Theory Term

(*  Title:      ZF/Induct/Term.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge
*)

section ‹Terms over an alphabet›

theory Term imports ZF begin

text ‹
  Illustrates the list functor (essentially the same type as in ‹Trees_Forest›).
›

consts
  "term" :: "i ⇒ i"

datatype "term(A)" = Apply ("a ∈ A", "l ∈ list(term(A))")
  monos list_mono
  type_elims list_univ [THEN subsetD, elim_format]

declare Apply [TC]

definition
  term_rec :: "[i, [i, i, i] ⇒ i] ⇒ i"  where
  "term_rec(t,d) ≡
    Vrec(t, λt g. term_case(λx zs. d(x, zs, map(λz. g`z, zs)), t))"

definition
  term_map :: "[i ⇒ i, i] ⇒ i"  where
  "term_map(f,t) ≡ term_rec(t, λx zs rs. Apply(f(x), rs))"

definition
  term_size :: "i ⇒ i"  where
  "term_size(t) ≡ term_rec(t, λx zs rs. succ(list_add(rs)))"

definition
  reflect :: "i ⇒ i"  where
  "reflect(t) ≡ term_rec(t, λx zs rs. Apply(x, rev(rs)))"

definition
  preorder :: "i ⇒ i"  where
  "preorder(t) ≡ term_rec(t, λx zs rs. Cons(x, flat(rs)))"

definition
  postorder :: "i ⇒ i"  where
  "postorder(t) ≡ term_rec(t, λx zs rs. flat(rs) @ [x])"

lemma term_unfold: "term(A) = A * list(term(A))"
  by (fast intro!: term.intros [unfolded term.con_defs]
    elim: term.cases [unfolded term.con_defs])

lemma term_induct2:
  "⟦t ∈ term(A);
      ⋀x.      ⟦x ∈ A⟧ ⟹ P(Apply(x,Nil));
      ⋀x z zs. ⟦x ∈ A;  z ∈ term(A);  zs: list(term(A));  P(Apply(x,zs))
⟧ ⟹ P(Apply(x, Cons(z,zs)))
⟧ ⟹ P(t)"
  ― ‹Induction on \<^term>‹term(A)› followed by induction on \<^term>‹list›.›
  apply (induct_tac t)
  apply (erule list.induct)
   apply (auto dest: list_CollectD)
  done

lemma term_induct_eqn [consumes 1, case_names Apply]:
  "⟦t ∈ term(A);
      ⋀x zs. ⟦x ∈ A;  zs: list(term(A));  map(f,zs) = map(g,zs)⟧ ⟹
              f(Apply(x,zs)) = g(Apply(x,zs))
⟧ ⟹ f(t) = g(t)"
  ― ‹Induction on \<^term>‹term(A)› to prove an equation.›
  apply (induct_tac t)
  apply (auto dest: map_list_Collect list_CollectD)
  done

text ‹
  \medskip Lemmas to justify using \<^term>‹term› in other recursive
  type definitions.
›

lemma term_mono: "A ⊆ B ⟹ term(A) ⊆ term(B)"
    unfolding term.defs
  apply (rule lfp_mono)
    apply (rule term.bnd_mono)+
  apply (rule univ_mono basic_monos| assumption)+
  done

lemma term_univ: "term(univ(A)) ⊆ univ(A)"
  ― ‹Easily provable by induction also›
    unfolding term.defs term.con_defs
  apply (rule lfp_lowerbound)
   apply (rule_tac [2] A_subset_univ [THEN univ_mono])
  apply safe
  apply (assumption | rule Pair_in_univ list_univ [THEN subsetD])+
  done

lemma term_subset_univ: "A ⊆ univ(B) ⟹ term(A) ⊆ univ(B)"
  apply (rule subset_trans)
   apply (erule term_mono)
  apply (rule term_univ)
  done

lemma term_into_univ: "⟦t ∈ term(A);  A ⊆ univ(B)⟧ ⟹ t ∈ univ(B)"
  by (rule term_subset_univ [THEN subsetD])

text ‹
  \medskip ‹term_rec› -- by ‹Vset› recursion.
›

lemma map_lemma: "⟦l ∈ list(A);  Ord(i);  rank(l)<i⟧
    ⟹ map(λz. (λx ∈ Vset(i).h(x)) ` z, l) = map(h,l)"
  ― ‹\<^term>‹map› works correctly on the underlying list of terms.›
  apply (induct set: list)
   apply simp
  apply (subgoal_tac "rank (a) <i ∧ rank (l) < i")
   apply (simp add: rank_of_Ord)
  apply (simp add: list.con_defs)
  apply (blast dest: rank_rls [THEN lt_trans])
  done

lemma term_rec [simp]: "ts ∈ list(A) ⟹
  term_rec(Apply(a,ts), d) = d(a, ts, map (λz. term_rec(z,d), ts))"
  ― ‹Typing premise is necessary to invoke ‹map_lemma›.›
  apply (rule term_rec_def [THEN def_Vrec, THEN trans])
    unfolding term.con_defs
  apply (simp add: rank_pair2 map_lemma)
  done

lemma term_rec_type:
  assumes t: "t ∈ term(A)"
    and a: "⋀x zs r. ⟦x ∈ A;  zs: list(term(A));
                   r ∈ list(⋃t ∈ term(A). C(t))⟧
                ⟹ d(x, zs, r): C(Apply(x,zs))"
  shows "term_rec(t,d) ∈ C(t)"
  ― ‹Slightly odd typing condition on ‹r› in the second premise!›
  using t
  apply induct
  apply (frule list_CollectD)
  apply (subst term_rec)
   apply (assumption | rule a)+
  apply (erule list.induct)
   apply auto
  done

