Finish all the BinOp warmup proofs

Changes

1 changed files (+25/-6)

• BinOp.lean
  • @@ -2,22 +2,41 @@
    import Mathlib - example {α} (f : α → α → α) (l r : α) (hl : ∀ x, f l x = x) (hr : ∀ x, f x r = x) : l = r := by + example {α} (f : α → α → α) l r (hl : ∀ x, f l x = x) (hr : ∀ x, f x r = x) : l = r := by have h₁ : f l r = l := by grind grind - example {α} (f : α → α → α) (h : ∀ x y z, f x z = y ∧ ∀ z' ≠ z, f x z' ≠ y) : ∃ g : α → α → α, ∀ x y, f x (g x y) = g x (f x y) ∧ f x (g x y) = y := by - sorry + example {α} [Nonempty α] (f : α → α → α) (h : ∀ x y, ∃ z, f x z = y ∧ ∀ z', f x z = f x z' → z = z') : ∃ g : α → α → α, ∀ x y, f x (g x y) = y ∧ g x (f x y) = y := by + let g x y := (f x).invFun y + use g + intro x y + constructor + have h₁ x : (f x).Surjective := by + rw [Function.Surjective] + intro y + specialize h x y + grind + apply Function.rightInverse_invFun (h₁ x) + have h₁ x : (f x).Injective := by + rw [Function.Injective] + intro a b + specialize h x (f x a) + obtain ⟨a', ⟨h₂, h₃⟩⟩ := h + have h₄ : a' = a := by + apply h₃ a h₂ + rw [← h₄] + exact h₃ b + apply Function.leftInverse_invFun (h₁ x) - example {α} (f g : α → α → α) (i j : α) (hid : ∀ x, f i x = x ∧ f x i = x ∧ g j x = x ∧ g x j = x) (h : ∀ x y z w, f (g x y) (g z w) = g (f x z) (f y w)) : f = g := by - have h₁ (x : α) : f x j = x := by + example {α} (f g : α → α → α) i j (hid : ∀ x, f i x = x ∧ f x i = x ∧ g j x = x ∧ g x j = x) (h : ∀ x y z w, f (g x y) (g z w) = g (f x z) (f y w)) : f = g := by + have h₁ x : f x j = x := by specialize h x j j i grind have h₂ : i = j := by specialize hid j grind - have h₃ (x y : α) : f x y = g x y := by + have h₃ x y : f x y = g x y := by specialize h x i i y grind grind