--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/Nominal/LFex.thy	Thu Jan 28 23:36:58 2010 +0100
@@ -0,0 +1,328 @@
+theory LFex
+imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "../QuotMain"
+begin
+
+atom_decl name
+atom_decl ident
+
+datatype kind = 
+    Type
+  | KPi "ty" "name" "kind"
+and ty =  
+    TConst "ident"
+  | TApp "ty" "trm"
+  | TPi "ty" "name" "ty"
+and trm = 
+    Const "ident"
+  | Var "name"
+  | App "trm" "trm"
+  | Lam "ty" "name" "trm" 
+
+fun
+    rfv_kind :: "kind \<Rightarrow> atom set"
+and rfv_ty   :: "ty \<Rightarrow> atom set"
+and rfv_trm  :: "trm \<Rightarrow> atom set"
+where
+  "rfv_kind (Type) = {}"
+| "rfv_kind (KPi A x K) = (rfv_ty A) \<union> ((rfv_kind K) - {atom x})"
+| "rfv_ty (TConst i) = {}"
+| "rfv_ty (TApp A M) = (rfv_ty A) \<union> (rfv_trm M)"
+| "rfv_ty (TPi A x B) = (rfv_ty A) \<union> ((rfv_ty B) - {atom x})"
+| "rfv_trm (Const i) = {}"
+| "rfv_trm (Var x) = {atom x}"
+| "rfv_trm (App M N) = (rfv_trm M) \<union> (rfv_trm N)"
+| "rfv_trm (Lam A x M) = (rfv_ty A) \<union> ((rfv_trm M) - {atom x})"
+
+instantiation kind and ty and trm :: pt
+begin
+
+primrec
+    permute_kind
+and permute_ty
+and permute_trm
+where
+  "permute_kind pi Type = Type"
+| "permute_kind pi (KPi t n k) = KPi (permute_ty pi t) (pi \<bullet> n) (permute_kind pi k)"
+| "permute_ty pi (TConst i) = TConst (pi \<bullet> i)"
+| "permute_ty pi (TApp A M) = TApp (permute_ty pi A) (permute_trm pi M)"
+| "permute_ty pi (TPi A x B) = TPi (permute_ty pi A) (pi \<bullet> x) (permute_ty pi B)"
+| "permute_trm pi (Const i) = Const (pi \<bullet> i)"
+| "permute_trm pi (Var x) = Var (pi \<bullet> x)"
+| "permute_trm pi (App M N) = App (permute_trm pi M) (permute_trm pi N)"
+| "permute_trm pi (Lam A x M) = Lam (permute_ty pi A) (pi \<bullet> x) (permute_trm pi M)"
+
+lemma rperm_zero_ok: "0 \<bullet> (x :: kind) = x \<and> 0 \<bullet> (y :: ty) = y \<and> 0 \<bullet> (z :: trm) = z"
+apply(induct_tac rule: kind_ty_trm.induct)
+apply(simp_all)
+done
+instance
+apply default
+apply (simp_all only:rperm_zero_ok)
+apply(induct_tac [!] x)
+apply(simp_all)
+apply(induct_tac ty)
+apply(simp_all)
+apply(induct_tac trm)
+apply(simp_all)
+apply(induct_tac trm)
+apply(simp_all)
+done
+
+end
+
+inductive
+    akind :: "kind \<Rightarrow> kind \<Rightarrow> bool" ("_ \<approx>ki _" [100, 100] 100)
+and aty   :: "ty \<Rightarrow> ty \<Rightarrow> bool"     ("_ \<approx>ty _" [100, 100] 100)
+and atrm  :: "trm \<Rightarrow> trm \<Rightarrow> bool"   ("_ \<approx>tr _" [100, 100] 100)
+where
+  a1: "(Type) \<approx>ki (Type)"
+| a2: "\<lbrakk>A \<approx>ty A'; \<exists>pi. (rfv_kind K - {atom x} = rfv_kind K' - {atom x'} \<and> (rfv_kind K - {atom x})\<sharp>* pi \<and> (pi \<bullet> K) \<approx>ki K' \<and> (pi \<bullet> x) = x')\<rbrakk> \<Longrightarrow> (KPi A x K) \<approx>ki (KPi A' x' K')"
+| a3: "i = j \<Longrightarrow> (TConst i) \<approx>ty (TConst j)"
+| a4: "\<lbrakk>A \<approx>ty A'; M \<approx>tr M'\<rbrakk> \<Longrightarrow> (TApp A M) \<approx>ty (TApp A' M')"
