62f14514da
This PR updates various error messages produced by or associated with built-in tactics and adapts their formatting to current conventions.
81 lines
2.9 KiB
Text
81 lines
2.9 KiB
Text
@[reducible]
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def swap {φ : α → β → Sort u₃} (f : ∀ x y, φ x y) : ∀ y x, φ x y := fun y x => f x y
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theorem forall_swap {p : α → β → Prop} : (∀ x y, p x y) ↔ ∀ y x, p x y := ⟨swap, swap⟩
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@[simp]
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theorem nonempty_Prop {p : Prop} : Nonempty p ↔ p :=
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Iff.intro (fun ⟨h⟩ ↦ h) fun h ↦ ⟨h⟩
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class IsEmpty (α : Sort _) : Prop where
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protected false : α → False
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@[elab_as_elim]
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def isEmptyElim [IsEmpty α] {p : α → Sort _} (a : α) : p a :=
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(IsEmpty.false a).elim
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@[elab_as_elim]
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protected def IsEmpty.elim {α : Sort u} (_ : IsEmpty α) {p : α → Sort _} (a : α) : p a :=
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(IsEmpty.false a).elim
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@[simp]
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theorem not_nonempty_iff : ¬Nonempty α ↔ IsEmpty α :=
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⟨fun h ↦ ⟨fun x ↦ h ⟨x⟩⟩, fun h1 h2 ↦ h2.elim h1.elim⟩
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@[simp]
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theorem isEmpty_Prop {p : Prop} : IsEmpty p ↔ ¬p := by
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simp only [← not_nonempty_iff, nonempty_Prop]
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class Preorder (α : Type u) extends LE α where
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le_refl : ∀ a : α, a ≤ a
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theorem le_refl [Preorder α] : ∀ a : α, a ≤ a :=
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Preorder.le_refl
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theorem le_of_eq [Preorder α] {a b : α} : a = b → a ≤ b := fun h => h ▸ le_refl a
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abbrev Eq.le := @le_of_eq
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@[simp] theorem le_of_subsingleton [Preorder α] [Subsingleton α] {a b : α} : a ≤ b := (Subsingleton.elim a b).le
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theorem iff_of_true' (ha : a) (hb : b) : a ↔ b := Iff.intro (fun _ => hb) (fun _ => ha)
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theorem iff_true_intro' (h : a) : a ↔ True := iff_of_true' h trivial
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@[simp]
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theorem IsEmpty.forall_iff [IsEmpty α] {p : α → Prop} : (∀ a, p a) ↔ True :=
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iff_true_intro' isEmptyElim
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@[simp] theorem and_imp' : (a ∧ b → c) ↔ (a → b → c) := ⟨fun h ha hb => h ⟨ha, hb⟩, fun h ⟨ha, hb⟩ => h ha hb⟩
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@[simp] theorem not_and'' : ¬(a ∧ b) ↔ (a → ¬b) := and_imp'
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set_option tactic.skipAssignedInstances false in
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/-- error: `simp` made no progress -/
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#guard_msgs in
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example [Preorder α] {a : α} {p : α → Prop} : ∀ (a_1 : α), a ≤ a_1 ∧ p a_1 → a ≤ a_1 := by
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simp only [isEmpty_Prop, not_and'', forall_swap, le_of_subsingleton, IsEmpty.forall_iff] -- should not loop
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theorem dec_and (p q : Prop) [Decidable (p ∧ q)] [Decidable p] [Decidable q] : decide (p ∧ q) = (p && q) := by
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by_cases p <;> by_cases q <;> simp [*]
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theorem dec_not (p : Prop) [Decidable (¬p)] [Decidable p] : decide (¬p) = !p := by
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by_cases p <;> simp [*]
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example [Decidable u] [Decidable v] : decide (u ∧ (v → False)) = (decide u && !decide v) := by
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simp only [imp_false]
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rw [dec_and]
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rw [dec_not]
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set_option tactic.skipAssignedInstances false in
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/--
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error: Tactic `rewrite` failed: failed to assign synthesized instance
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u v : Prop
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inst✝¹ : Decidable u
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inst✝ : Decidable v
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⊢ decide (u ∧ ¬v) = (decide u && !decide v)
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-/
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#guard_msgs in
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example [Decidable u] [Decidable v] : decide (u ∧ (v → False)) = (decide u && !decide v) := by
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simp only [imp_false]
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rw [dec_and]
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rw [dec_not]
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