--- a/Nominal/FSet.thy Wed Jun 23 15:21:04 2010 +0100
+++ b/Nominal/FSet.thy Wed Jun 23 15:40:00 2010 +0100
@@ -80,20 +80,20 @@
text {* Composition Quotient *}
-lemma list_rel_refl1:
- shows "(list_rel op \<approx>) r r"
- by (rule list_rel_refl) (metis equivp_def fset_equivp)
+lemma list_all2_refl1:
+ shows "(list_all2 op \<approx>) r r"
+ by (rule list_all2_refl) (metis equivp_def fset_equivp)
lemma compose_list_refl:
- shows "(list_rel op \<approx> OOO op \<approx>) r r"
+ shows "(list_all2 op \<approx> OOO op \<approx>) r r"
proof
have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
- show "list_rel op \<approx> r r" by (rule list_rel_refl1)
- with * show "(op \<approx> OO list_rel op \<approx>) r r" ..
+ show "list_all2 op \<approx> r r" by (rule list_all2_refl1)
+ with * show "(op \<approx> OO list_all2 op \<approx>) r r" ..
qed
lemma Quotient_fset_list:
- shows "Quotient (list_rel op \<approx>) (map abs_fset) (map rep_fset)"
+ shows "Quotient (list_all2 op \<approx>) (map abs_fset) (map rep_fset)"
by (fact list_quotient[OF Quotient_fset])
lemma set_in_eq: "(\<forall>e. ((e \<in> xs) \<longleftrightarrow> (e \<in> ys))) \<equiv> xs = ys"
@@ -105,32 +105,32 @@
lemma quotient_compose_list[quot_thm]:
- shows "Quotient ((list_rel op \<approx>) OOO (op \<approx>))
+ shows "Quotient ((list_all2 op \<approx>) OOO (op \<approx>))
(abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"
unfolding Quotient_def comp_def
proof (intro conjI allI)
fix a r s
show "abs_fset (map abs_fset (map rep_fset (rep_fset a))) = a"
by (simp add: abs_o_rep[OF Quotient_fset] Quotient_abs_rep[OF Quotient_fset] map_id)
- have b: "list_rel op \<approx> (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
- by (rule list_rel_refl1)
- have c: "(op \<approx> OO list_rel op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
+ have b: "list_all2 op \<approx> (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
+ by (rule list_all2_refl1)
+ have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)
- show "(list_rel op \<approx> OOO op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
- by (rule, rule list_rel_refl1) (rule c)
- show "(list_rel op \<approx> OOO op \<approx>) r s = ((list_rel op \<approx> OOO op \<approx>) r r \<and>
- (list_rel op \<approx> OOO op \<approx>) s s \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s))"
+ show "(list_all2 op \<approx> OOO op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
+ by (rule, rule list_all2_refl1) (rule c)
+ show "(list_all2 op \<approx> OOO op \<approx>) r s = ((list_all2 op \<approx> OOO op \<approx>) r r \<and>
+ (list_all2 op \<approx> OOO op \<approx>) s s \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s))"
proof (intro iffI conjI)
- show "(list_rel op \<approx> OOO op \<approx>) r r" by (rule compose_list_refl)
- show "(list_rel op \<approx> OOO op \<approx>) s s" by (rule compose_list_refl)
+ show "(list_all2 op \<approx> OOO op \<approx>) r r" by (rule compose_list_refl)
+ show "(list_all2 op \<approx> OOO op \<approx>) s s" by (rule compose_list_refl)
next
- assume a: "(list_rel op \<approx> OOO op \<approx>) r s"
+ assume a: "(list_all2 op \<approx> OOO op \<approx>) r s"
then have b: "map abs_fset r \<approx> map abs_fset s"
proof (elim pred_compE)
fix b ba
- assume c: "list_rel op \<approx> r b"
+ assume c: "list_all2 op \<approx> r b"
