(*  Title:      QuotMain.thy

    Author:     Cezary Kaliszyk and Christian Urban

*)

theory QuotMain

imports QuotBase

uses

  ("quotient_info.ML")

  ("quotient_typ.ML")

  ("quotient_def.ML")

  ("quotient_term.ML")

  ("quotient_tacs.ML")

begin

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

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

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

  assumes equivp: "equivp 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 homeier_lem9:

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

proof -

  have a: "R x x" using equivp by (simp add: equivp_reflp_symp_transp reflp_def)

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

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

    using equivp unfolding equivp_def by simp

qed

theorem homeier_thm10:

  shows "abs (rep a) = a"

  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 homeier_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 homeier_lem7:

  shows "(R x = R y) = (Abs (R x) = Abs (R y))" (is "?LHS = ?RHS")

proof -

  have "?RHS = (Rep (Abs (R x)) = Rep (Abs (R y)))" by (simp add: rep_inject)

  also have "\<dots> = ?LHS" by (simp add: abs_inverse)

  finally show "?LHS = ?RHS" by simp

qed

theorem homeier_thm11:

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

  unfolding abs_def

  by (simp only: equivp[simplified equivp_def] homeier_lem7)

lemma rep_refl:

  shows "R (rep a) (rep a)"

  unfolding rep_def

  by (simp add: equivp[simplified equivp_def])

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: homeier_thm10 homeier_thm11)

lemma Quotient:

  shows "Quotient R abs rep"

  unfolding Quotient_def

  apply(simp add: homeier_thm10)

  apply(simp add: rep_refl)

  apply(subst homeier_thm11[symmetric])

  apply(simp add: equivp[simplified equivp_def])

  done

end

section {* ML setup *}

text {* Auxiliary data for the quotient package *}

use "quotient_info.ML"

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

lemmas [quot_thm] = fun_quotient

lemmas [quot_respect] = quot_rel_rsp

lemmas [quot_equiv] = identity_equivp

text {* Lemmas about simplifying id's. *}

lemmas [id_simps] =

  id_def[symmetric]

  fun_map_id

  id_apply

  id_o

  o_id

  eq_comp_r

text {* Translation functions for the lifting process. *}

use "quotient_term.ML" 

text {* Definitions of the quotient types. *}

use "quotient_typ.ML"

text {* Definitions for quotient constants. *}

use "quotient_def.ML"

text {*

  An auxiliary constant for recording some information

  about the lifted theorem in a tactic.

*}

definition

  "Quot_True x \<equiv> True"

lemma 

  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"

  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"

  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"

  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"

  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"

  by (simp_all add: Quot_True_def ext)

lemma QT_imp: "Quot_True a \<equiv> Quot_True b"

  by (simp add: Quot_True_def)

text {* Tactics for proving the lifted theorems *}

use "quotient_tacs.ML"

section {* Methods / Interface *}

(* TODO inline *)

ML {*

fun mk_method1 tac thms ctxt =

  SIMPLE_METHOD (HEADGOAL (tac ctxt thms)) 

fun mk_method2 tac ctxt =

  SIMPLE_METHOD (HEADGOAL (tac ctxt)) 

*}

method_setup lifting =

  {* Attrib.thms >> (mk_method1 Quotient_Tacs.lift_tac) *}

  {* lifts theorems to quotient types *}

method_setup lifting_setup =

  {* Attrib.thm >> (mk_method1 Quotient_Tacs.procedure_tac) *}

  {* sets up the three goals for the quotient lifting procedure *}

method_setup regularize =

  {* Scan.succeed (mk_method2 Quotient_Tacs.regularize_tac) *}

  {* proves the regularization goals from the quotient lifting procedure *}

method_setup injection =

  {* Scan.succeed (mk_method2 Quotient_Tacs.all_injection_tac) *}

  {* proves the rep/abs injection goals from the quotient lifting procedure *}

method_setup cleaning =

  {* Scan.succeed (mk_method2 Quotient_Tacs.clean_tac) *}

  {* proves the cleaning goals from the quotient lifting procedure *}

attribute_setup quot_lifted =

  {* Scan.succeed Quotient_Tacs.lifted_attrib *}

  {* lifts theorems to quotient types *}

end