(* Title: ZF/Induct/Term.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
*)
section \<open>Terms over an alphabet\<close>
theory Term imports ZF begin
text \<open>
Illustrates the list functor (essentially the same type as in \<open>Trees_Forest\<close>).
\<close>
consts
"term" :: "i => i"
datatype "term(A)" = Apply ("a \<in> A", "l \<in> 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, \<lambda>t g. term_case(\<lambda>x zs. d(x, zs, map(\<lambda>z. g`z, zs)), t))"
definition
term_map :: "[i => i, i] => i" where
"term_map(f,t) == term_rec(t, \<lambda>x zs rs. Apply(f(x), rs))"
definition
term_size :: "i => i" where
"term_size(t) == term_rec(t, \<lambda>x zs rs. succ(list_add(rs)))"
definition
reflect :: "i => i" where
"reflect(t) == term_rec(t, \<lambda>x zs rs. Apply(x, rev(rs)))"
definition
preorder :: "i => i" where
"preorder(t) == term_rec(t, \<lambda>x zs rs. Cons(x, flat(rs)))"
definition
postorder :: "i => i" where
"postorder(t) == term_rec(t, \<lambda>x 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 \<in> term(A);
!!x. [| x \<in> A |] ==> P(Apply(x,Nil));
!!x z zs. [| x \<in> A; z \<in> term(A); zs: list(term(A)); P(Apply(x,zs))
|] ==> P(Apply(x, Cons(z,zs)))
|] ==> P(t)"
\<comment> \<open>Induction on \<^term>\<open>term(A)\<close> followed by induction on \<^term>\<open>list\<close>.\<close>
apply (induct_tac t)
apply (erule list.induct)
apply (auto dest: list_CollectD)
done
lemma term_induct_eqn [consumes 1, case_names Apply]:
"[| t \<in> term(A);
!!x zs. [| x \<in> A; zs: list(term(A)); map(f,zs) = map(g,zs) |] ==>
f(Apply(x,zs)) = g(Apply(x,zs))
|] ==> f(t) = g(t)"
\<comment> \<open>Induction on \<^term>\<open>term(A)\<close> to prove an equation.\<close>
apply (induct_tac t)
apply (auto dest: map_list_Collect list_CollectD)
done
text \<open>
\medskip Lemmas to justify using \<^term>\<open>term\<close> in other recursive
type definitions.
\<close>
lemma term_mono: "A \<subseteq> B ==> term(A) \<subseteq> 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)) \<subseteq> univ(A)"
\<comment> \<open>Easily provable by induction also\<close>
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 \<subseteq> univ(B) ==> term(A) \<subseteq> univ(B)"
apply (rule subset_trans)
apply (erule term_mono)
apply (rule term_univ)
done
lemma term_into_univ: "[| t \<in> term(A); A \<subseteq> univ(B) |] ==> t \<in> univ(B)"
by (rule term_subset_univ [THEN subsetD])
text \<open>
\medskip \<open>term_rec\<close> -- by \<open>Vset\<close> recursion.
\<close>
lemma map_lemma: "[| l \<in> list(A); Ord(i); rank(l)<i |]
==> map(\<lambda>z. (\<lambda>x \<in> Vset(i).h(x)) ` z, l) = map(h,l)"
\<comment> \<open>\<^term>\<open>map\<close> works correctly on the underlying list of terms.\<close>
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 \<in> list(A) ==>
term_rec(Apply(a,ts), d) = d(a, ts, map (\<lambda>z. term_rec(z,d), ts))"
\<comment> \<open>Typing premise is necessary to invoke \<open>map_lemma\<close>.\<close>
apply (rule term_rec_def [THEN def_Vrec, THEN trans])
apply (unfold term.con_defs)
apply (simp add: rank_pair2 map_lemma)
done
lemma term_rec_type:
assumes t: "t \<in> term(A)"
and a: "!!x zs r. [| x \<in> A; zs: list(term(A));
r \<in> list(\<Union>t \<in> term(A). C(t)) |]
==> d(x, zs, r): C(Apply(x,zs))"
shows "term_rec(t,d) \<in> C(t)"
\<comment> \<open>Slightly odd typing condition on \<open>r\<close> in the second premise!\<close>
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(\<lambda>Z. j(Z), ts))"
apply (simp only:)
apply (erule term_rec)
done
lemma term_rec_simple_type [TC]:
"[| t \<in> term(A);
!!x zs r. [| x \<in> A; zs: list(term(A)); r \<in> list(C) |]
==> d(x, zs, r): C
|] ==> term_rec(t,d) \<in> C"
apply (erule term_rec_type)
apply (drule subset_refl [THEN UN_least, THEN list_mono, THEN subsetD])
apply simp
done
text \<open>
\medskip \<^term>\<open>term_map\<close>.
\<close>
lemma term_map [simp]:
"ts \<in> 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 \<in> term(A); !!x. x \<in> A ==> f(x): B |] ==> term_map(f,t) \<in> term(B)"
apply (unfold term_map_def)
apply (erule term_rec_simple_type)
apply fast
done
lemma term_map_type2 [TC]:
"t \<in> term(A) ==> term_map(f,t) \<in> term({f(u). u \<in> A})"
apply (erule term_map_type)
apply (erule RepFunI)
done
text \<open>
\medskip \<^term>\<open>term_size\<close>.
\<close>
lemma term_size [simp]:
"ts \<in> 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 \<in> term(A) ==> term_size(t) \<in> nat"
by (auto simp add: term_size_def)
text \<open>
\medskip \<open>reflect\<close>.
\<close>
lemma reflect [simp]:
"ts \<in> 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 \<in> term(A) ==> reflect(t) \<in> term(A)"
by (auto simp add: reflect_def)
text \<open>
\medskip \<open>preorder\<close>.
\<close>
lemma preorder [simp]:
"ts \<in> 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 \<in> term(A) ==> preorder(t) \<in> list(A)"
by (simp add: preorder_def)
text \<open>
\medskip \<open>postorder\<close>.
\<close>
lemma postorder [simp]:
"ts \<in> list(A) ==> postorder(Apply(a, ts)) = flat(map(postorder, ts)) @ [a]"
by (rule postorder_def [THEN def_term_rec])
lemma postorder_type [TC]: "t \<in> term(A) ==> postorder(t) \<in> list(A)"
by (simp add: postorder_def)
text \<open>
\medskip Theorems about \<open>term_map\<close>.
\<close>
declare map_compose [simp]
lemma term_map_ident: "t \<in> term(A) ==> term_map(\<lambda>u. u, t) = t"
by (induct rule: term_induct_eqn) simp
lemma term_map_compose:
"t \<in> term(A) ==> term_map(f, term_map(g,t)) = term_map(\<lambda>u. f(g(u)), t)"
by (induct rule: term_induct_eqn) simp
lemma term_map_reflect:
"t \<in> 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 \<open>
\medskip Theorems about \<open>term_size\<close>.
\<close>
lemma term_size_term_map: "t \<in> term(A) ==> term_size(term_map(f,t)) = term_size(t)"
by (induct rule: term_induct_eqn) simp
lemma term_size_reflect: "t \<in> 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 \<in> term(A) ==> term_size(t) = length(preorder(t))"
by (induct rule: term_induct_eqn) (simp add: length_flat)
text \<open>
\medskip Theorems about \<open>reflect\<close>.
\<close>
lemma reflect_reflect_ident: "t \<in> term(A) ==> reflect(reflect(t)) = t"
by (induct rule: term_induct_eqn) (simp add: rev_map_distrib)
text \<open>
\medskip Theorems about preorder.
\<close>
lemma preorder_term_map:
"t \<in> 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 \<in> 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