ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean
𝕜 : Type u_1
inst✝²⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace 𝕜 E
H : Type u_3
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
Is : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹⁸ : NormedAddCommGroup E'
inst✝¹⁷ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝¹⁶ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹⁵ : TopologicalSpace M'
inst✝¹⁴ : ChartedSpace H' M'
I's : SmoothManifoldWithCorners I' M'
F : Type u_8
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜 F
G : Type u_9
inst✝¹¹ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹⁰ : TopologicalSpace N
inst✝⁹ : ChartedSpace G N
Js : SmoothManifoldWithCorners J N
F' : Type u_11
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
G' : Type u_12
inst✝⁶ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_13
inst✝⁵ : TopologicalSpace N'
inst✝⁴ : ChartedSpace G' N'
J's : SmoothManifoldWithCorners J' N'
F₁ : Type u_14
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpace 𝕜 F₁
F₂ : Type u_15
inst✝¹ : NormedAddCommGroup F₂
inst✝ : NormedSpace 𝕜 F₂
f✝ f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n : ℕ∞
x₀ : N
f : N → M → M'
g : N → M
hf : ContMDiffAt (J.prod I) I' n (uncurry f) (x₀, g x₀)
hg : ContMDiffAt J I m g x₀
hmn : m + 1 ≤ n
h4f✝ : ContinuousAt (fun x => f x (g x)) x₀
h4f : (fun x => f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source ∈ 𝓝 x₀
h3f : ∀ᶠ (x' : N × M) in 𝓝 (x₀, g x₀), ContMDiffAt (J.prod I) I' (↑One.one) (uncurry f) x'
h2f : ∀ᶠ (x₂ : N) in 𝓝 x₀, ContMDiffAt I I' 1 (f x₂) (g x₂)
h2g : g ⁻¹' (extChartAt I (g x₀)).source ∈ 𝓝 x₀
this✝² : ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f (↑(extChartAt J x₀).symm x) ∘ ↑(extChartAt I (g x₀)).symm) (range ↑I) (↑(extChartAt I (g x₀)) (g (↑(extChartAt J x₀).symm x)))) (range ↑J) (↑(extChartAt J x₀) x₀)
this✝¹ : ContMDiffAt J 𝓘(𝕜, E →L[𝕜] E') m (fun x => fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f x ∘ ↑(extChartAt I (g x₀)).symm) (range ↑I) (↑(extChartAt I (g x₀)) (g x))) x₀
this✝ : ContMDiffAt J 𝓘(𝕜, E →L[𝕜] E') m (fun x => fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ ↑(extChartAt I' (f x (g x))).symm ∘ writtenInExtChartAt I I' (g x) (f x) ∘ ↑(extChartAt I (g x)) ∘ ↑(extChartAt I (g x₀)).symm) (range ↑I) (↑(extChartAt I (g x₀)) (g x))) x₀
_inst✝ : (x : N) → NormedAddCommGroup (TangentSpace I (g x)) := fun x => inferInstanceAs (NormedAddCommGroup E)
_inst : (x : N) → NormedSpace 𝕜 (TangentSpace I (g x)) := fun x => inferInstanceAs (NormedSpace 𝕜 E)
this : ContMDiffAt J 𝓘(𝕜, E →L[𝕜] E') m (fun x => (fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ ↑(extChartAt I' (f x (g x))).symm) (range ↑I') (↑(extChartAt I' (f x (g x))) (f x (g x)))).comp ((mfderiv I I' (f x) (g x)).comp (fderivWithin 𝕜 (↑(extChartAt I (g x)) ∘ ↑(extChartAt I (g x₀)).symm) (range ↑I) (↑(extChartAt I (g x₀)) (g x))))) x₀
x : N
hx : x ∈ g ⁻¹' (extChartAt I (g x₀)).source
h2x : x ∈ (fun x => f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source

