3a408e0e54
This PR changes the signature of `Array.get` to take a Nat and a proof, rather than a `Fin`, for consistency with the rest of the (planned) Array API. Note that because of bootstrapping issues we can't provide `get_elem_tactic` as an autoparameter for the proof. As users will mostly use the `xs[i]` notation provided by `GetElem`, this hopefully isn't a problem. We may restore `Fin` based versions, either here or downstream, as needed, but they won't be the "main" functions. --------- Co-authored-by: David Thrane Christiansen <david@davidchristiansen.dk>
65 lines
2.2 KiB
Text
65 lines
2.2 KiB
Text
universe u
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def f1 (n m : Nat) (x : Fin n) (h : n = m) : Fin m :=
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h ▸ x
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def f2 (n m : Nat) (x : Fin n) (h : m = n) : Fin m :=
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h ▸ x
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theorem ex1 {α : Sort u} {a b c : α} (h₁ : a = b) (h₂ : b = c) : a = c :=
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h₂ ▸ h₁
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theorem ex2 {α : Sort u} {a b : α} (h : a = b) : b = a :=
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h ▸ rfl
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theorem ex3 {α : Sort u} {a b c : α} (r : α → α → Prop) (h₁ : r a b) (h₂ : b = c) : r a c :=
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h₂ ▸ h₁
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theorem ex3b {α : Sort u} {a b c : α} (r : α → α → Prop) (h₁ : r a b) (h₂ : b = c) : r a c :=
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h₂.symm ▸ h₁
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theorem ex3c {α : Sort u} {a b c : α} (r : α → α → Prop) (h₁ : r a b) (h₂ : b = c) : r a c :=
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h₂.symm.symm ▸ h₁
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theorem ex4 {α : Sort u} {a b c : α} (r : α → α → Prop) (h₁ : a = b) (h₂ : r b c) : r a c :=
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h₁ ▸ h₂
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theorem ex5 {p : Prop} (h : p = True) : p :=
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h ▸ trivial
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theorem ex6 {p : Prop} (h : p = False) : ¬p :=
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fun hp => h ▸ hp
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theorem ex7 {α} {a b c d : α} (h₁ : a = c) (h₂ : b = d) (h₃ : c ≠ d) : a ≠ b :=
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h₁ ▸ h₂ ▸ h₃
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theorem ex8 (n m k : Nat) (h : Nat.succ n + m = Nat.succ n + k) : Nat.succ (n + m) = Nat.succ (n + k) :=
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Nat.succ_add .. ▸ Nat.succ_add .. ▸ h
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theorem ex9 (a b : Nat) (h₁ : a = a + b) (h₂ : a = b) : a = b + a :=
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h₂ ▸ h₁
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theorem ex10 (a b : Nat) (h : a = b) : b = a :=
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h ▸ rfl
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def ex11 {α : Type u} {n : Nat} (a : Array α) (i : Nat) (h₁ : a.size = n) (h₂ : i < n) : α :=
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a[i]
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theorem ex12 {α : Type u} {n : Nat}
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(a b : Array α)
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(hsz₁ : a.size = n) (hsz₂ : b.size = n)
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(h : ∀ (i : Nat) (hi : i < n), a.getLit i hsz₁ hi = b.getLit i hsz₂ hi) : a = b :=
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Array.ext a b (hsz₁.trans hsz₂.symm) fun i hi₁ hi₂ => h i (hsz₁ ▸ hi₁)
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def toArrayLit {α : Type u} (a : Array α) (n : Nat) (hsz : a.size = n) : Array α :=
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List.toArray $ Array.toListLitAux a n hsz n (hsz ▸ Nat.le_refl _) []
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partial def isEqvAux {α} (a b : Array α) (hsz : a.size = b.size) (p : α → α → Bool) (i : Nat) : Bool :=
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if h : i < a.size then
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let aidx : Fin a.size := ⟨i, h⟩
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let bidx : Fin b.size := ⟨i, hsz ▸ h⟩
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match p a[aidx] b[bidx] with
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| true => isEqvAux a b hsz p (i+1)
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| false => false
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else
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true
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