--- a/Quot/Nominal/LFex.thy Wed Feb 24 17:32:43 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,236 +0,0 @@
-theory LFex
-imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv" "Rsp"
-begin
-
-atom_decl name
-atom_decl ident
-
-datatype rkind =
- Type
- | KPi "rty" "name" "rkind"
-and rty =
- TConst "ident"
- | TApp "rty" "rtrm"
- | TPi "rty" "name" "rty"
-and rtrm =
- Const "ident"
- | Var "name"
- | App "rtrm" "rtrm"
- | Lam "rty" "name" "rtrm"
-
-
-setup {* snd o define_raw_perms ["rkind", "rty", "rtrm"] ["LFex.rkind", "LFex.rty", "LFex.rtrm"] *}
-
-local_setup {*
- snd o define_fv_alpha "LFex.rkind"
- [[ [], [[], [(NONE, 1)], [(NONE, 1)]] ],
- [ [[]], [[], []], [[], [(NONE, 1)], [(NONE, 1)]] ],
- [ [[]], [[]], [[], []], [[], [(NONE, 1)], [(NONE, 1)]]]] *}
-notation
- 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>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
-
-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)"
- "t3 \<approx>tr s3 \<Longrightarrow> (pi \<bullet> t3) \<approx>tr (pi \<bullet> s3)"
-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
-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 assumption
-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
-
-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}))
-*}
-
-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
-
-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 *}
-
-lemmas kind_ty_trm_induct = rkind_rty_rtrm.induct[quot_lifted]
-
-thm rkind_rty_rtrm.inducts
-lemmas kind_ty_trm_inducts = rkind_rty_rtrm.inducts[quot_lifted]
-
-instantiation kind and ty and trm :: pt
-begin
-
-quotient_definition
- "permute_kind :: perm \<Rightarrow> kind \<Rightarrow> kind"
-is
- "permute :: perm \<Rightarrow> rkind \<Rightarrow> rkind"
-
-quotient_definition
- "permute_ty :: perm \<Rightarrow> ty \<Rightarrow> ty"
-is
- "permute :: perm \<Rightarrow> rty \<Rightarrow> rty"
-
-quotient_definition
- "permute_trm :: perm \<Rightarrow> trm \<Rightarrow> trm"
-is
- "permute :: perm \<Rightarrow> rtrm \<Rightarrow> rtrm"
-
-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
-
-(*
-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 permute_ktt[simp] = permute_rkind_permute_rty_permute_rtrm.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 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(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
-
-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 (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
-
-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
-
-
-
-