theory Lam_Funs
imports "../Nominal"
begin
text {*
Provides useful definitions for reasoning
with lambda-terms.
*}
atom_decl name
nominal_datatype lam =
Var "name"
| App "lam" "lam"
| Lam "«name»lam" ("Lam [_]._" [100,100] 100)
text {* The depth of a lambda-term. *}
nominal_primrec
depth :: "lam => 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 {*
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.
*}
nominal_primrec (invariant: "λs::name set. finite s")
frees :: "lam => name set"
where
"frees (Var a) = {a}"
| "frees (App t1 t2) = (frees t1) ∪ (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 {*
We can avoid the definition of frees by
using the build in notion of support.
*}
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 {* Parallel and single capture-avoiding substitution. *}
fun
lookup :: "(name×lam) list => name => lam"
where
"lookup [] x = Var x"
| "lookup ((y,e)#ϑ) x = (if x=y then e else lookup ϑ x)"
lemma lookup_eqvt[eqvt]:
fixes pi::"name prm"
and ϑ::"(name×lam) list"
and X::"name"
shows "pi•(lookup ϑ X) = lookup (pi•ϑ) (pi•X)"
by (induct ϑ) (auto simp add: eqvts)
nominal_primrec
psubst :: "(name×lam) list => lam => lam" ("_<_>" [95,95] 105)
where
"ϑ<(Var x)> = (lookup ϑ x)"
| "ϑ<(App e\<^isub>1 e\<^isub>2)> = App (ϑ<e\<^isub>1>) (ϑ<e\<^isub>2>)"
| "x\<sharp>ϑ ==> ϑ<(Lam [x].e)> = Lam [x].(ϑ<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•(ϑ<t>) = (pi•ϑ)<(pi•t)>"
by (nominal_induct t avoiding: ϑ rule: lam.strong_induct)
(simp_all add: eqvts fresh_bij)
abbreviation
subst :: "lam => name => lam => lam" ("_[_::=_]" [100,100,100] 100)
where
"t[x::=t'] ≡ ([(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') ==> (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]) ⊆ (((supp(t1)-{a})∪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 {*
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.
*}
nominal_datatype clam =
Hole ("\<box>" 1000)
| CAppL "clam" "lam"
| CAppR "lam" "clam"
| CLam "name" "clam" ("CLam [_]._" [100,100] 100)
text {* Filling a lambda-term into a context. *}
nominal_primrec
filling :: "clam => lam => lam" ("_[|_|]" [100,100] 100)
where
"\<box>[|t|] = t"
| "(CAppL E t')[|t|] = App (E[|t|]) t'"
| "(CAppR t' E)[|t|] = App t' (E[|t|])"
| "(CLam [x].E)[|t|] = Lam [x].(E[|t|])"
by (rule TrueI)+
text {* Composition od two contexts *}
nominal_primrec
clam_compose :: "clam => clam => clam" ("_ o _" [100,100] 100)
where
"\<box> o E' = E'"
| "(CAppL E t') o E' = CAppL (E o E') t'"
| "(CAppR t' E) o E' = CAppR t' (E o E')"
| "(CLam [x].E) o E' = CLam [x].(E o E')"
by (rule TrueI)+
lemma clam_compose:
shows "(E1 o E2)[|t|] = E1[|E2[|t|]|]"
by (induct E1 rule: clam.induct) (auto)
end