--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Manual/Term4.thy	Tue Mar 23 08:19:33 2010 +0100
@@ -0,0 +1,121 @@
+theory Term4
+imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv" "Rsp" "../Attic/Prove" "Quotient_List"
+begin
+
+atom_decl name
+
+section {*** lam with indirect list recursion ***}
+
+datatype rtrm4 =
+  rVr4 "name"
+| rAp4 "rtrm4" "rtrm4 list"
+| rLm4 "name" "rtrm4"  --"bind (name) in (trm)"
+print_theorems
+
+thm rtrm4.recs
+
+(* there cannot be a clause for lists, as *)
+(* permutations are  already defined in Nominal (also functions, options, and so on) *)
+setup {* snd o define_raw_perms (Datatype.the_info @{theory} "Term4.rtrm4") 1 *}
+
+(* "repairing" of the permute function *)
+lemma repaired:
+  fixes ts::"rtrm4 list"
+  shows "permute_rtrm4_list p ts = p \<bullet> ts"
+  apply(induct ts)
+  apply(simp_all)
+  done
+
+thm permute_rtrm4_permute_rtrm4_list.simps
+thm permute_rtrm4_permute_rtrm4_list.simps[simplified repaired]
+
+local_setup {* snd o define_fv_alpha (Datatype.the_info @{theory} "Term4.rtrm4")
+  [[[], [], [(NONE, 0,1)]], [[], []]  ] *}
+print_theorems
+
+lemma fix2: "alpha_rtrm4_list = list_rel alpha_rtrm4"
+apply (rule ext)+
+apply (induct_tac x xa rule: list_induct2')
+apply (simp_all add: alpha_rtrm4_alpha_rtrm4_list.intros)
+apply clarify apply (erule alpha_rtrm4_list.cases) apply(simp_all)
+apply clarify apply (erule alpha_rtrm4_list.cases) apply(simp_all)
+apply rule
+apply (erule alpha_rtrm4_list.cases)
+apply simp_all
+apply (rule alpha_rtrm4_alpha_rtrm4_list.intros)
+apply simp_all
+done
+
+(* We need sth like:
+lemma fix3: "fv_rtrm4_list = set o map fv_rtrm4" *)
+
+notation
+  alpha_rtrm4 ("_ \<approx>4 _" [100, 100] 100) and
+  alpha_rtrm4_list ("_ \<approx>4l _" [100, 100] 100)
+thm alpha_rtrm4_alpha_rtrm4_list.intros[simplified fix2]
+
+local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha4_inj}, []), (build_alpha_inj @{thms alpha_rtrm4_alpha_rtrm4_list.intros[simplified fix2]} @{thms rtrm4.distinct rtrm4.inject list.distinct list.inject} @{thms alpha_rtrm4.cases[simplified fix2] alpha_rtrm4_list.cases[simplified fix2]} ctxt)) ctxt)) *}
+thm alpha4_inj
+
+local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha4_inj_no}, []), (build_alpha_inj @{thms alpha_rtrm4_alpha_rtrm4_list.intros} @{thms rtrm4.distinct rtrm4.inject list.distinct list.inject} @{thms alpha_rtrm4.cases alpha_rtrm4_list.cases} ctxt)) ctxt)) *}
+thm alpha4_inj_no
+
+local_setup {*
+snd o build_eqvts @{binding fv_rtrm4_fv_rtrm4_list_eqvt} [@{term fv_rtrm4}, @{term fv_rtrm4_list}] [@{term "permute :: perm \<Rightarrow> rtrm4 \<Rightarrow> rtrm4"},@{term "permute :: perm \<Rightarrow> rtrm4 list \<Rightarrow> rtrm4 list"}] (@{thms fv_rtrm4_fv_rtrm4_list.simps permute_rtrm4_permute_rtrm4_list.simps[simplified repaired]}) @{thm rtrm4.induct}
