diff -r db158e995bfc -r 9df6144e281b Attic/Quot/Examples/LFex.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/Quot/Examples/LFex.thy	Thu Feb 25 07:57:17 2010 +0100
@@ -0,0 +1,297 @@
+theory LFex
+imports Nominal "../Quotient_List"
+begin
+
+atom_decl name ident
+
+nominal_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" 
+
+function
+    fv_kind :: "kind \<Rightarrow> name set"
+and fv_ty   :: "ty \<Rightarrow> name set"
+and fv_trm  :: "trm \<Rightarrow> name set"
+where
+  "fv_kind (Type) = {}"
+| "fv_kind (KPi A x K) = (fv_ty A) \<union> ((fv_kind K) - {x})"
+| "fv_ty (TConst i) = {}"
+| "fv_ty (TApp A M) = (fv_ty A) \<union> (fv_trm M)"
+| "fv_ty (TPi A x B) = (fv_ty A) \<union> ((fv_ty B) - {x})"
+| "fv_trm (Const i) = {}"
+| "fv_trm (Var x) = {x}"
+| "fv_trm (App M N) = (fv_trm M) \<union> (fv_trm N)"
+| "fv_trm (Lam A x M) = (fv_ty A) \<union> ((fv_trm M) - {x})"
+sorry
+
+termination fv_kind sorry
+
+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)"
+| a21: "\<lbrakk>A \<approx>ty A'; K \<approx>ki K'\<rbrakk> \<Longrightarrow> (KPi A x K) \<approx>ki (KPi A' x K')"
+| a22: "\<lbrakk>A \<approx>ty A'; K \<approx>ki ([(x,x')]\<bullet>K'); x \<notin> (fv_ty A'); x \<notin> ((fv_kind K') - {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')"
+| a51: "\<lbrakk>A \<approx>ty A'; B \<approx>ty B'\<rbrakk> \<Longrightarrow> (TPi A x B) \<approx>ty (TPi A' x B')"
+| a52: "\<lbrakk>A \<approx>ty A'; B \<approx>ty ([(x,x')]\<bullet>B'); x \<notin> (fv_ty B'); x \<notin> ((fv_ty B') - {x'})\<rbrakk> 
+        \<Longrightarrow> (TPi A x B) \<approx>ty (TPi A' x' B')"
+| a6:  "i = j \<Longrightarrow> (Const i) \<approx>trm (Const j)"
+| a7:  "x = y \<Longrightarrow> (Var x) \<approx>trm (Var y)"
+| a8:  "\<lbrakk>M \<approx>trm M'; N \<approx>tr N'\<rbrakk> \<Longrightarrow> (App M N) \<approx>tr (App M' N')"
+| a91: "\<lbrakk>A \<approx>ty A'; M \<approx>tr M'\<rbrakk> \<Longrightarrow> (Lam A x M) \<approx>tr (Lam A' x M')"
+| a92: "\<lbrakk>A \<approx>ty A'; M \<approx>tr ([(x,x')]\<bullet>M'); x \<notin> (fv_ty B'); x \<notin> ((fv_trm M') - {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)
+  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"
+is
+  "Type"
+
+quotient_definition
+   "KPI :: TY \<Rightarrow> name \<Rightarrow> KIND \<Rightarrow> KIND"
+is
+  "KPi"
+
+quotient_definition
+   "TCONST :: ident \<Rightarrow> TY"
+is
+  "TConst"
+
+quotient_definition
+   "TAPP :: TY \<Rightarrow> TRM \<Rightarrow> TY"
+is
+  "TApp"
+
+quotient_definition
+   "TPI :: TY \<Rightarrow> name \<Rightarrow> TY \<Rightarrow> TY"
+is
+  "TPi"
+
+(* FIXME: does not work with CONST *)
+quotient_definition
+   "CONS :: ident \<Rightarrow> TRM"
+is
+  "Const"
+
+quotient_definition
+   "VAR :: name \<Rightarrow> TRM"
+is
+  "Var"
+
+quotient_definition
+   "APP :: TRM \<Rightarrow> TRM \<Rightarrow> TRM"
+is
+  "App"
+
+quotient_definition
+   "LAM :: TY \<Rightarrow> name \<Rightarrow> TRM \<Rightarrow> TRM"
+is
+  "Lam"
+
+thm TYP_def
+thm KPI_def
+thm TCONST_def
+thm TAPP_def
+thm TPI_def
+thm VAR_def
+thm CONS_def
+thm APP_def
+thm LAM_def
+
+(* FIXME: print out a warning if the type contains a liftet type, like kind \<Rightarrow> name set *)
+quotient_definition
+   "FV_kind :: KIND \<Rightarrow> name set"
+is
+  "fv_kind"
+
+quotient_definition
+   "FV_ty :: TY \<Rightarrow> name set"
+is
+  "fv_ty"
+
+quotient_definition
+   "FV_trm :: TRM \<Rightarrow> name set"
+is
+  "fv_trm"
+
+thm FV_kind_def
+thm FV_ty_def
+thm FV_trm_def
+
+(* FIXME: does not work yet *)
+overloading
+    perm_kind \<equiv> "perm :: 'x prm \<Rightarrow> KIND \<Rightarrow> KIND"   (unchecked)
+    perm_ty   \<equiv> "perm :: 'x prm \<Rightarrow> TY \<Rightarrow> TY"       (unchecked)
+    perm_trm  \<equiv> "perm :: 'x prm \<Rightarrow> TRM \<Rightarrow> TRM"     (unchecked) 
+begin
+
+quotient_definition
+   "perm_kind :: 'x prm \<Rightarrow> KIND \<Rightarrow> KIND"
+is
+  "(perm::'x prm \<Rightarrow> kind \<Rightarrow> kind)"
+
+quotient_definition
+   "perm_ty :: 'x prm \<Rightarrow> TY \<Rightarrow> TY"
+is
+  "(perm::'x prm \<Rightarrow> ty \<Rightarrow> ty)"
+
+quotient_definition
+   "perm_trm :: 'x prm \<Rightarrow> TRM \<Rightarrow> TRM"
+is
+  "(perm::'x prm \<Rightarrow> trm \<Rightarrow> trm)"
+
+end
+
+(* TODO/FIXME: Think whether these RSP theorems are true. *)
+lemma kpi_rsp[quot_respect]: 
+  "(aty ===> op = ===> akind ===> akind) KPi KPi" sorry
+lemma tconst_rsp[quot_respect]: 
+  "(op = ===> aty) TConst TConst" sorry
+lemma tapp_rsp[quot_respect]: 
+  "(aty ===> atrm ===> aty) TApp TApp" sorry
+lemma tpi_rsp[quot_respect]: 
+  "(aty ===> op = ===> aty ===> aty) TPi TPi" sorry
+lemma var_rsp[quot_respect]: 
+  "(op = ===> atrm) Var Var" sorry
+lemma app_rsp[quot_respect]: 
+  "(atrm ===> atrm ===> atrm) App App" sorry
+lemma const_rsp[quot_respect]: 
+  "(op = ===> atrm) Const Const" sorry
+lemma lam_rsp[quot_respect]: 
+  "(aty ===> op = ===> atrm ===> atrm) Lam Lam" sorry
+
+lemma perm_kind_rsp[quot_respect]: 
+  "(op = ===> akind ===> akind) op \<bullet> op \<bullet>" sorry
+lemma perm_ty_rsp[quot_respect]: 
+  "(op = ===> aty ===> aty) op \<bullet> op \<bullet>" sorry
+lemma perm_trm_rsp[quot_respect]: 
+  "(op = ===> atrm ===> atrm) op \<bullet> op \<bullet>" sorry
+
+lemma fv_ty_rsp[quot_respect]: 
+  "(aty ===> op =) fv_ty fv_ty" sorry
+lemma fv_kind_rsp[quot_respect]: 
+  "(akind ===> op =) fv_kind fv_kind" sorry
+lemma fv_trm_rsp[quot_respect]: 
+  "(atrm ===> op =) fv_trm fv_trm" sorry
+
+
+thm akind_aty_atrm.induct
+thm kind_ty_trm.induct
+
+
+lemma 
+  assumes a0:
+  "P1 TYP TYP"
+  and a1: 
+  "\<And>A A' K K' x. \<lbrakk>(A::TY) = A'; P2 A A'; (K::KIND) = K'; P1 K K'\<rbrakk> 
+  \<Longrightarrow> P1 (KPI A x K) (KPI A' x K')"
+  and a2:    
+  "\<And>A A' K K' x x'. \<lbrakk>(A ::TY) = A'; P2 A A'; (K :: KIND) = ([(x, x')] \<bullet> K'); P1 K ([(x, x')] \<bullet> K'); 
+    x \<notin> FV_ty A'; x \<notin> FV_kind K' - {x'}\<rbrakk> \<Longrightarrow> P1 (KPI A x K) (KPI A' x' K')"
+  and a3: 
+  "\<And>i j. i = j \<Longrightarrow> P2 (TCONST i) (TCONST j)"
+  and a4:
+  "\<And>A A' M M'. \<lbrakk>(A ::TY) = A'; P2 A A'; (M :: TRM) = M'; P3 M M'\<rbrakk> \<Longrightarrow> P2 (TAPP A M) (TAPP A' M')"
+  and a5:
+  "\<And>A A' B B' x. \<lbrakk>(A ::TY) = A'; P2 A A'; (B ::TY) = B'; P2 B B'\<rbrakk> \<Longrightarrow> P2 (TPI A x B) (TPI A' x B')"
