--- a/src/FOL/IFOL.ML Wed Aug 25 20:42:01 1999 +0200
+++ b/src/FOL/IFOL.ML Wed Aug 25 20:45:19 1999 +0200
@@ -1,454 +1,28 @@
-(* Title: FOL/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)
-*)
-
-qed_goalw "TrueI" IFOL.thy [True_def] "True"
- (fn _ => [ (REPEAT (ares_tac [impI] 1)) ]);
-
-(*** Sequent-style elimination rules for & --> and ALL ***)
-
-qed_goal "conjE" IFOL.thy
- "[| P&Q; [| P; Q |] ==> R |] ==> R"
- (fn prems=>
- [ (REPEAT (resolve_tac prems 1
- ORELSE (resolve_tac [conjunct1, conjunct2] 1 THEN
- resolve_tac prems 1))) ]);
-
-qed_goal "impE" IFOL.thy
- "[| P-->Q; P; Q ==> R |] ==> R"
- (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
-
-qed_goal "allE" IFOL.thy
- "[| ALL x. P(x); P(x) ==> R |] ==> R"
- (fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
-
-(*Duplicates the quantifier; for use with eresolve_tac*)
-qed_goal "all_dupE" IFOL.thy
- "[| ALL x. P(x); [| P(x); ALL x. P(x) |] ==> R \
-\ |] ==> R"
- (fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
-
-
-(*** Negation rules, which translate between ~P and P-->False ***)
-
-qed_goalw "notI" IFOL.thy [not_def] "(P ==> False) ==> ~P"
- (fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]);
-
-qed_goalw "notE" IFOL.thy [not_def] "[| ~P; P |] ==> R"
- (fn prems=>
- [ (resolve_tac [mp RS FalseE] 1),
- (REPEAT (resolve_tac prems 1)) ]);
-
-qed_goal "rev_notE" IFOL.thy "!!P R. [| P; ~P |] ==> R"
- (fn _ => [REPEAT (ares_tac [notE] 1)]);
-
-(*This is useful with the special implication rules for each kind of P. *)
-qed_goal "not_to_imp" IFOL.thy
- "[| ~P; (P-->False) ==> Q |] ==> Q"
- (fn prems=> [ (REPEAT (ares_tac (prems@[impI,notE]) 1)) ]);
-
-(* For substitution into 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 *)
-qed_goal "rev_mp" IFOL.thy "[| P; P --> Q |] ==> Q"
- (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
-
-(*Contrapositive of an inference rule*)
-qed_goal "contrapos" IFOL.thy "[| ~Q; P==>Q |] ==> ~P"
- (fn [major,minor]=>
- [ (rtac (major RS notE RS notI) 1),
- (etac minor 1) ]);
-
-
-(*** Modus Ponens Tactics ***)
-
-(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
-fun mp_tac i = eresolve_tac [notE,impE] i THEN assume_tac i;
-
-(*Like mp_tac but instantiates no variables*)
-fun eq_mp_tac i = eresolve_tac [notE,impE] i THEN eq_assume_tac i;
-
-
-(*** If-and-only-if ***)
-
-qed_goalw "iffI" IFOL.thy [iff_def]
- "[| P ==> Q; Q ==> P |] ==> P<->Q"
- (fn prems=> [ (REPEAT (ares_tac (prems@[conjI, impI]) 1)) ]);
-
-
-(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
-qed_goalw "iffE" IFOL.thy [iff_def]
- "[| P <-> Q; [| P-->Q; Q-->P |] ==> R |] ==> R"
- (fn prems => [ (rtac conjE 1), (REPEAT (ares_tac prems 1)) ]);
-
-(* Destruct rules for <-> similar to Modus Ponens *)
-
-qed_goalw "iffD1" IFOL.thy [iff_def] "[| P <-> Q; P |] ==> Q"
- (fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
-
-qed_goalw "iffD2" IFOL.thy [iff_def] "[| P <-> Q; Q |] ==> P"
- (fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
-
-qed_goal "rev_iffD1" IFOL.thy "!!P. [| P; P <-> Q |] ==> Q"
- (fn _ => [etac iffD1 1, assume_tac 1]);
-
-qed_goal "rev_iffD2" IFOL.thy "!!P. [| Q; P <-> Q |] ==> P"
- (fn _ => [etac iffD2 1, assume_tac 1]);
-
-qed_goal "iff_refl" IFOL.thy "P <-> P"
- (fn _ => [ (REPEAT (ares_tac [iffI] 1)) ]);
-
-qed_goal "iff_sym" IFOL.thy "Q <-> P ==> P <-> Q"
- (fn [major] =>
- [ (rtac (major RS iffE) 1),
- (rtac iffI 1),
- (REPEAT (eresolve_tac [asm_rl,mp] 1)) ]);
-
-qed_goal "iff_trans" IFOL.thy
- "!!P Q R. [| P <-> Q; Q<-> R |] ==> P <-> R"
- (fn _ =>
- [ (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.
-***)
-
-qed_goalw "ex1I" IFOL.thy [ex1_def]
- "[| P(a); !!x. P(x) ==> x=a |] ==> EX! x. P(x)"
- (fn prems => [ (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)) ]);
-
-(*Sometimes easier to use: the premises have no shared variables. Safe!*)
-qed_goal "ex_ex1I" IFOL.thy
- "[| EX x. P(x); !!x y. [| P(x); P(y) |] ==> x=y |] ==> EX! x. P(x)"
- (fn [ex,eq] => [ (rtac (ex RS exE) 1),
- (REPEAT (ares_tac [ex1I,eq] 1)) ]);
-
-qed_goalw "ex1E" IFOL.thy [ex1_def]
- "[| EX! x. P(x); !!x. [| P(x); ALL y. P(y) --> y=x |] ==> R |] ==> 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);
-
-qed_goal "conj_cong" IFOL.thy
- "[| P <-> P'; P' ==> Q <-> Q' |] ==> (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)) ]);
-
-(*Reversed congruence rule! Used in ZF/Order*)
-qed_goal "conj_cong2" IFOL.thy
- "[| P <-> P'; P' ==> Q <-> Q' |] ==> (Q&P) <-> (Q'&P')"
- (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)) ]);
-
-qed_goal "disj_cong" IFOL.thy
- "[| P <-> P'; Q <-> Q' |] ==> (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)) ]);
-
-qed_goal "imp_cong" IFOL.thy
- "[| P <-> P'; P' ==> Q <-> Q' |] ==> (P-->Q) <-> (P'-->Q')"
- (fn prems =>
- [ (cut_facts_tac prems 1),
- (REPEAT (ares_tac [iffI,impI] 1
- ORELSE etac iffE 1
- ORELSE mp_tac 1 ORELSE iff_tac prems 1)) ]);
-
-qed_goal "iff_cong" IFOL.thy
- "[| P <-> P'; Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
- (fn prems =>
- [ (cut_facts_tac prems 1),
- (REPEAT (etac iffE 1
- ORELSE ares_tac [iffI] 1
- ORELSE mp_tac 1)) ]);
-
-qed_goal "not_cong" IFOL.thy
- "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)) ]);
-
-qed_goal "all_cong" IFOL.thy
- "(!!x. P(x) <-> Q(x)) ==> (ALL x. P(x)) <-> (ALL x. Q(x))"
- (fn prems =>
- [ (REPEAT (ares_tac [iffI,allI] 1
- ORELSE mp_tac 1
- ORELSE etac allE 1 ORELSE iff_tac prems 1)) ]);
-
-qed_goal "ex_cong" IFOL.thy
- "(!!x. P(x) <-> Q(x)) ==> (EX x. P(x)) <-> (EX x. Q(x))"
- (fn prems =>
- [ (REPEAT (etac exE 1 ORELSE ares_tac [iffI,exI] 1
- ORELSE mp_tac 1
- ORELSE iff_tac prems 1)) ]);
-
-qed_goal "ex1_cong" IFOL.thy
- "(!!x. P(x) <-> Q(x)) ==> (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 ***)
-
-qed_goal "sym" IFOL.thy "a=b ==> b=a"
- (fn [major] => [ (rtac (major RS subst) 1), (rtac refl 1) ]);
-
-qed_goal "trans" IFOL.thy "[| a=b; b=c |] ==> a=c"
- (fn [prem1,prem2] => [ (rtac (prem2 RS subst) 1), (rtac prem1 1) ]);
-
-(** ~ b=a ==> ~ a=b **)
-val [not_sym] = compose(sym,2,contrapos);
-
-
-(* Two theorms for rewriting only one instance of a definition:
- the first for definitions of formulae and the second for terms *)
-
-val prems = goal IFOL.thy "(A == B) ==> A <-> B";
-by (rewrite_goals_tac prems);
-by (rtac iff_refl 1);