+| a5: "\<lbrakk>A \<approx>ty A'; \<exists>pi. (rfv_ty B - {atom x} = rfv_ty B' - {atom x'} \<and> (rfv_ty B - {atom x})\<sharp>* pi \<and> (pi \<bullet> B) \<approx>ty B' \<and> (pi \<bullet> x) = x')\<rbrakk> \<Longrightarrow> (TPi A x B) \<approx>ty (TPi A' x' B')"
+| a6: "i = j \<Longrightarrow> (Const i) \<approx>tr (Const j)"
+| a7: "x = y \<Longrightarrow> (Var x) \<approx>tr (Var y)"
+| a8: "\<lbrakk>M \<approx>tr M'; N \<approx>tr N'\<rbrakk> \<Longrightarrow> (App M N) \<approx>tr (App M' N')"
+| a9: "\<lbrakk>A \<approx>ty A'; \<exists>pi. (rfv_trm M - {atom x} = rfv_trm M' - {atom x'} \<and> (rfv_trm M - {atom x})\<sharp>* pi \<and> (pi \<bullet> M) \<approx>tr M' \<and> (pi \<bullet> x) = x')\<rbrakk> \<Longrightarrow> (Lam A x M) \<approx>tr (Lam A' x' M')"
+
+lemma al_refl:
+  fixes K::"kind" 
+  and   A::"ty"
+  and   M::"trm"
+  shows "K \<approx>ki K"
+  and   "A \<approx>ty A"
+  and   "M \<approx>tr M"
+  apply(induct K and A and M rule: kind_ty_trm.inducts)
+  apply(auto intro: akind_aty_atrm.intros)
+  apply (rule a2)
+  apply auto
+  apply(rule_tac x="0" in exI)
+  apply(simp_all add: fresh_star_def fresh_zero_perm)
+  apply (rule a5)
+  apply auto
+  apply(rule_tac x="0" in exI)
+  apply(simp_all add: fresh_star_def fresh_zero_perm)
+  apply (rule a9)
+  apply auto
+  apply(rule_tac x="0" in exI)
+  apply(simp_all add: fresh_star_def fresh_zero_perm)
+  done
+
+lemma alpha_equivps:
+  shows "equivp akind"
+  and   "equivp aty"
+  and   "equivp atrm"
+sorry
+
+quotient_type KIND = kind / akind
+  by (rule alpha_equivps)
+
+quotient_type
+    TY = ty / aty and
+    TRM = trm / atrm
+  by (auto intro: alpha_equivps)
+
+quotient_definition
+   "TYP :: KIND"
+as
+  "Type"
+
+quotient_definition
+   "KPI :: TY \<Rightarrow> name \<Rightarrow> KIND \<Rightarrow> KIND"
+as
+  "KPi"
+
+quotient_definition
+   "TCONST :: ident \<Rightarrow> TY"
+as
+  "TConst"
+
+quotient_definition
+   "TAPP :: TY \<Rightarrow> TRM \<Rightarrow> TY"
+as
+  "TApp"
+
+quotient_definition
+   "TPI :: TY \<Rightarrow> name \<Rightarrow> TY \<Rightarrow> TY"
+as
+  "TPi"
+
+(* FIXME: does not work with CONST *)
+quotient_definition
+   "CONS :: ident \<Rightarrow> TRM"
+as
+  "Const"
+
+quotient_definition
+   "VAR :: name \<Rightarrow> TRM"
+as
+  "Var"
+
+quotient_definition
+   "APP :: TRM \<Rightarrow> TRM \<Rightarrow> TRM"
+as
+  "App"
+
+quotient_definition
+   "LAM :: TY \<Rightarrow> name \<Rightarrow> TRM \<Rightarrow> TRM"
+as
+  "Lam"
+
+(* FIXME: print out a warning if the type contains a liftet type, like kind \<Rightarrow> name set *)
+quotient_definition
+   "fv_kind :: KIND \<Rightarrow> atom set"
+as
+  "rfv_kind"
+
+quotient_definition
+   "fv_ty :: TY \<Rightarrow> atom set"
+as
+  "rfv_ty"
+
+quotient_definition
+   "fv_trm :: TRM \<Rightarrow> atom set"
+as
+  "rfv_trm"
+
+thm akind_aty_atrm.induct
+lemma alpha_rfv:
+  shows "(t \<approx>ki s \<longrightarrow> rfv_kind t = rfv_kind s) \<and>
+     (t1 \<approx>ty s1 \<longrightarrow> rfv_ty t1 = rfv_ty s1) \<and>