lemma def_term_rec:
  "⟦⋀t. j(t)≡term_rec(t,d);  ts: list(A)⟧ ⟹
    j(Apply(a,ts)) = d(a, ts, map(λZ. j(Z), ts))"
  apply (simp only:)
  apply (erule term_rec)
  done

lemma term_rec_simple_type [TC]:
  "⟦t ∈ term(A);
      ⋀x zs r. ⟦x ∈ A;  zs: list(term(A));  r ∈ list(C)⟧
                ⟹ d(x, zs, r): C
⟧ ⟹ term_rec(t,d) ∈ C"
  apply (erule term_rec_type)
  apply (drule subset_refl [THEN UN_least, THEN list_mono, THEN subsetD])
  apply simp
  done


text ‹
  \medskip \<^term>‹term_map›.
›

lemma term_map [simp]:
  "ts ∈ list(A) ⟹
    term_map(f, Apply(a, ts)) = Apply(f(a), map(term_map(f), ts))"
  by (rule term_map_def [THEN def_term_rec])

lemma term_map_type [TC]:
    "⟦t ∈ term(A);  ⋀x. x ∈ A ⟹ f(x): B⟧ ⟹ term_map(f,t) ∈ term(B)"
    unfolding term_map_def
  apply (erule term_rec_simple_type)
  apply fast
  done

lemma term_map_type2 [TC]:
    "t ∈ term(A) ⟹ term_map(f,t) ∈ term({f(u). u ∈ A})"
  apply (erule term_map_type)
  apply (erule RepFunI)
  done

text ‹
  \medskip \<^term>‹term_size›.
›

lemma term_size [simp]:
    "ts ∈ list(A) ⟹ term_size(Apply(a, ts)) = succ(list_add(map(term_size, ts)))"
  by (rule term_size_def [THEN def_term_rec])

lemma term_size_type [TC]: "t ∈ term(A) ⟹ term_size(t) ∈ nat"
  by (auto simp add: term_size_def)


text ‹
  \medskip ‹reflect›.
›

lemma reflect [simp]:
    "ts ∈ list(A) ⟹ reflect(Apply(a, ts)) = Apply(a, rev(map(reflect, ts)))"
  by (rule reflect_def [THEN def_term_rec])

lemma reflect_type [TC]: "t ∈ term(A) ⟹ reflect(t) ∈ term(A)"
  by (auto simp add: reflect_def)


text ‹
  \medskip ‹preorder›.
›

lemma preorder [simp]:
    "ts ∈ list(A) ⟹ preorder(Apply(a, ts)) = Cons(a, flat(map(preorder, ts)))"
  by (rule preorder_def [THEN def_term_rec])

lemma preorder_type [TC]: "t ∈ term(A) ⟹ preorder(t) ∈ list(A)"
  by (simp add: preorder_def)


text ‹
  \medskip ‹postorder›.
›

lemma postorder [simp]:
    "ts ∈ list(A) ⟹ postorder(Apply(a, ts)) = flat(map(postorder, ts)) @ [a]"
  by (rule postorder_def [THEN def_term_rec])

lemma postorder_type [TC]: "t ∈ term(A) ⟹ postorder(t) ∈ list(A)"
  by (simp add: postorder_def)


text ‹
  \medskip Theorems about ‹term_map›.
›

declare map_compose [simp]

lemma term_map_ident: "t ∈ term(A) ⟹ term_map(λu. u, t) = t"
  by (induct rule: term_induct_eqn) simp

lemma term_map_compose:
    "t ∈ term(A) ⟹ term_map(f, term_map(g,t)) = term_map(λu. f(g(u)), t)"
  by (induct rule: term_induct_eqn) simp

lemma term_map_reflect:
    "t ∈ term(A) ⟹ term_map(f, reflect(t)) = reflect(term_map(f,t))"
  by (induct rule: term_induct_eqn) (simp add: rev_map_distrib [symmetric])


text ‹
  \medskip Theorems about ‹term_size›.
›

lemma term_size_term_map: "t ∈ term(A) ⟹ term_size(term_map(f,t)) = term_size(t)"
  by (induct rule: term_induct_eqn) simp

lemma term_size_reflect: "t ∈ term(A) ⟹ term_size(reflect(t)) = term_size(t)"
  by (induct rule: term_induct_eqn) (simp add: rev_map_distrib [symmetric] list_add_rev)

lemma term_size_length: "t ∈ term(A) ⟹ term_size(t) = length(preorder(t))"
  by (induct rule: term_induct_eqn) (simp add: length_flat)


text ‹
  \medskip Theorems about ‹reflect›.
›

lemma reflect_reflect_ident: "t ∈ term(A) ⟹ reflect(reflect(t)) = t"
  by (induct rule: term_induct_eqn) (simp add: rev_map_distrib)


text ‹
  \medskip Theorems about preorder.
›

lemma preorder_term_map:
    "t ∈ term(A) ⟹ preorder(term_map(f,t)) = map(f, preorder(t))"
  by (induct rule: term_induct_eqn) (simp add: map_flat)

lemma preorder_reflect_eq_rev_postorder:
    "t ∈ term(A) ⟹ preorder(reflect(t)) = rev(postorder(t))"
  by (induct rule: term_induct_eqn)
    (simp add: rev_app_distrib rev_flat rev_map_distrib [symmetric])

end