imports Plain ATP_Linkup

text {*

  Basic definition for equivalence relations

  that are represented by predicates.

*}

definition

  "equivp E \<longleftrightarrow> (\<forall>x y. E x y = (E x = E y))"

definition

  "reflp E \<longleftrightarrow> (\<forall>x. E x x)"

definition

  "symp E \<longleftrightarrow> (\<forall>x y. E x y \<longrightarrow> E y x)"

definition

  "transp E \<longleftrightarrow> (\<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)"

  unfolding equivp_def reflp_def symp_def transp_def expand_fun_eq

  by blast

lemma equivp_reflp:

  shows "equivp E \<Longrightarrow> E x x"

  by (simp only: equivp_reflp_symp_transp reflp_def)

lemma equivp_symp:

  shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y x"

  by (metis equivp_reflp_symp_transp symp_def)

lemma equivp_transp:

  shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y z \<Longrightarrow> E x z"

  by (metis equivp_reflp_symp_transp transp_def)

lemma equivpI:

  assumes "reflp R" "symp R" "transp R"

  shows "equivp R"

  using assms by (simp add: equivp_reflp_symp_transp)

lemma eq_imp_rel:  

  shows "equivp R \<Longrightarrow> a = b \<longrightarrow> R a b" 

  by (simp add: equivp_reflp)

lemma identity_equivp:

  shows "equivp (op =)"

  unfolding equivp_def

  by auto

text {* Partial equivalences: not yet used anywhere *}

definition

  "part_equivp E \<longleftrightarrow> ((\<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_implies_part_equivp:

  assumes a: "equivp E"

  shows "part_equivp E"

  using a 

  unfolding equivp_def part_equivp_def

  by auto

text {* Composition of Relations *}

abbreviation 

  rel_conj (infixr "OOO" 75)

where

  "r1 OOO r2 \<equiv> r1 OO r2 OO r1"

lemma eq_comp_r: 

  shows "((op =) OOO R) = R"

  by (auto simp add: expand_fun_eq)

section {* Respects predicate *}

definition

  Respects

where

  "Respects R x \<longleftrightarrow> (R x x)"

lemma in_respects:

  shows "(x \<in> Respects R) = R x x"

  unfolding mem_def Respects_def 

  by simp

section {* Function map and function relation *}

definition

  fun_map (infixr "--->" 55)

where

[simp]: "fun_map f g h x = g (h (f x))"

definition

  fun_rel (infixr "===>" 55)

where

[simp]: "fun_rel E1 E2 f g = (\<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"

lemma fun_map_id:

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

  by (simp add: expand_fun_eq id_def)

lemma fun_rel_eq:

  shows "((op =) ===> (op =)) = (op =)"

  by (simp add: expand_fun_eq)

lemma fun_rel_id:

  assumes a: "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"

  shows "(R1 ===> R2) f g"

  using a by simp

lemma fun_rel_id_asm:

  assumes a: "\<And>x y. R1 x y \<Longrightarrow> (A \<longrightarrow> R2 (f x) (g y))"

  shows "A \<longrightarrow> (R1 ===> R2) f g"

  using a by auto

section {* Quotient Predicate *}

definition

  "Quotient E Abs Rep \<longleftrightarrow>

     (\<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_reflp:

  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_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 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_symp:

  assumes a: "Quotient E Abs Rep"

  shows "symp E"

  using a unfolding Quotient_def symp_def

  by metis

lemma Quotient_transp:

  assumes a: "Quotient E Abs Rep"

  shows "transp E"

  using a unfolding Quotient_def transp_def

  by metis

lemma identity_quotient:

  shows "Quotient (op =) id id"

  unfolding Quotient_def id_def

  by blast

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"

    using q1 q2 

    unfolding Quotient_def

    unfolding expand_fun_eq

    by simp

  moreover

  have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"

    using q1 q2 

    unfolding Quotient_def

    by (simp (no_asm)) (metis)

  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)"

    unfolding expand_fun_eq

    apply(auto)

    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

lemma abs_o_rep:

  assumes a: "Quotient R Abs Rep"

  shows "Abs o Rep = id"

  unfolding expand_fun_eq

  by (simp add: Quotient_abs_rep[OF a])

lemma equals_rsp:

  assumes q: "Quotient R Abs Rep"

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

  shows "R xa ya = R xb yb"

  using a Quotient_symp[OF q] Quotient_transp[OF q] 

  unfolding symp_def transp_def

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

  unfolding expand_fun_eq

  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]

  by simp

lemma rep_abs_rsp:

  assumes q: "Quotient R Abs Rep"

  and     a: "R x1 x2"

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

  using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]

  by metis

lemma rep_abs_rsp_left:

  assumes q: "Quotient R Abs Rep"

  and     a: "R x1 x2"

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

  using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]

  by metis

text{* 

  In the following theorem R1 can be instantiated with anything,

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

  so by solving Quotient assumptions we can get a unique R1 that

  will be provable; which is why we need to use apply_rsp and

  not the primed version *}

lemma apply_rsp:

  fixes f g::"'a \<Rightarrow> 'c"

  assumes q: "Quotient R1 Abs1 Rep1"

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

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

  using a by simp

lemma apply_rsp':

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

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

  using a by simp

section {* 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)"

  using a 

  unfolding equivp_def

  by (auto simp add: in_respects)

lemma bex_reg_eqv:

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

  assumes a: "equivp R"

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

  using a 

  unfolding equivp_def

  by (auto simp add: in_respects)

lemma ball_reg_right:

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

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

  using a by (metis COMBC_def Collect_def Collect_mem_eq)

lemma bex_reg_left:

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

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

  using a by (metis COMBC_def Collect_def Collect_mem_eq)

lemma ball_reg_left:

  assumes a: "equivp R"

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

  using a by (metis equivp_reflp in_respects)

lemma bex_reg_right:

  assumes a: "equivp R"

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

  using a by (metis equivp_reflp in_respects)

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: 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:

  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:

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

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

  using assms by auto

lemma bex_ex_comm:

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

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

  using assms by auto

section {* Bounded abstraction *}

definition

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

where

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

lemma babs_rsp:

  assumes q: "Quotient R1 Abs1 Rep1"

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

  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"

  apply (auto simp add: Babs_def in_respects)

  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")

  using a apply (simp add: Babs_def)

  apply (simp add: in_respects)

  using Quotient_rel[OF q]

  by metis

lemma babs_prs:

  assumes q1: "Quotient R1 Abs1 Rep1"

  and     q2: "Quotient R2 Abs2 Rep2"

  shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"

  apply (rule ext)

  apply (simp)

  apply (subgoal_tac "Rep1 x \<in> Respects R1")

  apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])

  apply (simp add: in_respects Quotient_rel_rep[OF q1])

  done

lemma babs_simp:

  assumes q: "Quotient R1 Abs Rep"

  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"

  apply(rule iffI)

  apply(simp_all only: babs_rsp[OF q])

  apply(auto simp add: Babs_def)

  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")

  apply(metis Babs_def)

  apply (simp add: in_respects)

  using Quotient_rel[OF q]

  by metis

(* If a user proves that a particular functional relation 

   is an equivalence this may be useful in regularising *)

lemma babs_reg_eqv:

  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"

  by (simp add: expand_fun_eq Babs_def in_respects equivp_reflp)

(* 3 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)

lemma bex1_rsp:

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

  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"

  using a 

  by (simp add: Ex1_def in_respects) auto

(* 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 Ball_def in_respects fun_map_def id_apply

  by metis

lemma ex_prs:

  assumes a: "Quotient R absf repf"

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

  using a unfolding Quotient_def Bex_def in_respects fun_map_def id_apply

  by metis

section {* Bex1_rel quantifier *}

definition

  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"

where

  "Bex1_rel R P \<longleftrightarrow> (\<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 bex1_rel_aux:

  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"

  unfolding Bex1_rel_def

  apply (erule conjE)+

  apply (erule bexE)

  apply rule

  apply (rule_tac x="xa" in bexI)

  apply metis

  apply metis

  apply rule+

  apply (erule_tac x="xaa" in ballE)

  prefer 2

  apply (metis)

  apply (erule_tac x="ya" in ballE)

  prefer 2

  apply (metis)

  apply (metis in_respects)

  done

lemma bex1_rel_aux2: