--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/Examples/LamEx.thy	Mon Dec 07 14:09:50 2009 +0100
@@ -0,0 +1,252 @@
+theory LamEx
+imports Nominal "../QuotMain"
+begin
+
+atom_decl name
+
+thm abs_fresh(1)
+
+nominal_datatype rlam =
+  rVar "name"
+| rApp "rlam" "rlam"
+| rLam "name" "rlam"
+
+print_theorems
+
+function
+  rfv :: "rlam \<Rightarrow> name set"
+where
+  rfv_var: "rfv (rVar a) = {a}"
+| rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"
+| rfv_lam: "rfv (rLam a t) = (rfv t) - {a}"
+sorry
+
+termination rfv sorry
+
+inductive
+  alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100)
+where
+  a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)"
+| a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2"
+| a3: "\<lbrakk>t \<approx> ([(a,b)]\<bullet>s); a \<notin> rfv (rLam b t)\<rbrakk> \<Longrightarrow> rLam a t \<approx> rLam b s"
+
+print_theorems
+
+lemma alpha_refl:
+  fixes t::"rlam"
+  shows "t \<approx> t"
+  apply(induct t rule: rlam.induct)
+  apply(simp add: a1)
+  apply(simp add: a2)
+  apply(rule a3)
+  apply(subst pt_swap_bij'')
+  apply(rule pt_name_inst)
+  apply(rule at_name_inst)
+  apply(simp)
+  apply(simp)
+  done
+
+lemma alpha_equivp:
+  shows "equivp alpha"
+sorry
+
+quotient lam = rlam / alpha
+  apply(rule alpha_equivp)
+  done
+
+print_quotients
+
+quotient_def 
+  Var :: "name \<Rightarrow> lam"
+where
+  "Var \<equiv> rVar"
+
+quotient_def 
+  App :: "lam \<Rightarrow> lam \<Rightarrow> lam"
+where
+  "App \<equiv> rApp"
+
+quotient_def 
+  Lam :: "name \<Rightarrow> lam \<Rightarrow> lam"
+where
+  "Lam \<equiv> rLam"
+
+thm Var_def
+thm App_def
+thm Lam_def
+
+quotient_def 
+  fv :: "lam \<Rightarrow> name set"
+where
+  "fv \<equiv> rfv"
+
+thm fv_def
+
+(* definition of overloaded permutation function *)
+(* for the lifted type lam                       *)
+overloading
+  perm_lam \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"   (unchecked)
+begin
+
+quotient_def 
+  perm_lam :: "'x prm \<Rightarrow> lam \<Rightarrow> lam"
+where
+  "perm_lam \<equiv> (perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam)"
+
+end
+
+(*quotient_def (for lam)
+  abs_fun_lam :: "'x prm \<Rightarrow> lam \<Rightarrow> lam"
+where
+  "perm_lam \<equiv> (perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam)"*)
+
+
+thm perm_lam_def
+
+(* lemmas that need to lift *)
+lemma pi_var_com:
+  fixes pi::"'x prm"
+  shows "(pi\<bullet>rVar a) \<approx> rVar (pi\<bullet>a)"
+  sorry
+
+lemma pi_app_com:
+  fixes pi::"'x prm"
+  shows "(pi\<bullet>rApp t1 t2) \<approx> rApp (pi\<bullet>t1) (pi\<bullet>t2)"
+  sorry
+
+lemma pi_lam_com:
+  fixes pi::"'x prm"
+  shows "(pi\<bullet>rLam a t) \<approx> rLam (pi\<bullet>a) (pi\<bullet>t)"
+  sorry
+
+
+
+lemma real_alpha:
+  assumes a: "t = [(a,b)]\<bullet>s" "a\<sharp>[b].s"
+  shows "Lam a t = Lam b s"
+using a
+unfolding fresh_def supp_def
+sorry
+
+lemma perm_rsp[quotient_rsp]:
+  "(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>"
+  apply(auto)
+  (* this is propably true if some type conditions are imposed ;o) *)
+  sorry
+
+lemma fresh_rsp:
+  "(op = ===> alpha ===> op =) fresh fresh"
+  apply(auto)
+  (* this is probably only true if some type conditions are imposed *)
+  sorry
+
+lemma rVar_rsp[quotient_rsp]:
+  "(op = ===> alpha) rVar rVar"
+by (auto intro:a1)
+
+lemma rApp_rsp[quotient_rsp]: "(alpha ===> alpha ===> alpha) rApp rApp"
+by (auto intro:a2)
+
+lemma rLam_rsp[quotient_rsp]: "(op = ===> alpha ===> alpha) rLam rLam"
+  apply(auto)
+  apply(rule a3)
+  apply(rule_tac t="[(x,x)]\<bullet>y" and s="y" in subst)
+  apply(rule sym)
+  apply(rule trans)
+  apply(rule pt_name3)
+  apply(rule at_ds1[OF at_name_inst])
+  apply(simp add: pt_name1)
+  apply(assumption)
+  apply(simp add: abs_fresh)
+  done
+
+lemma rfv_rsp[quotient_rsp]: "(alpha ===> op =) rfv rfv"
