
diff -r 11b8798dea5d -r 4b0563bc4b03 Quot/Nominal/LFex.thy
--- a/Quot/Nominal/LFex.thy	Wed Feb 24 17:32:22 2010 +0100
+++ b/Quot/Nominal/LFex.thy	Wed Feb 24 17:32:43 2010 +0100
@@ -1,169 +1,45 @@
 theory LFex
-imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv"
+imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv" "Rsp"
 begin
 
 atom_decl name
 atom_decl ident
 
-datatype kind =
+datatype rkind =
     Type
-  | KPi "ty" "name" "kind"
-and ty =
+  | KPi "rty" "name" "rkind"
+and rty =
     TConst "ident"
-  | TApp "ty" "trm"
-  | TPi "ty" "name" "ty"
-and trm =
+  | TApp "rty" "rtrm"
+  | TPi "rty" "name" "rty"
+and rtrm =
     Const "ident"
   | Var "name"
-  | App "trm" "trm"
-  | Lam "ty" "name" "trm"
-
-instantiation kind and ty and trm :: pt
-begin
+  | App "rtrm" "rtrm"
+  | Lam "rty" "name" "rtrm"
 
-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"
-  "0 \<bullet> (y :: ty) = y"
-  "0 \<bullet> (z :: trm) = z"
-apply(induct x and y and z rule: kind_ty_trm.inducts)
-apply(simp_all)
-done
+setup {* snd o define_raw_perms ["rkind", "rty", "rtrm"] ["LFex.rkind", "LFex.rty", "LFex.rtrm"] *}
 
-lemma rperm_plus_ok:
- "(p + q) \<bullet> (x :: kind) = p \<bullet> q \<bullet> x"
- "(p + q) \<bullet> (y :: ty) = p \<bullet> q \<bullet> y"
- "(p + q) \<bullet> (z :: trm) = p \<bullet> q \<bullet> z"
-apply(induct x and y and z rule: kind_ty_trm.inducts)
-apply(simp_all)
-done
-
-instance
-apply default
-apply (simp_all only:rperm_zero_ok rperm_plus_ok)
-done
-
-end
-
-(*
-setup {* snd o define_raw_perms ["kind", "ty", "trm"] ["LFex.kind", "LFex.ty", "LFex.trm"] *}
 local_setup {*
-  snd o define_fv_alpha "LFex.kind"
+  snd o define_fv_alpha "LFex.rkind"
   [[ [], [[], [(NONE, 1)], [(NONE, 1)]] ],
    [ [[]], [[], []], [[], [(NONE, 1)], [(NONE, 1)]] ],
    [ [[]], [[]], [[], []], [[], [(NONE, 1)], [(NONE, 1)]]]] *}
 notation
-    alpha_kind  ("_ \<approx>ki _" [100, 100] 100)
-and alpha_ty    ("_ \<approx>ty _" [100, 100] 100)
-and alpha_trm   ("_ \<approx>tr _" [100, 100] 100)
-thm fv_kind_fv_ty_fv_trm.simps alpha_kind_alpha_ty_alpha_trm.intros
-local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alphas_inj}, []), (build_alpha_inj @{thms alpha_kind_alpha_ty_alpha_trm.intros} @{thms kind.distinct ty.distinct trm.distinct kind.inject ty.inject trm.inject} @{thms alpha_kind.cases alpha_ty.cases alpha_trm.cases} ctxt)) ctxt)) *}
-thm alphas_inj
-
-lemma alphas_eqvt:
-  "t1 \<approx>ki s1 \<Longrightarrow> (pi \<bullet> t1) \<approx>ki (pi \<bullet> s1)"
-  "t2 \<approx>ty s2 \<Longrightarrow> (pi \<bullet> t2) \<approx>ty (pi \<bullet> s2)"
-  "t3 \<approx>tr s3 \<Longrightarrow> (pi \<bullet> t3) \<approx>tr (pi \<bullet> s3)"
-sorry
-
-local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alphas_equivp}, []),
-  (build_equivps [@{term alpha_kind}, @{term alpha_ty}, @{term alpha_trm}]
-     @{thm kind_ty_trm.induct} @{thm alpha_kind_alpha_ty_alpha_trm.induct} 
-     @{thms kind.inject ty.inject trm.inject} @{thms alphas_inj}
-     @{thms kind.distinct ty.distinct trm.distinct}
-     @{thms alpha_kind.cases alpha_ty.cases alpha_trm.cases}
-     @{thms alphas_eqvt} ctxt)) ctxt)) *}
-thm alphas_equivp
-*)
-
-primrec
-    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) = {atom 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) = {atom 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})"
-
-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. (({atom a}, K) \<approx>gen akind rfv_kind pi ({atom b}, K'))\<rbrakk> \<Longrightarrow> (KPi A a K) \<approx>ki (KPi A' b 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. (({atom a}, B) \<approx>gen aty rfv_ty pi ({atom b}, B'))\<rbrakk> \<Longrightarrow> (TPi A a B) \<approx>ty (TPi A' b 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. (({atom a}, M) \<approx>gen atrm rfv_trm pi ({atom b}, M'))\<rbrakk> \<Longrightarrow> (Lam A a M) \<approx>tr (Lam A' b M')"
-
-lemma akind_aty_atrm_inj:
-  "(Type) \<approx>ki (Type) = True"
-  "((KPi A a K) \<approx>ki (KPi A' b K')) = ((A \<approx>ty A') \<and> (\<exists>pi. ({atom a}, K) \<approx>gen akind rfv_kind pi ({atom b}, K')))"
-  "(TConst i) \<approx>ty (TConst j) = (i = j)"
-  "(TApp A M) \<approx>ty (TApp A' M') = (A \<approx>ty A' \<and> M \<approx>tr M')"
-  "((TPi A a B) \<approx>ty (TPi A' b B')) = ((A \<approx>ty A') \<and> (\<exists>pi. (({atom a}, B) \<approx>gen aty rfv_ty pi ({atom b}, B'))))"
-  "(Const i) \<approx>tr (Const j) = (i = j)"
-  "(Var x) \<approx>tr (Var y) = (x = y)"
-  "(App M N) \<approx>tr (App M' N') = (M \<approx>tr M' \<and> N \<approx>tr N')"
-  "((Lam A a M) \<approx>tr (Lam A' b M')) = ((A \<approx>ty A') \<and> (\<exists>pi. (({atom a}, M) \<approx>gen atrm rfv_trm pi ({atom b}, M'))))"
-apply -
-apply (simp add: akind_aty_atrm.intros)
-
-apply rule apply (erule akind.cases) apply simp apply blast
-apply (simp only: akind_aty_atrm.intros)
-
-apply rule apply (erule aty.cases) apply simp apply simp apply simp
-apply (simp only: akind_aty_atrm.intros)
-
-apply rule apply (erule aty.cases) apply simp apply simp apply simp
-apply (simp only: akind_aty_atrm.intros)
-
-apply rule apply (erule aty.cases) apply simp apply simp apply blast
-apply (simp only: akind_aty_atrm.intros)
-
-apply rule apply (erule atrm.cases) apply simp apply simp apply simp apply blast
-apply (simp only: akind_aty_atrm.intros)
-
-apply rule apply (erule atrm.cases) apply simp apply simp apply simp apply blast
-apply (simp only: akind_aty_atrm.intros)
-
-apply rule apply (erule atrm.cases) apply simp apply simp apply simp apply blast
-apply (simp only: akind_aty_atrm.intros)
-
-apply rule apply (erule atrm.cases) apply simp apply simp apply simp apply blast
-apply (simp only: akind_aty_atrm.intros)
-done
+    alpha_rkind  ("_ \<approx>ki _" [100, 100] 100)
+and alpha_rty    ("_ \<approx>ty _" [100, 100] 100)
+and alpha_rtrm   ("_ \<approx>tr _" [100, 100] 100)
+thm fv_rkind_fv_rty_fv_rtrm.simps alpha_rkind_alpha_rty_alpha_rtrm.intros
+local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha_rkind_alpha_rty_alpha_rtrm_inj}, []), (build_alpha_inj @{thms alpha_rkind_alpha_rty_alpha_rtrm.intros} @{thms rkind.distinct rty.distinct rtrm.distinct rkind.inject rty.inject rtrm.inject} @{thms alpha_rkind.cases alpha_rty.cases alpha_rtrm.cases} ctxt)) ctxt)) *}
+thm alpha_rkind_alpha_rty_alpha_rtrm_inj
 
 lemma rfv_eqvt[eqvt]:
-  "((pi\<bullet>rfv_kind t1) = rfv_kind (pi\<bullet>t1))"
-  "((pi\<bullet>rfv_ty t2) = rfv_ty (pi\<bullet>t2))"
-  "((pi\<bullet>rfv_trm t3) = rfv_trm (pi\<bullet>t3))"
-apply(induct t1 and t2 and t3 rule: kind_ty_trm.inducts)
-apply(simp_all add:  union_eqvt Diff_eqvt)
+  "((pi\<bullet>fv_rkind t1) = fv_rkind (pi\<bullet>t1))"
+  "((pi\<bullet>fv_rty t2) = fv_rty (pi\<bullet>t2))"
+  "((pi\<bullet>fv_rtrm t3) = fv_rtrm (pi\<bullet>t3))"
+apply(induct t1 and t2 and t3 rule: rkind_rty_rtrm.inducts)
+apply(simp_all add: union_eqvt Diff_eqvt)
 apply(simp_all add: permute_set_eq atom_eqvt)
 done
 
@@ -171,9 +47,9 @@
   "t1 \<approx>ki s1 \<Longrightarrow> (pi \<bullet> t1) \<approx>ki (pi \<bullet> s1)"
   "t2 \<approx>ty s2 \<Longrightarrow> (pi \<bullet> t2) \<approx>ty (pi \<bullet> s2)"
   "t3 \<approx>tr s3 \<Longrightarrow> (pi \<bullet> t3) \<approx>tr (pi \<bullet> s3)"
-apply(induct rule: akind_aty_atrm.inducts)
-apply (simp_all add: akind_aty_atrm.intros)
-apply (simp_all add: akind_aty_atrm_inj)
+apply(induct rule: alpha_rkind_alpha_rty_alpha_rtrm.inducts)
+apply (simp_all add: alpha_rkind_alpha_rty_alpha_rtrm.intros)
+apply (simp_all add: alpha_rkind_alpha_rty_alpha_rtrm_inj)
 apply (rule alpha_gen_atom_eqvt)
 apply (simp add: rfv_eqvt)
 apply assumption
@@ -185,387 +61,173 @@
 apply assumption
 done
 
-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 alpha_gen)
-  apply (rule a5)
-  apply auto
-  apply(rule_tac x="0" in exI)
-  apply(simp_all add: fresh_star_def fresh_zero_perm alpha_gen)
-  apply (rule a9)
-  apply auto
-  apply(rule_tac x="0" in exI)
-  apply(simp_all add: fresh_star_def fresh_zero_perm alpha_gen)
-  done
-
-lemma al_sym:
-  fixes K K'::"kind" and A A'::"ty" and M M'::"trm"
-  shows "K \<approx>ki K' \<Longrightarrow> K' \<approx>ki K"
-  and   "A \<approx>ty A' \<Longrightarrow> A' \<approx>ty A"
-  and   "M \<approx>tr M' \<Longrightarrow> M' \<approx>tr M"
-  apply(induct K K' and A A' and M M' rule: akind_aty_atrm.inducts)
-  apply(simp_all add: akind_aty_atrm.intros)
-  apply (simp_all add: akind_aty_atrm_inj)
-  apply(erule alpha_gen_compose_sym)
-  apply(erule alpha_eqvt)
-  apply(erule alpha_gen_compose_sym)
-  apply(erule alpha_eqvt)
-  apply(erule alpha_gen_compose_sym)
-  apply(erule alpha_eqvt)
-  done
+local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha_equivps}, []),
+  (build_equivps [@{term alpha_rkind}, @{term alpha_rty}, @{term alpha_rtrm}]
+     @{thm rkind_rty_rtrm.induct} @{thm alpha_rkind_alpha_rty_alpha_rtrm.induct} 
+     @{thms rkind.inject rty.inject rtrm.inject} @{thms alpha_rkind_alpha_rty_alpha_rtrm_inj}
+     @{thms rkind.distinct rty.distinct rtrm.distinct}
+     @{thms alpha_rkind.cases alpha_rty.cases alpha_rtrm.cases}
+     @{thms alpha_eqvt} ctxt)) ctxt)) *}
+thm alpha_equivps
 
-lemma al_trans:
-  fixes K K' K''::"kind" and A A' A''::"ty" and M M' M''::"trm"
-  shows "K \<approx>ki K' \<Longrightarrow> K' \<approx>ki K'' \<Longrightarrow> K \<approx>ki K''"
-  and   "A \<approx>ty A' \<Longrightarrow> A' \<approx>ty A'' \<Longrightarrow> A \<approx>ty A''"
-  and   "M \<approx>tr M' \<Longrightarrow> M' \<approx>tr M'' \<Longrightarrow> M \<approx>tr M''"
-  apply(induct K K' and A A' and M M' arbitrary: K'' and A'' and M'' rule: akind_aty_atrm.inducts)
-  apply(simp_all add: akind_aty_atrm.intros)
-  apply(erule akind.cases)
-  apply(simp_all add: akind_aty_atrm.intros)
-  apply(simp add: akind_aty_atrm_inj)
-  apply(erule alpha_gen_compose_trans)
-  apply(assumption)
-  apply(erule alpha_eqvt)
-  apply(rotate_tac 4)
-  apply(erule aty.cases)
-  apply(simp_all add: akind_aty_atrm.intros)
-  apply(rotate_tac 3)
-  apply(erule aty.cases)
-  apply(simp_all add: akind_aty_atrm.intros)
-  apply(simp add: akind_aty_atrm_inj)
-  apply(erule alpha_gen_compose_trans)
-  apply(assumption)
-  apply(erule alpha_eqvt)
-  apply(rotate_tac 4)
-  apply(erule atrm.cases)
-  apply(simp_all add: akind_aty_atrm.intros)
-  apply(rotate_tac 3)
-  apply(erule atrm.cases)
-  apply(simp_all add: akind_aty_atrm.intros)
-  apply(simp add: akind_aty_atrm_inj)
-  apply(erule alpha_gen_compose_trans)
-  apply(assumption)
-  apply(erule alpha_eqvt)
-  done
-
-lemma alpha_equivps:
-  shows "equivp akind"
-  and   "equivp aty"
-  and   "equivp atrm"
-  apply(rule equivpI)
-  unfolding reflp_def symp_def transp_def
-  apply(auto intro: al_refl al_sym al_trans)
-  apply(rule equivpI)
-  unfolding reflp_def symp_def transp_def
-  apply(auto intro: al_refl al_sym al_trans)
-  apply(rule equivpI)
-  unfolding reflp_def symp_def transp_def
-  apply(auto intro: al_refl al_sym al_trans)
-  done
-
-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"
+local_setup  {* define_quotient_type
+  [(([], @{binding kind}, NoSyn), (@{typ rkind}, @{term alpha_rkind})),
+   (([], @{binding ty},   NoSyn), (@{typ rty},   @{term alpha_rty}  )),
+   (([], @{binding trm},  NoSyn), (@{typ rtrm},  @{term alpha_rtrm} ))]
+  (ALLGOALS (resolve_tac @{thms alpha_equivps}))
+*}
 
-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"
-
-(* 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"
-is
-  "rfv_kind"
-
-quotient_definition
-   "fv_ty :: TY \<Rightarrow> atom set"
-is
-  "rfv_ty"
-
-quotient_definition
-   "fv_trm :: TRM \<Rightarrow> atom set"
-is
-  "rfv_trm"
-
-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 add: alpha_gen)
-  done
+local_setup {*
+(fn ctxt => ctxt
+ |> snd o (Quotient_Def.quotient_lift_const ("TYP", @{term Type}))
+ |> snd o (Quotient_Def.quotient_lift_const ("KPI", @{term KPi}))
+ |> snd o (Quotient_Def.quotient_lift_const ("TCONST", @{term TConst}))
+ |> snd o (Quotient_Def.quotient_lift_const ("TAPP", @{term TApp}))
+ |> snd o (Quotient_Def.quotient_lift_const ("TPI", @{term TPi}))
+ |> snd o (Quotient_Def.quotient_lift_const ("CONS", @{term Const}))
+ |> snd o (Quotient_Def.quotient_lift_const ("VAR", @{term Var}))
+ |> snd o (Quotient_Def.quotient_lift_const ("APP", @{term App}))
+ |> snd o (Quotient_Def.quotient_lift_const ("LAM", @{term Lam}))
+ |> snd o (Quotient_Def.quotient_lift_const ("fv_kind", @{term fv_rkind}))
+ |> snd o (Quotient_Def.quotient_lift_const ("fv_ty", @{term fv_rty}))
+ |> snd o (Quotient_Def.quotient_lift_const ("fv_trm", @{term fv_rtrm}))) *}
+print_theorems
 
-lemma perm_rsp[quot_respect]:
-  "(op = ===> akind ===> akind) permute permute"
-  "(op = ===> aty ===> aty) permute permute"
-  "(op = ===> atrm ===> atrm) permute permute"
-  by (simp_all add:alpha_eqvt)
-
-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 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 kpi_rsp[quot_respect]: 
-  "(aty ===> op = ===> akind ===> akind) KPi KPi"
-  apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9)
-  apply (rule a2) apply assumption
-  apply (rule_tac x="0" in exI)
-  apply (simp add: fresh_star_def fresh_zero_perm alpha_rfv alpha_gen)
-  done
+local_setup {* prove_const_rsp @{binding rfv_rsp} [@{term fv_rkind}, @{term fv_rty}, @{term fv_rtrm}]
+  (fn _ => fvbv_rsp_tac @{thm alpha_rkind_alpha_rty_alpha_rtrm.induct} @{thms fv_rkind_fv_rty_fv_rtrm.simps} 1) *}
+local_setup {* prove_const_rsp Binding.empty [@{term "permute :: perm \<Rightarrow> rkind \<Rightarrow> rkind"}]
+  (fn _ => asm_simp_tac (HOL_ss addsimps @{thms alpha_eqvt}) 1) *}
+local_setup {* prove_const_rsp Binding.empty [@{term "permute :: perm \<Rightarrow> rty \<Rightarrow> rty"}]
+  (fn _ => asm_simp_tac (HOL_ss addsimps @{thms alpha_eqvt}) 1) *}
+local_setup {* prove_const_rsp Binding.empty [@{term "permute :: perm \<Rightarrow> rtrm \<Rightarrow> rtrm"}]
+  (fn _ => asm_simp_tac (HOL_ss addsimps @{thms alpha_eqvt}) 1) *}
+ML {* fun const_rsp_tac _ = constr_rsp_tac @{thms alpha_rkind_alpha_rty_alpha_rtrm_inj}
+  @{thms rfv_rsp} @{thms alpha_equivps} 1 *}
+local_setup {* prove_const_rsp Binding.empty [@{term TConst}] const_rsp_tac *}
+local_setup {* prove_const_rsp Binding.empty [@{term TApp}] const_rsp_tac *}
+local_setup {* prove_const_rsp Binding.empty [@{term Var}] const_rsp_tac *}
+local_setup {* prove_const_rsp Binding.empty [@{term App}] const_rsp_tac *}
+local_setup {* prove_const_rsp Binding.empty [@{term Const}] const_rsp_tac *}
+local_setup {* prove_const_rsp Binding.empty [@{term KPi}] const_rsp_tac *}
+local_setup {* prove_const_rsp Binding.empty [@{term TPi}] const_rsp_tac *}
+local_setup {* prove_const_rsp Binding.empty [@{term Lam}] const_rsp_tac *}
 
-lemma tpi_rsp[quot_respect]: 
-  "(aty ===> op = ===> aty ===> aty) TPi TPi"
-  apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9)
-  apply (rule a5) apply assumption
-  apply (rule_tac x="0" in exI)
-  apply (simp add: fresh_star_def fresh_zero_perm alpha_rfv alpha_gen)
-  done
-lemma lam_rsp[quot_respect]: 
-  "(aty ===> op = ===> atrm ===> atrm) Lam Lam"
-  apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9)
-  apply (rule a9) apply assumption
-  apply (rule_tac x="0" in exI)
-  apply (simp add: fresh_star_def fresh_zero_perm alpha_rfv alpha_gen)
-  done
+lemmas kind_ty_trm_induct = rkind_rty_rtrm.induct[quot_lifted]
 
-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 rkind_rty_rtrm.inducts
+lemmas kind_ty_trm_inducts = rkind_rty_rtrm.inducts[quot_lifted]
 
-thm kind_ty_trm.induct
-lemmas KIND_TY_TRM_induct = kind_ty_trm.induct[quot_lifted]
-
-thm kind_ty_trm.inducts
-lemmas KIND_TY_TRM_inducts = kind_ty_trm.inducts[quot_lifted]
-
-instantiation KIND and TY and TRM :: pt
+instantiation kind and ty and trm :: pt
 begin
 
 quotient_definition
-  "permute_KIND :: perm \<Rightarrow> KIND \<Rightarrow> KIND"
+  "permute_kind :: perm \<Rightarrow> kind \<Rightarrow> kind"
 is
-  "permute :: perm \<Rightarrow> kind \<Rightarrow> kind"
-
-quotient_definition
-  "permute_TY :: perm \<Rightarrow> TY \<Rightarrow> TY"
-is
-  "permute :: perm \<Rightarrow> ty \<Rightarrow> ty"
+  "permute :: perm \<Rightarrow> rkind \<Rightarrow> rkind"
 
 quotient_definition
-  "permute_TRM :: perm \<Rightarrow> TRM \<Rightarrow> TRM"
+  "permute_ty :: perm \<Rightarrow> ty \<Rightarrow> ty"
 is
-  "permute :: perm \<Rightarrow> trm \<Rightarrow> trm"
-
-lemmas permute_ktt[simp] = permute_kind_permute_ty_permute_trm.simps[quot_lifted]
-
-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
+  "permute :: perm \<Rightarrow> rty \<Rightarrow> rty"
 
-lemma perm_add_ok:
-  "((p + q) \<bullet> (x1 :: KIND) = (p \<bullet> q \<bullet> x1))"
-  "((p + q) \<bullet> (x2 :: TY) = p \<bullet> q \<bullet> x2)"
-  "((p + q) \<bullet> (x3 :: TRM) = p \<bullet> q \<bullet> x3)"
-apply (induct x1 and x2 and x3 rule: KIND_TY_TRM_inducts)
-apply (simp_all)
-done
+quotient_definition
+  "permute_trm :: perm \<Rightarrow> trm \<Rightarrow> trm"
+is
+  "permute :: perm \<Rightarrow> rtrm \<Rightarrow> rtrm"
 
-instance
-apply default
-apply (simp_all add: perm_zero_ok perm_add_ok)
-done
+instance by default (simp_all add:
+  permute_rkind_permute_rty_permute_rtrm_zero[quot_lifted]
+  permute_rkind_permute_rty_permute_rtrm_append[quot_lifted])
 
 end
 
-lemmas AKIND_ATY_ATRM_inducts = akind_aty_atrm.inducts[unfolded alpha_gen, quot_lifted, folded alpha_gen]
+(*
+Lifts, but slow and not needed?.
+lemmas alpha_kind_alpha_ty_alpha_trm_induct = alpha_rkind_alpha_rty_alpha_rtrm.induct[unfolded alpha_gen, quot_lifted, folded alpha_gen]
+*)
 
-lemmas KIND_TY_TRM_INJECT = akind_aty_atrm_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]
+lemmas permute_ktt[simp] = permute_rkind_permute_rty_permute_rtrm.simps[quot_lifted]
 
-lemmas fv_kind_ty_trm = rfv_kind_rfv_ty_rfv_trm.simps[quot_lifted]
+lemmas kind_ty_trm_inj = alpha_rkind_alpha_rty_alpha_rtrm_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]
+
+lemmas fv_kind_ty_trm = fv_rkind_fv_rty_fv_rtrm.simps[quot_lifted]
 
 lemmas fv_eqvt = rfv_eqvt[quot_lifted]
 
-lemma supp_kind_ty_trm_easy:
- "supp TYP = {}"
- "supp (TCONST i) = {atom i}"
- "supp (TAPP A M) = supp A \<union> supp M"
- "supp (CONS i) = {atom i}"
- "supp (VAR x) = {atom x}"
- "supp (APP M N) = supp M \<union> supp N"
-apply (simp_all add: supp_def permute_ktt KIND_TY_TRM_INJECT)
-apply (simp_all only: supp_at_base[simplified supp_def])
-apply (simp_all add: Collect_imp_eq Collect_neg_eq)
-done
-
-lemma supp_bind:
-  "(supp (atom na, (ty, ki))) supports (KPI ty na ki)"
-  "(supp (atom na, (ty, ty2))) supports (TPI ty na ty2)"
-  "(supp (atom na, (ty, trm))) supports (LAM ty na trm)"
-apply(simp_all add: supports_def)
-apply(fold fresh_def)
-apply(simp_all add: fresh_Pair swap_fresh_fresh)
-apply(clarify)
-apply(subst swap_at_base_simps(3))
-apply(simp_all add: fresh_atom)
-apply(clarify)
-apply(subst swap_at_base_simps(3))
-apply(simp_all add: fresh_atom)
-apply(clarify)
-apply(subst swap_at_base_simps(3))
+lemma supports:
+  "{} supports TYP"
+  "(supp (atom i)) supports (TCONST i)"
+  "(supp A \<union> supp M) supports (TAPP A M)"
+  "(supp (atom i)) supports (CONS i)"
+  "(supp (atom x)) supports (VAR x)"
+  "(supp M \<union> supp N) supports (APP M N)"
+  "(supp ty \<union> supp (atom na) \<union> supp ki) supports (KPI ty na ki)"
+  "(supp ty \<union> supp (atom na) \<union> supp ty2) supports (TPI ty na ty2)"
+  "(supp ty \<union> supp (atom na) \<union> supp trm) supports (LAM ty na trm)"
+apply(simp_all add: supports_def fresh_def[symmetric] swap_fresh_fresh)
+apply(rule_tac [!] allI)+
+apply(rule_tac [!] impI)
+apply(tactic {* ALLGOALS (REPEAT o etac conjE) *})
 apply(simp_all add: fresh_atom)
 done
 
-lemma KIND_TY_TRM_fs:
-  "finite (supp (x\<Colon>KIND))"
-  "finite (supp (y\<Colon>TY))"
-  "finite (supp (z\<Colon>TRM))"
-apply(induct x and y and z rule: KIND_TY_TRM_inducts)
-apply(simp_all add: supp_kind_ty_trm_easy)
-apply(rule supports_finite)
-apply(rule supp_bind(1))
-apply(simp add: supp_Pair supp_atom)
-apply(rule supports_finite)
-apply(rule supp_bind(2))
-apply(simp add: supp_Pair supp_atom)
-apply(rule supports_finite)
-apply(rule supp_bind(3))
-apply(simp add: supp_Pair supp_atom)
+lemma kind_ty_trm_fs:
+  "finite (supp (x\<Colon>kind))"
+  "finite (supp (y\<Colon>ty))"
+  "finite (supp (z\<Colon>trm))"
+apply(induct x and y and z rule: kind_ty_trm_inducts)
+apply(tactic {* ALLGOALS (rtac @{thm supports_finite} THEN' resolve_tac @{thms supports}) *})
+apply(simp_all add: supp_atom)
+done
+
+instance kind and ty and trm :: fs
+apply(default)
+apply(simp_all only: kind_ty_trm_fs)
 done
 
-instance KIND and TY and TRM :: fs
-apply(default)
-apply(simp_all only: KIND_TY_TRM_fs)
-done
+lemma supp_eqs:
+  "supp TYP = {}"
+  "supp rkind = fv_kind rkind \<Longrightarrow> supp (KPI rty name rkind) = supp rty \<union> supp (Abs {atom name} rkind)"
+  "supp (TCONST i) = {atom i}"
+  "supp (TAPP A M) = supp A \<union> supp M"
+  "supp rty2 = fv_ty rty2 \<Longrightarrow> supp (TPI rty1 name rty2) = supp rty1 \<union> supp (Abs {atom name} rty2)"
+  "supp (CONS i) = {atom i}"
+  "supp (VAR x) = {atom x}"
+  "supp (APP M N) = supp M \<union> supp N"
+  "supp rtrm = fv_trm rtrm \<Longrightarrow> supp (LAM rty name rtrm) = supp rty \<union> supp (Abs {atom name} rtrm)"
+  apply(simp_all (no_asm) add: supp_def)
+  apply(simp_all only: kind_ty_trm_inj Abs_eq_iff alpha_gen)
+  apply(simp_all only: insert_eqvt empty_eqvt atom_eqvt supp_eqvt[symmetric] fv_eqvt[symmetric])
+  apply(simp_all add: Collect_imp_eq Collect_neg_eq[symmetric] Set.Un_commute)
+  apply(simp_all add: supp_at_base[simplified supp_def])
+  done
 
 lemma supp_fv:
- "supp t1 = fv_kind t1"
- "supp t2 = fv_ty t2"
- "supp t3 = fv_trm t3"
-apply(induct t1 and t2 and t3 rule: KIND_TY_TRM_inducts)
-apply (simp_all add: supp_kind_ty_trm_easy)
-apply (simp_all add: fv_kind_ty_trm)
-apply(subgoal_tac "supp (KPI ty name kind) = supp ty \<union> supp (Abs {atom name} kind)")
-apply(simp add: supp_Abs Set.Un_commute)
-apply(simp (no_asm) add: supp_def)
-apply(simp add: KIND_TY_TRM_INJECT)
-apply(simp add: Abs_eq_iff)
-apply(simp add: alpha_gen)
-apply(simp add: Collect_imp_eq Collect_neg_eq Set.Un_commute insert_eqvt empty_eqvt atom_eqvt)
-apply(simp add: supp_eqvt[symmetric] fv_eqvt[symmetric])
-apply(subgoal_tac "supp (TPI ty1 name ty2) = supp ty1 \<union> supp (Abs {atom name} ty2)")
-apply(simp add: supp_Abs Set.Un_commute)
-apply(simp (no_asm) add: supp_def)
-apply(simp add: KIND_TY_TRM_INJECT)
-apply(simp add: Abs_eq_iff)
-apply(simp add: alpha_gen)
-apply(simp add: supp_eqvt[symmetric] fv_eqvt[symmetric] insert_eqvt empty_eqvt atom_eqvt)
-apply(simp add: Collect_imp_eq Collect_neg_eq Set.Un_commute)
-apply(subgoal_tac "supp (LAM ty name trm) = supp ty \<union> supp (Abs {atom name} trm)")
-apply(simp add: supp_Abs Set.Un_commute)
-apply(simp (no_asm) add: supp_def)
-apply(simp add: KIND_TY_TRM_INJECT)
-apply(simp add: Abs_eq_iff)
-apply(simp add: alpha_gen)
-apply(simp add: supp_eqvt[symmetric] fv_eqvt[symmetric] insert_eqvt empty_eqvt atom_eqvt)
-apply(simp add: Collect_imp_eq Collect_neg_eq Set.Un_commute)
-done
+  "supp t1 = fv_kind t1"
+  "supp t2 = fv_ty t2"
+  "supp t3 = fv_trm t3"
+  apply(induct t1 and t2 and t3 rule: kind_ty_trm_inducts)
+  apply(simp_all (no_asm) only: supp_eqs fv_kind_ty_trm)
+  apply(simp_all)
+  apply(subst supp_eqs)
+  apply(simp_all add: supp_Abs)
+  apply(subst supp_eqs)
+  apply(simp_all add: supp_Abs)
+  apply(subst supp_eqs)
+  apply(simp_all add: supp_Abs)
+  done
 
-(* Not needed anymore *)
-lemma supp_kpi_pre: "supp (KPI A x K) = (supp (Abs {atom x} K)) \<union> supp A"
-apply (simp add: permute_set_eq supp_def Abs_eq_iff KIND_TY_TRM_INJECT)
-apply (simp add: alpha_gen supp_fv)
-apply (simp add: Collect_imp_eq Collect_neg_eq add: atom_eqvt Set.Un_commute)
-done
-
-lemma supp_kind_ty_trm:
- "supp TYP = {}"
- "supp (KPI A x K) = supp A \<union> (supp K - {atom x})"
- "supp (TCONST i) = {atom i}"
- "supp (TAPP A M) = supp A \<union> supp M"
- "supp (TPI A x B) = supp A \<union> (supp B - {atom x})"
- "supp (CONS i) = {atom i}"
- "supp (VAR x) = {atom x}"
- "supp (APP M N) = supp M \<union> supp N"
- "supp (LAM A x M) = supp A \<union> (supp M - {atom x})"
-apply (simp_all only: supp_kind_ty_trm_easy)
-apply (simp_all only: supp_fv fv_kind_ty_trm)
-done
+lemma supp_rkind_rty_rtrm:
+  "supp TYP = {}"
+  "supp (KPI A x K) = supp A \<union> (supp K - {atom x})"
+  "supp (TCONST i) = {atom i}"
+  "supp (TAPP A M) = supp A \<union> supp M"
+  "supp (TPI A x B) = supp A \<union> (supp B - {atom x})"
+  "supp (CONS i) = {atom i}"
+  "supp (VAR x) = {atom x}"
+  "supp (APP M N) = supp M \<union> supp N"
+  "supp (LAM A x M) = supp A \<union> (supp M - {atom x})"
+  by (simp_all only: supp_fv fv_kind_ty_trm)
 
 end
 




