--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/FOLP/ifolp.ML Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,444 @@
+(* Title: FOLP/ifol.ML
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1992 University of Cambridge
+
+Tactics and lemmas for ifol.thy (intuitionistic first-order logic)
+*)
+
+open IFOLP;
+
+signature IFOLP_LEMMAS =
+ sig
+ val allE: thm
+ val all_cong: thm
+ val all_dupE: thm
+ val all_impE: thm
+ val box_equals: thm
+ val conjE: thm
+ val conj_cong: thm
+ val conj_impE: thm
+ val contrapos: thm
+ val disj_cong: thm
+ val disj_impE: thm
+ val eq_cong: thm
+ val ex1I: thm
+ val ex1E: thm
+ val ex1_equalsE: thm
+(* val ex1_cong: thm****)
+ val ex_cong: thm
+ val ex_impE: thm
+ val iffD1: thm
+ val iffD2: thm
+ val iffE: thm
+ val iffI: thm
+ val iff_cong: thm
+ val iff_impE: thm
+ val iff_refl: thm
+ val iff_sym: thm
+ val iff_trans: thm
+ val impE: thm
+ val imp_cong: thm
+ val imp_impE: thm
+ val mp_tac: int -> tactic
+ val notE: thm
+ val notI: thm
+ val not_cong: thm
+ val not_impE: thm
+ val not_sym: thm
+ val not_to_imp: thm
+ val pred1_cong: thm
+ val pred2_cong: thm
+ val pred3_cong: thm
+ val pred_congs: thm list
+ val refl: thm
+ val rev_mp: thm
+ val simp_equals: thm
+ val subst: thm
+ val ssubst: thm
+ val subst_context: thm
+ val subst_context2: thm
+ val subst_context3: thm
+ val sym: thm
+ val trans: thm
+ val TrueI: thm
+ val uniq_assume_tac: int -> tactic
+ val uniq_mp_tac: int -> tactic
+ end;
+
+
+structure IFOLP_Lemmas : IFOLP_LEMMAS =
+struct
+
+val TrueI = TrueI;
+
+(*** Sequent-style elimination rules for & --> and ALL ***)
+
+val conjE = prove_goal IFOLP.thy
+ "[| p:P&Q; !!x y.[| x:P; y:Q |] ==> f(x,y):R |] ==> ?a:R"
+ (fn prems=>
+ [ (REPEAT (resolve_tac prems 1
+ ORELSE (resolve_tac [conjunct1, conjunct2] 1 THEN
+ resolve_tac prems 1))) ]);
+
+val impE = prove_goal IFOLP.thy
+ "[| p:P-->Q; q:P; !!x.x:Q ==> r(x):R |] ==> ?p:R"
+ (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
+
+val allE = prove_goal IFOLP.thy
+ "[| p:ALL x.P(x); !!y.y:P(x) ==> q(y):R |] ==> ?p:R"
+ (fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
+
+(*Duplicates the quantifier; for use with eresolve_tac*)
+val all_dupE = prove_goal IFOLP.thy
+ "[| p:ALL x.P(x); !!y z.[| y:P(x); z:ALL x.P(x) |] ==> q(y,z):R \
+\ |] ==> ?p:R"
+ (fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
+
+
+(*** Negation rules, which translate between ~P and P-->False ***)
+
+val notI = prove_goalw IFOLP.thy [not_def] "(!!x.x:P ==> q(x):False) ==> ?p:~P"
+ (fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]);
+
+val notE = prove_goalw IFOLP.thy [not_def] "[| p:~P; q:P |] ==> ?p:R"
+ (fn prems=>
+ [ (resolve_tac [mp RS FalseE] 1),
+ (REPEAT (resolve_tac prems 1)) ]);
+
+(*This is useful with the special implication rules for each kind of P. *)
+val not_to_imp = prove_goal IFOLP.thy
+ "[| p:~P; !!x.x:(P-->False) ==> q(x):Q |] ==> ?p:Q"
+ (fn prems=> [ (REPEAT (ares_tac (prems@[impI,notE]) 1)) ]);
+
+
+(* For substitution int an assumption P, reduce Q to P-->Q, substitute into
+ this implication, then apply impI to move P back into the assumptions.
