theory QuotMain

imports QuotScript QuotList Prove

uses ("quotient_info.ML") 

     ("quotient.ML")

     ("quotient_def.ML")

begin

locale QUOT_TYPE =

  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

  and   Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"

  and   Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"

  assumes equiv: "EQUIV R"

  and     rep_prop: "\<And>y. \<exists>x. Rep y = R x"

  and     rep_inverse: "\<And>x. Abs (Rep x) = x"

  and     abs_inverse: "\<And>x. (Rep (Abs (R x))) = (R x)"

  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"

begin

definition

  ABS::"'a \<Rightarrow> 'b"

where

  "ABS x \<equiv> Abs (R x)"

definition

  REP::"'b \<Rightarrow> 'a"

where

  "REP a = Eps (Rep a)"

lemma lem9:

  shows "R (Eps (R x)) = R x"

proof -

  have a: "R x x" using equiv by (simp add: EQUIV_REFL_SYM_TRANS REFL_def)

  then have "R x (Eps (R x))" by (rule someI)

  then show "R (Eps (R x)) = R x"

    using equiv unfolding EQUIV_def by simp

qed

theorem thm10:

  shows "ABS (REP a) \<equiv> a"

  apply  (rule eq_reflection)

  unfolding ABS_def REP_def

proof -

  from rep_prop

  obtain x where eq: "Rep a = R x" by auto

  have "Abs (R (Eps (Rep a))) = Abs (R (Eps (R x)))" using eq by simp

  also have "\<dots> = Abs (R x)" using lem9 by simp

  also have "\<dots> = Abs (Rep a)" using eq by simp

  also have "\<dots> = a" using rep_inverse by simp

  finally

  show "Abs (R (Eps (Rep a))) = a" by simp

qed

lemma REP_refl:

  shows "R (REP a) (REP a)"

unfolding REP_def

by (simp add: equiv[simplified EQUIV_def])

lemma lem7:

  shows "(R x = R y) = (Abs (R x) = Abs (R y))"

apply(rule iffI)

apply(simp)

apply(drule rep_inject[THEN iffD2])

apply(simp add: abs_inverse)

done

theorem thm11:

  shows "R r r' = (ABS r = ABS r')"

unfolding ABS_def

by (simp only: equiv[simplified EQUIV_def] lem7)

lemma REP_ABS_rsp:

  shows "R f (REP (ABS g)) = R f g"

  and   "R (REP (ABS g)) f = R g f"

by (simp_all add: thm10 thm11)

lemma QUOTIENT:

  "QUOTIENT R ABS REP"

apply(unfold QUOTIENT_def)

apply(simp add: thm10)

apply(simp add: REP_refl)

apply(subst thm11[symmetric])

apply(simp add: equiv[simplified EQUIV_def])

done

lemma R_trans:

  assumes ab: "R a b"

  and     bc: "R b c"

  shows "R a c"

proof -

  have tr: "TRANS R" using equiv EQUIV_REFL_SYM_TRANS[of R] by simp

  moreover have ab: "R a b" by fact

  moreover have bc: "R b c" by fact

  ultimately show "R a c" unfolding TRANS_def by blast

qed

lemma R_sym:

  assumes ab: "R a b"

  shows "R b a"

proof -

  have re: "SYM R" using equiv EQUIV_REFL_SYM_TRANS[of R] by simp

  then show "R b a" using ab unfolding SYM_def by blast

qed

lemma R_trans2:

  assumes ac: "R a c"

  and     bd: "R b d"

  shows "R a b = R c d"

using ac bd

by (blast intro: R_trans R_sym)

lemma REPS_same:

  shows "R (REP a) (REP b) \<equiv> (a = b)"

proof -

  have "R (REP a) (REP b) = (a = b)"

  proof

    assume as: "R (REP a) (REP b)"

    from rep_prop

    obtain x y

      where eqs: "Rep a = R x" "Rep b = R y" by blast

    from eqs have "R (Eps (R x)) (Eps (R y))" using as unfolding REP_def by simp

    then have "R x (Eps (R y))" using lem9 by simp

    then have "R (Eps (R y)) x" using R_sym by blast

    then have "R y x" using lem9 by simp

    then have "R x y" using R_sym by blast

    then have "ABS x = ABS y" using thm11 by simp

    then have "Abs (Rep a) = Abs (Rep b)" using eqs unfolding ABS_def by simp

    then show "a = b" using rep_inverse by simp

  next

    assume ab: "a = b"

    have "REFL R" using equiv EQUIV_REFL_SYM_TRANS[of R] by simp

    then show "R (REP a) (REP b)" unfolding REFL_def using ab by auto

  qed

  then show "R (REP a) (REP b) \<equiv> (a = b)" by simp

qed

end

section {* type definition for the quotient type *}

(* the auxiliary data for the quotient types *)

use "quotient_info.ML"

declare [[map list = (map, LIST_REL)]]

declare [[map * = (prod_fun, prod_rel)]]

declare [[map "fun" = (fun_map, FUN_REL)]]

ML {* maps_lookup @{theory} "List.list" *}

ML {* maps_lookup @{theory} "*" *}

ML {* maps_lookup @{theory} "fun" *}

(* definition of the quotient types *)

(* FIXME: should be called quotient_typ.ML *)

use "quotient.ML"

(* lifting of constants *)

use "quotient_def.ML"

section {* ATOMIZE *}

lemma atomize_eqv[atomize]: 

  shows "(Trueprop A \<equiv> Trueprop B) \<equiv> (A \<equiv> B)" 

proof

  assume "A \<equiv> B" 

  then show "Trueprop A \<equiv> Trueprop B" by unfold

next

  assume *: "Trueprop A \<equiv> Trueprop B"

  have "A = B"

  proof (cases A)

    case True

    have "A" by fact

    then show "A = B" using * by simp

  next

    case False

    have "\<not>A" by fact

    then show "A = B" using * by auto

  qed

  then show "A \<equiv> B" by (rule eq_reflection)

qed

ML {*

fun atomize_thm thm =

let

  val thm' = Thm.freezeT (forall_intr_vars thm)

  val thm'' = ObjectLogic.atomize (cprop_of thm')

in

  @{thm Pure.equal_elim_rule1} OF [thm'', thm']

end

*}

ML {* atomize_thm @{thm list.induct} *}

section {* REGULARIZE *}

(*

Regularizing a theorem means:

 - Quantifiers over a type that needs lifting are replaced by

   bounded quantifiers, for example:

      \<forall>x. P     \<Longrightarrow>     \<forall>x\<in>(Respects R). P

 - Abstractions over a type that needs lifting are replaced

   by bounded abstractions:

      \<lambda>x. P     \<Longrightarrow>     Ball (Respects R) (\<lambda>x. P)

 - Equalities over the type being lifted are replaced by

   appropriate relations:

      A = B     \<Longrightarrow>     A \<approx> B

   Example with more complicated types of A, B:

      A = B     \<Longrightarrow>     (op = \<Longrightarrow> op \<approx>) A B

Regularizing is done in 3 phases:

 - First a regularized term is created

 - Next we prove that the original theorem implies the new one

 - Finally using MP we get the new theorem.

To prove that the old theorem implies the new one, we first

atomize it and then try:

 - Reflexivity of the relation

 - Assumption

 - Elimnating quantifiers on both sides of toplevel implication

 - Simplifying implications on both sides of toplevel implication

 - Ball (Respects ?E) ?P = All ?P

 - (\<And>x. ?R x \<Longrightarrow> ?P x \<longrightarrow> ?Q x) \<Longrightarrow> All ?P \<longrightarrow> Ball ?R ?Q

*)

text {* tyRel takes a type and builds a relation that a quantifier over this

  type needs to respect. *}

ML {*

fun tyRel ty rty rel lthy =

  if Sign.typ_instance (ProofContext.theory_of lthy) (ty, rty)

  then rel

  else (case ty of

          Type (s, tys) =>

            let

              val tys_rel = map (fn ty => ty --> ty --> @{typ bool}) tys;

              val ty_out = ty --> ty --> @{typ bool};

              val tys_out = tys_rel ---> ty_out;

            in

            (case (maps_lookup (ProofContext.theory_of lthy) s) of

               SOME (info) => list_comb (Const (#relfun info, tys_out),

                              map (fn ty => tyRel ty rty rel lthy) tys)

             | NONE  => HOLogic.eq_const ty

            )

            end

        | _ => HOLogic.eq_const ty)

*}

(* 

ML {* cterm_of @{theory} 

  (tyRel @{typ "'a \<Rightarrow> 'a list \<Rightarrow> 't \<Rightarrow> 't"} (Logic.varifyT @{typ "'a list"}) 

         @{term "f::('a list \<Rightarrow> 'a list \<Rightarrow> bool)"} @{context}) 

*} 

*)

definition

  Babs :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"

where

  "(x \<in> p) \<Longrightarrow> (Babs p m x = m x)"

(* TODO: Consider defining it with an "if"; sth like:

   Babs p m = \<lambda>x. if x \<in> p then m x else undefined

*)

ML {*

fun needs_lift (rty as Type (rty_s, _)) ty =

  case ty of

    Type (s, tys) =>

      (s = rty_s) orelse (exists (needs_lift rty) tys)

  | _ => false

*}

ML {*

fun mk_babs ty ty' = Const (@{const_name "Babs"}, [ty' --> @{typ bool}, ty] ---> ty)

fun mk_ball ty = Const (@{const_name "Ball"}, [ty, ty] ---> @{typ bool})

fun mk_bex ty = Const (@{const_name "Bex"}, [ty, ty] ---> @{typ bool})

fun mk_resp ty = Const (@{const_name Respects}, [[ty, ty] ---> @{typ bool}, ty] ---> @{typ bool})

