(* Title: FOL/IFOL_lemmas.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Tactics and lemmas for theory IFOL (intuitionistic first-order logic).
*)
(* ML bindings *)
val refl = thm "refl";
val subst = thm "subst";
val conjI = thm "conjI";
val conjunct1 = thm "conjunct1";
val conjunct2 = thm "conjunct2";
val disjI1 = thm "disjI1";
val disjI2 = thm "disjI2";
val disjE = thm "disjE";
val impI = thm "impI";
val mp = thm "mp";
val FalseE = thm "FalseE";
val True_def = thm "True_def";
val not_def = thm "not_def";
val iff_def = thm "iff_def";
val ex1_def = thm "ex1_def";
val allI = thm "allI";
val spec = thm "spec";
val exI = thm "exI";
val exE = thm "exE";
val eq_reflection = thm "eq_reflection";
val iff_reflection = thm "iff_reflection";
Goalw [True_def] "True";
by (REPEAT (ares_tac [impI] 1)) ;
qed "TrueI";
(*** Sequent-style elimination rules for & --> and ALL ***)
val major::prems = Goal
"[| P&Q; [| P; Q |] ==> R |] ==> R";
by (resolve_tac prems 1);
by (rtac (major RS conjunct1) 1);
by (rtac (major RS conjunct2) 1);
qed "conjE";
val major::prems = Goal
"[| P-->Q; P; Q ==> R |] ==> R";
by (resolve_tac prems 1);
by (rtac (major RS mp) 1);
by (resolve_tac prems 1);
qed "impE";
val major::prems = Goal
"[| ALL x. P(x); P(x) ==> R |] ==> R";
by (resolve_tac prems 1);
by (rtac (major RS spec) 1);
qed "allE";
(*Duplicates the quantifier; for use with eresolve_tac*)
val major::prems = Goal
"[| ALL x. P(x); [| P(x); ALL x. P(x) |] ==> R \
\ |] ==> R";
by (resolve_tac prems 1);
by (rtac (major RS spec) 1);
by (rtac major 1);
qed "all_dupE";
(*** Negation rules, which translate between ~P and P-->False ***)
val prems = Goalw [not_def] "(P ==> False) ==> ~P";
by (REPEAT (ares_tac (prems@[impI]) 1)) ;
qed "notI";
Goalw [not_def] "[| ~P; P |] ==> R";
by (etac (mp RS FalseE) 1);
by (assume_tac 1);
qed "notE";
Goal "[| P; ~P |] ==> R";
by (etac notE 1);
by (assume_tac 1);
qed "rev_notE";
(*This is useful with the special implication rules for each kind of P. *)
val prems = Goal
"[| ~P; (P-->False) ==> Q |] ==> Q";
by (REPEAT (ares_tac (prems@[impI,notE]) 1)) ;
qed "not_to_imp";
(* 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 *)
Goal "[| P; P --> Q |] ==> Q";
by (etac mp 1);
by (assume_tac 1);
qed "rev_mp";
(*Contrapositive of an inference rule*)
val [major,minor]= Goal "[| ~Q; P==>Q |] ==> ~P";
by (rtac (major RS notE RS notI) 1);
by (etac minor 1) ;
qed "contrapos";
(*** 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 ***)
val prems = Goalw [iff_def]
"[| P ==> Q; Q ==> P |] ==> P<->Q";
by (REPEAT (ares_tac (prems@[conjI, impI]) 1)) ;
qed "iffI";
(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
val prems = Goalw [iff_def]
"[| P <-> Q; [| P-->Q; Q-->P |] ==> R |] ==> R";
by (rtac conjE 1);
by (REPEAT (ares_tac prems 1)) ;
qed "iffE";
(* Destruct rules for <-> similar to Modus Ponens *)
Goalw [iff_def] "[| P <-> Q; P |] ==> Q";
by (etac (conjunct1 RS mp) 1);
by (assume_tac 1);
qed "iffD1";
val prems = Goalw [iff_def] "[| P <-> Q; Q |] ==> P";
by (etac (conjunct2 RS mp) 1);
by (assume_tac 1);
qed "iffD2";
Goal "[| P; P <-> Q |] ==> Q";
by (etac iffD1 1);
by (assume_tac 1);
qed "rev_iffD1";
Goal "[| Q; P <-> Q |] ==> P";
by (etac iffD2 1);
by (assume_tac 1);
qed "rev_iffD2";
Goal "P <-> P";
by (REPEAT (ares_tac [iffI] 1)) ;
qed "iff_refl";
Goal "Q <-> P ==> P <-> Q";
by (etac iffE 1);
by (rtac iffI 1);
by (REPEAT (eresolve_tac [asm_rl,mp] 1)) ;
qed "iff_sym";
Goal "[| P <-> Q; Q<-> R |] ==> P <-> R";
by (rtac iffI 1);
by (REPEAT (eresolve_tac [asm_rl,iffE] 1 ORELSE mp_tac 1)) ;
qed "iff_trans";
(*** 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 prems = Goalw [ex1_def]
"[| P(a); !!x. P(x) ==> x=a |] ==> EX! x. P(x)";
by (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)) ;
qed "ex1I";
(*Sometimes easier to use: the premises have no shared variables. Safe!*)
val [ex,eq] = Goal
"[| EX x. P(x); !!x y. [| P(x); P(y) |] ==> x=y |] ==> EX! x. P(x)";
by (rtac (ex RS exE) 1);
by (REPEAT (ares_tac [ex1I,eq] 1)) ;
qed "ex_ex1I";
val prems = Goalw [ex1_def]
"[| EX! x. P(x); !!x. [| P(x); ALL y. P(y) --> y=x |] ==> R |] ==> R";
by (cut_facts_tac prems 1);
by (REPEAT (eresolve_tac [exE,conjE] 1 ORELSE ares_tac prems 1)) ;
qed "ex1E";
(*** <-> 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 prems = Goal
"[| P <-> P'; P' ==> Q <-> Q' |] ==> (P&Q) <-> (P'&Q')";
by (cut_facts_tac prems 1);
by (REPEAT (ares_tac [iffI,conjI] 1
ORELSE eresolve_tac [iffE,conjE,mp] 1
ORELSE iff_tac prems 1)) ;
qed "conj_cong";
(*Reversed congruence rule! Used in ZF/Order*)
val prems = Goal
"[| P <-> P'; P' ==> Q <-> Q' |] ==> (Q&P) <-> (Q'&P')";
by (cut_facts_tac prems 1);
by (REPEAT (ares_tac [iffI,conjI] 1
ORELSE eresolve_tac [iffE,conjE,mp] 1 ORELSE iff_tac prems 1)) ;
qed "conj_cong2";
val prems = Goal
"[| P <-> P'; Q <-> Q' |] ==> (P|Q) <-> (P'|Q')";
by (cut_facts_tac prems 1);
by (REPEAT (eresolve_tac [iffE,disjE,disjI1,disjI2] 1
ORELSE ares_tac [iffI] 1 ORELSE mp_tac 1)) ;
qed "disj_cong";
val prems = Goal
"[| P <-> P'; P' ==> Q <-> Q' |] ==> (P-->Q) <-> (P'-->Q')";
by (cut_facts_tac prems 1);
by (REPEAT (ares_tac [iffI,impI] 1
ORELSE etac iffE 1 ORELSE mp_tac 1 ORELSE iff_tac prems 1)) ;
qed "imp_cong";
val prems = Goal
"[| P <-> P'; Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')";
by (cut_facts_tac prems 1);
by (REPEAT (etac iffE 1 ORELSE ares_tac [iffI] 1 ORELSE mp_tac 1)) ;
qed "iff_cong";
val prems = Goal
"P <-> P' ==> ~P <-> ~P'";
by (cut_facts_tac prems 1);
by (REPEAT (ares_tac [iffI,notI] 1
ORELSE mp_tac 1 ORELSE eresolve_tac [iffE,notE] 1)) ;
qed "not_cong";
val prems = Goal
"(!!x. P(x) <-> Q(x)) ==> (ALL x. P(x)) <-> (ALL x. Q(x))";
by (REPEAT (ares_tac [iffI,allI] 1
ORELSE mp_tac 1 ORELSE etac allE 1 ORELSE iff_tac prems 1)) ;
qed "all_cong";
val prems = Goal
"(!!x. P(x) <-> Q(x)) ==> (EX x. P(x)) <-> (EX x. Q(x))";
by (REPEAT (etac exE 1 ORELSE ares_tac [iffI,exI] 1
ORELSE mp_tac 1 ORELSE iff_tac prems 1)) ;
qed "ex_cong";
val prems = Goal
"(!!x. P(x) <-> Q(x)) ==> (EX! x. P(x)) <-> (EX! x. Q(x))";
by (REPEAT (eresolve_tac [ex1E, spec RS mp] 1
ORELSE ares_tac [iffI,ex1I] 1 ORELSE mp_tac 1
ORELSE iff_tac prems 1)) ;
qed "ex1_cong";
(*** Equality rules ***)
Goal "a=b ==> b=a";
by (etac subst 1);
by (rtac refl 1) ;
qed "sym";
Goal "[| a=b; b=c |] ==> a=c";
by (etac subst 1 THEN assume_tac 1) ;
qed "trans";
(** ~ b=a ==> ~ a=b **)
bind_thm ("not_sym", hd (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 (the_context()) "(A == B) ==> A <-> B";
by (rewrite_goals_tac prems);
by (rtac iff_refl 1);
qed "def_imp_iff";
val prems = goal (the_context()) "(A == B) ==> A = B";
by (rewrite_goals_tac prems);
by (rtac refl 1);
qed "meta_eq_to_obj_eq";
(*substitution*)
bind_thm ("ssubst", sym RS subst);
