99 lines
4.2 KiB
Text
99 lines
4.2 KiB
Text
import data.nat data.sum data.sigma data.bool
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open nat sigma algebra
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inductive tree (A : Type) : Type :=
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| node : A → forest A → tree A
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with forest : Type :=
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| nil : forest A
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| cons : tree A → forest A → forest A
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namespace manual
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check tree.rec_on
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definition tree.height {A : Type} (t : tree A) : nat :=
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tree.rec_on t
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(λ (a : A) (f : forest A) (ih : nat), succ ih)
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zero
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(λ (t : tree A) (f : forest A) (ih₁ : nat) (ih₂ : nat), succ (max ih₁ ih₂))
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definition forest.height {A : Type} (f : forest A) : nat :=
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forest.rec_on f
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(λ (a : A) (f : forest A) (ih : nat), succ ih)
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zero
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(λ (t : tree A) (f : forest A) (ih₁ : nat) (ih₂ : nat), succ (max ih₁ ih₂))
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definition tree_forest (A : Type) := sum (tree A) (forest A)
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definition tree_forest_height {A : Type} (t : tree_forest A) : nat :=
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sum.rec_on t (λ t, tree.height t) (λ f, forest.height f)
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definition tree_forest.subterm {A : Type} : tree_forest A → tree_forest A → Prop :=
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inv_image lt tree_forest_height
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definition tree_forest.subterm.wf [instance] (A : Type) : well_founded (@tree_forest.subterm A) :=
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inv_image.wf tree_forest_height lt.wf
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local infix `≺`:50 := tree_forest.subterm
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definition tree_forest.height_lt.node {A : Type} (a : A) (f : forest A) : sum.inr f ≺ sum.inl (tree.node a f) :=
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have aux : forest.height f < tree.height (tree.node a f), from
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lt.base (forest.height f),
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aux
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definition tree_forest.height_lt.cons₁ {A : Type} (t : tree A) (f : forest A) : sum.inl t ≺ sum.inr (forest.cons t f) :=
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have aux : tree.height t < forest.height (forest.cons t f), from
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lt_succ_of_le (le_max_left _ _),
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aux
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definition tree_forest.height_lt.cons₂ {A : Type} (t : tree A) (f : forest A) : sum.inr f ≺ sum.inr (forest.cons t f) :=
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have aux : forest.height f < forest.height (forest.cons t f), from
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lt_succ_of_le (le_max_right _ _),
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aux
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definition kind {A : Type} (t : tree_forest A) : bool :=
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sum.cases_on t (λ t, bool.tt) (λ f, bool.ff)
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definition map.res {A : Type} (B : Type) (t : tree_forest A) :=
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Σ r : tree_forest B, kind r = kind t
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set_option find_decl.expensive true
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find_decl bool.ff ≠ bool.tt
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-- map using well-founded recursion. We could have used the default recursor.
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-- this is just a test for the definitional package
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definition map.F {A B : Type₁} (f : A → B) (tf₁ : tree_forest A) : (Π tf₂ : tree_forest A, tf₂ ≺ tf₁ → map.res B tf₂) → map.res B tf₁ :=
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sum.cases_on tf₁
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(λ t : tree A, tree.cases_on t
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(λ a₁ f₁ (r : Π (tf₂ : tree_forest A), tf₂ ≺ sum.inl (tree.node a₁ f₁) → map.res B tf₂),
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show map.res B (sum.inl (tree.node a₁ f₁)), from
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have rf₁ : map.res B (sum.inr f₁), from r (sum.inr f₁) (tree_forest.height_lt.node a₁ f₁),
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have nf₁ : forest B, from sum.cases_on (pr₁ rf₁)
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(λf (h : kind (sum.inl f) = kind (sum.inr f₁)), absurd (eq.symm h) bool.ff_ne_tt)
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(λf h, f)
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(pr₂ rf₁),
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sigma.mk (sum.inl (tree.node (f a₁) nf₁)) rfl))
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(λ f : forest A, forest.cases_on f
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(λ r : Π (tf₂ : tree_forest A), tf₂ ≺ sum.inr (forest.nil A) → map.res B tf₂,
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show map.res B (sum.inr (forest.nil A)), from
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sigma.mk (sum.inr (forest.nil B)) rfl)
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(λ t₁ f₁ (r : Π (tf₂ : tree_forest A), tf₂ ≺ sum.inr (forest.cons t₁ f₁) → map.res B tf₂),
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show map.res B (sum.inr (forest.cons t₁ f₁)), from
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have rt₁ : map.res B (sum.inl t₁), from r (sum.inl t₁) (tree_forest.height_lt.cons₁ t₁ f₁),
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have rf₁ : map.res B (sum.inr f₁), from r (sum.inr f₁) (tree_forest.height_lt.cons₂ t₁ f₁),
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have nt₁ : tree B, from sum.cases_on (pr₁ rt₁)
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(λ t h, t)
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(λ f h, absurd h bool.ff_ne_tt)
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(pr₂ rt₁),
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have nf₁ : forest B, from sum.cases_on (pr₁ rf₁)
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(λf (h : kind (sum.inl f) = kind (sum.inr f₁)), absurd (eq.symm h) bool.ff_ne_tt)
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(λf h, f)
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(pr₂ rf₁),
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sigma.mk (sum.inr (forest.cons nt₁ nf₁)) rfl))
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definition map {A B : Type₁} (f : A → B) (tf : tree_forest A) : map.res B tf :=
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well_founded.fix (@map.F A B f) tf
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eval map succ (sum.inl (tree.node 2 (forest.cons (tree.node 1 (forest.nil nat)) (forest.nil nat))))
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end manual
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