theory QuotScript
imports Main
begin

definition 
  "EQUIV E \<equiv> \<forall>x y. E x y = (E x = E y)" 

definition
  "REFL E \<equiv> \<forall>x. E x x"

definition 
  "SYM E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x"

definition
  "TRANS E \<equiv> \<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z"

lemma EQUIV_REFL_SYM_TRANS:
  shows "EQUIV E = (REFL E \<and> SYM E \<and> TRANS E)"
unfolding EQUIV_def REFL_def SYM_def TRANS_def expand_fun_eq
by (blast)

lemma EQUIV_REFL:
  shows "EQUIV E ==> REFL E"
  by (simp add: EQUIV_REFL_SYM_TRANS)

definition
  "PART_EQUIV E \<equiv> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))"

lemma EQUIV_IMP_PART_EQUIV:
  assumes a: "EQUIV E"
  shows "PART_EQUIV E"
using a unfolding EQUIV_def PART_EQUIV_def
by auto

definition
  "QUOTIENT E Abs Rep \<equiv> (\<forall>a. Abs (Rep a) = a) \<and> 
                        (\<forall>a. E (Rep a) (Rep a)) \<and> 
                        (\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"

lemma QUOTIENT_ABS_REP:
  assumes a: "QUOTIENT E Abs Rep"
  shows "Abs (Rep a) = a" 
using a unfolding QUOTIENT_def
by simp

lemma QUOTIENT_REP_REFL:
  assumes a: "QUOTIENT E Abs Rep"
  shows "E (Rep a) (Rep a)" 
using a unfolding QUOTIENT_def
by blast

lemma QUOTIENT_REL:
  assumes a: "QUOTIENT E Abs Rep"
  shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"
using a unfolding QUOTIENT_def
by blast

lemma QUOTIENT_REL_ABS:
  assumes a: "QUOTIENT E Abs Rep"
  shows "E r s \<Longrightarrow> Abs r = Abs s"
using a unfolding QUOTIENT_def
by blast

lemma QUOTIENT_REL_ABS_EQ:
  assumes a: "QUOTIENT E Abs Rep"
  shows "E r r \<Longrightarrow> E s s \<Longrightarrow> E r s = (Abs r = Abs s)"
using a unfolding QUOTIENT_def
by blast

lemma QUOTIENT_REL_REP:
  assumes a: "QUOTIENT E Abs Rep"
  shows "E (Rep a) (Rep b) = (a = b)"
using a unfolding QUOTIENT_def
by metis

lemma QUOTIENT_REP_ABS:
  assumes a: "QUOTIENT E Abs Rep"
  shows "E r r \<Longrightarrow> E (Rep (Abs r)) r"
using a unfolding QUOTIENT_def
by blast

lemma IDENTITY_EQUIV:
  shows "EQUIV (op =)"
unfolding EQUIV_def
by auto

lemma IDENTITY_QUOTIENT:
  shows "QUOTIENT (op =) id id"
unfolding QUOTIENT_def id_def
by blast

lemma QUOTIENT_SYM:
  assumes a: "QUOTIENT E Abs Rep"
  shows "SYM E"
using a unfolding QUOTIENT_def SYM_def
by metis

lemma QUOTIENT_TRANS:
  assumes a: "QUOTIENT E Abs Rep"
  shows "TRANS E"
using a unfolding QUOTIENT_def TRANS_def
by metis

fun
  prod_rel
where
  "prod_rel r1 r2 = (\<lambda>(a,b) (c,d). r1 a c \<and> r2 b d)"

fun
  fun_map
where
  "fun_map f g h x = g (h (f x))"


abbreviation
  fun_map_syn (infixr "--->" 55)
where
  "f ---> g \<equiv> fun_map f g"

lemma FUN_MAP_I:
  shows "(id ---> id) = id"
by (simp add: expand_fun_eq id_def)

lemma IN_FUN:
  shows "x \<in> ((f ---> g) s) = g (f x \<in> s)"
by (simp add: mem_def)

fun
  FUN_REL 
where
  "FUN_REL E1 E2 f g = (\<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"

abbreviation
  FUN_REL_syn (infixr "===>" 55)
where
  "E1 ===> E2 \<equiv> FUN_REL E1 E2"

lemma FUN_REL_EQ:
  "(op =) ===> (op =) = (op =)"
by (simp add: expand_fun_eq)

lemma FUN_QUOTIENT:
  assumes q1: "QUOTIENT R1 abs1 rep1"
  and     q2: "QUOTIENT R2 abs2 rep2"
  shows "QUOTIENT (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
proof -
  have "\<forall>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
    apply(simp add: expand_fun_eq)
    using q1 q2
    apply(simp add: QUOTIENT_def)
    done
  moreover
  have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
    apply(auto)
    using q1 q2 unfolding QUOTIENT_def
    apply(metis)
    done
  moreover
  have "\<forall>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and> 
        (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
    apply(auto simp add: expand_fun_eq)
    using q1 q2 unfolding QUOTIENT_def
    apply(metis)
    using q1 q2 unfolding QUOTIENT_def
    apply(metis)
    using q1 q2 unfolding QUOTIENT_def
    apply(metis)
    using q1 q2 unfolding QUOTIENT_def
    apply(metis)
    done
  ultimately
  show "QUOTIENT (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
    unfolding QUOTIENT_def by blast
qed

definition
  Respects
where
  "Respects R x \<equiv> (R x x)"

lemma IN_RESPECTS:
  shows "(x \<in> Respects R) = R x x"
unfolding mem_def Respects_def by simp

lemma RESPECTS_THM:
  shows "Respects (R1 ===> R2) f = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (f y))"
unfolding Respects_def
by (simp add: expand_fun_eq) 

lemma RESPECTS_MP:
  assumes a: "Respects (R1 ===> R2) f"
  and     b: "R1 x y"
  shows "R2 (f x) (f y)"
using a b unfolding Respects_def
by simp

lemma RESPECTS_REP_ABS:
  assumes a: "QUOTIENT R1 Abs1 Rep1"
  and     b: "Respects (R1 ===> R2) f"
  and     c: "R1 x x"
  shows "R2 (f (Rep1 (Abs1 x))) (f x)"
using a b[simplified RESPECTS_THM] c unfolding QUOTIENT_def
by blast

lemma RESPECTS_o:
  assumes a: "Respects (R2 ===> R3) f"
  and     b: "Respects (R1 ===> R2) g"
  shows "Respects (R1 ===> R3) (f o g)"
using a b unfolding Respects_def
by simp

(*
definition
  "RES_EXISTS_EQUIV R P \<equiv> (\<exists>x \<in> Respects R. P x) \<and> 
                          (\<forall>x\<in> Respects R. \<forall>y\<in> Respects R. P x \<and> P y \<longrightarrow> R x y)"
*)

lemma FUN_REL_EQ_REL:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  shows "(R1 ===> R2) f g = ((Respects (R1 ===> R2) f) \<and> (Respects (R1 ===> R2) g) 
                             \<and> ((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g))"
using FUN_QUOTIENT[OF q1 q2] unfolding Respects_def QUOTIENT_def expand_fun_eq
by blast

(* q1 and q2 not used; see next lemma *)
lemma FUN_REL_MP:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  shows "(R1 ===> R2) f g \<Longrightarrow> R1 x y \<Longrightarrow> R2 (f x) (g y)"
by simp

lemma FUN_REL_IMP:
  shows "(R1 ===> R2) f g \<Longrightarrow> R1 x y \<Longrightarrow> R2 (f x) (g y)"
by simp

lemma FUN_REL_EQUALS:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  and     r1: "Respects (R1 ===> R2) f"
  and     r2: "Respects (R1 ===> R2) g" 
  shows "((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g) = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"
apply(rule_tac iffI)
using FUN_QUOTIENT[OF q1 q2] r1 r2 unfolding QUOTIENT_def Respects_def
apply(metis FUN_REL_IMP)
using r1 unfolding Respects_def expand_fun_eq
apply(simp (no_asm_use))
apply(metis QUOTIENT_REL[OF q2] QUOTIENT_REL_REP[OF q1])
done

(* ask Peter: FUN_REL_IMP used twice *) 
lemma FUN_REL_IMP2:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  and     r1: "Respects (R1 ===> R2) f"
  and     r2: "Respects (R1 ===> R2) g" 
  and     a:  "(Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g"
  shows "R1 x y \<Longrightarrow> R2 (f x) (g y)"
using q1 q2 r1 r2 a
by (simp add: FUN_REL_EQUALS)

lemma EQUALS_PRS:
  assumes q: "QUOTIENT R Abs Rep"
  shows "(x = y) = R (Rep x) (Rep y)"
by (simp add: QUOTIENT_REL_REP[OF q]) 

lemma EQUALS_RSP:
  assumes q: "QUOTIENT R Abs Rep"
  and     a: "R x1 x2" "R y1 y2"
  shows "R x1 y1 = R x2 y2"
using QUOTIENT_SYM[OF q] QUOTIENT_TRANS[OF q] unfolding SYM_def TRANS_def
using a by blast

lemma LAMBDA_PRS:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
unfolding expand_fun_eq
using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2]
by simp

lemma LAMBDA_PRS1:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  shows "(\<lambda>x. f x) = (Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x)"
unfolding expand_fun_eq
using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2]
by (simp)

lemma APP_PRS:
  assumes q1: "QUOTIENT R1 abs1 rep1"
  and     q2: "QUOTIENT R2 abs2 rep2"
  shows "abs2 ((abs1 ---> rep2) f (rep1 x)) = f x"
unfolding expand_fun_eq
using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2]
by simp

