# Theory Term

theory Term
imports ZF
```(*  Title:      ZF/Induct/Term.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

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)"
apply (unfold 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›
apply (unfold 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 (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])
apply (unfold term.con_defs)
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)"
apply (unfold 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"

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)"

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)"

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)"

text ‹
›

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 ‹
›

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)"

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

text ‹
›

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

text ‹