--- a/src/HOL/Quotient.thy Fri Mar 23 14:21:41 2012 +0100
+++ b/src/HOL/Quotient.thy Fri Mar 23 14:25:31 2012 +0100
@@ -9,6 +9,7 @@
keywords
"print_quotmaps" "print_quotients" "print_quotconsts" :: diag and
"quotient_type" :: thy_goal and "/" and
+ "setup_lifting" :: thy_decl and
"quotient_definition" :: thy_goal
uses
("Tools/Quotient/quotient_info.ML")
@@ -137,6 +138,18 @@
unfolding Quotient_def
by blast
+lemma Quotient_refl1:
+ assumes a: "Quotient R Abs Rep"
+ shows "R r s \<Longrightarrow> R r r"
+ using a unfolding Quotient_def
+ by fast
+
+lemma Quotient_refl2:
+ assumes a: "Quotient R Abs Rep"
+ shows "R r s \<Longrightarrow> R s s"
+ using a unfolding Quotient_def
+ by fast
+
lemma Quotient_rel_rep:
assumes a: "Quotient R Abs Rep"
shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
@@ -263,6 +276,15 @@
shows "R2 (f x) (g y)"
using a by (auto elim: fun_relE)
+lemma apply_rsp'':
+ assumes "Quotient R Abs Rep"
+ and "(R ===> S) f f"
+ shows "S (f (Rep x)) (f (Rep x))"
+proof -
+ from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
+ then show ?thesis using assms(2) by (auto intro: apply_rsp')
+qed
+
subsection {* lemmas for regularisation of ball and bex *}
lemma ball_reg_eqv:
@@ -679,6 +701,153 @@
end
+subsection {* Quotient composition *}
+
+lemma OOO_quotient:
+ fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
+ fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
+ fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
+ fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
+ fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
+ assumes R1: "Quotient R1 Abs1 Rep1"
+ assumes R2: "Quotient R2 Abs2 Rep2"
+ assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
+ assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
+ shows "Quotient (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
+apply (rule QuotientI)
+ apply (simp add: o_def Quotient_abs_rep [OF R2] Quotient_abs_rep [OF R1])
+ apply simp
+ apply (rule_tac b="Rep1 (Rep2 a)" in pred_compI)
+ apply (rule Quotient_rep_reflp [OF R1])
+ apply (rule_tac b="Rep1 (Rep2 a)" in pred_compI [rotated])
+ apply (rule Quotient_rep_reflp [OF R1])
+ apply (rule Rep1)
+ apply (rule Quotient_rep_reflp [OF R2])
+ apply safe
+ apply (rename_tac x y)
+ apply (drule Abs1)
+ apply (erule Quotient_refl2 [OF R1])
+ apply (erule Quotient_refl1 [OF R1])
+ apply (drule Quotient_refl1 [OF R2], drule Rep1)
+ apply (subgoal_tac "R1 r (Rep1 (Abs1 x))")
+ apply (rule_tac b="Rep1 (Abs1 x)" in pred_compI, assumption)
+ apply (erule pred_compI)
+ apply (erule Quotient_symp [OF R1, THEN sympD])
+ apply (rule Quotient_rel[symmetric, OF R1, THEN iffD2])
+ apply (rule conjI, erule Quotient_refl1 [OF R1])
+ apply (rule conjI, rule Quotient_rep_reflp [OF R1])
+ apply (subst Quotient_abs_rep [OF R1])
+ apply (erule Quotient_rel_abs [OF R1])
+ apply (rename_tac x y)
+ apply (drule Abs1)
+ apply (erule Quotient_refl2 [OF R1])
+ apply (erule Quotient_refl1 [OF R1])
+ apply (drule Quotient_refl2 [OF R2], drule Rep1)
+ apply (subgoal_tac "R1 s (Rep1 (Abs1 y))")
+ apply (rule_tac b="Rep1 (Abs1 y)" in pred_compI, assumption)
+ apply (erule pred_compI)
+ apply (erule Quotient_symp [OF R1, THEN sympD])
+ apply (rule Quotient_rel[symmetric, OF R1, THEN iffD2])
+ apply (rule conjI, erule Quotient_refl2 [OF R1])
+ apply (rule conjI, rule Quotient_rep_reflp [OF R1])
+ apply (subst Quotient_abs_rep [OF R1])
+ apply (erule Quotient_rel_abs [OF R1, THEN sym])
+ apply simp