+  and a6:
+  "\<And>A A' B x x' B'. \<lbrakk>(A ::TY) = A'; P2 A A'; (B ::TY) = ([(x, x')] \<bullet> B'); P2 B ([(x, x')] \<bullet> B'); 
+  x \<notin> FV_ty B'; x \<notin> FV_ty B' - {x'}\<rbrakk> \<Longrightarrow> P2 (TPI A x B) (TPI A' x' B')"
+  and a7:
+  "\<And>i j m. i = j \<Longrightarrow> P3 (CONS i) (m (CONS j))"
+  and a8:
+  "\<And>x y m. x = y \<Longrightarrow> P3 (VAR x) (m (VAR y))"
+  and a9:
+  "\<And>M m M' N N'. \<lbrakk>(M :: TRM) = m M'; P3 M (m M'); (N :: TRM) = N'; P3 N N'\<rbrakk> \<Longrightarrow> P3 (APP M N) (APP M' N')"
+  and a10: 
+  "\<And>A A' M M' x. \<lbrakk>(A ::TY) = A'; P2 A A'; (M :: TRM) = M'; P3 M M'\<rbrakk> \<Longrightarrow> P3 (LAM A x M) (LAM A' x M')"
+  and a11:
+  "\<And>A A' M x x' M' B'. \<lbrakk>(A ::TY) = A'; P2 A A'; (M :: TRM) = ([(x, x')] \<bullet> M'); P3 M ([(x, x')] \<bullet> M'); 
+  x \<notin> FV_ty B'; x \<notin> FV_trm M' - {x'}\<rbrakk> \<Longrightarrow> P3 (LAM A x M) (LAM A' x' M')"
+  shows "((x1 :: KIND) = x2 \<longrightarrow> P1 x1 x2) \<and>
+         ((x3 ::TY) = x4 \<longrightarrow> P2 x3 x4) \<and> 
+         ((x5 :: TRM) = x6 \<longrightarrow> P3 x5 x6)"
+using a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11
+apply(lifting akind_aty_atrm.induct)
+(*
+Profiling:
+ML_prf {* fun ith i =  (#concl (fst (Subgoal.focus @{context} i (#goal (Isar.goal ()))))) *}
+ML_prf {* profile 2 Seq.list_of ((clean_tac @{context} quot defs 1) (ith 3)) *}
+ML_prf {* profile 2 Seq.list_of ((regularize_tac @{context} @{thms alpha_equivps} 1) (ith 1)) *}
+ML_prf {* PolyML.profiling 1 *}
+ML_prf {* profile 2 Seq.list_of ((all_inj_repabs_tac @{context} quot rel_refl trans2 1) (#goal (Isar.goal ()))) *}
+*)
+ done
+
+(* Does not work:
+lemma
+  assumes a0: "P1 TYP"
+  and     a1: "\<And>ty name kind. \<lbrakk>P2 ty; P1 kind\<rbrakk> \<Longrightarrow> P1 (KPI ty name kind)"
+  and     a2: "\<And>id. P2 (TCONST id)"
+  and     a3: "\<And>ty trm. \<lbrakk>P2 ty; P3 trm\<rbrakk> \<Longrightarrow> P2 (TAPP ty trm)"
+  and     a4: "\<And>ty1 name ty2. \<lbrakk>P2 ty1; P2 ty2\<rbrakk> \<Longrightarrow> P2 (TPI ty1 name ty2)"
+  and     a5: "\<And>id. P3 (CONS id)"
+  and     a6: "\<And>name. P3 (VAR name)"
+  and     a7: "\<And>trm1 trm2. \<lbrakk>P3 trm1; P3 trm2\<rbrakk> \<Longrightarrow> P3 (APP trm1 trm2)"
+  and     a8: "\<And>ty name trm. \<lbrakk>P2 ty; P3 trm\<rbrakk> \<Longrightarrow> P3 (LAM ty name trm)"
+  shows "P1 mkind \<and> P2 mty \<and> P3 mtrm"
+using a0 a1 a2 a3 a4 a5 a6 a7 a8
+*)
+
+
+lemma "\<lbrakk>P TYP;
+  \<And>ty name kind. \<lbrakk>Q ty; P kind\<rbrakk> \<Longrightarrow> P (KPI ty name kind);
+  \<And>id. Q (TCONST id);
+  \<And>ty trm. \<lbrakk>Q ty; R trm\<rbrakk> \<Longrightarrow> Q (TAPP ty trm);
+  \<And>ty1 name ty2. \<lbrakk>Q ty1; Q ty2\<rbrakk> \<Longrightarrow> Q (TPI ty1 name ty2);
+  \<And>id. R (CONS id); \<And>name. R (VAR name);
+  \<And>trm1 trm2. \<lbrakk>R trm1; R trm2\<rbrakk> \<Longrightarrow> R (APP trm1 trm2);
+  \<And>ty name trm. \<lbrakk>Q ty; R trm\<rbrakk> \<Longrightarrow> R (LAM ty name trm)\<rbrakk>
+  \<Longrightarrow> P mkind \<and> Q mty \<and> R mtrm"
+apply(lifting kind_ty_trm.induct)
+done
+
+end
+
+
+
+