mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.UniformSpace.UniformApproximation

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
f : α → β
s : Set α
x : α
hx : x ∈ s
L : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∃ F, ContinuousWithinAt F s x ∧ ∀ y ∈ t, (f y, F y) ∈ u
u₀ : Set (β × β)
hu₀ : u₀ ∈ 𝓤 β
u₁ : Set (β × β)
h₁ : u₁ ∈ 𝓤 β
u₁₀ : u₁ ○ u₁ ⊆ u₀
u₂ : Set (β × β)
h₂ : u₂ ∈ 𝓤 β
hsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂
u₂₁ : u₂ ○ u₂ ⊆ u₁
t : Set α
tx : t ∈ 𝓝[s] x
F : α → β
hFc : ContinuousWithinAt F s x
hF : ∀ y ∈ t, (f y, F y) ∈ u₂
A : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F y) ∈ u₂
B : ∀ᶠ (y : α) in 𝓝[s] x, (F y, F x) ∈ u₂
⊢ u₀ ∈ map (fun x_1 ↦ (f x_1, f x)) (𝓝[s] x)

Tactic:
have C : ∀ᶠ y in 𝓝[s] x, (f y, F x) ∈ u₁ :=
    (A.and B).mono fun y hy => u₂₁ (prodMk_mem_comp hy.1 hy.2)

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
f : α → β
s : Set α
x : α
hx : x ∈ s
L : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∃ F, ContinuousWithinAt F s x ∧ ∀ y ∈ t, (f y, F y) ∈ u
u₀ : Set (β × β)
hu₀ : u₀ ∈ 𝓤 β
u₁ : Set (β × β)
h₁ : u₁ ∈ 𝓤 β
u₁₀ : u₁ ○ u₁ ⊆ u₀
u₂ : Set (β × β)
h₂ : u₂ ∈ 𝓤 β
hsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂
u₂₁ : u₂ ○ u₂ ⊆ u₁
t : Set α
tx : t ∈ 𝓝[s] x
F : α → β
hFc : ContinuousWithinAt F s x
hF : ∀ y ∈ t, (f y, F y) ∈ u₂
A : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F y) ∈ u₂
B : ∀ᶠ (y : α) in 𝓝[s] x, (F y, F x) ∈ u₂
C : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F x) ∈ u₁
⊢ u₀ ∈ map (fun x_1 ↦ (f x_1, f x)) (𝓝[s] x)
Module: Mathlib.Algebra.Order.Group.MinMax

Goal state before:
pp: α : Type u_1
inst✝² : Group α
inst✝¹ : LinearOrder α
inst✝ : MulLeftMono α
a : α
⊢ max a⁻¹ 1 = a⁻¹ * max a 1

Tactic:
rw [eq_inv_mul_iff_mul_eq, ← eq_div_iff_mul_eq', max_one_div_max_inv_one_eq_self]

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: case mp
ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
F : ι → X → α
S : Set X
x₀ : X
hx₀ : x₀ ∈ S
H : EquicontinuousWithinAt F S x₀
U : Set (α × α)
hU : U ∈ 𝓤 α
⊢ ∃ V ∈ 𝓝[S] x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U

Tactic:
rcases comp_symm_mem_uniformity_sets hU with ⟨V, hV, hVsymm, hVU⟩
    refine ⟨_, H V hV, fun x hx y hy i => hVU (SetRel.prodMk_mem_comp ?_ (hy i))⟩
    exact SetRel.symm V (hx i)

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
α' : Type u_7
β : Type u_8
uβ : UniformSpace β
u : κ → UniformSpace α'
F : ι → β → α'
S : Set β
⊢ UniformEquicontinuousOn F S ↔ ∀ (k : κ), UniformEquicontinuousOn F S

Tactic:
simp_rw [uniformEquicontinuousOn_iff_uniformContinuousOn (uα := _)]
  unfold UniformContinuousOn
  rw [UniformFun.iInf_eq, iInf_uniformity, tendsto_iInf]

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
p : κ → Prop
s : κ → Set (α × α)
F : ι → X → α
S : Set X
x₀ : X
hα : (𝓤 α).HasBasis p s
⊢ EquicontinuousWithinAt F S x₀ ↔ ∀ (k : κ), p k → ∀ᶠ (x : X) in 𝓝[S] x₀, ∀ (i : ι), (F i x₀, F i x) ∈ s k

Tactic:
by
  rw [equicontinuousWithinAt_iff_continuousWithinAt, ContinuousWithinAt,
    (UniformFun.hasBasis_nhds_of_basis ι α _ hα).tendsto_right_iff]
  rfl

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
s : Set α
inst✝ : UniformSpace β
𝔖 : Set (Set α)
hs : s ∈ 𝔖
V : Set (β × β)
hV : V ∈ 𝓤 β
⊢ UniformOnFun.gen 𝔖 s V ∈ 𝓤 (α →ᵤ[𝔖] β)

Tactic:
by
  rw [UniformOnFun.uniformity_eq]
  apply_rules [mem_iInf_of_mem, mem_principal_self]

Goal state after:
Module: Mathlib.Algebra.Order.Group.Pointwise.Interval

Goal state before:
pp: G₀ : Type u_2
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
c a : G₀
h : 0 < c
⊢ (fun x ↦ x * c) ⁻¹' Iio a = Iio (a / c)

Tactic:
by
  simpa only [division_def] using! (OrderIso.mulRight₀ c h).preimage_Iio a

Goal state after:
Module: Mathlib.Topology.Algebra.ConstMulAction

Goal state before:
pp: Γ : Type u_4
inst✝³ : Group Γ
T : Type u_5
inst✝² : TopologicalSpace T
inst✝¹ : MulAction Γ T
inst✝ : ContinuousConstSMul Γ T
U : Set T
hU : IsOpen[inst✝²] U
⊢ IsOpen[instTopologicalSpaceQuotient] (Quotient.mk' '' U)

Tactic:
by
  rw [isOpen_coinduced, MulAction.quotient_preimage_image_eq_union_mul U]
  exact isOpen_iUnion fun γ => isOpenMap_smul γ U hU

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Pointwise

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
s t : Set G
hs : IsOpen[inst✝²] s
⊢ IsOpen[inst✝²] (⋃ a ∈ t, (fun x ↦ x / a) '' s)

Tactic:
exact isOpen_biUnion fun a _ => isOpenMap_div_right a s hs

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

Goal state before:
pp: G : Type w
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
H : Type u_1
M : Type u_2
inst✝⁵ : CommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : ContinuousMul M
inst✝² : Group H
inst✝¹ : TopologicalSpace H
inst✝ : IsTopologicalGroup H
f : G →* H →* M
hf : ContinuousAt (fun x ↦ (f x.1) x.2) (1, 1)
hl : ∀ (x : G), ContinuousAt (⇑(f x)) 1
hr : ∀ (y : H), ContinuousAt (fun x ↦ (f x) y) 1
x✝ : G × H
x : G
y : H
⊢ ContinuousAt (fun x ↦ (f x.1) x.2) (x, y)

Tactic:
by
  simp only [ContinuousAt, nhds_prod_eq, ← map_mul_left_nhds_one x, ← map_mul_left_nhds_one y,
    prod_map_map_eq, tendsto_map'_iff, Function.comp_def, map_mul, MonoidHom.mul_apply] at *
  refine ((tendsto_const_nhds.mul ((hr y).comp tendsto_fst)).mul
    (((hl x).comp tendsto_snd).mul hf)).mono_right (le_of_eq ?_)
  simp only [map_one, mul_one, MonoidHom.one_apply]

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

Goal state before:
pp: case e'_3
α : Type u
G : Type u_1
inst✝² : GroupWithZero G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousDiv G
b : G
hb : b ≠ 0
c : G
f : α → G
l : Filter α
h : Tendsto (fun x ↦ f x / b) l (𝓝 (c / b))
k : α
⊢ f k = f k / b / b⁻¹

Tactic:
rw [← div_mul_eq_div_div_swap, inv_mul_cancel₀ hb, div_one]

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

Goal state before:
pp: case e'_5
α : Type u
G : Type u_1
inst✝² : GroupWithZero G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousDiv G
b : G
hb : b ≠ 0
c : G
f : α → G
l : Filter α
h : Tendsto (fun x ↦ f x / b) l (𝓝 (c / b))
⊢ c = c / b / b⁻¹