+ apply (rule Quotient_rel_abs [OF R2])
+ apply (rule Quotient_rel_abs [OF R1, THEN ssubst], assumption)
+ apply (rule Quotient_rel_abs [OF R1, THEN subst], assumption)
+ apply (erule Abs1)
+ apply (erule Quotient_refl2 [OF R1])
+ apply (erule Quotient_refl1 [OF R1])
+ apply (rename_tac a b c d)
+ apply simp
+ apply (rule_tac b="Rep1 (Abs1 r)" in pred_compI)
+ apply (rule Quotient_rel[symmetric, OF R1, THEN iffD2])
+ apply (rule conjI, erule Quotient_refl1 [OF R1])
+ apply (simp add: Quotient_abs_rep [OF R1] Quotient_rep_reflp [OF R1])
+ apply (rule_tac b="Rep1 (Abs1 s)" in pred_compI [rotated])
+ apply (rule Quotient_rel[symmetric, OF R1, THEN iffD2])
+ apply (simp add: Quotient_abs_rep [OF R1] Quotient_rep_reflp [OF R1])
+ apply (erule Quotient_refl2 [OF R1])
+ apply (rule Rep1)
+ apply (drule Abs1)
+ apply (erule Quotient_refl2 [OF R1])
+ apply (erule Quotient_refl1 [OF R1])
+ apply (drule Abs1)
+ apply (erule Quotient_refl2 [OF R1])
+ apply (erule Quotient_refl1 [OF R1])
+ apply (drule Quotient_rel_abs [OF R1])
+ apply (drule Quotient_rel_abs [OF R1])
+ apply (drule Quotient_rel_abs [OF R1])
+ apply (drule Quotient_rel_abs [OF R1])
+ apply simp
+ apply (rule Quotient_rel[symmetric, OF R2, THEN iffD2])
+ apply simp
+done
+
+lemma OOO_eq_quotient:
+ fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
+ fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
+ fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
+ assumes R1: "Quotient R1 Abs1 Rep1"
+ assumes R2: "Quotient op= Abs2 Rep2"
+ shows "Quotient (R1 OOO op=) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
+using assms
+by (rule OOO_quotient) auto
+
+subsection {* Invariant *}
+
+definition invariant :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
+ where "invariant R = (\<lambda>x y. R x \<and> x = y)"
+
+lemma invariant_to_eq:
+ assumes "invariant P x y"
+ shows "x = y"
+using assms by (simp add: invariant_def)
+
+lemma fun_rel_eq_invariant:
+ shows "((invariant R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
+by (auto simp add: invariant_def fun_rel_def)
+
+lemma invariant_same_args:
+ shows "invariant P x x \<equiv> P x"
+using assms by (auto simp add: invariant_def)
+
+lemma copy_type_to_Quotient:
+ assumes "type_definition Rep Abs UNIV"
+ shows "Quotient (op =) Abs Rep"
+proof -
+ interpret type_definition Rep Abs UNIV by fact
+ from Abs_inject Rep_inverse show ?thesis by (auto intro!: QuotientI)
+qed
+
+lemma copy_type_to_equivp:
+ fixes Abs :: "'a \<Rightarrow> 'b"
+ and Rep :: "'b \<Rightarrow> 'a"
+ assumes "type_definition Rep Abs (UNIV::'a set)"
+ shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
+by (rule identity_equivp)
+
+lemma invariant_type_to_Quotient:
+ assumes "type_definition Rep Abs {x. P x}"
+ shows "Quotient (invariant P) Abs Rep"
+proof -
+ interpret type_definition Rep Abs "{x. P x}" by fact
+ from Rep Abs_inject Rep_inverse show ?thesis by (auto intro!: QuotientI simp: invariant_def)
+qed
+
+lemma invariant_type_to_part_equivp:
+ assumes "type_definition Rep Abs {x. P x}"
+ shows "part_equivp (invariant P)"
+proof (intro part_equivpI)
+ interpret type_definition Rep Abs "{x. P x}" by fact
+ show "\<exists>x. invariant P x x" using Rep by (auto simp: invariant_def)
+next
+ show "symp (invariant P)" by (auto intro: sympI simp: invariant_def)
+next
+ show "transp (invariant P)" by (auto intro: transpI simp: invariant_def)
+qed
+
subsection {* ML setup *}
text {* Auxiliary data for the quotient package *}