# Theory NestedDatatype

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theory NestedDatatype
imports Main
begin

    header {* Nested datatypes *}

    theory NestedDatatype imports Main begin

    subsection {* Terms and substitution *}

    datatype ('a, 'b) "term" =
        Var 'a
      | App 'b "('a, 'b) term list"

    consts
      subst_term :: "('a => ('a, 'b) term) => ('a, 'b) term => ('a, 'b) term"
      subst_term_list ::
        "('a => ('a, 'b) term) => ('a, 'b) term list => ('a, 'b) term list"

    primrec (subst)
      "subst_term f (Var a) = f a"
      "subst_term f (App b ts) = App b (subst_term_list f ts)"
      "subst_term_list f [] = []"
      "subst_term_list f (t # ts) = subst_term f t # subst_term_list f ts"

    text {*
     \medskip A simple lemma about composition of substitutions.
    *}

    lemma
       "subst_term (subst_term f1 o f2) t =
          subst_term f1 (subst_term f2 t) &
        subst_term_list (subst_term f1 o f2) ts =
          subst_term_list f1 (subst_term_list f2 ts)"
      by (induct t and ts) simp_all

    lemma "subst_term (subst_term f1 o f2) t =
      subst_term f1 (subst_term f2 t)"
    proof -
      let "?P t" = ?thesis
      let ?Q = "λts. subst_term_list (subst_term f1 o f2) ts =
        subst_term_list f1 (subst_term_list f2 ts)"
      show ?thesis
      proof (induct t)
        fix a show "?P (Var a)" by simp
      next
        fix b ts assume "?Q ts"
        thus "?P (App b ts)" by (simp add: o_def)
      next
        show "?Q []" by simp
      next
        fix t ts
        assume "?P t" "?Q ts" thus "?Q (t # ts)" by simp
      qed
    qed

    subsection {* Alternative induction *}

    theorem term_induct' [case_names Var App]:
      "(!!a. P (Var a)) ==>
       (!!b ts. list_all P ts ==> P (App b ts)) ==> P t"
    proof -
      assume var: "!!a. P (Var a)"
      assume app: "!!b ts. list_all P ts ==> P (App b ts)"
      show ?thesis
      proof (induct t)
        fix a show "P (Var a)" by (rule var)
      next
        fix b t ts assume "list_all P ts"
        thus "P (App b ts)" by (rule app)
      next
        show "list_all P []" by simp
      next
        fix t ts assume "P t" "list_all P ts"
        thus "list_all P (t # ts)" by simp
      qed
    qed

    lemma
      "subst_term (subst_term f1 o f2) t = subst_term f1 (subst_term f2 t)"
      (is "?P t")
    proof (induct t rule: term_induct')
      case (Var a)
      show "?P (Var a)" by (simp add: o_def)
    next
      case (App b ts)
      thus "?P (App b ts)" by (induct ts) simp_all
    qed

    end

### Terms and substitution

lemma

      subst_term (subst_term f1.0 o f2.0) t = subst_term f1.0 (subst_term f2.0 t) ∧
      subst_term_list (subst_term f1.0 o f2.0) ts =
      subst_term_list f1.0 (subst_term_list f2.0 ts)

lemma

      subst_term (subst_term f1.0 o f2.0) t = subst_term f1.0 (subst_term f2.0 t)

### Alternative induction

theorem term_induct':

      [| !!a. P (Var a); !!b ts. list_all P ts ==> P (App b ts) |] ==> P t

lemma

      subst_term (subst_term f1.0 o f2.0) t = subst_term f1.0 (subst_term f2.0 t)