(* 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";
qed_goalw "TrueI" (the_context ()) [True_def] "True"
(fn _ => [ (REPEAT (ares_tac [impI] 1)) ]);
(*** Sequent-style elimination rules for & --> and ALL ***)
qed_goal "conjE" (the_context ())
"[| 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" (the_context ())
"[| P-->Q; P; Q ==> R |] ==> R"
(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
qed_goal "allE" (the_context ())
"[| 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" (the_context ())
"[| 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" (the_context ()) [not_def] "(P ==> False) ==> ~P"
(fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]);
qed_goalw "notE" (the_context ()) [not_def] "[| ~P; P |] ==> R"
(fn prems=>
[ (resolve_tac [mp RS FalseE] 1),
(REPEAT (resolve_tac prems 1)) ]);
qed_goal "rev_notE" (the_context ()) "!!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" (the_context ())
"[| ~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" (the_context ()) "[| P; P --> Q |] ==> Q"
(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
(*Contrapositive of an inference rule*)
qed_goal "contrapos" (the_context ()) "[| ~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" (the_context ()) [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" (the_context ()) [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" (the_context ()) [iff_def] "[| P <-> Q; P |] ==> Q"
(fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
qed_goalw "iffD2" (the_context ()) [iff_def] "[| P <-> Q; Q |] ==> P"
(fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
qed_goal "rev_iffD1" (the_context ()) "!!P. [| P; P <-> Q |] ==> Q"
(fn _ => [etac iffD1 1, assume_tac 1]);
qed_goal "rev_iffD2" (the_context ()) "!!P. [| Q; P <-> Q |] ==> P"
(fn _ => [etac iffD2 1, assume_tac 1]);
qed_goal "iff_refl" (the_context ()) "P <-> P"
(fn _ => [ (REPEAT (ares_tac [iffI] 1)) ]);
qed_goal "iff_sym" (the_context ()) "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" (the_context ())
"!!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" (the_context ()) [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" (the_context ())
"[| 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" (the_context ()) [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" (the_context ())
"[| 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" (the_context ())
"[| 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" (the_context ())
"[| 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" (the_context ())
"[| 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" (the_context ())
"[| 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" (the_context ())
"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" (the_context ())
"(!!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" (the_context ())
"(!!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" (the_context ())
"(!!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" (the_context ()) "a=b ==> b=a"
(fn [major] => [ (rtac (major RS subst) 1), (rtac refl 1) ]);
qed_goal "trans" (the_context ()) "[| a=b; b=c |] ==> a=c"
(fn [prem1,prem2] => [ (rtac (prem2 RS subst) 1), (rtac prem1 1) ]);
(** ~ 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: 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" (the_context ())
"[| 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" (the_context ())
"[| a=b |] ==> t(a)=t(b)"
(fn prems=>
[ (resolve_tac (prems RL [ssubst]) 1),
(rtac refl 1) ]);
qed_goal "subst_context2" (the_context ())
"[| a=b; c=d |] ==> t(a,c)=t(b,d)"
(fn prems=>
[ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
qed_goal "subst_context3" (the_context ())
"[| 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" (the_context ())
"[| 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" (the_context ())
"[| 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" (the_context ())
"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" (the_context ())
"[| 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" (the_context ())
"[| 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!*)
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) ***)
qed_goal "conj_impE" (the_context ())
"[| (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" (the_context ())
"[| (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" (the_context ())
"[| (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" (the_context ())
"[| ~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" (the_context ())
"[| (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" (the_context ())
"[| (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" (the_context ())
"[| (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 (the_context ()) "[| 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;