+  sorry
+
+lemma rvar_inject: "rVar a \<approx> rVar b = (a = b)"
+apply (auto)
+apply (erule alpha.cases)
+apply (simp_all add: rlam.inject alpha_refl)
+done
+
+ML {* val qty = @{typ "lam"} *}
+ML {* val rsp_thms = @{thms perm_rsp fresh_rsp rVar_rsp rApp_rsp rLam_rsp rfv_rsp} *}
+
+ML {* val (rty, rel, rel_refl, rel_eqv) = lookup_quot_data @{context} qty *}
+ML {* val (trans2, reps_same, absrep, quot) = lookup_quot_thms @{context} "lam" *}
+ML {* fun lift_tac_lam lthy t = lift_tac lthy t *}
+
+lemma pi_var: "(pi\<Colon>('x \<times> 'x) list) \<bullet> Var a = Var (pi \<bullet> a)"
+apply (tactic {* lift_tac_lam @{context} @{thm pi_var_com} 1 *})
+done
+
+lemma pi_app: "(pi\<Colon>('x \<times> 'x) list) \<bullet> App (x\<Colon>lam) (xa\<Colon>lam) = App (pi \<bullet> x) (pi \<bullet> xa)"
+apply (tactic {* lift_tac_lam @{context} @{thm pi_app_com} 1 *})
+done
+
+lemma pi_lam: "(pi\<Colon>('x \<times> 'x) list) \<bullet> Lam (a\<Colon>name) (x\<Colon>lam) = Lam (pi \<bullet> a) (pi \<bullet> x)"
+apply (tactic {* lift_tac_lam @{context} @{thm pi_lam_com} 1 *})
+done
+
+lemma fv_var: "fv (Var (a\<Colon>name)) = {a}"
+apply (tactic {* lift_tac_lam @{context} @{thm rfv_var} 1 *})
+done
+
+lemma fv_app: "fv (App (x\<Colon>lam) (xa\<Colon>lam)) = fv x \<union> fv xa"
+apply (tactic {* lift_tac_lam @{context} @{thm rfv_app} 1 *})
+done
+
+lemma fv_lam: "fv (Lam (a\<Colon>name) (x\<Colon>lam)) = fv x - {a}"
+apply (tactic {* lift_tac_lam @{context} @{thm rfv_lam} 1 *})
+done
+
+lemma a1: "(a\<Colon>name) = (b\<Colon>name) \<Longrightarrow> Var a = Var b"
+apply (tactic {* lift_tac_lam @{context} @{thm a1} 1 *})
+done
+
+lemma a2: "\<lbrakk>(x\<Colon>lam) = (xa\<Colon>lam); (xb\<Colon>lam) = (xc\<Colon>lam)\<rbrakk> \<Longrightarrow> App x xb = App xa xc"
+apply (tactic {* lift_tac_lam @{context} @{thm a2} 1 *})
+done
+
+lemma a3: "\<lbrakk>(x\<Colon>lam) = [(a\<Colon>name, b\<Colon>name)] \<bullet> (xa\<Colon>lam); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> Lam a x = Lam b xa"
+apply (tactic {* lift_tac_lam @{context} @{thm a3} 1 *})
+done
+
+lemma alpha_cases: "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;
+     \<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P;
+     \<And>x a b xa. \<lbrakk>a1 = Lam a x; a2 = Lam b xa; x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> P\<rbrakk>
+    \<Longrightarrow> P"
+apply (tactic {* lift_tac_lam @{context} @{thm alpha.cases} 1 *})
+done
+
+lemma alpha_induct: "\<lbrakk>(qx\<Colon>lam) = (qxa\<Colon>lam); \<And>(a\<Colon>name) b\<Colon>name. a = b \<Longrightarrow> (qxb\<Colon>lam \<Rightarrow> lam \<Rightarrow> bool) (Var a) (Var b);
+     \<And>(x\<Colon>lam) (xa\<Colon>lam) (xb\<Colon>lam) xc\<Colon>lam. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc);
+     \<And>(x\<Colon>lam) (a\<Colon>name) (b\<Colon>name) xa\<Colon>lam.
+        \<lbrakk>x = [(a, b)] \<bullet> xa; qxb x ([(a, b)] \<bullet> xa); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> qxb (Lam a x) (Lam b xa)\<rbrakk>
+    \<Longrightarrow> qxb qx qxa"
+apply (tactic {* lift_tac_lam @{context} @{thm alpha.induct} 1 *})
+done
+
+lemma var_inject: "(Var a = Var b) = (a = b)"
+apply (tactic {* lift_tac_lam @{context} @{thm rvar_inject} 1 *})
+done
+
+lemma lam_induct:" \<lbrakk>\<And>name. P (Var name); \<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2);
+              \<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk> \<Longrightarrow> P lam"
+apply (tactic {* lift_tac_lam @{context} @{thm rlam.induct} 1 *})
+done
+
+lemma var_supp:
+  shows "supp (Var a) = ((supp a)::name set)"
+  apply(simp add: supp_def)
+  apply(simp add: pi_var)
+  apply(simp add: var_inject)
+  done
+
+lemma var_fresh:
+  fixes a::"name"
+  shows "(a\<sharp>(Var b)) = (a\<sharp>b)"
+  apply(simp add: fresh_def)
+  apply(simp add: var_supp)
+  done
+