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 ⌀ |
|---|---|---|---|---|---|---|
bihimp_eq_sup_himp_inf : a ⇔ b = a ⊔ b ⇨ a ⊓ b := by simp [himp_inf_distrib, bihimp]
@[deprecated (since := "2025-06-05")] alias bihimp_eq_inf_himp_inf := bihimp_eq_sup_himp_inf | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_eq_sup_himp_inf | null |
Codisjoint.bihimp_eq_inf {a b : α} (h : Codisjoint a b) : a ⇔ b = a ⊓ b := by
rw [bihimp, h.himp_eq_left, h.himp_eq_right] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | Codisjoint.bihimp_eq_inf | null |
himp_bihimp : a ⇨ b ⇔ c = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c) := by
rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | himp_bihimp | null |
sup_himp_bihimp : a ⊔ b ⇨ a ⇔ b = a ⇔ b := by
rw [himp_bihimp]
simp [bihimp]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | sup_himp_bihimp | null |
bihimp_himp_eq_inf : a ⇔ (a ⇨ b) = a ⊓ b :=
@symmDiff_sdiff_eq_sup αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_himp_eq_inf | null |
himp_bihimp_eq_inf : (b ⇨ a) ⇔ b = a ⊓ b :=
@sdiff_symmDiff_eq_sup αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | himp_bihimp_eq_inf | null |
bihimp_inf_sup : a ⇔ b ⊓ (a ⊔ b) = a ⊓ b :=
@symmDiff_sup_inf αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_inf_sup | null |
sup_inf_bihimp : (a ⊔ b) ⊓ a ⇔ b = a ⊓ b :=
@inf_sup_symmDiff αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | sup_inf_bihimp | null |
bihimp_bihimp_sup : a ⇔ b ⇔ (a ⊔ b) = a ⊓ b :=
@symmDiff_symmDiff_inf αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_bihimp_sup | null |
sup_bihimp_bihimp : (a ⊔ b) ⇔ (a ⇔ b) = a ⊓ b :=
@inf_symmDiff_symmDiff αᵒᵈ _ _ _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | sup_bihimp_bihimp | null |
bihimp_triangle : a ⇔ b ⊓ b ⇔ c ≤ a ⇔ c :=
@symmDiff_triangle αᵒᵈ _ _ _ _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_triangle | null |
@[simp]
symmDiff_top' : a ∆ ⊤ = ￢a := by simp [symmDiff]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_top' | null |
top_symmDiff' : ⊤ ∆ a = ￢a := by simp [symmDiff]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | top_symmDiff' | null |
hnot_symmDiff_self : (￢a) ∆ a = ⊤ := by
rw [eq_top_iff, symmDiff, hnot_sdiff, sup_sdiff_self]
exact Codisjoint.top_le codisjoint_hnot_left
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | hnot_symmDiff_self | null |
symmDiff_hnot_self : a ∆ (￢a) = ⊤ := by rw [symmDiff_comm, hnot_symmDiff_self] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_hnot_self | null |
IsCompl.symmDiff_eq_top {a b : α} (h : IsCompl a b) : a ∆ b = ⊤ := by
rw [h.eq_hnot, hnot_symmDiff_self] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | IsCompl.symmDiff_eq_top | null |
@[simp]
bihimp_bot : a ⇔ ⊥ = aᶜ := by simp [bihimp]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_bot | null |
bot_bihimp : ⊥ ⇔ a = aᶜ := by simp [bihimp]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bot_bihimp | null |
compl_bihimp_self : aᶜ ⇔ a = ⊥ :=
@hnot_symmDiff_self αᵒᵈ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | compl_bihimp_self | null |
bihimp_hnot_self : a ⇔ aᶜ = ⊥ :=
@symmDiff_hnot_self αᵒᵈ _ _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_hnot_self | null |
IsCompl.bihimp_eq_bot {a b : α} (h : IsCompl a b) : a ⇔ b = ⊥ := by
rw [h.eq_compl, compl_bihimp_self] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | IsCompl.bihimp_eq_bot | null |
@[simp]
sup_sdiff_symmDiff : (a ⊔ b) \ a ∆ b = a ⊓ b :=
sdiff_eq_symm inf_le_sup (by rw [symmDiff_eq_sup_sdiff_inf]) | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | sup_sdiff_symmDiff | null |
