fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
monotone_map_iff {f : α → β} : Monotone (WithBot.map f) ↔ Monotone f :=
monotone_iff.trans <| by simp [Monotone]
alias ⟨_, _root_.Monotone.withBot_map⟩ := monotone_map_iff | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | monotone_map_iff | null |
strictMono_iff {f : WithBot α → β} :
StrictMono f ↔ StrictMono (fun a => f a : α → β) ∧ ∀ x : α, f ⊥ < f x :=
⟨fun h => ⟨h.comp WithBot.coe_strictMono, fun _ => h (bot_lt_coe _)⟩, fun h =>
WithBot.forall.2
⟨WithBot.forall.2 ⟨flip absurd (lt_irrefl _), fun x _ => h.2 x⟩, fun _ =>
WithBot.forall.2... | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | strictMono_iff | null |
strictAnti_iff {f : WithBot α → β} :
StrictAnti f ↔ StrictAnti (fun a ↦ f a : α → β) ∧ ∀ x : α, f x < f ⊥ :=
strictMono_iff (β := βᵒᵈ)
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | strictAnti_iff | null |
strictMono_map_iff {f : α → β} :
StrictMono (WithBot.map f) ↔ StrictMono f :=
strictMono_iff.trans <| by simp [StrictMono, bot_lt_coe]
alias ⟨_, _root_.StrictMono.withBot_map⟩ := strictMono_map_iff | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | strictMono_map_iff | null |
map_le_iff (f : α → β) (mono_iff : ∀ {a b}, f a ≤ f b ↔ a ≤ b) :
x.map f ≤ y.map f ↔ x ≤ y := by cases x <;> cases y <;> simp [mono_iff] | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map_le_iff | null |
le_coe_unbotD (x : WithBot α) (b : α) : x ≤ x.unbotD b := by cases x <;> simp
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | le_coe_unbotD | null |
lt_coe_bot [OrderBot α] : x < (⊥ : α) ↔ x = ⊥ := by cases x <;> simp | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | lt_coe_bot | null |
eq_bot_iff_forall_lt : x = ⊥ ↔ ∀ b : α, x < b := by
cases x <;> simp; simpa using ⟨_, lt_irrefl _⟩ | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | eq_bot_iff_forall_lt | null |
eq_bot_iff_forall_le [NoBotOrder α] : x = ⊥ ↔ ∀ b : α, x ≤ b := by
refine ⟨by simp +contextual, fun h ↦ (x.eq_bot_iff_forall_ne).2 fun y => ?_⟩
rintro rfl
exact not_isBot y fun z => coe_le_coe.1 (h z)
@[deprecated (since := "2025-03-19")] alias forall_lt_iff_eq_bot := eq_bot_iff_forall_lt
@[deprecated (since := "... | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | eq_bot_iff_forall_le | null |
forall_le_coe_iff_le [NoBotOrder α] : (∀ a : α, y ≤ a → x ≤ a) ↔ x ≤ y := by
obtain _ | y := y
· simp [WithBot.none_eq_bot, eq_bot_iff_forall_le]
· exact ⟨fun h ↦ h _ le_rfl, fun hmn a ham ↦ hmn.trans ham⟩ | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | forall_le_coe_iff_le | null |
eq_of_forall_le_coe_iff (h : ∀ a : α, x ≤ a ↔ y ≤ a) : x = y :=
le_antisymm (forall_le_coe_iff_le.mp fun a ↦ (h a).2) (forall_le_coe_iff_le.mp fun a ↦ (h a).1) | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | eq_of_forall_le_coe_iff | null |
semilatticeSup [SemilatticeSup α] : SemilatticeSup (WithBot α) where
sup
| ⊥, ⊥ => ⊥
| (a : α), ⊥ => a
| ⊥, (b : α) => b
| (a : α), (b : α) => ↑(a ⊔ b)
le_sup_left x y := by cases x <;> cases y <;> simp
le_sup_right x y := by cases x <;> cases y <;> simp
sup_le x y z := by cases x <;> cases y <;... | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | semilatticeSup | null |
coe_sup [SemilatticeSup α] (a b : α) : ((a ⊔ b : α) : WithBot α) = (a : WithBot α) ⊔ b :=
rfl | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | coe_sup | null |
semilatticeInf [SemilatticeInf α] : SemilatticeInf (WithBot α) where
inf := .map₂ (· ⊓ ·)
