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 ⌀ |
|---|---|---|---|---|---|---|
coheight_int (n : ℤ) : coheight n = ⊤ := coheight_of_noMaxOrder .. | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_int | null |
krullDim_int : krullDim ℤ = ⊤ := krullDim_of_noMaxOrder ..
@[simp] lemma height_coe_withBot (x : α) : height (x : WithBot α) = height x + 1 := by
apply le_antisymm
· apply height_le
intro p hlast
wlog hlenpos : p.length ≠ 0
· simp_all
let p' : LTSeries α := {
length := p.length - 1
toFun... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_int | null |
krullDim_enat : krullDim ℕ∞ = ⊤ := by
change (krullDim (WithTop ℕ) = ⊤)
simp [← WithBot.coe_top, ← WithBot.coe_one, ← WithBot.coe_add]
@[simp] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_enat | null |
height_enat (n : ℕ∞) : height n = n := by
cases n with
| top => simp only [← WithBot.coe_eq_coe, height_top_eq_krullDim, krullDim_enat, WithBot.coe_top]
| coe n => exact (height_coe_withTop _).trans (height_nat _)
@[simp] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_enat | null |
coheight_coe_enat (n : ℕ) : coheight (n : ℕ∞) = ⊤ := by
apply (coheight_coe_withTop _).trans
simp only [coheight_nat, top_add] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_coe_enat | null |
height_le_of_krullDim_preimage_le (x : α) :
Order.height x ≤ (m + 1) * Order.height (f x) + m := by
generalize h' : Order.height (f x) = n
cases n with | top => simp | coe n =>
induction n using Nat.strong_induction_on generalizing x with | h n ih =>
refine height_le_iff.mpr fun p hp ↦ le_of_not_gt fun ... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_le_of_krullDim_preimage_le | null |
coheight_le_of_krullDim_preimage_le (x : α) :
Order.coheight x ≤ (m + 1) * Order.coheight (f x) + m := by
rw [Order.coheight, Order.coheight]
apply height_le_of_krullDim_preimage_le (f := f.dual)
exact fun x ↦ le_of_eq_of_le (krullDim_orderDual (α := f ⁻¹' {x})) (h x)
include f h in | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_le_of_krullDim_preimage_le | null |
krullDim_le_of_krullDim_preimage_le :
Order.krullDim α ≤ (m + 1) * Order.krullDim β + m := by
rw [Order.krullDim_eq_iSup_height, Order.krullDim_eq_iSup_height]
apply iSup_le fun x ↦ (le_trans (WithBot.coe_mono (height_le_of_krullDim_preimage_le f h x)) ?_)
push_cast
apply add_le_add_right <| mul_le_mul_of_n... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_le_of_krullDim_preimage_le | null |
krullDim_le_of_krullDim_preimage_le' (f : α → β) (h_mono : Monotone f)
(h : ∀ (x : β), Order.krullDim (f ⁻¹' {x}) ≤ m) :
Order.krullDim α ≤ (m + 1) * Order.krullDim β + m :=
Order.krullDim_le_of_krullDim_preimage_le ⟨f, h_mono⟩ h | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_le_of_krullDim_preimage_le' | Another version when the `OrderHom` is unbundled |
SemilatticeSup (α : Type u) extends PartialOrder α where
/-- The binary supremum, used to derive `Max α` -/
sup : α → α → α
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ sup a b
/-- The supremum is an upper bound on the second argument -/
protected le_sup... | class | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | SemilatticeSup | A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. |
SemilatticeSup.toMax [SemilatticeSup α] : Max α where max a b := SemilatticeSup.sup a b | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | SemilatticeSup.toMax | null |
SemilatticeSup.mk' {α : Type*} [Max α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by rw [← hbc, ← sup_assoc, hab]
le_a... | def | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | SemilatticeSup.mk' | A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`. |
@[simp]
le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | le_sup_left | null |
le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | le_sup_right | null |
le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | le_sup_of_le_left | null |
le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | le_sup_of_le_right | null |
lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | lt_sup_of_lt_left | null |
lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | lt_sup_of_lt_right | null |
sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_le | null |
sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_le_iff | null |
sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans <| by simp
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_eq_left | null |
sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans <| by simp
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_eq_right | null |
left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | left_eq_sup | null |
right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
attribute [simp] sup_of_le_left sup_of_le_right
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | right_eq_sup | null |
left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans <| not_congr left_eq_sup
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | left_lt_sup | null |
right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans <| not_congr right_eq_sup | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | right_lt_sup | null |
left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_ge.symm.imp left_lt_sup.2 right_lt_sup.2 | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | left_or_right_lt_sup | null |
le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
@[gcongr] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | le_iff_exists_sup | null |
sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂) | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_le_sup | null |
sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_le_sup_left | null |
sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_le_sup_right | null |
sup_idem (a : α) : a ⊔ a = a := by simp | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_idem | null |
sup_comm (a b : α) : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_comm | null |
sup_assoc (a b c : α) : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff fun x => by simp only [sup_le_iff]; rw [and_assoc] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_assoc | null |
sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, sup_comm a, sup_assoc] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_left_right_swap | null |
sup_left_idem (a b : α) : a ⊔ (a ⊔ b) = a ⊔ b := by simp | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_left_idem | null |
sup_right_idem (a b : α) : a ⊔ b ⊔ b = a ⊔ b := by simp | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_right_idem | null |
sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_left_comm | null |
sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, sup_comm b] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_right_comm | null |
sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_sup_sup_comm | null |
sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_sup_distrib_left | null |
sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_sup_distrib_right | null |
sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm <| sup_le le_sup_left hc | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_congr_left | null |
sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm <| sup_le hb le_sup_right | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_congr_right | null |
sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_eq_sup_iff_left | null |
sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_eq_sup_iff_right | null |
Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_ge.imp left_lt_sup.2 right_lt_sup.2 | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | Ne.lt_sup_or_lt_sup | null |
Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | Monotone.forall_le_of_antitone | If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. |
SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | SemilatticeSup.ext_sup | null |
SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases PartialOrder.ext H
congr
ext; apply SemilatticeSup.ext_sup H | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | SemilatticeSup.ext | null |
ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | ite_le_sup | null |
SemilatticeInf (α : Type u) extends PartialOrder α where
/-- The binary infimum, used to derive `Min α` -/
inf : α → α → α
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, inf a b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_righ... | class | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | SemilatticeInf | A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. |
SemilatticeInf.toMin [SemilatticeInf α] : Min α where min a b := SemilatticeInf.inf a b | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | SemilatticeInf.toMin | null |
OrderDual.instSemilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
sup := @SemilatticeInf.inf α _
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | OrderDual.instSemilatticeSup | null |
OrderDual.instSemilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
inf := @SemilatticeSup.sup α _
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | OrderDual.instSemilatticeInf | null |
SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.instSemilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext fun _ _ => Iff.rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | SemilatticeSup.dual_dual | null |
@[simp]
inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_le_left | null |
inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_le_right | null |
le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | le_inf | null |
inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_le_of_left_le | null |
inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_le_of_right_le | null |
inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_lt_of_left_lt | null |
inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_lt_of_right_lt | null |
le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | le_inf_iff | null |
inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans <| by simp
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_eq_left | null |
inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans <| by simp
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_eq_right | null |
left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | left_eq_inf | null |
right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | right_eq_inf | null |
inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_lt_left | null |
inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_lt_right | null |
inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
@[gcongr] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_lt_left_or_right | null |
inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_le_inf | null |
inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_le_inf_right | null |
inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_le_inf_left | null |
inf_idem (a : α) : a ⊓ a = a := by simp | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_idem | null |
inf_comm (a b : α) : a ⊓ b = b ⊓ a := @sup_comm αᵒᵈ _ _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_comm | null |
inf_assoc (a b c : α) : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) := @sup_assoc αᵒᵈ _ _ _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_assoc | null |
inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_left_right_swap | null |
inf_left_idem (a b : α) : a ⊓ (a ⊓ b) = a ⊓ b := by simp | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_left_idem | null |
inf_right_idem (a b : α) : a ⊓ b ⊓ b = a ⊓ b := by simp | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_right_idem | null |
inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_left_comm | null |
inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_right_comm | null |
inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_inf_inf_comm | null |
inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_inf_distrib_left | null |
inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_inf_distrib_right | null |
inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_congr_left | null |
inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2 | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_congr_right | null |
inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_eq_inf_iff_left | null |
inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_eq_inf_iff_right | null |
Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | Ne.inf_lt_or_inf_lt | null |
SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | SemilatticeInf.ext_inf | null |
SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases PartialOrder.ext H
congr
ext; apply SemilatticeInf.ext_inf H | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | SemilatticeInf.ext | null |
SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.instSemilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext fun _ _ => Iff.rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | SemilatticeInf.dual_dual | null |
inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_le_ite | null |
SemilatticeInf.mk' {α : Type*} [Min α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.instSemilatticeInf αᵒᵈ
... | def | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | SemilatticeInf.mk' | A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`. |
Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α | class | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | Lattice | A lattice is a join-semilattice which is also a meet-semilattice. |
OrderDual.instLattice (α) [Lattice α] : Lattice αᵒᵈ where | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | OrderDual.instLattice | null |
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Max α] [Min α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c))... | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder | The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). |
Lattice.mk' {α : Type*} [Max α] [Min α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : La... | def | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | Lattice.mk' | A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`. |
inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_le_sup | null |
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