mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : CompleteLattice β
f : CompleteLatticeHom α β
carrier : Set α
sSupClosed' : ∀ ⦃s : Set α⦄, s ⊆ carrier → sSup s ∈ carrier
sInfClosed' : ∀ ⦃s : Set α⦄, s ⊆ carrier → sInf s ∈ carrier
x : α
hx : x ∈ carrier
y : α
hy : y ∈ carrier
this : x ⊓ y = sInf {x, y}
z : α
hz : z ∈ {x, y}
⊢ z ∈ carrier

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : CompleteLattice β
f : CompleteLatticeHom α β
carrier : Set α
sSupClosed' : ∀ ⦃s : Set α⦄, s ⊆ carrier → sSup s ∈ carrier
sInfClosed' : ∀ ⦃s : Set α⦄, s ⊆ carrier → sInf s ∈ carrier
x : α
hx : x ∈ carrier
y : α
hy : y ∈ carrier
this : x ⊓ y = sInf {x, y}
⊢ x ⊓ y ∈ carrier

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteLattice.PiLex

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : LinearOrder ι
inst✝¹ : (i : ι) → CompleteLinearOrder (α i)
inst✝ : WellFoundedLT ι
s : Set (Lex ((i : ι) → (fun i ↦ α i) i))
⊢ IsGLB s (sInf s)

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : CompleteLattice β
f : CompleteLatticeHom α β
carrier : Set α
sSupClosed' : ∀ ⦃s : Set α⦄, s ⊆ carrier → sSup s ∈ carrier
sInfClosed' : ∀ ⦃s : Set α⦄, s ⊆ carrier → sInf s ∈ carrier
x : α
hx : x ∈ carrier
y : α
hy : y ∈ carrier
⊢ x ⊓ y ∈ carrier

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteLattice.PiLex

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : LinearOrder ι
inst✝¹ : (i : ι) → CompleteLinearOrder (α i)
inst✝ : WellFoundedLT ι
s : Set (Lex ((i : ι) → (fun i ↦ α i) i))
⊢ IsLUB s (sSup s)

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteLattice.PiLex

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : LinearOrder ι
inst✝¹ : (i : ι) → CompleteLinearOrder (α i)
inst✝ : WellFoundedLT ι
x✝ : Set (Lex ((i : ι) → (fun i ↦ α i) i))
⊢ IsLUB x✝ (sSup x✝)

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteLattice.PiLex

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : LinearOrder ι
inst✝¹ : (i : ι) → CompleteLinearOrder (α i)
inst✝ : WellFoundedLT ι
x✝ : Set (Lex ((i : ι) → (fun i ↦ α i) i))
⊢ IsGLB x✝ (sInf x✝)

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : CompleteLattice β
f : CompleteLatticeHom α β
L✝ L M : CompleteSublattice α
h : (fun L ↦ L.carrier) L = (fun L ↦ L.carrier) M
⊢ L = M

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteLattice.PiLex

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : LinearOrder ι
inst✝¹ : (i : ι) → CompleteLinearOrder (α i)
inst✝ : WellFoundedGT ι
x✝ : Set (Colex ((i : ι) → α i))
⊢ IsLUB x✝ (sSup x✝)

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
L : CompleteSublattice α
⊢ ⊤ ∈ L

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteLattice.PiLex

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : LinearOrder ι
inst✝¹ : (i : ι) → CompleteLinearOrder (α i)
inst✝ : WellFoundedGT ι
x✝ : Set (Colex ((i : ι) → α i))
⊢ IsGLB x✝ (sInf x✝)

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
L : CompleteSublattice α
⊢ ⊥ ∈ L

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
L : CompleteSublattice α
S : Set ↥L
⊢ ↑(sSup S) = ⨆ N ∈ S, ↑N

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
L : CompleteSublattice α
S : Set ↥L
⊢ ↑(sInf S) = ⨅ N ∈ S, ↑N

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.WellApproximable

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p ↦ Nat.Prime p
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p ↦ ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n ↦ approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
hE₀ : NullMeasurableSet E μ
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∧ B ↑p =ᵐ[μ] ∅ → E =ᵐ[μ] C ↑p
⊢ ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∨ A ↑p =ᵐ[μ] univ

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
L : CompleteSublattice α
ι : Sort u_3
f : ι → ↥L
⊢ ↑(iSup f) = ⨆ i, ↑(f i)

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
L : CompleteSublattice α
ι : Sort u_3
f : ι → ↥L
⊢ ↑(iInf f) = ⨅ i, ↑(f i)

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : CompleteLattice β
f : CompleteLatticeHom α β
L✝ L : CompleteSublattice α
s : Set β
hs : s ⊆ ⇑f '' ↑L
⊢ sSup s ∈ ⇑f '' ↑L

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : CompleteLattice β
f : CompleteLatticeHom α β
L✝ L : CompleteSublattice α
s : Set β
hs : s ⊆ ⇑f '' ↑L
⊢ sInf s ∈ ⇑f '' ↑L

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : CompleteLattice β
f : CompleteLatticeHom α β
L✝ : CompleteSublattice α
L : CompleteSublattice β
s : Set α
hs : s ⊆ ⇑f ⁻¹' ↑L
⊢ sSup s ∈ ⇑f ⁻¹' ↑L

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : CompleteLattice β
f : CompleteLatticeHom α β
L✝ : CompleteSublattice α
L : CompleteSublattice β
s : Set α
hs : s ⊆ ⇑f ⁻¹' ↑L
⊢ sInf s ∈ ⇑f ⁻¹' ↑L

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
L : CompleteSublattice α
a b : ↥L
⊢ Disjoint a b ↔ Disjoint ↑a ↑b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
a : α
b : β
⊢ b ∈ upperPolar r {a} ↔ r a b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
a : α
b : β
⊢ a ∈ lowerPolar r {b} ↔ r a b

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
L : CompleteSublattice α
a b : ↥L
⊢ Codisjoint a b ↔ Codisjoint ↑a ↑b

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
L : CompleteSublattice α
a b : ↥L
⊢ IsCompl a b ↔ IsCompl ↑a ↑b

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompleteSublattice

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
L : CompleteSublattice α
⊢ ComplementedLattice ↥L ↔ ∀ a ∈ L, ∃ b ∈ L, IsCompl a b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
s s' : Set α
⊢ IsExtent r s → IsExtent r s' → IsExtent r (s ∩ s')

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
s s' : Set α
h : IsExtent r s
hs' : s' ⊆ s
⊢ lowerPolar r (upperPolar r s') ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.WellApproximable

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this✝ : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p ↦ Nat.Prime p
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p ↦ ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n ↦ approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
hE₀ : NullMeasurableSet E μ
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∧ B ↑p =ᵐ[μ] ∅ → E =ᵐ[μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∨ A ↑p =ᵐ[μ] univ
p : ℕ
hp : Nat.Prime p
x : 𝕊 := u ⟨p, hp⟩
f : 𝕊 → 𝕊 := fun y ↦ p • y + x
this : f '' B p ⊆ blimsup (fun n ↦ approxAddOrderOf 𝕊 n (↑p * δ n)) atTop fun n ↦ 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
⊢ B ↑⟨p, hp⟩ =ᵐ[μ] ∅ ∨ B ↑⟨p, hp⟩ =ᵐ[μ] univ

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
t t' : Set β
h : IsIntent r t
ht' : t' ⊆ t
⊢ upperPolar r (lowerPolar r t') ⊆ t

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
c d : Concept α β r
h : c.extent = d.extent
⊢ c = d

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
c d : Concept α β r
h : c.intent = d.intent
⊢ c = d

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
c : Concept α β r
e : Set α
i : Set β
he : e = c.extent
hi : i = c.intent
⊢ c.copy e i he hi = c

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
c : Concept α β r
x : α
y : β
hx : x ∈ c.extent
hy : y ∈ c.intent
⊢ r x y

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
r' : α → α → Prop
c' : Concept α α r'
inst✝ : Std.Irrefl r'
⊢ Disjoint c'.extent c'.intent

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
r' : α → α → Prop
c' : Concept α α r'
inst✝ : IsTrans α r'
x y : α
hy : r' y x
hx : x ∈ c'.extent
⊢ y ∈ c'.extent

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
r' : α → α → Prop
c' : Concept α α r'
inst✝ : IsTrans α r'
x y : α
hy : r' x y
hx : x ∈ c'.intent
⊢ y ∈ c'.intent

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
f : α → β
hf : Monotone f
A : DedekindCut α
⊢ f '' A.left ⊆ lowerBounds (f '' A.right)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
f : α → β
hf : Monotone f
A : DedekindCut α
⊢ f '' A.right ⊆ upperBounds (f '' A.left)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
a : α
⊢ Iic a = (ofObject (fun x1 x2 ↦ x1 ≤ x2) a).extent

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
a : α
⊢ Ici a = (ofObject (fun x1 x2 ↦ x1 ≤ x2) a).intent

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
r' : α → α → Prop
c' : Concept α α r'
inst✝¹ : Std.Trichotomous r'
inst✝ : IsTrans α r'
⊢ Codisjoint c'.extent c'.intent

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a : α
⊢ ofAttribute (fun x1 x2 ↦ x1 ≤ x2) a = principal a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a b : α
⊢ principal a ≤ principal b ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a b : α
⊢ principal a < principal b ↔ a < b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a : α
c : DedekindCut α
⊢ principal a ≤ c ↔ a ∈ c.left

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a : α
c : DedekindCut α
⊢ c ≤ principal a ↔ a ∈ c.right

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
a b : α
⊢ principal a = principal b ↔ a = b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
c d : Concept α β r
⊢ c.intent ⊆ d.intent ↔ d ≤ c

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
A : DedekindCut α
⊢ principal (sSup A.left) = A

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
c d : Concept α β r
⊢ c.intent ⊂ d.intent ↔ d < c

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
A : DedekindCut α
⊢ principal (sInf A.right) = A

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_6
inst✝ : PartialOrder α
c : Concept α α fun x1 x2 ↦ x1 < x2
⊢ IsLowerSet c.extent

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : PartialOrder β
f : β ↪o α
⊢ ∀ (a b : DedekindCut β), sSup (⇑f '' a.left) ≤ sSup (⇑f '' b.left) ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_6
inst✝ : PartialOrder α
c : Concept α α fun x1 x2 ↦ x1 < x2
⊢ IsUpperSet c.intent

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.WellApproximable

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p ↦ Nat.Prime p
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p ↦ ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n ↦ approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
hE₀ : NullMeasurableSet E μ
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∧ B ↑p =ᵐ[μ] ∅ → E =ᵐ[μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∨ A ↑p =ᵐ[μ] univ
p : ℕ
hp : Nat.Prime p
x : 𝕊 := u ⟨p, hp⟩
f : 𝕊 → 𝕊 := fun y ↦ p • y + x
q : ℕ
hn : 0 < p * q
h_ndiv : ¬p * p ∣ p * q
⊢ ¬p ∣ q

Tactic:
exact fun contra => h_ndiv (mul_dvd_mul_left p contra)

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : PartialOrder β
f : β ↪o α
x : β
⊢ sSup (⇑f '' (principal x).left) ≤ f x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s : Set α
t : Set β
a : α
b : β
r' : α → α → Prop
c d : Concept α β r
c' : Concept α α r'
⊢ ∀ {x y : Concept α β r}, (⇑toDual ∘ intent) x ≤ (⇑toDual ∘ intent) y ↔ x ≤ y

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.WellApproximable

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p ↦ Nat.Prime p
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p ↦ ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n ↦ approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
hE₀ : NullMeasurableSet E μ
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∧ B ↑p =ᵐ[μ] ∅ → E =ᵐ[μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∨ A ↑p =ᵐ[μ] univ
p : ℕ
hp : Nat.Prime p
x : 𝕊 := u ⟨p, hp⟩
f : 𝕊 → 𝕊 := fun y ↦ p • y + x
n : ℕ
hn : 0 < n
h_div : p ∣ n
h_ndiv : ¬p * p ∣ n
⊢ (addOrderOf x).Coprime (n / p)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s : Set α
t : Set β
a : α
b : β
r' : α → α → Prop
c d : Concept α β r
c' : Concept α α r'
⊢ ∀ {x y : Concept α β r}, (⇑toDual ∘ intent) x < (⇑toDual ∘ intent) y ↔ x < y

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : PartialOrder β
f : β ↪o α
x : β
⊢ (factorEmbedding f) (principal x) = f x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
s : Set α
c : Concept α β r
⊢ ofObjects r s ≤ c ↔ s ⊆ c.extent

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
s : Set α
c : Concept α β r
h : c.extent ⊆ s
⊢ c ≤ ofObjects r s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : PartialOrder β
f : β ↪o α
⊢ RelEmbedding.trans principalEmbedding (factorEmbedding f) = f

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : PartialOrder β
x : DedekindCut α
⊢ principalEmbedding.toFun ((factorEmbedding (RelIso.toRelEmbedding (OrderIso.refl α))) x) = x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
A : DedekindCut α
⊢ sSup (⇑(RelIso.toRelEmbedding (OrderIso.refl α)) '' A.left) = sSup A.left

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
t : Set β
c : Concept α β r
⊢ c ≤ ofAttributes r t ↔ t ⊆ c.intent

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
t : Set β
c : Concept α β r
h : c.intent ⊆ t
⊢ ofAttributes r t ≤ c

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
a : α
b : β
⊢ ofObject r a ≤ ofAttribute r b ↔ r a b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
a b : DedekindCut α
⊢ a < b ↔ ∃ c, a < principal c ∧ principal c ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s✝ : Set α
t : Set β
a : α
b : β
r' : α → α → Prop
c d : Concept α β r
c' : Concept α α r'
s : Set (Concept α β r)
⊢ IsLUB s (sSup s)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
a b : DedekindCut α
⊢ a < b ↔ ∃ c, a ≤ principal c ∧ principal c < b

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.WellApproximable

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p ↦ Nat.Prime p
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p ↦ ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n ↦ approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
hE₀ : NullMeasurableSet E μ
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∧ B ↑p =ᵐ[μ] ∅ → E =ᵐ[μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∨ A ↑p =ᵐ[μ] univ
p : ℕ
hp : Nat.Prime p
x : 𝕊 := u ⟨p, hp⟩
f : 𝕊 → 𝕊 := fun y ↦ p • y + x
n : ℕ
hn : 0 < n
h_ndiv : ¬p * p ∣ n
h_cop : (addOrderOf x).Coprime (n / p)
h_div : n / p * p = n
⊢ f = (fun y ↦ x + y) ∘ fun y ↦ p • y

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: ι : Sort u_1
α : Type u_2
s : Set α
inst✝² : CompleteLattice α
inst✝¹ : Countable ι
inst✝ : Nonempty ι
hs : CountableSupClosed s
A : ι → α
hA : ∀ (n : ι), A n ∈ s
i₀ : ι := ⋯.some
⊢ range A ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
r : α → β → Prop
f : ι → Concept α β r
⊢ (⨆ i, f i).extent = lowerPolar r (⋂ i, (f i).intent)

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: ι : Sort u_1
α : Type u_2
s : Set α
inst✝² : CompleteLattice α
inst✝¹ : Countable ι
inst✝ : Nonempty ι
hs : CountableSupClosed s
A : ι → α
hA : ∀ (n : ι), A n ∈ s
i₀ : ι := ⋯.some
⊢ A i₀ ∈ range A

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: ι : Sort u_1
α : Type u_2
s : Set α
inst✝² : CompleteLattice α
inst✝¹ : Countable ι
inst✝ : Nonempty ι
hs : CountableSupClosed s
A : ι → α
hA : ∀ (n : ι), A n ∈ s
⊢ ⨆ n, A n ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : DenselyOrdered α
a b : DedekindCut α
h : a < b
c : α
hac : a < principal c
hcb : principal c ≤ b
d : α
had : a ≤ principal d
hdc : d < c
u : α
left✝ : d < u
right✝ : u < c
⊢ principal d < principal u

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
r : α → β → Prop
f : ι → Concept α β r
⊢ (⨆ i, f i).intent = ⋂ i, (f i).intent

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : DenselyOrdered α
a b : DedekindCut α
h : a < b
c : α
hac : a < principal c
hcb : principal c ≤ b
d : α
had : a ≤ principal d
hdc : d < c
u : α
left✝ : d < u
right✝ : u < c
⊢ principal u < principal c

Tactic:
by

Goal state after:
Module: Mathlib.Order.Concept

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
r : α → β → Prop
f : ι → Concept α β r
⊢ (⨅ i, f i).extent = ⋂ i, (f i).extent

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : DenselyOrdered α
a b : DedekindCut α
h : a < b
⊢ ∃ a_1, a < a_1 ∧ a_1 < b

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
s : Set α
inst✝ : CompleteLattice α
hs : ∀ (A : ℕ → α), (∀ (n : ℕ), A n ∈ s) → ⨆ n, A n ∈ s
x : α
f : ℕ → α
hAs : range f ⊆ s
hA_ne : (range f).Nonempty
hAc : (range f).Countable
hx : IsLUB (range f) x
⊢ ∀ (n : ℕ), f n ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
s : Set α
inst✝ : CompleteLattice α
hs : ∀ (A : ℕ → α), (∀ (n : ℕ), A n ∈ s) → ⨆ n, A n ∈ s
A : Set α
hAs : A ⊆ s
hA_ne : A.Nonempty
hAc : A.Countable
x : α
hx : IsLUB A x
⊢ x ∈ s

Tactic:
by

Goal state after: