src/FOL/IFOL.thy

ML context: full generic context, tuned signature;

(*  Title:      FOL/IFOL.thy
    ID:         $Id$
    Author:     Lawrence C Paulson and Markus Wenzel
*)

header {* Intuitionistic first-order logic *}

theory IFOL
imports Pure
uses ("fologic.ML") ("hypsubstdata.ML") ("intprover.ML")
begin


subsection {* Syntax and axiomatic basis *}

global

classes "term"
defaultsort "term"

typedecl o

judgment
  Trueprop      :: "o => prop"                  ("(_)" 5)

consts
  True          :: o
  False         :: o

  (* Connectives *)

  "op ="        :: "['a, 'a] => o"              (infixl "=" 50)

  Not           :: "o => o"                     ("~ _" [40] 40)
  "op &"        :: "[o, o] => o"                (infixr "&" 35)
  "op |"        :: "[o, o] => o"                (infixr "|" 30)
  "op -->"      :: "[o, o] => o"                (infixr "-->" 25)
  "op <->"      :: "[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)


abbreviation
  not_equal :: "['a, 'a] => o"  (infixl "~=" 50) where
  "x ~= y == ~ (x = y)"

notation (xsymbols)
  not_equal  (infixl "\<noteq>" 50)

notation (HTML output)
  not_equal  (infixl "\<noteq>" 50)

notation (xsymbols)
  Not       ("\<not> _" [40] 40) and
  "op &"    (infixr "\<and>" 35) and
  "op |"    (infixr "\<or>" 30) and
  All       (binder "\<forall>" 10) and
  Ex        (binder "\<exists>" 10) and
  Ex1       (binder "\<exists>!" 10) and
  "op -->"  (infixr "\<longrightarrow>" 25) and
  "op <->"  (infixr "\<longleftrightarrow>" 25)

notation (HTML output)
  Not       ("\<not> _" [40] 40) and
  "op &"    (infixr "\<and>" 35) and
  "op |"    (infixr "\<or>" 30) and
  All       (binder "\<forall>" 10) and
  Ex        (binder "\<exists>" 10) and
  Ex1       (binder "\<exists>!" 10)

local

finalconsts
  False All Ex
  "op ="
  "op &"
  "op |"
  "op -->"

axioms

  (* Equality *)

  refl:         "a=a"

  (* Propositional logic *)

  conjI:        "[| P;  Q |] ==> P&Q"
  conjunct1:    "P&Q ==> P"
  conjunct2:    "P&Q ==> Q"

  disjI1:       "P ==> P|Q"
  disjI2:       "Q ==> P|Q"
  disjE:        "[| P|Q;  P ==> R;  Q ==> R |] ==> R"

  impI:         "(P ==> Q) ==> P-->Q"
  mp:           "[| P-->Q;  P |] ==> Q"

  FalseE:       "False ==> P"

  (* Quantifiers *)

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

  exI:          "P(x) ==> (EX x. P(x))"
  exE:          "[| EX x. P(x);  !!x. P(x) ==> R |] ==> R"

  (* Reflection *)

  eq_reflection:  "(x=y)   ==> (x==y)"
  iff_reflection: "(P<->Q) ==> (P==Q)"


lemmas strip = impI allI


text{*Thanks to Stephan Merz*}
theorem subst:
  assumes eq: "a = b" and p: "P(a)"
  shows "P(b)"
proof -
  from eq have meta: "a \<equiv> b"
    by (rule eq_reflection)
  from p show ?thesis
    by (unfold meta)
qed


defs
  (* Definitions *)

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

  (* Unique existence *)

  ex1_def:      "Ex1(P) == EX x. P(x) & (ALL y. P(y) --> y=x)"


subsection {* Lemmas and proof tools *}

lemma TrueI: True
  unfolding True_def by (rule impI)


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

lemma conjE:
  assumes major: "P & Q"
    and r: "[| P; Q |] ==> R"
  shows R
  apply (rule r)
   apply (rule major [THEN conjunct1])
  apply (rule major [THEN conjunct2])
  done

lemma impE:
  assumes major: "P --> Q"
    and P
  and r: "Q ==> R"
  shows R
  apply (rule r)
  apply (rule major [THEN mp])
  apply (rule `P`)
  done

lemma allE:
  assumes major: "ALL x. P(x)"
    and r: "P(x) ==> R"
  shows R
  apply (rule r)
  apply (rule major [THEN spec])
  done

(*Duplicates the quantifier; for use with eresolve_tac*)
lemma all_dupE:
  assumes major: "ALL x. P(x)"
    and r: "[| P(x); ALL x. P(x) |] ==> R"
  shows R
  apply (rule r)
   apply (rule major [THEN spec])
  apply (rule major)
  done