*}

(* applies f to the subterm of an abstractions, otherwise to the given term *)

ML {*

fun apply_subt f trm =

  case trm of

    Abs (x, T, t) => 

       let 

         val (x', t') = Term.dest_abs (x, T, t)

       in

         Term.absfree (x', T, f t') 

       end

  | _ => f trm

*}

(* FIXME: if there are more than one quotient, then you have to look up the relation *)

ML {*

fun my_reg lthy rel rty trm =

  case trm of

    Abs (x, T, t) =>

       if (needs_lift rty T) then

         let

            val rrel = tyRel T rty rel lthy

         in

           (mk_babs (fastype_of trm) T) $ (mk_resp T $ rrel) $ (apply_subt (my_reg lthy rel rty) trm)

         end

       else

         Abs(x, T, (apply_subt (my_reg lthy rel rty) t))

  | Const (@{const_name "All"}, ty) $ (t as Abs (x, T, _)) =>

       let

          val ty1 = domain_type ty

          val ty2 = domain_type ty1

          val rrel = tyRel T rty rel lthy

       in

         if (needs_lift rty T) then

           (mk_ball ty1) $ (mk_resp ty2 $ rrel) $ (apply_subt (my_reg lthy rel rty) t)

         else

           Const (@{const_name "All"}, ty) $ apply_subt (my_reg lthy rel rty) t

       end

  | Const (@{const_name "Ex"}, ty) $ (t as Abs (x, T, _)) =>

       let

          val ty1 = domain_type ty

          val ty2 = domain_type ty1

          val rrel = tyRel T rty rel lthy

       in

         if (needs_lift rty T) then

           (mk_bex ty1) $ (mk_resp ty2 $ rrel) $ (apply_subt (my_reg lthy rel rty) t)

         else

           Const (@{const_name "Ex"}, ty) $ apply_subt (my_reg lthy rel rty) t

       end

  | Const (@{const_name "op ="}, ty) $ t =>

      if needs_lift rty (fastype_of t) then

        (tyRel (fastype_of t) rty rel lthy) $ t   (* FIXME: t should be regularised too *)

      else Const (@{const_name "op ="}, ty) $ (my_reg lthy rel rty t)

  | t1 $ t2 => (my_reg lthy rel rty t1) $ (my_reg lthy rel rty t2)

  | _ => trm

*}

(* For polymorphic types we need to find the type of the Relation term. *)

(* TODO: we assume that the relation is a Constant. Is this always true? *)

ML {*

fun my_reg_inst lthy rel rty trm =

  case rel of

    Const (n, _) => Syntax.check_term lthy

      (my_reg lthy (Const (n, dummyT)) (Logic.varifyT rty) trm)

*}

(*

ML {*

  val r = Free ("R", dummyT);

  val t = (my_reg_inst @{context} r @{typ "'a list"} @{term "\<forall>(x::'b list). P x"});

  val t2 = Syntax.check_term @{context} t;

  cterm_of @{theory} t2

*}

*)

text {* Assumes that the given theorem is atomized *}

ML {*

  fun build_regularize_goal thm rty rel lthy =

     Logic.mk_implies

       ((prop_of thm),

       (my_reg_inst lthy rel rty (prop_of thm)))

*}

lemma universal_twice:

  assumes *: "\<And>x. (P x \<longrightarrow> Q x)"

  shows "(\<forall>x. P x) \<longrightarrow> (\<forall>x. Q x)"

using * by auto

lemma implication_twice:

  assumes a: "c \<longrightarrow> a"

  assumes b: "a \<Longrightarrow> b \<longrightarrow> d"

  shows "(a \<longrightarrow> b) \<longrightarrow> (c \<longrightarrow> d)"

using a b by auto

ML {*

fun regularize thm rty rel rel_eqv rel_refl lthy =

  let

    val goal = build_regularize_goal thm rty rel lthy;

    fun tac ctxt =

      (ObjectLogic.full_atomize_tac) THEN'

      REPEAT_ALL_NEW (FIRST' [

        rtac rel_refl,

        atac,

        rtac @{thm universal_twice},

        (rtac @{thm impI} THEN' atac),

        rtac @{thm implication_twice},

        EqSubst.eqsubst_tac ctxt [0]

          [(@{thm equiv_res_forall} OF [rel_eqv]),

           (@{thm equiv_res_exists} OF [rel_eqv])],

        (* For a = b \<longrightarrow> a \<approx> b *)

        (rtac @{thm impI} THEN' (asm_full_simp_tac HOL_ss) THEN' rtac rel_refl),