(* Ask Peter: assumption q1 and q2 not used and lemma is the 'identity' *)
lemma LAMBDA_RSP:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  and     a: "(R1 ===> R2) f1 f2"
  shows "(R1 ===> R2) (\<lambda>x. f1 x) (\<lambda>y. f2 y)"
by (rule a)

(* ASK Peter about next four lemmas in quotientScript
lemma ABSTRACT_PRS:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  shows "f = (Rep1 ---> Abs2) ???"
*)

lemma LAMBDA_REP_ABS_RSP:
  assumes r1: "\<And>r r'. R1 r r' \<Longrightarrow>R1 r (Rep1 (Abs1 r'))"
  and     r2: "\<And>r r'. R2 r r' \<Longrightarrow>R2 r (Rep2 (Abs2 r'))"
  shows "(R1 ===> R2) f1 f2 \<Longrightarrow> (R1 ===> R2) f1 ((Abs1 ---> Rep2) ((Rep1 ---> Abs2) f2))"
using r1 r2 by auto

lemma REP_ABS_RSP:
  assumes q: "QUOTIENT R Abs Rep"
  and     a: "R x1 x2"
  shows "R x1 (Rep (Abs x2))"
  and   "R (Rep (Abs x1)) x2"
proof -
  show "R x1 (Rep (Abs x2))" 
    using q a by (metis QUOTIENT_REL[OF q] QUOTIENT_ABS_REP[OF q] QUOTIENT_REP_REFL[OF q])
next
  show "R (Rep (Abs x1)) x2"
    using q a by (metis QUOTIENT_REL[OF q] QUOTIENT_ABS_REP[OF q] QUOTIENT_REP_REFL[OF q] QUOTIENT_SYM[of q])
qed

(* ----------------------------------------------------- *)
(* Quantifiers: FORALL, EXISTS, EXISTS_UNIQUE,           *)
(*              RES_FORALL, RES_EXISTS, RES_EXISTS_EQUIV *)
(* ----------------------------------------------------- *)

(* what is RES_FORALL *)
(*--`!R (abs:'a -> 'b) rep. QUOTIENT R abs rep ==>
         !f. $! f = RES_FORALL (respects R) ((abs --> I) f)`--*)
(*as peter here *)

(* bool theory: COND, LET *)

lemma IF_PRS:
  assumes q: "QUOTIENT R Abs Rep"
  shows "If a b c = Abs (If a (Rep b) (Rep c))"
using QUOTIENT_ABS_REP[OF q] by auto

(* ask peter: no use of q *)
lemma IF_RSP:
  assumes q: "QUOTIENT R Abs Rep"
  and     a: "a1 = a2" "R b1 b2" "R c1 c2"
  shows "R (If a1 b1 c1) (If a2 b2 c2)"
using a by auto

lemma LET_PRS:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  shows "Let x f = Abs2 (Let (Rep1 x) ((Abs1 ---> Rep2) f))"
using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2] by auto

lemma LET_RSP:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  and     a1: "(R1 ===> R2) f g"
  and     a2: "R1 x y"
  shows "R2 (Let x f) (Let y g)"
using FUN_REL_MP[OF q1 q2 a1] a2
by auto


(* ask peter what are literal_case *)
(* literal_case_PRS *)
(* literal_case_RSP *)


(* FUNCTION APPLICATION *)

lemma APPLY_PRS:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  shows "f x = Abs2 (((Abs1 ---> Rep2) f) (Rep1 x))"
using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2] by auto

(* ask peter: no use of q1 q2 *)
lemma APPLY_RSP:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  and     a: "(R1 ===> R2) f g" "R1 x y"
  shows "R2 (f x) (g y)"
using a by (rule FUN_REL_IMP)

lemma APPLY_RSP2:
  assumes a: "(R1 ===> R2) f g" "R1 x y"
  shows "R2 (f x) (g y)"
using a by (rule FUN_REL_IMP)


(* combinators: I, K, o, C, W *)

lemma I_PRS:
  assumes q: "QUOTIENT R Abs Rep"
  shows "id e = Abs (id (Rep e))"
using QUOTIENT_ABS_REP[OF q] by auto

lemma I_RSP:
  assumes q: "QUOTIENT R Abs Rep"
  and     a: "R e1 e2"
  shows "R (id e1) (id e2)"
using a by auto

lemma o_PRS:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  and     q3: "QUOTIENT R3 Abs3 Rep3"
  shows "f o g = (Rep1 ---> Abs3) (((Abs2 ---> Rep3) f) o ((Abs1 ---> Rep2) g))"
using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2] QUOTIENT_ABS_REP[OF q3]
unfolding o_def expand_fun_eq
by simp

lemma o_RSP:
  assumes q1: "QUOTIENT R1 Abs1 Rep1"
  and     q2: "QUOTIENT R2 Abs2 Rep2"
  and     q3: "QUOTIENT R3 Abs3 Rep3"
  and     a1: "(R2 ===> R3) f1 f2"
  and     a2: "(R1 ===> R2) g1 g2"
  shows "(R1 ===> R3) (f1 o g1) (f2 o g2)"
using a1 a2 unfolding o_def expand_fun_eq
by (auto)


(* TODO: Put the following lemmas in proper places *)

lemma equiv_res_forall:
  fixes P :: "'a \<Rightarrow> bool"
  assumes a: "EQUIV E"
  shows "Ball (Respects E) P = (All P)"
  using a by (metis EQUIV_def IN_RESPECTS a)

lemma equiv_res_exists:
  fixes P :: "'a \<Rightarrow> bool"
  assumes a: "EQUIV E"
  shows "Bex (Respects E) P = (Ex P)"
  using a by (metis EQUIV_def IN_RESPECTS a)

lemma FORALL_REGULAR:
  assumes a: "!x :: 'a. (P x --> Q x)"
  and     b: "All P"
  shows "All Q"
  using a b by (metis)

lemma EXISTS_REGULAR:
  assumes a: "!x :: 'a. (P x --> Q x)"
  and     b: "Ex P"
  shows "Ex Q"
  using a b by (metis)

lemma RES_FORALL_REGULAR:
  assumes a: "!x :: 'a. (R x --> P x --> Q x)"
  and     b: "Ball R P"
  shows "Ball R Q"
  using a b by (metis COMBC_def Collect_def Collect_mem_eq)

lemma RES_EXISTS_REGULAR:
  assumes a: "!x :: 'a. (R x --> P x --> Q x)"
  and     b: "Bex R P"
  shows "Bex R Q"
  using a b by (metis COMBC_def Collect_def Collect_mem_eq)

lemma LEFT_RES_FORALL_REGULAR:
  assumes a: "!x. (R x \<and> (Q x --> P x))"
  shows "Ball R Q ==> All P"
  using a
  apply (metis COMBC_def Collect_def Collect_mem_eq a)
  done

lemma RIGHT_RES_FORALL_REGULAR:
  assumes a: "\<And>x :: 'a. (R x \<Longrightarrow> P x \<longrightarrow> Q x)"
  shows "All P \<longrightarrow> Ball R Q"
  using a
  apply (metis COMBC_def Collect_def Collect_mem_eq a)
  done

lemma LEFT_RES_EXISTS_REGULAR:
  assumes a: "!x :: 'a. (R x --> Q x --> P x)"
  shows "Bex R Q ==> Ex P"
  using a by (metis COMBC_def Collect_def Collect_mem_eq)

lemma RIGHT_RES_EXISTS_REGULAR:
  assumes a: "!x :: 'a. (R x \<and> (P x --> Q x))"
  shows "Ex P \<Longrightarrow> Bex R Q"
  using a by (metis COMBC_def Collect_def Collect_mem_eq)

(* TODO: Add similar *)
lemma RES_FORALL_RSP:
  shows "\<And>f g. (R ===> (op =)) f g ==>
        (Ball (Respects R) f = Ball (Respects R) g)"
  apply (simp add: FUN_REL.simps Ball_def IN_RESPECTS)
  done

lemma RES_EXISTS_RSP:
  shows "\<And>f g. (R ===> (op =)) f g ==>
        (Bex (Respects R) f = Bex (Respects R) g)"
  apply (simp add: FUN_REL.simps Bex_def IN_RESPECTS)
  done


lemma FORALL_PRS:
  assumes a: "QUOTIENT R absf repf"
  shows "All f = Ball (Respects R) ((absf ---> id) f)"
  using a
  unfolding QUOTIENT_def
  by (metis IN_RESPECTS fun_map.simps id_apply)

lemma EXISTS_PRS:
  assumes a: "QUOTIENT R absf repf"
  shows "Ex f = Bex (Respects R) ((absf ---> id) f)"
  using a
  unfolding QUOTIENT_def
  by (metis COMBC_def Collect_def Collect_mem_eq IN_RESPECTS fun_map.simps id_apply mem_def)

lemma COND_PRS:
  assumes a: "QUOTIENT R absf repf"
  shows "(if a then b else c) = absf (if a then repf b else repf c)"
  using a unfolding QUOTIENT_def by auto

(* These are the fixed versions, general ones need to be proved. *)
lemma ho_all_prs:
  shows "((op = ===> op =) ===> op =) All All"
  by auto

lemma ho_ex_prs:
  shows "((op = ===> op =) ===> op =) Ex Ex"
  by auto

end