import logic data.nat.basic open nat inductive inftree (A : Type) := | leaf : A → inftree A | node : (nat → inftree A) → inftree A namespace inftree inductive dsub {A : Type} : inftree A → inftree A → Prop := intro : Π (f : nat → inftree A) (a : nat), dsub (f a) (node f) definition dsub.node.acc {A : Type} (f : nat → inftree A) (H : ∀a, acc dsub (f a)) : acc dsub (node f) := acc.intro (node f) (λ (y : inftree A) (hlt : dsub y (node f)), have aux : ∀ z, dsub y z → node f = z → acc dsub y, from λ z hlt, dsub.rec_on hlt (λ (f₁ : nat → inftree A) (a : nat) (eq₁ : node f = node f₁), inftree.no_confusion eq₁ (λe, eq.rec_on e (H a))), aux (node f) hlt rfl) definition dsub.leaf.acc {A : Type} (a : A) : acc dsub (leaf a) := acc.intro (leaf a) (λ (y : inftree A) (hlt : dsub y (leaf a)), have aux : ∀ z, dsub y z → leaf a = z → acc dsub y, from λz hlt, dsub.rec_on hlt (λ f n (heq : leaf a = node f), inftree.no_confusion heq), aux (leaf a) hlt rfl) definition dsub.wf (A : Type) : well_founded (@dsub A) := well_founded.intro (λ (t : inftree A), inftree.rec_on t (λ a, dsub.leaf.acc a) (λ f (ih :∀a, acc dsub (f a)), dsub.node.acc f ih)) end inftree