src/FOLP/IFOLP.thy

major reorganization of document structure;

(*  Title:      FOLP/IFOLP.thy
    ID:         $Id$
    Author:     Martin D Coen, Cambridge University Computer Laboratory
    Copyright   1992  University of Cambridge
*)

header {* Intuitionistic First-Order Logic with Proofs *}

theory IFOLP
imports Pure
uses ("hypsubst.ML") ("intprover.ML")
begin

setup PureThy.old_appl_syntax_setup

global

classes "term"
defaultsort "term"

typedecl p
typedecl o

consts
      (*** Judgements ***)
 "@Proof"       ::   "[p,o]=>prop"      ("(_ /: _)" [51,10] 5)
 Proof          ::   "[o,p]=>prop"
 EqProof        ::   "[p,p,o]=>prop"    ("(3_ /= _ :/ _)" [10,10,10] 5)

      (*** Logical Connectives -- Type Formers ***)
 "="            ::      "['a,'a] => o"  (infixl 50)
 True           ::      "o"
 False          ::      "o"
 Not            ::      "o => o"        ("~ _" [40] 40)
 "&"            ::      "[o,o] => o"    (infixr 35)
 "|"            ::      "[o,o] => o"    (infixr 30)
 "-->"          ::      "[o,o] => o"    (infixr 25)
 "<->"          ::      "[o,o] => o"    (infixr 25)
      (*Quantifiers*)
 All            ::      "('a => o) => o"        (binder "ALL " 10)
 Ex             ::      "('a => o) => o"        (binder "EX " 10)
 Ex1            ::      "('a => o) => o"        (binder "EX! " 10)
      (*Rewriting gadgets*)
 NORM           ::      "o => o"
 norm           ::      "'a => 'a"

      (*** Proof Term Formers: precedence must exceed 50 ***)
 tt             :: "p"
 contr          :: "p=>p"
 fst            :: "p=>p"
 snd            :: "p=>p"
 pair           :: "[p,p]=>p"           ("(1<_,/_>)")
 split          :: "[p, [p,p]=>p] =>p"
 inl            :: "p=>p"
 inr            :: "p=>p"
 when           :: "[p, p=>p, p=>p]=>p"
 lambda         :: "(p => p) => p"      (binder "lam " 55)
 "`"            :: "[p,p]=>p"           (infixl 60)
 alll           :: "['a=>p]=>p"         (binder "all " 55)
 "^"            :: "[p,'a]=>p"          (infixl 55)
 exists         :: "['a,p]=>p"          ("(1[_,/_])")
 xsplit         :: "[p,['a,p]=>p]=>p"
 ideq           :: "'a=>p"
 idpeel         :: "[p,'a=>p]=>p"
 nrm            :: p
 NRM            :: p

local

ML {*

(*show_proofs:=true displays the proof terms -- they are ENORMOUS*)
val show_proofs = ref false;

fun proof_tr [p,P] = Const (@{const_name Proof}, dummyT) $ P $ p;

fun proof_tr' [P,p] =
    if !show_proofs then Const("@Proof",dummyT) $ p $ P
    else P  (*this case discards the proof term*);
*}

parse_translation {* [("@Proof", proof_tr)] *}
print_translation {* [("Proof", proof_tr')] *}

axioms

(**** Propositional logic ****)

(*Equality*)
(* Like Intensional Equality in MLTT - but proofs distinct from terms *)

ieqI:      "ideq(a) : a=a"
ieqE:      "[| p : a=b;  !!x. f(x) : P(x,x) |] ==> idpeel(p,f) : P(a,b)"

(* Truth and Falsity *)

TrueI:     "tt : True"
FalseE:    "a:False ==> contr(a):P"

(* Conjunction *)

conjI:     "[| a:P;  b:Q |] ==> <a,b> : P&Q"
conjunct1: "p:P&Q ==> fst(p):P"
conjunct2: "p:P&Q ==> snd(p):Q"

(* Disjunction *)

disjI1:    "a:P ==> inl(a):P|Q"
disjI2:    "b:Q ==> inr(b):P|Q"
disjE:     "[| a:P|Q;  !!x. x:P ==> f(x):R;  !!x. x:Q ==> g(x):R
           |] ==> when(a,f,g):R"

