mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.IsCountablyGenerated
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a : ι → ℝ
f : ℝ → E
ha : Tendsto a l atBot
⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in Iic (a i), f x ∂μ) l (𝓝 0)

Tactic:
by_cases! h : ∀ b, ¬ IntegrableOn f (Iic b) μ
  · exact tendsto_const_nhds.congr (fun i => (integral_undef (h (a i))).symm)
  obtain ⟨b, hb⟩ := h
  have : ∀ᶠ i in l, ∫ x in Iic b, f x ∂μ - ∫ x in a i..b, f x ∂μ = ∫ x in Iic (a i), f x ∂μ := by
    filter_upwards [ha.eventually_mem (Iic_mem_atBot b)] with i hi
    rw [sub_eq_iff_comm,
      intervalIntegral.integral_Iic_sub_Iic (hb.mono_set (Iic_subset_Iic.2 hi)) hb]
  rw [← sub_self (∫ x in Iic b, f x ∂μ)]
  exact Tendsto.congr' this (Tendsto.const_sub _ <| intervalIntegral_tendsto_integral_Iic b hb ha)

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Polynomial

Goal state before:
pp: p : ℝ[X]
hp' : derivative p ≠ 0
⊢ derivative p ≠ derivative 0

Tactic:
by rwa [derivative_zero]

Goal state after:
Module: Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts

Goal state before:
pp: E : Type u_1
F : Type u_2
G : Type u_3
W : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace ℝ G
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
f f' : E → F
g g' : E → G
v : E
B : F →L[ℝ] G →L[ℝ] W
hf'g : Integrable (fun x ↦ (B (f' x)) (g x)) μ
hfg' : Integrable (fun x ↦ (B (f x)) (g' x)) μ
hfg : Integrable (fun x ↦ (B (f x)) (g x)) μ
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x v
hg : ∀ x ∈ tsupport f, HasLineDerivAt ℝ g (g' x) x v
hW : CompleteSpace W
hv : v ≠ 0
this✝ : Nontrivial E
n : ℕ := finrank ℝ E
E' : Type := Fin (n - 1) → ℝ
this : finrank ℝ (E' × ℝ) = n
L₀ : E ≃L[ℝ] E' × ℝ
M : (E' × ℝ) ≃L[ℝ] E' × ℝ
hM : M (L₀ v) = (0, 1)
⊢ ∃ L, L v = (0, 1)

Tactic:
exact ⟨L₀.trans M, by simp [hM]⟩

Goal state after:
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
e e' : OpenPartialHomeomorph M H
he : e ∈ StructureGroupoid.maximalAtlas M G
he' : e' ∈ atlas H M
s : Set M
hs : IsOpen[inst✝²] s
⊢ (e.restr s).symm ≫ₕ e' ∈ G

Tactic:
have hs'' : IsOpen (e '' (e.source ∩ s)) := by
    rw [isOpen_image_iff_of_subset_source _ inter_subset_left]
    exact e.open_source.inter hs

Goal state after:
H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
e e' : OpenPartialHomeomorph M H
he : e ∈ StructureGroupoid.maximalAtlas M G
he' : e' ∈ atlas H M
s : Set M
hs : IsOpen[inst✝²] s
hs'' : IsOpen[inst✝³] (↑e '' (e.source ∩ s))
⊢ (e.restr s).symm ≫ₕ e' ∈ G
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
G : StructureGroupoid H
inst✝¹ : HasGroupoid M G
s : Opens M
inst✝ : ClosedUnderRestriction G
⊢ ∀ {e e' : OpenPartialHomeomorph (↥s) H}, e ∈ atlas H ↥s → e' ∈ atlas H ↥s → e.symm ≫ₕ e' ∈ G

Tactic:
rintro e e' ⟨_, ⟨x, hc⟩, he⟩ ⟨_, ⟨x', hc'⟩, he'⟩
    rw [hc.symm, mem_singleton_iff] at he
    rw [hc'.symm, mem_singleton_iff] at he'
    rw [he, he']
    refine G.mem_of_eqOnSource ?_
      (subtypeRestr_symm_trans_subtypeRestr (s := s) _ (chartAt H x) (chartAt H x'))
    apply closedUnderRestriction'
    · exact G.compatible (chart_mem_atlas _ _) (chart_mem_atlas _ _)
    · exact isOpen_inter_preimage_symm (chartAt _ _) s.2

Goal state after:
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case mp
H : Type u_1
inst✝ : TopologicalSpace H
G : StructureGroupoid H
⊢ ClosedUnderRestriction G → idRestrGroupoid ≤ G

