theory Tutorial5
imports Tutorial4
begin

section {* Type-Preservation and Progress Lemma*}

text {*
  The point of this tutorial is to prove the
  type-preservation and progress lemma. Since
  we now know that \<Down>, \<longrightarrow>cbv* and the machine
  correspond to each other, we only need to
  prove this property for one of them. We chose
  \<longrightarrow>cbv.

  First we need to establish two elimination
  properties and two auxiliary lemmas about contexts.
*}


lemma valid_elim:
  assumes a: "valid ((x, T) # \<Gamma>)"
  shows "atom x \<sharp> \<Gamma> \<and> valid \<Gamma>"
using a by (cases) (auto)

lemma valid_insert:
  assumes a: "valid (\<Delta> @ [(x, T)] @ \<Gamma>)"
  shows "valid (\<Delta> @ \<Gamma>)" 
using a
by (induct \<Delta>)
   (auto simp add: fresh_append fresh_Cons dest!: valid_elim)

lemma fresh_list: 
  shows "atom y \<sharp> xs = (\<forall>x \<in> set xs. atom y \<sharp> x)"
by (induct xs) (simp_all add: fresh_Nil fresh_Cons)

lemma context_unique:
  assumes a1: "valid \<Gamma>"
  and     a2: "(x, T) \<in> set \<Gamma>"
  and     a3: "(x, U) \<in> set \<Gamma>"
  shows "T = U" 
using a1 a2 a3
by (induct) (auto simp add: fresh_list fresh_Pair fresh_at_base)


section {* EXERCISE 16 *}

text {*
  Next we want to show the type substitution lemma. Unfortunately,
  we have to prove a slightly more general version of it, where
  the variable being substituted occurs somewhere inside the 
  context.
*}

lemma type_substitution_aux:
  assumes a: "\<Delta> @ [(x, T')] @ \<Gamma> \<turnstile> e : T"
  and     b: "\<Gamma> \<turnstile> e' : T'"
  shows "\<Delta> @ \<Gamma> \<turnstile> e[x ::= e'] : T" 
using a b 
proof (nominal_induct \<Gamma>'\<equiv>"\<Delta> @ [(x, T')] @ \<Gamma>" e T avoiding: x e' \<Delta> rule: typing.strong_induct)
  case (t_Var y T x e' \<Delta>)
  have a1: "valid (\<Delta> @ [(x, T')] @ \<Gamma>)" by fact
  have a2: "(y,T) \<in> set (\<Delta> @ [(x, T')] @ \<Gamma>)" by fact 
  have a3: "\<Gamma> \<turnstile> e' : T'" by fact
  
  from a1 have a4: "valid (\<Delta> @ \<Gamma>)" by (rule valid_insert)
  { assume eq: "x = y"
    
    have "\<Delta> @ \<Gamma> \<turnstile> Var y[x ::= e'] : T" sorry
  }
  moreover
  { assume ineq: "x \<noteq> y"
    from a2 have "(y, T) \<in> set (\<Delta> @ \<Gamma>)" using ineq by simp
    then have "\<Delta> @ \<Gamma> \<turnstile> Var y[x ::= e'] : T" using ineq a4 by auto
  }
  ultimately show "\<Delta> @ \<Gamma> \<turnstile> Var y[x::=e'] : T" by blast
next
  case (t_Lam y T1 t T2 x e' \<Delta>)
  have a1: "atom y \<sharp> e'" by fact
  have a2: "atom y \<sharp> \<Delta> @ [(x, T')] @ \<Gamma>" by fact
  have a3: "\<Gamma> \<turnstile> e' : T'" by fact 
  have ih: "\<Gamma> \<turnstile> e' : T' \<Longrightarrow> ((y, T1) # \<Delta>) @ \<Gamma> \<turnstile> t [x ::= e'] : T2" 
    using t_Lam(6)[of "(y, T1) # \<Delta>"] by auto 
  

  show "\<Delta> @ \<Gamma> \<turnstile> (Lam [y]. t)[x ::= e'] : T1 \<rightarrow> T2" sorry
next
  case (t_App t1 T1 T2 t2 x e' \<Delta>)
  have ih1: "\<Gamma> \<turnstile> e' : T' \<Longrightarrow> \<Delta> @ \<Gamma> \<turnstile> t1 [x ::= e'] : T1 \<rightarrow> T2" using t_App(2) by auto 
  have ih2: "\<Gamma> \<turnstile> e' : T' \<Longrightarrow> \<Delta> @ \<Gamma> \<turnstile> t2 [x ::= e'] : T1" using t_App(4) by auto 
  have a: "\<Gamma> \<turnstile> e' : T'" by fact

  show "\<Delta> @ \<Gamma> \<turnstile> App t1 t2 [x ::= e'] : T2" sorry
qed 

text {*
  From this we can derive the usual version of the substitution
  lemma.
*}