⊢ g x ∈ (chartAt H (g x₀)).source
-- Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean
𝕜 : Type u_1
inst✝²⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace 𝕜 E
H : Type u_3
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
Is : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹⁸ : NormedAddCommGroup E'
inst✝¹⁷ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝¹⁶ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹⁵ : TopologicalSpace M'
inst✝¹⁴ : ChartedSpace H' M'
I's : SmoothManifoldWithCorners I' M'
F : Type u_8
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜 F
G : Type u_9
inst✝¹¹ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹⁰ : TopologicalSpace N
inst✝⁹ : ChartedSpace G N
Js : SmoothManifoldWithCorners J N
F' : Type u_11
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
G' : Type u_12
inst✝⁶ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_13
inst✝⁵ : TopologicalSpace N'
inst✝⁴ : ChartedSpace G' N'
J's : SmoothManifoldWithCorners J' N'
F₁ : Type u_14
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpace 𝕜 F₁
F₂ : Type u_15
inst✝¹ : NormedAddCommGroup F₂
inst✝ : NormedSpace 𝕜 F₂
f✝ f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n : ℕ∞
x₀ : N
f : N → M → M'
g : N → M
hf : ContMDiffAt (J.prod I) I' n (uncurry f) (x₀, g x₀)
hg : ContMDiffAt J I m g x₀
hmn : m + 1 ≤ n
h4f✝ : ContinuousAt (fun x => f x (g x)) x₀
h4f : (fun x => f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source ∈ 𝓝 x₀
h3f : ∀ᶠ (x' : N × M) in 𝓝 (x₀, g x₀), ContMDiffAt (J.prod I) I' (↑One.one) (uncurry f) x'
h2f : ∀ᶠ (x₂ : N) in 𝓝 x₀, ContMDiffAt I I' 1 (f x₂) (g x₂)
h2g : g ⁻¹' (extChartAt I (g x₀)).source ∈ 𝓝 x₀
this✝² : ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f (↑(extChartAt J x₀).symm x) ∘ ↑(extChartAt I (g x₀)).symm) (range ↑I) (↑(extChartAt I (g x₀)) (g (↑(extChartAt J x₀).symm x)))) (range ↑J) (↑(extChartAt J x₀) x₀)
this✝¹ : ContMDiffAt J 𝓘(𝕜, E →L[𝕜] E') m (fun x => fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f x ∘ ↑(extChartAt I (g x₀)).symm) (range ↑I) (↑(extChartAt I (g x₀)) (g x))) x₀
this✝ : ContMDiffAt J 𝓘(𝕜, E →L[𝕜] E') m (fun x => fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ ↑(extChartAt I' (f x (g x))).symm ∘ writtenInExtChartAt I I' (g x) (f x) ∘ ↑(extChartAt I (g x)) ∘ ↑(extChartAt I (g x₀)).symm) (range ↑I) (↑(extChartAt I (g x₀)) (g x))) x₀
_inst✝ : (x : N) → NormedAddCommGroup (TangentSpace I (g x)) := fun x => inferInstanceAs (NormedAddCommGroup E)
_inst : (x : N) → NormedSpace 𝕜 (TangentSpace I (g x)) := fun x => inferInstanceAs (NormedSpace 𝕜 E)
this : ContMDiffAt J 𝓘(𝕜, E →L[𝕜] E') m (fun x => (fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ ↑(extChartAt I' (f x (g x))).symm) (range ↑I') (↑(extChartAt I' (f x (g x))) (f x (g x)))).comp ((mfderiv I I' (f x) (g x)).comp (fderivWithin 𝕜 (↑(extChartAt I (g x)) ∘ ↑(extChartAt I (g x₀)).symm) (range ↑I) (↑(extChartAt I (g x₀)) (g x))))) x₀
x : N
hx : x ∈ g ⁻¹' (extChartAt I (g x₀)).source
h2x : x ∈ (fun x => f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source

⊢ f x (g x) ∈ (chartAt H' (f x₀ (g x₀))).source
-- Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
g : K → C
e : J ≃ K
w : (j : J) → g (e j) ≅ f j
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct g

⊢ (whiskerEquiv e w).hom = lift fun k => π f (e.symm k) ≫ (w (e.symm k)).inv ≫ eqToHom ⋯
-- Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
g : K → C
e : J ≃ K
w : (j : J) → g (e j) ≅ f j
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct g

⊢ (desc fun j => (w j).inv ≫ ι g (e j)) = lift fun k => π f (e.symm k) ≫ (w (e.symm k)).inv ≫ eqToHom ⋯
-- Mathlib/MeasureTheory/Function/Jacobian.lean
E : Type u_1
F : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s✝ : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0

⊢ ∃ t A, Pairwise (Disjoint on t) ∧ (∀ (n : ℕ), MeasurableSet (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
-- Mathlib/MeasureTheory/Function/Jacobian.lean
E : Type u_1
F : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s✝ : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
t : ℕ → Set E
A : ℕ → E →L[ℝ] F
t_closed : ∀ (n : ℕ), IsClosed (t n)
st : s ⊆ ⋃ n, t n
t_approx : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))
ht : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y