disjoint_symmDiff_inf : Disjoint (a ∆ b) (a ⊓ b) := by
rw [symmDiff_eq_sup_sdiff_inf]
exact disjoint_sdiff_self_left | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | disjoint_symmDiff_inf | null |
inf_symmDiff_distrib_left : a ⊓ b ∆ c = (a ⊓ b) ∆ (a ⊓ c) := by
rw [symmDiff_eq_sup_sdiff_inf, inf_sdiff_distrib_left, inf_sup_left, inf_inf_distrib_left,
symmDiff_eq_sup_sdiff_inf] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | inf_symmDiff_distrib_left | null |
inf_symmDiff_distrib_right : a ∆ b ⊓ c = (a ⊓ c) ∆ (b ⊓ c) := by
simp_rw [inf_comm _ c, inf_symmDiff_distrib_left] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | inf_symmDiff_distrib_right | null |
sdiff_symmDiff : c \ a ∆ b = c ⊓ a ⊓ b ⊔ c \ a ⊓ c \ b := by
simp only [(· ∆ ·), sdiff_sdiff_sup_sdiff'] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | sdiff_symmDiff | null |
sdiff_symmDiff' : c \ a ∆ b = c ⊓ a ⊓ b ⊔ c \ (a ⊔ b) := by
rw [sdiff_symmDiff, sdiff_sup]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | sdiff_symmDiff' | null |
symmDiff_sdiff_left : a ∆ b \ a = b \ a := by
rw [symmDiff_def, sup_sdiff, sdiff_idem, sdiff_sdiff_self, bot_sup_eq]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_sdiff_left | null |
symmDiff_sdiff_right : a ∆ b \ b = a \ b := by rw [symmDiff_comm, symmDiff_sdiff_left]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_sdiff_right | null |
sdiff_symmDiff_left : a \ a ∆ b = a ⊓ b := by simp [sdiff_symmDiff]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | sdiff_symmDiff_left | null |
sdiff_symmDiff_right : b \ a ∆ b = a ⊓ b := by
rw [symmDiff_comm, inf_comm, sdiff_symmDiff_left] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | sdiff_symmDiff_right | null |
symmDiff_eq_sup : a ∆ b = a ⊔ b ↔ Disjoint a b := by
refine ⟨fun h => ?_, Disjoint.symmDiff_eq_sup⟩
rw [symmDiff_eq_sup_sdiff_inf, sdiff_eq_self_iff_disjoint] at h
exact h.of_disjoint_inf_of_le le_sup_left
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_eq_sup | null |
le_symmDiff_iff_left : a ≤ a ∆ b ↔ Disjoint a b := by
refine ⟨fun h => ?_, fun h => h.symmDiff_eq_sup.symm ▸ le_sup_left⟩
rw [symmDiff_eq_sup_sdiff_inf] at h
exact disjoint_iff_inf_le.mpr (le_sdiff_right.1 <| inf_le_of_left_le h).le
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | le_symmDiff_iff_left | null |
le_symmDiff_iff_right : b ≤ a ∆ b ↔ Disjoint a b := by
rw [symmDiff_comm, le_symmDiff_iff_left, disjoint_comm] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | le_symmDiff_iff_right | null |
symmDiff_symmDiff_left :
a ∆ b ∆ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) ⊔ c \ (a ⊔ b) ⊔ a ⊓ b ⊓ c :=
calc
a ∆ b ∆ c = a ∆ b \ c ⊔ c \ a ∆ b := symmDiff_def _ _
_ = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) ⊔ (c \ (a ⊔ b) ⊔ c ⊓ a ⊓ b) := by
{ rw [sdiff_symmDiff', sup_comm (c ⊓ a ⊓ b), symmDiff_sdiff] }
_ = a \ (b ⊔ c) ... | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_symmDiff_left | null |
symmDiff_symmDiff_right :
a ∆ (b ∆ c) = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) ⊔ c \ (a ⊔ b) ⊔ a ⊓ b ⊓ c :=
calc
a ∆ (b ∆ c) = a \ b ∆ c ⊔ b ∆ c \ a := symmDiff_def _ _
_ = a \ (b ⊔ c) ⊔ a ⊓ b ⊓ c ⊔ (b \ (c ⊔ a) ⊔ c \ (b ⊔ a)) := by
{ rw [sdiff_symmDiff', sup_comm (a ⊓ b ⊓ c), symmDiff_sdiff] }
_ = a \ (b ... | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_symmDiff_right | null |
symmDiff_assoc : a ∆ b ∆ c = a ∆ (b ∆ c) := by
rw [symmDiff_symmDiff_left, symmDiff_symmDiff_right] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_assoc | null |
symmDiff_isAssociative : Std.Associative (α := α) (· ∆ ·) :=
⟨symmDiff_assoc⟩ | instance | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_isAssociative | null |
symmDiff_left_comm : a ∆ (b ∆ c) = b ∆ (a ∆ c) := by
simp_rw [← symmDiff_assoc, symmDiff_comm] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_left_comm | null |
symmDiff_right_comm : a ∆ b ∆ c = a ∆ c ∆ b := by simp_rw [symmDiff_assoc, symmDiff_comm] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_right_comm | null |
symmDiff_symmDiff_symmDiff_comm : a ∆ b ∆ (c ∆ d) = a ∆ c ∆ (b ∆ d) := by
simp_rw [symmDiff_assoc, symmDiff_left_comm]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_symmDiff_symmDiff_comm | null |
symmDiff_symmDiff_cancel_left : a ∆ (a ∆ b) = b := by simp [← symmDiff_assoc]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_symmDiff_cancel_left | null |
symmDiff_symmDiff_cancel_right : b ∆ a ∆ a = b := by simp [symmDiff_assoc]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_symmDiff_cancel_right | null |
symmDiff_symmDiff_self' : a ∆ b ∆ a = b := by
rw [symmDiff_comm, symmDiff_symmDiff_cancel_left] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_symmDiff_self' | null |
symmDiff_left_involutive (a : α) : Involutive (· ∆ a) :=
symmDiff_symmDiff_cancel_right _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_left_involutive | null |
symmDiff_right_involutive (a : α) : Involutive (a ∆ ·) :=
symmDiff_symmDiff_cancel_left _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_right_involutive | null |
symmDiff_left_injective (a : α) : Injective (· ∆ a) :=
Function.Involutive.injective (symmDiff_left_involutive a) | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_left_injective | null |
symmDiff_right_injective (a : α) : Injective (a ∆ ·) :=
Function.Involutive.injective (symmDiff_right_involutive _) | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_right_injective | null |
symmDiff_left_surjective (a : α) : Surjective (· ∆ a) :=
Function.Involutive.surjective (symmDiff_left_involutive _) | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_left_surjective | null |
symmDiff_right_surjective (a : α) : Surjective (a ∆ ·) :=
Function.Involutive.surjective (symmDiff_right_involutive _)
variable {a b c}
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_right_surjective | null |
symmDiff_left_inj : a ∆ b = c ∆ b ↔ a = c :=
(symmDiff_left_injective _).eq_iff
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_left_inj | null |
symmDiff_right_inj : a ∆ b = a ∆ c ↔ b = c :=
(symmDiff_right_injective _).eq_iff
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_right_inj | null |
symmDiff_eq_left : a ∆ b = a ↔ b = ⊥ :=
calc
a ∆ b = a ↔ a ∆ b = a ∆ ⊥ := by rw [symmDiff_bot]
_ ↔ b = ⊥ := by rw [symmDiff_right_inj]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_eq_left | null |
symmDiff_eq_right : a ∆ b = b ↔ a = ⊥ := by rw [symmDiff_comm, symmDiff_eq_left] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_eq_right | null |
protected Disjoint.symmDiff_left (ha : Disjoint a c) (hb : Disjoint b c) :
Disjoint (a ∆ b) c := by
rw [symmDiff_eq_sup_sdiff_inf]
exact (ha.sup_left hb).disjoint_sdiff_left | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | Disjoint.symmDiff_left | null |
protected Disjoint.symmDiff_right (ha : Disjoint a b) (hb : Disjoint a c) :
Disjoint a (b ∆ c) :=
(ha.symm.symmDiff_left hb.symm).symm | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | Disjoint.symmDiff_right | null |
symmDiff_eq_iff_sdiff_eq (ha : a ≤ c) : a ∆ b = c ↔ c \ a = b := by
rw [← symmDiff_of_le ha]
exact ((symmDiff_right_involutive a).toPerm _).apply_eq_iff_eq_symm_apply.trans eq_comm | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_eq_iff_sdiff_eq | null |
@[simp]
inf_himp_bihimp : a ⇔ b ⇨ a ⊓ b = a ⊔ b :=
@sup_sdiff_symmDiff αᵒᵈ _ _ _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | inf_himp_bihimp | null |
codisjoint_bihimp_sup : Codisjoint (a ⇔ b) (a ⊔ b) :=
@disjoint_symmDiff_inf αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | codisjoint_bihimp_sup | null |
himp_bihimp_left : a ⇨ a ⇔ b = a ⇨ b :=
@symmDiff_sdiff_left αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | himp_bihimp_left | null |
himp_bihimp_right : b ⇨ a ⇔ b = b ⇨ a :=
@symmDiff_sdiff_right αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | himp_bihimp_right | null |
bihimp_himp_left : a ⇔ b ⇨ a = a ⊔ b :=
@sdiff_symmDiff_left αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_himp_left | null |
bihimp_himp_right : a ⇔ b ⇨ b = a ⊔ b :=
@sdiff_symmDiff_right αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_himp_right | null |
bihimp_eq_inf : a ⇔ b = a ⊓ b ↔ Codisjoint a b :=
@symmDiff_eq_sup αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_eq_inf | null |
bihimp_le_iff_left : a ⇔ b ≤ a ↔ Codisjoint a b :=
@le_symmDiff_iff_left αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_le_iff_left | null |
bihimp_le_iff_right : a ⇔ b ≤ b ↔ Codisjoint a b :=
@le_symmDiff_iff_right αᵒᵈ _ _ _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_le_iff_right | null |
bihimp_assoc : a ⇔ b ⇔ c = a ⇔ (b ⇔ c) :=
@symmDiff_assoc αᵒᵈ _ _ _ _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_assoc | null |
bihimp_isAssociative : Std.Associative (α := α) (· ⇔ ·) :=
⟨bihimp_assoc⟩ | instance | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_isAssociative | null |
bihimp_left_comm : a ⇔ (b ⇔ c) = b ⇔ (a ⇔ c) := by simp_rw [← bihimp_assoc, bihimp_comm] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_left_comm | null |
bihimp_right_comm : a ⇔ b ⇔ c = a ⇔ c ⇔ b := by simp_rw [bihimp_assoc, bihimp_comm] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_right_comm | null |
bihimp_bihimp_bihimp_comm : a ⇔ b ⇔ (c ⇔ d) = a ⇔ c ⇔ (b ⇔ d) := by
simp_rw [bihimp_assoc, bihimp_left_comm]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_bihimp_bihimp_comm | null |
bihimp_bihimp_cancel_left : a ⇔ (a ⇔ b) = b := by simp [← bihimp_assoc]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_bihimp_cancel_left | null |
bihimp_bihimp_cancel_right : b ⇔ a ⇔ a = b := by simp [bihimp_assoc]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_bihimp_cancel_right | null |
bihimp_bihimp_self : a ⇔ b ⇔ a = b := by rw [bihimp_comm, bihimp_bihimp_cancel_left] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_bihimp_self | null |
bihimp_left_involutive (a : α) : Involutive (· ⇔ a) :=
bihimp_bihimp_cancel_right _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_left_involutive | null |
bihimp_right_involutive (a : α) : Involutive (a ⇔ ·) :=
bihimp_bihimp_cancel_left _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_right_involutive | null |
bihimp_left_injective (a : α) : Injective (· ⇔ a) :=
@symmDiff_left_injective αᵒᵈ _ _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_left_injective | null |
bihimp_right_injective (a : α) : Injective (a ⇔ ·) :=
@symmDiff_right_injective αᵒᵈ _ _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_right_injective | null |
bihimp_left_surjective (a : α) : Surjective (· ⇔ a) :=
@symmDiff_left_surjective αᵒᵈ _ _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_left_surjective | null |
bihimp_right_surjective (a : α) : Surjective (a ⇔ ·) :=
@symmDiff_right_surjective αᵒᵈ _ _
variable {a b c}
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_right_surjective | null |
bihimp_left_inj : a ⇔ b = c ⇔ b ↔ a = c :=
(bihimp_left_injective _).eq_iff
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_left_inj | null |
bihimp_right_inj : a ⇔ b = a ⇔ c ↔ b = c :=
(bihimp_right_injective _).eq_iff
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_right_inj | null |
bihimp_eq_left : a ⇔ b = a ↔ b = ⊤ :=
@symmDiff_eq_left αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_eq_left | null |
bihimp_eq_right : a ⇔ b = b ↔ a = ⊤ :=
@symmDiff_eq_right αᵒᵈ _ _ _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_eq_right | null |
protected Codisjoint.bihimp_left (ha : Codisjoint a c) (hb : Codisjoint b c) :
Codisjoint (a ⇔ b) c :=
(ha.inf_left hb).mono_left inf_le_bihimp | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | Codisjoint.bihimp_left | null |
protected Codisjoint.bihimp_right (ha : Codisjoint a b) (hb : Codisjoint a c) :
Codisjoint a (b ⇔ c) :=
(ha.inf_right hb).mono_right inf_le_bihimp | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | Codisjoint.bihimp_right | null |
symmDiff_eq : a ∆ b = a ⊓ bᶜ ⊔ b ⊓ aᶜ := by simp only [(· ∆ ·), sdiff_eq] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_eq | null |
bihimp_eq : a ⇔ b = (a ⊔ bᶜ) ⊓ (b ⊔ aᶜ) := by simp only [(· ⇔ ·), himp_eq] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_eq | null |
symmDiff_eq' : a ∆ b = (a ⊔ b) ⊓ (aᶜ ⊔ bᶜ) := by
rw [symmDiff_eq_sup_sdiff_inf, sdiff_eq, compl_inf] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_eq' | null |
bihimp_eq' : a ⇔ b = a ⊓ b ⊔ aᶜ ⊓ bᶜ :=
@symmDiff_eq' αᵒᵈ _ _ _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_eq' | null |
symmDiff_top : a ∆ ⊤ = aᶜ :=
symmDiff_top' _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_top | null |
top_symmDiff : ⊤ ∆ a = aᶜ :=
top_symmDiff' _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | top_symmDiff | null |
compl_symmDiff : (a ∆ b)ᶜ = a ⇔ b := by
simp_rw [symmDiff, compl_sup_distrib, compl_sdiff, bihimp, inf_comm]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | compl_symmDiff | null |
compl_bihimp : (a ⇔ b)ᶜ = a ∆ b :=
@compl_symmDiff αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | compl_bihimp | null |
compl_symmDiff_compl : aᶜ ∆ bᶜ = a ∆ b :=
(sup_comm _ _).trans <| by simp_rw [compl_sdiff_compl, sdiff_eq, symmDiff_eq]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | compl_symmDiff_compl | null |
compl_bihimp_compl : aᶜ ⇔ bᶜ = a ⇔ b :=
@compl_symmDiff_compl αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | compl_bihimp_compl | null |
symmDiff_eq_top : a ∆ b = ⊤ ↔ IsCompl a b := by
rw [symmDiff_eq', ← compl_inf, inf_eq_top_iff, compl_eq_top, isCompl_iff, disjoint_iff,
codisjoint_iff, and_comm]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_eq_top | null |
bihimp_eq_bot : a ⇔ b = ⊥ ↔ IsCompl a b := by
rw [bihimp_eq', ← compl_sup, sup_eq_bot_iff, compl_eq_bot, isCompl_iff, disjoint_iff,
codisjoint_iff]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_eq_bot | null |
compl_symmDiff_self : aᶜ ∆ a = ⊤ :=
hnot_symmDiff_self _
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | compl_symmDiff_self | null |
symmDiff_compl_self : a ∆ aᶜ = ⊤ :=
symmDiff_hnot_self _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_compl_self | null |
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