inf_le_left x y := by cases x <;> cases y <;> simp
inf_le_right x y := by cases x <;> cases y <;> simp
le_inf x y z := by cases x <;> cases y <;> cases z <;> simp; simpa using le_inf | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | semilatticeInf | null |
coe_inf [SemilatticeInf α] (a b : α) : ((a ⊓ b : α) : WithBot α) = (a : WithBot α) ⊓ b :=
rfl | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | coe_inf | null |
lattice [Lattice α] : Lattice (WithBot α) :=
{ WithBot.semilatticeSup, WithBot.semilatticeInf with } | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | lattice | null |
distribLattice [DistribLattice α] : DistribLattice (WithBot α) where
le_sup_inf x y z := by
cases x <;> cases y <;> cases z <;> simp [← coe_inf, ← coe_sup]
simpa [← coe_inf, ← coe_sup] using le_sup_inf | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | distribLattice | null |
decidableEq [DecidableEq α] : DecidableEq (WithBot α) :=
inferInstanceAs <| DecidableEq (Option α) | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | decidableEq | null |
decidableLE [LE α] [DecidableLE α] : DecidableLE (WithBot α)
| ⊥, _ => isTrue <| by simp
| (a : α), ⊥ => isFalse <| by simp
| (a : α), (b : α) => decidable_of_iff' _ coe_le_coe | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | decidableLE | null |
decidableLT [LT α] [DecidableLT α] : DecidableLT (WithBot α)
| _, ⊥ => isFalse <| by simp
| ⊥, (a : α) => isTrue <| by simp
| (a : α), (b : α) => decidable_of_iff' _ coe_lt_coe | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | decidableLT | null |
isTotal_le [LE α] [IsTotal α (· ≤ ·)] : IsTotal (WithBot α) (· ≤ ·) where
total x y := by cases x <;> cases y <;> simp; simpa using IsTotal.total .. | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | isTotal_le | null |
linearOrder : LinearOrder (WithBot α) := Lattice.toLinearOrder _
@[simp, norm_cast] lemma coe_min (a b : α) : ↑(min a b) = min (a : WithBot α) b := rfl
@[simp, norm_cast] lemma coe_max (a b : α) : ↑(max a b) = max (a : WithBot α) b := rfl
variable [DenselyOrdered α] [NoMinOrder α] | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | linearOrder | null |
le_of_forall_lt_iff_le : (∀ z : α, x < z → y ≤ z) ↔ y ≤ x := by
cases x <;> cases y <;> simp [exists_lt, forall_gt_imp_ge_iff_le_of_dense] | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | le_of_forall_lt_iff_le | null |
ge_of_forall_gt_iff_ge : (∀ z : α, z < x → z ≤ y) ↔ x ≤ y := by
cases x <;> cases y <;> simp [exists_lt, forall_lt_imp_le_iff_le_of_dense] | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | ge_of_forall_gt_iff_ge | null |
instWellFoundedLT [LT α] [WellFoundedLT α] : WellFoundedLT (WithBot α) where
wf := .intro fun
| ⊥ => ⟨_, by simp⟩
| (a : α) => (wellFounded_lt.1 a).rec fun _ _ ih ↦ .intro _ fun
| ⊥, _ => ⟨_, by simp⟩
| (b : α), hlt => ih _ (coe_lt_coe.1 hlt) | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | instWellFoundedLT | null |
_root_.WithBot.instWellFoundedGT [LT α] [WellFoundedGT α] : WellFoundedGT (WithBot α) where
wf :=
have acc_some (a : α) : Acc ((· > ·) : WithBot α → WithBot α → Prop) a :=
(wellFounded_gt.1 a).rec fun _ _ ih =>
.intro _ fun
| (b : α), hlt => ih _ (coe_lt_coe.1 hlt)
.intro fun
| (a : α) => ac... | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | _root_.WithBot.instWellFoundedGT | null |
denselyOrdered_iff [LT α] [NoMinOrder α] :
DenselyOrdered (WithBot α) ↔ DenselyOrdered α := by
constructor <;> intro h <;> constructor
· intro a b hab
obtain ⟨c, hc⟩ := exists_between (WithBot.coe_lt_coe.mpr hab)
induction c with
| bot => simp at hc
| coe c => exact ⟨c, by simpa using hc⟩
· si... | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | denselyOrdered_iff | null |
denselyOrdered [LT α] [DenselyOrdered α] [NoMinOrder α] : DenselyOrdered (WithBot α) :=
denselyOrdered_iff.mpr inferInstance | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | denselyOrdered | null |
lt_iff_exists_coe_btwn [Preorder α] [DenselyOrdered α] [NoMinOrder α] {a b : WithBot α} :
a < b ↔ ∃ x : α, a < ↑x ∧ ↑x < b :=
⟨fun h =>
let ⟨_, hy⟩ := exists_between h
let ⟨x, hx⟩ := lt_iff_exists_coe.1 hy.1
⟨x, hx.1 ▸ hy⟩,
fun ⟨_, hx⟩ => lt_trans hx.1 hx.2⟩ | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | lt_iff_exists_coe_btwn | null |
noTopOrder [LE α] [NoTopOrder α] [Nonempty α] : NoTopOrder (WithBot α) where
exists_not_le := fun
| ⊥ => ‹Nonempty α›.elim fun a ↦ ⟨a, by simp⟩
| (a : α) => let ⟨b, hba⟩ := exists_not_le a; ⟨b, mod_cast hba⟩ | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | noTopOrder | null |
noMaxOrder [LT α] [NoMaxOrder α] [Nonempty α] : NoMaxOrder (WithBot α) where
exists_gt := fun
| ⊥ => ‹Nonempty α›.elim fun a ↦ ⟨a, by simp⟩
| (a : α) => let ⟨b, hba⟩ := exists_gt a; ⟨b, mod_cast hba⟩ | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | noMaxOrder | null |
nontrivial [Nonempty α] : Nontrivial (WithTop α) :=
Option.nontrivial
open Function | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | nontrivial | null |
coe_injective : Injective ((↑) : α → WithTop α) :=
Option.some_injective _
@[norm_cast] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | coe_injective | null |
coe_inj : (a : WithTop α) = b ↔ a = b :=
Option.some_inj | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | coe_inj | null |
none_eq_top : (none : WithTop α) = (⊤ : WithTop α) :=
rfl | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | none_eq_top | null |
some_eq_coe (a : α) : (Option.some a : WithTop α) = (↑a : WithTop α) :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | some_eq_coe | null |
top_ne_coe : ⊤ ≠ (a : WithTop α) :=
nofun
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | top_ne_coe | null |
coe_ne_top : (a : WithTop α) ≠ ⊤ :=
nofun | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | coe_ne_top | null |
protected toDual : WithTop α ≃ WithBot αᵒᵈ :=
Equiv.refl _ | def | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | toDual | `WithTop.toDual` is the equivalence sending `⊤` to `⊥` and any `a : α` to `toDual a : αᵒᵈ`.
See `WithTop.toDualBotEquiv` for the related order-iso. |
protected ofDual : WithTop αᵒᵈ ≃ WithBot α :=
Equiv.refl _ | def | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | ofDual | `WithTop.ofDual` is the equivalence sending `⊤` to `⊥` and any `a : αᵒᵈ` to `ofDual a : α`.
See `WithTop.toDualBotEquiv` for the related order-iso. |
protected _root_.WithBot.toDual : WithBot α ≃ WithTop αᵒᵈ :=
Equiv.refl _ | def | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | _root_.WithBot.toDual | `WithBot.toDual` is the equivalence sending `⊥` to `⊤` and any `a : α` to `toDual a : αᵒᵈ`.
See `WithBot.toDual_top_equiv` for the related order-iso. |
protected _root_.WithBot.ofDual : WithBot αᵒᵈ ≃ WithTop α :=
Equiv.refl _
@[simp] | def | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | _root_.WithBot.ofDual | `WithBot.ofDual` is the equivalence sending `⊥` to `⊤` and any `a : αᵒᵈ` to `ofDual a : α`.
See `WithBot.ofDual_top_equiv` for the related order-iso. |
toDual_symm_apply (a : WithBot αᵒᵈ) : WithTop.toDual.symm a = WithBot.ofDual a :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | toDual_symm_apply | null |
ofDual_symm_apply (a : WithBot α) : WithTop.ofDual.symm a = WithBot.toDual a :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | ofDual_symm_apply | null |
toDual_apply_top : WithTop.toDual (⊤ : WithTop α) = ⊥ :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | toDual_apply_top | null |
ofDual_apply_top : WithTop.ofDual (⊤ : WithTop α) = ⊥ :=
rfl
open OrderDual
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | ofDual_apply_top | null |
toDual_apply_coe (a : α) : WithTop.toDual (a : WithTop α) = toDual a :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | toDual_apply_coe | null |
ofDual_apply_coe (a : αᵒᵈ) : WithTop.ofDual (a : WithTop αᵒᵈ) = ofDual a :=
rfl | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | ofDual_apply_coe | null |
untopD (d : α) (x : WithTop α) : α :=
recTopCoe d id x
@[simp] | def | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | untopD | Specialization of `Option.getD` to values in `WithTop α` that respects API boundaries. |
untopD_top {α} (d : α) : untopD d ⊤ = d :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | untopD_top | null |
untopD_coe {α} (d x : α) : untopD d x = x :=
rfl
@[simp, norm_cast] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | untopD_coe | null |
coe_eq_coe : (a : WithTop α) = b ↔ a = b :=
Option.some_inj | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | coe_eq_coe | null |
untopD_eq_iff {d y : α} {x : WithTop α} : untopD d x = y ↔ x = y ∨ x = ⊤ ∧ y = d :=
WithBot.unbotD_eq_iff
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | untopD_eq_iff | null |
untopD_eq_self_iff {d : α} {x : WithTop α} : untopD d x = d ↔ x = d ∨ x = ⊤ :=
WithBot.unbotD_eq_self_iff | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | untopD_eq_self_iff | null |
untopD_eq_untopD_iff {d : α} {x y : WithTop α} :
untopD d x = untopD d y ↔ x = y ∨ x = d ∧ y = ⊤ ∨ x = ⊤ ∧ y = d :=
WithBot.unbotD_eq_unbotD_iff | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | untopD_eq_untopD_iff | null |
map (f : α → β) : WithTop α → WithTop β :=
Option.map f
@[simp] | def | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map | Lift a map `f : α → β` to `WithTop α → WithTop β`. Implemented using `Option.map`. |
map_top (f : α → β) : map f ⊤ = ⊤ :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map_top | null |
map_coe (f : α → β) (a : α) : map f a = f a :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map_coe | null |
map_eq_top_iff {f : α → β} {a : WithTop α} :
map f a = ⊤ ↔ a = ⊤ := Option.map_eq_none_iff | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map_eq_top_iff | null |
map_eq_some_iff {f : α → β} {y : β} {v : WithTop α} :
WithTop.map f v = .some y ↔ ∃ x, v = .some x ∧ f x = y := Option.map_eq_some_iff | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map_eq_some_iff | null |
some_eq_map_iff {f : α → β} {y : β} {v : WithTop α} :
.some y = WithTop.map f v ↔ ∃ x, v = .some x ∧ f x = y := by
cases v <;> simp [eq_comm] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | some_eq_map_iff | null |
map_id : map (id : α → α) = id :=
Option.map_id
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map_id | null |
map_map (h : β → γ) (g : α → β) (a : WithTop α) : map h (map g a) = map (h ∘ g) a :=
Option.map_map h g a | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map_map | null |
comp_map (h : β → γ) (g : α → β) (x : WithTop α) : x.map (h ∘ g) = (x.map g).map h :=
(map_map ..).symm
@[simp] theorem map_comp_map (f : α → β) (g : β → γ) :
WithTop.map g ∘ WithTop.map f = WithTop.map (g ∘ f) :=
Option.map_comp_map f g | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | comp_map | null |
map_comm {f₁ : α → β} {f₂ : α → γ} {g₁ : β → δ} {g₂ : γ → δ}
(h : g₁ ∘ f₁ = g₂ ∘ f₂) (a : α) : map g₁ (map f₁ a) = map g₂ (map f₂ a) :=
Option.map_comm h _ | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map_comm | null |
map_injective {f : α → β} (Hf : Injective f) : Injective (WithTop.map f) :=
Option.map_injective Hf | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map_injective | null |
map₂ : (α → β → γ) → WithTop α → WithTop β → WithTop γ := Option.map₂ | def | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map₂ | The image of a binary function `f : α → β → γ` as a function
`WithTop α → WithTop β → WithTop γ`.
Mathematically this should be thought of as the image of the corresponding function `α × β → γ`. |
map₂_coe_coe (f : α → β → γ) (a : α) (b : β) : map₂ f a b = f a b := rfl
@[simp] lemma map₂_top_left (f : α → β → γ) (b) : map₂ f ⊤ b = ⊤ := rfl
@[simp] lemma map₂_top_right (f : α → β → γ) (a) : map₂ f a ⊤ = ⊤ := by cases a <;> rfl
@[simp] lemma map₂_coe_left (f : α → β → γ) (a : α) (b) : map₂ f a b = b.map fun b ↦ f ... | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map₂_coe_coe | null |
map_toDual (f : αᵒᵈ → βᵒᵈ) (a : WithBot α) :
map f (WithBot.toDual a) = a.map (toDual ∘ f) :=
rfl | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map_toDual | null |
map_ofDual (f : α → β) (a : WithBot αᵒᵈ) : map f (WithBot.ofDual a) = a.map (ofDual ∘ f) :=
rfl | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | map_ofDual | null |
toDual_map (f : α → β) (a : WithTop α) :
WithTop.toDual (map f a) = WithBot.map (toDual ∘ f ∘ ofDual) (WithTop.toDual a) :=
rfl | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | toDual_map | null |
ofDual_map (f : αᵒᵈ → βᵒᵈ) (a : WithTop αᵒᵈ) :
WithTop.ofDual (map f a) = WithBot.map (ofDual ∘ f ∘ toDual) (WithTop.ofDual a) :=
rfl | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | ofDual_map | null |
ne_top_iff_exists {x : WithTop α} : x ≠ ⊤ ↔ ∃ a : α, ↑a = x := Option.ne_none_iff_exists | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | ne_top_iff_exists | null |
eq_top_iff_forall_ne {x : WithTop α} : x = ⊤ ↔ ∀ a : α, ↑a ≠ x :=
Option.eq_none_iff_forall_some_ne
@[deprecated (since := "2025-03-19")] alias forall_ne_iff_eq_top := eq_top_iff_forall_ne | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | eq_top_iff_forall_ne | null |
untop : ∀ x : WithTop α, x ≠ ⊤ → α | (x : α), _ => x
@[simp] lemma coe_untop : ∀ (x : WithTop α) hx, x.untop hx = x | (x : α), _ => rfl
@[simp] | def | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | untop | Deconstruct a `x : WithTop α` to the underlying value in `α`, given a proof that `x ≠ ⊤`. |
untop_coe (x : α) (h : (x : WithTop α) ≠ ⊤ := coe_ne_top) : (x : WithTop α).untop h = x :=
rfl | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | untop_coe | null |
canLift : CanLift (WithTop α) α (↑) fun r => r ≠ ⊤ where
prf x h := ⟨x.untop h, coe_untop _ _⟩ | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | canLift | null |
instBot [Bot α] : Bot (WithTop α) where
bot := (⊥ : α)
@[simp, norm_cast] lemma coe_bot [Bot α] : ((⊥ : α) : WithTop α) = ⊥ := rfl
@[simp, norm_cast] lemma coe_eq_bot [Bot α] {a : α} : (a : WithTop α) = ⊥ ↔ a = ⊥ := coe_eq_coe
@[simp, norm_cast] lemma bot_eq_coe [Bot α] {a : α} : (⊥ : WithTop α) = a ↔ ⊥ = a := coe_eq... | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | instBot | null |
untop_eq_iff {a : WithTop α} {b : α} (h : a ≠ ⊤) :
a.untop h = b ↔ a = b :=
WithBot.unbot_eq_iff (α := αᵒᵈ) h | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | untop_eq_iff | null |
eq_untop_iff {a : α} {b : WithTop α} (h : b ≠ ⊤) :
a = b.untop h ↔ a = b :=
WithBot.eq_unbot_iff (α := αᵒᵈ) h | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | eq_untop_iff | null |
@[simps] _root_.Equiv.withTopSubtypeNe : {y : WithTop α // y ≠ ⊤} ≃ α where
toFun := fun ⟨x,h⟩ => WithTop.untop x h
invFun x := ⟨x, WithTop.coe_ne_top⟩
left_inv _ := by simp
right_inv _:= by simp | def | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | _root_.Equiv.withTopSubtypeNe | The equivalence between the non-top elements of `WithTop α` and `α`. |
@[simps apply]
withTopCongr (e : α ≃ β) : WithTop α ≃ WithTop β where
toFun := WithTop.map e
invFun := WithTop.map e.symm
left_inv x := by cases x <;> simp
right_inv x := by cases x <;> simp
attribute [grind =] withTopCongr_apply
@[simp] | def | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | withTopCongr | A universe-polymorphic version of `EquivFunctor.mapEquiv WithTop e`. |
withTopCongr_refl : withTopCongr (Equiv.refl α) = Equiv.refl _ :=
Equiv.ext <| congr_fun WithBot.map_id
@[simp, grind =] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | withTopCongr_refl | null |
withTopCongr_symm (e : α ≃ β) : withTopCongr e.symm = (withTopCongr e).symm :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | withTopCongr_symm | null |
withTopCongr_trans (e₁ : α ≃ β) (e₂ : β ≃ γ) :
withTopCongr (e₁.trans e₂) = (withTopCongr e₁).trans (withTopCongr e₂) := by
ext x
simp | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | withTopCongr_trans | null |
le_def : x ≤ y ↔ ∀ b : α, y = ↑b → ∃ a : α, x = ↑a ∧ a ≤ b := .rfl
@[simp, norm_cast] lemma coe_le_coe : (a : WithTop α) ≤ b ↔ a ≤ b := by simp [le_def] | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | le_def | null |
not_top_le_coe (a : α) : ¬ ⊤ ≤ (a : WithTop α) := by simp [le_def] | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | not_top_le_coe | null |
orderTop : OrderTop (WithTop α) where le_top := by simp [le_def] | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | orderTop | null |
orderBot [OrderBot α] : OrderBot (WithTop α) where bot_le x := by cases x <;> simp [le_def] | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | orderBot | null |
boundedOrder [OrderBot α] : BoundedOrder (WithTop α) :=
{ WithTop.orderTop, WithTop.orderBot with } | instance | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | boundedOrder | null |
@[simp]
protected top_le_iff : ∀ {a : WithTop α}, ⊤ ≤ a ↔ a = ⊤
| (a : α) => by simp [not_top_le_coe _]
| ⊤ => by simp | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | top_le_iff | There is a general version `top_le_iff`, but this lemma does not require a `PartialOrder`. |
le_coe : ∀ {o : Option α}, a ∈ o → (@LE.le (WithTop α) _ o b ↔ a ≤ b)
| _, rfl => coe_le_coe | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | le_coe | null |
le_coe_iff : x ≤ b ↔ ∃ a : α, x = a ∧ a ≤ b := by simp [le_def] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | le_coe_iff | null |
coe_le_iff : ↑a ≤ x ↔ ∀ b : α, x = ↑b → a ≤ b := by simp [le_def] | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | coe_le_iff | null |
protected _root_.IsMin.withTop (h : IsMin a) : IsMin (a : WithTop α) :=
fun x ↦ by cases x <;> simp; simpa using @h _ | theorem | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | _root_.IsMin.withTop | null |
untop_le_iff (hx : x ≠ ⊤) : untop x hx ≤ b ↔ x ≤ b := by lift x to α using id hx; simp | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | untop_le_iff | null |
le_untop_iff (hy : y ≠ ⊤) : a ≤ untop y hy ↔ a ≤ y := by lift y to α using id hy; simp | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | le_untop_iff | null |
le_untopD_iff (hy : y = ⊤ → a ≤ b) : a ≤ y.untopD b ↔ a ≤ y := by cases y <;> simp [hy] | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | le_untopD_iff | null |
lt_def : x < y ↔ ∃ a : α, x = ↑a ∧ ∀ b : α, y = ↑b → a < b := .rfl
@[simp, norm_cast] lemma coe_lt_coe : (a : WithTop α) < b ↔ a < b := by simp [lt_def]
@[simp] lemma coe_lt_top (a : α) : (a : WithTop α) < ⊤ := by simp [lt_def]
@[simp] protected lemma not_top_lt (a : WithTop α) : ¬⊤ < a := by simp [lt_def] | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | lt_def | null |
lt_iff_exists_coe : x < y ↔ ∃ a : α, x = a ∧ a < y := by cases x <;> simp | lemma | Order | [
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Order.TypeTags",
"Mathlib.Data.Option.NAry",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.Lift",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.Lattice",
"Mathlib.Order.BoundedOrder.Basic"
] | Mathlib/Order/WithBot.lean | lt_iff_exists_coe | null |
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