34bd6e8bfd
This PR improves the error messages produced by the `split` tactic, including suggesting syntax fixes and related tactics with which it might be confused. Note that, to avoid clashing with the new error message styling conventions used in these messages, this PR also updates the formatting of the message produced by `throwTacticEx`. Closes #6224
83 lines
2.9 KiB
Text
83 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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/--
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error: simp made no progress
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-/
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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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