assume d: "b \<approx> ba"
- assume e: "list_rel op \<approx> ba s"
+ assume e: "list_all2 op \<approx> ba s"
have f: "map abs_fset r = map abs_fset b"
using Quotient_rel[OF Quotient_fset_list] c by blast
have "map abs_fset ba = map abs_fset s"
@@ -141,20 +141,20 @@
then show "abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"
using Quotient_rel[OF Quotient_fset] by blast
next
- assume a: "(list_rel op \<approx> OOO op \<approx>) r r \<and> (list_rel op \<approx> OOO op \<approx>) s s
+ assume a: "(list_all2 op \<approx> OOO op \<approx>) r r \<and> (list_all2 op \<approx> OOO op \<approx>) s s
\<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"
- then have s: "(list_rel op \<approx> OOO op \<approx>) s s" by simp
+ then have s: "(list_all2 op \<approx> OOO op \<approx>) s s" by simp
have d: "map abs_fset r \<approx> map abs_fset s"
by (subst Quotient_rel[OF Quotient_fset]) (simp add: a)
have b: "map rep_fset (map abs_fset r) \<approx> map rep_fset (map abs_fset s)"
by (rule map_rel_cong[OF d])
- have y: "list_rel op \<approx> (map rep_fset (map abs_fset s)) s"
- by (fact rep_abs_rsp_left[OF Quotient_fset_list, OF list_rel_refl1[of s]])
- have c: "(op \<approx> OO list_rel op \<approx>) (map rep_fset (map abs_fset r)) s"
+ have y: "list_all2 op \<approx> (map rep_fset (map abs_fset s)) s"
+ by (fact rep_abs_rsp_left[OF Quotient_fset_list, OF list_all2_refl1[of s]])
+ have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (map abs_fset r)) s"
by (rule pred_compI) (rule b, rule y)
- have z: "list_rel op \<approx> r (map rep_fset (map abs_fset r))"
- by (fact rep_abs_rsp[OF Quotient_fset_list, OF list_rel_refl1[of r]])
- then show "(list_rel op \<approx> OOO op \<approx>) r s"
+ have z: "list_all2 op \<approx> r (map rep_fset (map abs_fset r))"
+ by (fact rep_abs_rsp[OF Quotient_fset_list, OF list_all2_refl1[of r]])
+ then show "(list_all2 op \<approx> OOO op \<approx>) r s"
using a c pred_compI by simp
qed
qed
@@ -342,27 +342,27 @@
by (simp add: memb_def[symmetric] ffold_raw_rsp_pre)
lemma concat_rsp_pre:
- assumes a: "list_rel op \<approx> x x'"
+ assumes a: "list_all2 op \<approx> x x'"
and b: "x' \<approx> y'"
- and c: "list_rel op \<approx> y' y"
+ and c: "list_all2 op \<approx> y' y"
and d: "\<exists>x\<in>set x. xa \<in> set x"
shows "\<exists>x\<in>set y. xa \<in> set x"
proof -
obtain xb where e: "xb \<in> set x" and f: "xa \<in> set xb" using d by auto
- have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_rel_find_element[OF e a])
+ have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_all2_find_element[OF e a])
then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto
have "ya \<in> set y'" using b h by simp
- then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" using c by (rule list_rel_find_element)
+ then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" using c by (rule list_all2_find_element)
then show ?thesis using f i by auto
qed
lemma concat_rsp[quot_respect]:
- shows "(list_rel op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"
+ shows "(list_all2 op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"
proof (rule fun_relI, elim pred_compE)
fix a b ba bb
- assume a: "list_rel op \<approx> a ba"
+ assume a: "list_all2 op \<approx> a ba"
assume b: "ba \<approx> bb"
- assume c: "list_rel op \<approx> bb b"
+ assume c: "list_all2 op \<approx> bb b"
