--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/ExTySch.thy Tue Mar 23 08:51:43 2010 +0100
@@ -0,0 +1,166 @@
+theory ExTySch
+imports "Parser"
+begin
+
+(* Type Schemes *)
+atom_decl name
+
+nominal_datatype t =
+ Var "name"
+| Fun "t" "t"
+and tyS =
+ All xs::"name fset" ty::"t" bind xs in ty
+
+lemma size_eqvt_raw:
+ "size (pi \<bullet> t :: t_raw) = size t"
+ "size (pi \<bullet> ts :: tyS_raw) = size ts"
+ apply (induct rule: t_raw_tyS_raw.inducts)
+ apply simp_all
+ done
+
+instantiation t and tyS :: size begin
+
+quotient_definition
+ "size_t :: t \<Rightarrow> nat"
+is
+ "size :: t_raw \<Rightarrow> nat"
+
+quotient_definition
+ "size_tyS :: tyS \<Rightarrow> nat"
+is
+ "size :: tyS_raw \<Rightarrow> nat"
+
+lemma size_rsp:
+ "alpha_t_raw x y \<Longrightarrow> size x = size y"
+ "alpha_tyS_raw a b \<Longrightarrow> size a = size b"
+ apply (induct rule: alpha_t_raw_alpha_tyS_raw.inducts)
+ apply (simp_all only: t_raw_tyS_raw.size)
+ apply (simp_all only: alpha_gen)
+ apply clarify
+ apply (simp_all only: size_eqvt_raw)
+ done
+
+lemma [quot_respect]:
+ "(alpha_t_raw ===> op =) size size"
+ "(alpha_tyS_raw ===> op =) size size"
+ by (simp_all add: size_rsp)
+
+lemma [quot_preserve]:
+ "(rep_t ---> id) size = size"
+ "(rep_tyS ---> id) size = size"
+ by (simp_all add: size_t_def size_tyS_def)
+
+instance
+ by default
+
+end
+
+thm t_raw_tyS_raw.size(4)[quot_lifted]
+thm t_raw_tyS_raw.size(5)[quot_lifted]
+thm t_raw_tyS_raw.size(6)[quot_lifted]
+
+
+thm t_tyS.fv
+thm t_tyS.eq_iff
+thm t_tyS.bn
+thm t_tyS.perm
+thm t_tyS.inducts
+thm t_tyS.distinct
+ML {* Sign.of_sort @{theory} (@{typ t}, @{sort fs}) *}
+
+lemmas t_tyS_supp = t_tyS.fv[simplified t_tyS.supp]
+
+lemma induct:
+ assumes a1: "\<And>name b. P b (Var name)"
+ and a2: "\<And>t1 t2 b. \<lbrakk>\<And>c. P c t1; \<And>c. P c t2\<rbrakk> \<Longrightarrow> P b (Fun t1 t2)"
+ and a3: "\<And>fset t b. \<lbrakk>\<And>c. P c t; fset_to_set (fmap atom fset) \<sharp>* b\<rbrakk> \<Longrightarrow> P' b (All fset t)"
+ shows "P (a :: 'a :: pt) t \<and> P' (d :: 'b :: {fs}) ts "
+proof -
+ have " (\<forall>p a. P a (p \<bullet> t)) \<and> (\<forall>p d. P' d (p \<bullet> ts))"
+ apply (rule t_tyS.induct)
+ apply (simp add: a1)
+ apply (simp)
+ apply (rule allI)+
+ apply (rule a2)
+ apply simp
+ apply simp
+ apply (rule allI)
+ apply (rule allI)
+ apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (fset_to_set (fmap atom (p \<bullet> fset)))) \<sharp>* d \<and> supp (p \<bullet> TySch.All fset t) \<sharp>* pa)")
+ apply clarify
+ apply(rule_tac t="p \<bullet> TySch.All fset t" and
+ s="pa \<bullet> (p \<bullet> TySch.All fset t)" in subst)
+ apply (rule supp_perm_eq)
+ apply assumption
+ apply (simp only: t_tyS.perm)
+ apply (rule a3)
+ apply(erule_tac x="(pa + p)" in allE)
+ apply simp
+ apply (simp add: eqvts eqvts_raw)
+ apply (rule at_set_avoiding2)
+ apply (simp add: fin_fset_to_set)
+ apply (simp add: finite_supp)
+ apply (simp add: eqvts finite_supp)
+ apply (subst atom_eqvt_raw[symmetric])
+ apply (subst fmap_eqvt[symmetric])
+ apply (subst fset_to_set_eqvt[symmetric])
+ apply (simp only: fresh_star_permute_iff)
+ apply (simp add: fresh_star_def)
+ apply clarify
+ apply (simp add: fresh_def)
+ apply (simp add: t_tyS_supp)
+ done
+ then have "P a (0 \<bullet> t) \<and> P' d (0 \<bullet> ts)" by blast
+ then show ?thesis by simp
+qed
+
+lemma
+ shows "All {|a, b|} (Fun (Var a) (Var b)) = All {|b, a|} (Fun (Var a) (Var b))"
+ apply(simp add: t_tyS.eq_iff)
+ apply(rule_tac x="0::perm" in exI)
+ apply(simp add: alpha_gen)
+ apply(auto)
+ apply(simp add: fresh_star_def fresh_zero_perm)
+ done
+
+lemma
+ shows "All {|a, b|} (Fun (Var a) (Var b)) = All {|a, b|} (Fun (Var b) (Var a))"
+ apply(simp add: t_tyS.eq_iff)
+ apply(rule_tac x="(atom a \<rightleftharpoons> atom b)" in exI)
+ apply(simp add: alpha_gen fresh_star_def eqvts)
+ apply auto
+ done
+
+lemma
+ shows "All {|a, b, c|} (Fun (Var a) (Var b)) = All {|a, b|} (Fun (Var a) (Var b))"
+ apply(simp add: t_tyS.eq_iff)
+ apply(rule_tac x="0::perm" in exI)
+ apply(simp add: alpha_gen fresh_star_def eqvts t_tyS.eq_iff)
+oops
+
+lemma
+ assumes a: "a \<noteq> b"
+ shows "\<not>(All {|a, b|} (Fun (Var a) (Var b)) = All {|c|} (Fun (Var c) (Var c)))"
+ using a
+ apply(simp add: t_tyS.eq_iff)
+ apply(clarify)
+ apply(simp add: alpha_gen fresh_star_def eqvts t_tyS.eq_iff)
+ apply auto
+ done
+
+(* PROBLEM:
+Type schemes with separate datatypes
+
+nominal_datatype T =
+ TVar "name"
+| TFun "T" "T"
+nominal_datatype TyS =
+ TAll xs::"name list" ty::"T" bind xs in ty
+
+*** exception Datatype raised
+*** (line 218 of "/usr/local/src/Isabelle_16-Mar-2010/src/HOL/Tools/Datatype/datatype_aux.ML")
+*** At command "nominal_datatype".
+*)
+
+
+end