--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Unification/Fresh.thy	Sun Apr 29 11:29:56 2012 +0100
@@ -0,0 +1,90 @@
+
+
+theory Fresh = Main + Swap + Terms + Disagreement:
+
+types fresh_envs = "(string \<times> string)set"
+
+consts 
+  fresh :: "(fresh_envs \<times> string \<times> trm) set"
+syntax 
+  "_fresh_judge" :: "fresh_envs \<Rightarrow> string \<Rightarrow> trm \<Rightarrow> bool" (" _ \<turnstile> _ \<sharp> _" [80,80,80] 80)
+translations 
+  "nabla \<turnstile> a \<sharp> t"  \<rightleftharpoons> "(nabla,a,t) \<in> fresh"
+
+inductive fresh  
+intros
+  fresh_abst_ab[intro!]: "\<lbrakk>(nabla\<turnstile>a\<sharp>t); a\<noteq>b\<rbrakk> \<Longrightarrow> (nabla\<turnstile>a\<sharp>Abst b t)"
+  fresh_abst_aa[intro!]: "(nabla\<turnstile>a\<sharp>Abst a t)"
+  fresh_unit[intro!]:    "(nabla\<turnstile>a\<sharp>Unit)"
+  fresh_atom[intro!]:    "a\<noteq>b \<Longrightarrow> (nabla\<turnstile>a\<sharp>Atom b)"
+  fresh_susp[intro!]:    "(swapas (rev pi) a,X)\<in>nabla \<Longrightarrow> (nabla\<turnstile>a\<sharp>Susp pi X)"
+  fresh_paar[intro!]:    "\<lbrakk>(nabla\<turnstile>a\<sharp>t1);(nabla\<turnstile>a\<sharp>t2)\<rbrakk> \<Longrightarrow> (nabla\<turnstile>a\<sharp>Paar t1 t2)"
+  fresh_func[intro!]:    "(nabla\<turnstile>a\<sharp>t) \<Longrightarrow> (nabla\<turnstile>a\<sharp>Func F t)"
+
+lemma fresh_atom_elim[elim!]: "(nabla\<turnstile>a\<sharp>Atom b) \<Longrightarrow> a\<noteq>b"
+apply(ind_cases "(nabla \<turnstile>a\<sharp>Atom b)")
+apply(auto)
+done
+lemma fresh_susp_elim[elim!]: "(nabla\<turnstile>a\<sharp>Susp pi X) \<Longrightarrow> (swapas (rev pi) a,X)\<in>nabla"
+apply(ind_cases "(nabla\<turnstile>a\<sharp>Susp pi X)")
+apply(auto)
+done
+lemma fresh_paar_elim[elim!]: "(nabla\<turnstile>a\<sharp>Paar t1 t2) \<Longrightarrow> (nabla\<turnstile>a\<sharp>t1)\<and>(nabla \<turnstile>a\<sharp>t2)"
+apply(ind_cases "(nabla\<turnstile>a\<sharp>Paar t1 t2)")
+apply(auto)
+done
+lemma fresh_func_elim[elim!]: "(nabla\<turnstile>a\<sharp>Func F t) \<Longrightarrow> (nabla\<turnstile>a\<sharp>t)"
+apply(ind_cases "nabla\<turnstile>a\<sharp>Func F t")
+apply(auto)
+done
+lemma fresh_abst_ab_elim[elim!]: "\<lbrakk>nabla\<turnstile>a\<sharp>Abst b t;a\<noteq>b\<rbrakk> \<Longrightarrow> (nabla\<turnstile>a\<sharp>t)"
+apply(ind_cases "nabla\<turnstile>a\<sharp>Abst b t", auto)
+done
+
+lemma fresh_swap_left: "(nabla\<turnstile>a\<sharp>swap pi t) \<longrightarrow> (nabla\<turnstile>swapas (rev pi) a\<sharp>t)"
+apply(induct t)
+apply(simp_all)
+-- Abst
+apply(rule impI)
+apply(case_tac "swapas (rev pi) a = list")
+apply(force)
+apply(force dest!: fresh_abst_ab_elim)
+--Susp
+apply(force dest!: fresh_susp_elim simp add: swapas_append[THEN sym]) 
+--Unit
+apply(force)
+--Atom
+apply(force dest!: fresh_atom_elim)
+--Paar
+apply(force dest!: fresh_paar_elim)
+-- Func
+apply(force dest!: fresh_func_elim)
+done
+
+lemma fresh_swap_right: "(nabla\<turnstile>swapas (rev pi) a\<sharp>t) \<longrightarrow> (nabla\<turnstile>a\<sharp>swap pi t)"
+apply(induct t)
+apply(simp_all)
+-- Abst
+apply(rule impI)
+apply(case_tac "a = swapas pi list")
+apply(force)
+apply(force dest!: fresh_abst_ab_elim)
+--Susp
+apply(force dest!: fresh_susp_elim simp add: swapas_append[THEN sym])
+--Unit
+apply(force)
+--Atom
+apply(force dest!: fresh_atom_elim)
+--Paar
+apply(force dest!: fresh_paar_elim)
+-- Func
+apply(force dest!: fresh_func_elim)
+done
+
+lemma fresh_weak: "nabla1\<turnstile>a\<sharp>t\<longrightarrow>(nabla1\<union>nabla2)\<turnstile>a\<sharp>t"
+apply(rule impI)
+apply(erule fresh.induct)
+apply(auto)
+done
+
+end