theory QuotScriptimports Mainbegindefinition "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_eqby (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_defby autodefinition "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_defby simplemma QUOTIENT_REP_REFL: assumes a: "QUOTIENT E Abs Rep" shows "E (Rep a) (Rep a)" using a unfolding QUOTIENT_defby blastlemma 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_defby blastlemma QUOTIENT_REL_ABS: assumes a: "QUOTIENT E Abs Rep" shows "E r s \<Longrightarrow> Abs r = Abs s"using a unfolding QUOTIENT_defby blastlemma 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_defby blastlemma QUOTIENT_REL_REP: assumes a: "QUOTIENT R Abs Rep" shows "R (Rep a) (Rep b) = (a = b)"using a unfolding QUOTIENT_defby metislemma QUOTIENT_REP_ABS: assumes a: "QUOTIENT R Abs Rep" shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"using a unfolding QUOTIENT_defby blastlemma IDENTITY_EQUIV: shows "EQUIV (op =)"unfolding EQUIV_defby autolemma IDENTITY_QUOTIENT: shows "QUOTIENT (op =) id id"unfolding QUOTIENT_def id_defby blastlemma QUOTIENT_SYM: assumes a: "QUOTIENT E Abs Rep" shows "SYM E"using a unfolding QUOTIENT_def SYM_defby metislemma QUOTIENT_TRANS: assumes a: "QUOTIENT E Abs Rep" shows "TRANS E"using a unfolding QUOTIENT_def TRANS_defby metisfun prod_relwhere "prod_rel r1 r2 = (\<lambda>(a,b) (c,d). r1 a c \<and> r2 b d)"fun fun_mapwhere "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 blastqeddefinition Respectswhere "Respects R x \<equiv> (R x x)"lemma IN_RESPECTS: shows "(x \<in> Respects R) = R x x"unfolding mem_def Respects_def by simplemma RESPECTS_THM: shows "Respects (R1 ===> R2) f = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (f y))"unfolding Respects_defby (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_defby simplemma 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_defby blastlemma 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_defby 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_eqby 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 simplemma FUN_REL_IMP: shows "(R1 ===> R2) f g \<Longrightarrow> R1 x y \<Longrightarrow> R2 (f x) (g y)"by simplemma 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_defapply(metis FUN_REL_IMP)using r1 unfolding Respects_def expand_fun_eqapply(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 aby (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_defusing a by blastlemma 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_equsing QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2]by simplemma 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_equsing 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_equsing 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 quotientScriptlemma 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 autolemma 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 autolemma 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 autolemma 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] a2by 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 autolemma I_RSP: assumes q: "QUOTIENT R Abs Rep" and a: "R e1 e2" shows "R (id e1) (id e2)"using a by autolemma 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_eqby simplemma 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_eqby (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) donelemma 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) donelemma 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 autolemma 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)end