+ To specify P use something like
+ eres_inst_tac [ ("P","ALL y. ?S(x,y)") ] rev_mp 1 *)
+val rev_mp = prove_goal IFOLP.thy "[| p:P; q:P --> Q |] ==> ?p:Q"
+ (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
+
+
+(*Contrapositive of an inference rule*)
+val contrapos = prove_goal IFOLP.thy "[| p:~Q; !!y.y:P==>q(y):Q |] ==> ?a:~P"
+ (fn [major,minor]=>
+ [ (rtac (major RS notE RS notI) 1),
+ (etac minor 1) ]);
+
+(** Unique assumption tactic.
+ Ignores proof objects.
+ Fails unless one assumption is equal and exactly one is unifiable
+**)
+
+local
+ fun discard_proof (Const("Proof",_) $ P $ _) = P;
+in
+val uniq_assume_tac =
+ SUBGOAL
+ (fn (prem,i) =>
+ let val hyps = map discard_proof (Logic.strip_assums_hyp prem)
+ and concl = discard_proof (Logic.strip_assums_concl prem)
+ in
+ if exists (fn hyp => hyp aconv concl) hyps
+ then case distinct (filter (fn hyp=> could_unify(hyp,concl)) hyps) of
+ [_] => assume_tac i
+ | _ => no_tac
+ else no_tac
+ end);
+end;
+
+
+(*** Modus Ponens Tactics ***)
+
+(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
+fun mp_tac i = eresolve_tac [notE,make_elim mp] i THEN assume_tac i;
+
+(*Like mp_tac but instantiates no variables*)
+fun uniq_mp_tac i = eresolve_tac [notE,impE] i THEN uniq_assume_tac i;
+
+
+(*** If-and-only-if ***)
+
+val iffI = prove_goalw IFOLP.thy [iff_def]
+ "[| !!x.x:P ==> q(x):Q; !!x.x:Q ==> r(x):P |] ==> ?p:P<->Q"
+ (fn prems=> [ (REPEAT (ares_tac (prems@[conjI, impI]) 1)) ]);
+
+
+(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
+val iffE = prove_goalw IFOLP.thy [iff_def]
+ "[| p:P <-> Q; !!x y.[| x:P-->Q; y:Q-->P |] ==> q(x,y):R |] ==> ?p:R"
+ (fn prems => [ (resolve_tac [conjE] 1), (REPEAT (ares_tac prems 1)) ]);
+
+(* Destruct rules for <-> similar to Modus Ponens *)
+
+val iffD1 = prove_goalw IFOLP.thy [iff_def] "[| p:P <-> Q; q:P |] ==> ?p:Q"
+ (fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
+
+val iffD2 = prove_goalw IFOLP.thy [iff_def] "[| p:P <-> Q; q:Q |] ==> ?p:P"
+ (fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
+
+val iff_refl = prove_goal IFOLP.thy "?p:P <-> P"
+ (fn _ => [ (REPEAT (ares_tac [iffI] 1)) ]);
+
+val iff_sym = prove_goal IFOLP.thy "p:Q <-> P ==> ?p:P <-> Q"
+ (fn [major] =>
+ [ (rtac (major RS iffE) 1),
+ (rtac iffI 1),
+ (REPEAT (eresolve_tac [asm_rl,mp] 1)) ]);
+
+val iff_trans = prove_goal IFOLP.thy "[| p:P <-> Q; q:Q<-> R |] ==> ?p:P <-> R"
+ (fn prems =>
+ [ (cut_facts_tac prems 1),
+ (rtac iffI 1),
+ (REPEAT (eresolve_tac [asm_rl,iffE] 1 ORELSE mp_tac 1)) ]);
+
+
+(*** Unique existence. NOTE THAT the following 2 quantifications
+ EX!x such that [EX!y such that P(x,y)] (sequential)
+ EX!x,y such that P(x,y) (simultaneous)
+ do NOT mean the same thing. The parser treats EX!x y.P(x,y) as sequential.
