--- a/src/HOL/Library/Quotient_List.thy Wed Jun 23 10:20:33 2010 +0200
+++ b/src/HOL/Library/Quotient_List.thy Wed Jun 23 10:20:54 2010 +0200
@@ -8,15 +8,7 @@
imports Main Quotient_Syntax
begin
-fun
- list_rel
-where
- "list_rel R [] [] = True"
-| "list_rel R (x#xs) [] = False"
-| "list_rel R [] (x#xs) = False"
-| "list_rel R (x#xs) (y#ys) = (R x y \<and> list_rel R xs ys)"
-
-declare [[map list = (map, list_rel)]]
+declare [[map list = (map, list_all2)]]
lemma split_list_all:
shows "(\<forall>x. P x) \<longleftrightarrow> P [] \<and> (\<forall>x xs. P (x#xs))"
@@ -33,52 +25,47 @@
apply(simp_all)
done
+lemma list_all2_reflp:
+ shows "equivp R \<Longrightarrow> list_all2 R xs xs"
+ by (induct xs, simp_all add: equivp_reflp)
-lemma list_rel_reflp:
- shows "equivp R \<Longrightarrow> list_rel R xs xs"
- apply(induct xs)
- apply(simp_all add: equivp_reflp)
- done
-
-lemma list_rel_symp:
+lemma list_all2_symp:
assumes a: "equivp R"
- shows "list_rel R xs ys \<Longrightarrow> list_rel R ys xs"
- apply(induct xs ys rule: list_induct2')
+ and b: "list_all2 R xs ys"
+ shows "list_all2 R ys xs"
+ using list_all2_lengthD[OF b] b
+ apply(induct xs ys rule: list_induct2)
apply(simp_all)
apply(rule equivp_symp[OF a])
apply(simp)
done
-lemma list_rel_transp:
+thm list_induct3
+
+lemma list_all2_transp:
assumes a: "equivp R"
- shows "list_rel R xs1 xs2 \<Longrightarrow> list_rel R xs2 xs3 \<Longrightarrow> list_rel R xs1 xs3"
- using a
- apply(induct R xs1 xs2 arbitrary: xs3 rule: list_rel.induct)
- apply(simp)
- apply(simp)
- apply(simp)
- apply(case_tac xs3)
- apply(clarify)
- apply(simp (no_asm_use))
- apply(clarify)
- apply(simp (no_asm_use))
- apply(auto intro: equivp_transp)
+ and b: "list_all2 R xs1 xs2"
+ and c: "list_all2 R xs2 xs3"
+ shows "list_all2 R xs1 xs3"
+ using list_all2_lengthD[OF b] list_all2_lengthD[OF c] b c
+ apply(induct rule: list_induct3)
+ apply(simp_all)
+ apply(auto intro: equivp_transp[OF a])
done
lemma list_equivp[quot_equiv]:
assumes a: "equivp R"
- shows "equivp (list_rel R)"
- apply(rule equivpI)
+ shows "equivp (list_all2 R)"
+ apply (intro equivpI)
unfolding reflp_def symp_def transp_def
- apply(subst split_list_all)
- apply(simp add: equivp_reflp[OF a] list_rel_reflp[OF a])
- apply(blast intro: list_rel_symp[OF a])
- apply(blast intro: list_rel_transp[OF a])
+ apply(simp add: list_all2_reflp[OF a])
+ apply(blast intro: list_all2_symp[OF a])
+ apply(blast intro: list_all2_transp[OF a])
done
-lemma list_rel_rel:
+lemma list_all2_rel:
assumes q: "Quotient R Abs Rep"
- shows "list_rel R r s = (list_rel R r r \<and> list_rel R s s \<and> (map Abs r = map Abs s))"
+ shows "list_all2 R r s = (list_all2 R r r \<and> list_all2 R s s \<and> (map Abs r = map Abs s))"
apply(induct r s rule: list_induct2')
apply(simp_all)
using Quotient_rel[OF q]
@@ -87,21 +74,16 @@
lemma list_quotient[quot_thm]:
assumes q: "Quotient R Abs Rep"
- shows "Quotient (list_rel R) (map Abs) (map Rep)"
+ shows "Quotient (list_all2 R) (map Abs) (map Rep)"
unfolding Quotient_def
apply(subst split_list_all)
apply(simp add: Quotient_abs_rep[OF q] abs_o_rep[OF q] map_id)
- apply(rule conjI)
- apply(rule allI)
+ apply(intro conjI allI)
apply(induct_tac a)
- apply(simp)
- apply(simp)
- apply(simp add: Quotient_rep_reflp[OF q])
- apply(rule allI)+
- apply(rule list_rel_rel[OF q])
+ apply(simp_all add: Quotient_rep_reflp[OF q])
+ apply(rule list_all2_rel[OF q])
done
-
lemma cons_prs_aux:
assumes q: "Quotient R Abs Rep"
shows "(map Abs) ((Rep h) # (map Rep t)) = h # t"
@@ -115,7 +97,7 @@
lemma cons_rsp[quot_respect]:
assumes q: "Quotient R Abs Rep"
- shows "(R ===> list_rel R ===> list_rel R) (op #) (op #)"
+ shows "(R ===> list_all2 R ===> list_all2 R) (op #) (op #)"
by (auto)
lemma nil_prs[quot_preserve]:
@@ -125,7 +107,7 @@
lemma nil_rsp[quot_respect]:
assumes q: "Quotient R Abs Rep"
- shows "list_rel R [] []"
+ shows "list_all2 R [] []"
by simp
lemma map_prs_aux:
@@ -146,8 +128,8 @@
lemma map_rsp[quot_respect]:
assumes q1: "Quotient R1 Abs1 Rep1"
and q2: "Quotient R2 Abs2 Rep2"
- shows "((R1 ===> R2) ===> (list_rel R1) ===> list_rel R2) map map"
- and "((R1 ===> op =) ===> (list_rel R1) ===> op =) map map"
+ shows "((R1 ===> R2) ===> (list_all2 R1) ===> list_all2 R2) map map"
+ and "((R1 ===> op =) ===> (list_all2 R1) ===> op =) map map"
apply simp_all
apply(rule_tac [!] allI)+
apply(rule_tac [!] impI)
@@ -183,53 +165,45 @@
by (simp only: expand_fun_eq fun_map_def foldl_prs_aux[OF a b])
(simp)
-lemma list_rel_empty:
- shows "list_rel R [] b \<Longrightarrow> length b = 0"
+lemma list_all2_empty:
+ shows "list_all2 R [] b \<Longrightarrow> length b = 0"
by (induct b) (simp_all)
-lemma list_rel_len:
- shows "list_rel R a b \<Longrightarrow> length a = length b"
- apply (induct a arbitrary: b)
- apply (simp add: list_rel_empty)
- apply (case_tac b)
- apply simp_all
- done
-
(* induct_tac doesn't accept 'arbitrary', so we manually 'spec' *)
lemma foldl_rsp[quot_respect]:
assumes q1: "Quotient R1 Abs1 Rep1"
and q2: "Quotient R2 Abs2 Rep2"
- shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_rel R2 ===> R1) foldl foldl"
+ shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_all2 R2 ===> R1) foldl foldl"
apply(auto)
- apply (subgoal_tac "R1 xa ya \<longrightarrow> list_rel R2 xb yb \<longrightarrow> R1 (foldl x xa xb) (foldl y ya yb)")
+ apply (subgoal_tac "R1 xa ya \<longrightarrow> list_all2 R2 xb yb \<longrightarrow> R1 (foldl x xa xb) (foldl y ya yb)")
apply simp
apply (rule_tac x="xa" in spec)
apply (rule_tac x="ya" in spec)
apply (rule_tac xs="xb" and ys="yb" in list_induct2)
- apply (rule list_rel_len)
+ apply (rule list_all2_lengthD)
apply (simp_all)
done
lemma foldr_rsp[quot_respect]:
assumes q1: "Quotient R1 Abs1 Rep1"
and q2: "Quotient R2 Abs2 Rep2"
- shows "((R1 ===> R2 ===> R2) ===> list_rel R1 ===> R2 ===> R2) foldr foldr"
+ shows "((R1 ===> R2 ===> R2) ===> list_all2 R1 ===> R2 ===> R2) foldr foldr"
apply auto
- apply(subgoal_tac "R2 xb yb \<longrightarrow> list_rel R1 xa ya \<longrightarrow> R2 (foldr x xa xb) (foldr y ya yb)")
+ apply(subgoal_tac "R2 xb yb \<longrightarrow> list_all2 R1 xa ya \<longrightarrow> R2 (foldr x xa xb) (foldr y ya yb)")
apply simp
apply (rule_tac xs="xa" and ys="ya" in list_induct2)
- apply (rule list_rel_len)
+ apply (rule list_all2_lengthD)
apply (simp_all)
done
-lemma list_rel_rsp:
+lemma list_all2_rsp:
assumes r: "\<forall>x y. R x y \<longrightarrow> (\<forall>a b. R a b \<longrightarrow> S x a = T y b)"
- and l1: "list_rel R x y"
- and l2: "list_rel R a b"
- shows "list_rel S x a = list_rel T y b"
+ and l1: "list_all2 R x y"
+ and l2: "list_all2 R a b"
+ shows "list_all2 S x a = list_all2 T y b"
proof -
- have a: "length y = length x" by (rule list_rel_len[OF l1, symmetric])
- have c: "length a = length b" by (rule list_rel_len[OF l2])
+ have a: "length y = length x" by (rule list_all2_lengthD[OF l1, symmetric])
+ have c: "length a = length b" by (rule list_all2_lengthD[OF l2])
show ?thesis proof (cases "length x = length a")
case True
have b: "length x = length a" by fact
@@ -243,20 +217,20 @@
next
case False
have d: "length x \<noteq> length a" by fact
- then have e: "\<not>list_rel S x a" using list_rel_len by auto
+ then have e: "\<not>list_all2 S x a" using list_all2_lengthD by auto
have "length y \<noteq> length b" using d a c by simp
- then have "\<not>list_rel T y b" using list_rel_len by auto
+ then have "\<not>list_all2 T y b" using list_all2_lengthD by auto
then show ?thesis using e by simp
qed
qed
lemma[quot_respect]:
- "((R ===> R ===> op =) ===> list_rel R ===> list_rel R ===> op =) list_rel list_rel"
- by (simp add: list_rel_rsp)
+ "((R ===> R ===> op =) ===> list_all2 R ===> list_all2 R ===> op =) list_all2 list_all2"
+ by (simp add: list_all2_rsp)
lemma[quot_preserve]:
assumes a: "Quotient R abs1 rep1"
- shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_rel = list_rel"
+ shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_all2 = list_all2"
apply (simp add: expand_fun_eq)
apply clarify
apply (induct_tac xa xb rule: list_induct2')
@@ -265,29 +239,29 @@
lemma[quot_preserve]:
assumes a: "Quotient R abs1 rep1"
- shows "(list_rel ((rep1 ---> rep1 ---> id) R) l m) = (l = m)"
+ shows "(list_all2 ((rep1 ---> rep1 ---> id) R) l m) = (l = m)"
by (induct l m rule: list_induct2') (simp_all add: Quotient_rel_rep[OF a])
-lemma list_rel_eq[id_simps]:
- shows "(list_rel (op =)) = (op =)"
+lemma list_all2_eq[id_simps]:
+ shows "(list_all2 (op =)) = (op =)"
unfolding expand_fun_eq
apply(rule allI)+
apply(induct_tac x xa rule: list_induct2')
apply(simp_all)
done
-lemma list_rel_find_element:
+lemma list_all2_find_element:
assumes a: "x \<in> set a"
- and b: "list_rel R a b"
+ and b: "list_all2 R a b"
shows "\<exists>y. (y \<in> set b \<and> R x y)"
proof -
- have "length a = length b" using b by (rule list_rel_len)
+ have "length a = length b" using b by (rule list_all2_lengthD)
then show ?thesis using a b by (induct a b rule: list_induct2) auto
qed
-lemma list_rel_refl:
+lemma list_all2_refl:
assumes a: "\<And>x y. R x y = (R x = R y)"
- shows "list_rel R x x"
+ shows "list_all2 R x x"
by (induct x) (auto simp add: a)
end
--- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Wed Jun 23 10:20:33 2010 +0200
+++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Wed Jun 23 10:20:54 2010 +0200
@@ -975,12 +975,12 @@
apply (subst matrix_vector_mul[OF lf])
unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
-section {* lambda_skolem on cartesian products *}
+section {* lambda skolemization on cartesian products *}
(* FIXME: rename do choice_cart *)
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
- (\<exists>x::'a ^ 'n. \<forall>i. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")
+ (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
proof-
let ?S = "(UNIV :: 'n set)"
{assume H: "?rhs"
@@ -991,7 +991,7 @@
let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
{fix i
from f have "P i (f i)" by metis
- then have "P i (?x$i)" by auto
+ then have "P i (?x $ i)" by auto
}
hence "\<forall>i. P i (?x$i)" by metis
hence ?rhs by metis }
@@ -1830,7 +1830,7 @@
unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding algebra_simps by (auto intro: mult_pos_pos)
qed
-subsection {* Lemmas for working on real^1 *}
+subsection {* Lemmas for working on @{typ "real^1"} *}
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
by (metis num1_eq_iff)
--- a/src/HOL/Quotient.thy Wed Jun 23 10:20:33 2010 +0200
+++ b/src/HOL/Quotient.thy Wed Jun 23 10:20:54 2010 +0200
@@ -59,7 +59,7 @@
unfolding equivp_def
by auto
-text {* Partial equivalences: not yet used anywhere *}
+text {* Partial equivalences *}
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)))"
@@ -71,6 +71,23 @@
unfolding equivp_def part_equivp_def
by auto
+lemma part_equivp_symp:
+ assumes e: "part_equivp R"
+ and a: "R x y"
+ shows "R y x"
+ using e[simplified part_equivp_def] a
+ by (metis)
+
+lemma part_equivp_typedef:
+ shows "part_equivp R \<Longrightarrow> \<exists>d. d \<in> (\<lambda>c. \<exists>x. R x x \<and> c = R x)"
+ unfolding part_equivp_def mem_def
+ apply clarify
+ apply (intro exI)
+ apply (rule conjI)
+ apply assumption
+ apply (rule refl)
+ done
+
text {* Composition of Relations *}
abbreviation
@@ -630,10 +647,10 @@
fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
and Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"
and Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"
- assumes equivp: "equivp R"
- and rep_prop: "\<And>y. \<exists>x. Rep y = R x"
+ assumes equivp: "part_equivp R"
+ and rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = R x"
and rep_inverse: "\<And>x. Abs (Rep x) = x"
- and abs_inverse: "\<And>x. (Rep (Abs (R x))) = (R x)"
+ and abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = R x))) \<Longrightarrow> (Rep (Abs c)) = c"
and rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
begin
@@ -647,64 +664,46 @@
where
"rep a = Eps (Rep a)"
-lemma homeier_lem9:
- shows "R (Eps (R x)) = R x"
-proof -
- have a: "R x x" using equivp by (simp add: equivp_reflp_symp_transp reflp_def)
- then have "R x (Eps (R x))" by (rule someI)
- then show "R (Eps (R x)) = R x"
- using equivp unfolding equivp_def by simp
-qed
-
-theorem homeier_thm10:
- shows "abs (rep a) = a"
- unfolding abs_def rep_def
-proof -
- from rep_prop
- obtain x where eq: "Rep a = R x" by auto
- have "Abs (R (Eps (Rep a))) = Abs (R (Eps (R x)))" using eq by simp
- also have "\<dots> = Abs (R x)" using homeier_lem9 by simp
- also have "\<dots> = Abs (Rep a)" using eq by simp
- also have "\<dots> = a" using rep_inverse by simp
- finally
- show "Abs (R (Eps (Rep a))) = a" by simp
-qed
+lemma homeier5:
+ assumes a: "R r r"
+ shows "Rep (Abs (R r)) = R r"
+ apply (subst abs_inverse)
+ using a by auto
-lemma homeier_lem7:
- shows "(R x = R y) = (Abs (R x) = Abs (R y))" (is "?LHS = ?RHS")
-proof -
- have "?RHS = (Rep (Abs (R x)) = Rep (Abs (R y)))" by (simp add: rep_inject)
- also have "\<dots> = ?LHS" by (simp add: abs_inverse)
- finally show "?LHS = ?RHS" by simp
-qed
+theorem homeier6:
+ assumes a: "R r r"
+ and b: "R s s"
+ shows "Abs (R r) = Abs (R s) \<longleftrightarrow> R r = R s"
+ by (metis a b homeier5)
-theorem homeier_thm11:
- shows "R r r' = (abs r = abs r')"
- unfolding abs_def
- by (simp only: equivp[simplified equivp_def] homeier_lem7)
-
-lemma rep_refl:
