theory Height
  imports "../Nominal2"
begin

text {*  
  A small problem suggested by D. Wang. It shows how
  the height of a lambda-terms behaves under substitution.
*}

atom_decl name

nominal_datatype lam = 
    Var "name"
  | App "lam" "lam"
  | Lam x::"name" l::"lam" bind x in l ("Lam [_]._" [100,100] 100)


text {* Definition of the height-function on lambda-terms. *} 

function
  height :: "lam \<Rightarrow> int"
where
  "height (Var x) = 1"
| "height (App t1 t2) = (max (height t1) (height t2)) + 1"
| "height (Lam [a].t) = (height t) + 1"
  apply(rule_tac y="x" in lam.exhaust)
  apply(simp_all)[3]
  apply(simp add: lam.eq_iff)
  apply(simp add: lam.distinct)
  apply(simp add: lam.distinct)
  apply(simp add: lam.eq_iff)
  apply(simp add: lam.distinct)
  apply(simp add: lam.eq_iff)
  sorry

termination
  apply(relation "measure size")
  apply(simp_all add: lam.size)
  done

text {* Definition of capture-avoiding substitution. *}

function
  subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam"  ("_[_::=_]" [100,100,100] 100)
where
  "(Var x)[y::=t'] = (if x=y then t' else (Var x))"
| "(App t1 t2)[y::=t'] = App (t1[y::=t']) (t2[y::=t'])"
| "\<lbrakk>atom x \<sharp> y; atom x \<sharp> t'\<rbrakk> \<Longrightarrow> (Lam [x].t)[y::=t'] = Lam [x].(t[y::=t'])"
apply(case_tac x)
apply(simp)
apply(rule_tac y="a" and c="(b, c)" in lam.strong_exhaust)
apply(simp_all)[3]
apply(blast)
apply(blast)
apply(simp add: fresh_star_def fresh_Pair)
apply(blast)
apply(simp add: lam.eq_iff)
apply(simp add: lam.distinct)
apply(simp add: lam.distinct)
apply(simp add: lam.eq_iff)
apply(simp add: lam.distinct)
apply(simp add: lam.eq_iff)
sorry

termination
  apply(relation "measure (size o fst)")
  apply(simp_all add: lam.size)
  done

text{* The next lemma is needed in the Var-case of the theorem below. *}

lemma height_ge_one: 
  shows "1 \<le> (height e)"
apply(nominal_induct e rule: lam.strong_induct) 
apply(simp_all)
done

text {* 
  Unlike the proplem suggested by Wang, however, the 
  theorem is here formulated entirely by using functions. 
*}

theorem height_subst:
  shows "height (e[x::=e']) \<le> ((height e) - 1) + (height e')"
proof (nominal_induct e avoiding: x e' rule: lam.strong_induct)
  case (Var y)
  have "1 \<le> height e'" by (rule height_ge_one)
  then show "height (Var y[x::=e']) \<le> height (Var y) - 1 + height e'" by simp
next
  case (Lam y e1)
  have ih: "height (e1[x::=e']) \<le> ((height e1) - 1) + (height e')" by fact
  moreover
  have vc: "atom y \<sharp> x" "atom y \<sharp> e'" by fact+ (* usual variable convention *)
  ultimately show "height ((Lam [y].e1)[x::=e']) \<le> height (Lam [y].e1) - 1 + height e'" by simp
next    
  case (App e1 e2)
  have ih1: "height (e1[x::=e']) \<le> ((height e1) - 1) + (height e')" 
  and  ih2: "height (e2[x::=e']) \<le> ((height e2) - 1) + (height e')" by fact+
  then show "height ((App e1 e2)[x::=e']) \<le> height (App e1 e2) - 1 + height e'"  by simp 
qed

end