mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: G : Type u
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : FirstCountableTopology G
N : Subgroup G
inst✝¹ : N.Normal
inst✝ : CompleteSpace G
this✝¹ : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this✝ : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : (𝓤 (G ⧸ N)).IsCountablyGenerated
u : ℕ → Set G
hu : (𝓝 1).HasAntitoneBasis u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : (𝓝 1).HasBasis (fun x ↦ True) fun x ↦ mk '' u x
v_anti : Antitone fun x ↦ mk '' u x
x : ℕ → G ⧸ N
h𝓤GN : (𝓤 (G ⧸ N)).HasBasis (fun x ↦ True) fun i ↦ {x | x.2 / x.1 ∈ mk '' u i}
hx : ∀ (i : ℕ), ∃ N_1, ∀ (m : ℕ), N_1 ≤ m → ∀ (n : ℕ), N_1 ≤ n → ∃ x_1 ∈ u i, ↑x_1 = x n / x m
i j M : ℕ
hM : ∀ (m : ℕ), M ≤ m → ∀ (n : ℕ), M ≤ n → ∃ x_1 ∈ u i, ↑x_1 = x n / x m
a b : ℕ
ha : max j M + 1 ≤ a
hb : max j M + 1 ≤ b
g : G
hg : x b = ↑g
y : G
y_mem : y ∈ u i
hy : ↑y = x b / x a
⊢ g / (y⁻¹ * g) ∈ u i

Tactic:
simpa using

Goal state after:
G : Type u
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : FirstCountableTopology G
N : Subgroup G
inst✝¹ : N.Normal
inst✝ : CompleteSpace G
this✝¹ : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this✝ : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : (𝓤 (G ⧸ N)).IsCountablyGenerated
u : ℕ → Set G
hu : (𝓝 1).HasAntitoneBasis u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : (𝓝 1).HasBasis (fun x ↦ True) fun x ↦ mk '' u x
v_anti : Antitone fun x ↦ mk '' u x
x : ℕ → G ⧸ N
h𝓤GN : (𝓤 (G ⧸ N)).HasBasis (fun x ↦ True) fun i ↦ {x | x.2 / x.1 ∈ mk '' u i}
hx : ∀ (i : ℕ), ∃ N_1, ∀ (m : ℕ), N_1 ≤ m → ∀ (n : ℕ), N_1 ≤ n → ∃ x_1 ∈ u i, ↑x_1 = x n / x m
i j M : ℕ
hM : ∀ (m : ℕ), M ≤ m → ∀ (n : ℕ), M ≤ n → ∃ x_1 ∈ u i, ↑x_1 = x n / x m
a b : ℕ
ha : max j M + 1 ≤ a
hb : max j M + 1 ≤ b
g : G
hg : x b = ↑g
y : G
y_mem : y ∈ u i
hy : ↑y = x b / x a
⊢ y ∈ u i
Module: Mathlib.Topology.Algebra.Order.Group

Goal state before:
pp: case inr.inl
G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
a : G
h : DenseRange fun x ↦ a ^ x
ha₀ : 1 < a
b : G
m : ℤ
hm : b < a ^ m
hm' : a ^ m < b * a * a
hne : b ≠ a ^ (m - 1)
hlt : b < a ^ (m - 1)
⊢ b * a * a < a ^ (m + 1)

Tactic:
simpa using

Goal state after:
case inr.inl
G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
a : G
h : DenseRange fun x ↦ a ^ x
ha₀ : 1 < a
b : G
m : ℤ
hm : b < a ^ m
hm' : a ^ m < b * a * a
hne : b ≠ a ^ (m - 1)
hlt : b < a ^ (m - 1)
⊢ b * a < a ^ m
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Semifield 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : 0 < C
hf : Tendsto f l (𝓝 C)
hg : Tendsto g l atTop
⊢ Tendsto (fun x ↦ f x * g x) l atTop

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Semifield 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : 0 < C
hf : Tendsto f l (𝓝 C)
hg : Tendsto g l atTop
⊢ Tendsto (fun x ↦ f x * g x) l atTop
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : Semifield 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
N : ℕ
hn : 0 < N
h : ↑N = -n
⊢ Tendsto (fun x ↦ x ^ n) atTop (𝓝 0)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁴ : Semifield 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
N : ℕ
hn : 0 < N
h : ↑N = -n
⊢ Tendsto (fun x ↦ x ^ n) atTop (𝓝 0)
Module: Mathlib.Topology.Order.LeftRightNhds

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}
y : α → α
hy : ∀ x ∈ t, y x > x
h'y : ∀ x ∈ t, s ∩ Ioo x (y x) = ∅
a : α
ha : a ∈ t
b : α
hb : b ∈ t
hab : a ≠ b
H : a < b
this : y a ≤ b
⊢ ∀ ⦃a_1 : α⦄, a_1 ∈ Ioo a (y a) → ∀ ⦃b_1 : α⦄, b_1 ∈ Ioo b (y b) → a_1 ≠ b_1

Tactic:
exact fun u hu v hv ↦ ((hu.2.trans_le this).trans hv.1).ne

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: G : Type u
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : FirstCountableTopology G
N : Subgroup G
inst✝¹ : N.Normal
inst✝ : CompleteSpace G
this✝¹ : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this✝ : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : (𝓤 (G ⧸ N)).IsCountablyGenerated
u : ℕ → Set G
hu : (𝓝 1).HasAntitoneBasis u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : (𝓝 1).HasBasis (fun x ↦ True) fun x ↦ mk '' u x
v_anti : Antitone fun x ↦ mk '' u x
x : ℕ → G ⧸ N
hx : CauchySeq x
key₀ : ∀ (i j : ℕ), ∃ M, j < M ∧ ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = ↑g → ∃ g', g / g' ∈ u i ∧ x a = ↑g'
φ : ℕ → ℕ := fun n ↦ Nat.recOn n (Classical.choose ⋯) fun k yk ↦ Classical.choose ⋯
hφ :
  ∀ (n : ℕ),
    φ n < φ (n + 1) ∧
      ∀ (a b : ℕ), φ (n + 1) ≤ a → φ (n + 1) ≤ b → ∀ (g : G), x b = ↑g → ∃ g', g / g' ∈ u (n + 1) ∧ x a = ↑g'
x' : (n : ℕ) → (g : G) ×' x (φ (n + 1)) = ↑g :=
  fun n ↦ Nat.recOn n ⟨Classical.choose ⋯, ⋯⟩ fun k hk ↦ ⟨Classical.choose ⋯, ⋯⟩
hx' : ∀ (n : ℕ), (x' n).fst / (x' (n + 1)).fst ∈ u (n + 1)
h𝓤G : (𝓤 G).HasBasis (fun x ↦ True) fun i ↦ {x | x.2 / x.1 ∈ u i}
m n : ℕ
hmn : n ≥ m
⊢ (x' n).fst / (x' n).fst ∈ u n

Tactic:
simpa using

Goal state after:
G : Type u
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : FirstCountableTopology G
N : Subgroup G
inst✝¹ : N.Normal
inst✝ : CompleteSpace G
this✝¹ : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this✝ : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : (𝓤 (G ⧸ N)).IsCountablyGenerated
u : ℕ → Set G
hu : (𝓝 1).HasAntitoneBasis u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : (𝓝 1).HasBasis (fun x ↦ True) fun x ↦ mk '' u x
v_anti : Antitone fun x ↦ mk '' u x
x : ℕ → G ⧸ N
hx : CauchySeq x
key₀ : ∀ (i j : ℕ), ∃ M, j < M ∧ ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = ↑g → ∃ g', g / g' ∈ u i ∧ x a = ↑g'
φ : ℕ → ℕ := fun n ↦ Nat.recOn n (Classical.choose ⋯) fun k yk ↦ Classical.choose ⋯
hφ :
  ∀ (n : ℕ),
    φ n < φ (n + 1) ∧
      ∀ (a b : ℕ), φ (n + 1) ≤ a → φ (n + 1) ≤ b → ∀ (g : G), x b = ↑g → ∃ g', g / g' ∈ u (n + 1) ∧ x a = ↑g'
x' : (n : ℕ) → (g : G) ×' x (φ (n + 1)) = ↑g :=
  fun n ↦ Nat.recOn n ⟨Classical.choose ⋯, ⋯⟩ fun k hk ↦ ⟨Classical.choose ⋯, ⋯⟩
hx' : ∀ (n : ℕ), (x' n).fst / (x' (n + 1)).fst ∈ u (n + 1)
h𝓤G : (𝓤 G).HasBasis (fun x ↦ True) fun i ↦ {x | x.2 / x.1 ∈ u i}
m n : ℕ
hmn : n ≥ m
⊢ 1 ∈ u n
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
⊢ (𝓝 0).HasBasis (fun x ↦ 0 < x) fun ε ↦ {x | |x| < ε}

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
⊢ (𝓝 0).HasBasis (fun x ↦ 0 < x) fun ε ↦ {x | |x| < ε}
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : C < 0
hf : Tendsto f l atTop
hg : Tendsto g l (𝓝 C)
this : Tendsto (fun x ↦ f x * -g x) l atTop
⊢ Tendsto (fun x ↦ f x * g x) l atBot

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : C < 0
hf : Tendsto f l atTop
hg : Tendsto g l (𝓝 C)
this : Tendsto (fun x ↦ f x * -g x) l atTop
⊢ Tendsto (fun x ↦ f x * g x) l atBot
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : C < 0
hf : Tendsto f l (𝓝 C)
hg : Tendsto g l atTop
⊢ Tendsto (fun x ↦ f x * g x) l atBot

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : C < 0
hf : Tendsto f l (𝓝 C)
hg : Tendsto g l atTop
⊢ Tendsto (fun x ↦ f x * g x) l atBot
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : 0 < C
hf : Tendsto f l atBot
hg : Tendsto g l (𝓝 C)
this : Tendsto (fun x ↦ (Neg.neg ∘ f) x * g x) l atTop
⊢ Tendsto (fun x ↦ f x * g x) l atBot

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : 0 < C
hf : Tendsto f l atBot
hg : Tendsto g l (𝓝 C)
this : Tendsto (fun x ↦ (Neg.neg ∘ f) x * g x) l atTop
⊢ Tendsto (fun x ↦ f x * g x) l atBot
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : C < 0
hf : Tendsto f l atBot
hg : Tendsto g l (𝓝 C)
this : Tendsto (fun x ↦ (Neg.neg ∘ f) x * g x) l atBot
⊢ Tendsto (fun x ↦ f x * g x) l atTop

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : C < 0
hf : Tendsto f l atBot
hg : Tendsto g l (𝓝 C)
this : Tendsto (fun x ↦ (Neg.neg ∘ f) x * g x) l atBot
⊢ Tendsto (fun x ↦ f x * g x) l atTop
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : 0 < C
hf : Tendsto f l (𝓝 C)
hg : Tendsto g l atBot
⊢ Tendsto (fun x ↦ f x * g x) l atBot

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : 0 < C
hf : Tendsto f l (𝓝 C)
hg : Tendsto g l atBot
⊢ Tendsto (fun x ↦ f x * g x) l atBot
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : C < 0
hf : Tendsto f l (𝓝 C)
hg : Tendsto g l atBot
⊢ Tendsto (fun x ↦ f x * g x) l atTop

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : C < 0
hf : Tendsto f l (𝓝 C)
hg : Tendsto g l atBot
⊢ Tendsto (fun x ↦ f x * g x) l atTop
Module: Mathlib.Topology.Order.LeftRightNhds

Goal state before:
pp: α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : CommGroup α
inst✝³ : LinearOrder α
inst✝² : IsOrderedMonoid α
inst✝¹ : OrderTopology α
inst✝ : NoMaxOrder α
⊢ (𝓝 1).HasBasis (fun ε ↦ 1 < ε) fun ε ↦ {b | |b|ₘ < ε}

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : CommGroup α
inst✝³ : LinearOrder α
inst✝² : IsOrderedMonoid α
inst✝¹ : OrderTopology α
inst✝ : NoMaxOrder α
⊢ (𝓝 1).HasBasis (fun ε ↦ 1 < ε) fun ε ↦ {b | |b|ₘ < ε}
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case refine_1.ofNat
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
c d : 𝕜
hc : c ≠ 0
n : ℕ
h : Tendsto (fun x ↦ c * x ^ Int.ofNat n) atTop (𝓝 d)
⊢ Int.ofNat n = 0 ∧ c = d

Tactic:
simpa using

Goal state after:
case refine_1.ofNat
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
c d : 𝕜
hc : c ≠ 0
n : ℕ
h : Tendsto (fun x ↦ c * x ^ Int.ofNat n) atTop (𝓝 d)
⊢ n = 0 ∧ c = d
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
h : e.source = univ
⊢ Injective ↑e

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
e : PartialEquiv α β
h : e.source = univ
⊢ Injective ↑e
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: α : Type u
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTop α
inst✝¹ : OrderTopology α
inst✝ : Nontrivial α
this : ∃ x, x < ⊤
⊢ (𝓝 ⊤).HasBasis (fun a ↦ a < ⊤) fun a ↦ Ioi a

Tactic:
simpa using

Goal state after:
α : Type u
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTop α
inst✝¹ : OrderTopology α
inst✝ : Nontrivial α
this : ∃ x, x < ⊤
⊢ (𝓝 ⊤).HasBasis (fun a ↦ a < ⊤) fun a ↦ Ioi a
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: case h
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
s : Set α
hs : s ∈ 𝓝[≥] a
ha : IsMax a
⊢ a ≤ a ∧ Icc a a ∈ 𝓝[≥] a ∧ Icc a a ⊆ s

Tactic:
simpa using

Goal state after:
case h
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
s : Set α
hs : s ∈ 𝓝[≥] a
ha : IsMax a
⊢ a ∈ s
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: case inr.inr
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : Nontrivial α
s : Set α
hs : IsOpen[inst✝³] s
x : α
hx : x ∈ s
y : α
hy : y ≠ x
H : y < x
l : α
lx : l < x
hl : Ioc l x ⊆ s
⊢ ∃ a b, a < b ∧ Ioo a b ⊆ s

Tactic:
exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩

Goal state after:
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : WellFoundedLT α
s : Set α
h : IsLowerSet s
⊢ IsOpen[inst✝³] s

Tactic:
simpa using

Goal state after:
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : WellFoundedLT α
s : Set α
h : IsLowerSet s
⊢ IsOpen[inst✝³] s
Module: Mathlib.Algebra.Module.CharacterModule

Goal state before:
pp: A : Type uA
inst✝ : AddCommGroup A
a : A
ne_zero : a ≠ 0
c : CharacterModule A
hc : (dual (ℤ ∙ a).subtype) c = ofSpanSingleton a
h : c a = 0
⊢ (ofSpanSingleton a) ⟨a, ⋯⟩ = 0

Tactic:
by rwa [← hc]

Goal state after:
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
a✝ : Nontrivial α
s : Set α := ⋯
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen[inst✝³] a
t : Set α := ⋯
x : α
hx : x ∈ t
⊢ ∃ l, l < x

Tactic:
simpa using

Goal state after:
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
a✝ : Nontrivial α
s : Set α := ⋯
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen[inst✝³] a
t : Set α := ⋯
x : α
hx : x ∈ t
⊢ ∃ l, l < x
Module: Mathlib.Algebra.Module.CharacterModule

Goal state before:
pp: R : Type uR
inst✝⁴ : CommRing R
A : Type uA
inst✝³ : AddCommGroup A
A' : Type u_1
inst✝² : AddCommGroup A'
inst✝¹ : Module R A
inst✝ : Module R A'
f : A →ₗ[R] A'
hf : Function.Injective ⇑(dual f)
c : CharacterModule (A' ⧸ f.range)
b : A'
⊢ c ((QuotientAddGroup.mk' f.range.toAddSubgroup) b) = 0

Tactic:
suffices eq : dual (Submodule.mkQ _) c = 0 from congr($eq b)

Goal state after:
R : Type uR
inst✝⁴ : CommRing R
A : Type uA
inst✝³ : AddCommGroup A
A' : Type u_1
inst✝² : AddCommGroup A'
inst✝¹ : Module R A
inst✝ : Module R A'
f : A →ₗ[R] A'
hf : Function.Injective ⇑(dual f)
c : CharacterModule (A' ⧸ f.range)
b : A'
⊢ (dual f.range.mkQ) c = 0
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : AddCommGroup 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsOrderedAddMonoid 𝕜
hp : 0 < p
n : ℤ
hx : n ≤ 0
⊢ n • p ≤ 0

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : AddCommGroup 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsOrderedAddMonoid 𝕜
hp : 0 < p
n : ℤ
hx : n ≤ 0
⊢ 0 ≤ -n • p
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
⊢ {x | ∃ y < x, Ioo y x = ∅}.Countable

Tactic:
simpa using

Goal state after:
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
⊢ {x | ∃ y, y ⋖ x}.Countable
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
a : AddCircle p
b : ↑(Ico 0 (0 + p)) := (QuotientAddGroup.equivIcoMod ⋯ 0) a
⊢ ↑b ∈ Ico 0 p

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
a : AddCircle p
b : ↑(Ico 0 (0 + p)) := (QuotientAddGroup.equivIcoMod ⋯ 0) a
⊢ ↑b ∈ Ico 0 p
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
a : AddCircle p
b : ↑(Ioc 0 (0 + p)) := (QuotientAddGroup.equivIocMod ⋯ 0) a
⊢ ↑b ∈ Ioc 0 p

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
a : AddCircle p
b : ↑(Ioc 0 (0 + p)) := (QuotientAddGroup.equivIocMod ⋯ 0) a
⊢ ↑b ∈ Ioc 0 p
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
ha : a ∈ Ico 0 p
⊢ 0 ∈ Ico 0 (0 + p)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
ha : a ∈ Ico 0 p
⊢ 0 < p
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : AddCommGroup 𝕜
p : 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
x : AddCircle p
hx : x ≠ ↑a
⊢ ContinuousAt (⇑(equivIco p a)) x

Tactic:
induction x using QuotientAddGroup.induction_on
  rw [ContinuousAt, Filter.Tendsto, QuotientAddGroup.nhds_eq, Filter.map_map]
  exact (continuousAt_toIcoMod hp.out a <| not_modEq_iff_ne_mod_zmultiples.mpr hx).codRestrict _

Goal state after:
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
inst✝ : SecondCountableTopology α
t : Set β
f : β → α
a✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | x ∈ t ∧ ∃ z, f x < z ∧ ∀ y ∈ t, x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ y ∈ t, x < y → z x ≤ f y
I : InjOn f s
A : (f '' s).PairwiseDisjoint fun x ↦ Ioo x (z (invFunOn f s x))
y : β
ys : y ∈ s
⊢ f y < z (invFunOn f s (f y))

Tactic:
simpa using

Goal state after:
α : Type u
β : Type v
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
inst✝ : SecondCountableTopology α
t : Set β
f : β → α
a✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | x ∈ t ∧ ∃ z, f x < z ∧ ∀ y ∈ t, x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ y ∈ t, x < y → z x ≤ f y
I : InjOn f s
A : (f '' s).PairwiseDisjoint fun x ↦ Ioo x (z (invFunOn f s x))
y : β
ys : y ∈ s
⊢ f y < z y
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
⊢ {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}.Countable

Tactic:
simpa using

Goal state after:
α : Type u
β : Type v
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
⊢ {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}.Countable
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
a : 𝕜
⊢ IsOfFinAddOrder ↑a ↔ ∃ q, ↑q = a / p

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
a : 𝕜
⊢ IsOfFinAddOrder ↑a ↔ ∃ q, ↑q = a / p
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
h : ¬∃ u, ¬IsOfFinAddOrder u
⊢ IsEmpty { u // ¬IsOfFinAddOrder u }

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
h : ¬∃ u, ¬IsOfFinAddOrder u
⊢ ∀ (x : AddCircle p), IsOfFinAddOrder x
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
inst✝³ : AddCommGroup 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
f : 𝕜 → B
hf : f a = f (a + p)
q : 𝕋
⊢ ?m.37

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
B : Type u_2
inst✝³ : AddCommGroup 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
f : 𝕜 → B
hf : f a = f (a + p)
q : 𝕋
⊢ ?m.37
Module: Mathlib.Algebra.Category.ModuleCat.Subobject

Goal state before:
pp: case e'_3
R : Type u
inst✝ : Ring R
M : ModuleCat R
S T : Submodule R ↑M
h :
  { toFun := fun N ↦ mk (↟N.subtype), invFun := fun S ↦ (Hom.hom S.arrow).range, left_inv := ⋯, right_inv := ⋯ } S ≤
    { toFun := fun N ↦ mk (↟N.subtype), invFun := fun S ↦ (Hom.hom S.arrow).range, left_inv := ⋯, right_inv := ⋯ } T
⊢ S = (Hom.hom (↟T.subtype) ∘ₗ Hom.hom (ofMkLEMk (↟S.subtype) (↟T.subtype) h)).range

Tactic:
rw [← hom_comp, ofMkLEMk_comp]

Goal state after:
case e'_3
R : Type u
inst✝ : Ring R
M : ModuleCat R
S T : Submodule R ↑M
h :
  { toFun := fun N ↦ mk (↟N.subtype), invFun := fun S ↦ (Hom.hom S.arrow).range, left_inv := ⋯, right_inv := ⋯ } S ≤
    { toFun := fun N ↦ mk (↟N.subtype), invFun := fun S ↦ (Hom.hom S.arrow).range, left_inv := ⋯, right_inv := ⋯ } T
⊢ S = (Hom.hom (↟S.subtype)).range
Module: Mathlib.Algebra.Category.ModuleCat.AB

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
inst✝ : Small.{v, u} R
X Y : ModuleCat R
f g : X ⟶ Y
h : ∀ (h : of R (Shrink.{v, u} R) ⟶ X) (x : Shrink.{v, u} R), (Hom.hom f) ((Hom.hom h) x) = (Hom.hom g) ((Hom.hom h) x)
x : ↑X
⊢ (Hom.hom f) x = (Hom.hom g) x

Tactic:
simpa using

Goal state after:
R : Type u
inst✝¹ : Ring R
inst✝ : Small.{v, u} R
X Y : ModuleCat R
f g : X ⟶ Y
h : ∀ (h : of R (Shrink.{v, u} R) ⟶ X) (x : Shrink.{v, u} R), (Hom.hom f) ((Hom.hom h) x) = (Hom.hom g) ((Hom.hom h) x)
x : ↑X
⊢ (Hom.hom f) x = (Hom.hom g) x