- shows "R (rep a) (rep a)"
- unfolding rep_def
- by (simp add: equivp[simplified equivp_def])
-
-
-lemma rep_abs_rsp:
- shows "R f (rep (abs g)) = R f g"
- and "R (rep (abs g)) f = R g f"
- by (simp_all add: homeier_thm10 homeier_thm11)
+theorem homeier8:
+ assumes "R r r"
+ shows "R (Eps (R r)) = R r"
+ using assms equivp[simplified part_equivp_def]
+ apply clarify
+ by (metis assms exE_some)
lemma Quotient:
shows "Quotient R abs rep"
- unfolding Quotient_def
- apply(simp add: homeier_thm10)
- apply(simp add: rep_refl)
- apply(subst homeier_thm11[symmetric])
- apply(simp add: equivp[simplified equivp_def])
- done
+ unfolding Quotient_def abs_def rep_def
+ proof (intro conjI allI)
+ fix a r s
+ show "Abs (R (Eps (Rep a))) = a"
+ by (metis equivp exE_some part_equivp_def rep_inverse rep_prop)
+ show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (R r) = Abs (R s))"
+ by (metis homeier6 equivp[simplified part_equivp_def])
+ show "R (Eps (Rep a)) (Eps (Rep a))" proof -
+ obtain x where r: "R x x" and rep: "Rep a = R x" using rep_prop[of a] by auto
+ have "R (Eps (R x)) x" using homeier8 r by simp
+ then have "R x (Eps (R x))" using part_equivp_symp[OF equivp] by fast
+ then have "R (Eps (R x)) (Eps (R x))" using homeier8[OF r] by simp
+ then show "R (Eps (Rep a)) (Eps (Rep a))" using rep by simp
+ qed
+ qed
end
+
subsection {* ML setup *}
text {* Auxiliary data for the quotient package *}
--- a/src/HOL/Quotient_Examples/FSet.thy Wed Jun 23 10:20:33 2010 +0200
+++ b/src/HOL/Quotient_Examples/FSet.thy Wed Jun 23 10:20:54 2010 +0200
@@ -80,20 +80,20 @@
text {* Composition Quotient *}
-lemma list_rel_refl:
- shows "(list_rel op \<approx>) r r"
- by (rule list_rel_refl) (metis equivp_def fset_equivp)
+lemma list_all2_refl:
+ 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_refl)
- with * show "(op \<approx> OO list_rel op \<approx>) r r" ..
+ show "list_all2 op \<approx> r r" by (rule list_all2_refl)
+ 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"
@@ -104,32 +104,32 @@
by (simp only: set_map set_in_eq)
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_refl)
- 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_refl)
+ 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_refl) (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_refl) (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"
@@ -140,20 +140,20 @@
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_refl[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_refl[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_refl[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_refl[of r]])
+ then show "(list_all2 op \<approx> OOO op \<approx>) r s"
using a c pred_compI by simp
qed
qed
@@ -336,27 +336,27 @@
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 [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
@@ -364,9 +364,9 @@
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
@@ -581,14 +581,14 @@
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)
@@ -607,59 +607,59 @@
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_refl)
+ by simp_all (rule list_all2_refl)
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_refl)
+ apply (rule list_all2_refl)
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_refl)
+ apply (rule list_all2_refl)
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
--- a/src/HOL/Tools/Quotient/quotient_tacs.ML Wed Jun 23 10:20:33 2010 +0200
+++ b/src/HOL/Tools/Quotient/quotient_tacs.ML Wed Jun 23 10:20:54 2010 +0200
@@ -147,6 +147,14 @@
finally jump back to 1
*)
+fun reflp_get ctxt =
+ map_filter (fn th => if prems_of th = [] then SOME (OF1 @{thm equivp_reflp} th) else NONE
+ handle THM _ => NONE) (equiv_rules_get ctxt)
+
+val eq_imp_rel = @{lemma "equivp R ==> a = b --> R a b" by (simp add: equivp_reflp)}
+
+fun eq_imp_rel_get ctxt = map (OF1 eq_imp_rel) (equiv_rules_get ctxt)
+
fun regularize_tac ctxt =
let
val thy = ProofContext.theory_of ctxt
@@ -157,8 +165,7 @@
addsimps @{thms ball_reg_eqv bex_reg_eqv babs_reg_eqv babs_simp}
addsimprocs [simproc]
addSolver equiv_solver addSolver quotient_solver
- val eq_imp_rel = @{lemma "equivp R ==> a = b --> R a b" by (simp add: equivp_reflp)}
- val eq_eqvs = map (OF1 eq_imp_rel) (equiv_rules_get ctxt)
+ val eq_eqvs = eq_imp_rel_get ctxt
in
simp_tac simpset THEN'
REPEAT_ALL_NEW (CHANGED o FIRST'
@@ -254,7 +261,7 @@
val inst_thm = Drule.instantiate' ty_inst
([NONE, NONE, NONE] @ t_inst) @{thm apply_rsp}
in
- (rtac inst_thm THEN' quotient_tac context) 1
+ (rtac inst_thm THEN' SOLVED' (quotient_tac context)) 1
end
| _ => no_tac
end)
@@ -406,7 +413,7 @@
fun injection_tac ctxt =
let
- val rel_refl = map (OF1 @{thm equivp_reflp}) (equiv_rules_get ctxt)
+ val rel_refl = reflp_get ctxt
in
injection_step_tac ctxt rel_refl
end
--- a/src/HOL/Tools/Quotient/quotient_typ.ML Wed Jun 23 10:20:33 2010 +0200
+++ b/src/HOL/Tools/Quotient/quotient_typ.ML Wed Jun 23 10:20:54 2010 +0200
@@ -7,13 +7,13 @@
signature QUOTIENT_TYPE =
sig
- val add_quotient_type: ((string list * binding * mixfix) * (typ * term)) * thm
+ val add_quotient_type: ((string list * binding * mixfix) * (typ * term * bool)) * thm
-> Proof.context -> (thm * thm) * local_theory
- val quotient_type: ((string list * binding * mixfix) * (typ * term)) list
+ val quotient_type: ((string list * binding * mixfix) * (typ * term * bool)) list
-> Proof.context -> Proof.state
- val quotient_type_cmd: ((((string list * binding) * mixfix) * string) * string) list
+ val quotient_type_cmd: ((((string list * binding) * mixfix) * string) * (bool * string)) list
-> Proof.context -> Proof.state
end;
@@ -64,15 +64,15 @@
|> map Free
in
lambda c (HOLogic.exists_const rty $
- lambda x (HOLogic.mk_eq (c, (rel $ x))))
+ lambda x (HOLogic.mk_conj ((rel $ x $ x), (HOLogic.mk_eq (c, (rel $ x))))))
end
(* makes the new type definitions and proves non-emptyness *)
-fun typedef_make (vs, qty_name, mx, rel, rty) lthy =
+fun typedef_make (vs, qty_name, mx, rel, rty) equiv_thm lthy =
let
val typedef_tac =
- EVERY1 (map rtac [@{thm exI}, mem_def2, @{thm exI}, @{thm refl}])
+ EVERY1 (map rtac [@{thm part_equivp_typedef}, equiv_thm])
in
(* FIXME: purely local typedef causes at the moment
problems with type variables
@@ -93,14 +93,14 @@
let
val rep_thm = #Rep typedef_info RS mem_def1
val rep_inv = #Rep_inverse typedef_info
- val abs_inv = mem_def2 RS #Abs_inverse typedef_info
+ val abs_inv = #Abs_inverse typedef_info
val rep_inj = #Rep_inject typedef_info
in
(rtac @{thm quot_type.intro} THEN' RANGE [
rtac equiv_thm,
rtac rep_thm,
rtac rep_inv,
- EVERY' (map rtac [abs_inv, @{thm exI}, @{thm refl}]),
+ rtac abs_inv THEN' rtac mem_def2 THEN' atac,
rtac rep_inj]) 1
end
