src/FOL/ifol.ML

Initial revision

(*  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)
*)

open IFOL;

signature IFOL_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 eq_mp_tac: int -> tactic
  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 rev_mp: thm
  val simp_equals: 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
  end;


structure IFOL_Lemmas : IFOL_LEMMAS =
struct

val TrueI = prove_goalw IFOL.thy [True_def] "True"
 (fn _ => [ (REPEAT (ares_tac [impI] 1)) ]);

(*** Sequent-style elimination rules for & --> and ALL ***)

val conjE = prove_goal 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))) ]);

val impE = prove_goal IFOL.thy 
    "[| P-->Q;  P;  Q ==> R |] ==> R"
 (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);

val allE = prove_goal 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*)
val all_dupE = prove_goal 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 ***)

val notI = prove_goalw IFOL.thy [not_def] "(P ==> False) ==> ~P"
 (fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]);

val notE = prove_goalw IFOL.thy [not_def] "[| ~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 IFOL.thy 
    "[| ~P;  (P-->False) ==> Q |] ==> 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 IFOL.thy "[| P;  P --> Q |] ==> Q"
 (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);


(*Contrapositive of an inference rule*)
val contrapos = prove_goal 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 ***)

val iffI = prove_goalw 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) *)
val iffE = prove_goalw IFOL.thy [iff_def]
    "[| P <-> Q;  [| P-->Q; Q-->P |] ==> R |] ==> R"
 (fn prems => [ (resolve_tac [conjE] 1), (REPEAT (ares_tac prems 1)) ]);

(* Destruct rules for <-> similar to Modus Ponens *)

val iffD1 = prove_goalw IFOL.thy [iff_def] "[| P <-> Q;  P |] ==> Q"
 (fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);

val iffD2 = prove_goalw IFOL.thy [iff_def] "[| P <-> Q;  Q |] ==> P"
 (fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);

val iff_refl = prove_goal IFOL.thy "P <-> P"
 (fn _ => [ (REPEAT (ares_tac [iffI] 1)) ]);

val iff_sym = prove_goal IFOL.thy "Q <-> 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 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.
***)

val ex1I = prove_goalw 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)) ]);

val ex1E = prove_goalw 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);

val conj_cong = prove_goal 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)) ]);

val disj_cong = prove_goal 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)) ]);

val imp_cong = prove_goal 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  eresolve_tac [iffE] 1
      ORELSE  mp_tac 1 ORELSE iff_tac prems 1)) ]);

val iff_cong = prove_goal IFOL.thy 
    "[| P <-> P';  Q <-> Q' |] ==> (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 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)) ]);

val all_cong = prove_goal 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   eresolve_tac [allE] 1 ORELSE iff_tac prems 1)) ]);

val ex_cong = prove_goal IFOL.thy 
    "(!!x.P(x) <-> Q(x)) ==> (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)) ]);

val ex1_cong = prove_goal 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 ***)

val sym = prove_goal IFOL.thy "a=b ==> b=a"
 (fn [major] => [ (rtac (major RS subst) 1), (rtac refl 1) ]);

val trans = prove_goal 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);

(*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 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 **)

val subst_context = prove_goal IFOL.thy 
   "[| a=b |]  ==>  t(a)=t(b)"
 (fn prems=>
  [ (resolve_tac (prems RL [ssubst]) 1),
    (resolve_tac [refl] 1) ]);

val subst_context2 = prove_goal IFOL.thy 
   "[| a=b;  c=d |]  ==>  t(a,c)=t(b,d)"
 (fn prems=>
  [ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);

val subst_context3 = prove_goal 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	*)
val box_equals = prove_goal IFOL.thy
    "[| a=b;  a=c;  b=d |] ==> 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 IFOL.thy
    "[| a=c;  b=d;  c=d |] ==> 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 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)) ]);

val pred2_cong = prove_goal 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)) ]);

val pred3_cong = prove_goal 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)  ***)

val conj_impE = prove_goal IFOL.thy 
    "[| (P&Q)-->S;  P-->(Q-->S) ==> R |] ==> R"
 (fn major::prems=>
  [ (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ]);

val disj_impE = prove_goal 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.  *)
val imp_impE = prove_goal 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.  *)
val not_impE = prove_goal 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.  *)
val iff_impE = prove_goal 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*)
val all_impE = prove_goal 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.  *)
val ex_impE = prove_goal 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)) ]);

end;

open IFOL_Lemmas;