,Q)) = (I # dest_disj) (dest_pforall tm) ?
failwith `expecting "!x. P \\/ Q"` in
let oP = occs_in x P and oQ = occs_in x Q in
if (oP & oQ) then
failwith `bound pair occurs in both conjuncts`
else if ((not oP) & (not oQ)) then
let th1 =
INST_TYPE
[(type_of x, mk_vartype `*`)]
BOTH_FORALL_OR_THM in
let th2 = SPECL [P;Q] th1 in
let th3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th2 in
let th4 =
CONV_RULE
(RAND_CONV (RATOR_CONV (RAND_CONV (RAND_CONV
(PALPHA_CONV x)))))
th3 in
let th5 =
CONV_RULE
(RAND_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th4
in
th5
else if oP then % not free in Q %
let f = mk_pabs(x,P) in
let th1 = ISPECL [Q;f] LEFT_FORALL_OR_THM in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th1 in
let th3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV
(PABS_CONV (RATOR_CONV (RAND_CONV PBETA_CONV))))))
th2 in
let th4 =
CONV_RULE
(RAND_CONV
(RATOR_CONV (RAND_CONV (RAND_CONV ETA_CONV))))
th3
in
th4
else % not free in P%
let g = mk_pabs(x,Q) in
let th1 = ISPECL [P;g] RIGHT_FORALL_OR_THM in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th1 in
let th3 =
(CONV_RULE (RATOR_CONV (RAND_CONV (RAND_CONV
(PABS_CONV (RAND_CONV PBETA_CONV))))))
th2 in
let th4 =
(CONV_RULE (RAND_CONV (RAND_CONV (RAND_CONV ETA_CONV))))
th3
in
th4
) ?\st failwith `PFORALL_OR_CONV: `^st;;
% ------------------------------------------------------------------------- %
% OR_PFORALL_CONV "(!x.P) \/ (!x.Q)" = |- (!x.P) \/ (!x.Q) = (!x. P \/ Q) %
% ------------------------------------------------------------------------- %
let OR_PFORALL_CONV tm =
(let ((x,P),(y,Q)) = (dest_pforall # dest_pforall) (dest_disj tm)
? failwith `expecting "(!x.P) \\/ (!x.Q)"`
in
if not (x=y) then
failwith `expecting "(!x.P) \\/ (!x.Q)"`
else if (occs_in x P) or (occs_in x Q) then
failwith `"` ^ (fst(dest_var x)) ^ `" free in disjuncts(s)`
else
SYM (PFORALL_OR_CONV (mk_pforall (x,mk_disj(P,Q))))
) ?\st failwith `OR_FORALL_CONV: ` ^ st;;
% ------------------------------------------------------------------------- %
% LEFT_OR_PFORALL_CONV "(!x.P) \/ Q" = %
% |- (!x.P) \/ Q = (!x'. P[x'/x] \/ Q) %
% ------------------------------------------------------------------------- %
let LEFT_OR_PFORALL_CONV tm =
(let ((x,P),Q) = (dest_pforall # I) (dest_disj tm) in
let f = mk_pabs(x,P) in
let th1 = SYM (ISPECL [Q;f] LEFT_FORALL_OR_THM) in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RATOR_CONV (RAND_CONV (RAND_CONV
ETA_CONV)))))
th1 in
let th3 = CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x))) th2 in
let th4 =
CONV_RULE
(RAND_CONV (RAND_CONV (PABS_CONV (RATOR_CONV (RAND_CONV
PBETA_CONV)))))
th3
in
th4
) ? failwith `LEFT_OR_PFORALL_CONV: expecting "(!x.P) \\/ Q"`;;
% ------------------------------------------------------------------------- %
% RIGHT_OR_PFORALL_CONV "P \/ (!x.Q)" = %
% |- P \/ (!x.Q) = (!x'. P \/ (Q[x'/x]) %
% ------------------------------------------------------------------------- %
let RIGHT_OR_PFORALL_CONV tm =
(let (P,(x,Q)) = (I # dest_pforall) (dest_disj tm) in
let g = mk_pabs(x,Q) in
let th1 = SYM (ISPECL [P;g] RIGHT_FORALL_OR_THM) in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (RAND_CONV ETA_CONV))))
th1 in
let th3 = CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x))) th2 in
let th4 =
CONV_RULE
(RAND_CONV (RAND_CONV (PABS_CONV (RAND_CONV PBETA_CONV))))
th3
in
th4
) ? failwith `RIGHT_OR_FORALL_CONV: expecting "P \\/ (!x.Q)"`;;
% ------------------------------------------------------------------------- %
% BOTH_FORALL_IMP_THM = |- (!x. P ==> Q) = ((?x.P) ==> (!x.Q)) %
% ------------------------------------------------------------------------- %
let BOTH_FORALL_IMP_THM =
let x = "x:*" in
let P = "P:bool" in
let Q = "Q:bool" in
let tm = mk_forall(x, mk_imp (P, Q)) in
let asm = mk_exists(x,P) in
let th1 = GEN x (CHOOSE(x,ASSUME asm)(UNDISCH(SPEC x (ASSUME tm)))) in
let imp1 = DISCH tm (DISCH asm th1) in
let cncl = rand(concl imp1) in
let th2 = SPEC x (MP (ASSUME cncl) (EXISTS (asm,x) (ASSUME P))) in
let imp2 = DISCH cncl (GEN x (DISCH P th2)) in
GENL [P;Q] (IMP_ANTISYM_RULE imp1 imp2) ;;
% ------------------------------------------------------------------------- %
% LEFT_FORALL_IMP_THM = |- (!x. P[x]==>Q) = ((?x.P[x]) ==> Q) %
% ------------------------------------------------------------------------- %
let LEFT_FORALL_IMP_THM =
let x = "x:*" in
let f = "f:*->bool" in
let P = mk_comb(f,x) in
let Q = "Q:bool" in
let tm = mk_forall(x, mk_imp (P, Q)) in
let asm = mk_exists(x,P) in
let th1 = CHOOSE(x,ASSUME asm)(UNDISCH(SPEC x (ASSUME tm))) in
let imp1 = DISCH tm (DISCH asm th1) in
let cncl = rand(concl imp1) in
let th2 = MP (ASSUME cncl) (EXISTS (asm,x) (ASSUME P)) in
let imp2 = DISCH cncl (GEN x (DISCH P th2)) in
GENL [Q;f] (IMP_ANTISYM_RULE imp1 imp2) ;;
% ------------------------------------------------------------------------- %
% RIGHT_FORALL_IMP_THM = |- (!x. P==>Q[x]) = (P ==> (!x.Q[x])) %
% ------------------------------------------------------------------------- %
let RIGHT_FORALL_IMP_THM =
let x = "x:*" in
let P = "P:bool" in
let g = "g:*->bool" in
let Q = mk_comb (g,x) in
let tm = mk_forall(x, mk_imp (P, Q)) in
let imp1 = DISCH P(GEN x(UNDISCH(SPEC x(ASSUME tm)))) in
let cncl = concl imp1 in
let imp2 = GEN x (DISCH P(SPEC x(UNDISCH (ASSUME cncl)))) in
GENL [P;g] (IMP_ANTISYM_RULE (DISCH tm imp1) (DISCH cncl imp2)) ;;
% ------------------------------------------------------------------------- %
% BOTH_EXISTS_IMP_THM = |- (?x. P ==> Q) = ((!x.P) ==> (?x.Q)) %
% ------------------------------------------------------------------------- %
let BOTH_EXISTS_IMP_THM =
let x = "x:*" in
let P = "P:bool" in
let Q = "Q:bool" in
let tm = mk_exists(x, mk_imp (P, Q)) in
let eQ = mk_exists(x,Q) in
