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 \<Longrightarrow> (\<And>x. E x x)"

  by (simp add: EQUIV_REFL_SYM_TRANS REFL_def)

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 R Abs Rep"

  shows "R (Rep a) (Rep b) = (a = b)"

using a unfolding QUOTIENT_def

by metis

lemma QUOTIENT_REP_ABS:

  assumes a: "QUOTIENT R Abs Rep"

  shows "R r r \<Longrightarrow> R (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

(* TODO: it is the same as APPLY_RSP *)

(* 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)

(* We don't use it, it is exactly the same as QUOTIENT_REL_REP but wrong way *)

lemma EQUALS_PRS:

  assumes q: "QUOTIENT R Abs Rep"

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

by (rule QUOTIENT_REL_REP[OF q, symmetric])

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

(* Not used since applic_prs proves a version for an arbitrary number of arguments *)

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))"

using q a by (metis QUOTIENT_REL[OF q] QUOTIENT_ABS_REP[OF q] QUOTIENT_REP_REFL[OF q])

(* Not used *)

lemma REP_ABS_RSP_LEFT:

  assumes q: "QUOTIENT R Abs Rep"

  and     a: "R x1 x2"

  and   "R (Rep (Abs x1)) x2"

  shows "R x1 (Rep (Abs x2))"

using q a by (metis QUOTIENT_REL[OF q] QUOTIENT_ABS_REP[OF q] QUOTIENT_REP_REFL[OF q] QUOTIENT_SYM[of q])

(* ----------------------------------------------------- *)

(* Quantifiers: FORALL, EXISTS, EXISTS_UNIQUE,           *)

(*              RES_FORALL, RES_EXISTS, RES_EXISTS_EQUIV *)

(* ----------------------------------------------------- *)

(* 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 *)

(* We use id_simps which includes id_apply; so these 2 theorems can be removed *)

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)

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

(* Set of lemmas for regularisation of ball and bex *)

lemma ball_reg_eqv:

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

  assumes a: "EQUIV R"

  shows "Ball (Respects R) P = (All P)"

  by (metis EQUIV_def IN_RESPECTS a)

lemma bex_reg_eqv:

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

  assumes a: "EQUIV R"

  shows "Bex (Respects R) P = (Ex P)"

  by (metis EQUIV_def IN_RESPECTS a)

lemma ball_reg_right:

  assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"

  shows "All P \<longrightarrow> Ball R Q"

  by (metis COMBC_def Collect_def Collect_mem_eq a)

lemma bex_reg_left:

  assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"

  shows "Bex R Q \<longrightarrow> Ex P"

  by (metis COMBC_def Collect_def Collect_mem_eq a)

lemma ball_reg_left:

  assumes a: "EQUIV R"

  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"

  by (metis EQUIV_REFL IN_RESPECTS a)

lemma bex_reg_right:

  assumes a: "EQUIV R"

  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"

  by (metis EQUIV_REFL IN_RESPECTS a)

lemma ball_reg_eqv_range:

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

  and x::"'a"

  assumes a: "EQUIV R2"

  shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"

  apply(rule iffI)

  apply(rule allI)

  apply(drule_tac x="\<lambda>y. f x" in bspec)

  apply(simp add: Respects_def IN_RESPECTS)

  apply(rule impI)

  using a EQUIV_REFL_SYM_TRANS[of "R2"]

  apply(simp add: REFL_def)

  apply(simp)

  apply(simp)

  done

lemma bex_reg_eqv_range:

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

  and x::"'a"

  assumes a: "EQUIV R2"

  shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"

  apply(auto)

  apply(rule_tac x="\<lambda>y. f x" in bexI)

  apply(simp)

  apply(simp add: Respects_def IN_RESPECTS)

  apply(rule impI)

  using a EQUIV_REFL_SYM_TRANS[of "R2"]

  apply(simp add: REFL_def)

  done

lemma all_reg:

  assumes a: "!x :: 'a. (P x --> Q x)"

  and     b: "All P"

  shows "All Q"

  using a b by (metis)

lemma ex_reg:

  assumes a: "!x :: 'a. (P x --> Q x)"

  and     b: "Ex P"