(* $Id$ *)
theory Lam_Funs
imports "../Nominal"
begin
atom_decl name
nominal_datatype lam = Var "name"
| App "lam" "lam"
| Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
text {* depth of a lambda-term *}
consts
depth :: "lam \<Rightarrow> nat"
nominal_primrec
"depth (Var x) = (1::nat)"
"depth (App t1 t2) = (max (depth t1) (depth t2)) + 1"
"depth (Lam [a].t) = (depth t) + (1::nat)"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: fresh_nat)
apply(fresh_guess)+
done
text {* free variables of a lambda-term *}
consts
frees :: "lam \<Rightarrow> name set"
nominal_primrec (invariant: "\<lambda>s::name set. finite s")
"frees (Var a) = {a}"
"frees (App t1 t2) = (frees t1) \<union> (frees t2)"
"frees (Lam [a].t) = (frees t) - {a}"
apply(finite_guess)+
apply(simp)+
apply(simp add: fresh_def)
apply(simp add: supp_of_fin_sets[OF pt_name_inst, OF at_name_inst, OF fs_at_inst[OF at_name_inst]])
apply(simp add: supp_atm)
apply(force)
apply(fresh_guess)+
done
lemma frees_equals_support:
shows "frees t = supp t"
by (nominal_induct t rule: lam.induct)
(simp_all add: lam.supp supp_atm abs_supp)
text {* capture-avoiding substitution *}
consts
subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_[_::=_]" [100,100,100] 100)
nominal_primrec
"(Var x)[y::=t'] = (if x=y then t' else (Var x))"
"(App t1 t2)[y::=t'] = App (t1[y::=t']) (t2[y::=t'])"
"x\<sharp>(y,t') \<Longrightarrow> (Lam [x].t)[y::=t'] = Lam [x].(t[y::=t'])"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: abs_fresh)
apply(fresh_guess)+
done
lemma subst_eqvt[eqvt]:
fixes pi:: "name prm"
and t1:: "lam"
and t2:: "lam"
and a :: "name"
shows "pi\<bullet>(t1[b::=t2]) = (pi\<bullet>t1)[(pi\<bullet>b)::=(pi\<bullet>t2)]"
apply(nominal_induct t1 avoiding: b t2 rule: lam.induct)
apply(auto simp add: perm_bij fresh_prod fresh_atm fresh_bij)
done
lemma subst_supp:
shows "supp(t1[a::=t2]) \<subseteq> (((supp(t1)-{a})\<union>supp(t2))::name set)"
apply(nominal_induct t1 avoiding: a t2 rule: lam.induct)
apply(auto simp add: lam.supp supp_atm fresh_prod abs_supp)
apply(blast)+
done
text{* parallel substitution *}
consts
psubst :: "(name\<times>lam) list \<Rightarrow> lam \<Rightarrow> lam" ("_<_>" [100,100] 900)
fun
lookup :: "(name\<times>lam) list \<Rightarrow> name \<Rightarrow> lam"
where
"lookup [] x = Var x"
| "lookup ((y,T)#\<theta>) x = (if x=y then T else lookup \<theta> x)"
lemma lookup_eqvt[eqvt]:
fixes pi::"name prm"
and \<theta>::"(name\<times>lam) list"
and x::"name"
shows "pi\<bullet>(lookup \<theta> x) = lookup (pi\<bullet>\<theta>) (pi\<bullet>x)"
by (induct \<theta>) (auto simp add: perm_bij)
lemma lookup_fresh:
fixes z::"name"
assumes "z\<sharp>\<theta>" "z\<sharp>x"
shows "z\<sharp> lookup \<theta> x"
using assms
by (induct rule: lookup.induct) (auto simp add: fresh_list_cons)
lemma lookup_fresh':
assumes "z\<sharp>\<theta>"
shows "lookup \<theta> z = Var z"
using assms
by (induct rule: lookup.induct)
(auto simp add: fresh_list_cons fresh_prod fresh_atm)
nominal_primrec
"\<theta><(Var x)> = (lookup \<theta> x)"
"\<theta><(App t1 t2)> = App (\<theta><t1>) (\<theta><t2>)"
"x\<sharp>\<theta>\<Longrightarrow>\<theta><(Lam [x].t)> = Lam [x].(\<theta><t>)"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: abs_fresh)
apply(fresh_guess)+
done
end