let aP = mk_forall(x,P) in
let thm1 = EXISTS(eQ,x)(UNDISCH(ASSUME(mk_imp(P,Q)))) in
let thm2 = DISCH aP (PROVE_HYP (SPEC x (ASSUME aP)) thm1) in
let imp1 = DISCH tm (CHOOSE(x,ASSUME tm) thm2) in
let thm2 = CHOOSE(x,UNDISCH (ASSUME (rand(concl imp1)))) (ASSUME Q) in
let thm3 = DISCH P (PROVE_HYP (GEN x (ASSUME P)) thm2) in
let imp2 = DISCH (rand(concl imp1)) (EXISTS(tm,x) thm3) in
GENL [P;Q] (IMP_ANTISYM_RULE imp1 imp2) ;;
% ------------------------------------------------------------------------- %
% LEFT_EXISTS_IMP_THM = |- (?x. P[x] ==> Q) = ((!x.P[x]) ==> Q) %
% ------------------------------------------------------------------------- %
let LEFT_EXISTS_IMP_THM =
let x = "x:*" in
let f = "f:*->bool" in
let P = mk_comb(f,x) in
let Q = "Q:bool" in
let tm = mk_exists(x, mk_imp (P, Q)) in
let allp = mk_forall(x,P) in
let th1 = SPEC x (ASSUME allp) in
let thm1 = MP (ASSUME(mk_imp(P,Q))) th1 in
let imp1 = DISCH tm (CHOOSE(x,ASSUME tm)(DISCH allp thm1)) in
let otm = rand(concl imp1) in
let thm2 = EXISTS(tm,x)(DISCH P (UNDISCH(ASSUME otm))) in
let nex = mk_exists(x,mk_neg P) in
let asm1 = EXISTS (nex, x) (ASSUME (mk_neg P)) in
%WW% let th2 = CCONTR P (MP (NOT_ELIM(ASSUME (mk_neg nex))) asm1) in
%WW% let th3 = CCONTR nex (MP (NOT_ELIM(ASSUME(mk_neg allp)))(GEN x th2)) in
let thm4 = DISCH P (CONTR Q (UNDISCH (ASSUME (mk_neg P)))) in
let thm5 = CHOOSE(x,th3)(EXISTS(tm,x)thm4) in
let thm6 = DISJ_CASES (SPEC allp EXCLUDED_MIDDLE) thm2 thm5 in
let imp2 = DISCH otm thm6 in
GENL [Q; f] (IMP_ANTISYM_RULE imp1 imp2) ;;
% ------------------------------------------------------------------------- %
% RIGHT_EXISTS_IMP_THM = |- (?x. P ==> Q[x]) = (P ==> (?x.Q[x])) %
% ------------------------------------------------------------------------- %
let RIGHT_EXISTS_IMP_THM =
let x = "x:*" in
let P = "P:bool" in
let g = "g:*->bool" in
let Q = mk_comb (g,x) in
let tm = mk_exists(x, mk_imp (P, Q)) in
let thm1 = EXISTS (mk_exists(x,Q),x) (UNDISCH(ASSUME(mk_imp(P,Q)))) in
let imp1 = DISCH tm (CHOOSE(x,ASSUME tm) (DISCH P thm1)) in
let thm2 = UNDISCH (ASSUME (rand(concl imp1))) in
let thm3 = CHOOSE (x,thm2) (EXISTS (tm,x) (DISCH P (ASSUME Q))) in
let thm4 = EXISTS(tm,x)(DISCH P(CONTR Q(UNDISCH(ASSUME(mk_neg P))))) in
let thm5 = DISJ_CASES (SPEC P EXCLUDED_MIDDLE) thm3 thm4 in
let imp2 = (DISCH(rand(concl imp1)) thm5) in
GENL [P;g] (IMP_ANTISYM_RULE imp1 imp2) ;;
% ------------------------------------------------------------------------- %
% PFORALL_IMP_CONV, implements the following axiom schemes. %
% %
% |- (!x. P==>Q[x]) = (P ==> (!x.Q[x])) [x not free in P] %
% %
% |- (!x. P[x]==>Q) = ((?x.P[x]) ==> Q) [x not free in Q] %
% %
% |- (!x. P==>Q) = ((?x.P) ==> (!x.Q)) [x not free in P==>Q] %
% ------------------------------------------------------------------------- %
let PFORALL_IMP_CONV tm =
(let (x,(P,Q)) = (I # dest_imp) (dest_pforall tm) ?