(* Implication *)

impI:      "(!!x. x:P ==> f(x):Q) ==> lam x. f(x):P-->Q"
mp:        "[| f:P-->Q;  a:P |] ==> f`a:Q"

(*Quantifiers*)

allI:      "(!!x. f(x) : P(x)) ==> all x. f(x) : ALL x. P(x)"
spec:      "(f:ALL x. P(x)) ==> f^x : P(x)"

exI:       "p : P(x) ==> [x,p] : EX x. P(x)"
exE:       "[| p: EX x. P(x);  !!x u. u:P(x) ==> f(x,u) : R |] ==> xsplit(p,f):R"

(**** Equality between proofs ****)

prefl:     "a : P ==> a = a : P"
psym:      "a = b : P ==> b = a : P"
ptrans:    "[| a = b : P;  b = c : P |] ==> a = c : P"

idpeelB:   "[| !!x. f(x) : P(x,x) |] ==> idpeel(ideq(a),f) = f(a) : P(a,a)"

fstB:      "a:P ==> fst(<a,b>) = a : P"
sndB:      "b:Q ==> snd(<a,b>) = b : Q"
pairEC:    "p:P&Q ==> p = <fst(p),snd(p)> : P&Q"

whenBinl:  "[| a:P;  !!x. x:P ==> f(x) : Q |] ==> when(inl(a),f,g) = f(a) : Q"
whenBinr:  "[| b:P;  !!x. x:P ==> g(x) : Q |] ==> when(inr(b),f,g) = g(b) : Q"
plusEC:    "a:P|Q ==> when(a,%x. inl(x),%y. inr(y)) = a : P|Q"

applyB:     "[| a:P;  !!x. x:P ==> b(x) : Q |] ==> (lam x. b(x)) ` a = b(a) : Q"
funEC:      "f:P ==> f = lam x. f`x : P"

specB:      "[| !!x. f(x) : P(x) |] ==> (all x. f(x)) ^ a = f(a) : P(a)"


(**** Definitions ****)

not_def:              "~P == P-->False"
iff_def:         "P<->Q == (P-->Q) & (Q-->P)"

(*Unique existence*)
ex1_def:   "EX! x. P(x) == EX x. P(x) & (ALL y. P(y) --> y=x)"

(*Rewriting -- special constants to flag normalized terms and formulae*)
norm_eq: "nrm : norm(x) = x"
NORM_iff:        "NRM : NORM(P) <-> P"

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

lemma conjE:
  assumes "p:P&Q"
    and "!!x y.[| x:P; y:Q |] ==> f(x,y):R"
  shows "?a:R"
  apply (rule assms(2))
   apply (rule conjunct1 [OF assms(1)])
  apply (rule conjunct2 [OF assms(1)])
  done

lemma impE:
  assumes "p:P-->Q"
    and "q:P"
    and "!!x. x:Q ==> r(x):R"
  shows "?p:R"
  apply (rule assms mp)+
  done

lemma allE:
  assumes "p:ALL x. P(x)"
    and "!!y. y:P(x) ==> q(y):R"
  shows "?p:R"
  apply (rule assms spec)+
  done

(*Duplicates the quantifier; for use with eresolve_tac*)
lemma all_dupE:
  assumes "p:ALL x. P(x)"
    and "!!y z.[| y:P(x); z:ALL x. P(x) |] ==> q(y,z):R"
  shows "?p:R"
  apply (rule assms spec)+
  done


(*** Negation rules, which translate between ~P and P-->False ***)

lemma notI:
  assumes "!!x. x:P ==> q(x):False"
  shows "?p:~P"
  unfolding not_def
  apply (assumption | rule assms impI)+
  done

lemma notE: "p:~P \<Longrightarrow> q:P \<Longrightarrow> ?p:R"
  unfolding not_def
  apply (drule (1) mp)
  apply (erule FalseE)
  done

(*This is useful with the special implication rules for each kind of P. *)
lemma not_to_imp:
  assumes "p:~P"
    and "!!x. x:(P-->False) ==> q(x):Q"
  shows "?p:Q"
  apply (assumption | rule assms impI notE)+
  done

(* 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   *)
lemma rev_mp: "[| p:P;  q:P --> Q |] ==> ?p:Q"
  apply (assumption | rule mp)+
  done


