--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/Quot/Examples/SigmaEx.thy	Thu Feb 25 07:57:17 2010 +0100
@@ -0,0 +1,253 @@
+theory SigmaEx
+imports Nominal "../Quotient" "../Quotient_List" "../Quotient_Product"
+begin
+
+atom_decl name
+
+datatype robj =
+  rVar "name"
+| rObj "(string \<times> rmethod) list"
+| rInv "robj" "string"
+| rUpd "robj" "string" "rmethod"
+and rmethod =
+  rSig "name" "robj"
+
+inductive
+    alpha_obj :: "robj \<Rightarrow> robj \<Rightarrow> bool" ("_ \<approx>o _" [100, 100] 100)
+and alpha_method :: "rmethod \<Rightarrow> rmethod \<Rightarrow> bool" ("_ \<approx>m _" [100, 100] 100)
+where
+  a1: "a = b \<Longrightarrow> (rVar a) \<approx>o (rVar b)"
+| a2: "rObj [] \<approx>o rObj []"
+| a3: "rObj t1 \<approx>o rObj t2 \<Longrightarrow> m1 \<approx>m r2 \<Longrightarrow> rObj ((l1, m1) # t1) \<approx>o rObj ((l2, m2) # t2)"
+| a4: "x \<approx>o y \<Longrightarrow> rInv x l1 \<approx>o rInv y l2"
+| a5: "\<exists>pi::name prm. (rfv t - {a} = rfv s - {b} \<and> (rfv t - {a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx>o s \<and> (pi \<bullet> a) = b)
+       \<Longrightarrow> rSig a t \<approx>m rSig b s"
+
+lemma alpha_equivps:
+  shows "equivp alpha_obj"
+  and   "equivp alpha_method"
+sorry
+
+quotient_type
+    obj = robj / alpha_obj
+and method = rmethod / alpha_method
+  by (auto intro: alpha_equivps)
+
+quotient_definition
+  "Var :: name \<Rightarrow> obj"
+is
+  "rVar"
+
+quotient_definition
+  "Obj :: (string \<times> method) list \<Rightarrow> obj"
+is
+  "rObj"
+
+quotient_definition
+  "Inv :: obj \<Rightarrow> string \<Rightarrow> obj"
+is
+  "rInv"
+
+quotient_definition
+  "Upd :: obj \<Rightarrow> string \<Rightarrow> method \<Rightarrow> obj"
+is
+  "rUpd"
+
+quotient_definition
+  "Sig :: name \<Rightarrow> obj \<Rightarrow> method"
+is
+  "rSig"
+
+overloading
+  perm_obj \<equiv> "perm :: 'x prm \<Rightarrow> obj \<Rightarrow> obj" (unchecked)
+  perm_method   \<equiv> "perm :: 'x prm \<Rightarrow> method \<Rightarrow> method" (unchecked)
+begin
+
+quotient_definition
+  "perm_obj :: 'x prm \<Rightarrow> obj \<Rightarrow> obj"
+is
+  "(perm::'x prm \<Rightarrow> robj \<Rightarrow> robj)"
+
+quotient_definition
+  "perm_method :: 'x prm \<Rightarrow> method \<Rightarrow> method"
+is
+  "(perm::'x prm \<Rightarrow> rmethod \<Rightarrow> rmethod)"
+
+end
+
+
+
+lemma tolift:
+"\<forall> fvar.
+ \<forall> fobj\<in>Respects (op = ===> list_rel (prod_rel (op =) alpha_method) ===> op =).
+ \<forall> fnvk\<in>Respects (op = ===> alpha_obj ===> op =).
+ \<forall> fupd\<in>Respects (op = ===> op = ===> alpha_obj ===> op = ===> alpha_method ===> op =).
+ \<forall> fcns\<in>Respects (op = ===> op = ===> prod_rel (op =) alpha_method ===> list_rel (prod_rel (op =) alpha_method) ===> op =).
+ \<forall> fnil.
+ \<forall> fpar\<in>Respects (op = ===> op = ===> alpha_method ===> op =).
+ \<forall> fsgm\<in>Respects (op = ===> (op = ===> alpha_obj) ===> op =).
+
+ Ex1 (\<lambda>x.
+(x \<in> (Respects (prod_rel (alpha_obj ===> op =)
+     (prod_rel (list_rel (prod_rel (op =) alpha_method) ===> op =)
+       (prod_rel ((prod_rel (op =) alpha_method) ===> op =)
+         (alpha_method ===> op =)
+       )
+     )))) \<and>
+(\<lambda> (hom_o\<Colon>robj \<Rightarrow> 'a, hom_d\<Colon>(char list \<times> rmethod) list \<Rightarrow> 'b, hom_e\<Colon>char list \<times> rmethod \<Rightarrow> 'c, hom_m\<Colon>rmethod \<Rightarrow> 'd).
+
+((\<forall>x. hom_o (rVar x) = fvar x) \<and>
+ (\<forall>d. hom_o (rObj d) = fobj (hom_d d) d) \<and>
+ (\<forall>a l. hom_o (rInv a l) = fnvk (hom_o a) a l) \<and>
+ (\<forall>a l m. hom_o (rUpd a l m) = fupd (hom_o a) (hom_m m) a l m) \<and>
+ (\<forall>e d. hom_d (e # d) = fcns (hom_e e) (hom_d d) e d) \<and>
+ (hom_d [] = fnil) \<and>
+ (\<forall>l m. hom_e (l, m) = fpar (hom_m m) l m) \<and>
+ (\<forall>x a. hom_m (rSig x a) = fsgm (\<lambda>y. hom_o ([(x, y)] \<bullet> a)) (\<lambda>y. [(x, y)] \<bullet> a))
+)) x) "
+sorry
+
+lemma test_to: "Ex1 (\<lambda>x. (x \<in> (Respects alpha_obj)) \<and> P x)"
+ML_prf {* prop_of (#goal (Isar.goal ())) *}
+sorry
+lemma test_tod: "Ex1 (P :: obj \<Rightarrow> bool)"
+apply (lifting test_to)
+done
+
+
+
+
+(*syntax
+  "_expttrn"        :: "pttrn => bool => bool"      ("(3\<exists>\<exists> _./ _)" [0, 10] 10)
+
+translations
+  "\<exists>\<exists> x. P"   == "Ex (%x. P)"
+*)
+
+lemma rvar_rsp[quot_respect]: "(op = ===> alpha_obj) rVar rVar"
+  by (simp add: a1)
+
+lemma robj_rsp[quot_respect]: "(list_rel (prod_rel op = alpha_method) ===> alpha_obj) rObj rObj"
+sorry
+lemma rinv_rsp[quot_respect]: "(alpha_obj ===> op = ===> alpha_obj) rInv rInv"
+sorry
+lemma rupd_rsp[quot_respect]: "(alpha_obj ===> op = ===> alpha_method ===> alpha_obj) rUpd rUpd"
+sorry
+lemma rsig_rsp[quot_respect]: "(op = ===> alpha_obj ===> alpha_method) rSig rSig"
+sorry
+lemma operm_rsp[quot_respect]: "(op = ===> alpha_obj ===> alpha_obj) op \<bullet> op \<bullet>"
+sorry
+
+  
+lemma bex1_bex1reg: "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
+apply (simp add: Ex1_def Bex1_rel_def in_respects)
+apply clarify
+apply auto
+apply (rule bexI)
+apply assumption
+apply (simp add: in_respects)
+apply (simp add: in_respects)
+apply auto
+done
+
+lemma liftd: "
+Ex1 (\<lambda>(hom_o, hom_d, hom_e, hom_m).
