nominal2: comparison Quot/QuotScript.thy

equal deleted inserted replaced

-:93dce7c71929
+:d705d7ae2410
 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: "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 identity_equivp:
 shows "equivp (op =)"
 unfolding equivp_def
 by auto
 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)"
 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 *)
 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
 (* Bounded abstraction *)
 definition
 Babs :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
 where
 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
-(* needed for regularising? *)
+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"
 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 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
 lemma 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)+
 done
+lemma 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 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 "absf (if a then repf b else repf c) = (if a then b else c)"
+using a unfolding Quotient_def by auto
+lemma 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 auto
+lemma 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 auto
+lemma 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) *)
 (******************************************)
-(* Always safe to apply, but not too helpful *)
-lemma app_prs2:
-assumes q1: "Quotient R1 abs1 rep1"
-and     q2: "Quotient R2 abs2 rep2"
-shows  "((abs1 ---> rep2) ((rep1 ---> abs2) f) (rep1 x)) = rep2 (((rep1 ---> abs2) f) x)"
-unfolding expand_fun_eq
-using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
-by simp
-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:
+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:
+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
+using a b[simplified respects_thm] c unfolding Quotient_def
 by blast
-lemma RESPECTS_MP:
+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:
+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:
+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 let_babs:
-lemma APP_PRS:
+"v \<in> r \<Longrightarrow> Let v (Babs r lam) = Let v lam"
-assumes q1: "Quotient R1 abs1 rep1"
+by (simp add: Babs_def)
-and     q2: "Quotient R2 abs2 rep2"
-shows "abs2 ((abs1 ---> rep2) f (rep1 x)) = f x"
+lemma fun_rel_equals:
-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
 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)
+by (simp add: fun_rel_equals)
-lemma LAMBDA_REP_ABS_RSP:
+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 (Rep (Abs x1)) x2"
-using q a by (metis Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q])
-(* 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     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
 (* ask peter what are literal_case *)
 (* literal_case_PRS *)
 (* literal_case_RSP *)
+(* Cez: !f x. literal_case f x = f x *)
-(* combinators: I, K, o, C, W *)
 (* We use id_simps which includes id_apply; so these 2 theorems can be removed *)
+lemma id_prs:
-lemma I_PRS:
+assumes q: "Quotient R Abs Rep"
-assumes q: "Quotient R Abs Rep"
+shows "Abs (id (Rep e)) = id e"
-shows "id e = Abs (id (Rep e))"
+using Quotient_abs_rep[OF q] by auto
-using Quotient_abs_rep[OF q] by auto
+lemma id_rsp:
-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
 end

changeset 724	d705d7ae2410
parent 721	19032e71fb1c
child 807	a5495a323b49