@@ -137,10 +137,12 @@
(* main function for constructing a quotient type *)
-fun add_quotient_type (((vs, qty_name, mx), (rty, rel)), equiv_thm) lthy =
+fun add_quotient_type (((vs, qty_name, mx), (rty, rel, partial)), equiv_thm) lthy =
let
+ val part_equiv = if partial then equiv_thm else equiv_thm RS @{thm equivp_implies_part_equivp}
+
(* generates the typedef *)
- val ((qty_full_name, typedef_info), lthy1) = typedef_make (vs, qty_name, mx, rel, rty) lthy
+ val ((qty_full_name, typedef_info), lthy1) = typedef_make (vs, qty_name, mx, rel, rty) part_equiv lthy
(* abs and rep functions from the typedef *)
val Abs_ty = #abs_type (#1 typedef_info)
@@ -162,7 +164,7 @@
val ((rep, rep_def), lthy3) = define (rep_name, NoSyn, rep_trm) lthy2
(* quot_type theorem *)
- val quot_thm = typedef_quot_type_thm (rel, Abs_const, Rep_const, equiv_thm, typedef_info) lthy3
+ val quot_thm = typedef_quot_type_thm (rel, Abs_const, Rep_const, part_equiv, typedef_info) lthy3
(* quotient theorem *)
val quotient_thm = typedef_quotient_thm (rel, abs, rep, abs_def, rep_def, quot_thm) lthy3
@@ -179,12 +181,12 @@
in
lthy4
|> note (quotient_thm_name, quotient_thm, [intern_attr quotient_rules_add])
- ||>> note (equiv_thm_name, equiv_thm, [intern_attr equiv_rules_add])
+ ||>> note (equiv_thm_name, equiv_thm, if partial then [] else [intern_attr equiv_rules_add])
end
(* sanity checks for the quotient type specifications *)
-fun sanity_check ((vs, qty_name, _), (rty, rel)) =
+fun sanity_check ((vs, qty_name, _), (rty, rel, _)) =
let
val rty_tfreesT = map fst (Term.add_tfreesT rty [])
val rel_tfrees = map fst (Term.add_tfrees rel [])
@@ -223,7 +225,7 @@
end
(* check for existence of map functions *)
-fun map_check ctxt (_, (rty, _)) =
+fun map_check ctxt (_, (rty, _, _)) =
let
val thy = ProofContext.theory_of ctxt
@@ -263,11 +265,12 @@
val _ = List.app sanity_check quot_list
val _ = List.app (map_check lthy) quot_list
- fun mk_goal (rty, rel) =
+ fun mk_goal (rty, rel, partial) =
let
val equivp_ty = ([rty, rty] ---> @{typ bool}) --> @{typ bool}
+ val const = if partial then @{const_name part_equivp} else @{const_name equivp}
in
- HOLogic.mk_Trueprop (Const (@{const_name equivp}, equivp_ty) $ rel)
+ HOLogic.mk_Trueprop (Const (const, equivp_ty) $ rel)
end
val goals = map (mk_goal o snd) quot_list
@@ -280,7 +283,7 @@
fun quotient_type_cmd specs lthy =
let
- fun parse_spec ((((vs, qty_name), mx), rty_str), rel_str) lthy =
+ fun parse_spec ((((vs, qty_name), mx), rty_str), (partial, rel_str)) lthy =
let
val rty = Syntax.read_typ lthy rty_str
val lthy1 = Variable.declare_typ rty lthy
@@ -290,7 +293,7 @@
|> Syntax.check_term lthy1
val lthy2 = Variable.declare_term rel lthy1
in
- (((vs, qty_name, mx), (rty, rel)), lthy2)
+ (((vs, qty_name, mx), (rty, rel, partial)), lthy2)
end
val (spec', lthy') = fold_map parse_spec specs lthy
@@ -298,11 +301,13 @@
quotient_type spec' lthy'
end
+val partial = Scan.optional (Parse.reserved "partial" -- Parse.$$$ ":" >> K true) false
+
val quotspec_parser =
- Parse.and_list1
- ((Parse.type_args -- Parse.binding) --
- Parse.opt_mixfix -- (Parse.$$$ "=" |-- Parse.typ) --
- (Parse.$$$ "/" |-- Parse.term))
+ Parse.and_list1
+ ((Parse.type_args -- Parse.binding) --
+ Parse.opt_mixfix -- (Parse.$$$ "=" |-- Parse.typ) --
+ (Parse.$$$ "/" |-- (partial -- Parse.term)))
val _ = Keyword.keyword "/"