83 lines
2 KiB
Text
83 lines
2 KiB
Text
set_option tactic.simp.trace true
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set_option trace.Meta.Tactic.simp.rewrite true
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def f (x : α) := x
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example (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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example (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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| [] => default
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| a::_ => a
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example [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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example (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 := Id.run <| do
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let x := x
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return x
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example : g x = x := by
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simp [g, bind, pure]
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rfl
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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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example : 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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example (x : Nat) : bla x = x + x := by
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simp [bla, h]
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example (x : Nat) (h : 1 ≤ x) : x - 1 + 1 + 2 = x + 2 := by
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simp [h, Nat.sub_add_cancel]
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example (x : Nat) : (if h : 1 ≤ x then x - 1 + 1 else 0) = (if _h : 1 ≤ x then x else 0) := by
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simp (config := {contextual := true}) [h, Nat.sub_add_cancel]
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theorem my_thm : a ∧ a ↔ a := ⟨fun h => h.1, fun h => ⟨h, h⟩⟩
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example : a ∧ (b ∧ b) ↔ a ∧ b := by simp [my_thm]
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example : (a ∧ (b ∧ b)) = (a ∧ b) := by simp only [my_thm]
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example : x - 1 + 1 = x := by simp (discharger := sorry) [Nat.sub_add_cancel]
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