⊢ ∃ t A, Pairwise (Disjoint on t) ∧ (∀ (n : ℕ), MeasurableSet (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
-- Mathlib/MeasureTheory/Function/Jacobian.lean
E : Type u_1
F : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s✝ : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
t : ℕ → Set E
A : ℕ → E →L[ℝ] F
t_closed : ∀ (n : ℕ), IsClosed (t n)
st : s ⊆ ⋃ n, t n
t_approx : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))
ht : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y

⊢ s ⊆ ⋃ n, disjointed t n
-- Mathlib/MeasureTheory/Function/Jacobian.lean
E : Type u_1
F : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s✝ : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
t : ℕ → Set E
A : ℕ → E →L[ℝ] F
t_closed : ∀ (n : ℕ), IsClosed (t n)
st : s ⊆ ⋃ n, t n
t_approx : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))
ht : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y

⊢ ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ disjointed t n) (r (A n))
-- Mathlib/Algebra/Order/Field/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedSemifield α
a b c d e : α
m n : ℤ
s : Set α
ha : 0 ≤ a
hs : IsGLB s b

⊢ IsGLB ((fun b => b * a) '' s) (b * a)
-- Mathlib/RingTheory/Flat/Stability.lean
R : Type u
S : Type v
M : Type w
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : Flat R S
I : Ideal R

⊢ ↑R (Module.Flat.auxLTensor R S M I) = ↑(TensorProduct.rid R M) ∘ₗ lTensor M (Submodule.subtype I)
-- Mathlib/RingTheory/Flat/Stability.lean
R : Type u
S : Type v
M : Type w
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : Flat R S
I : Ideal R

⊢ ∀ (x : M) (y : ↥I), (↑R (Module.Flat.auxLTensor R S M I)) (x ⊗ₜ[R] y) = (↑(TensorProduct.rid R M) ∘ₗ lTensor M (Submodule.subtype I)) (x ⊗ₜ[R] y)
-- Mathlib/RingTheory/Flat/Stability.lean
R : Type u
S : Type v
M : Type w
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : Flat R S
I : Ideal R
m : M
x : ↥I

⊢ (↑R (Module.Flat.auxLTensor R S M I)) (m ⊗ₜ[R] x) = (↑(TensorProduct.rid R M) ∘ₗ lTensor M (Submodule.subtype I)) (m ⊗ₜ[R] x)
-- Mathlib/RingTheory/Flat/Stability.lean
R : Type u
S : Type v
M : Type w
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : Flat R S
I : Ideal R
m : M
x : ↥I

⊢ (Submodule.subtype (Module.Flat.J R S I)) (↑(LinearEquiv.refl S M) (((m, x).1, 1).1, ((TensorProduct.mk R S ↥I) ((m, x).1, 1).2) (↑(AlgebraTensorModule.rid S S M).symm (m, x).1, ↑(LinearEquiv.refl R ↥I).symm (m, x).2).2).1, ↑(Module.Flat.auxIso R S) (((m, x).1, 1).1, ((TensorProduct.mk R S ↥I) ((m, x).1, 1).2) (↑(AlgebraTensorModule.rid S S M).symm (m, x).1, ↑(LinearEquiv.refl R ↥I).symm (m, x).2).2).2).2 • LinearMap.id (↑(LinearEquiv.refl S M) (((m, x).1, 1).1, ((TensorProduct.mk R S ↥I) ((m, x).1, 1).2) (↑(AlgebraTensorModule.rid S S M).symm (m, x).1, ↑(LinearEquiv.refl R ↥I).symm (m, x).2).2).1, ↑(Module.Flat.auxIso R S) (((m, x).1, 1).1, ((TensorProduct.mk R S ↥I) ((m, x).1, 1).2) (↑(AlgebraTensorModule.rid S S M).symm (m, x).1, ↑(LinearEquiv.refl R ↥I).symm (m, x).2).2).2).1 = (↑(TensorProduct.rid R M) ∘ₗ lTensor M (Submodule.subtype I)) (m ⊗ₜ[R] x)
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
s✝ : Set E
x : E
r : ℝ
hr : 0 ≤ r
s : ℝ
hs : 0 ≤ s

⊢ μ (closedBall x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 s)
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
s✝ : Set E
x : E
r : ℝ
hr : 0 ≤ r
s : ℝ
hs : 0 ≤ s
this : closedBall 0 (r * s) = r • closedBall 0 s

⊢ μ (closedBall x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 s)
-- Mathlib/Analysis/Convex/Side.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁶ : StrictOrderedCommRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f

⊢ (map f s).WSameSide (f x) (f y) ↔ s.WSameSide x y
-- Mathlib/Analysis/Convex/Side.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁶ : StrictOrderedCommRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
h : (map f s).WSameSide (f x) (f y)

⊢ s.WSameSide x y
-- Mathlib/Analysis/Convex/Side.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁶ : StrictOrderedCommRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
fp₁ : P'
hfp₁ : fp₁ ∈ map f s
fp₂ : P'
hfp₂ : fp₂ ∈ map f s
h : SameRay R (f x -ᵥ fp₁) (f y -ᵥ fp₂)

⊢ s.WSameSide x y
-- Mathlib/Analysis/Convex/Side.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁶ : StrictOrderedCommRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
fp₁ : P'
hfp₁ : ∃ y ∈ s, f y = fp₁
fp₂ : P'
hfp₂ : ∃ y ∈ s, f y = fp₂
h : SameRay R (f x -ᵥ fp₁) (f y -ᵥ fp₂)

⊢ s.WSameSide x y
-- Mathlib/Analysis/Convex/Side.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁶ : StrictOrderedCommRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : SameRay R (f x -ᵥ f p₁) (f y -ᵥ f p₂)

⊢ SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
-- Mathlib/Analysis/Convex/Side.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁶ : StrictOrderedCommRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

⊢ SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
this : ↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B) = Algebra.linearMap B L

⊢ Submodule.map (↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B)) (differentIdeal A B) = 1 / Submodule.traceDual A K 1
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
this : ↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B) = Algebra.linearMap B L
H : (algebraMap (FractionRing A) (FractionRing B)).comp ↑(FractionRing.algEquiv A K).symm.toRingEquiv = (↑(FractionRing.algEquiv B L).symm.toRingEquiv).comp (algebraMap K L)

⊢ Submodule.map (↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B)) (differentIdeal A B) = 1 / Submodule.traceDual A K 1
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
this✝ : ↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B) = Algebra.linearMap B L
H : (algebraMap (FractionRing A) (FractionRing B)).comp ↑(FractionRing.algEquiv A K).symm.toRingEquiv = (↑(FractionRing.algEquiv B L).symm.toRingEquiv).comp (algebraMap K L)
this : Algebra.IsSeparable (FractionRing A) (FractionRing B)

⊢ Submodule.map (↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B)) (differentIdeal A B) = 1 / Submodule.traceDual A K 1
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
this✝¹ : ↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B) = Algebra.linearMap B L
H : (algebraMap (FractionRing A) (FractionRing B)).comp ↑(FractionRing.algEquiv A K).symm.toRingEquiv = (↑(FractionRing.algEquiv B L).symm.toRingEquiv).comp (algebraMap K L)
this✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
this : FiniteDimensional (FractionRing A) (FractionRing B)

⊢ Submodule.map (↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B)) (differentIdeal A B) = 1 / Submodule.traceDual A K 1
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
this✝² : ↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B) = Algebra.linearMap B L
H : (algebraMap (FractionRing A) (FractionRing B)).comp ↑(FractionRing.algEquiv A K).symm.toRingEquiv = (↑(FractionRing.algEquiv B L).symm.toRingEquiv).comp (algebraMap K L)
this✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : Algebra.IsIntegral A B

⊢ Submodule.map (↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B)) (differentIdeal A B) = 1 / Submodule.traceDual A K 1
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
this✝² : ↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B) = Algebra.linearMap B L
H : (algebraMap (FractionRing A) (FractionRing B)).comp ↑(FractionRing.algEquiv A K).symm.toRingEquiv = (↑(FractionRing.algEquiv B L).symm.toRingEquiv).comp (algebraMap K L)
this✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : Algebra.IsIntegral A B

⊢ Submodule.map (FractionRing.algEquiv B L).toLinearMap (Submodule.map (Algebra.linearMap B (FractionRing B)) (differentIdeal A B)) = 1 / Submodule.traceDual A K 1
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : IsIntegralClosure B A L
inst✝⁵ : IsFractionRing B L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
this✝² : ↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B) = Algebra.linearMap B L
H : (algebraMap (FractionRing A) (FractionRing B)).comp ↑(FractionRing.algEquiv A K).symm.toRingEquiv = (↑(FractionRing.algEquiv B L).symm.toRingEquiv).comp (algebraMap K L)
this✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
this✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : Algebra.IsIntegral A B

⊢ 1 / Submodule.map (↑(FractionRing.algEquiv B L)).toLinearMap (Submodule.traceDual A (FractionRing A) 1) = 1 / Submodule.traceDual A K 1