39 lines
1.7 KiB
Text
39 lines
1.7 KiB
Text
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import logic data.nat.basic
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open nat
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inductive inftree (A : Type) : Type :=
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leaf : A → inftree A,
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node : (nat → inftree A) → 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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intro₁ : Π (f : nat → inftree A) (a : nat) (t : inftree A), dsub (f a) (node f t),
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intro₂ : Π (f : nat → inftree A) (a : nat) (t : inftree A), dsub t (node f t)
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definition dsub.node.acc {A : Type} (f : nat → inftree A) (hf : ∀a, acc dsub (f a))
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(t : inftree A) (ht : acc dsub t) : acc dsub (node f t) :=
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acc.intro (node f t) (λ (y : inftree A) (hlt : dsub y (node f t)),
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have aux : ∀ z, dsub y z → node f t = z → acc dsub y, from
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λ z hlt, dsub.rec_on hlt
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(λ f₁ n t₁ (heq : (node f t = node f₁ t₁)),
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inftree.no_confusion heq (λ e₁ e₂, eq.rec_on e₁ (hf n)))
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(λ f₁ n t₁ (heq : (node f t = node f₁ t₁)),
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inftree.no_confusion heq (λ e₁ e₂, eq.rec_on e₂ ht)),
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aux (node f t) 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
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(λ f n t (heq : leaf a = node f t), inftree.no_confusion heq)
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(λ f n t (heq : leaf a = node f t), 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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rec_on t
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(λ a, dsub.leaf.acc a)
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(λ f t (ihf :∀a, acc dsub (f a)) (iht : acc dsub t), dsub.node.acc f ihf t iht))
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end inftree
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