diff -r 2406350c358f -r 8b5a1ad60487 Nominal/ExTySch.thy
--- /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