+***)
+
+val ex1I = prove_goalw IFOLP.thy [ex1_def]
+ "[| p:P(a); !!x u.u:P(x) ==> f(u) : x=a |] ==> ?p:EX! x. P(x)"
+ (fn prems => [ (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)) ]);
+
+val ex1E = prove_goalw IFOLP.thy [ex1_def]
+ "[| p:EX! x.P(x); \
+\ !!x u v. [| u:P(x); v:ALL y. P(y) --> y=x |] ==> f(x,u,v):R |] ==>\
+\ ?a : R"
+ (fn prems =>
+ [ (cut_facts_tac prems 1),
+ (REPEAT (eresolve_tac [exE,conjE] 1 ORELSE ares_tac prems 1)) ]);
+
+
+(*** <-> congruence rules for simplification ***)
+
+(*Use iffE on a premise. For conj_cong, imp_cong, all_cong, ex_cong*)
+fun iff_tac prems i =
+ resolve_tac (prems RL [iffE]) i THEN
+ REPEAT1 (eresolve_tac [asm_rl,mp] i);
+
+val conj_cong = prove_goal IFOLP.thy
+ "[| p:P <-> P'; !!x.x:P' ==> q(x):Q <-> Q' |] ==> ?p:(P&Q) <-> (P'&Q')"
+ (fn prems =>
+ [ (cut_facts_tac prems 1),
+ (REPEAT (ares_tac [iffI,conjI] 1
+ ORELSE eresolve_tac [iffE,conjE,mp] 1
+ ORELSE iff_tac prems 1)) ]);
+
+val disj_cong = prove_goal IFOLP.thy
+ "[| p:P <-> P'; q:Q <-> Q' |] ==> ?p:(P|Q) <-> (P'|Q')"
+ (fn prems =>
+ [ (cut_facts_tac prems 1),
+ (REPEAT (eresolve_tac [iffE,disjE,disjI1,disjI2] 1
+ ORELSE ares_tac [iffI] 1
+ ORELSE mp_tac 1)) ]);
+
+val imp_cong = prove_goal IFOLP.thy
+ "[| p:P <-> P'; !!x.x:P' ==> q(x):Q <-> Q' |] ==> ?p:(P-->Q) <-> (P'-->Q')"
+ (fn prems =>
+ [ (cut_facts_tac prems 1),
+ (REPEAT (ares_tac [iffI,impI] 1
+ ORELSE eresolve_tac [iffE] 1
+ ORELSE mp_tac 1 ORELSE iff_tac prems 1)) ]);
+
+val iff_cong = prove_goal IFOLP.thy
+ "[| p:P <-> P'; q:Q <-> Q' |] ==> ?p:(P<->Q) <-> (P'<->Q')"
+ (fn prems =>
+ [ (cut_facts_tac prems 1),
+ (REPEAT (eresolve_tac [iffE] 1
+ ORELSE ares_tac [iffI] 1
+ ORELSE mp_tac 1)) ]);
+
+val not_cong = prove_goal IFOLP.thy
+ "p:P <-> P' ==> ?p:~P <-> ~P'"
+ (fn prems =>
+ [ (cut_facts_tac prems 1),
+ (REPEAT (ares_tac [iffI,notI] 1
+ ORELSE mp_tac 1
+ ORELSE eresolve_tac [iffE,notE] 1)) ]);
+
+val all_cong = prove_goal IFOLP.thy
+ "(!!x.f(x):P(x) <-> Q(x)) ==> ?p:(ALL x.P(x)) <-> (ALL x.Q(x))"
+ (fn prems =>
+ [ (REPEAT (ares_tac [iffI,allI] 1
+ ORELSE mp_tac 1
+ ORELSE eresolve_tac [allE] 1 ORELSE iff_tac prems 1)) ]);
+
+val ex_cong = prove_goal IFOLP.thy
+ "(!!x.f(x):P(x) <-> Q(x)) ==> ?p:(EX x.P(x)) <-> (EX x.Q(x))"
+ (fn prems =>
+ [ (REPEAT (eresolve_tac [exE] 1 ORELSE ares_tac [iffI,exI] 1
+ ORELSE mp_tac 1
+ ORELSE iff_tac prems 1)) ]);
+
+(*NOT PROVED
+val ex1_cong = prove_goal IFOLP.thy
+ "(!!x.f(x):P(x) <-> Q(x)) ==> ?p:(EX! x.P(x)) <-> (EX! x.Q(x))"
+ (fn prems =>
+ [ (REPEAT (eresolve_tac [ex1E, spec RS mp] 1 ORELSE ares_tac [iffI,ex1I] 1
+ ORELSE mp_tac 1
+ ORELSE iff_tac prems 1)) ]);
+*)
+
+(*** Equality rules ***)
+
+val refl = ieqI;
+
+val subst = prove_goal IFOLP.thy "[| p:a=b; q:P(a) |] ==> ?p : P(b)"
+ (fn [prem1,prem2] => [ rtac (prem2 RS rev_mp) 1, (rtac (prem1 RS ieqE) 1),
+ rtac impI 1, atac 1 ]);
+
+val sym = prove_goal IFOLP.thy "q:a=b ==> ?c:b=a"
+ (fn [major] => [ (rtac (major RS subst) 1), (rtac refl 1) ]);
+
+val trans = prove_goal IFOLP.thy "[| p:a=b; q:b=c |] ==> ?d:a=c"
+ (fn [prem1,prem2] => [ (rtac (prem2 RS subst) 1), (rtac prem1 1) ]);
+
+(** ~ b=a ==> ~ a=b **)
+val not_sym = prove_goal IFOLP.thy "p:~ b=a ==> ?q:~ a=b"
+ (fn [prem] => [ (rtac (prem RS contrapos) 1), (etac sym 1) ]);
+
+(*calling "standard" reduces maxidx to 0*)
+val ssubst = standard (sym RS subst);
+
+(*A special case of ex1E that would otherwise need quantifier expansion*)
+val ex1_equalsE = prove_goal IFOLP.thy
+ "[| p:EX! x.P(x); q:P(a); r:P(b) |] ==> ?d:a=b"
+ (fn prems =>
+ [ (cut_facts_tac prems 1),
+ (etac ex1E 1),
+ (rtac trans 1),
+ (rtac sym 2),
+ (REPEAT (eresolve_tac [asm_rl, spec RS mp] 1)) ]);
+
+(** Polymorphic congruence rules **)
+
+val subst_context = prove_goal IFOLP.thy
+ "[| p:a=b |] ==> ?d:t(a)=t(b)"
+ (fn prems=>
+ [ (resolve_tac (prems RL [ssubst]) 1),
+ (resolve_tac [refl] 1) ]);
+
+val subst_context2 = prove_goal IFOLP.thy
+ "[| p:a=b; q:c=d |] ==> ?p:t(a,c)=t(b,d)"
+ (fn prems=>
+ [ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
+
+val subst_context3 = prove_goal IFOLP.thy
+ "[| p:a=b; q:c=d; r:e=f |] ==> ?p:t(a,c,e)=t(b,d,f)"
+ (fn prems=>
+ [ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
+
+(*Useful with eresolve_tac for proving equalties from known equalities.
+ a = b
+ | |
+ c = d *)
+val box_equals = prove_goal IFOLP.thy
+ "[| p:a=b; q:a=c; r:b=d |] ==> ?p:c=d"
+ (fn prems=>
+ [ (resolve_tac [trans] 1),
+ (resolve_tac [trans] 1),
+ (resolve_tac [sym] 1),
+ (REPEAT (resolve_tac prems 1)) ]);
+
+(*Dual of box_equals: for proving equalities backwards*)
+val simp_equals = prove_goal IFOLP.thy
+ "[| p:a=c; q:b=d; r:c=d |] ==> ?p:a=b"
+ (fn prems=>
+ [ (resolve_tac [trans] 1),
+ (resolve_tac [trans] 1),
+ (REPEAT (resolve_tac (prems @ (prems RL [sym])) 1)) ]);
+
+(** Congruence rules for predicate letters **)
+
+val pred1_cong = prove_goal IFOLP.thy
+ "p:a=a' ==> ?p:P(a) <-> P(a')"
+ (fn prems =>
+ [ (cut_facts_tac prems 1),
+ (rtac iffI 1),
+ (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
+
+val pred2_cong = prove_goal IFOLP.thy
+ "[| p:a=a'; q:b=b' |] ==> ?p:P(a,b) <-> P(a',b')"
+ (fn prems =>
+ [ (cut_facts_tac prems 1),
+ (rtac iffI 1),
+ (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
+
+val pred3_cong = prove_goal IFOLP.thy
+ "[| p:a=a'; q:b=b'; r:c=c' |] ==> ?p:P(a,b,c) <-> P(a',b',c')"
+ (fn prems =>
+ [ (cut_facts_tac prems 1),
+ (rtac iffI 1),
+ (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
+
+(*special cases for free variables P, Q, R, S -- up to 3 arguments*)
+
+val pred_congs =
+ flat (map (fn c =>
+ map (fn th => read_instantiate [("P",c)] th)
+ [pred1_cong,pred2_cong,pred3_cong])
+ (explode"PQRS"));
+
+(*special case for the equality predicate!*)
+val eq_cong = read_instantiate [("P","op =")] pred2_cong;
+
+
+(*** Simplifications of assumed implications.
+ Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
+ used with mp_tac (restricted to atomic formulae) is COMPLETE for
+ intuitionistic propositional logic. See
+ R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
+ (preprint, University of St Andrews, 1991) ***)
+
+val conj_impE = prove_goal IFOLP.thy
+ "[| p:(P&Q)-->S; !!x.x:P-->(Q-->S) ==> q(x):R |] ==> ?p:R"
+ (fn major::prems=>
+ [ (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ]);
+
+val disj_impE = prove_goal IFOLP.thy
+ "[| p:(P|Q)-->S; !!x y.[| x:P-->S; y:Q-->S |] ==> q(x,y):R |] ==> ?p:R"
+ (fn major::prems=>
+ [ (DEPTH_SOLVE (ares_tac ([disjI1, disjI2, impI, major RS mp]@prems) 1)) ]);
+
+(*Simplifies the implication. Classical version is stronger.
+ Still UNSAFE since Q must be provable -- backtracking needed. *)
+val imp_impE = prove_goal IFOLP.thy
+ "[| p:(P-->Q)-->S; !!x y.[| x:P; y:Q-->S |] ==> q(x,y):Q; !!x.x:S ==> r(x):R |] ==> \
+\ ?p:R"
+ (fn major::prems=>
+ [ (REPEAT (ares_tac ([impI, major RS mp]@prems) 1)) ]);
+
+(*Simplifies the implication. Classical version is stronger.
+ Still UNSAFE since ~P must be provable -- backtracking needed. *)
+val not_impE = prove_goal IFOLP.thy
+ "[| p:~P --> S; !!y.y:P ==> q(y):False; !!y.y:S ==> r(y):R |] ==> ?p:R"
+ (fn major::prems=>
+ [ (REPEAT (ares_tac ([notI, impI, major RS mp]@prems) 1)) ]);
+
+(*Simplifies the implication. UNSAFE. *)
+val iff_impE = prove_goal IFOLP.thy
+ "[| p:(P<->Q)-->S; !!x y.[| x:P; y:Q-->S |] ==> q(x,y):Q; \
+\ !!x y.[| x:Q; y:P-->S |] ==> r(x,y):P; !!x.x:S ==> s(x):R |] ==> ?p:R"
+ (fn major::prems=>
+ [ (REPEAT (ares_tac ([iffI, impI, major RS mp]@prems) 1)) ]);
+
+(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
+val all_impE = prove_goal IFOLP.thy
+ "[| p:(ALL x.P(x))-->S; !!x.q:P(x); !!y.y:S ==> r(y):R |] ==> ?p:R"
+ (fn major::prems=>
+ [ (REPEAT (ares_tac ([allI, impI, major RS mp]@prems) 1)) ]);
+
+(*Unsafe: (EX x.P(x))-->S is equivalent to ALL x.P(x)-->S. *)
+val ex_impE = prove_goal IFOLP.thy
+ "[| p:(EX x.P(x))-->S; !!y.y:P(a)-->S ==> q(y):R |] ==> ?p:R"
+ (fn major::prems=>
+ [ (REPEAT (ares_tac ([exI, impI, major RS mp]@prems) 1)) ]);
+
+end;
+
+open IFOLP_Lemmas;
+