66 lines
1.4 KiB
Text
66 lines
1.4 KiB
Text
def f (x : α) := x
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theorem ex1 (a : α) (b : List α) : f (a::b = []) = False :=
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by simp [f]
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def length : List α → Nat
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| [] => 0
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| a::as => length as + 1
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theorem ex2 (a b c : α) (as : List α) : length (a :: b :: as) > length as := by
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simp [length]
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apply Nat.lt.step
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apply Nat.lt_succ_self
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def fact : Nat → Nat
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| 0 => 1
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| x+1 => (x+1) * fact x
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theorem ex3 : fact x > 0 := by
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induction x with
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| zero => decide
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| succ x ih =>
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simp [fact]
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apply Nat.mul_pos
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apply Nat.zero_lt_succ
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apply ih
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def head [Inhabited α] : List α → α
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| [] => arbitrary
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| a::_ => a
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theorem ex4 [Inhabited α] (a : α) (as : List α) : head (a::as) = a :=
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by simp [head]
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def foo := 10
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theorem ex5 (x : Nat) : foo + x = 10 + x := by
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simp [foo]
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done
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def g (x : Nat) : Nat := do
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let x := x
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return x
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theorem ex6 : g x = x := by
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simp [g, bind, pure]
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def f1 : StateM Nat Unit := do
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modify fun x => g x
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def f2 : StateM Nat Unit := do
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let s ← get
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set <| g s
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theorem ex7 : f1 = f2 := by
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simp [f1, f2, bind, StateT.bind, get, getThe, MonadStateOf.get, StateT.get, pure, set, StateT.set, modify, modifyGet, MonadStateOf.modifyGet, StateT.modifyGet]
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def h (x : Nat) : Sum (Nat × Nat) Nat := Sum.inl (x, x)
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def bla (x : Nat) :=
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match h x with
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| Sum.inl (y, z) => y + z
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| Sum.inr _ => 0
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theorem ex8 (x : Nat) : bla x = x + x := by
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simp [bla, h]
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