nominal2: comparison Quot/QuotBase.thy

equal deleted inserted replaced

-:4a4c714ff795
+:68b8ebcc5240
 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"
 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
-using q1 q2
+unfolding Quotient_def
-apply(simp add: Quotient_def)
+unfolding expand_fun_eq
-done
+by simp
 moreover
 have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
-apply(auto)
+using q1 q2
-using q1 q2 unfolding Quotient_def
+unfolding Quotient_def
-apply(metis)
+by (simp (no_asm)) (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)
+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
 qed
 lemma abs_o_rep:
 assumes a: "Quotient R Abs Rep"
 shows "Abs o Rep = id"
-apply(rule ext)
+unfolding expand_fun_eq
-apply(simp add: Quotient_abs_rep[OF a])
+by (simp add: Quotient_abs_rep[OF a])
-done
 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 Quotient_symp[OF q] Quotient_transp[OF q]
-using a by blast
+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)"
 lemma rep_abs_rsp:
 assumes q: "Quotient R Abs Rep"
 and     a: "R x1 x2"
 shows "R x1 (Rep (Abs x2))"
-using a
+using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
-by (metis 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
+using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
-by (metis Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q])
+by metis
-(* In the following theorem R1 can be instantiated with anything,
+text{*
-but we know some of the types of the Rep and Abs functions;
+In the following theorem R1 can be instantiated with anything,
-so by solving Quotient assumptions we can get a unique R1 that
+but we know some of the types of the Rep and Abs functions;
-will be provable; which is why we need to use apply_rsp and
+so by solving Quotient assumptions we can get a unique R1 that
-not the primed version *)
+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)"
 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)
+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)"
-by (metis equivp_def in_respects a)
+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"
-by (metis COMBC_def Collect_def Collect_mem_eq a)
+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"
-by (metis COMBC_def Collect_def Collect_mem_eq a)
+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"
-by (metis equivp_reflp in_respects a)
+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"
-by (metis equivp_reflp in_respects a)
+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"
 lemma ex_reg:
 assumes a: "!x :: 'a. (P x --> Q x)"
 and     b: "Ex P"
 shows "Ex Q"
-using a b by (metis)
+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"
 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))"
+assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
-by auto
+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:
-"((\<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))"
+assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
-by auto
+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"
 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 (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

changeset 1125	68b8ebcc5240
parent 1123	41f89d4f9548