theory QuotScriptimports Plain ATP_Linkup Predicatebegindefinition "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_reflp: shows "equivp E \<Longrightarrow> (\<And>x. E x x)" by (simp only: equivp_reflp_symp_transp reflp_def)lemma equivp_symp: shows "equivp E \<Longrightarrow> (\<And>x y. E x y \<Longrightarrow> E y x)" by (metis equivp_reflp_symp_transp symp_def)lemma equivp_transp: shows "equivp E \<Longrightarrow> (\<And>x y z. 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)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 autoabbreviation rel_conj (infixr "OOO" 75)where "r1 OOO r2 \<equiv> r1 OO r2 OO r1"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)))"(* TESTlemma fixes Abs1::"'b \<Rightarrow> 'c" and Abs2::"'a \<Rightarrow> 'b" and Rep1::"'c \<Rightarrow> 'b" and Rep2::"'b \<Rightarrow> 'a" assumes "Quotient R1 Abs1 Rep1" and "Quotient R2 Abs2 Rep2" shows "Quotient (f R2 R1) (Abs1 \<circ> Abs2) (Rep2 \<circ> Rep1)"*)lemma Quotient_abs_rep: assumes a: "Quotient E Abs Rep" shows "Abs (Rep a) \<equiv> a" using a unfolding Quotient_def by simplemma Quotient_rep_reflp: assumes a: "Quotient E Abs Rep" shows "E (Rep a) (Rep a)" using a unfolding Quotient_def by 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_def by blastlemma 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 metislemma 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 blastlemma Quotient_rel_abs: assumes a: "Quotient E Abs Rep" shows "E r s \<Longrightarrow> Abs r = Abs s" using a unfolding Quotient_def by blastlemma identity_equivp: shows "equivp (op =)" unfolding equivp_def by autolemma identity_quotient: shows "Quotient (op =) id id" unfolding Quotient_def id_def by blastlemma Quotient_symp: assumes a: "Quotient E Abs Rep" shows "symp E" using a unfolding Quotient_def symp_def by metislemma Quotient_transp: assumes a: "Quotient E Abs Rep" shows "transp E" using a unfolding Quotient_def transp_def by metisfun 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_id: shows "(id ---> id) = id" by (simp add: expand_fun_eq id_def)fun fun_relwhere "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 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 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 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_eq using 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 "(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 simplemma 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_reflp[OF q])lemma rep_abs_rsp_left: assumes q: "Quotient R Abs Rep" and a: "R x1 x2" shows "R (Rep (Abs x1)) x2"using q a by (metis Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q])(* 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 simplemma apply_rsp': assumes a: "(R1 ===> R2) f g" "R1 x y" shows "R2 (f x) (g y)" using a by simp(* 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: in_respects) apply(rule impI) using a equivp_reflp_symp_transp[of "R2"] apply(simp add: reflp_def) apply(simp) apply(simp) donelemma 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) 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(* 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)"(* 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 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) 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 metislemma babs_prs: assumes q1: "Quotient R1 Abs1 Rep1" and q2: "Quotient R2 Abs2 Rep2" shows "(Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f)) \<equiv> f" apply(rule eq_reflection) 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]) donelemma 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)(* 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 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 simplemma 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 autolemma quot_rel_rsp: assumes a: "Quotient R Abs Rep" shows "(R ===> R ===> op =) R R" apply(rule fun_rel_id)+ apply(rule equals_rsp[OF a]) apply(assumption)+ donelemma o_prs: assumes q1: "Quotient R1 Abs1 Rep1" and q2: "Quotient R2 Abs2 Rep2" and q3: "Quotient R3 Abs3 Rep3" shows "(Rep1 ---> Abs3) (((Abs2 ---> Rep3) f) o ((Abs1 ---> Rep2) g)) = f o g" using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3] unfolding o_def expand_fun_eq by 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_eq by (auto)lemma cond_prs: assumes a: "Quotient R absf repf" shows "absf (if a then repf b else repf c) = (if a then b else c)" using a unfolding Quotient_def by autolemma if_prs: assumes q: "Quotient R Abs Rep" shows "Abs (If a (Rep b) (Rep c)) = If a b c"using Quotient_abs_rep[OF q] by auto(* q not used *)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 "Abs2 (Let (Rep1 x) ((Abs1 ---> Rep2) f)) = Let x f" using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] by autolemma let_rsp: assumes q1: "Quotient R1 Abs1 Rep1" and a1: "(R1 ===> R2) f g" and a2: "R1 x y" shows "R2 ((Let x f)::'c) ((Let y g)::'c)" using apply_rsp[OF q1 a1] a2 by auto(******************************************)(* REST OF THE FILE IS UNUSED (until now) *)(******************************************)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 blastlemma 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 simplemma 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 simplemma abs_o_rep: assumes a: "Quotient r absf repf" shows "absf o repf = id" apply(rule ext) apply(simp add: Quotient_abs_rep[OF a]) donelemma eq_comp_r: "op = OO R OO op = \<equiv> R" apply (rule eq_reflection) apply (rule ext)+ apply auto donelemma 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 blastlemma let_babs: "v \<in> r \<Longrightarrow> Let v (Babs r lam) = Let v lam" by (simp add: Babs_def)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 apply_rsp') 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(* ask peter what are literal_case *)(* literal_case_PRS *)(* literal_case_RSP *)(* Cez: !f x. literal_case f x = f x *)(* We use id_simps which includes id_apply; so these 2 theorems can be removed *)lemma id_prs: assumes q: "Quotient R Abs Rep" shows "Abs (id (Rep e)) = id e" using Quotient_abs_rep[OF q] by autolemma id_rsp: assumes q: "Quotient R Abs Rep" and a: "R e1 e2" shows "R (id e1) (id e2)" using a by autoend