Tactic:
intro _i

Goal state after:
case mp
H : Type u_1
inst✝ : TopologicalSpace H
G : StructureGroupoid H
_i : ClosedUnderRestriction G
⊢ idRestrGroupoid ≤ G
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
s : Set E
hs : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s

Tactic:
intro u hu v hv a b ha hb hab

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
s : Set E
hs : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
u : E
hu : u ∈ s
v : E
hv : v ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • u + b • v ∈ s
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: case mp
H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e' : OpenPartialHomeomorph H' H'
he' : e' ∈ G'
hfs : ContinuousWithinAt f s x
hxe' : f x ∈ e'.source
h2f : f ⁻¹' e'.source ∈ 𝓝[s] x
h3f : ↑e' ∘ f ⁻¹' e'.symm.source ∈ 𝓝[s] x
h : P (↑e' ∘ f) (s ∩ ↑e' ∘ f ⁻¹' e'.symm.source) x
h2 : P (↑e'.symm ∘ ↑e' ∘ f) s x
⊢ P f s x

Tactic:
refine hG.congr_nhdsWithin ?_ (e'.left_inv hxe') h2

Goal state after:
case mp
H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e' : OpenPartialHomeomorph H' H'
he' : e' ∈ G'
hfs : ContinuousWithinAt f s x
hxe' : f x ∈ e'.source
h2f : f ⁻¹' e'.source ∈ 𝓝[s] x
h3f : ↑e' ∘ f ⁻¹' e'.symm.source ∈ 𝓝[s] x
h : P (↑e' ∘ f) (s ∩ ↑e' ∘ f ⁻¹' e'.symm.source) x
h2 : P (↑e'.symm ∘ ↑e' ∘ f) s x
⊢ ↑e'.symm ∘ ↑e' ∘ f =ᶠ[𝓝[s] x] f
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
x : M
hx : x ∈ f.source
⊢ ↑f ⁻¹' f.target ∈ 𝓝 x

Tactic:
exact (f.continuousAt hx).preimage_mem_nhds (f.open_target.mem_nhds (f.map_source hx))

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ interior (f.extend I).target ⊆ interior (range ↑I)

Tactic:
rw [f.extend_target, interior_inter, (f.open_target.preimage I.continuous_symm).interior_eq]
  exact inter_subset_right

Goal state after:
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: case refine_2
H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
this : P ((f ∘ ↑e.symm) ∘ ↑e) (↑e ⁻¹' ↑e.symm ⁻¹' s) x
⊢ ↑e ⁻¹' ↑e.symm ⁻¹' s =ᶠ[𝓝 x] s

Tactic:
eventuallyEq_set

Goal state after:
case refine_2
H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
this : P ((f ∘ ↑e.symm) ∘ ↑e) (↑e ⁻¹' ↑e.symm ⁻¹' s) x
⊢ ∀ᶠ (x : H) in 𝓝 x, x ∈ ↑e ⁻¹' ↑e.symm ⁻¹' s ↔ x ∈ s
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
s : Set M
y : M
hy : y ∈ f.source
⊢ map (↑(f.extend I)) (𝓝[s] y) = 𝓝[↑(f.extend I).symm ⁻¹' s ∩ range ↑I] ↑(f.extend I) y

Tactic:
map_extend_nhdsWithin_eq_image f hy,

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
s : Set M
y : M
hy : y ∈ f.source
⊢ 𝓝[↑(f.extend I) '' ((f.extend I).source ∩ s)] ↑(f.extend I) y = 𝓝[↑(f.extend I).symm ⁻¹' s ∩ range ↑I] ↑(f.extend I) y
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
s : Set M
y : M
hy : y ∈ f.source
⊢ ↑(f.extend I).symm ∘ ↑(f.extend I) =ᶠ[𝓝[s] y] id

Tactic:
exact (f.extend I).leftInvOn.eqOn.eventuallyEq_of_mem (extend_source_mem_nhdsWithin _ hy)

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
W :
  ∀ ⦃x : E⦄, x ∈ range ↑I → ∀ ⦃y : E⦄, y ∈ range ↑I → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ range ↑I
⊢ ∀ ⦃x : E⦄, x ∈ range ↑I → ∀ ⦃y : E⦄, y ∈ range ↑I → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ range ↑I

