
diff -r a9f3600c9ae6 -r 589b1a0c75e6 Nominal/Ex/SFT/Theorem.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/SFT/Theorem.thy	Fri Jun 24 10:30:06 2011 +0900
@@ -0,0 +1,70 @@
+header {* The main lemma about Num and the Second Fixed Point Theorem *}
+theory Theorem imports Consts begin
+
+(*<*)
+lemmas [simp] = b3[OF bI] b1 b4 b5 supp_Num[unfolded Num_def supp_ltgt] Num_def lam.fresh[unfolded fresh_def] fresh_def b6
+lemmas app = Ltgt1_app
+(*>*)
+
+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>".
+next
+  case (Ap 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 .
+  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> \<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)
+  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
+  also have "... \<approx> Abs \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using app .
+  also have "... \<approx> \<guillemotleft>[A1,A2,A3]\<guillemotright> \<cdot> Umn 2 0 \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using Abs_app .
+  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 "... = \<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
+
+lemmas [simp] = Ap Num
+lemmas [simp del] = fresh_def Num_def
+
+theorem SFP:
+  fixes F :: lam
+  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 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 "... \<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
+qed
+(*>*)




