nominal2: comparison Quot/Nominal/LFex.thy

equal deleted inserted replaced

-:59dc48db4a84
+:1240d5eb275d
 begin
 atom_decl name
 atom_decl ident
-datatype kind =
+datatype rkind =
 Type
-| KPi "ty" "name" "kind"
+| KPi "rty" "name" "rkind"
-and ty =
+and rty =
 TConst "ident"
-| TApp "ty" "trm"
+| TApp "rty" "rtrm"
-| TPi "ty" "name" "ty"
+| TPi "rty" "name" "rty"
-and trm =
+and rtrm =
 Const "ident"
 | Var "name"
-| App "trm" "trm"
+| App "rtrm" "rtrm"
-| Lam "ty" "name" "trm"
+| Lam "rty" "name" "rtrm"
-instantiation kind and ty and trm :: pt
+instantiation rkind and rty and rtrm :: pt
 begin
 primrec
-permute_kind
+permute_rkind
-and permute_ty
+and permute_rty
-and permute_trm
+and permute_rtrm
 where
-"permute_kind pi Type = Type"
+"permute_rkind pi Type = Type"
-| "permute_kind pi (KPi t n k) = KPi (permute_ty pi t) (pi \<bullet> n) (permute_kind pi k)"
+| "permute_rkind pi (KPi t n k) = KPi (permute_rty pi t) (pi \<bullet> n) (permute_rkind pi k)"
-| "permute_ty pi (TConst i) = TConst (pi \<bullet> i)"
+| "permute_rty pi (TConst i) = TConst (pi \<bullet> i)"
-| "permute_ty pi (TApp A M) = TApp (permute_ty pi A) (permute_trm pi M)"
+| "permute_rty pi (TApp A M) = TApp (permute_rty pi A) (permute_rtrm pi M)"
-| "permute_ty pi (TPi A x B) = TPi (permute_ty pi A) (pi \<bullet> x) (permute_ty pi B)"
+| "permute_rty pi (TPi A x B) = TPi (permute_rty pi A) (pi \<bullet> x) (permute_rty pi B)"
-| "permute_trm pi (Const i) = Const (pi \<bullet> i)"
+| "permute_rtrm pi (Const i) = Const (pi \<bullet> i)"
-| "permute_trm pi (Var x) = Var (pi \<bullet> x)"
+| "permute_rtrm pi (Var x) = Var (pi \<bullet> x)"
-| "permute_trm pi (App M N) = App (permute_trm pi M) (permute_trm pi N)"
+| "permute_rtrm pi (App M N) = App (permute_rtrm pi M) (permute_rtrm pi N)"
-| "permute_trm pi (Lam A x M) = Lam (permute_ty pi A) (pi \<bullet> x) (permute_trm pi M)"
+| "permute_rtrm pi (Lam A x M) = Lam (permute_rty pi A) (pi \<bullet> x) (permute_rtrm pi M)"
 lemma rperm_zero_ok:
-"0 \<bullet> (x :: kind) = x"
+"0 \<bullet> (x :: rkind) = x"
-"0 \<bullet> (y :: ty) = y"
+"0 \<bullet> (y :: rty) = y"
-"0 \<bullet> (z :: trm) = z"
+"0 \<bullet> (z :: rtrm) = z"
-apply(induct x and y and z rule: kind_ty_trm.inducts)
+apply(induct x and y and z rule: rkind_rty_rtrm.inducts)
 apply(simp_all)
 done
 lemma rperm_plus_ok:
-"(p + q) \<bullet> (x :: kind) = p \<bullet> q \<bullet> x"
+"(p + q) \<bullet> (x :: rkind) = p \<bullet> q \<bullet> x"
-"(p + q) \<bullet> (y :: ty) = p \<bullet> q \<bullet> y"
+"(p + q) \<bullet> (y :: rty) = p \<bullet> q \<bullet> y"
-"(p + q) \<bullet> (z :: trm) = p \<bullet> q \<bullet> z"
+"(p + q) \<bullet> (z :: rtrm) = p \<bullet> q \<bullet> z"
-apply(induct x and y and z rule: kind_ty_trm.inducts)
+apply(induct x and y and z rule: rkind_rty_rtrm.inducts)
 apply(simp_all)
 done
 instance
 apply default
 done
 end
 (*
-setup {* snd o define_raw_perms ["kind", "ty", "trm"] ["LFex.kind", "LFex.ty", "LFex.trm"] *}
+setup {* snd o define_raw_perms ["rkind", "rty", "rtrm"] ["LFex.rkind", "LFex.rty", "LFex.rtrm"] *}
 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)
+alpha_rkind  ("_ \<approx>ki _" [100, 100] 100)
-and alpha_ty    ("_ \<approx>ty _" [100, 100] 100)
+and alpha_rty    ("_ \<approx>rty _" [100, 100] 100)
-and alpha_trm   ("_ \<approx>tr _" [100, 100] 100)
+and alpha_rtrm   ("_ \<approx>tr _" [100, 100] 100)
-thm fv_kind_fv_ty_fv_trm.simps alpha_kind_alpha_ty_alpha_trm.intros
+thm fv_rkind_fv_rty_fv_rtrm.simps alpha_rkind_alpha_rty_alpha_rtrm.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)) *}
+local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alphas_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 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)"
+"t2 \<approx>rty s2 \<Longrightarrow> (pi \<bullet> t2) \<approx>rty (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}]
+(build_equivps [@{term alpha_rkind}, @{term alpha_rty}, @{term alpha_rtrm}]
-@{thm kind_ty_trm.induct} @{thm alpha_kind_alpha_ty_alpha_trm.induct}
+@{thm rkind_rty_rtrm.induct} @{thm alpha_rkind_alpha_rty_alpha_rtrm.induct}
-@{thms kind.inject ty.inject trm.inject} @{thms alphas_inj}
+@{thms rkind.inject rty.inject rtrm.inject} @{thms alphas_inj}
-@{thms kind.distinct ty.distinct trm.distinct}
+@{thms rkind.distinct rty.distinct rtrm.distinct}
-@{thms alpha_kind.cases alpha_ty.cases alpha_trm.cases}
+@{thms alpha_rkind.cases alpha_rty.cases alpha_rtrm.cases}
 @{thms alphas_eqvt} ctxt)) ctxt)) *}
 thm alphas_equivp
 *)
 primrec
-rfv_kind :: "kind \<Rightarrow> atom set"
+rfv_rkind :: "rkind \<Rightarrow> atom set"
-and rfv_ty   :: "ty \<Rightarrow> atom set"
+and rfv_rty   :: "rty \<Rightarrow> atom set"
-and rfv_trm  :: "trm \<Rightarrow> atom set"
+and rfv_rtrm  :: "rtrm \<Rightarrow> atom set"
 where
-"rfv_kind (Type) = {}"
+"rfv_rkind (Type) = {}"
-| "rfv_kind (KPi A x K) = (rfv_ty A) \<union> ((rfv_kind K) - {atom x})"
+| "rfv_rkind (KPi A x K) = (rfv_rty A) \<union> ((rfv_rkind K) - {atom x})"
-| "rfv_ty (TConst i) = {atom i}"
+| "rfv_rty (TConst i) = {atom i}"
-| "rfv_ty (TApp A M) = (rfv_ty A) \<union> (rfv_trm M)"
+| "rfv_rty (TApp A M) = (rfv_rty A) \<union> (rfv_rtrm M)"
-| "rfv_ty (TPi A x B) = (rfv_ty A) \<union> ((rfv_ty B) - {atom x})"
+| "rfv_rty (TPi A x B) = (rfv_rty A) \<union> ((rfv_rty B) - {atom x})"
-| "rfv_trm (Const i) = {atom i}"
+| "rfv_rtrm (Const i) = {atom i}"
-| "rfv_trm (Var x) = {atom x}"
+| "rfv_rtrm (Var x) = {atom x}"
-| "rfv_trm (App M N) = (rfv_trm M) \<union> (rfv_trm N)"
+| "rfv_rtrm (App M N) = (rfv_rtrm M) \<union> (rfv_rtrm N)"
-| "rfv_trm (Lam A x M) = (rfv_ty A) \<union> ((rfv_trm M) - {atom x})"
+| "rfv_rtrm (Lam A x M) = (rfv_rty A) \<union> ((rfv_rtrm M) - {atom x})"
 inductive
-akind :: "kind \<Rightarrow> kind \<Rightarrow> bool" ("_ \<approx>ki _" [100, 100] 100)
+arkind :: "rkind \<Rightarrow> rkind \<Rightarrow> bool" ("_ \<approx>ki _" [100, 100] 100)
-and aty   :: "ty \<Rightarrow> ty \<Rightarrow> bool"     ("_ \<approx>ty _" [100, 100] 100)
+and arty   :: "rty \<Rightarrow> rty \<Rightarrow> bool"     ("_ \<approx>rty _" [100, 100] 100)
-and atrm  :: "trm \<Rightarrow> trm \<Rightarrow> bool"   ("_ \<approx>tr _" [100, 100] 100)
+and artrm  :: "rtrm \<Rightarrow> rtrm \<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')"
+| a2: "\<lbrakk>A \<approx>rty A'; \<exists>pi. (({atom a}, K) \<approx>gen arkind rfv_rkind 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)"
+| a3: "i = j \<Longrightarrow> (TConst i) \<approx>rty (TConst j)"
-| a4: "\<lbrakk>A \<approx>ty A'; M \<approx>tr M'\<rbrakk> \<Longrightarrow> (TApp A M) \<approx>ty (TApp A' M')"
+| a4: "\<lbrakk>A \<approx>rty A'; M \<approx>tr M'\<rbrakk> \<Longrightarrow> (TApp A M) \<approx>rty (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')"
+| a5: "\<lbrakk>A \<approx>rty A'; \<exists>pi. (({atom a}, B) \<approx>gen arty rfv_rty pi ({atom b}, B'))\<rbrakk> \<Longrightarrow> (TPi A a B) \<approx>rty (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')"
+| a9: "\<lbrakk>A \<approx>rty A'; \<exists>pi. (({atom a}, M) \<approx>gen artrm rfv_rtrm pi ({atom b}, M'))\<rbrakk> \<Longrightarrow> (Lam A a M) \<approx>tr (Lam A' b M')"
-lemma akind_aty_atrm_inj:
+lemma arkind_arty_artrm_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')))"
+"((KPi A a K) \<approx>ki (KPi A' b K')) = ((A \<approx>rty A') \<and> (\<exists>pi. ({atom a}, K) \<approx>gen arkind rfv_rkind pi ({atom b}, K')))"
-"(TConst i) \<approx>ty (TConst j) = (i = j)"
+"(TConst i) \<approx>rty (TConst j) = (i = j)"
-"(TApp A M) \<approx>ty (TApp A' M') = (A \<approx>ty A' \<and> M \<approx>tr M')"
+"(TApp A M) \<approx>rty (TApp A' M') = (A \<approx>rty 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'))))"
+"((TPi A a B) \<approx>rty (TPi A' b B')) = ((A \<approx>rty A') \<and> (\<exists>pi. (({atom a}, B) \<approx>gen arty rfv_rty 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'))))"
+"((Lam A a M) \<approx>tr (Lam A' b M')) = ((A \<approx>rty A') \<and> (\<exists>pi. (({atom a}, M) \<approx>gen artrm rfv_rtrm pi ({atom b}, M'))))"
 apply -
-apply (simp add: akind_aty_atrm.intros)
+apply (simp add: arkind_arty_artrm.intros)
-apply rule apply (erule akind.cases) apply simp apply blast
+apply rule apply (erule arkind.cases) apply simp apply blast
-apply (simp only: akind_aty_atrm.intros)
+apply (simp only: arkind_arty_artrm.intros)
-apply rule apply (erule aty.cases) apply simp apply simp apply simp
+apply rule apply (erule arty.cases) apply simp apply simp apply simp
-apply (simp only: akind_aty_atrm.intros)
+apply (simp only: arkind_arty_artrm.intros)
-apply rule apply (erule aty.cases) apply simp apply simp apply simp
+apply rule apply (erule arty.cases) apply simp apply simp apply simp
-apply (simp only: akind_aty_atrm.intros)
+apply (simp only: arkind_arty_artrm.intros)
-apply rule apply (erule aty.cases) apply simp apply simp apply blast
+apply rule apply (erule arty.cases) apply simp apply simp apply blast
-apply (simp only: akind_aty_atrm.intros)
+apply (simp only: arkind_arty_artrm.intros)
-apply rule apply (erule atrm.cases) apply simp apply simp apply simp apply blast
+apply rule apply (erule artrm.cases) apply simp apply simp apply simp apply blast
-apply (simp only: akind_aty_atrm.intros)
+apply (simp only: arkind_arty_artrm.intros)
-apply rule apply (erule atrm.cases) apply simp apply simp apply simp apply blast
+apply rule apply (erule artrm.cases) apply simp apply simp apply simp apply blast
-apply (simp only: akind_aty_atrm.intros)
+apply (simp only: arkind_arty_artrm.intros)
-apply rule apply (erule atrm.cases) apply simp apply simp apply simp apply blast
+apply rule apply (erule artrm.cases) apply simp apply simp apply simp apply blast
-apply (simp only: akind_aty_atrm.intros)
+apply (simp only: arkind_arty_artrm.intros)
-apply rule apply (erule atrm.cases) apply simp apply simp apply simp apply blast
+apply rule apply (erule artrm.cases) apply simp apply simp apply simp apply blast
-apply (simp only: akind_aty_atrm.intros)
+apply (simp only: arkind_arty_artrm.intros)
 done
 lemma rfv_eqvt[eqvt]:
-"((pi\<bullet>rfv_kind t1) = rfv_kind (pi\<bullet>t1))"
+"((pi\<bullet>rfv_rkind t1) = rfv_rkind (pi\<bullet>t1))"
-"((pi\<bullet>rfv_ty t2) = rfv_ty (pi\<bullet>t2))"
+"((pi\<bullet>rfv_rty t2) = rfv_rty (pi\<bullet>t2))"
-"((pi\<bullet>rfv_trm t3) = rfv_trm (pi\<bullet>t3))"
+"((pi\<bullet>rfv_rtrm t3) = rfv_rtrm (pi\<bullet>t3))"
-apply(induct t1 and t2 and t3 rule: kind_ty_trm.inducts)
+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
 lemma alpha_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)"
+"t2 \<approx>rty s2 \<Longrightarrow> (pi \<bullet> t2) \<approx>rty (pi \<bullet> s2)"
 "t3 \<approx>tr s3 \<Longrightarrow> (pi \<bullet> t3) \<approx>tr (pi \<bullet> s3)"
-apply(induct rule: akind_aty_atrm.inducts)
+apply(induct rule: arkind_arty_artrm.inducts)
-apply (simp_all add: akind_aty_atrm.intros)
+apply (simp_all add: arkind_arty_artrm.intros)
-apply (simp_all add: akind_aty_atrm_inj)
+apply (simp_all add: arkind_arty_artrm_inj)
 apply (rule alpha_gen_atom_eqvt)
 apply (simp add: rfv_eqvt)
 apply assumption
 apply (rule alpha_gen_atom_eqvt)
 apply (simp add: rfv_eqvt)
 apply (simp add: rfv_eqvt)
 apply assumption
 done
 lemma al_refl:
-fixes K::"kind"
+fixes K::"rkind"
-and   A::"ty"
+and   A::"rty"
-and   M::"trm"
+and   M::"rtrm"
 shows "K \<approx>ki K"
-and   "A \<approx>ty A"
+and   "A \<approx>rty A"
 and   "M \<approx>tr M"
-apply(induct K and A and M rule: kind_ty_trm.inducts)
+apply(induct K and A and M rule: rkind_rty_rtrm.inducts)
-apply(auto intro: akind_aty_atrm.intros)
+apply(auto intro: arkind_arty_artrm.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(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"
+fixes K K'::"rkind" and A A'::"rty" and M M'::"rtrm"
 shows "K \<approx>ki K' \<Longrightarrow> K' \<approx>ki K"
-and   "A \<approx>ty A' \<Longrightarrow> A' \<approx>ty A"
+and   "A \<approx>rty A' \<Longrightarrow> A' \<approx>rty 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(induct K K' and A A' and M M' rule: arkind_arty_artrm.inducts)
-apply(simp_all add: akind_aty_atrm.intros)
+apply(simp_all add: arkind_arty_artrm.intros)
-apply (simp_all add: akind_aty_atrm_inj)
+apply (simp_all add: arkind_arty_artrm_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
 lemma al_trans:
-fixes K K' K''::"kind" and A A' A''::"ty" and M M' M''::"trm"
+fixes K K' K''::"rkind" and A A' A''::"rty" and M M' M''::"rtrm"
 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   "A \<approx>rty A' \<Longrightarrow> A' \<approx>rty A'' \<Longrightarrow> A \<approx>rty 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(induct K K' and A A' and M M' arbitrary: K'' and A'' and M'' rule: arkind_arty_artrm.inducts)
-apply(simp_all add: akind_aty_atrm.intros)
+apply(simp_all add: arkind_arty_artrm.intros)
-apply(erule akind.cases)
+apply(erule arkind.cases)
-apply(simp_all add: akind_aty_atrm.intros)
+apply(simp_all add: arkind_arty_artrm.intros)
-apply(simp add: akind_aty_atrm_inj)
+apply(simp add: arkind_arty_artrm_inj)
 apply(erule alpha_gen_compose_trans)
 apply(assumption)
 apply(erule alpha_eqvt)
 apply(rotate_tac 4)
-apply(erule aty.cases)
+apply(erule arty.cases)
-apply(simp_all add: akind_aty_atrm.intros)
+apply(simp_all add: arkind_arty_artrm.intros)
 apply(rotate_tac 3)
-apply(erule aty.cases)
+apply(erule arty.cases)
-apply(simp_all add: akind_aty_atrm.intros)
+apply(simp_all add: arkind_arty_artrm.intros)
-apply(simp add: akind_aty_atrm_inj)
+apply(simp add: arkind_arty_artrm_inj)
 apply(erule alpha_gen_compose_trans)
 apply(assumption)
 apply(erule alpha_eqvt)
 apply(rotate_tac 4)
-apply(erule atrm.cases)
+apply(erule artrm.cases)
-apply(simp_all add: akind_aty_atrm.intros)
+apply(simp_all add: arkind_arty_artrm.intros)
 apply(rotate_tac 3)
-apply(erule atrm.cases)
+apply(erule artrm.cases)
-apply(simp_all add: akind_aty_atrm.intros)
+apply(simp_all add: arkind_arty_artrm.intros)
-apply(simp add: akind_aty_atrm_inj)
+apply(simp add: arkind_arty_artrm_inj)
 apply(erule alpha_gen_compose_trans)
 apply(assumption)
 apply(erule alpha_eqvt)
 done
 lemma alpha_equivps:
-shows "equivp akind"
+shows "equivp arkind"
-and   "equivp aty"
+and   "equivp arty"
-and   "equivp atrm"
+and   "equivp artrm"
 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(rule equivpI)
 unfolding reflp_def symp_def transp_def
 apply(auto intro: al_refl al_sym al_trans)
 done
-quotient_type KIND = kind / akind
+quotient_type RKIND = rkind / arkind
 by (rule alpha_equivps)
 quotient_type
-TY = ty / aty and
+RTY = rty / arty and
-TRM = trm / atrm
+RTRM = rtrm / artrm
 by (auto intro: alpha_equivps)
 quotient_definition
-"TYP :: KIND"
+"TYP :: RKIND"
 is
 "Type"
 quotient_definition
-"KPI :: TY \<Rightarrow> name \<Rightarrow> KIND \<Rightarrow> KIND"
+"KPI :: RTY \<Rightarrow> name \<Rightarrow> RKIND \<Rightarrow> RKIND"
 is
 "KPi"
 quotient_definition
-"TCONST :: ident \<Rightarrow> TY"
+"TCONST :: ident \<Rightarrow> RTY"
 is
 "TConst"
 quotient_definition
-"TAPP :: TY \<Rightarrow> TRM \<Rightarrow> TY"
+"TAPP :: RTY \<Rightarrow> RTRM \<Rightarrow> RTY"
 is
 "TApp"
 quotient_definition
-"TPI :: TY \<Rightarrow> name \<Rightarrow> TY \<Rightarrow> TY"
+"TPI :: RTY \<Rightarrow> name \<Rightarrow> RTY \<Rightarrow> RTY"
 is
 "TPi"
 (* FIXME: does not work with CONST *)
 quotient_definition
-"CONS :: ident \<Rightarrow> TRM"
+"CONS :: ident \<Rightarrow> RTRM"
 is
 "Const"
 quotient_definition
-"VAR :: name \<Rightarrow> TRM"
+"VAR :: name \<Rightarrow> RTRM"
 is
 "Var"
 quotient_definition
-"APP :: TRM \<Rightarrow> TRM \<Rightarrow> TRM"
+"APP :: RTRM \<Rightarrow> RTRM \<Rightarrow> RTRM"
 is
 "App"
 quotient_definition
-"LAM :: TY \<Rightarrow> name \<Rightarrow> TRM \<Rightarrow> TRM"
+"LAM :: RTY \<Rightarrow> name \<Rightarrow> RTRM \<Rightarrow> RTRM"
 is
 "Lam"
-(* FIXME: print out a warning if the type contains a liftet type, like kind \<Rightarrow> name set *)
+(* FIXME: print out a warning if the type contains a liftet type, like rkind \<Rightarrow> name set *)
 quotient_definition
-"fv_kind :: KIND \<Rightarrow> atom set"
+"fv_rkind :: RKIND \<Rightarrow> atom set"
 is
-"rfv_kind"
+"rfv_rkind"
 quotient_definition
-"fv_ty :: TY \<Rightarrow> atom set"
+"fv_rty :: RTY \<Rightarrow> atom set"
 is
-"rfv_ty"
+"rfv_rty"
 quotient_definition
-"fv_trm :: TRM \<Rightarrow> atom set"
+"fv_rtrm :: RTRM \<Rightarrow> atom set"
 is
-"rfv_trm"
+"rfv_rtrm"
 lemma alpha_rfv:
-shows "(t \<approx>ki s \<longrightarrow> rfv_kind t = rfv_kind s) \<and>
+shows "(t \<approx>ki s \<longrightarrow> rfv_rkind t = rfv_rkind s) \<and>
-(t1 \<approx>ty s1 \<longrightarrow> rfv_ty t1 = rfv_ty s1) \<and>
+(t1 \<approx>rty s1 \<longrightarrow> rfv_rty t1 = rfv_rty s1) \<and>
-(t2 \<approx>tr s2 \<longrightarrow> rfv_trm t2 = rfv_trm s2)"
+(t2 \<approx>tr s2 \<longrightarrow> rfv_rtrm t2 = rfv_rtrm s2)"
-apply(rule akind_aty_atrm.induct)
+apply(rule arkind_arty_artrm.induct)
 apply(simp_all add: alpha_gen)
 done
 lemma perm_rsp[quot_respect]:
-"(op = ===> akind ===> akind) permute permute"
+"(op = ===> arkind ===> arkind) permute permute"
-"(op = ===> aty ===> aty) permute permute"
+"(op = ===> arty ===> arty) permute permute"
-"(op = ===> atrm ===> atrm) permute permute"
+"(op = ===> artrm ===> artrm) permute permute"
 by (simp_all add:alpha_eqvt)
 lemma tconst_rsp[quot_respect]:
-"(op = ===> aty) TConst TConst"
+"(op = ===> arty) 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"
+"(arty ===> artrm ===> arty) TApp TApp"
 apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) done
 lemma var_rsp[quot_respect]:
-"(op = ===> atrm) Var Var"
+"(op = ===> artrm) 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"
+"(artrm ===> artrm ===> artrm) App App"
 apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) done
 lemma const_rsp[quot_respect]:
-"(op = ===> atrm) Const Const"
+"(op = ===> artrm) 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"
+"(arty ===> op = ===> arkind ===> arkind) 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
 lemma tpi_rsp[quot_respect]:
-"(aty ===> op = ===> aty ===> aty) TPi TPi"
+"(arty ===> op = ===> arty ===> arty) 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"
+"(arty ===> op = ===> artrm ===> artrm) 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
-lemma rfv_ty_rsp[quot_respect]:
+lemma rfv_rty_rsp[quot_respect]:
-"(aty ===> op =) rfv_ty rfv_ty"
+"(arty ===> op =) rfv_rty rfv_rty"
 by (simp add: alpha_rfv)
-lemma rfv_kind_rsp[quot_respect]:
+lemma rfv_rkind_rsp[quot_respect]:
-"(akind ===> op =) rfv_kind rfv_kind"
+"(arkind ===> op =) rfv_rkind rfv_rkind"
 by (simp add: alpha_rfv)
-lemma rfv_trm_rsp[quot_respect]:
+lemma rfv_rtrm_rsp[quot_respect]:
-"(atrm ===> op =) rfv_trm rfv_trm"
+"(artrm ===> op =) rfv_rtrm rfv_rtrm"
 by (simp add: alpha_rfv)
-thm kind_ty_trm.induct
+thm rkind_rty_rtrm.induct
-lemmas KIND_TY_TRM_induct = kind_ty_trm.induct[quot_lifted]
+lemmas RKIND_RTY_RTRM_induct = rkind_rty_rtrm.induct[quot_lifted]
-thm kind_ty_trm.inducts
+thm rkind_rty_rtrm.inducts
-lemmas KIND_TY_TRM_inducts = kind_ty_trm.inducts[quot_lifted]
+lemmas RKIND_RTY_RTRM_inducts = rkind_rty_rtrm.inducts[quot_lifted]
-instantiation KIND and TY and TRM :: pt
+instantiation RKIND and RTY and RTRM :: pt
 begin
 quotient_definition
-"permute_KIND :: perm \<Rightarrow> KIND \<Rightarrow> KIND"
+"permute_RKIND :: perm \<Rightarrow> RKIND \<Rightarrow> RKIND"
 is
-"permute :: perm \<Rightarrow> kind \<Rightarrow> kind"
+"permute :: perm \<Rightarrow> rkind \<Rightarrow> rkind"
 quotient_definition
-"permute_TY :: perm \<Rightarrow> TY \<Rightarrow> TY"
+"permute_RTY :: perm \<Rightarrow> RTY \<Rightarrow> RTY"
 is
-"permute :: perm \<Rightarrow> ty \<Rightarrow> ty"
+"permute :: perm \<Rightarrow> rty \<Rightarrow> rty"
 quotient_definition
-"permute_TRM :: perm \<Rightarrow> TRM \<Rightarrow> TRM"
+"permute_RTRM :: perm \<Rightarrow> RTRM \<Rightarrow> RTRM"
 is
-"permute :: perm \<Rightarrow> trm \<Rightarrow> trm"
+"permute :: perm \<Rightarrow> rtrm \<Rightarrow> rtrm"
-lemmas permute_ktt[simp] = permute_kind_permute_ty_permute_trm.simps[quot_lifted]
+lemmas permute_ktt[simp] = permute_rkind_permute_rty_permute_rtrm.simps[quot_lifted]
-lemma perm_zero_ok: "0 \<bullet> (x :: KIND) = x \<and> 0 \<bullet> (y :: TY) = y \<and> 0 \<bullet> (z :: TRM) = z"
+lemma perm_zero_ok: "0 \<bullet> (x :: RKIND) = x \<and> 0 \<bullet> (y :: RTY) = y \<and> 0 \<bullet> (z :: RTRM) = z"
-apply (induct rule: KIND_TY_TRM_induct)
+apply (induct rule: RKIND_RTY_RTRM_induct)
 apply (simp_all)
 done
 lemma perm_add_ok:
-"((p + q) \<bullet> (x1 :: KIND) = (p \<bullet> q \<bullet> x1))"
+"((p + q) \<bullet> (x1 :: RKIND) = (p \<bullet> q \<bullet> x1))"
-"((p + q) \<bullet> (x2 :: TY) = p \<bullet> q \<bullet> x2)"
+"((p + q) \<bullet> (x2 :: RTY) = p \<bullet> q \<bullet> x2)"
-"((p + q) \<bullet> (x3 :: TRM) = p \<bullet> q \<bullet> x3)"
+"((p + q) \<bullet> (x3 :: RTRM) = p \<bullet> q \<bullet> x3)"
-apply (induct x1 and x2 and x3 rule: KIND_TY_TRM_inducts)
+apply (induct x1 and x2 and x3 rule: RKIND_RTY_RTRM_inducts)
 apply (simp_all)
 done
 instance
 apply default
 apply (simp_all add: perm_zero_ok perm_add_ok)
 done
 end
-lemmas AKIND_ATY_ATRM_inducts = akind_aty_atrm.inducts[unfolded alpha_gen, quot_lifted, folded alpha_gen]
+lemmas ARKIND_ARTY_ARTRM_inducts = arkind_arty_artrm.inducts[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 RKIND_RTY_RTRM_INJECT = arkind_arty_artrm_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]
-lemmas fv_kind_ty_trm = rfv_kind_rfv_ty_rfv_trm.simps[quot_lifted]
+lemmas fv_rkind_rty_rtrm = rfv_rkind_rfv_rty_rfv_rtrm.simps[quot_lifted]
 lemmas fv_eqvt = rfv_eqvt[quot_lifted]
-lemma supp_kind_ty_trm_easy:
+lemma supp_rkind_rty_rtrm_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 add: supp_def permute_ktt RKIND_RTY_RTRM_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)"
+"(supp (atom na, (ty, rtrm))) supports (LAM ty na rtrm)"
 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(clarify)
 apply(subst swap_at_base_simps(3))
 apply(simp_all add: fresh_atom)
 done
-lemma KIND_TY_TRM_fs:
+lemma RKIND_RTY_RTRM_fs:
-"finite (supp (x\<Colon>KIND))"
+"finite (supp (x\<Colon>RKIND))"
-"finite (supp (y\<Colon>TY))"
+"finite (supp (y\<Colon>RTY))"
-"finite (supp (z\<Colon>TRM))"
+"finite (supp (z\<Colon>RTRM))"
-apply(induct x and y and z rule: KIND_TY_TRM_inducts)
+apply(induct x and y and z rule: RKIND_RTY_RTRM_inducts)
-apply(simp_all add: supp_kind_ty_trm_easy)
+apply(simp_all add: supp_rkind_rty_rtrm_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(rule supports_finite)
 apply(rule supp_bind(3))
 apply(simp add: supp_Pair supp_atom)
 done
-instance KIND and TY and TRM :: fs
+instance RKIND and RTY and RTRM :: fs
 apply(default)
-apply(simp_all only: KIND_TY_TRM_fs)
+apply(simp_all only: RKIND_RTY_RTRM_fs)
 done
 lemma supp_fv:
-"supp t1 = fv_kind t1"
+"supp t1 = fv_rkind t1"
-"supp t2 = fv_ty t2"
+"supp t2 = fv_rty t2"
-"supp t3 = fv_trm t3"
+"supp t3 = fv_rtrm t3"
-apply(induct t1 and t2 and t3 rule: KIND_TY_TRM_inducts)
+apply(induct t1 and t2 and t3 rule: RKIND_RTY_RTRM_inducts)
-apply (simp_all add: supp_kind_ty_trm_easy)
+apply (simp_all add: supp_rkind_rty_rtrm_easy)
-apply (simp_all add: fv_kind_ty_trm)
+apply (simp_all add: fv_rkind_rty_rtrm)
-apply(subgoal_tac "supp (KPI ty name kind) = supp ty \<union> supp (Abs {atom name} kind)")
+apply(subgoal_tac "supp (KPI rty name rkind) = supp rty \<union> supp (Abs {atom name} rkind)")
 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: RKIND_RTY_RTRM_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(subgoal_tac "supp (TPI rty1 name rty2) = supp rty1 \<union> supp (Abs {atom name} rty2)")
 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: RKIND_RTY_RTRM_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(subgoal_tac "supp (LAM rty name rtrm) = supp rty \<union> supp (Abs {atom name} rtrm)")
 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: RKIND_RTY_RTRM_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
 (* 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: permute_set_eq supp_def Abs_eq_iff RKIND_RTY_RTRM_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:
+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})"
-apply (simp_all only: supp_kind_ty_trm_easy)
+apply (simp_all only: supp_rkind_rty_rtrm_easy)
-apply (simp_all only: supp_fv fv_kind_ty_trm)
+apply (simp_all only: supp_fv fv_rkind_rty_rtrm)
 done
 end

changeset 1234	1240d5eb275d
parent 1219	c74c8ba46db7
child 1236	ca3c69545a78