Tactic:
intro u hu v hv a b ha hb hab

Goal state after:
case pos
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
W :
  ∀ ⦃x : E⦄, x ∈ range ↑I → ∀ ⦃y : E⦄, y ∈ range ↑I → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ range ↑I
u : E
hu : u ∈ range ↑I
v : E
hv : v ∈ range ↑I
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • u + b • v ∈ range ↑I
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
H' : Type u_3
M' : Type u_4
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace H'
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
G : StructureGroupoid H
G' : StructureGroupoid H'
f : OpenPartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
g : H → M'
x : H
s : Set H
hf : f ∈ maximalAtlas M' G'
xf : g x ∈ f.source
hgs : ContinuousWithinAt g s x
hcont : ContinuousWithinAt (↑(chartAt H' (g x)) ∘ g) s x
⊢ (↑(chartAt H' (g x)) ∘ g) x ∈ ((chartAt H' (g x)).symm ≫ₕ f).source

Tactic:
by simp [xf, mfld_simps]

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type w
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type v'
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type w'
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
⊢ {x | x.1 ∈ I.source ∧ x.2 ∈ I'.source} = univ

Tactic:
by simp only [ofPred_true, mfld_simps]

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
⊢ contDiffGroupoid 0 I = continuousGroupoid H

Tactic:
apply le_antisymm le_top

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
⊢ ⊤ ≤ contDiffGroupoid 0 I
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜✝ : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝¹⁷ : NormedAddCommGroup E✝
inst✝¹⁶ : NormedSpace 𝕜✝ E✝
H✝ : Type u_3
inst✝¹⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners 𝕜✝ E✝ H✝
n : ℕ∞ω
M✝ : Type u_4
inst✝¹⁴ : TopologicalSpace M✝
inst✝¹³ : ChartedSpace H✝ M✝
𝕜 : Type u_5
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H : Type u_8
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_9
inst✝⁶ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_10
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : IsManifold I n M
M' : Type u_11
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
inst✝ : IsManifold I' n M'
⊢ ∀ {e e' : OpenPartialHomeomorph (M × M') (ModelProd H H')},
    e ∈ atlas (ModelProd H H') (M × M') →
      e' ∈ atlas (ModelProd H H') (M × M') → e.symm ≫ₕ e' ∈ contDiffGroupoid n (I.prod I')

Tactic:
rintro f g ⟨f1, hf1, f2, hf2, rfl⟩ ⟨g1, hg1, g2, hg2, rfl⟩

Goal state after:
𝕜✝ : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝¹⁷ : NormedAddCommGroup E✝
inst✝¹⁶ : NormedSpace 𝕜✝ E✝
H✝ : Type u_3
inst✝¹⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners 𝕜✝ E✝ H✝
n : ℕ∞ω
M✝ : Type u_4
inst✝¹⁴ : TopologicalSpace M✝
inst✝¹³ : ChartedSpace H✝ M✝
𝕜 : Type u_5
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H : Type u_8
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_9
inst✝⁶ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_10
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : IsManifold I n M
M' : Type u_11
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
inst✝ : IsManifold I' n M'
f1 : OpenPartialHomeomorph M H
hf1 : f1 ∈ atlas H M
f2 : OpenPartialHomeomorph M' H'
hf2 : f2 ∈ atlas H' M'
g1 : OpenPartialHomeomorph M H
hg1 : g1 ∈ atlas H M
g2 : OpenPartialHomeomorph M' H'
hg2 : g2 ∈ atlas H' M'
⊢ (f1.prod f2).symm ≫ₕ g1.prod g2 ∈ contDiffGroupoid n (I.prod I')
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜✝ : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝¹⁷ : NormedAddCommGroup E✝
inst✝¹⁶ : NormedSpace 𝕜✝ E✝
H✝ : Type u_3
inst✝¹⁵ : TopologicalSpace H✝
I✝ : ModelWithCorners 𝕜✝ E✝ H✝
n : ℕ∞ω
M✝ : Type u_4
inst✝¹⁴ : TopologicalSpace M✝
inst✝¹³ : ChartedSpace H✝ M✝
𝕜 : Type u_5
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H : Type u_8
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_9
inst✝⁶ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_10
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : IsManifold I n M
M' : Type u_11
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
inst✝ : IsManifold I' n M'
⊢ ∀ {e e' : OpenPartialHomeomorph (M × M') (ModelProd H H')},
    e ∈ atlas (ModelProd H H') (M × M') →
      e' ∈ atlas (ModelProd H H') (M × M') → e.symm ≫ₕ e' ∈ contDiffGroupoid n (I.prod I')

Tactic:
rintro f g ⟨f1, hf1, f2, hf2, rfl⟩ ⟨g1, hg1, g2, hg2, rfl⟩
    rw [OpenPartialHomeomorph.prod_symm, OpenPartialHomeomorph.prod_trans]
    have h1 := (contDiffGroupoid n I).compatible hf1 hg1
    have h2 := (contDiffGroupoid n I').compatible hf2 hg2
    exact contDiffGroupoid_prod h1 h2

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Basic

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
n : ℕ∞ω
f : M → M
hf : ContMDiffOn I I n f s
hmaps : MapsTo f s s
⊢ ContMDiffOn I I n f^[0] s

Tactic:
| zero =>

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
e : M → H
h : IsOpenEmbedding e
n : ℕ∞ω
inst✝ : Nonempty M
this : IsManifold I ω M
⊢ Continuous[inst✝¹, inst✝²] e ∧
    ∀ (x : M) (y : H),
      ContDiffOn 𝕜 n (↑(extChartAt I y) ∘ e ∘ ↑(extChartAt I x).symm)
        ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' e ⁻¹' (extChartAt I y).source)

Tactic:
use h.continuous

Goal state after:
case right
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
e : M → H
h : IsOpenEmbedding e
n : ℕ∞ω
inst✝ : Nonempty M
this : IsManifold I ω M
⊢ ∀ (x : M) (y : H),
    ContDiffOn 𝕜 n (↑(extChartAt I y) ∘ e ∘ ↑(extChartAt I x).symm)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' e ⁻¹' (extChartAt I y).source)
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
e' : OpenPartialHomeomorph M' H'
f : M → M'
s : Set M
x : M
n : ℕ∞ω
he' : e' ∈ maximalAtlas I' n M'
hy : f x ∈ e'.source
⊢ ContinuousWithinAt (f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x) ∧
      ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') n (↑(e'.extend I') ∘ f ∘ ↑(extChartAt I x).symm)
        (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x) ↔
    ContinuousWithinAt f s x ∧
      ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
        (↑(extChartAt I x) x)

Tactic:
apply and_congr continuousWithinAt_iff_source.symm

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
e' : OpenPartialHomeomorph M' H'
f : M → M'
s : Set M
x : M
n : ℕ∞ω
he' : e' ∈ maximalAtlas I' n M'
hy : f x ∈ e'.source
⊢ ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') n (↑(e'.extend I') ∘ f ∘ ↑(extChartAt I x).symm)
      (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x) ↔
    ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
      (↑(extChartAt I x) x)
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
x x' : M
y : M'
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hx : x' ∈ (chartAt H x).source
hy : f x' ∈ (chartAt H' y).source
⊢ ContinuousWithinAt f univ x' ∧
      ContDiffWithinAt 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
        (↑(extChartAt I x).symm ⁻¹' univ ∩ range ↑I) (↑(extChartAt I x) x') ↔
    ContinuousAt f x' ∧
      ContDiffWithinAt 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm) (range ↑I) (↑(extChartAt I x) x')

Tactic:
rw [continuousWithinAt_univ, preimage_univ, univ_inter]

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s✝ : Set M
x : M
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
s : Set M
hxs : x ∈ s
h : ContMDiffWithinAt I I' n f s x
v : Set E
hmem :
  v ∈
    𝓝[(extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source)] ↑(extChartAt I x) x
hsub : v ⊆ (extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source)
hv : ContDiffOn 𝕜 n (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm) v
hxs' :
  ↑(extChartAt I x) x ∈ (extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source)
⊢ ContMDiffOn I I' n f (↑(extChartAt I x).symm '' v)

Tactic:
· have hv₁ : (extChartAt I x).symm '' v ⊆ (extChartAt I x).source :=
      image_subset_iff.2 fun y hy ↦ (extChartAt I x).map_target (hsub hy).1
    have hv₂ : MapsTo f ((extChartAt I x).symm '' v) (extChartAt I' (f x)).source := by
      rintro _ ⟨y, hy, rfl⟩
      exact (hsub hy).2.2
    rwa [contMDiffOn_iff_of_subset_source' hv₁ hv₂, PartialEquiv.image_symm_image_of_subset_target]
    exact hsub.trans inter_subset_left

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f f₁ : M → M'
n : ℕ∞ω
h₁ : ∀ (y : M), f₁ y = f y
⊢ ContMDiffOn I I' n f₁ univ ↔ ContMDiffOn I I' n f univ

Tactic:
exact contMDiffOn_congr fun y _ ↦ h₁ y

Goal state after: