113 lines
4.8 KiB
Text
113 lines
4.8 KiB
Text
import data.unit data.prod
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inductive ftree (A : Type) (B : Type) : Type :=
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leafa : ftree A B,
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node : (A → B → ftree A B) → (B → ftree A B) → ftree A B
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set_option pp.universes true
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check @ftree
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namespace ftree
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namespace manual
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definition below.{l l₁ l₂} {A : Type.{l₁}} {B : Type.{l₂}} (C : ftree A B → Type.{l+1}) (t : ftree A B)
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: Type.{max l₁ l₂ (l+1)} :=
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@rec.{(max l₁ l₂ (l+1))+1 l₁ l₂}
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A
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B
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(λ t : ftree A B, Type.{max l₁ l₂ (l+1)})
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unit.{max l₁ l₂ (l+1)}
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(λ (f₁ : A → B → ftree A B)
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(f₂ : B → ftree A B)
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(r₁ : Π (a : A) (b : B), Type.{max l₁ l₂ (l+1)})
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(r₂ : Π (b : B), Type.{max l₁ l₂ (l+1)}),
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let fc₁ : Type.{max l₁ l₂ (l+1)} := Π (a : A) (b : B), C (f₁ a b) in
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let fc₂ : Type.{max l₂ l+1} := Π (b : B), C (f₂ b) in
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let fr₁ : Type.{max l₁ l₂ (l+1)} := Π (a : A) (b : B), r₁ a b in
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let fr₂ : Type.{max l₁ l₂ (l+1)} := Π (b : B), r₂ b in
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let p₁ : Type.{max l₁ l₂ (l+1)} := prod.{(max l₁ l₂ (l+1)) (max l₁ l₂ (l+1))} fc₁ fr₁ in
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let p₂ : Type.{max l₁ l₂ (l+1)} := prod.{(max l₂ (l+1)) (max l₁ l₂ (l+1))} fc₂ fr₂ in
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prod.{(max l₁ l₂ (l+1)) (max l₁ l₂ (l+1))} p₁ p₂)
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t
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definition pbelow.{l₁ l₂} {A : Type.{l₁}} {B : Type.{l₂}} (C : ftree A B → Prop) (t : ftree A B)
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: Prop :=
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@rec.{1 l₁ l₂}
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A
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B
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(λ t : ftree A B, Prop)
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true
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(λ (f₁ : A → B → ftree A B)
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(f₂ : B → ftree A B)
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(r₁ : Π (a : A) (b : B), Prop)
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(r₂ : Π (b : B), Prop),
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let fc₁ : Prop := ∀ (a : A) (b : B), C (f₁ a b) in
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let fc₂ : Prop := ∀ (b : B), C (f₂ b) in
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let fr₁ : Prop := ∀ (a : A) (b : B), r₁ a b in
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let fr₂ : Prop := ∀ (b : B), r₂ b in
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let p₁ : Prop := fc₁ ∧ fr₁ in
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let p₂ : Prop := fc₂ ∧ fr₂ in
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p₁ ∧ p₂)
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t
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definition brec_on.{l l₁ l₂} {A : Type.{l₁}} {B : Type.{l₂}} {C : ftree A B → Type.{l+1}}
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(t : ftree A B)
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(F : Π (t : ftree A B), @below A B C t → C t)
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: C t :=
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have gen : prod.{(l+1) (max l₁ l₂ (l+1))} (C t) (@below A B C t), from
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@rec.{(max l₁ l₂ (l+1)) l₁ l₂}
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A
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B
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(λ t : ftree A B, prod.{(l+1) (max l₁ l₂ (l+1))} (C t) (@below A B C t))
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(have b : @below A B C (leafa A B), from
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unit.star.{max l₁ l₂ (l+1)},
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have c : C (leafa A B), from
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F (leafa A B) b,
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prod.mk.{(l+1) (max l₁ l₂ (l+1))} c b)
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(λ (f₁ : A → B → ftree A B)
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(f₂ : B → ftree A B)
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(r₁ : Π (a : A) (b : B), prod.{(l+1) (max l₁ l₂ (l+1))} (C (f₁ a b)) (@below A B C (f₁ a b)))
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(r₂ : Π (b : B), prod.{(l+1) (max l₁ l₂ (l+1))} (C (f₂ b)) (@below A B C (f₂ b))),
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let fc₁ : Π (a : A) (b : B), C (f₁ a b) := λ (a : A) (b : B), prod.pr1 (r₁ a b) in
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let fr₁ : Π (a : A) (b : B), @below A B C (f₁ a b) := λ (a : A) (b : B), prod.pr2 (r₁ a b) in
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let fc₂ : Π (b : B), C (f₂ b) := λ (b : B), prod.pr1 (r₂ b) in
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let fr₂ : Π (b : B), @below A B C (f₂ b) := λ (b : B), prod.pr2 (r₂ b) in
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have b : @below A B C (node f₁ f₂), from
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prod.mk (prod.mk fc₁ fr₁) (prod.mk fc₂ fr₂),
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have c : C (node f₁ f₂), from
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F (node f₁ f₂) b,
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prod.mk c b)
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t,
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prod.pr1 gen
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definition binduction_on.{l₁ l₂} {A : Type.{l₁}} {B : Type.{l₂}} {C : ftree A B → Prop}
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(t : ftree A B)
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(F : Π (t : ftree A B), @ibelow A B C t → C t)
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: C t :=
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have gen : C t ∧ @ibelow A B C t, from
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@rec.{0 l₁ l₂}
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A
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B
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(λ t : ftree A B, C t ∧ @ibelow A B C t)
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(have b : @ibelow A B C (leafa A B), from
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true.intro,
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have c : C (leafa A B), from
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F (leafa A B) b,
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and.intro c b)
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(λ (f₁ : A → B → ftree A B)
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(f₂ : B → ftree A B)
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(r₁ : ∀ (a : A) (b : B), C (f₁ a b) ∧ @ibelow A B C (f₁ a b))
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(r₂ : ∀ (b : B), C (f₂ b) ∧ @ibelow A B C (f₂ b)),
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let fc₁ : ∀ (a : A) (b : B), C (f₁ a b) := λ (a : A) (b : B), and.elim_left (r₁ a b) in
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let fr₁ : ∀ (a : A) (b : B), @ibelow A B C (f₁ a b) := λ (a : A) (b : B), and.elim_right (r₁ a b) in
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let fc₂ : ∀ (b : B), C (f₂ b) := λ (b : B), and.elim_left (r₂ b) in
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let fr₂ : ∀ (b : B), @ibelow A B C (f₂ b) := λ (b : B), and.elim_right (r₂ b) in
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have b : @ibelow A B C (node f₁ f₂), from
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and.intro (and.intro fc₁ fr₁) (and.intro fc₂ fr₂),
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have c : C (node f₁ f₂), from
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F (node f₁ f₂) b,
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and.intro c b)
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t,
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and.elim_left gen
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end manual
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end ftree
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