--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/activities/Lambda.thy	Wed Mar 30 17:27:34 2016 +0100
@@ -0,0 +1,87 @@
+theory Lambda 
+  imports "Nominal" 
+begin
+
+atom_decl name
+
+section {* Alpha-Equated Lambda-Terms *}
+
+nominal_datatype lam =
+  Var "name"
+| App "lam" "lam" 
+| Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._")
+
+section {* Capture-Avoiding Substitution *}
+
+consts subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam"  ("_[_::=_]")
+
+nominal_primrec
+  "(Var x)[y::=s] = (if x=y then s else (Var x))"
+  "(App t1 t2)[y::=s] = App (t1[y::=s]) (t2[y::=s])"
+  "x\<sharp>(y,s) \<Longrightarrow> (Lam [x].t)[y::=s] = Lam [x].(t[y::=s])"
+apply(finite_guess)+
+apply(rule TrueI)+
+apply(simp add: abs_fresh)+
+apply(fresh_guess)+
+done
+
+lemma  subst_eqvt[eqvt]:
+  fixes pi::"name prm"
+  shows "pi\<bullet>(t1[x::=t2]) = (pi\<bullet>t1)[(pi\<bullet>x)::=(pi\<bullet>t2)]"
+by (nominal_induct t1 avoiding: x t2 rule: lam.strong_induct)
+   (auto simp add: perm_bij fresh_atm fresh_bij)
+
+lemma forget: 
+  assumes a: "x\<sharp>L"
+  shows "L[x::=P] = L"
+  using a 
+by (nominal_induct L avoiding: x P rule: lam.strong_induct)
+   (auto simp add: abs_fresh fresh_atm)
+
+lemma fresh_fact:
+  fixes z::"name"
+  shows "\<lbrakk>z\<sharp>s; (z=y \<or> z\<sharp>t)\<rbrakk> \<Longrightarrow> z\<sharp>t[y::=s]"
+by (nominal_induct t avoiding: z y s rule: lam.strong_induct)
+   (auto simp add: abs_fresh fresh_prod fresh_atm)
+
+lemma subst_rename: 
+  assumes a: "y\<sharp>t"
+  shows "t[x::=s] = ([(y,x)]\<bullet>t)[y::=s]"
+using a 
+by (nominal_induct t avoiding: x y s rule: lam.strong_induct)
+     (auto simp add: calc_atm fresh_atm abs_fresh)
+
+text {* 
+  The purpose of the two lemmas below is to work
+  around some quirks in Isabelle's handling of 
+  meta_quantifiers and meta_implications. 
+  *} 
+
+lemma meta_impCE:
+  assumes major: "P ==> PROP Q"
+    and 1: "~ P ==> R"
+    and 2: "PROP Q ==> R"
+  shows R
+proof (cases P)
+  assume P
+  then have "PROP Q" by (rule major)
+  then show R by (rule 2)
+next
+  assume "~ P"
+  then show R by (rule 1)
+qed
+
+declare meta_allE [elim]
+    and meta_impCE [elim!]
+
+end    
+   
+
+
+
+  
+
+  
+  
+
+