theory QuotScript

imports Plain ATP_Linkup

begin

definition

  "reflp E \<equiv> \<forall>x. E x x"

  "symp E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x"

definition

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

lemma equivp_reflp_symp_transp:

  shows "equivp E = (reflp E \<and> symp E \<and> transp E)"

lemma equivp_refl:

  shows "equivp R \<Longrightarrow> (\<And>x. R x x)"

  by (simp add: equivp_reflp_symp_transp reflp_def)

lemma equivp_reflp:

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

  by (simp add: equivp_reflp_symp_transp reflp_def)

definition

  "part_equivp 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 equivp_IMP_part_equivp:

  assumes a: "equivp E"

  shows "part_equivp E"

definition

                        (\<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"

lemma Quotient_rep_reflp:

  assumes a: "Quotient E Abs Rep"

  shows "E (Rep a) (Rep a)"

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))"

lemma Quotient_rel_rep:

  assumes a: "Quotient R Abs Rep"

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

lemma Quotient_rep_abs:

  assumes a: "Quotient R Abs Rep"

  shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"

lemma identity_equivp:

  shows "equivp (op =)"

lemma identity_quotient:

  shows "Quotient (op =) id id"

lemma Quotient_symp:

  assumes a: "Quotient E Abs Rep"

  shows "symp E"

lemma Quotient_transp:

  assumes a: "Quotient E Abs Rep"

  shows "transp E"

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_id:

  shows "(id ---> id) = id"

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 =) \<equiv> (op =)"

by (rule eq_reflection) (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"

lemma equals_rsp:

  assumes q: "Quotient R Abs Rep"

  and     a: "R xa xb" "R ya yb"

  shows "R xa ya = R xb yb"

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)"

lemma lambda_prs1:

  assumes q1: "Quotient R1 Abs1 Rep1"

  and     q2: "Quotient R2 Abs2 Rep2"

  shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"

lemma rep_abs_rsp:

  assumes q: "Quotient R Abs Rep"

  and     a: "R x1 x2"

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

(* In the following theorem R1 can be instantiated with anything,

   but we know some of the types of the Rep and Abs functions;

lemma apply_rsp:

  assumes q: "Quotient R1 Abs1 Rep1"

  and     a: "(R1 ===> R2) f g" "R1 x y"

  shows "R2 ((f::'a\<Rightarrow>'c) x) ((g::'a\<Rightarrow>'c) y)"

lemma apply_rsp':

  assumes a: "(R1 ===> R2) f g" "R1 x y"

  shows "R2 (f x) (g y)"

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

lemma ball_reg_eqv:

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

  assumes a: "equivp R"

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

  by (metis equivp_def in_respects a)

lemma bex_reg_eqv:

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

  assumes a: "equivp R"

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

  by (metis equivp_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: "equivp R"

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

  by (metis equivp_reflp in_respects a)

lemma bex_reg_right:

  assumes a: "equivp R"

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

  by (metis equivp_reflp in_respects a)

lemma ball_reg_eqv_range:

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

  and x::"'a"

  assumes a: "equivp 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 equivp_reflp_symp_transp[of "R2"]

  apply(simp add: reflp_def)

  apply(simp)

  apply(simp)

  done

lemma bex_reg_eqv_range:

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

  and x::"'a"

  assumes a: "equivp 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 equivp_reflp_symp_transp[of "R2"]

  apply(simp add: reflp_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"

  shows "Ex Q"

  using a b by (metis)

lemma ball_reg:

  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 bex_reg:

  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 ball_all_comm:

  "(\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)) \<Longrightarrow> ((\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y))"

by auto

lemma bex_ex_comm:

  "((\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)) \<Longrightarrow> ((\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y))"

by auto

(* 2 lemmas needed for proving repabs_inj *)

lemma ball_rsp:

  assumes a: "(R ===> (op =)) f g"

  shows "Ball (Respects R) f = Ball (Respects R) g"

  using a by (simp add: Ball_def in_respects)

lemma bex_rsp:

  assumes a: "(R ===> (op =)) f g"

  shows "(Bex (Respects R) f = Bex (Respects R) g)"

  using a by (simp add: Bex_def in_respects)

(* 2 lemmas needed for cleaning of quantifiers *)

lemma all_prs:

  assumes a: "Quotient R absf repf"

  shows "Ball (Respects R) ((absf ---> id) f) = All f"

  using a unfolding Quotient_def

  by (metis in_respects fun_map.simps id_apply)

lemma ex_prs:

  assumes a: "Quotient R absf repf"

  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"

  using a unfolding Quotient_def

  by (metis COMBC_def Collect_def Collect_mem_eq in_respects fun_map.simps id_apply)

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 in_fun:

  shows "x \<in> ((f ---> g) s) = g (f x \<in> s)"

by (simp add: mem_def)

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_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_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_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

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

(* Not used since in the end we just unfold fun_map *)

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 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

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 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

(* Not used *)

lemma rep_abs_rsp_left:

  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_reflp[OF q])