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

lemma notI: "(P ==> False) ==> ~P"
  unfolding not_def by (erule impI)

lemma notE: "[| ~P;  P |] ==> R"
  unfolding not_def by (erule mp [THEN FalseE])

lemma rev_notE: "[| P; ~P |] ==> R"
  by (erule notE)

(*This is useful with the special implication rules for each kind of P. *)
lemma not_to_imp:
  assumes "~P"
    and r: "P --> False ==> Q"
  shows Q
  apply (rule r)
  apply (rule impI)
  apply (erule notE [OF `~P`])
  done

(* 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   *)
lemma rev_mp: "[| P;  P --> Q |] ==> Q"
  by (erule mp)

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


(*** Modus Ponens Tactics ***)

(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
ML {*
  local
    val notE = thm "notE"
    val impE = thm "impE"
  in
    fun mp_tac i = eresolve_tac [notE,impE] i  THEN  assume_tac i
    fun eq_mp_tac i = eresolve_tac [notE,impE] i  THEN  eq_assume_tac i
  end
*}


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

lemma iffI: "[| P ==> Q; Q ==> P |] ==> P<->Q"
  apply (unfold iff_def)
  apply (rule conjI)
   apply (erule impI)
  apply (erule impI)
  done


(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
lemma iffE:
  assumes major: "P <-> Q"
    and r: "P-->Q ==> Q-->P ==> R"
  shows R
  apply (insert major, unfold iff_def)
  apply (erule conjE)
  apply (erule r)
  apply assumption
  done

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

lemma iffD1: "[| P <-> Q;  P |] ==> Q"
  apply (unfold iff_def)
  apply (erule conjunct1 [THEN mp])
  apply assumption
  done

lemma iffD2: "[| P <-> Q;  Q |] ==> P"
  apply (unfold iff_def)
  apply (erule conjunct2 [THEN mp])
  apply assumption
  done

lemma rev_iffD1: "[| P; P <-> Q |] ==> Q"
  apply (erule iffD1)
  apply assumption
  done

lemma rev_iffD2: "[| Q; P <-> Q |] ==> P"
  apply (erule iffD2)
  apply assumption
  done

lemma iff_refl: "P <-> P"
  by (rule iffI)

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

lemma iff_trans: "[| P <-> Q;  Q<-> R |] ==> P <-> R"
  apply (rule iffI)
  apply (assumption | erule iffE | erule (1) notE 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(a)"
    and "!!x. P(x) ==> x=a"
  shows "EX! x. P(x)"
  apply (unfold ex1_def)
  apply (assumption | rule assms exI conjI allI impI)+
  done

(*Sometimes easier to use: the premises have no shared variables.  Safe!*)
lemma ex_ex1I:
  assumes ex: "EX x. P(x)"
    and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
  shows "EX! x. P(x)"
  apply (rule ex [THEN exE])
  apply (assumption | rule ex1I eq)+
  done

lemma ex1E:
  assumes ex1: "EX! x. P(x)"
    and r: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
  shows R
  apply (insert ex1, unfold ex1_def)
  apply (assumption | erule exE conjE)+
  done


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

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

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

(*Reversed congruence rule!   Used in ZF/Order*)
lemma conj_cong2:
  assumes "P <-> P'"
    and "P' ==> Q <-> Q'"
  shows "(Q&P) <-> (Q'&P')"
  apply (insert assms)
  apply (assumption | rule iffI conjI | erule iffE conjE mp |
    tactic {* iff_tac (thms "assms") 1 *})+
  done

lemma disj_cong:
  assumes "P <-> P'" and "Q <-> Q'"
  shows "(P|Q) <-> (P'|Q')"
  apply (insert assms)
  apply (erule iffE disjE disjI1 disjI2 | assumption | rule iffI | erule (1) notE impE)+
  done

lemma imp_cong:
  assumes "P <-> P'"
    and "P' ==> Q <-> Q'"
  shows "(P-->Q) <-> (P'-->Q')"
  apply (insert assms)
  apply (assumption | rule iffI impI | erule iffE | erule (1) notE impE |
    tactic {* iff_tac (thms "assms") 1 *})+
  done

lemma iff_cong: "[| P <-> P'; Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
  apply (erule iffE | assumption | rule iffI | erule (1) notE impE)+
  done

lemma not_cong: "P <-> P' ==> ~P <-> ~P'"
  apply (assumption | rule iffI notI | erule (1) notE impE | erule iffE notE)+
  done

