3f548edcd7
When updating Std, be careful that not every lemma has been upstreamed, so we need to be careful to only delete things that have already been declared.
78 lines
2.4 KiB
Text
78 lines
2.4 KiB
Text
universe u
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def concat {α} : List α → α → List α
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| [], a => [a]
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| x::xs, a => x :: concat xs a
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def last {α} : (xs : List α) → xs ≠ [] → α
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| [], h => by contradiction
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| [a], h => a
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| _::a::as, h => last (a::as) (fun h => by injection h)
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def dropLast {α} : List α → List α
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| [] => []
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| [a] => []
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| a::as => a :: dropLast as
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variable {α}
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theorem concatEq (xs : List α) (h : xs ≠ []) : concat (dropLast xs) (last xs h) = xs := by
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match xs, h with
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| [], h => contradiction
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| [x], h => rfl
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| x₁::x₂::xs, h => simp [concat, last, concatEq (x₂::xs) List.noConfusion]
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theorem lengthCons {α} (x : α) (xs : List α) : (x::xs).length = xs.length + 1 :=
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rfl
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theorem eqNilOfLengthZero {α} : (xs : List α) → xs.length = 0 → xs = []
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| [], h => rfl
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| x::xs, h => by rw [lengthCons] at h; contradiction
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theorem dropLastLen {α} (xs : List α) : (n : Nat) → xs.length = n+1 → (dropLast xs).length = n := by
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match xs with
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| [] => intros; contradiction
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| [a] =>
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intro n h
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have : 1 = n + 1 := h
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have : 0 = n := by injection this
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subst this
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rfl
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| x₁::x₂::xs =>
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intro n h
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cases n with
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| zero =>
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simp [lengthCons] at h
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| succ n =>
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have : (x₁ :: x₂ :: xs).length = xs.length + 2 := by simp [lengthCons]
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have : xs.length = n := by rw [this] at h; injection h with h; injection h
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simp [dropLast, lengthCons, dropLastLen (x₂::xs) xs.length (lengthCons ..), this]
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@[inline]
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def concatElim {α}
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(motive : List α → Sort u)
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(base : Unit → motive [])
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(ind : (xs : List α) → (a : α) → motive xs → motive (concat xs a))
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(xs : List α)
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: motive xs :=
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let rec @[specialize] aux : (n : Nat) → (xs : List α) → xs.length = n → motive xs
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| 0, xs, h => by
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have aux := eqNilOfLengthZero _ h
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subst aux
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apply base ()
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| n+1, xs, h => by
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have notNil : xs ≠ [] := by intro h1; subst h1; injection h
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let ih := aux n (dropLast xs) (dropLastLen _ _ h)
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let aux := ind (dropLast xs) (last xs notNil) ih
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rw [concatEq] at aux
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exact aux
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aux xs.length xs rfl
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-- The generated code is tail recursive
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def test (xs : List Nat) : IO Unit :=
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concatElim (motive := fun _ => IO Unit)
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(fun _ => pure ())
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(fun xs x r => do IO.println s!"step xs: {xs} x: {x}"; r)
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xs
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#eval test [1, 2, 3, 4]
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