+
+ (\<forall>x. hom_o (Var x) = fvar x) \<and>
+ (\<forall>d. hom_o (Obj d) = fobj (hom_d d) d) \<and>
+ (\<forall>a l. hom_o (Inv a l) = fnvk (hom_o a) a l) \<and>
+ (\<forall>a l m. hom_o (Upd a l m) = fupd (hom_o a) (hom_m m) a l m) \<and>
+ (\<forall>e d. hom_d (e # d) = fcns (hom_e e) (hom_d d) e d) \<and>
+ (hom_d [] = fnil) \<and>
+ (\<forall>l m. hom_e (l, m) = fpar (hom_m m) l m) \<and>
+ (\<forall>x a. hom_m (Sig x a) = fsgm (\<lambda>y. hom_o ([(x, y)] \<bullet> a)) (\<lambda>y. [(x, y)] \<bullet> a))
+)"
+apply (lifting tolift)
+done
+
+lemma tolift':
+"\<forall> fvar.
+ \<forall> fobj\<in>Respects (op = ===> list_rel (prod_rel (op =) alpha_method) ===> op =).
+ \<forall> fnvk\<in>Respects (op = ===> alpha_obj ===> op =).
+ \<forall> fupd\<in>Respects (op = ===> op = ===> alpha_obj ===> op = ===> alpha_method ===> op =).
+ \<forall> fcns\<in>Respects (op = ===> op = ===> prod_rel (op =) alpha_method ===> list_rel (prod_rel (op =) alpha_method) ===> op =).
+ \<forall> fnil.
+ \<forall> fpar\<in>Respects (op = ===> op = ===> alpha_method ===> op =).
+ \<forall> fsgm\<in>Respects (op = ===> (op = ===> alpha_obj) ===> op =).
+
+ \<exists> hom_o\<Colon>robj \<Rightarrow> 'a \<in> Respects (alpha_obj ===> op =).
+ \<exists> hom_d\<Colon>(char list \<times> rmethod) list \<Rightarrow> 'b \<in> Respects (list_rel (prod_rel (op =) alpha_method) ===> op =).
+ \<exists> hom_e\<Colon>char list \<times> rmethod \<Rightarrow> 'c \<in> Respects ((prod_rel (op =) alpha_method) ===> op =).
+ \<exists> hom_m\<Colon>rmethod \<Rightarrow> 'd \<in> Respects (alpha_method ===> op =).
+(
+ (\<forall>x. hom_o (rVar x) = fvar x) \<and>
+ (\<forall>d. hom_o (rObj d) = fobj (hom_d d) d) \<and>
+ (\<forall>a l. hom_o (rInv a l) = fnvk (hom_o a) a l) \<and>
+ (\<forall>a l m. hom_o (rUpd a l m) = fupd (hom_o a) (hom_m m) a l m) \<and>
+ (\<forall>e d. hom_d (e # d) = fcns (hom_e e) (hom_d d) e d) \<and>
+ (hom_d [] = fnil) \<and>
+ (\<forall>l m. hom_e (l, m) = fpar (hom_m m) l m) \<and>
+ (\<forall>x a. hom_m (rSig x a) = fsgm (\<lambda>y. hom_o ([(x, y)] \<bullet> a)) (\<lambda>y. [(x, y)] \<bullet> a))
+)"
+sorry
+
+lemma liftd': "
+\<exists>hom_o. \<exists>hom_d. \<exists>hom_e. \<exists>hom_m.
+(
+ (\<forall>x. hom_o (Var x) = fvar x) \<and>
+ (\<forall>d. hom_o (Obj d) = fobj (hom_d d) d) \<and>
+ (\<forall>a l. hom_o (Inv a l) = fnvk (hom_o a) a l) \<and>
+ (\<forall>a l m. hom_o (Upd a l m) = fupd (hom_o a) (hom_m m) a l m) \<and>
+ (\<forall>e d. hom_d (e # d) = fcns (hom_e e) (hom_d d) e d) \<and>
+ (hom_d [] = fnil) \<and>
+ (\<forall>l m. hom_e (l, m) = fpar (hom_m m) l m) \<and>
+ (\<forall>x a. hom_m (Sig x a) = fsgm (\<lambda>y. hom_o ([(x, y)] \<bullet> a)) (\<lambda>y. [(x, y)] \<bullet> a))
+)"
+apply (lifting tolift')
+done
+
+lemma tolift'':
+"\<forall> fvar.