have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" proof
fix x
show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" proof
@@ -370,9 +370,9 @@
show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])
next
assume e: "\<exists>xa\<in>set b. x \<in> set xa"
- have a': "list_rel op \<approx> ba a" by (rule list_rel_symp[OF list_eq_equivp, OF a])
+ have a': "list_all2 op \<approx> ba a" by (rule list_all2_symp[OF list_eq_equivp, OF a])
have b': "bb \<approx> ba" by (rule equivp_symp[OF list_eq_equivp, OF b])
- have c': "list_rel op \<approx> b bb" by (rule list_rel_symp[OF list_eq_equivp, OF c])
+ have c': "list_all2 op \<approx> b bb" by (rule list_all2_symp[OF list_eq_equivp, OF c])
show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])
qed
qed
@@ -382,12 +382,12 @@
lemma concat_rsp_unfolded:
- "\<lbrakk>list_rel op \<approx> a ba; ba \<approx> bb; list_rel op \<approx> bb b\<rbrakk> \<Longrightarrow> concat a \<approx> concat b"
+ "\<lbrakk>list_all2 op \<approx> a ba; ba \<approx> bb; list_all2 op \<approx> bb b\<rbrakk> \<Longrightarrow> concat a \<approx> concat b"
proof -
fix a b ba bb
- assume a: "list_rel op \<approx> a ba"
+ assume a: "list_all2 op \<approx> a ba"
assume b: "ba \<approx> bb"
- assume c: "list_rel op \<approx> bb b"
+ assume c: "list_all2 op \<approx> bb b"
have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" proof
fix x
show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" proof
@@ -395,9 +395,9 @@
show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])
next
assume e: "\<exists>xa\<in>set b. x \<in> set xa"
- have a': "list_rel op \<approx> ba a" by (rule list_rel_symp[OF list_eq_equivp, OF a])
+ have a': "list_all2 op \<approx> ba a" by (rule list_all2_symp[OF list_eq_equivp, OF a])
have b': "bb \<approx> ba" by (rule equivp_symp[OF list_eq_equivp, OF b])
- have c': "list_rel op \<approx> b bb" by (rule list_rel_symp[OF list_eq_equivp, OF c])
+ have c': "list_all2 op \<approx> b bb" by (rule list_all2_symp[OF list_eq_equivp, OF c])
show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])
qed
qed
@@ -614,14 +614,14 @@
text {* Compositional Respectfullness and Preservation *}
-lemma [quot_respect]: "(list_rel op \<approx> OOO op \<approx>) [] []"
+lemma [quot_respect]: "(list_all2 op \<approx> OOO op \<approx>) [] []"
by (fact compose_list_refl)
lemma [quot_preserve]: "(abs_fset \<circ> map f) [] = abs_fset []"
by simp
lemma [quot_respect]:
- "(op \<approx> ===> list_rel op \<approx> OOO op \<approx> ===> list_rel op \<approx> OOO op \<approx>) op # op #"
+ "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) op # op #"
apply auto
apply (simp add: set_in_eq)
apply (rule_tac b="x # b" in pred_compI)
@@ -645,59 +645,59 @@
by (simp add: expand_fun_eq Quotient_abs_rep[OF Quotient_fset]
abs_o_rep[OF Quotient_fset] map_id sup_fset_def)
-lemma list_rel_app_l:
+lemma list_all2_app_l:
assumes a: "reflp R"
- and b: "list_rel R l r"
- shows "list_rel R (z @ l) (z @ r)"
+ and b: "list_all2 R l r"
+ shows "list_all2 R (z @ l) (z @ r)"
by (induct z) (simp_all add: b rev_iffD1[OF a meta_eq_to_obj_eq[OF reflp_def]])
lemma append_rsp2_pre0:
- assumes a:"list_rel op \<approx> x x'"
- shows "list_rel op \<approx> (x @ z) (x' @ z)"
+ assumes a:"list_all2 op \<approx> x x'"
+ shows "list_all2 op \<approx> (x @ z) (x' @ z)"
