# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1295645024 -3600
# Node ID abb6c3ac2df2080527e94b636a9491da6f922ebb
# Parent  f325eefe803e2b2e72892f59b04bc2be7be557a9
separated type preservation and progress into a separate file

diff -r f325eefe803e -r abb6c3ac2df2 Tutorial/Tutorial3.thy
--- a/Tutorial/Tutorial3.thy	Fri Jan 21 22:02:34 2011 +0100
+++ b/Tutorial/Tutorial3.thy	Fri Jan 21 22:23:44 2011 +0100
@@ -6,12 +6,12 @@
 
 text {*
   Barendregt's proof needs in the variable case a case distinction.
-  One way to do this in Isar is to use blocks. A block is some sequent
-  or reasoning steps enclosed in curly braces
+  One way to do this in Isar is to use blocks. A block consist of some
+  assumptions and reasoning steps enclosed in curly braces, like
 
   { \<dots>
-
     have "statement"
+    have "last_statement_in_the_block"
   }
 
   Such a block can contain local assumptions like
@@ -25,7 +25,7 @@
   Where "C" is the last have-statement in this block. The behaviour 
   of such a block to the 'outside' is the implication
 
-   \<lbrakk>A; B\<rbrakk> \<Longrightarrow> "C" 
+   A \<Longrightarrow> B \<Longrightarrow> C 
 
   Now if we want to prove a property "smth" using the case-distinctions
   P1, P2 and P3 then we can use the following reasoning:
@@ -52,17 +52,12 @@
     P2 \<Longrightarrow> smth
     P3 \<Longrightarrow> smth
 
-  If we know that P1, P2 and P3 cover all the cases, that is P1 \<or> P2 \<or> P3 is
-  true, then we have 'ultimately' established the property "smth" 
+  If we know that P1, P2 and P3 cover all the cases, that is P1 \<or> P2 \<or> P3 
+  holds, then we have 'ultimately' established the property "smth" 
   
 *}
 
-section {* EXERCISE 7 *}
-
-text {*
-  Fill in the cases 1.2 and 1.3 and the equational reasoning 
-  in the lambda-case.
-*}
+subsection {* Two preliminary facts *}
 
 lemma forget:
   shows "atom x \<sharp> t \<Longrightarrow> t[x ::= s] = t"
@@ -78,6 +73,14 @@
    (auto simp add: lam.fresh fresh_at_base)
 
 
+
+section {* EXERCISE 7 *}
+
+text {*
+  Fill in the cases 1.2 and 1.3 and the equational reasoning 
+  in the lambda-case.
+*}
+
 lemma 
   assumes a: "x \<noteq> y"
   and     b: "atom x \<sharp> L"
@@ -111,12 +114,12 @@
   qed
 next
   case (Lam z M1) -- {* case 2: lambdas *}
-  have ih: "\<lbrakk>x \<noteq> y; atom x \<sharp> L\<rbrakk> \<Longrightarrow> M1[x::=N][y::=L] = M1[y::=L][x::=N[y::=L]]" by fact
+  have ih: "\<lbrakk>x \<noteq> y; atom x \<sharp> L\<rbrakk> \<Longrightarrow> M1[x ::= N][y ::= L] = M1[y ::= L][x ::= N[y ::= L]]" by fact
   have a1: "x \<noteq> y" by fact
   have a2: "atom x \<sharp> L" by fact
-  have fs: "atom z \<sharp> x" "atom z \<sharp> y" "atom z \<sharp> N" "atom z \<sharp> L" by fact+
+  have fs: "atom z \<sharp> x" "atom z \<sharp> y" "atom z \<sharp> N" "atom z \<sharp> L" by fact+   -- {* !! *}
   then have b: "atom z \<sharp> N[y::=L]" by (simp add: fresh_fact)
-  show "(Lam [z].M1)[x::=N][y::=L] = (Lam [z].M1)[y::=L][x::=N[y::=L]]" (is "?LHS=?RHS") 
+  show "(Lam [z].M1)[x ::= N][y ::= L] = (Lam [z].M1)[y ::= L][x ::= N[y ::= L]]" (is "?LHS=?RHS") 
   proof - 
     have "?LHS = \<dots>" sorry
 
@@ -130,7 +133,7 @@
 
 text {* 
   Again the strong induction principle enables Isabelle to find
-  the proof of the substitution lemma automatically. 
+  the proof of the substitution lemma completely automatically. 
 *}
 
 lemma substitution_lemma_version:  
diff -r f325eefe803e -r abb6c3ac2df2 Tutorial/Tutorial4.thy
--- a/Tutorial/Tutorial4.thy	Fri Jan 21 22:02:34 2011 +0100
+++ b/Tutorial/Tutorial4.thy	Fri Jan 21 22:23:44 2011 +0100
@@ -1,5 +1,6 @@
+
 theory Tutorial4
-imports Tutorial1 Tutorial2
+imports Tutorial1 Tutorial2 Tutorial3
 begin
 
 section {* The CBV Reduction Relation (Small-Step Semantics) *}
@@ -163,6 +164,7 @@
   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"
@@ -234,137 +236,8 @@
   then show "t1 \<Down> t2" using b cbvs_implies_eval by simp
 qed
 
-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)
-
-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"
-    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)
-  }
-  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
-qed (force simp add: fresh_append fresh_Cons)+
-
-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>="[]"]
-by auto
-
-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
 
-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)
-
-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)
-
-text {*
-  Done!
-*}
 
 end
 
diff -r f325eefe803e -r abb6c3ac2df2 Tutorial/Tutorial5.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Tutorial/Tutorial5.thy	Fri Jan 21 22:23:44 2011 +0100
@@ -0,0 +1,142 @@
+theory Tutorial5
+imports Tutorial4
+begin
+
+
+section {* Type Preservation (fixme separate file) *}
+
+
+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)
+
+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"
+    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)
+  }
+  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
+qed (force simp add: fresh_append fresh_Cons)+
+
+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>="[]"]
+by auto
+
+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
+
+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)
+
+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)
+
+text {*
+  Done! Congratulations!
+*}
+
+end
+