Tactic:
rw [← div_mul_eq_div_div_swap, inv_mul_cancel₀ hb, div_one]

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

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : MulOneClass G
inst✝ : ContinuousMul G
K U : Set G
hK : IsCompact K
hU : IsOpen[inst✝²] U
hKU : K ⊆ U
V : Set Gᵐᵒᵖ
hV : V ∈ 𝓝 (op 1)
hV' : op '' K * V ⊆ op '' U
⊢ op ⁻¹' V * K ⊆ U

Tactic:
rwa [← image_preimage_eq V op_surjective, ← image_op_mul, image_subset_iff,
    preimage_image_eq _ op_injective] at hV'

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
⊢ IsTopologicalGroup.rightUniformSpace G ≤ UniformSpace.comap (⇑(Equiv.inv G)) (IsTopologicalGroup.leftUniformSpace G)

Tactic:
← comap_inv_leftUniformSpace

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
⊢ UniformSpace.comap (⇑(Equiv.inv G)) (IsTopologicalGroup.leftUniformSpace G) ≤
    UniformSpace.comap (⇑(Equiv.inv G)) (IsTopologicalGroup.leftUniformSpace G)
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: case right
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹⁴ : TopologicalSpace α
inst✝¹³ : AddCommGroup α
inst✝¹² : IsTopologicalAddGroup α
inst✝¹¹ : TopologicalSpace β
inst✝¹⁰ : AddCommGroup β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : AddCommGroup γ
inst✝⁷ : IsTopologicalAddGroup γ
inst✝⁶ : TopologicalSpace δ
inst✝⁵ : AddCommGroup δ
inst✝⁴ : UniformSpace G
inst✝³ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
f : δ →+ γ
df : IsDenseInducing ⇑f
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝⁴.toTopologicalSpace] fun p ↦ (φ p.1) p.2
inst✝² : IsUniformAddGroup G
inst✝¹ : T0Space G
inst✝ : CompleteSpace G
x₀ : α
y₀ : γ
⊢ map (fun p ↦ (fun p ↦ (φ p.1) p.2) p.2 - (fun p ↦ (φ p.1) p.2) p.1)
      (comap (fun p ↦ ((e p.1.1, f p.1.2), e p.2.1, f p.2.2)) (𝓝 (x₀, y₀) ×ˢ 𝓝 (x₀, y₀))) ≤
    𝓝 0

Tactic:
intro W' W'_nhds

Goal state after:
case right
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹⁴ : TopologicalSpace α
inst✝¹³ : AddCommGroup α
inst✝¹² : IsTopologicalAddGroup α
inst✝¹¹ : TopologicalSpace β
inst✝¹⁰ : AddCommGroup β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : AddCommGroup γ
inst✝⁷ : IsTopologicalAddGroup γ
inst✝⁶ : TopologicalSpace δ
inst✝⁵ : AddCommGroup δ
inst✝⁴ : UniformSpace G
inst✝³ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
f : δ →+ γ
df : IsDenseInducing ⇑f
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝⁴.toTopologicalSpace] fun p ↦ (φ p.1) p.2
inst✝² : IsUniformAddGroup G
inst✝¹ : T0Space G
inst✝ : CompleteSpace G
x₀ : α
y₀ : γ
W' : Set G
W'_nhds : W' ∈ 𝓝 0
⊢ W' ∈
    map (fun p ↦ (fun p ↦ (φ p.1) p.2) p.2 - (fun p ↦ (φ p.1) p.2) p.1)
      (comap (fun p ↦ ((e p.1.1, f p.1.2), e p.2.1, f p.2.2)) (𝓝 (x₀, y₀) ×ˢ 𝓝 (x₀, y₀)))
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:
by
  simpa only [mul_comm] using hg.atTop_mul_pos hC hf

Goal state after:
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)
⊢ Tendsto (fun x ↦ f x * g x) l atBot

Tactic:
have := hf.atTop_mul_pos (neg_pos.2 hC) hg.neg
  simpa only [Function.comp_def, neg_mul_eq_mul_neg, neg_neg] using
    tendsto_neg_atTop_atBot.comp this

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
s : Set α
⊢ ↑e '' (s ∩ e.source) = e.target ∩ ↑e.symm ⁻¹' (s ∩ e.source)

Tactic:
e.leftInvOn.image_inter,

Goal state after:
α : Type u_1
β : Type u_2
e : PartialEquiv α β
s : Set α
⊢ ↑e.symm ⁻¹' (s ∩ e.source) ∩ ↑e '' e.source = e.target ∩ ↑e.symm ⁻¹' (s ∩ e.source)
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: case pos
α : Type u
β : Type v
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : ∀ {x y : α}, f x < f y ↔ x < y
H₁ : ∀ {a : α} {b : β} {x : α}, b < f a → ¬b < f x → ∃ y < a, b ≤ f y
H₂ : ∀ {a : α} {b : β} {x : α}, f a < b → ¬f x < b → ∃ y, a < y ∧ f y ≤ b
x✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
a : α
b : β
hb : b ∈ Iio (f a)
h : ∃ x, ¬b < f x
⊢ ⨅ b ∈ Iio a, 𝓟 (Ioi b) ≤ 𝓟 (f ⁻¹' Ioi b)

pp: case neg
α : Type u
β : Type v
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : ∀ {x y : α}, f x < f y ↔ x < y
H₁ : ∀ {a : α} {b : β} {x : α}, b < f a → ¬b < f x → ∃ y < a, b ≤ f y
H₂ : ∀ {a : α} {b : β} {x : α}, f a < b → ¬f x < b → ∃ y, a < y ∧ f y ≤ b
x✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
a : α
b : β
hb : b ∈ Iio (f a)
h : ∀ (x : α), b < f x
⊢ ⨅ b ∈ Iio a, 𝓟 (Ioi b) ≤ 𝓟 (f ⁻¹' Ioi b)

Tactic:
· rcases h with ⟨x, hx⟩
      rcases H₁ hb hx with ⟨y, hya, hyb⟩
      exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))

Goal state after:
case neg
α : Type u
β : Type v
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : ∀ {x y : α}, f x < f y ↔ x < y
H₁ : ∀ {a : α} {b : β} {x : α}, b < f a → ¬b < f x → ∃ y < a, b ≤ f y
H₂ : ∀ {a : α} {b : β} {x : α}, f a < b → ¬f x < b → ∃ y, a < y ∧ f y ≤ b
x✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
a : α
b : β
hb : b ∈ Iio (f a)
h : ∀ (x : α), b < f x
⊢ ⨅ b ∈ Iio a, 𝓟 (Ioi b) ≤ 𝓟 (f ⁻¹' Ioi b)
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: case refine_1
α : Type u
β : Type v
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : ∀ {x y : α}, f x < f y ↔ x < y
H₁ : ∀ {a : α} {b : β} {x : α}, b < f a → ¬b < f x → ∃ y < a, b ≤ f y
H₂ : ∀ {a : α} {b : β} {x : α}, f a < b → ¬f x < b → ∃ y, a < y ∧ f y ≤ b
x✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
a : α
b : β
hb : b ∈ Iio (f a)
⊢ ⨅ b ∈ Iio a, 𝓟 (Ioi b) ≤ 𝓟 (f ⁻¹' Ioi b)

pp: case refine_2
α : Type u
β : Type v
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : ∀ {x y : α}, f x < f y ↔ x < y
H₁ : ∀ {a : α} {b : β} {x : α}, b < f a → ¬b < f x → ∃ y < a, b ≤ f y
H₂ : ∀ {a : α} {b : β} {x : α}, f a < b → ¬f x < b → ∃ y, a < y ∧ f y ≤ b
x✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
a : α
b : β
hb : b ∈ Ioi (f a)
⊢ ⨅ b ∈ Ioi a, 𝓟 (Iio b) ≤ 𝓟 (f ⁻¹' Iio b)

Tactic:
· by_cases! h : ∃ x, ¬(b < f x)
    · rcases h with ⟨x, hx⟩
      rcases H₁ hb hx with ⟨y, hya, hyb⟩
      exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))
    · exact le_principal_iff.2 (univ_mem' h)

Goal state after:
case refine_2
α : Type u
β : Type v
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : ∀ {x y : α}, f x < f y ↔ x < y
H₁ : ∀ {a : α} {b : β} {x : α}, b < f a → ¬b < f x → ∃ y < a, b ≤ f y
H₂ : ∀ {a : α} {b : β} {x : α}, f a < b → ¬f x < b → ∃ y, a < y ∧ f y ≤ b
x✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
a : α
b : β
hb : b ∈ Ioi (f a)
⊢ ⨅ b ∈ Ioi a, 𝓟 (Iio b) ≤ 𝓟 (f ⁻¹' Iio b)
Module: Mathlib.Algebra.Module.CharacterModule

Goal state before:
pp: A : Type uA
inst✝ : AddCommGroup A
a : A
n : ℤ
hn : (n • 1).den = (↑(if addOrderOf a = 0 then 2 else addOrderOf a))⁻¹.den
⊢ a = 0

Tactic:
rw [zsmul_one, Rat.den_intCast, Rat.inv_natCast_den_of_pos] at hn

Goal state after:
A : Type uA
inst✝ : AddCommGroup A
a : A
n : ℤ
hn : 1 = if addOrderOf a = 0 then 2 else addOrderOf a
⊢ a = 0

A : Type uA
inst✝ : AddCommGroup A
a : A
n : ℤ
hn : 1 = (↑(if addOrderOf a = 0 then 2 else addOrderOf a))⁻¹.den
⊢ 0 < if addOrderOf a = 0 then 2 else addOrderOf a
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
inst✝ : Field 𝕜
p q : 𝕜
hp : p ≠ 0
hq : q ≠ 0
⊢ map (↑(AddAut.mulRight ((Units.mk0 p hp)⁻¹ * Units.mk0 q hq))) (zmultiples p) = zmultiples q

Tactic:
by
    rw [AddMonoidHom.map_zmultiples, AddMonoidHom.coe_coe, AddAut.mulRight_apply, Units.val_mul,
      Units.val_mk0, Units.val_inv_eq_inv_val, Units.val_mk0, mul_inv_cancel_left₀ hp]

Goal state after:
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:
by simpa only [mem_image] using coe_image_Ico_eq p a ▸ mem_univ q

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Refinements

Goal state before:
pp: case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h :
  ∀ ⦃A : C⦄ (x₂ : A ⟶ S₁.X₂),
    x₂ ≫ S₁.g = 0 → ∀ (y₁ : A ⟶ S₂.X₁), x₂ ≫ φ.τ₂ = y₁ ≫ S₂.f → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S₁.f
A : C
γ : A ⟶ S₁.homology
hγ : γ ≫ homologyMap φ = 0
A₁ : C
π₁ : A₁ ⟶ A
hπ₁ : Epi π₁
z : A₁ ⟶ S₁.X₂
hz : z ≫ S₁.g = 0
fac : π₁ ≫ γ = S₁.liftCycles z hz ≫ S₁.homologyπ
⊢ π₁ ≫ γ = π₁ ≫ 0

Tactic:
fac,

Goal state after:
case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h :
  ∀ ⦃A : C⦄ (x₂ : A ⟶ S₁.X₂),
    x₂ ≫ S₁.g = 0 → ∀ (y₁ : A ⟶ S₂.X₁), x₂ ≫ φ.τ₂ = y₁ ≫ S₂.f → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ S₁.f
A : C
γ : A ⟶ S₁.homology
hγ : γ ≫ homologyMap φ = 0
A₁ : C
π₁ : A₁ ⟶ A
hπ₁ : Epi π₁
z : A₁ ⟶ S₁.X₂
hz : z ≫ S₁.g = 0
fac : π₁ ≫ γ = S₁.liftCycles z hz ≫ S₁.homologyπ
⊢ S₁.liftCycles z hz ≫ S₁.homologyπ = π₁ ≫ 0
Module: Mathlib.Algebra.Homology.ShortComplex.SnakeLemma

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : SnakeInput C
⊢ Hom.τ₃ 0 = 0

Tactic:
zero_τ₃

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : SnakeInput C
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.ShortComplex.SnakeLemma

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : SnakeInput C
⊢ S.L₀X₂ToP ≫ S.φ₁ ≫ S.v₂₃.τ₁ = 0

Tactic:
L₀X₂ToP_comp_φ₁_assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : SnakeInput C
⊢ 0 ≫ S.v₂₃.τ₁ = 0
Module: Mathlib.Algebra.Homology.ShortComplex.SnakeLemma

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : SnakeInput C
f : S₁ ⟶ S₂
⊢ pullback.map S₁.L₁.g S₁.v₀₁.τ₃ S₂.L₁.g S₂.v₀₁.τ₃ f.f₁.τ₂ f.f₀.τ₃ f.f₁.τ₃ ⋯ ⋯ ≫ pullback.snd S₂.L₁.g S₂.v₀₁.τ₃ ≫ S₂.δ =
    pullback.snd S₁.L₁.g S₁.v₀₁.τ₃ ≫ f.f₀.τ₃ ≫ S₂.δ

Tactic:
pullback.lift_snd_assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : SnakeInput C
f : S₁ ⟶ S₂
⊢ (pullback.snd S₁.L₁.g S₁.v₀₁.τ₃ ≫ f.f₀.τ₃) ≫ S₂.δ = pullback.snd S₁.L₁.g S₁.v₀₁.τ₃ ≫ f.f₀.τ₃ ≫ S₂.δ
Module: Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type w
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
X₁ X₂ : C
inst✝¹ : HasBinaryProduct X₁ X₂
inst✝ : PreservesLimit (pair X₁ X₂) (forget C)
x : ToType (X₁ ⨯ X₂)
⊢ ((prodEquiv X₁ X₂) x).2 = (hom prod.snd) x

Tactic:
simpa using! congr_hom (prodComparison_snd (forget C) X₁ X₂) x

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer

Goal state before:
pp: J : MulticospanShape
⊢ ∀ {W X Y Z : WalkingMulticospan J} (f : W.Hom X) (g : X.Hom Y) (h : Y.Hom Z), (f.comp g).comp h = f.comp (g.comp h)

Tactic:
rintro (_ | _) (_ | _) (_ | _) (_ | _) (_ | _ | _) (_ | _ | _) (_ | _ | _) <;> rfl

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory

Goal state before:
pp: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type w
inst✝⁶ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝⁵ : ConcreteCategory C FC
inst✝⁴ : HasForget₂ C Ab
inst✝³ : Preadditive C
inst✝² : (forget₂ C Ab).Additive
inst✝¹ : (forget₂ C Ab).PreservesHomology
inst✝ : HasZeroObject C
X Y : C
f : X ⟶ Y
⊢ Epi ((forget₂ C Ab).map f) → Epi f

Tactic:
· apply Functor.epi_of_epi_map

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type w
inst✝⁶ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝⁵ : ConcreteCategory C FC
inst✝⁴ : HasForget₂ C Ab
inst✝³ : Preadditive C
inst✝² : (forget₂ C Ab).Additive
inst✝¹ : (forget₂ C Ab).PreservesHomology
S : ShortComplex C
inst✝ : HasZeroObject C
hS : S.ShortExact
⊢ Epi S.g

Tactic:
exact hS.epi_g

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer

Goal state before:
pp: J : MultispanShape
⊢ ∀ {W X Y Z : WalkingMultispan J} (f : W.Hom X) (g : X.Hom Y) (h : Y.Hom Z), (f.comp g).comp h = f.comp (g.comp h)

Tactic:
rintro (_ | _) (_ | _) (_ | _) (_ | _) (_ | _ | _) (_ | _ | _) (_ | _ | _) <;> rfl

Goal state after:
Module: Mathlib.CategoryTheory.ComposableArrows.Basic

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : ComposableArrows C 0
f : F ⟶ G
⊢ IsIso f ↔ IsIso (f.app 0)

Tactic:
by
  rw [NatTrans.isIso_iff_isIso_app]
  exact ⟨fun h ↦ h 0, fun _ i ↦ by fin_cases i; assumption⟩

Goal state after:
Module: Mathlib.CategoryTheory.Monad.Limits

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
T : Monad C
J : Type u
inst✝² : Category.{v, u} J
D : J ⥤ T.Algebra
c : Cocone (D ⋙ T.forget)
t : IsColimit c
inst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor
inst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor
⊢ T.μ.app c.pt ≫ lambda c t = T.map (lambda c t) ≫ lambda c t

Tactic:
refine (isColimitOfPreserves _ (isColimitOfPreserves _ t)).hom_ext fun j => ?_
    rw [Functor.mapCocone_ι_app, Functor.mapCocone_ι_app,
      show (T : C ⥤ C).map ((T : C ⥤ C).map _) ≫ _ ≫ _ = _ from T.μ.naturality_assoc _ _, ←
      Functor.map_comp_assoc, commuting, Functor.map_comp, Category.assoc, commuting]
    apply (D.obj j).assoc_assoc _

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Reflexive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
F G : WalkingReflexivePair ⥤ C
e₀ : F.obj zero ⟶ G.obj zero
e₁ : F.obj one ⟶ G.obj one
h₁ : F.map left ≫ e₀ = e₁ ≫ G.map left
h₂ : F.map right ≫ e₀ = e₁ ≫ G.map right
h₃ : F.map reflexion ≫ e₁ = e₀ ≫ G.map reflexion
D : Type u₂
inst✝ : Category.{v₂, u₂} D
H : C ⥤ D
⊢ (F ⋙ H).map right ≫ H.map e₀ = H.map e₁ ≫ (G ⋙ H).map right

Tactic:
simp only [Functor.comp_obj, Functor.comp_map, ← Functor.map_comp, h₂]

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Submodule

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid N
inst✝⁶ : Module R M
inst✝⁵ : Module R N
inst✝⁴ : Algebra R S
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
p : Submonoid R
inst✝¹ : IsLocalization p S
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
M'✝ M'' M' : Submodule R M
N' : Submodule S N
h : M' ≤ (fun x ↦ comap f (restrictScalars R x)) N'
⊢ ⇑f '' ↑M' ⊆ ↑N'

Tactic:
apply map_le_iff_le_comap.mpr h

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Submodule

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
N : Type u_4
inst✝⁷ : CommSemiring R
inst✝⁶ : CommSemiring S
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Algebra R S
inst✝² : Module S N
inst✝¹ : IsScalarTower R S N
p : Submonoid R
inst✝ : IsLocalization p S
I : Submodule R R
N' : Submodule S N
r : R
hr : r ∈ I
n : N
hn : n ∈ restrictScalars R N'
⊢ r • n ∈ restrictScalars R (localized' S p (Algebra.linearMap R S) I • N')

Tactic:
rw [← algebraMap_smul S, Submodule.restrictScalars_mem]

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
N : Type u_4
inst✝⁷ : CommSemiring R
inst✝⁶ : CommSemiring S
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Algebra R S
inst✝² : Module S N
inst✝¹ : IsScalarTower R S N
p : Submonoid R
inst✝ : IsLocalization p S
I : Submodule R R
N' : Submodule S N
r : R
hr : r ∈ I
n : N
hn : n ∈ restrictScalars R N'
⊢ (algebraMap R S) r • n ∈ localized' S p (Algebra.linearMap R S) I • N'
Module: Mathlib.RingTheory.Ideal.MinimalPrime.Basic

Goal state before:
pp: case refine_1
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
I p : Ideal R
P : Ideal S
inst✝¹ : P.IsPrime
inst✝ : P.LiesOver p
hI : p ∈ I.minimalPrimes
J : Ideal S
hJP : J ≤ P
hJ : map (Quotient.mk (map (algebraMap R S) p)) P ∈ (map (Quotient.mk (map (algebraMap R S) p)) J).minimalPrimes
⊢ map (algebraMap R S) I ≤ P

Tactic:
rw [Ideal.map_le_iff_le_comap, ← Ideal.under_def, ← Ideal.over_def P p]

Goal state after:
case refine_1
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
I p : Ideal R
P : Ideal S
inst✝¹ : P.IsPrime
inst✝ : P.LiesOver p
hI : p ∈ I.minimalPrimes
J : Ideal S
hJP : J ≤ P
hJ : map (Quotient.mk (map (algebraMap R S) p)) P ∈ (map (Quotient.mk (map (algebraMap R S) p)) J).minimalPrimes
⊢ I ≤ p