diff -r 3750c08f627e -r c95afd0dc594 Nominal/Ex/SFT/Theorem.thy
--- a/Nominal/Ex/SFT/Theorem.thy	Wed Dec 21 14:25:05 2011 +0900
+++ b/Nominal/Ex/SFT/Theorem.thy	Wed Dec 21 15:43:58 2011 +0900
@@ -8,31 +8,31 @@
 lemma Num:
   shows "Num \<cdot> \<lbrace>M\<rbrace> \<approx> \<lbrace>\<lbrace>M\<rbrace>\<rbrace>"
 proof (induct M rule: lam.induct)
-  case (V n)
-  have "Num \<cdot> \<lbrace>V n\<rbrace> = Num \<cdot> (Var \<cdot> V n)" by simp
-  also have "... = \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright> \<cdot> (Var \<cdot> V n)" by simp
-  also have "... \<approx> Var \<cdot> V n \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using app .
-  also have "... \<approx> \<guillemotleft>[A1,A2,A3]\<guillemotright> \<cdot> Umn 2 2 \<cdot> V n \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using Var_app .
-  also have "... \<approx> A1 \<cdot> V n \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using U_app by simp
-  also have "... \<approx> F1 \<cdot> V n" using A_app(1) .
-  also have "... \<approx> App \<cdot> \<lbrace>Var\<rbrace> \<cdot> (Var \<cdot> V n)" using F_app(1) .
-  also have "... = \<lbrace>\<lbrace>V n\<rbrace>\<rbrace>" by simp
-  finally show "Num \<cdot> \<lbrace>V n\<rbrace> \<approx> \<lbrace>\<lbrace>V n\<rbrace>\<rbrace>".
+  case (Var n)
+  have "Num \<cdot> \<lbrace>Var n\<rbrace> = Num \<cdot> (VAR \<cdot> Var n)" by simp
+  also have "... = \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright> \<cdot> (VAR \<cdot> Var n)" by simp
+  also have "... \<approx> VAR \<cdot> Var n \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using app .
+  also have "... \<approx> \<guillemotleft>[A1,A2,A3]\<guillemotright> \<cdot> Umn 2 2 \<cdot> Var n \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using VAR_app .
+  also have "... \<approx> A1 \<cdot> Var n \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using U_app by simp
+  also have "... \<approx> F1 \<cdot> Var n" using A_app(1) .
+  also have "... \<approx> APP \<cdot> \<lbrace>VAR\<rbrace> \<cdot> (VAR \<cdot> Var n)" using F_app(1) .
+  also have "... = \<lbrace>\<lbrace>Var n\<rbrace>\<rbrace>" by simp
+  finally show "Num \<cdot> \<lbrace>Var n\<rbrace> \<approx> \<lbrace>\<lbrace>Var n\<rbrace>\<rbrace>".
 next
-  case (Ap M N)
+  case (App M N)
   assume IH: "Num \<cdot> \<lbrace>M\<rbrace> \<approx> \<lbrace>\<lbrace>M\<rbrace>\<rbrace>" "Num \<cdot> \<lbrace>N\<rbrace> \<approx> \<lbrace>\<lbrace>N\<rbrace>\<rbrace>"
-  have "Num \<cdot> \<lbrace>M \<cdot> N\<rbrace> = Num \<cdot> (App \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace>)" by simp
-  also have "... = \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright> \<cdot> (App \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace>)" by simp
-  also have "... \<approx> App \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using app .
-  also have "... \<approx> \<guillemotleft>[A1,A2,A3]\<guillemotright> \<cdot> Umn 2 1 \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using App_app .
+  have "Num \<cdot> \<lbrace>M \<cdot> N\<rbrace> = Num \<cdot> (APP \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace>)" by simp
+  also have "... = \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright> \<cdot> (APP \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace>)" by simp
+  also have "... \<approx> APP \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using app .
+  also have "... \<approx> \<guillemotleft>[A1,A2,A3]\<guillemotright> \<cdot> Umn 2 1 \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using APP_app .
   also have "... \<approx> A2 \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using U_app by simp
   also have "... \<approx> F2 \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> Num" using A_app(2) by simp
-  also have "... \<approx> App \<cdot> (App \<cdot> \<lbrace>App\<rbrace> \<cdot> (Num \<cdot> \<lbrace>M\<rbrace>)) \<cdot> (Num \<cdot> \<lbrace>N\<rbrace>)" using F_app(2) .
-  also have "... \<approx> App \<cdot> (App \<cdot> \<lbrace>App\<rbrace> \<cdot> (\<lbrace>\<lbrace>M\<rbrace>\<rbrace>)) \<cdot> (Num \<cdot> \<lbrace>N\<rbrace>)" using IH by simp
+  also have "... \<approx> APP \<cdot> (APP \<cdot> \<lbrace>APP\<rbrace> \<cdot> (Num \<cdot> \<lbrace>M\<rbrace>)) \<cdot> (Num \<cdot> \<lbrace>N\<rbrace>)" using F_app(2) .
+  also have "... \<approx> APP \<cdot> (APP \<cdot> \<lbrace>APP\<rbrace> \<cdot> (\<lbrace>\<lbrace>M\<rbrace>\<rbrace>)) \<cdot> (Num \<cdot> \<lbrace>N\<rbrace>)" using IH by simp
   also have "... \<approx> \<lbrace>\<lbrace>M \<cdot> N\<rbrace>\<rbrace>" using IH by simp
   finally show "Num \<cdot> \<lbrace>M \<cdot> N\<rbrace> \<approx> \<lbrace>\<lbrace>M \<cdot> N\<rbrace>\<rbrace>".
 next
-  case (Lm x P)
+  case (Lam x P)
   assume IH: "Num \<cdot> \<lbrace>P\<rbrace> \<approx> \<lbrace>\<lbrace>P\<rbrace>\<rbrace>"
   have "Num \<cdot> \<lbrace>\<integral> x. P\<rbrace> = Num \<cdot> (Abs \<cdot> \<integral> x. \<lbrace>P\<rbrace>)" by simp
   also have "... = \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright> \<cdot> (Abs \<cdot> \<integral> x. \<lbrace>P\<rbrace>)" by simp
@@ -41,9 +41,9 @@
   also have "... \<approx> A3 \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using U_app by simp
   also have "... \<approx> F3 \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright>" using A_app(3) .
   also have "... = F3 \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> Num" by simp
-  also have "... \<approx> App \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> \<integral> x. (Num \<cdot> ((\<integral> x. \<lbrace>P\<rbrace>) \<cdot> V x)))" by (rule F3_app) simp_all
-  also have "... \<approx> App \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> \<integral> x. (Num \<cdot> \<lbrace>P\<rbrace>))" using beta_app by simp
-  also have "... \<approx> App \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> \<integral> x. \<lbrace>\<lbrace>P\<rbrace>\<rbrace>)" using IH by simp
+  also have "... \<approx> APP \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> \<integral> x. (Num \<cdot> ((\<integral> x. \<lbrace>P\<rbrace>) \<cdot> Var x)))" by (rule F3_app) simp_all
+  also have "... \<approx> APP \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> \<integral> x. (Num \<cdot> \<lbrace>P\<rbrace>))" using beta_app by simp
+  also have "... \<approx> APP \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> \<integral> x. \<lbrace>\<lbrace>P\<rbrace>\<rbrace>)" using IH by simp
   also have "... = \<lbrace>\<lbrace>\<integral> x. P\<rbrace>\<rbrace>" by simp
   finally show "Num \<cdot> \<lbrace>\<integral> x. P\<rbrace> \<approx> \<lbrace>\<lbrace>\<integral> x. P\<rbrace>\<rbrace>" .
 qed
@@ -56,12 +56,12 @@
   shows "\<exists>X. X \<approx> F \<cdot> \<lbrace>X\<rbrace>"
 proof -
   obtain x :: var where [simp]:"atom x \<sharp> F" using obtain_fresh by blast
-  def W \<equiv> "\<integral>x. (F \<cdot> (App \<cdot> V x \<cdot> (Num \<cdot> V x)))"
+  def W \<equiv> "\<integral>x. (F \<cdot> (APP \<cdot> Var x \<cdot> (Num \<cdot> Var x)))"
   def X \<equiv> "W \<cdot> \<lbrace>W\<rbrace>"
   have a: "X = W \<cdot> \<lbrace>W\<rbrace>" unfolding X_def ..
-  also have "... = (\<integral>x. (F \<cdot> (App \<cdot> V x \<cdot> (Num \<cdot> V x)))) \<cdot> \<lbrace>W\<rbrace>" unfolding W_def ..
-  also have "... \<approx> F \<cdot> (App \<cdot> \<lbrace>W\<rbrace> \<cdot> (Num \<cdot> \<lbrace>W\<rbrace>))" by simp
-  also have "... \<approx> F \<cdot> (App \<cdot> \<lbrace>W\<rbrace> \<cdot> \<lbrace>\<lbrace>W\<rbrace>\<rbrace>)" by simp
+  also have "... = (\<integral>x. (F \<cdot> (APP \<cdot> Var x \<cdot> (Num \<cdot> Var x)))) \<cdot> \<lbrace>W\<rbrace>" unfolding W_def ..
+  also have "... \<approx> F \<cdot> (APP \<cdot> \<lbrace>W\<rbrace> \<cdot> (Num \<cdot> \<lbrace>W\<rbrace>))" by simp
+  also have "... \<approx> F \<cdot> (APP \<cdot> \<lbrace>W\<rbrace> \<cdot> \<lbrace>\<lbrace>W\<rbrace>\<rbrace>)" by simp
   also have "... \<approx> F \<cdot> \<lbrace>W \<cdot> \<lbrace>W\<rbrace>\<rbrace>" by simp
   also have "... = F \<cdot> \<lbrace>X\<rbrace>" unfolding X_def ..
   finally show ?thesis by blast