# Theory Tree_Forest

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

section ‹Trees and forests, a mutually recursive type definition›

theory Tree_Forest imports ZF begin

subsection ‹Datatype definition›

consts
tree :: "i => i"
forest :: "i => i"
tree_forest :: "i => i"

datatype "tree(A)" = Tcons ("a ∈ A", "f ∈ forest(A)")
and "forest(A)" = Fnil | Fcons ("t ∈ tree(A)", "f ∈ forest(A)")

(* FIXME *)
lemmas tree'induct =
tree_forest.mutual_induct [THEN conjunct1, THEN spec, THEN [2] rev_mp, of concl: _ t, consumes 1]
and forest'induct =
tree_forest.mutual_induct [THEN conjunct2, THEN spec, THEN [2] rev_mp, of concl: _ f, consumes 1]
for t

declare tree_forest.intros [simp, TC]

lemma tree_def: "tree(A) == Part(tree_forest(A), Inl)"
by (simp only: tree_forest.defs)

lemma forest_def: "forest(A) == Part(tree_forest(A), Inr)"
by (simp only: tree_forest.defs)

text ‹
\medskip @{term "tree_forest(A)"} as the union of @{term "tree(A)"}
and @{term "forest(A)"}.
›

lemma tree_subset_TF: "tree(A) ⊆ tree_forest(A)"
apply (unfold tree_forest.defs)
apply (rule Part_subset)
done

lemma treeI [TC]: "x ∈ tree(A) ==> x ∈ tree_forest(A)"
by (rule tree_subset_TF [THEN subsetD])

lemma forest_subset_TF: "forest(A) ⊆ tree_forest(A)"
apply (unfold tree_forest.defs)
apply (rule Part_subset)
done

lemma treeI' [TC]: "x ∈ forest(A) ==> x ∈ tree_forest(A)"
by (rule forest_subset_TF [THEN subsetD])

lemma TF_equals_Un: "tree(A) ∪ forest(A) = tree_forest(A)"
apply (insert tree_subset_TF forest_subset_TF)
apply (auto intro!: equalityI tree_forest.intros elim: tree_forest.cases)
done

lemma tree_forest_unfold:
"tree_forest(A) = (A × forest(A)) + ({0} + tree(A) × forest(A))"
― ‹NOT useful, but interesting \dots›
supply rews = tree_forest.con_defs tree_def forest_def
apply (unfold tree_def forest_def)
apply (fast intro!: tree_forest.intros [unfolded rews, THEN PartD1]
elim: tree_forest.cases [unfolded rews])
done

lemma tree_forest_unfold':
"tree_forest(A) =
A × Part(tree_forest(A), λw. Inr(w)) +
{0} + Part(tree_forest(A), λw. Inl(w)) * Part(tree_forest(A), λw. Inr(w))"
by (rule tree_forest_unfold [unfolded tree_def forest_def])

lemma tree_unfold: "tree(A) = {Inl(x). x ∈ A × forest(A)}"
apply (unfold tree_def forest_def)
apply (rule Part_Inl [THEN subst])
apply (rule tree_forest_unfold' [THEN subst_context])
done

lemma forest_unfold: "forest(A) = {Inr(x). x ∈ {0} + tree(A)*forest(A)}"
apply (unfold tree_def forest_def)
apply (rule Part_Inr [THEN subst])
apply (rule tree_forest_unfold' [THEN subst_context])
done

text ‹
\medskip Type checking for recursor: Not needed; possibly interesting?
›

lemma TF_rec_type:
"[| z ∈ tree_forest(A);
!!x f r. [| x ∈ A;  f ∈ forest(A);  r ∈ C(f)
|] ==> b(x,f,r) ∈ C(Tcons(x,f));
c ∈ C(Fnil);
!!t f r1 r2. [| t ∈ tree(A);  f ∈ forest(A);  r1 ∈ C(t); r2 ∈ C(f)
|] ==> d(t,f,r1,r2) ∈ C(Fcons(t,f))
|] ==> tree_forest_rec(b,c,d,z) ∈ C(z)"
by (induct_tac z) simp_all

lemma tree_forest_rec_type:
"[| !!x f r. [| x ∈ A;  f ∈ forest(A);  r ∈ D(f)
|] ==> b(x,f,r) ∈ C(Tcons(x,f));
c ∈ D(Fnil);
!!t f r1 r2. [| t ∈ tree(A);  f ∈ forest(A);  r1 ∈ C(t); r2 ∈ D(f)
|] ==> d(t,f,r1,r2) ∈ D(Fcons(t,f))
|] ==> (∀t ∈ tree(A).    tree_forest_rec(b,c,d,t) ∈ C(t)) ∧
(∀f ∈ forest(A). tree_forest_rec(b,c,d,f) ∈ D(f))"
― ‹Mutually recursive version.›
apply (unfold Ball_def)
apply (rule tree_forest.mutual_induct)
apply simp_all
done

subsection ‹Operations›

consts
map :: "[i => i, i] => i"
size :: "i => i"
preorder :: "i => i"
list_of_TF :: "i => i"
of_list :: "i => i"
reflect :: "i => i"

primrec
"list_of_TF (Tcons(x,f)) = [Tcons(x,f)]"
"list_of_TF (Fnil) = []"
"list_of_TF (Fcons(t,tf)) = Cons (t, list_of_TF(tf))"

primrec
"of_list([]) = Fnil"
"of_list(Cons(t,l)) = Fcons(t, of_list(l))"

primrec
"map (h, Tcons(x,f)) = Tcons(h(x), map(h,f))"
"map (h, Fnil) = Fnil"
"map (h, Fcons(t,tf)) = Fcons (map(h, t), map(h, tf))"

primrec
"size (Tcons(x,f)) = succ(size(f))"
"size (Fnil) = 0"
"size (Fcons(t,tf)) = size(t) #+ size(tf)"

primrec
"preorder (Tcons(x,f)) = Cons(x, preorder(f))"
"preorder (Fnil) = Nil"
"preorder (Fcons(t,tf)) = preorder(t) @ preorder(tf)"

primrec
"reflect (Tcons(x,f)) = Tcons(x, reflect(f))"
"reflect (Fnil) = Fnil"
"reflect (Fcons(t,tf)) =
of_list (list_of_TF (reflect(tf)) @ Cons(reflect(t), Nil))"

text ‹
\medskip ‹list_of_TF› and ‹of_list›.
›

lemma list_of_TF_type [TC]:
"z ∈ tree_forest(A) ==> list_of_TF(z) ∈ list(tree(A))"
by (induct set: tree_forest) simp_all

lemma of_list_type [TC]: "l ∈ list(tree(A)) ==> of_list(l) ∈ forest(A)"
by (induct set: list) simp_all

text ‹
\medskip ‹map›.
›

lemma
assumes "!!x. x ∈ A ==> h(x): B"
shows map_tree_type: "t ∈ tree(A) ==> map(h,t) ∈ tree(B)"
and map_forest_type: "f ∈ forest(A) ==> map(h,f) ∈ forest(B)"
using assms
by (induct rule: tree'induct forest'induct) simp_all

text ‹
\medskip ‹size›.
›

lemma size_type [TC]: "z ∈ tree_forest(A) ==> size(z) ∈ nat"
by (induct set: tree_forest) simp_all

text ‹
\medskip ‹preorder›.
›

lemma preorder_type [TC]: "z ∈ tree_forest(A) ==> preorder(z) ∈ list(A)"
by (induct set: tree_forest) simp_all

text ‹
\medskip Theorems about ‹list_of_TF› and ‹of_list›.
›

lemma forest_induct [consumes 1, case_names Fnil Fcons]:
"[| f ∈ forest(A);
R(Fnil);
!!t f. [| t ∈ tree(A);  f ∈ forest(A);  R(f) |] ==> R(Fcons(t,f))
|] ==> R(f)"
― ‹Essentially the same as list induction.›
apply (erule tree_forest.mutual_induct
[THEN conjunct2, THEN spec, THEN [2] rev_mp])
apply (rule TrueI)
apply simp
apply simp
done

lemma forest_iso: "f ∈ forest(A) ==> of_list(list_of_TF(f)) = f"
by (induct rule: forest_induct) simp_all

lemma tree_list_iso: "ts: list(tree(A)) ==> list_of_TF(of_list(ts)) = ts"
by (induct set: list) simp_all

text ‹
›

lemma map_ident: "z ∈ tree_forest(A) ==> map(λu. u, z) = z"
by (induct set: tree_forest) simp_all

lemma map_compose:
"z ∈ tree_forest(A) ==> map(h, map(j,z)) = map(λu. h(j(u)), z)"
by (induct set: tree_forest) simp_all

text ‹
›

lemma size_map: "z ∈ tree_forest(A) ==> size(map(h,z)) = size(z)"
by (induct set: tree_forest) simp_all

lemma size_length: "z ∈ tree_forest(A) ==> size(z) = length(preorder(z))"
by (induct set: tree_forest) (simp_all add: length_app)

text ‹