theory Tutorial4imports Tutorial1beginsection {* 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 valequivariance cbvnominal_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 simpqedtext {* 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 aproof (induct E) case Hole have "t \<longrightarrow>cbv t'" by fact then show "\<box>\<lbrakk>t\<rbrakk> \<longrightarrow>cbv \<box>\<lbrakk>t'\<rbrakk>" by simpnext 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 autonext 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 autoqedsection {* 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 simpnext 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 simpnext 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 blastqedtext {* 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 qedtext {* 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 aproof(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 autoqed (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 aby (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 simpqedend