+*}
+print_theorems
+
+local_setup {*
+(fn ctxt => snd (Local_Theory.note ((@{binding alpha4_eqvt_no}, []),
+  build_alpha_eqvts [@{term alpha_rtrm4}, @{term alpha_rtrm4_list}] [@{term "permute :: perm \<Rightarrow> rtrm4 \<Rightarrow> rtrm4"},@{term "permute :: perm \<Rightarrow> rtrm4 list \<Rightarrow> rtrm4 list"}] @{thms permute_rtrm4_permute_rtrm4_list.simps[simplified repaired] alpha4_inj_no} @{thm alpha_rtrm4_alpha_rtrm4_list.induct} ctxt) ctxt))
+*}
+lemmas alpha4_eqvt = alpha4_eqvt_no[simplified fix2]
+
+local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha4_equivp_no}, []),
+  (build_equivps [@{term alpha_rtrm4}, @{term alpha_rtrm4_list}] @{thm rtrm4.induct} @{thm alpha_rtrm4_alpha_rtrm4_list.induct} @{thms rtrm4.inject list.inject} @{thms alpha4_inj_no} @{thms rtrm4.distinct list.distinct} @{thms alpha_rtrm4_list.cases alpha_rtrm4.cases} @{thms alpha4_eqvt_no} ctxt)) ctxt)) *}
+lemmas alpha4_equivp = alpha4_equivp_no[simplified fix2]
+
+(*lemma fv_rtrm4_rsp:
+  "xa \<approx>4 ya \<Longrightarrow> fv_rtrm4 xa = fv_rtrm4 ya"
+  "x \<approx>4l y \<Longrightarrow> fv_rtrm4_list x = fv_rtrm4_list y"
+  apply (induct rule: alpha_rtrm4_alpha_rtrm4_list.inducts)
+  apply (simp_all add: alpha_gen)
+done*)
+
+
+quotient_type 
+  trm4 = rtrm4 / alpha_rtrm4
+(*and
+  trm4list = "rtrm4 list" / alpha_rtrm4_list*)
+  by (simp_all add: alpha4_equivp)
+
+local_setup {*
+(fn ctxt => ctxt
+ |> snd o (Quotient_Def.quotient_lift_const ("Vr4", @{term rVr4}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Ap4", @{term rAp4}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Lm4", @{term rLm4})))
+*}
+print_theorems
+
+local_setup {* snd o prove_const_rsp @{binding fv_rtrm4_rsp} [@{term fv_rtrm4}]
+  (fn _ => fvbv_rsp_tac @{thm alpha_rtrm4_alpha_rtrm4_list.inducts(1)} @{thms fv_rtrm4_fv_rtrm4_list.simps} 1) *}
+print_theorems
+
+local_setup {* snd o prove_const_rsp @{binding rVr4_rsp} [@{term rVr4}]
+  (fn _ => constr_rsp_tac @{thms alpha4_inj} @{thms fv_rtrm4_rsp} @{thms alpha4_equivp} 1) *}
+lemma "(alpha_rtrm4 ===> list_rel alpha_rtrm4 ===> alpha_rtrm4) rAp4 rAp4"
+apply simp
+apply clarify
+apply (simp add: alpha4_inj)
+
+
+local_setup {* snd o prove_const_rsp @{binding rLm4_rsp} [@{term rLm4}]
+  (fn _ => constr_rsp_tac @{thms alpha4_inj} @{thms fv_rtrm4_rsp} @{thms alpha4_equivp} 1) *}
+local_setup {* snd o prove_const_rsp @{binding permute_rtrm4_rsp}
+  [@{term "permute :: perm \<Rightarrow> rtrm4 \<Rightarrow> rtrm4"}, @{term "permute :: perm \<Rightarrow> rtrm4 list \<Rightarrow> rtrm4 list"}] 
+  (fn _ => asm_simp_tac (HOL_ss addsimps @{thms alpha4_eqvt}) 1) *}
+
+thm rtrm4.induct
+lemmas trm1_bp_induct = rtrm4.induct[quot_lifted]
+
+end