+     (t2 \<approx>tr s2 \<longrightarrow> rfv_trm t2 = rfv_trm s2)"
+  apply(rule akind_aty_atrm.induct)
+  apply(simp_all)
+  done
+
+lemma perm_rsp[quot_respect]:
+  "(op = ===> akind ===> akind) permute permute"
+  "(op = ===> aty ===> aty) permute permute"
+  "(op = ===> atrm ===> atrm) permute permute"
+apply simp_all
+sorry (* by eqvt *)
+
+lemma kpi_rsp[quot_respect]: 
+  "(aty ===> op = ===> akind ===> akind) KPi KPi"
+  apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) sorry
+lemma tconst_rsp[quot_respect]: 
+  "(op = ===> aty) TConst TConst"
+  apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) done
+lemma tapp_rsp[quot_respect]: 
+  "(aty ===> atrm ===> aty) TApp TApp" 
+  apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) done
+lemma tpi_rsp[quot_respect]: 
+  "(aty ===> op = ===> aty ===> aty) TPi TPi"
+  apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) sorry
+lemma var_rsp[quot_respect]: 
+  "(op = ===> atrm) Var Var"
+  apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) done
+lemma app_rsp[quot_respect]: 
+  "(atrm ===> atrm ===> atrm) App App"
+  apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) done
+lemma const_rsp[quot_respect]: 
+  "(op = ===> atrm) Const Const"
+  apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) done
+lemma lam_rsp[quot_respect]: 
+  "(aty ===> op = ===> atrm ===> atrm) Lam Lam"
+  apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) sorry
+lemma rfv_ty_rsp[quot_respect]: 
+  "(aty ===> op =) rfv_ty rfv_ty"
+  by (simp add: alpha_rfv)
+lemma rfv_kind_rsp[quot_respect]:
+  "(akind ===> op =) rfv_kind rfv_kind"
+  by (simp add: alpha_rfv)
+lemma rfv_trm_rsp[quot_respect]:
+  "(atrm ===> op =) rfv_trm rfv_trm"
+  by (simp add: alpha_rfv)
+
+thm akind_aty_atrm.induct
+thm kind_ty_trm.induct
+
+lemma KIND_TY_TRM_induct: "\<lbrakk>P10 TYP; \<And>ty name kind. \<lbrakk>P20 ty; P10 kind\<rbrakk> \<Longrightarrow> P10 (KPI ty name kind); \<And>ident. P20 (TCONST ident);
+ \<And>ty trm. \<lbrakk>P20 ty; P30 trm\<rbrakk> \<Longrightarrow> P20 (TAPP ty trm);
+ \<And>ty1 name ty2. \<lbrakk>P20 ty1; P20 ty2\<rbrakk> \<Longrightarrow> P20 (TPI ty1 name ty2); \<And>ident. P30 (CONS ident);
+ \<And>name. P30 (VAR name); \<And>trm1 trm2. \<lbrakk>P30 trm1; P30 trm2\<rbrakk> \<Longrightarrow> P30 (APP trm1 trm2);
+ \<And>ty name trm. \<lbrakk>P20 ty; P30 trm\<rbrakk> \<Longrightarrow> P30 (LAM ty name trm)\<rbrakk>
+\<Longrightarrow> P10 kind \<and> P20 ty \<and> P30 trm"
+by (lifting kind_ty_trm.induct)
+
+instantiation KIND and TY and TRM :: pt
+begin
+
+quotient_definition
+  "permute_KIND :: perm \<Rightarrow> KIND \<Rightarrow> KIND"
+as
+  "permute :: perm \<Rightarrow> kind \<Rightarrow> kind"
+
+quotient_definition
+  "permute_TY :: perm \<Rightarrow> TY \<Rightarrow> TY"
+as
+  "permute :: perm \<Rightarrow> ty \<Rightarrow> ty"
+
+quotient_definition
+  "permute_TRM :: perm \<Rightarrow> TRM \<Rightarrow> TRM"
+as
+  "permute :: perm \<Rightarrow> trm \<Rightarrow> trm"
+
+term "permute_TRM"
+thm permute_kind_permute_ty_permute_trm.simps
+lemma [simp]:
+shows "pi \<bullet> TYP = TYP"
+and   "pi \<bullet> (KPI t n k) = KPI (pi \<bullet> t) (pi \<bullet> n) (pi \<bullet> k)"
+and   "pi \<bullet> (TCONST i) = TCONST (pi \<bullet> i)"
+and   "pi \<bullet> (TAPP A M) = TAPP (pi \<bullet> A) (pi \<bullet> M)"
+and   "pi \<bullet> (TPI A x B) = TPI (pi \<bullet> A) (pi \<bullet> x) (pi \<bullet> B)"
+and   "pi \<bullet> (CONS i) = CONS (pi \<bullet> i)"
+and   "pi \<bullet> (VAR x) = VAR (pi \<bullet> x)"
+and   "pi \<bullet> (APP M N) = APP (pi \<bullet> M) (pi \<bullet> N)"
+and   "pi \<bullet> (LAM A x M) = LAM (pi \<bullet> A) (pi \<bullet> x) (pi \<bullet> M)"
+apply (lifting permute_kind_permute_ty_permute_trm.simps)
+done
+
+lemma perm_zero_ok: "0 \<bullet> (x :: KIND) = x \<and> 0 \<bullet> (y :: TY) = y \<and> 0 \<bullet> (z :: TRM) = z"
+apply (induct rule: KIND_TY_TRM_induct)
+apply simp_all
+done
+
+lemma perm_add_ok: "((p1 + q1) \<bullet> (x1 :: KIND) = (p1 \<bullet> q1 \<bullet> (x1 :: KIND))) \<and> ((p2 + q2) \<bullet> (x2 :: TY) = p2 \<bullet> q2 \<bullet> x2) \<and> ((p3 + q3) \<bullet> (x3 :: TRM) = p3 \<bullet> q3 \<bullet> x3)"
+apply(induct_tac [!] rule: KIND_TY_TRM_induct)
+apply (simp_all)
+(* Sth went wrong... *)
+sorry
+
+instance
+apply default
+apply (simp_all add: perm_zero_ok perm_add_ok)
+done
+
+end
+
+lemma "\<lbrakk>P10 TYP TYP;
+ \<And>A A' K x K' x'.
+    \<lbrakk>(A :: TY) = A'; P20 A A';
+     \<exists>pi. fv_kind K - {atom x} = fv_kind K' - {atom x'} \<and>
+          (fv_kind K - {atom x}) \<sharp>* pi \<and> ((pi \<bullet> K) = K' \<and> P10 (pi \<bullet> K) K') \<and> pi \<bullet> x = x'\<rbrakk>
+    \<Longrightarrow> P10 (KPI A x K) (KPI A' x' K');
+ \<And>i j. i = j \<Longrightarrow> P20 (TCONST i) (TCONST j);
+ \<And>A A' M M'. \<lbrakk>A = A'; P20 A A'; M = M'; P30 M M'\<rbrakk> \<Longrightarrow> P20 (TAPP A M) (TAPP A' M');
+ \<And>A A' B x B' x'.
+    \<lbrakk>A = A'; P20 A A';
+     \<exists>pi. fv_ty B - {atom x} = fv_ty B' - {atom x'} \<and>
+          (fv_ty B - {atom x}) \<sharp>* pi \<and> ((pi \<bullet> B) = B' \<and> P20 (pi \<bullet> B) B') \<and> pi \<bullet> x = x'\<rbrakk>
+    \<Longrightarrow> P20 (TPI A x B) (TPI A' x' B');
+ \<And>i j. i = j \<Longrightarrow> P30 (CONS i) (CONS j); \<And>x y. x = y \<Longrightarrow> P30 (VAR x) (VAR y);
+ \<And>M M' N N'. \<lbrakk>M = M'; P30 M M'; N = N'; P30 N N'\<rbrakk> \<Longrightarrow> P30 (APP M N) (APP M' N');
+ \<And>A A' M x M' x'.
+    \<lbrakk>A = A'; P20 A A';
+     \<exists>pi. fv_trm M - {atom x} = fv_trm M' - {atom x'} \<and>
+          (fv_trm M - {atom x}) \<sharp>* pi \<and> ((pi \<bullet> M) = M' \<and> P30 (pi \<bullet> M) M') \<and> pi \<bullet> x = x'\<rbrakk>
+    \<Longrightarrow> P30 (LAM A x M) (LAM A' x' M')\<rbrakk>
+\<Longrightarrow> (x10 = x20 \<longrightarrow> P10 x10 x20) \<and>
+   (x30 = x40 \<longrightarrow> P20 x30 x40) \<and> (x50 = x60 \<longrightarrow> P30 x50 x60)"
+apply (lifting akind_aty_atrm.induct)
+done
+
+end
+
+
+
+