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

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

lemma ex1_cong:
  assumes "!!x. P(x) <-> Q(x)"
  shows "(EX! x. P(x)) <-> (EX! x. Q(x))"
  apply (erule ex1E spec [THEN mp] | assumption | rule iffI ex1I | erule (1) notE impE |
    tactic {* iff_tac (thms "assms") 1 *})+
  done

(*** Equality rules ***)

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

lemma trans: "[| a=b;  b=c |] ==> a=c"
  apply (erule subst, assumption)
  done

(**  **)
lemma not_sym: "b ~= a ==> a ~= b"
  apply (erule contrapos)
  apply (erule sym)
  done
  
(* Two theorms for rewriting only one instance of a definition:
   the first for definitions of formulae and the second for terms *)

lemma def_imp_iff: "(A == B) ==> A <-> B"
  apply unfold
  apply (rule iff_refl)
  done

lemma meta_eq_to_obj_eq: "(A == B) ==> A = B"
  apply unfold
  apply (rule refl)
  done

lemma meta_eq_to_iff: "x==y ==> x<->y"
  by unfold (rule iff_refl)

(*substitution*)
lemma ssubst: "[| b = a; P(a) |] ==> P(b)"
  apply (drule sym)
  apply (erule (1) subst)
  done

(*A special case of ex1E that would otherwise need quantifier expansion*)
lemma ex1_equalsE:
    "[| EX! x. P(x);  P(a);  P(b) |] ==> 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: "[| a=b |]  ==>  t(a)=t(b)"
  apply (erule ssubst)
  apply (rule refl)
  done

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

lemma subst_context3: "[| a=b;  c=d;  e=f |]  ==>  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: "[| a=b;  a=c;  b=d |] ==> 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: "[| a=c;  b=d;  c=d |] ==> a=b"
  apply (rule trans)
   apply (rule trans)
    apply assumption+
  apply (erule sym)
  done

(** Congruence rules for predicate letters **)

lemma pred1_cong: "a=a' ==> P(a) <-> P(a')"
  apply (rule iffI)
   apply (erule (1) subst)
  apply (erule (1) ssubst)
  done

lemma pred2_cong: "[| a=a';  b=b' |] ==> P(a,b) <-> P(a',b')"
  apply (rule iffI)
   apply (erule subst)+
   apply assumption
  apply (erule ssubst)+
  apply assumption
  done

lemma pred3_cong: "[| a=a';  b=b';  c=c' |] ==> P(a,b,c) <-> P(a',b',c')"
  apply (rule iffI)
   apply (erule subst)+
   apply assumption
  apply (erule ssubst)+
  apply assumption
  done

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

ML {*
bind_thms ("pred_congs",
  List.concat (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!*)
lemma eq_cong: "[| a = a'; b = b' |] ==> a = b <-> a' = b'"
  apply (erule (1) pred2_cong)
  done


(*** 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&Q)-->S"
    and r: "P-->(Q-->S) ==> R"
  shows R
  by (assumption | rule conjI impI major [THEN mp] r)+

lemma disj_impE:
  assumes major: "(P|Q)-->S"
    and r: "[| P-->S; Q-->S |] ==> R"
  shows R
  by (assumption | rule disjI1 disjI2 impI major [THEN mp] r)+

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

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

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

(*** Courtesy of Krzysztof Grabczewski ***)

lemma disj_imp_disj:
  assumes major: "P|Q"
    and "P==>R" and "Q==>S"
  shows "R|S"
  apply (rule disjE [OF major])
  apply (rule disjI1) apply assumption
  apply (rule disjI2) apply assumption
  done

ML {*
structure ProjectRule = ProjectRuleFun
(struct
  val conjunct1 = thm "conjunct1";
  val conjunct2 = thm "conjunct2";
  val mp = thm "mp";
end)
*}

use "fologic.ML"

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

use "hypsubstdata.ML"
setup hypsubst_setup
use "intprover.ML"


subsection {* Intuitionistic Reasoning *}

lemma impE':
  assumes 1: "P --> Q"
    and 2: "Q ==> R"
    and 3: "P --> Q ==> P"
  shows R
proof -
  from 3 and 1 have P .
  with 1 have Q by (rule impE)
  with 2 show R .
qed

lemma allE':
  assumes 1: "ALL x. P(x)"
    and 2: "P(x) ==> ALL x. P(x) ==> Q"
  shows Q
proof -
  from 1 have "P(x)" by (rule spec)
  from this and 1 show Q by (rule 2)
qed