+ \<forall> fobj\<in>Respects (op = ===> list_rel (prod_rel (op =) alpha_method) ===> op =).
+ \<forall> fnvk\<in>Respects (op = ===> alpha_obj ===> op =).
+ \<forall> fupd\<in>Respects (op = ===> op = ===> alpha_obj ===> op = ===> alpha_method ===> op =).
+ \<forall> fcns\<in>Respects (op = ===> op = ===> prod_rel (op =) alpha_method ===> list_rel (prod_rel (op =) alpha_method) ===> op =).
+ \<forall> fnil.
+ \<forall> fpar\<in>Respects (op = ===> op = ===> alpha_method ===> op =).
+ \<forall> fsgm\<in>Respects (op = ===> (op = ===> alpha_obj) ===> op =).
+
+ Bex1_rel (alpha_obj ===> op =) (\<lambda>hom_o\<Colon>robj \<Rightarrow> 'a .
+ Bex1_rel (list_rel (prod_rel (op =) alpha_method) ===> op =) (\<lambda>hom_d\<Colon>(char list \<times> rmethod) list \<Rightarrow> 'b.
+ Bex1_rel ((prod_rel (op =) alpha_method) ===> op =) (\<lambda>hom_e\<Colon>char list \<times> rmethod \<Rightarrow> 'c.
+ Bex1_rel (alpha_method ===> op =) (\<lambda>hom_m\<Colon>rmethod \<Rightarrow> 'd.
+(
+ (\<forall>x. hom_o (rVar x) = fvar x) \<and>
+ (\<forall>d. hom_o (rObj d) = fobj (hom_d d) d) \<and>
+ (\<forall>a l. hom_o (rInv a l) = fnvk (hom_o a) a l) \<and>
+ (\<forall>a l m. hom_o (rUpd a l m) = fupd (hom_o a) (hom_m m) a l m) \<and>
+ (\<forall>e d. hom_d (e # d) = fcns (hom_e e) (hom_d d) e d) \<and>
+ (hom_d [] = fnil) \<and>
+ (\<forall>l m. hom_e (l, m) = fpar (hom_m m) l m) \<and>
+ (\<forall>x a. hom_m (rSig x a) = fsgm (\<lambda>y. hom_o ([(x, y)] \<bullet> a)) (\<lambda>y. [(x, y)] \<bullet> a))
+)
+))))"
+sorry
+
+lemma liftd'': "
+\<exists>!hom_o. \<exists>!hom_d. \<exists>!hom_e. \<exists>!hom_m.
+(
+ (\<forall>x. hom_o (Var x) = fvar x) \<and>
+ (\<forall>d. hom_o (Obj d) = fobj (hom_d d) d) \<and>
+ (\<forall>a l. hom_o (Inv a l) = fnvk (hom_o a) a l) \<and>
+ (\<forall>a l m. hom_o (Upd a l m) = fupd (hom_o a) (hom_m m) a l m) \<and>
+ (\<forall>e d. hom_d (e # d) = fcns (hom_e e) (hom_d d) e d) \<and>
+ (hom_d [] = fnil) \<and>
+ (\<forall>l m. hom_e (l, m) = fpar (hom_m m) l m) \<and>
+ (\<forall>x a. hom_m (Sig x a) = fsgm (\<lambda>y. hom_o ([(x, y)] \<bullet> a)) (\<lambda>y. [(x, y)] \<bullet> a))
+)"
+apply (lifting tolift'')
+done
+
+
+end
+