-qed "def_imp_iff";
-
-val prems = goal IFOL.thy "(A == B) ==> A = B";
-by (rewrite_goals_tac prems);
-by (rtac refl 1);
-qed "meta_eq_to_obj_eq";
-
-
-(*Substitution: rule and tactic*)
-bind_thm ("ssubst", sym RS subst);
-
-(*Apply an equality or definition ONCE.
- Fails unless the substitution has an effect*)
-fun stac th =
- let val th' = th RS meta_eq_to_obj_eq handle THM _ => th
- in CHANGED_GOAL (rtac (th' RS ssubst))
- end;
-
-(*A special case of ex1E that would otherwise need quantifier expansion*)
-qed_goal "ex1_equalsE" IFOL.thy
- "[| EX! x. P(x); P(a); P(b) |] ==> 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 **)
-
-qed_goal "subst_context" IFOL.thy
- "[| a=b |] ==> t(a)=t(b)"
- (fn prems=>
- [ (resolve_tac (prems RL [ssubst]) 1),
- (rtac refl 1) ]);
-
-qed_goal "subst_context2" IFOL.thy
- "[| a=b; c=d |] ==> t(a,c)=t(b,d)"
- (fn prems=>
- [ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
-
-qed_goal "subst_context3" IFOL.thy
- "[| a=b; c=d; e=f |] ==> 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 *)
-qed_goal "box_equals" IFOL.thy
- "[| a=b; a=c; b=d |] ==> c=d"
- (fn prems=>
- [ (rtac trans 1),
- (rtac trans 1),
- (rtac sym 1),
- (REPEAT (resolve_tac prems 1)) ]);
-
-(*Dual of box_equals: for proving equalities backwards*)
-qed_goal "simp_equals" IFOL.thy
- "[| a=c; b=d; c=d |] ==> a=b"
- (fn prems=>
- [ (rtac trans 1),
- (rtac trans 1),
- (REPEAT (resolve_tac (prems @ (prems RL [sym])) 1)) ]);
-
-(** Congruence rules for predicate letters **)
-
-qed_goal "pred1_cong" IFOL.thy
- "a=a' ==> P(a) <-> P(a')"
- (fn prems =>
- [ (cut_facts_tac prems 1),
- (rtac iffI 1),
- (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
-
-qed_goal "pred2_cong" IFOL.thy
- "[| a=a'; b=b' |] ==> 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)) ]);
-
-qed_goal "pred3_cong" IFOL.thy
- "[| a=a'; b=b'; c=c' |] ==> 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) ***)
-
-qed_goal "conj_impE" IFOL.thy
- "[| (P&Q)-->S; P-->(Q-->S) ==> R |] ==> R"
- (fn major::prems=>
- [ (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ]);
-
-qed_goal "disj_impE" IFOL.thy
- "[| (P|Q)-->S; [| P-->S; Q-->S |] ==> R |] ==> 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. *)
-qed_goal "imp_impE" IFOL.thy
- "[| (P-->Q)-->S; [| P; Q-->S |] ==> Q; S ==> R |] ==> 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. *)
-qed_goal "not_impE" IFOL.thy
- "[| ~P --> S; P ==> False; S ==> R |] ==> R"
- (fn major::prems=>
- [ (REPEAT (ares_tac ([notI, impI, major RS mp]@prems) 1)) ]);
-
-(*Simplifies the implication. UNSAFE. *)
-qed_goal "iff_impE" IFOL.thy
- "[| (P<->Q)-->S; [| P; Q-->S |] ==> Q; [| Q; P-->S |] ==> P; \