failwith `expecting "?x. P ==> Q"` in
let oP = occs_in x P and oQ = occs_in x Q in
if (oP & oQ) then
failwith `bound pair occurs in both sides of "==>"`
else if ((not oP) & (not oQ)) then
let th1 =
INST_TYPE
[(type_of x, mk_vartype `*`)]
BOTH_FORALL_IMP_THM in
let th2 = SPECL [P;Q] th1 in
let th3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th2 in
let th4 =
CONV_RULE
(RAND_CONV (RATOR_CONV (RAND_CONV (RAND_CONV
(PALPHA_CONV x)))))
th3 in
let th5 =
CONV_RULE
(RAND_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th4
in
th5
else if oP then % not free in Q %
let f = mk_pabs(x,P) in
let th1 = ISPECL [Q;f] LEFT_FORALL_IMP_THM in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th1 in
let th3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV
(PABS_CONV (RATOR_CONV (RAND_CONV PBETA_CONV))))))
th2 in
let th4 =
CONV_RULE
(RAND_CONV
(RATOR_CONV (RAND_CONV (RAND_CONV ETA_CONV))))
th3
in
th4
else % not free in P%
let g = mk_pabs(x,Q) in
let th1 = ISPECL [P;g] RIGHT_FORALL_IMP_THM in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th1 in
let th3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PABS_CONV
(RAND_CONV PBETA_CONV)))))
th2 in
let th4 =
CONV_RULE (RAND_CONV (RAND_CONV (RAND_CONV ETA_CONV))) th3
in
th4
) ?\st failwith `PFORALL_IMP_CONV: `^st;;
% ------------------------------------------------------------------------- %
% LEFT_IMP_PEXISTS_CONV "(?x.P) ==> Q" = %
% |- ((?x.P) ==> Q) = (!x'. P[x'/x] ==> Q) %
% ------------------------------------------------------------------------- %
let LEFT_IMP_PEXISTS_CONV tm =
(let (x,P),Q = (dest_pexists # I) (dest_imp tm) in
let f = mk_pabs(x,P) in
let th = SYM (ISPECL [Q;f] LEFT_FORALL_IMP_THM) in
let th1 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RATOR_CONV (RAND_CONV
(RAND_CONV ETA_CONV)))))
th in
let th2 = CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x))) th1
in
CONV_RULE
(RAND_CONV (RAND_CONV (PABS_CONV
(RATOR_CONV (RAND_CONV PBETA_CONV)))))
th2
) ? failwith `LEFT_IMP_PEXISTS_CONV: expecting "(?p.P) ==> Q"`;;
% ------------------------------------------------------------------------- %
% RIGHT_IMP_PFORALL_CONV "P ==> (!x.Q)" = %
% |- (P ==> (!x.Q)) = (!x'. P ==> (Q[x'/x]) %
% ------------------------------------------------------------------------- %
let RIGHT_IMP_PFORALL_CONV tm =
(let (P,(x,Q)) = (I # dest_pforall) (dest_imp tm) in
let g = mk_pabs(x,Q) in
let th1 = SYM (ISPECL [P;g] RIGHT_FORALL_IMP_THM) in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (RAND_CONV ETA_CONV))))
th1 in
let th3 = CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x))) th2 in
let th4 =
CONV_RULE
(RAND_CONV (RAND_CONV (PABS_CONV (RAND_CONV PBETA_CONV))))
th3
in
th4
) ? failwith `RIGHT_IMP_FORALL_CONV: expecting "P ==> (!x.Q)"`;;
% ------------------------------------------------------------------------- %
% PEXISTS_IMP_CONV, implements the following axiom schemes. %
% %
% |- (?x. P==>Q[x]) = (P ==> (?x.Q[x])) [x not free in P] %
% %
% |- (?x. P[x]==>Q) = ((!x.P[x]) ==> Q) [x not free in Q] %
% %
% |- (?x. P==>Q) = ((!x.P) ==> (?x.Q)) [x not free in P==>Q] %
% ------------------------------------------------------------------------- %
let PEXISTS_IMP_CONV tm =
(let (x,(P,Q)) = (I # dest_imp) (dest_pexists tm) ?
failwith `expecting "?x. P ==> Q"` in
let oP = occs_in x P and oQ = occs_in x Q in
if (oP & oQ) then
failwith `bound pair occurs in both sides of "==>"`
else if ((not oP) & (not oQ)) then
let th1 =
INST_TYPE
[(type_of x, mk_vartype `*`)]
BOTH_EXISTS_IMP_THM in
let th2 = SPECL [P;Q] th1 in
let th3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th2 in
let th4 =
CONV_RULE
(RAND_CONV (RATOR_CONV (RAND_CONV (RAND_CONV
(PALPHA_CONV x)))))
th3 in
let th5 =
CONV_RULE
(RAND_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th4
in
th5
else if oP then % not free in Q %
let f = mk_pabs(x,P) in
let th1 = ISPECL [Q;f] LEFT_EXISTS_IMP_THM in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th1 in
let th3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV
(PABS_CONV (RATOR_CONV (RAND_CONV PBETA_CONV))))))
th2 in
let th4 =
CONV_RULE
(RAND_CONV (RATOR_CONV (RAND_CONV (RAND_CONV
ETA_CONV))))
th3
in
th4
else % not free in P%
let g = mk_pabs(x,Q) in
let th1 = ISPECL [P;g] RIGHT_EXISTS_IMP_THM in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (PALPHA_CONV x))))
th1 in
let th3 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV
(PABS_CONV (RAND_CONV PBETA_CONV)))))
th2 in
let th4 =
CONV_RULE (RAND_CONV (RAND_CONV (RAND_CONV ETA_CONV))) th3
in
th4
) ?\st failwith `PEXISTS_IMP_CONV: `^st;;
% ------------------------------------------------------------------------- %
% LEFT_IMP_PFORALL_CONV "(!x. t1[x]) ==> t2" = %
% |- (!x. t1[x]) ==> t2 = (?x'. t1[x'] ==> t2) %
% ------------------------------------------------------------------------- %
let LEFT_IMP_PFORALL_CONV tm =
(let ((x,P),Q) = (dest_pforall # I) (dest_imp tm) in
let f = mk_pabs(x,P) in
let th1 = SYM (ISPECL [Q;f] LEFT_EXISTS_IMP_THM) in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RATOR_CONV (RAND_CONV (RAND_CONV
ETA_CONV)))))
th1 in
let th3 = CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x))) th2 in
let th4 =
CONV_RULE
(RAND_CONV (RAND_CONV (PABS_CONV
(RATOR_CONV (RAND_CONV PBETA_CONV)))))
th3
in
th4
) ? failwith `LEFT_IMP_PFORALL_CONV: expecting "(!x.P) ==> Q"`;;
% ------------------------------------------------------------------------- %
% RIGHT_IMP_EXISTS_CONV "t1 ==> (?x. t2)" = %
% |- (t1 ==> ?x. t2) = (?x'. t1 ==> t2[x'/x]) %
% ------------------------------------------------------------------------- %
let RIGHT_IMP_PEXISTS_CONV tm =
(let (P,(x,Q)) = (I # dest_pexists) (dest_imp tm) in
let g = mk_pabs(x,Q) in
let th1 = SYM (ISPECL [P;g] RIGHT_EXISTS_IMP_THM) in
let th2 =
CONV_RULE
(RATOR_CONV (RAND_CONV (RAND_CONV (RAND_CONV ETA_CONV))))
th1 in
let th3 = CONV_RULE (RAND_CONV (RAND_CONV (PALPHA_CONV x))) th2 in
let th4 =
CONV_RULE
(RAND_CONV (RAND_CONV (PABS_CONV (RAND_CONV PBETA_CONV))))
th3
in
th4
) ? failwith `RIGHT_IMP_PEXISTS_CONV: expecting "P ==> (!x.Q)"`;;
(
NOT_PFORALL_CONV,
NOT_PEXISTS_CONV,
PEXISTS_NOT_CONV,
PFORALL_NOT_CONV,
PFORALL_AND_CONV,
PEXISTS_OR_CONV,
AND_PFORALL_CONV,
LEFT_AND_PFORALL_CONV,
RIGHT_AND_PFORALL_CONV,
OR_PEXISTS_CONV,
LEFT_OR_PEXISTS_CONV,
RIGHT_OR_PEXISTS_CONV,
PEXISTS_AND_CONV,
AND_PEXISTS_CONV,
LEFT_AND_PEXISTS_CONV,
RIGHT_AND_PEXISTS_CONV,
PFORALL_OR_CONV,
OR_PFORALL_CONV,
LEFT_OR_PFORALL_CONV,
RIGHT_OR_PFORALL_CONV,
PFORALL_IMP_CONV,
LEFT_IMP_PEXISTS_CONV,
RIGHT_IMP_PFORALL_CONV,
PEXISTS_IMP_CONV,
LEFT_IMP_PFORALL_CONV,
RIGHT_IMP_PEXISTS_CONV
);;
end_section convsec;;
let (
NOT_PFORALL_CONV,
NOT_PEXISTS_CONV,
PEXISTS_NOT_CONV,
PFORALL_NOT_CONV,
PFORALL_AND_CONV,
PEXISTS_OR_CONV,
AND_PFORALL_CONV,
LEFT_AND_PFORALL_CONV,
RIGHT_AND_PFORALL_CONV,
OR_PEXISTS_CONV,
LEFT_OR_PEXISTS_CONV,
RIGHT_OR_PEXISTS_CONV,
PEXISTS_AND_CONV,
AND_PEXISTS_CONV,
LEFT_AND_PEXISTS_CONV,
RIGHT_AND_PEXISTS_CONV,
PFORALL_OR_CONV,
OR_PFORALL_CONV,
LEFT_OR_PFORALL_CONV,
RIGHT_OR_PFORALL_CONV,
PFORALL_IMP_CONV,
LEFT_IMP_PEXISTS_CONV,
RIGHT_IMP_PFORALL_CONV,
PEXISTS_IMP_CONV,
LEFT_IMP_PFORALL_CONV,
RIGHT_IMP_PEXISTS_CONV
) = it;;
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