--- a/Nominal/Ex/SFT/Utils.thy	Sat May 12 21:39:09 2012 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,102 +0,0 @@
-header {* Utilities for defining constants and functions *}
-
-theory Utils imports LambdaTerms begin
-
-lemma beta_app:
-  "(\<integral> x. M) \<cdot> Var 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 :: name 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 :: name where a: "atom x \<sharp> (A, B)" using obtain_fresh by blast
-  obtain y :: name where b: "atom y \<sharp> (x, A, B)" using obtain_fresh by blast
-  obtain z :: name 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 :: name where a: "atom x \<sharp> (A, B, C)" using obtain_fresh by blast
-  obtain y :: name where b: "atom y \<sharp> (x, A, B, C)" using obtain_fresh by blast
-  obtain z :: name 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> name" where "cn n \<equiv> Abs_name (Atom (Sort ''LambdaTerms.name'' []) n)"
-
-lemma cnd[simp]: "m \<noteq> n \<Longrightarrow> cn m \<noteq> cn n"
-  unfolding cn_def using Abs_name_inject by simp
-
-definition cx :: name where "cx \<equiv> cn 0"
-definition cy :: name where "cy \<equiv> cn 1"
-definition cz :: name 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
-
-
-