theory TySch

imports "Parser" "../Attic/Prove" "FSet"

begin

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

end