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