(*Contrapositive of an inference rule*)
lemma contrapos:
  assumes major: "p:~Q"
    and minor: "!!y. y:P==>q(y):Q"
  shows "?a:~P"
  apply (rule major [THEN notE, THEN notI])
  apply (erule minor)
  done

(** Unique assumption tactic.
    Ignores proof objects.
    Fails unless one assumption is equal and exactly one is unifiable
**)

ML {*
local
  fun discard_proof (Const (@{const_name 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 (op =) (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 *)
ML {*
  fun mp_tac i = eresolve_tac [@{thm notE}, make_elim @{thm mp}] i  THEN  assume_tac i
*}

(*Like mp_tac but instantiates no variables*)
ML {*
  fun int_uniq_mp_tac i = eresolve_tac [@{thm notE}, @{thm impE}] i  THEN  uniq_assume_tac i
*}


(*** If-and-only-if ***)

lemma iffI:
  assumes "!!x. x:P ==> q(x):Q"
    and "!!x. x:Q ==> r(x):P"
  shows "?p:P<->Q"
  unfolding iff_def
  apply (assumption | rule assms conjI impI)+
  done


(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
  
lemma iffE:
  assumes "p:P <-> Q"
    and "!!x y.[| x:P-->Q; y:Q-->P |] ==> q(x,y):R"
  shows "?p:R"
  apply (rule conjE)
   apply (rule assms(1) [unfolded iff_def])
  apply (rule assms(2))
   apply assumption+
  done

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

lemma iffD1: "[| p:P <-> Q; q:P |] ==> ?p:Q"
  unfolding iff_def
  apply (rule conjunct1 [THEN mp], assumption+)
  done

lemma iffD2: "[| p:P <-> Q; q:Q |] ==> ?p:P"
  unfolding iff_def
  apply (rule conjunct2 [THEN mp], assumption+)
  done

lemma iff_refl: "?p:P <-> P"
  apply (rule iffI)
   apply assumption+
  done

lemma iff_sym: "p:Q <-> P ==> ?p:P <-> Q"
  apply (erule iffE)
  apply (rule iffI)
   apply (erule (1) mp)+
  done

lemma iff_trans: "[| p:P <-> Q; q:Q<-> R |] ==> ?p:P <-> R"
  apply (rule iffI)
   apply (assumption | erule iffE | erule (1) impE)+
  done

(*** 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.
***)

lemma ex1I:
  assumes "p:P(a)"
    and "!!x u. u:P(x) ==> f(u) : x=a"
  shows "?p:EX! x. P(x)"
  unfolding ex1_def
  apply (assumption | rule assms exI conjI allI impI)+
  done

lemma ex1E:
  assumes "p:EX! x. P(x)"
    and "!!x u v. [| u:P(x);  v:ALL y. P(y) --> y=x |] ==> f(x,u,v):R"
  shows "?a : R"
  apply (insert assms(1) [unfolded ex1_def])
  apply (erule exE conjE | assumption | rule assms(1))+
  done


(*** <-> congruence rules for simplification ***)

(*Use iffE on a premise.  For conj_cong, imp_cong, all_cong, ex_cong*)
ML {*
fun iff_tac prems i =
    resolve_tac (prems RL [@{thm iffE}]) i THEN
    REPEAT1 (eresolve_tac [asm_rl, @{thm mp}] i)
*}

lemma conj_cong:
  assumes "p:P <-> P'"
    and "!!x. x:P' ==> q(x):Q <-> Q'"
  shows "?p:(P&Q) <-> (P'&Q')"
  apply (insert assms(1))
  apply (assumption | rule iffI conjI |
    erule iffE conjE mp | tactic {* iff_tac @{thms assms} 1 *})+
  done

lemma disj_cong:
  "[| p:P <-> P'; q:Q <-> Q' |] ==> ?p:(P|Q) <-> (P'|Q')"
  apply (erule iffE disjE disjI1 disjI2 | assumption | rule iffI | tactic {* mp_tac 1 *})+
  done

lemma imp_cong:
  assumes "p:P <-> P'"
    and "!!x. x:P' ==> q(x):Q <-> Q'"
  shows "?p:(P-->Q) <-> (P'-->Q')"
  apply (insert assms(1))
  apply (assumption | rule iffI impI | erule iffE | tactic {* mp_tac 1 *} |
    tactic {* iff_tac @{thms assms} 1 *})+
  done

