theory QuotScript

imports Plain ATP_Linkup

begin

definition 

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

definition

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

definition 

  "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)"

unfolding equivp_def reflp_def symp_def transp_def expand_fun_eq

by (blast)

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"

using a unfolding equivp_def part_equivp_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_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) \<equiv> (a = b)"

apply (rule eq_reflection)

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

  shows "equivp (op =)"

unfolding equivp_def

by auto

  shows "Quotient (op =) id id"

unfolding Quotient_def id_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

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"

by (simp add: expand_fun_eq id_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 =) \<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)"

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

unfolding mem_def Respects_def by simp

(* 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 equals_rsp:

  assumes q: "Quotient R Abs Rep"

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

  shows "R xa ya = R xb yb"

using Quotient_symp[OF q] Quotient_transp[OF q] unfolding symp_def transp_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 "(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

  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])

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

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

(*              Ball, Bex, 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 *)

(* 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 R2 that

   will be provable; which is why we need to use APPLY_RSP *)

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)"

using a by (rule fun_rel_IMP)

lemma apply_rsp':

  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: "equivp R"

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

lemma bex_reg_eqv:

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

  assumes a: "equivp R"

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

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"

lemma bex_reg_right:

  assumes a: "equivp R"

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

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(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(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"

lemma bex_rsp:

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

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

(* 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

lemma ex_prs:

  assumes a: "Quotient R absf repf"

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

  using a unfolding Quotient_def

(* UNUSED *)

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