theory Lam_Funs
imports "HOL-Nominal.Nominal"
begin
text \<open>
Provides useful definitions for reasoning
with lambda-terms.
\<close>
atom_decl name
nominal_datatype lam =
Var "name"
| App "lam" "lam"
| Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
text \<open>The depth of a lambda-term.\<close>
nominal_primrec
depth :: "lam \<Rightarrow> nat"
where
"depth (Var x) = 1"
| "depth (App t1 t2) = (max (depth t1) (depth t2)) + 1"
| "depth (Lam [a].t) = (depth t) + 1"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: fresh_nat)
apply(fresh_guess)+
done
text \<open>
The free variables of a lambda-term. A complication in this
function arises from the fact that it returns a name set, which
is not a finitely supported type. Therefore we have to prove
the invariant that frees always returns a finite set of names.
\<close>
nominal_primrec (invariant: "\<lambda>s::name set. finite s")
frees :: "lam \<Rightarrow> name set"
where
"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(blast)
apply(fresh_guess)+
done
text \<open>
We can avoid the definition of frees by
using the build in notion of support.
\<close>
lemma frees_equals_support:
shows "frees t = supp t"
by (nominal_induct t rule: lam.strong_induct)
(simp_all add: lam.supp supp_atm abs_supp)
text \<open>Parallel and single capture-avoiding substitution.\<close>
fun
lookup :: "(name\<times>lam) list \<Rightarrow> name \<Rightarrow> lam"
where
"lookup [] x = Var x"
| "lookup ((y,e)#\<theta>) x = (if x=y then e 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: eqvts)
nominal_primrec
psubst :: "(name\<times>lam) list \<Rightarrow> lam \<Rightarrow> lam" ("_<_>" [95,95] 105)
where
"\<theta><(Var x)> = (lookup \<theta> x)"
| "\<theta><(App e\<^sub>1 e\<^sub>2)> = App (\<theta><e\<^sub>1>) (\<theta><e\<^sub>2>)"
| "x\<sharp>\<theta> \<Longrightarrow> \<theta><(Lam [x].e)> = Lam [x].(\<theta><e>)"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: abs_fresh)+
apply(fresh_guess)+
done
lemma psubst_eqvt[eqvt]:
fixes pi::"name prm"
and t::"lam"
shows "pi\<bullet>(\<theta><t>) = (pi\<bullet>\<theta>)<(pi\<bullet>t)>"
by (nominal_induct t avoiding: \<theta> rule: lam.strong_induct)
(simp_all add: eqvts fresh_bij)
abbreviation
subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_[_::=_]" [100,100,100] 100)
where
"t[x::=t'] \<equiv> ([(x,t')])<t>"
lemma subst[simp]:
shows "(Var x)[y::=t'] = (if x=y then t' else (Var x))"
and "(App t1 t2)[y::=t'] = App (t1[y::=t']) (t2[y::=t'])"
and "x\<sharp>(y,t') \<Longrightarrow> (Lam [x].t)[y::=t'] = Lam [x].(t[y::=t'])"
by (simp_all add: fresh_list_cons fresh_list_nil)
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.strong_induct)
apply(auto simp add: lam.supp supp_atm fresh_prod abs_supp)
apply(blast)+
done
text \<open>
Contexts - lambda-terms with a single hole.
Note that the lambda case in contexts does not bind a
name, even if we introduce the notation [_]._ for CLam.
\<close>
nominal_datatype clam =
Hole ("\<box>" 1000)
| CAppL "clam" "lam"
| CAppR "lam" "clam"
| CLam "name" "clam" ("CLam [_]._" [100,100] 100)
text \<open>Filling a lambda-term into a context.\<close>
nominal_primrec
filling :: "clam \<Rightarrow> lam \<Rightarrow> lam" ("_\<lbrakk>_\<rbrakk>" [100,100] 100)
where
"\<box>\<lbrakk>t\<rbrakk> = t"
| "(CAppL E t')\<lbrakk>t\<rbrakk> = App (E\<lbrakk>t\<rbrakk>) t'"
| "(CAppR t' E)\<lbrakk>t\<rbrakk> = App t' (E\<lbrakk>t\<rbrakk>)"
| "(CLam [x].E)\<lbrakk>t\<rbrakk> = Lam [x].(E\<lbrakk>t\<rbrakk>)"
by (rule TrueI)+
text \<open>Composition od two contexts\<close>
nominal_primrec
clam_compose :: "clam \<Rightarrow> clam \<Rightarrow> clam" ("_ \<circ> _" [100,100] 100)
where
"\<box> \<circ> E' = E'"
| "(CAppL E t') \<circ> E' = CAppL (E \<circ> E') t'"
| "(CAppR t' E) \<circ> E' = CAppR t' (E \<circ> E')"
| "(CLam [x].E) \<circ> E' = CLam [x].(E \<circ> E')"
by (rule TrueI)+
lemma clam_compose:
shows "(E1 \<circ> E2)\<lbrakk>t\<rbrakk> = E1\<lbrakk>E2\<lbrakk>t\<rbrakk>\<rbrakk>"
by (induct E1 rule: clam.induct) (auto)
end