(* Title: ZF/Induct/Tree_Forest.thy Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1994 University of Cambridge *) 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 ‹ \medskip Theorems about ‹map›. › 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 ‹ \medskip Theorems about ‹size›. › 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 ‹ \medskip Theorems about ‹preorder›. › lemma preorder_map: "z ∈ tree_forest(A) ==> preorder(map(h,z)) = List_ZF.map(h, preorder(z))" by (induct set: tree_forest) (simp_all add: map_app_distrib) end