a23292f049
When using `set_option tactic.skipAssignedInstances false`, `simp` and `rw` will synthesize instance implicit arguments even if they have assigned by unification. If the synthesized argument does not match the assigned one the rewrite is not performed. This option has been added for backward compatibility.
82 lines
2.9 KiB
Text
82 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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