84 lines
2.5 KiB
Text
84 lines
2.5 KiB
Text
theorem zeroLtOfLt : {a b : Nat} → a < b → 0 < b
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| 0, _, h => h
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| a+1, b, h =>
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have a < b from Nat.ltTrans (Nat.ltSuccSelf _) h
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zeroLtOfLt this
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def fold {m α β} [Monad m] (as : Array α) (b : β) (f : α → β → m β) : m β := do
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let rec loop : (i : Nat) → i ≤ as.size → β → m β
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| 0, h, b => b
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| i+1, h, b => do
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have h' : i < as.size from Nat.ltOfLtOfLe (Nat.ltSuccSelf i) h
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have as.size - 1 < as.size from Nat.subLt (zeroLtOfLt h') (by decide)
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have as.size - 1 - i < as.size from Nat.ltOfLeOfLt (Nat.subLe (as.size - 1) i) this
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let b ← f (as.get ⟨as.size - 1 - i, this⟩) b
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loop i (Nat.leOfLt h') b
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loop as.size (Nat.leRefl _) b
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#eval Id.run $ fold #[1, 2, 3, 4] 0 (pure $ · + ·)
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theorem ex : (Id.run $ fold #[1, 2, 3, 4] 0 (pure $ · + ·)) = 10 :=
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rfl
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def fold2 {m α β} [Monad m] (as : Array α) (b : β) (f : α → β → m β) : m β :=
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let rec loop (i : Nat) (h : i ≤ as.size) (b : β) : m β := do
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match i, h with
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| 0, h => return b
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| i+1, h =>
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have h' : i < as.size from Nat.ltOfLtOfLe (Nat.ltSuccSelf i) h
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have as.size - 1 < as.size from Nat.subLt (zeroLtOfLt h') (by decide)
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have as.size - 1 - i < as.size from Nat.ltOfLeOfLt (Nat.subLe (as.size - 1) i) this
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let b ← f (as.get ⟨as.size - 1 - i, this⟩) b
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loop i (Nat.leOfLt h') b
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loop as.size (Nat.leRefl _) b
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def f (x : Nat) (ref : IO.Ref Nat) : IO Nat := do
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let mut x := x
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if x == 0 then
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x ← ref.get
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IO.println x
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return x + 1
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def fTest : IO Unit := do
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unless (← f 0 (← IO.mkRef 10)) == 11 do throw $ IO.userError "unexpected"
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unless (← f 1 (← IO.mkRef 10)) == 2 do throw $ IO.userError "unexpected"
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def g (x y : Nat) (ref : IO.Ref (Nat × Nat)) : IO (Nat × Nat) := do
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let mut (x, y) := (x, y)
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if x == 0 then
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(x, y) ← ref.get
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IO.println ("x: " ++ toString x ++ ", y: " ++ toString y)
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return (x, y)
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def gTest : IO Unit := do
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unless (← g 2 1 (← IO.mkRef (10, 20))) == (2, 1) do throw $ IO.userError "unexpected"
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unless (← g 0 1 (← IO.mkRef (10, 20))) == (10, 20) do throw $ IO.userError "unexpected"
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return ()
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#eval gTest
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macro "ret!" x:term : doElem => `(return $x)
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def f1 (x : Nat) : Nat := do
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let mut x := x
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if x == 0 then
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ret! 100
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x := x + 1
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ret! x
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theorem ex1 : f1 0 = 100 := rfl
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theorem ex2 : f1 1 = 2 := rfl
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theorem ex3 : f1 3 = 4 := rfl
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syntax "inc!" ident : doElem
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macro_rules
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| `(doElem| inc! $x) => `(doElem| $x:ident := $x + 1)
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def f2 (x : Nat) : Nat := do
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let mut x := x
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inc! x
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ret! x
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theorem ex4 : f2 0 = 1 := rfl
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theorem ex5 : f2 3 = 4 := rfl
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