(*A special case of ex1E that would otherwise need quantifier expansion*)
val prems = Goal
"[| EX! x. P(x); P(a); P(b) |] ==> a=b";
by (cut_facts_tac prems 1);
by (etac ex1E 1);
by (rtac trans 1);
by (rtac sym 2);
by (REPEAT (eresolve_tac [asm_rl, spec RS mp] 1)) ;
qed "ex1_equalsE";
(** Polymorphic congruence rules **)
Goal "[| a=b |] ==> t(a)=t(b)";
by (etac ssubst 1);
by (rtac refl 1) ;
qed "subst_context";
Goal "[| a=b; c=d |] ==> t(a,c)=t(b,d)";
by (REPEAT (etac ssubst 1));
by (rtac refl 1) ;
qed "subst_context2";
Goal "[| a=b; c=d; e=f |] ==> t(a,c,e)=t(b,d,f)";
by (REPEAT (etac ssubst 1));
by (rtac refl 1) ;
qed "subst_context3";
(*Useful with eresolve_tac for proving equalties from known equalities.
a = b
| |
c = d *)
Goal "[| a=b; a=c; b=d |] ==> c=d";
by (rtac trans 1);
by (rtac trans 1);
by (rtac sym 1);
by (REPEAT (assume_tac 1));
qed "box_equals";
(*Dual of box_equals: for proving equalities backwards*)
Goal "[| a=c; b=d; c=d |] ==> a=b";
by (rtac trans 1);
by (rtac trans 1);
by (REPEAT (assume_tac 1));
by (etac sym 1);
qed "simp_equals";
(** Congruence rules for predicate letters **)
Goal "a=a' ==> P(a) <-> P(a')";
by (rtac iffI 1);
by (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ;
qed "pred1_cong";
Goal "[| a=a'; b=b' |] ==> P(a,b) <-> P(a',b')";
by (rtac iffI 1);
by (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ;
qed "pred2_cong";
Goal "[| a=a'; b=b'; c=c' |] ==> P(a,b,c) <-> P(a',b',c')";
by (rtac iffI 1);
by (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ;
qed "pred3_cong";
(*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!*)
bind_thm ("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 major::prems= Goal
"[| (P&Q)-->S; P-->(Q-->S) ==> R |] ==> R";
by (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ;
qed "conj_impE";
val major::prems= Goal
"[| (P|Q)-->S; [| P-->S; Q-->S |] ==> R |] ==> R";
by (DEPTH_SOLVE (ares_tac ([disjI1, disjI2, impI, major RS mp]@prems) 1)) ;
qed "disj_impE";
(*Simplifies the implication. Classical version is stronger.
Still UNSAFE since Q must be provable -- backtracking needed. *)
val major::prems= Goal
"[| (P-->Q)-->S; [| P; Q-->S |] ==> Q; S ==> R |] ==> R";
by (REPEAT (ares_tac ([impI, major RS mp]@prems) 1)) ;
qed "imp_impE";
(*Simplifies the implication. Classical version is stronger.
Still UNSAFE since ~P must be provable -- backtracking needed. *)
val major::prems= Goal
"[| ~P --> S; P ==> False; S ==> R |] ==> R";
by (REPEAT (ares_tac ([notI, impI, major RS mp]@prems) 1)) ;
qed "not_impE";
(*Simplifies the implication. UNSAFE. *)
val major::prems= Goal
"[| (P<->Q)-->S; [| P; Q-->S |] ==> Q; [| Q; P-->S |] ==> P; \
\ S ==> R |] ==> R";
by (REPEAT (ares_tac ([iffI, impI, major RS mp]@prems) 1)) ;
qed "iff_impE";
(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
val major::prems= Goal
"[| (ALL x. P(x))-->S; !!x. P(x); S ==> R |] ==> R";
by (REPEAT (ares_tac ([allI, impI, major RS mp]@prems) 1)) ;
qed "all_impE";
(*Unsafe: (EX x.P(x))-->S is equivalent to ALL x.P(x)-->S. *)
val major::prems= Goal
"[| (EX x. P(x))-->S; P(x)-->S ==> R |] ==> R";
by (REPEAT (ares_tac ([exI, impI, major RS mp]@prems) 1)) ;
qed "ex_impE";
(*** Courtesy of Krzysztof Grabczewski ***)
val major::prems = Goal "[| 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));
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;