lean2/tests/lean/run/inf_tree2.lean
2015-02-11 14:09:25 -08:00

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import logic data.nat.basic
open nat
inductive inftree (A : Type) : Type :=
leaf : A → inftree A,
node : (nat → inftree A) → inftree A → inftree A
namespace inftree
inductive dsub {A : Type} : inftree A → inftree A → Prop :=
intro₁ : Π (f : nat → inftree A) (a : nat) (t : inftree A), dsub (f a) (node f t),
intro₂ : Π (f : nat → inftree A) (a : nat) (t : inftree A), dsub t (node f t)
definition dsub.node.acc {A : Type} (f : nat → inftree A) (hf : ∀a, acc dsub (f a))
(t : inftree A) (ht : acc dsub t) : acc dsub (node f t) :=
acc.intro (node f t) (λ (y : inftree A) (hlt : dsub y (node f t)),
by cases hlt; apply (hf a); apply ht)
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)),
by cases hlt)
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 t (ihf :∀a, acc dsub (f a)) (iht : acc dsub t), dsub.node.acc f ihf t iht))
end inftree