lemma notE':
  assumes 1: "~ P"
    and 2: "~ P ==> P"
  shows R
proof -
  from 2 and 1 have P .
  with 1 show R by (rule notE)
qed

lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
  and [Pure.elim 2] = allE notE' impE'
  and [Pure.intro] = exI disjI2 disjI1

setup {* ContextRules.addSWrapper (fn tac => hyp_subst_tac ORELSE' tac) *}


lemma iff_not_sym: "~ (Q <-> P) ==> ~ (P <-> Q)"
  by iprover

lemmas [sym] = sym iff_sym not_sym iff_not_sym
  and [Pure.elim?] = iffD1 iffD2 impE


lemma eq_commute: "a=b <-> b=a"
apply (rule iffI) 
apply (erule sym)+
done


subsection {* Atomizing meta-level rules *}

lemma atomize_all [atomize]: "(!!x. P(x)) == Trueprop (ALL x. P(x))"
proof
  assume "!!x. P(x)"
  show "ALL x. P(x)" ..
next
  assume "ALL x. P(x)"
  thus "!!x. P(x)" ..
qed

lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
proof
  assume "A ==> B"
  thus "A --> B" ..
next
  assume "A --> B" and A
  thus B by (rule mp)
qed

lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
proof
  assume "x == y"
  show "x = y" by (unfold prems) (rule refl)
next
  assume "x = y"
  thus "x == y" by (rule eq_reflection)
qed

lemma atomize_iff [atomize]: "(A == B) == Trueprop (A <-> B)"
proof
  assume "A == B"
  show "A <-> B" by (unfold prems) (rule iff_refl)
next
  assume "A <-> B"
  thus "A == B" by (rule iff_reflection)
qed

lemma atomize_conj [atomize]:
  includes meta_conjunction_syntax
  shows "(A && B) == Trueprop (A & B)"
proof
  assume conj: "A && B"
  show "A & B"
  proof (rule conjI)
    from conj show A by (rule conjunctionD1)
    from conj show B by (rule conjunctionD2)
  qed
next
  assume conj: "A & B"
  show "A && B"
  proof -
    from conj show A ..
    from conj show B ..
  qed
qed

lemmas [symmetric, rulify] = atomize_all atomize_imp
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq atomize_iff


subsection {* Calculational rules *}

lemma forw_subst: "a = b ==> P(b) ==> P(a)"
  by (rule ssubst)

lemma back_subst: "P(a) ==> a = b ==> P(b)"
  by (rule subst)

text {*
  Note that this list of rules is in reverse order of priorities.
*}

lemmas basic_trans_rules [trans] =
  forw_subst
  back_subst
  rev_mp
  mp
  trans

subsection {* ``Let'' declarations *}

nonterminals letbinds letbind

constdefs
  Let :: "['a::{}, 'a => 'b] => ('b::{})"
    "Let(s, f) == f(s)"

syntax
  "_bind"       :: "[pttrn, 'a] => letbind"           ("(2_ =/ _)" 10)
  ""            :: "letbind => letbinds"              ("_")
  "_binds"      :: "[letbind, letbinds] => letbinds"  ("_;/ _")
  "_Let"        :: "[letbinds, 'a] => 'a"             ("(let (_)/ in (_))" 10)

translations
  "_Let(_binds(b, bs), e)"  == "_Let(b, _Let(bs, e))"
  "let x = a in e"          == "Let(a, %x. e)"


lemma LetI: 
  assumes "!!x. x=t ==> P(u(x))"
  shows "P(let x=t in u(x))"
  apply (unfold Let_def)
  apply (rule refl [THEN assms])
  done


subsection {* ML bindings *}

ML {*
val refl = thm "refl"
val trans = thm "trans"
val sym = thm "sym"
val subst = thm "subst"
val ssubst = thm "ssubst"
val conjI = thm "conjI"
val conjE = thm "conjE"
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 impE = thm "impE"
val mp = thm "mp"
val rev_mp = thm "rev_mp"
val TrueI = thm "TrueI"
val FalseE = thm "FalseE"
val iff_refl = thm "iff_refl"
val iff_trans = thm "iff_trans"
val iffI = thm "iffI"
val iffE = thm "iffE"
val iffD1 = thm "iffD1"
val iffD2 = thm "iffD2"
val notI = thm "notI"
val notE = thm "notE"
val allI = thm "allI"
val allE = thm "allE"
val spec = thm "spec"
val exI = thm "exI"
val exE = thm "exE"
val eq_reflection = thm "eq_reflection"
val iff_reflection = thm "iff_reflection"
val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq"
val meta_eq_to_iff = thm "meta_eq_to_iff"
*}

end