--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/TypeSchemes.thy	Thu Apr 08 13:04:49 2010 +0200
@@ -0,0 +1,171 @@
+theory TypeSchemes
+imports "../Parser"
+begin
+
+section {*** Type Schemes ***}
+
+atom_decl name
+
+ML {* val _ = alpha_type := AlphaRes *}
+
+nominal_datatype ty =
+  Var "name"
+| Fun "ty" "ty"
+and tys =
+  All xs::"name fset" ty::"ty" bind xs in ty
+
+lemmas ty_tys_supp = ty_tys.fv[simplified ty_tys.supp]
+
+(* below we define manually the function for size *)
+
+lemma size_eqvt_raw:
+  "size (pi \<bullet> t  :: ty_raw)  = size t"
+  "size (pi \<bullet> ts :: tys_raw) = size ts"
+  apply (induct rule: ty_raw_tys_raw.inducts)
+  apply simp_all
+  done
+
+instantiation ty and tys :: size 
+begin
+
+quotient_definition
+  "size_ty :: ty \<Rightarrow> nat"
+is
+  "size :: ty_raw \<Rightarrow> nat"
+
+quotient_definition
+  "size_tys :: tys \<Rightarrow> nat"
+is
+  "size :: tys_raw \<Rightarrow> nat"
+
+lemma size_rsp:
+  "alpha_ty_raw x y \<Longrightarrow> size x = size y"
+  "alpha_tys_raw a b \<Longrightarrow> size a = size b"
+  apply (induct rule: alpha_ty_raw_alpha_tys_raw.inducts)
+  apply (simp_all only: ty_raw_tys_raw.size)
+  apply (simp_all only: alphas)
+  apply clarify
+  apply (simp_all only: size_eqvt_raw)
+  done
+
+lemma [quot_respect]:
+  "(alpha_ty_raw ===> op =) size size"
+  "(alpha_tys_raw ===> op =) size size"
+  by (simp_all add: size_rsp)
+
+lemma [quot_preserve]:
+  "(rep_ty ---> id) size = size"
+  "(rep_tys ---> id) size = size"
+  by (simp_all add: size_ty_def size_tys_def)
+
+instance
+  by default
+
+end
+
+thm ty_raw_tys_raw.size(4)[quot_lifted]
+thm ty_raw_tys_raw.size(5)[quot_lifted]
+thm ty_raw_tys_raw.size(6)[quot_lifted]
+
+
+thm ty_tys.fv
+thm ty_tys.eq_iff
+thm ty_tys.bn
+thm ty_tys.perm
+thm ty_tys.inducts
+thm ty_tys.distinct
+
+ML {* Sign.of_sort @{theory} (@{typ ty}, @{sort fs}) *}
+
+lemma strong_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 ty_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> All fset ty) \<sharp>* pa)")
+    apply clarify
+    apply(rule_tac t="p \<bullet> All fset ty" and 
+                   s="pa \<bullet> (p \<bullet> All fset ty)" in subst)
+    apply (rule supp_perm_eq)
+    apply assumption
+    apply (simp only: ty_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: ty_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: ty_tys.eq_iff)
+  apply(rule_tac x="0::perm" in exI)
+  apply(simp add: alphas)
+  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: ty_tys.eq_iff)
+  apply(rule_tac x="(atom a \<rightleftharpoons> atom b)" in exI)
+  apply(simp add: alphas fresh_star_def eqvts)
+  done
+
+lemma
+  shows "All {|a, b, c|} (Fun (Var a) (Var b)) = All {|a, b|} (Fun (Var a) (Var b))"
+  apply(simp add: ty_tys.eq_iff)
+  apply(rule_tac x="0::perm" in exI)
+  apply(simp add: alphas fresh_star_def eqvts ty_tys.eq_iff)
+done
+
+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: ty_tys.eq_iff)
+  apply(clarify)
+  apply(simp add: alphas fresh_star_def eqvts ty_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