import logic open eq.ops inductive tree (A : Type) := | leaf : A → tree A | node : tree A → tree A → tree A namespace tree inductive direct_subterm {A : Type} : tree A → tree A → Prop := | node_l : Π (l r : tree A), direct_subterm l (node l r) | node_r : Π (l r : tree A), direct_subterm r (node l r) definition direct_subterm.wf {A : Type} : well_founded (@direct_subterm A) := well_founded.intro (λ t : tree A, tree.rec_on t (λ (a : A), acc.intro (leaf a) (λ (s : tree A) (H : direct_subterm s (leaf a)), have gen : ∀ r : tree A, direct_subterm s r → r = leaf a → acc direct_subterm s, from λ r H, direct_subterm.rec_on H (λ l r e, tree.no_confusion e) (λ l r e, tree.no_confusion e), gen (leaf a) H rfl)) (λ (l r : tree A) (ihl : acc direct_subterm l) (ihr : acc direct_subterm r), acc.intro (node l r) (λ (s : tree A) (H : direct_subterm s (node l r)), have gen : ∀ n₁ : tree A, direct_subterm s n₁ → node l r = n₁ → acc direct_subterm s, from λ n₁ H, direct_subterm.rec_on H (λ (l' r' : tree A) (Heq : node l r = node l' r'), tree.no_confusion Heq (λ leq req, eq.rec_on leq ihl)) (λ (l' r' : tree A) (Heq : node l r = node l' r'), tree.no_confusion Heq (λ leq req, eq.rec_on req ihr)), gen (node l r) H rfl))) definition direct_subterm.wf₂ {A : Type} : well_founded (@direct_subterm A) := begin constructor, intro t, induction t, repeat (constructor; intro y hlt; cases hlt; repeat assumption) end definition subterm {A : Type} : tree A → tree A → Prop := tc (@direct_subterm A) definition subterm.wf {A : Type} : well_founded (@subterm A) := tc.wf (@direct_subterm.wf A) example : subterm (leaf 2) (node (leaf 1) (leaf 2)) := !tc.base !direct_subterm.node_r example : subterm (leaf 2) (node (node (leaf 1) (leaf 2)) (leaf 3)) := have s₁ : subterm (leaf 2) (node (leaf 1) (leaf 2)), from !tc.base !direct_subterm.node_r, have s₂ : subterm (node (leaf 1) (leaf 2)) (node (node (leaf 1) (leaf 2)) (leaf 3)), from !tc.base !direct_subterm.node_l, !tc.trans s₁ s₂ end tree