corollary type_substitution:
  assumes a: "(x, T') # \<Gamma> \<turnstile> e : T"
  and     b: "\<Gamma> \<turnstile> e' : T'"
  shows "\<Gamma> \<turnstile> e[x ::= e'] : T"
using a b type_substitution_aux[of "[]"]
by auto


section {* Type Preservation *}

text {*
  Finally we are in a position to establish the type preservation
  property. We just need the following two inversion rules for
  particualr typing instances.
*}

lemma t_App_elim:
  assumes a: "\<Gamma> \<turnstile> App t1 t2 : T"
  obtains T' where "\<Gamma> \<turnstile> t1 : T' \<rightarrow> T" "\<Gamma> \<turnstile> t2 : T'"
using a
by (cases) (auto simp add: lam.eq_iff lam.distinct)

text {* we have not yet generated strong elimination rules *}
lemma t_Lam_elim:
  assumes ty: "\<Gamma> \<turnstile> Lam [x].t : T" 
  and     fc: "atom x \<sharp> \<Gamma>" 
  obtains T1 T2 where "T = T1 \<rightarrow> T2" "(x, T1) # \<Gamma> \<turnstile> t : T2"
using ty fc
apply(cases)
apply(auto simp add: lam.eq_iff lam.distinct ty.eq_iff)
apply(auto simp add: Abs1_eq_iff)
apply(rotate_tac 3)
apply(drule_tac p="(x \<leftrightarrow> xa)" in permute_boolI)
apply(perm_simp)
apply(auto simp add: flip_def swap_fresh_fresh ty_fresh)
done


section {* EXERCISE 17 *}

text {*
  Fill in the gaps in the t_Lam case. You will need
  the type substitution lemma proved above. 
*}

theorem cbv_type_preservation:
  assumes a: "t \<longrightarrow>cbv t'"
  and     b: "\<Gamma> \<turnstile> t : T" 
  shows "\<Gamma> \<turnstile> t' : T"
using a b
proof (nominal_induct avoiding: \<Gamma> T rule: cbv.strong_induct)
  case (cbv1 v x t \<Gamma> T) 
  have fc: "atom x \<sharp> \<Gamma>" by fact
  have "\<Gamma> \<turnstile> App (Lam [x]. t) v : T" by fact
  then obtain T' where 
      *: "\<Gamma> \<turnstile> Lam [x]. t : T' \<rightarrow> T" and 
     **: "\<Gamma> \<turnstile> v : T'" by (rule t_App_elim)
  have "(x, T') # \<Gamma> \<turnstile> t : T" using * fc by (rule t_Lam_elim) (simp add: ty.eq_iff)

  show "\<Gamma> \<turnstile> t [x ::= v] : T " sorry
qed (auto elim!: t_App_elim)

text {*
  We can easily extend this to sequences of cbv* reductions.
*}

corollary cbvs_type_preservation:
  assumes a: "t \<longrightarrow>cbv* t'"
  and     b: "\<Gamma> \<turnstile> t : T" 
  shows "\<Gamma> \<turnstile> t' : T"
using a b
by (induct) (auto intro: cbv_type_preservation)

text {* 
  The type-preservation property for the machine and 
  evaluation relation. 
*}

theorem machine_type_preservation:
  assumes a: "<t, []> \<mapsto>* <t', []>"
  and     b: "\<Gamma> \<turnstile> t : T" 
  shows "\<Gamma> \<turnstile> t' : T"
proof -
  have "t \<longrightarrow>cbv* t'" using a machines_implies_cbvs by simp
  then show "\<Gamma> \<turnstile> t' : T" using b cbvs_type_preservation by simp
qed

theorem eval_type_preservation:
  assumes a: "t \<Down> t'"
  and     b: "\<Gamma> \<turnstile> t : T" 
  shows "\<Gamma> \<turnstile> t' : T"
proof -
  have "<t, []> \<mapsto>* <t', []>" using a eval_implies_machines by simp
  then show "\<Gamma> \<turnstile> t' : T" using b machine_type_preservation by simp
qed

text {* The Progress Property *}

lemma canonical_tArr:
  assumes a: "[] \<turnstile> t : T1 \<rightarrow> T2"
  and     b: "val t"
  obtains x t' where "t = Lam [x].t'"
using b a by (induct) (auto) 

theorem progress:
  assumes a: "[] \<turnstile> t : T"
  shows "(\<exists>t'. t \<longrightarrow>cbv t') \<or> (val t)"
using a
by (induct \<Gamma>\<equiv>"[]::ty_ctx" t T)
   (auto elim: canonical_tArr simp add: val.simps)

text {*
  Done! Congratulations!
*}

end