--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/SFT/Utils.thy	Fri Jun 24 10:30:06 2011 +0900
@@ -0,0 +1,101 @@
+header {* Utilities for defining constants and functions *}
+theory Utils imports Lambda begin
+
+lemma beta_app:
+  "(\<integral> x. M) \<cdot> V x \<approx> M"
+  by (rule b3, rule bI)
+     (simp add: b1)
+
+lemma lam1_fast_app:
+  assumes leq: "\<And>a. (L = \<integral> a. (F (V a)))"
+      and su: "\<And>x. atom x \<sharp> A \<Longrightarrow> F (V x) [x ::= A] = F A"
+  shows "L \<cdot> A \<approx> F A"
+proof -
+  obtain x :: var where a: "atom x \<sharp> A" using obtain_fresh by blast
+  show ?thesis
+    by (simp add: leq[of x], rule b3, rule bI, simp add: su b1 a)
+qed
+
+lemma lam2_fast_app:
+  assumes leq: "\<And>a b. a \<noteq> b \<Longrightarrow> L = \<integral> a. \<integral> b. (F (V a) (V b))"
+     and su: "\<And>x y. atom x \<sharp> A \<Longrightarrow> atom y \<sharp> A \<Longrightarrow> atom x \<sharp> B \<Longrightarrow> atom y \<sharp> B \<Longrightarrow>
+       x \<noteq> y \<Longrightarrow> F (V x) (V y) [x ::= A] [y ::= B] = F A B"
+  shows "L \<cdot> A \<cdot> B \<approx> F A B"
+proof -
+  obtain x :: var where a: "atom x \<sharp> (A, B)" using obtain_fresh by blast
+  obtain y :: var where b: "atom y \<sharp> (x, A, B)" using obtain_fresh by blast
+  obtain z :: var where c: "atom z \<sharp> (x, y, A, B)" using obtain_fresh by blast
+  have *: "x \<noteq> y" "x \<noteq> z" "y \<noteq> z"
+    using a b c by (simp_all add: fresh_Pair fresh_at_base) blast+
+  have ** : "atom y \<sharp> z" "atom x \<sharp> z" "atom y \<sharp> x"
+            "atom z \<sharp> y" "atom z \<sharp> x" "atom x \<sharp> y"
+            "atom x \<sharp> A" "atom y \<sharp> A" "atom z \<sharp> A"
+            "atom x \<sharp> B" "atom y \<sharp> B" "atom z \<sharp> B"
+    using a b c by (simp_all add: fresh_Pair fresh_at_base) blast+
+  show ?thesis
+    apply (simp add: leq[OF *(1)])
+    apply (rule b3) apply (rule b5) apply (rule bI)
+    apply (simp add: ** fresh_Pair)
+    apply (rule b3) apply (rule bI) apply (simp add: su b1 ** *)
+    done
+  qed
+
+lemma lam3_fast_app:
+  assumes leq: "\<And>a b c. a \<noteq> b \<Longrightarrow> b \<noteq> c \<Longrightarrow> c \<noteq> a \<Longrightarrow>
+       L = \<integral> a. \<integral> b. \<integral> c. (F (V a) (V b) (V c))"
+     and su: "\<And>x y z. atom x \<sharp> A \<Longrightarrow> atom y \<sharp> A \<Longrightarrow> atom z \<sharp> A \<Longrightarrow>
+                     atom x \<sharp> B \<Longrightarrow> atom y \<sharp> B \<Longrightarrow> atom z \<sharp> B \<Longrightarrow>
+                     atom y \<sharp> B \<Longrightarrow> atom y \<sharp> B \<Longrightarrow> atom z \<sharp> B \<Longrightarrow>
+                     x \<noteq> y \<Longrightarrow> y \<noteq> z \<Longrightarrow> z \<noteq> x \<Longrightarrow>
+      ((F (V x) (V y) (V z))[x ::= A] [y ::= B] [z ::= C] = F A B C)"
+  shows "L \<cdot> A \<cdot> B \<cdot> C \<approx> F A B C"
+proof -
+  obtain x :: var where a: "atom x \<sharp> (A, B, C)" using obtain_fresh by blast
+  obtain y :: var where b: "atom y \<sharp> (x, A, B, C)" using obtain_fresh by blast
+  obtain z :: var where c: "atom z \<sharp> (x, y, A, B, C)" using obtain_fresh by blast
+  have *: "x \<noteq> y" "y \<noteq> z" "z \<noteq> x"
+    using a b c by (simp_all add: fresh_Pair fresh_at_base) blast+
+  have ** : "atom y \<sharp> z" "atom x \<sharp> z" "atom y \<sharp> x"
+            "atom z \<sharp> y" "atom z \<sharp> x" "atom x \<sharp> y"
+            "atom x \<sharp> A" "atom y \<sharp> A" "atom z \<sharp> A"
+            "atom x \<sharp> B" "atom y \<sharp> B" "atom z \<sharp> B"
+            "atom x \<sharp> C" "atom y \<sharp> C" "atom z \<sharp> C"
+    using a b c by (simp_all add: fresh_Pair fresh_at_base) blast+
+  show ?thesis
+    apply (simp add: leq[OF *(1) *(2) *(3)])
+    apply (rule b3) apply (rule b5) apply (rule b5) apply (rule bI)
+    apply (simp add: ** fresh_Pair)
+    apply (rule b3) apply (rule b5) apply (rule bI)
+    apply (simp add: ** fresh_Pair)
+    apply (rule b3) apply (rule bI) apply (simp add: su b1 ** *)
+    done
+  qed
+
+definition cn :: "nat \<Rightarrow> var" where "cn n \<equiv> Abs_var (Atom (Sort ''Lambda.var'' []) n)"
+
+lemma cnd[simp]: "m \<noteq> n \<Longrightarrow> cn m \<noteq> cn n"
+  unfolding cn_def using Abs_var_inject by simp
+
+definition cx :: var where "cx \<equiv> cn 0"
+definition cy :: var where "cy \<equiv> cn 1"
+definition cz :: var where "cz \<equiv> cn 2"
+
+lemma cx_cy_cz[simp]:
+  "cx \<noteq> cy" "cx \<noteq> cz" "cz \<noteq> cy"
+  unfolding cx_def cy_def cz_def
+  by simp_all
+
+lemma noteq_fresh: "atom x \<sharp> y = (x \<noteq> y)"
+  by (simp add: fresh_at_base(2))
+
+lemma fresh_fun_eqvt_app2:
+  assumes a: "eqvt f"
+  and b: "a \<sharp> x" "a \<sharp> y"
+  shows "a \<sharp> f x y"
+  using fresh_fun_eqvt_app[OF a b(1)] a b
+  by (metis fresh_fun_app)
+
+end
+
+
+