# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1295744388 21600
# Node ID 7b2691911fbcd4afeabf853ae285986fe6576ea3
# Parent  e0391947b7daf2916ddac26ec6764a4db05b2b8e
cleaning up

diff -r e0391947b7da -r 7b2691911fbc Tutorial/Lambda.thy
--- a/Tutorial/Lambda.thy	Sat Jan 22 16:37:00 2011 -0600
+++ b/Tutorial/Lambda.thy	Sat Jan 22 18:59:48 2011 -0600
@@ -105,18 +105,13 @@
   shows "(p \<bullet> t[x ::= s]) = (p \<bullet> t)[(p \<bullet> x) ::= (p \<bullet> s)]"
 by (induct t x s rule: subst.induct) (simp_all)
 
-
-subsection {* Single-Step Beta-Reduction *}
-
-inductive 
-  beta :: "lam \<Rightarrow> lam \<Rightarrow> bool" (" _ \<longrightarrow>b _" [80,80] 80)
-where
-  b1[intro]: "t1 \<longrightarrow>b t2 \<Longrightarrow> App t1 s \<longrightarrow>b App t2 s"
-| b2[intro]: "s1 \<longrightarrow>b s2 \<Longrightarrow> App t s1 \<longrightarrow>b App t s2"
-| b3[intro]: "t1 \<longrightarrow>b t2 \<Longrightarrow> Lam [x]. t1 \<longrightarrow>b Lam [x]. t2"
-| b4[intro]: "App (Lam [x]. t) s \<longrightarrow>b t[x ::= s]"
-
-
+lemma fresh_fact:
+  assumes a: "atom z \<sharp> s"
+  and b: "z = y \<or> atom z \<sharp> t"
+  shows "atom z \<sharp> t[y ::= s]"
+using a b
+by (nominal_induct t avoiding: z y s rule: lam.strong_induct)
+   (auto simp add: lam.fresh fresh_at_base)
 
 
 end
diff -r e0391947b7da -r 7b2691911fbc Tutorial/Tutorial4.thy
--- a/Tutorial/Tutorial4.thy	Sat Jan 22 16:37:00 2011 -0600
+++ b/Tutorial/Tutorial4.thy	Sat Jan 22 18:59:48 2011 -0600
@@ -1,6 +1,6 @@
 
 theory Tutorial4
-imports Tutorial1 Tutorial2 Tutorial3
+imports Tutorial1 Tutorial2
 begin
 
 section {* The CBV Reduction Relation (Small-Step Semantics) *}
@@ -21,14 +21,14 @@
 
 declare cbv.intros[intro] cbv_star.intros[intro]
 
-subsection {* EXERCISE 3 *}
+subsection {* EXERCISE 11 *}
 
 text {*
   Show that cbv* is transitive, by filling the gaps in the 
   proof below.
 *}
 
-lemma 
+lemma cbvs3 [intro]:
   assumes a: "e1 \<longrightarrow>cbv* e2" "e2 \<longrightarrow>cbv* e3"
   shows "e1 \<longrightarrow>cbv* e3"
 using a 
@@ -46,17 +46,9 @@
   show "e1 \<longrightarrow>cbv* e3" sorry
 qed 
 
-lemma cbvs3 [intro]:
-  assumes a: "e1 \<longrightarrow>cbv* e2" "e2 \<longrightarrow>cbv* e3"
-  shows "e1 \<longrightarrow>cbv* e3"
-using a by (induct) (auto) 
-
-
-
-
 
 text {*
-  In order to help establishing the property that the CK Machine
+  In order to help establishing the property that the machine
   calculates a nomrmalform that corresponds to the evaluation 
   relation, we introduce the call-by-value small-step semantics.
 *}
@@ -98,15 +90,9 @@
   finally show "App (Lam [x].t) v \<longrightarrow>cbv t[x ::= v]" by simp
 qed
 
-text {*
-  The transitive closure of the cbv-reduction relation: 
-*}
 
 
-
-
-
-subsection {* EXERCISE 8 *}
+subsection {* EXERCISE 12 *}
 
 text {*  
   If more simple exercises are needed, then complete the following proof. 
@@ -119,26 +105,22 @@
 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
+  show "\<box>\<lbrakk>t\<rbrakk> \<longrightarrow>cbv \<box>\<lbrakk>t'\<rbrakk>" sorry
 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
+
+  show "(CAppL E s)\<lbrakk>t\<rbrakk> \<longrightarrow>cbv (CAppL E s)\<lbrakk>t'\<rbrakk>" sorry
 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
+  
+  show "(CAppR s E)\<lbrakk>t\<rbrakk> \<longrightarrow>cbv (CAppR s E)\<lbrakk>t'\<rbrakk>" sorry
 qed
 
-section {* EXERCISE 9 *} 
+section {* EXERCISE 13 *} 
  
 text {*
   The point of the cbv-reduction was that we can easily relatively 
@@ -167,7 +149,8 @@
 
 text {* 
   It is not difficult to extend the lemma above to
-  arbitrary reductions sequences of the CK machine. *}
+  arbitrary reductions sequences of the machine. 
+*}
 
 lemma machines_implies_cbvs_ctx:
   assumes a: "<e, Es> \<mapsto>* <e', Es'>"
@@ -176,7 +159,7 @@
 by (induct) (blast)+
 
 text {* 
-  So whenever we let the CL machine start in an initial
+  So whenever we let the machine start in an initial
   state and it arrives at a final state, then there exists
   a corresponding cbv-reduction sequence. 
 *}
@@ -192,14 +175,10 @@
 
 text {*
   We now want to relate the cbv-reduction to the evaluation
-  relation. For this we need two auxiliary lemmas. 
+  relation. For this we need one auxiliary lemma about
+  inverting the e_App rule. 
 *}
 
-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"
@@ -207,7 +186,7 @@
 using a by (cases) (auto simp add: lam.eq_iff lam.distinct) 
 
 
-subsection {* EXERCISE *}
+subsection {* EXERCISE 13 *}
 
 text {*
   Complete the first and second case in the 
@@ -222,9 +201,8 @@
   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 
+
+  show "App (Lam [x].t) v \<Down> t3" sorry
 next
   case (cbv2 t t' t2 t3)
   have ih: "\<And>t3. t' \<Down> t3 \<Longrightarrow> t \<Down> t3" by fact
@@ -233,14 +211,15 @@
     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
+  
+  show "App t t2 \<Down> t3" sorry
 qed (auto elim!: e_App_elim)
 
 
 text {* 
   Next we extend the lemma above to arbitray initial
-  sequences of cbv-reductions. *}
+  sequences of cbv-reductions. 
+*}
 
 lemma cbvs_eval:
   assumes a: "t1 \<longrightarrow>cbv* t2" "t2 \<Down> t3"
@@ -250,30 +229,42 @@
 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. 
+
+  This proof is not by induction. So we have to set up the proof
+  with
+
+    proof -
+    
+  in order to prevent Isabelle from applying a default introduction   
+  rule.
 *}
 
 lemma cbvs_implies_eval:
-  assumes a: "t \<longrightarrow>cbv* v" "val v"
+  assumes a: "t \<longrightarrow>cbv* v" 
+  and     b: "val v"
   shows "t \<Down> v"
-using a
-by (induct) (auto intro: eval_val cbvs_eval)
+proof - 
+  have "v \<Down> v" using b eval_val by simp
+  then show "t \<Down> v" using a cbvs_eval by auto
+qed
+
+section {* EXERCISE 15 *}
 
 text {* 
-  All facts tied together give us the desired property about
-  machines. 
+  All facts tied together give us the desired property 
+  about machines: we know that a machines transitions
+  correspond to cbvs transitions, and with the lemma
+  above they correspond to an eval judgement.
 *}
 
 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
+proof - 
+  
+  show "t1 \<Down> t2" sorry
 qed
 
-
-
-
 end
 
diff -r e0391947b7da -r 7b2691911fbc Tutorial/Tutorial5.thy
--- a/Tutorial/Tutorial5.thy	Sat Jan 22 16:37:00 2011 -0600
+++ b/Tutorial/Tutorial5.thy	Sat Jan 22 18:59:48 2011 -0600
@@ -1,9 +1,22 @@
+
+
 theory Tutorial5
 imports Tutorial4
 begin
 
+section {* Type-Preservation and Progress Lemma*}
 
-section {* Type Preservation (fixme separate file) *}
+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:
@@ -30,6 +43,16 @@
 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'"
@@ -40,10 +63,11 @@
   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"
-    from a1 a2 have "T = T'" using eq by (auto intro: context_unique)
-    with a3 have "\<Delta> @ \<Gamma> \<turnstile> Var y[x ::= e'] : T" using eq a4 by (auto intro: weakening)
+    
+    have "\<Delta> @ \<Gamma> \<turnstile> Var y[x ::= e'] : T" sorry
   }
   moreover
   { assume ineq: "x \<noteq> y"
@@ -51,15 +75,46 @@
     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
-qed (force simp add: fresh_append fresh_Cons)+
+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[where \<Delta>="[]"]
+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'"
@@ -81,13 +136,34 @@
 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
-by (nominal_induct avoiding: \<Gamma> T rule: cbv.strong_induct)
-   (auto elim!: t_Lam_elim t_App_elim simp add: type_substitution ty.eq_iff)
+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'"