38 lines
1.5 KiB
Text
38 lines
1.5 KiB
Text
import logic data.nat.basic
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open nat
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inductive inftree (A : Type) :=
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| leaf : A → inftree A
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| node : (nat → inftree A) → inftree A
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namespace inftree
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inductive dsub {A : Type} : inftree A → inftree A → Prop :=
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intro : Π (f : nat → inftree A) (a : nat), dsub (f a) (node f)
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definition dsub.node.acc {A : Type} (f : nat → inftree A) (H : ∀a, acc dsub (f a)) : acc dsub (node f) :=
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acc.intro (node f) (λ (y : inftree A) (hlt : dsub y (node f)),
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have aux : ∀ z, dsub y z → node f = z → acc dsub y, from
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λ z hlt, dsub.rec_on hlt (λ (f₁ : nat → inftree A) (a : nat) (eq₁ : node f = node f₁),
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inftree.no_confusion eq₁ (λe, eq.rec_on e (H a))),
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aux (node f) hlt rfl)
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definition dsub.leaf.acc {A : Type} (a : A) : acc dsub (leaf a) :=
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acc.intro (leaf a) (λ (y : inftree A) (hlt : dsub y (leaf a)),
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have aux : ∀ z, dsub y z → leaf a = z → acc dsub y, from
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λz hlt, dsub.rec_on hlt (λ f n (heq : leaf a = node f), inftree.no_confusion heq),
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aux (leaf a) hlt rfl)
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definition dsub.wf (A : Type) : well_founded (@dsub A) :=
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well_founded.intro (λ (t : inftree A),
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inftree.rec_on t
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(λ a, dsub.leaf.acc a)
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(λ f (ih :∀a, acc dsub (f a)), dsub.node.acc f ih))
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theorem dsub.wf₂ (A : Type) : well_founded (@dsub A) :=
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begin
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constructor, intro t, induction t with a f ih ,
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{constructor, intro y hlt, cases hlt},
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{constructor, intro y hlt, cases hlt, exact ih a}
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end
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end inftree
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