theory Tutorial4s
imports Tutorial1s
begin

section {* The CBV Reduction Relation (Small-Step Semantics) *}

text {*
  In order to help establishing the property that the CK Machine
  calculates a nomrmalform that corresponds to the evaluation 
  relation, we introduce the call-by-value small-step semantics.
*}

inductive
  cbv :: "lam \<Rightarrow> lam \<Rightarrow> bool" ("_ \<longrightarrow>cbv _" [60, 60] 60) 
where
  cbv1: "\<lbrakk>val v; atom x \<sharp> v\<rbrakk> \<Longrightarrow> App (Lam [x].t) v \<longrightarrow>cbv t[x ::= v]"
| cbv2[intro]: "t \<longrightarrow>cbv t' \<Longrightarrow> App t t2 \<longrightarrow>cbv App t' t2"
| cbv3[intro]: "t \<longrightarrow>cbv t' \<Longrightarrow> App t2 t \<longrightarrow>cbv App t2 t'"

equivariance val
equivariance cbv
nominal_inductive cbv
  avoids cbv1: "x"
  unfolding fresh_star_def
  by (simp_all add: lam.fresh Abs_fresh_iff fresh_Pair fresh_fact)

text {*
  In order to satisfy the vc-condition we have to formulate
  this relation with the additional freshness constraint
  atom x \<sharp> v. Although this makes the definition vc-ompatible, it
  makes the definition less useful. We can with a little bit of 
  pain show that the more restricted rule is equivalent to the
  usual rule. 
*}

lemma subst_rename: 
  assumes a: "atom y \<sharp> t"
  shows "t[x ::= s] = ((y \<leftrightarrow> x) \<bullet> t)[y ::= s]"
using a 
by (nominal_induct t avoiding: x y s rule: lam.strong_induct)
   (auto simp add: lam.fresh fresh_at_base)


lemma better_cbv1 [intro]: 
  assumes a: "val v" 
  shows "App (Lam [x].t) v \<longrightarrow>cbv t[x::=v]"
proof -
  obtain y::"name" where fs: "atom y \<sharp> (x, t, v)" by (rule obtain_fresh)
  have "App (Lam [x].t) v = App (Lam [y].((y \<leftrightarrow> x) \<bullet> t)) v" using fs
    by (auto simp add: lam.eq_iff Abs1_eq_iff' flip_def fresh_Pair fresh_at_base)
  also have "\<dots> \<longrightarrow>cbv ((y \<leftrightarrow> x) \<bullet> t)[y ::= v]" using fs a cbv1 by auto
  also have "\<dots> = t[x ::= v]" using fs subst_rename[symmetric] by simp
  finally show "App (Lam [x].t) v \<longrightarrow>cbv t[x ::= v]" by simp
qed

text {*
  The transitive closure of the cbv-reduction relation: 
*}

inductive 
  "cbvs" :: "lam \<Rightarrow> lam \<Rightarrow> bool" (" _ \<longrightarrow>cbv* _" [60, 60] 60)
where
  cbvs1[intro]: "e \<longrightarrow>cbv* e"
| cbvs2[intro]: "\<lbrakk>e1\<longrightarrow>cbv e2; e2 \<longrightarrow>cbv* e3\<rbrakk> \<Longrightarrow> e1 \<longrightarrow>cbv* e3"

lemma cbvs3 [intro]:
  assumes a: "e1 \<longrightarrow>cbv* e2" "e2 \<longrightarrow>cbv* e3"
  shows "e1 \<longrightarrow>cbv* e3"
using a by (induct) (auto) 


subsection {* EXERCISE 8 *}

text {*  
  If more simple exercises are needed, then complete the following proof. 
*}

lemma cbv_in_ctx:
  assumes a: "t \<longrightarrow>cbv t'"
  shows "E\<lbrakk>t\<rbrakk> \<longrightarrow>cbv E\<lbrakk>t'\<rbrakk>"
using a
proof (induct E)
  case Hole
  have "t \<longrightarrow>cbv t'" by fact
  then show "\<box>\<lbrakk>t\<rbrakk> \<longrightarrow>cbv \<box>\<lbrakk>t'\<rbrakk>" by simp
next
  case (CAppL E s)
  have ih: "t \<longrightarrow>cbv t' \<Longrightarrow> E\<lbrakk>t\<rbrakk> \<longrightarrow>cbv E\<lbrakk>t'\<rbrakk>" by fact
  moreover
  have "t \<longrightarrow>cbv t'" by fact
  ultimately 
  have "E\<lbrakk>t\<rbrakk> \<longrightarrow>cbv E\<lbrakk>t'\<rbrakk>" by simp
  then show "(CAppL E s)\<lbrakk>t\<rbrakk> \<longrightarrow>cbv (CAppL E s)\<lbrakk>t'\<rbrakk>" by auto
next
  case (CAppR s E)
  have ih: "t \<longrightarrow>cbv t' \<Longrightarrow> E\<lbrakk>t\<rbrakk> \<longrightarrow>cbv E\<lbrakk>t'\<rbrakk>" by fact
  moreover
  have a: "t \<longrightarrow>cbv t'" by fact
  ultimately 
  have "E\<lbrakk>t\<rbrakk> \<longrightarrow>cbv E\<lbrakk>t'\<rbrakk>" by simp
  then show "(CAppR s E)\<lbrakk>t\<rbrakk> \<longrightarrow>cbv (CAppR s E)\<lbrakk>t'\<rbrakk>" by auto
qed

section {* EXERCISE 9 *} 
 
text {*
  The point of the cbv-reduction was that we can easily relatively 
  establish the follwoing property:
*}

lemma machine_implies_cbvs_ctx:
  assumes a: "<e, Es> \<mapsto> <e', Es'>"
  shows "(Es\<down>)\<lbrakk>e\<rbrakk> \<longrightarrow>cbv* (Es'\<down>)\<lbrakk>e'\<rbrakk>"
using a 
proof (induct)
  case (m1 t1 t2 Es)
thm machine.intros thm cbv2
  have "Es\<down>\<lbrakk>App t1 t2\<rbrakk> = (Es\<down> \<odot> CAppL \<box> t2)\<lbrakk>t1\<rbrakk>" 
    using ctx_compose ctx_composes.simps filling.simps by simp
  then show "Es\<down>\<lbrakk>App t1 t2\<rbrakk> \<longrightarrow>cbv* ((CAppL \<box> t2) # Es)\<down>\<lbrakk>t1\<rbrakk>" using cbvs.intros by simp
next
  case (m2 v t2 Es)
  have "val v" by fact
  have "((CAppL \<box> t2) # Es)\<down>\<lbrakk>v\<rbrakk> = (CAppR v \<box> # Es)\<down>\<lbrakk>t2\<rbrakk>"  
    using ctx_compose ctx_composes.simps filling.simps by simp
  then show "((CAppL \<box> t2) # Es)\<down>\<lbrakk>v\<rbrakk> \<longrightarrow>cbv* (CAppR v \<box> # Es)\<down>\<lbrakk>t2\<rbrakk>" using cbvs.intros by simp
next
  case (m3 v x t Es)
  have aa: "val v" by fact
  have "(((CAppR (Lam [x].t) \<box>) # Es)\<down>)\<lbrakk>v\<rbrakk> = Es\<down>\<lbrakk>App (Lam [x]. t) v\<rbrakk>" 
    using ctx_compose ctx_composes.simps filling.simps by simp
  then have "(((CAppR (Lam [x].t) \<box>) # Es)\<down>)\<lbrakk>v\<rbrakk> \<longrightarrow>cbv (Es\<down>)\<lbrakk>(t[x ::= v])\<rbrakk>" 
    using better_cbv1[OF aa] cbv_in_ctx by simp
  then show "(((CAppR (Lam [x].t) \<box>) # Es)\<down>)\<lbrakk>v\<rbrakk> \<longrightarrow>cbv* (Es\<down>)\<lbrakk>(t[x ::= v])\<rbrakk>" using cbvs.intros by blast
qed

text {* 
  It is not difficult to extend the lemma above to
  arbitrary reductions sequences of the CK machine. *}

lemma machines_implies_cbvs_ctx:
  assumes a: "<e, Es> \<mapsto>* <e', Es'>"
  shows "(Es\<down>)\<lbrakk>e\<rbrakk> \<longrightarrow>cbv* (Es'\<down>)\<lbrakk>e'\<rbrakk>"
using a machine_implies_cbvs_ctx 
by (induct) (blast)+

text {* 
  So whenever we let the CL machine start in an initial
  state and it arrives at a final state, then there exists
  a corresponding cbv-reduction sequence. 
*}

corollary machines_implies_cbvs:
  assumes a: "<e, []> \<mapsto>* <e', []>"
  shows "e \<longrightarrow>cbv* e'"
proof - 
  have "[]\<down>\<lbrakk>e\<rbrakk> \<longrightarrow>cbv* []\<down>\<lbrakk>e'\<rbrakk>" 
     using a machines_implies_cbvs_ctx by blast
  then show "e \<longrightarrow>cbv* e'" by simp  
qed

text {*
  We now want to relate the cbv-reduction to the evaluation
  relation. For this we need two auxiliary lemmas. 
*}

lemma eval_val:
  assumes a: "val t"
  shows "t \<Down> t"
using a by (induct) (auto)


lemma e_App_elim:
  assumes a: "App t1 t2 \<Down> v"
  obtains x t v' where "t1 \<Down> Lam [x].t" "t2 \<Down> v'" "t[x::=v'] \<Down> v"
using a by (cases) (auto simp add: lam.eq_iff lam.distinct) 


subsection {* EXERCISE *}

text {*
  Complete the first and second case in the 
  proof below. 
*}

lemma cbv_eval:
  assumes a: "t1 \<longrightarrow>cbv t2" "t2 \<Down> t3"
  shows "t1 \<Down> t3"
using a
proof(induct arbitrary: t3)
  case (cbv1 v x t t3)
  have a1: "val v" by fact
  have a2: "t[x ::= v] \<Down> t3" by fact
  have a3: "Lam [x].t \<Down> Lam [x].t" by auto
  have a4: "v \<Down> v" using a1 eval_val by auto
  show "App (Lam [x].t) v \<Down> t3" using a3 a4 a2 by auto 
next
  case (cbv2 t t' t2 t3)
  have ih: "\<And>t3. t' \<Down> t3 \<Longrightarrow> t \<Down> t3" by fact
  have "App t' t2 \<Down> t3" by fact
  then obtain x t'' v' 
    where a1: "t' \<Down> Lam [x].t''" 
      and a2: "t2 \<Down> v'" 
      and a3: "t''[x ::= v'] \<Down> t3" by (rule e_App_elim) 
  have "t \<Down>  Lam [x].t''" using ih a1 by auto 
  then show "App t t2 \<Down> t3" using a2 a3 by auto
qed (auto elim!: e_App_elim)


text {* 
  Next we extend the lemma above to arbitray initial
  sequences of cbv-reductions. *}

lemma cbvs_eval:
  assumes a: "t1 \<longrightarrow>cbv* t2" "t2 \<Down> t3"
  shows "t1 \<Down> t3"
using a by (induct) (auto intro: cbv_eval)

text {* 
  Finally, we can show that if from a term t we reach a value 
  by a cbv-reduction sequence, then t evaluates to this value. 
*}

lemma cbvs_implies_eval:
  assumes a: "t \<longrightarrow>cbv* v" "val v"
  shows "t \<Down> v"
using a
by (induct) (auto intro: eval_val cbvs_eval)

text {* 
  All facts tied together give us the desired property about
  machines. 
*}

theorem machines_implies_eval:
  assumes a: "<t1, []> \<mapsto>* <t2, []>" 
  and     b: "val t2" 
  shows "t1 \<Down> t2"
proof -
  have "t1 \<longrightarrow>cbv* t2" using a machines_implies_cbvs by simp
  then show "t1 \<Down> t2" using b cbvs_implies_eval by simp
qed




end