using a apply (induct x x' rule: list_induct2')
- by simp_all (rule list_rel_refl1)
+ by simp_all (rule list_all2_refl1)
lemma append_rsp2_pre1:
- assumes a:"list_rel op \<approx> x x'"
- shows "list_rel op \<approx> (z @ x) (z @ x')"
+ assumes a:"list_all2 op \<approx> x x'"
+ shows "list_all2 op \<approx> (z @ x) (z @ x')"
using a apply (induct x x' arbitrary: z rule: list_induct2')
- apply (rule list_rel_refl1)
+ apply (rule list_all2_refl1)
apply (simp_all del: list_eq.simps)
- apply (rule list_rel_app_l)
+ apply (rule list_all2_app_l)
apply (simp_all add: reflp_def)
done
lemma append_rsp2_pre:
- assumes a:"list_rel op \<approx> x x'"
- and b: "list_rel op \<approx> z z'"
- shows "list_rel op \<approx> (x @ z) (x' @ z')"
- apply (rule list_rel_transp[OF fset_equivp])
+ assumes a:"list_all2 op \<approx> x x'"
+ and b: "list_all2 op \<approx> z z'"
+ shows "list_all2 op \<approx> (x @ z) (x' @ z')"
+ apply (rule list_all2_transp[OF fset_equivp])
apply (rule append_rsp2_pre0)
apply (rule a)
using b apply (induct z z' rule: list_induct2')
apply (simp_all only: append_Nil2)
- apply (rule list_rel_refl1)
+ apply (rule list_all2_refl1)
apply simp_all
apply (rule append_rsp2_pre1)
apply simp
done
lemma [quot_respect]:
- "(list_rel op \<approx> OOO op \<approx> ===> list_rel op \<approx> OOO op \<approx> ===> list_rel op \<approx> OOO op \<approx>) op @ op @"
+ "(list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) op @ op @"
proof (intro fun_relI, elim pred_compE)
fix x y z w x' z' y' w' :: "'a list list"
- assume a:"list_rel op \<approx> x x'"
+ assume a:"list_all2 op \<approx> x x'"
and b: "x' \<approx> y'"
- and c: "list_rel op \<approx> y' y"
- assume aa: "list_rel op \<approx> z z'"
+ and c: "list_all2 op \<approx> y' y"
+ assume aa: "list_all2 op \<approx> z z'"
and bb: "z' \<approx> w'"
- and cc: "list_rel op \<approx> w' w"
- have a': "list_rel op \<approx> (x @ z) (x' @ z')" using a aa append_rsp2_pre by auto
+ and cc: "list_all2 op \<approx> w' w"
+ have a': "list_all2 op \<approx> (x @ z) (x' @ z')" using a aa append_rsp2_pre by auto
have b': "x' @ z' \<approx> y' @ w'" using b bb by simp
- have c': "list_rel op \<approx> (y' @ w') (y @ w)" using c cc append_rsp2_pre by auto
- have d': "(op \<approx> OO list_rel op \<approx>) (x' @ z') (y @ w)"
+ have c': "list_all2 op \<approx> (y' @ w') (y @ w)" using c cc append_rsp2_pre by auto
+ have d': "(op \<approx> OO list_all2 op \<approx>) (x' @ z') (y @ w)"
by (rule pred_compI) (rule b', rule c')
- show "(list_rel op \<approx> OOO op \<approx>) (x @ z) (y @ w)"
+ show "(list_all2 op \<approx> OOO op \<approx>) (x @ z) (y @ w)"
by (rule pred_compI) (rule a', rule d')
qed
@@ -1416,49 +1416,49 @@
apply auto
done
-lemma list_rel_refl:
+lemma list_all2_refl:
assumes q: "equivp R"
- shows "(list_rel R) r r"
- by (rule list_rel_refl) (metis equivp_def fset_equivp q)
+ shows "(list_all2 R) r r"
+ by (rule list_all2_refl) (metis equivp_def fset_equivp q)
lemma compose_list_refl2:
assumes q: "equivp R"
- shows "(list_rel R OOO op \<approx>) r r"
+ shows "(list_all2 R OOO op \<approx>) r r"
proof
have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
- show "list_rel R r r" by (rule list_rel_refl[OF q])
- with * show "(op \<approx> OO list_rel R) r r" ..
+ show "list_all2 R r r" by (rule list_all2_refl[OF q])
+ with * show "(op \<approx> OO list_all2 R) r r" ..
qed
lemma quotient_compose_list_g:
assumes q: "Quotient R Abs Rep"
and e: "equivp R"
- shows "Quotient ((list_rel R) OOO (op \<approx>))
+ shows "Quotient ((list_all2 R) OOO (op \<approx>))
(abs_fset \<circ> (map Abs)) ((map Rep) \<circ> rep_fset)"
unfolding Quotient_def comp_def
proof (intro conjI allI)
fix a r s
show "abs_fset (map Abs (map Rep (rep_fset a))) = a"
by (simp add: abs_o_rep[OF q] Quotient_abs_rep[OF Quotient_fset] map_id)
- have b: "list_rel R (map Rep (rep_fset a)) (map Rep (rep_fset a))"
- by (rule list_rel_refl[OF e])
- have c: "(op \<approx> OO list_rel R) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
+ have b: "list_all2 R (map Rep (rep_fset a)) (map Rep (rep_fset a))"
+ by (rule list_all2_refl[OF e])
+ have c: "(op \<approx> OO list_all2 R) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)
- show "(list_rel R OOO op \<approx>) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
- by (rule, rule list_rel_refl[OF e]) (rule c)
- show "(list_rel R OOO op \<approx>) r s = ((list_rel R OOO op \<approx>) r r \<and>
- (list_rel R OOO op \<approx>) s s \<and> abs_fset (map Abs r) = abs_fset (map Abs s))"
+ show "(list_all2 R OOO op \<approx>) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
+ by (rule, rule list_all2_refl[OF e]) (rule c)
+ show "(list_all2 R OOO op \<approx>) r s = ((list_all2 R OOO op \<approx>) r r \<and>
+ (list_all2 R OOO op \<approx>) s s \<and> abs_fset (map Abs r) = abs_fset (map Abs s))"
proof (intro iffI conjI)
- show "(list_rel R OOO op \<approx>) r r" by (rule compose_list_refl2[OF e])
- show "(list_rel R OOO op \<approx>) s s" by (rule compose_list_refl2[OF e])
+ show "(list_all2 R OOO op \<approx>) r r" by (rule compose_list_refl2[OF e])
+ show "(list_all2 R OOO op \<approx>) s s" by (rule compose_list_refl2[OF e])
next
- assume a: "(list_rel R OOO op \<approx>) r s"
+ assume a: "(list_all2 R OOO op \<approx>) r s"
then have b: "map Abs r \<approx> map Abs s"
proof (elim pred_compE)
fix b ba
- assume c: "list_rel R r b"
+ assume c: "list_all2 R r b"
assume d: "b \<approx> ba"
- assume e: "list_rel R ba s"
+ assume e: "list_all2 R ba s"
have f: "map Abs r = map Abs b"
using Quotient_rel[OF list_quotient[OF q]] c by blast
have "map Abs ba = map Abs s"
@@ -1469,20 +1469,20 @@
then show "abs_fset (map Abs r) = abs_fset (map Abs s)"
using Quotient_rel[OF Quotient_fset] by blast
next
- assume a: "(list_rel R OOO op \<approx>) r r \<and> (list_rel R OOO op \<approx>) s s
+ assume a: "(list_all2 R OOO op \<approx>) r r \<and> (list_all2 R OOO op \<approx>) s s
\<and> abs_fset (map Abs r) = abs_fset (map Abs s)"
- then have s: "(list_rel R OOO op \<approx>) s s" by simp
+ then have s: "(list_all2 R OOO op \<approx>) s s" by simp
have d: "map Abs r \<approx> map Abs s"
by (subst Quotient_rel[OF Quotient_fset]) (simp add: a)
have b: "map Rep (map Abs r) \<approx> map Rep (map Abs s)"
by (rule map_rel_cong[OF d])
- have y: "list_rel R (map Rep (map Abs s)) s"
- by (fact rep_abs_rsp_left[OF list_quotient[OF q], OF list_rel_refl[OF e, of s]])
- have c: "(op \<approx> OO list_rel R) (map Rep (map Abs r)) s"
+ have y: "list_all2 R (map Rep (map Abs s)) s"
+ by (fact rep_abs_rsp_left[OF list_quotient[OF q], OF list_all2_refl[OF e, of s]])
+ have c: "(op \<approx> OO list_all2 R) (map Rep (map Abs r)) s"
by (rule pred_compI) (rule b, rule y)
- have z: "list_rel R r (map Rep (map Abs r))"
- by (fact rep_abs_rsp[OF list_quotient[OF q], OF list_rel_refl[OF e, of r]])
- then show "(list_rel R OOO op \<approx>) r s"
+ have z: "list_all2 R r (map Rep (map Abs r))"
+ by (fact rep_abs_rsp[OF list_quotient[OF q], OF list_all2_refl[OF e, of r]])
+ then show "(list_all2 R OOO op \<approx>) r s"
using a c pred_compI by simp
qed
qed