56 lines
2.4 KiB
Text
56 lines
2.4 KiB
Text
instance {α : Type u} : HAppend (Fin m → α) (Fin n → α) (Fin (m + n) → α) where
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hAppend a b i := if h : i < m then a ⟨i, h⟩ else b ⟨i - m, sorry⟩
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def empty : Fin 0 → Nat := (nomatch ·)
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theorem append_empty (x : Fin i → Nat) : x ++ empty = x :=
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funext fun i => dif_pos _
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opaque f : (Fin 0 → Nat) → Prop
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example : f (empty ++ empty) = f empty := by simp only [append_empty] -- should work
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@[congr] theorem Array.get_congr (as bs : Array α) (h : as = bs) (i : Fin as.size) (j : Nat) (hi : i = j) :
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as.get i = bs.get ⟨j, h ▸ hi ▸ i.2⟩ := by
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subst bs; subst j; rfl
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example (as : Array Nat) (h : 0 + x < as.size) :
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as.get ⟨0 + x, h⟩ = as.get ⟨x, Nat.zero_add x ▸ h⟩ := by
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simp -- should work
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example (as : Array (Nat → Nat)) (h : 0 + x < as.size) :
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as.get ⟨0 + x, h⟩ i = as.get ⟨x, Nat.zero_add x ▸ h⟩ i := by
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simp -- should also work
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example (as : Array (Nat → Nat)) (h : 0 + x < as.size) :
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as.get ⟨0 + x, h⟩ i = as.get ⟨x, Nat.zero_add x ▸ h⟩ (0+i) := by
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simp -- should also work
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example [Decidable p] : decide (p ∧ True) = decide p := by simp -- should work
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def Pi.single [DecidableEq ι] {f : ι → Type u} [∀ i, Inhabited (f i)] (i : ι) (x : f i) : ∀ i, f i :=
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fun j => if h : j = i then h ▸ x else default
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structure Set (α : Type u) where of :: mem : α → Prop
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instance : CoeSort (Set α) (Type u) where coe s := Subtype s.mem
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@[congr]
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theorem dep_congr [DecidableEq ι] {p : ι → Set α} [∀ i, Inhabited (p i)] :
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∀ {i j} (h : i = j) (x : p i) (y : α) (hx : x = y), Pi.single (f := (p ·)) i x = Pi.single (f := (p ·)) j ⟨y, hx ▸ h ▸ x.2⟩
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| _, _, rfl, _, _, rfl => rfl
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theorem aux {p : Nat → Set Nat} {i j y : Nat} (x : p j) (h₁ : x = y) (h₂ : i = j) : Set.mem (p i) y := by
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have := x.2
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subst h₁ h₂
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assumption
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example {p : Nat → Set Nat} [∀ i, Inhabited (p i)] (i : Nat) (x : p (0 + i)) (y : Nat) : Pi.single (f := (p ·)) (0 + i) x = Pi.single (f := (p ·)) i ⟨x, aux x rfl (Nat.zero_add i).symm ⟩ := by
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simp
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def Submodule (α : Type u) [OfNat α 0] := { s : Set α // s.mem 0 }
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instance [OfNat α 0] : CoeSort (Submodule α) (Type u) where coe s := s.1
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instance [OfNat α 0] (p : Submodule α) : Inhabited p where default := ⟨0, p.2⟩
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example (p : Nat → Submodule Nat) :
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Pi.single (f := (p ·)) (x - x) ⟨0, (p ..).2⟩ = Pi.single 0 ⟨0, (p ..).2⟩ := by
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simp only [Nat.sub_self] -- should work
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