lemma iff_cong:
  "[| p:P <-> P'; q:Q <-> Q' |] ==> ?p:(P<->Q) <-> (P'<->Q')"
  apply (erule iffE | assumption | rule iffI | tactic {* mp_tac 1 *})+
  done

lemma not_cong:
  "p:P <-> P' ==> ?p:~P <-> ~P'"
  apply (assumption | rule iffI notI | tactic {* mp_tac 1 *} | erule iffE notE)+
  done

lemma all_cong:
  assumes "!!x. f(x):P(x) <-> Q(x)"
  shows "?p:(ALL x. P(x)) <-> (ALL x. Q(x))"
  apply (assumption | rule iffI allI | tactic {* mp_tac 1 *} | erule allE |
    tactic {* iff_tac @{thms assms} 1 *})+
  done

lemma ex_cong:
  assumes "!!x. f(x):P(x) <-> Q(x)"
  shows "?p:(EX x. P(x)) <-> (EX x. Q(x))"
  apply (erule exE | assumption | rule iffI exI | tactic {* mp_tac 1 *} |
    tactic {* iff_tac @{thms assms} 1 *})+
  done

(*NOT PROVED
bind_thm ("ex1_cong", prove_goal (the_context ())
    "(!!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 ***)

lemmas refl = ieqI

lemma subst:
  assumes prem1: "p:a=b"
    and prem2: "q:P(a)"
  shows "?p : P(b)"
  apply (rule prem2 [THEN rev_mp])
  apply (rule prem1 [THEN ieqE])
  apply (rule impI)
  apply assumption
  done

lemma sym: "q:a=b ==> ?c:b=a"
  apply (erule subst)
  apply (rule refl)
  done

lemma trans: "[| p:a=b;  q:b=c |] ==> ?d:a=c"
  apply (erule (1) subst)
  done

(** ~ b=a ==> ~ a=b **)
lemma not_sym: "p:~ b=a ==> ?q:~ a=b"
  apply (erule contrapos)
  apply (erule sym)
  done

(*calling "standard" reduces maxidx to 0*)
lemmas ssubst = sym [THEN subst, standard]

(*A special case of ex1E that would otherwise need quantifier expansion*)
lemma ex1_equalsE: "[| p:EX! x. P(x);  q:P(a);  r:P(b) |] ==> ?d:a=b"
  apply (erule ex1E)
  apply (rule trans)
   apply (rule_tac [2] sym)
   apply (assumption | erule spec [THEN mp])+
  done

(** Polymorphic congruence rules **)

lemma subst_context: "[| p:a=b |]  ==>  ?d:t(a)=t(b)"
  apply (erule ssubst)
  apply (rule refl)
  done

lemma subst_context2: "[| p:a=b;  q:c=d |]  ==>  ?p:t(a,c)=t(b,d)"
  apply (erule ssubst)+
  apply (rule refl)
  done

lemma subst_context3: "[| p:a=b;  q:c=d;  r:e=f |]  ==>  ?p:t(a,c,e)=t(b,d,f)"
  apply (erule ssubst)+
  apply (rule refl)
  done

(*Useful with eresolve_tac for proving equalties from known equalities.
        a = b
        |   |
        c = d   *)
lemma box_equals: "[| p:a=b;  q:a=c;  r:b=d |] ==> ?p:c=d"
  apply (rule trans)
   apply (rule trans)
    apply (rule sym)
    apply assumption+
  done

(*Dual of box_equals: for proving equalities backwards*)
lemma simp_equals: "[| p:a=c;  q:b=d;  r:c=d |] ==> ?p:a=b"
  apply (rule trans)
   apply (rule trans)
    apply (assumption | rule sym)+
  done

(** Congruence rules for predicate letters **)

lemma pred1_cong: "p:a=a' ==> ?p:P(a) <-> P(a')"
  apply (rule iffI)
   apply (tactic {* DEPTH_SOLVE (atac 1 ORELSE eresolve_tac [@{thm subst}, @{thm ssubst}] 1) *})
  done

lemma pred2_cong: "[| p:a=a';  q:b=b' |] ==> ?p:P(a,b) <-> P(a',b')"
  apply (rule iffI)
   apply (tactic {* DEPTH_SOLVE (atac 1 ORELSE eresolve_tac [@{thm subst}, @{thm ssubst}] 1) *})
  done

lemma pred3_cong: "[| p:a=a';  q:b=b';  r:c=c' |] ==> ?p:P(a,b,c) <-> P(a',b',c')"
  apply (rule iffI)
   apply (tactic {* DEPTH_SOLVE (atac 1 ORELSE eresolve_tac [@{thm subst}, @{thm ssubst}] 1) *})
  done

(*special cases for free variables P, Q, R, S -- up to 3 arguments*)

ML {*
  bind_thms ("pred_congs",
    flat (map (fn c =>
               map (fn th => read_instantiate [("P",c)] th)
                   [@{thm pred1_cong}, @{thm pred2_cong}, @{thm pred3_cong}])
               (explode"PQRS")))
*}

(*special case for the equality predicate!*)
lemmas eq_cong = pred2_cong [where P = "op =", standard]


(*** 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)  ***)

lemma conj_impE:
  assumes major: "p:(P&Q)-->S"
    and minor: "!!x. x:P-->(Q-->S) ==> q(x):R"
  shows "?p:R"
  apply (assumption | rule conjI impI major [THEN mp] minor)+
  done

lemma disj_impE:
  assumes major: "p:(P|Q)-->S"
    and minor: "!!x y.[| x:P-->S; y:Q-->S |] ==> q(x,y):R"
  shows "?p:R"
  apply (tactic {* DEPTH_SOLVE (atac 1 ORELSE
      resolve_tac [@{thm disjI1}, @{thm disjI2}, @{thm impI},
        @{thm major} RS @{thm mp}, @{thm minor}] 1) *})
  done

(*Simplifies the implication.  Classical version is stronger.
  Still UNSAFE since Q must be provable -- backtracking needed.  *)
lemma imp_impE:
  assumes major: "p:(P-->Q)-->S"
    and r1: "!!x y.[| x:P; y:Q-->S |] ==> q(x,y):Q"
    and r2: "!!x. x:S ==> r(x):R"
  shows "?p:R"
  apply (assumption | rule impI major [THEN mp] r1 r2)+
  done

(*Simplifies the implication.  Classical version is stronger.
  Still UNSAFE since ~P must be provable -- backtracking needed.  *)
lemma not_impE:
  assumes major: "p:~P --> S"
    and r1: "!!y. y:P ==> q(y):False"
    and r2: "!!y. y:S ==> r(y):R"
  shows "?p:R"
  apply (assumption | rule notI impI major [THEN mp] r1 r2)+
  done

(*Simplifies the implication.   UNSAFE.  *)
lemma iff_impE:
  assumes major: "p:(P<->Q)-->S"
    and r1: "!!x y.[| x:P; y:Q-->S |] ==> q(x,y):Q"
    and r2: "!!x y.[| x:Q; y:P-->S |] ==> r(x,y):P"
    and r3: "!!x. x:S ==> s(x):R"
  shows "?p:R"
  apply (assumption | rule iffI impI major [THEN mp] r1 r2 r3)+
  done

(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
lemma all_impE:
  assumes major: "p:(ALL x. P(x))-->S"
    and r1: "!!x. q:P(x)"
    and r2: "!!y. y:S ==> r(y):R"
  shows "?p:R"
  apply (assumption | rule allI impI major [THEN mp] r1 r2)+
  done

(*Unsafe: (EX x.P(x))-->S  is equivalent to  ALL x.P(x)-->S.  *)
lemma ex_impE:
  assumes major: "p:(EX x. P(x))-->S"
    and r: "!!y. y:P(a)-->S ==> q(y):R"
  shows "?p:R"
  apply (assumption | rule exI impI major [THEN mp] r)+
  done


lemma rev_cut_eq:
  assumes "p:a=b"
    and "!!x. x:a=b ==> f(x):R"
  shows "?p:R"
  apply (rule assms)+
  done

lemma thin_refl: "!!X. [|p:x=x; PROP W|] ==> PROP W" .

use "hypsubst.ML"

ML {*

(*** Applying HypsubstFun to generate hyp_subst_tac ***)

structure Hypsubst_Data =
struct
  (*Take apart an equality judgement; otherwise raise Match!*)
  fun dest_eq (Const (@{const_name Proof}, _) $
    (Const (@{const_name "op ="}, _)  $ t $ u) $ _) = (t, u);

  val imp_intr = @{thm impI}

  (*etac rev_cut_eq moves an equality to be the last premise. *)
  val rev_cut_eq = @{thm rev_cut_eq}

  val rev_mp = @{thm rev_mp}
  val subst = @{thm subst}
  val sym = @{thm sym}
  val thin_refl = @{thm thin_refl}
end;

structure Hypsubst = HypsubstFun(Hypsubst_Data);
open Hypsubst;
*}

use "intprover.ML"


(*** Rewrite rules ***)

lemma conj_rews:
  "?p1 : P & True <-> P"
  "?p2 : True & P <-> P"
  "?p3 : P & False <-> False"
  "?p4 : False & P <-> False"
  "?p5 : P & P <-> P"
  "?p6 : P & ~P <-> False"
  "?p7 : ~P & P <-> False"
  "?p8 : (P & Q) & R <-> P & (Q & R)"
  apply (tactic {* fn st => IntPr.fast_tac 1 st *})+
  done

lemma disj_rews:
  "?p1 : P | True <-> True"
  "?p2 : True | P <-> True"
  "?p3 : P | False <-> P"
  "?p4 : False | P <-> P"
  "?p5 : P | P <-> P"
  "?p6 : (P | Q) | R <-> P | (Q | R)"
  apply (tactic {* IntPr.fast_tac 1 *})+
  done

lemma not_rews:
  "?p1 : ~ False <-> True"
  "?p2 : ~ True <-> False"
  apply (tactic {* IntPr.fast_tac 1 *})+
  done

lemma imp_rews:
  "?p1 : (P --> False) <-> ~P"
  "?p2 : (P --> True) <-> True"
  "?p3 : (False --> P) <-> True"
  "?p4 : (True --> P) <-> P"
  "?p5 : (P --> P) <-> True"
  "?p6 : (P --> ~P) <-> ~P"
  apply (tactic {* IntPr.fast_tac 1 *})+
  done

lemma iff_rews:
  "?p1 : (True <-> P) <-> P"
  "?p2 : (P <-> True) <-> P"
  "?p3 : (P <-> P) <-> True"
  "?p4 : (False <-> P) <-> ~P"
  "?p5 : (P <-> False) <-> ~P"
  apply (tactic {* IntPr.fast_tac 1 *})+
  done

lemma quant_rews:
  "?p1 : (ALL x. P) <-> P"
  "?p2 : (EX x. P) <-> P"
  apply (tactic {* IntPr.fast_tac 1 *})+
  done

(*These are NOT supplied by default!*)
lemma distrib_rews1:
  "?p1 : ~(P|Q) <-> ~P & ~Q"
  "?p2 : P & (Q | R) <-> P&Q | P&R"
  "?p3 : (Q | R) & P <-> Q&P | R&P"
  "?p4 : (P | Q --> R) <-> (P --> R) & (Q --> R)"
  apply (tactic {* IntPr.fast_tac 1 *})+
  done

lemma distrib_rews2:
  "?p1 : ~(EX x. NORM(P(x))) <-> (ALL x. ~NORM(P(x)))"
  "?p2 : ((EX x. NORM(P(x))) --> Q) <-> (ALL x. NORM(P(x)) --> Q)"
  "?p3 : (EX x. NORM(P(x))) & NORM(Q) <-> (EX x. NORM(P(x)) & NORM(Q))"
  "?p4 : NORM(Q) & (EX x. NORM(P(x))) <-> (EX x. NORM(Q) & NORM(P(x)))"
  apply (tactic {* IntPr.fast_tac 1 *})+
  done

lemmas distrib_rews = distrib_rews1 distrib_rews2

lemma P_Imp_P_iff_T: "p:P ==> ?p:(P <-> True)"
  apply (tactic {* IntPr.fast_tac 1 *})
  done

lemma not_P_imp_P_iff_F: "p:~P ==> ?p:(P <-> False)"
  apply (tactic {* IntPr.fast_tac 1 *})
  done

end