-\ S ==> R |] ==> 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*)
-qed_goal "all_impE" IFOL.thy
- "[| (ALL x. P(x))-->S; !!x. P(x); S ==> R |] ==> 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. *)
-qed_goal "ex_impE" IFOL.thy
- "[| (EX x. P(x))-->S; P(x)-->S ==> R |] ==> R"
- (fn major::prems=>
- [ (REPEAT (ares_tac ([exI, impI, major RS mp]@prems) 1)) ]);
-
-(*** Courtesy of Krzysztof Grabczewski ***)
-
-val major::prems = goal IFOL.thy "[| P|Q; P==>R; Q==>S |] ==> R|S";
-by (rtac (major RS disjE) 1);
-by (REPEAT (eresolve_tac (prems RL [disjI1, disjI2]) 1));
-qed "disj_imp_disj";
-
-
-(** strip ALL and --> from proved goal while preserving ALL-bound var names **)
-
-fun make_new_spec rl =
- (* Use a crazy name to avoid forall_intr failing because of
- duplicate variable name *)
- let val myspec = read_instantiate [("P","?wlzickd")] rl
- val _ $ (_ $ (vx as Var(_,vxT))) = concl_of myspec;
- val cvx = cterm_of (#sign(rep_thm myspec)) vx
- in (vxT, forall_intr cvx myspec) end;
-
-local
-
-val (specT, spec') = make_new_spec spec
-
-in
-
-fun RSspec th =
- (case concl_of th of
- _ $ (Const("All",_) $ Abs(a,_,_)) =>
- let val ca = cterm_of (#sign(rep_thm th)) (Var((a,0),specT))
- in th RS forall_elim ca spec' end
- | _ => raise THM("RSspec",0,[th]));
-
-fun RSmp th =
- (case concl_of th of
- _ $ (Const("op -->",_)$_$_) => th RS mp
- | _ => raise THM("RSmp",0,[th]));
-
-fun normalize_thm funs =
- let fun trans [] th = th
- | trans (f::fs) th = (trans funs (f th)) handle THM _ => trans fs th
- in trans funs end;
-
-fun qed_spec_mp name =
- let val thm = normalize_thm [RSspec,RSmp] (result())
- in bind_thm(name, thm) end;
-
-fun qed_goal_spec_mp name thy s p =
- bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goal thy s p));
-
-fun qed_goalw_spec_mp name thy defs s p =
- bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goalw thy defs s p));
-
+structure IFOL =
+struct
+ val thy = the_context ();
+ val refl = refl;
+ val subst = subst;
+ val conjI = conjI;
+ val conjunct1 = conjunct1;
+ val conjunct2 = conjunct2;
+ val disjI1 = disjI1;
+ val disjI2 = disjI2;
+ val disjE = disjE;
+ val impI = impI;
+ val mp = mp;
+ val FalseE = FalseE;
+ val True_def = True_def;
+ val not_def = not_def;
+ val iff_def = iff_def;
+ val ex1_def = ex1_def;
+ val allI = allI;
+ val spec = spec;
+ val exI = exI;
+ val exE = exE;
+ val eq_reflection = eq_reflection;
+ val iff_reflection = iff_reflection;
end;
-
-(* attributes *)
-
-local
-
-fun gen_rulify x = Attrib.no_args (Drule.rule_attribute (K (normalize_thm [RSspec, RSmp]))) x;
-
-in
-
-val attrib_setup =
- [Attrib.add_attributes
- [("rulify", (gen_rulify, gen_rulify), "put theorem into standard rule form")]];
-
-end;
+open IFOL;