file_path
stringlengths 11
79
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stringclasses 4
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Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Hom.toFun_eq_coe
|
[] |
[
521,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
520,
1
] |
Mathlib/LinearAlgebra/Pi.lean
|
Submodule.pi_empty
|
[] |
[
288,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean
|
CategoryTheory.Limits.WidePushout.hom_eq_desc
|
[
{
"state_after": "J : Type w\nC : Type u\ninst✝² : Category C\nD : Type ?u.254030\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : D\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\ng : widePushout B (fun j => objs j) arrows ⟶ X\nj : J\n⊢ (arrows j ≫ ι arrows j) ≫ g = head arrows ≫ g",
"state_before": "J : Type w\nC : Type u\ninst✝² : Category C\nD : Type ?u.254030\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : D\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\ng : widePushout B (fun j => objs j) arrows ⟶ X\nj : J\n⊢ arrows j ≫ (fun j => ι arrows j ≫ g) j = head arrows ≫ g",
"tactic": "rw [← Category.assoc]"
},
{
"state_after": "no goals",
"state_before": "J : Type w\nC : Type u\ninst✝² : Category C\nD : Type ?u.254030\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : D\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\ng : widePushout B (fun j => objs j) arrows ⟶ X\nj : J\n⊢ (arrows j ≫ ι arrows j) ≫ g = head arrows ≫ g",
"tactic": "simp"
},
{
"state_after": "case a\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : D\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\ng : widePushout B (fun j => objs j) arrows ⟶ X\n⊢ ∀ (j : J), ι arrows j ≫ g = ι arrows j ≫ g\n\ncase a\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : D\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\ng : widePushout B (fun j => objs j) arrows ⟶ X\n⊢ head arrows ≫ g = head arrows ≫ g",
"state_before": "J : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : D\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\ng : widePushout B (fun j => objs j) arrows ⟶ X\n⊢ g =\n desc (head arrows ≫ g) (fun j => ι arrows j ≫ g)\n (_ : ∀ (j : J), arrows j ≫ (fun j => ι arrows j ≫ g) j = head arrows ≫ g)",
"tactic": "apply eq_desc_of_comp_eq"
},
{
"state_after": "case a\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : D\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\ng : widePushout B (fun j => objs j) arrows ⟶ X\n⊢ head arrows ≫ g = head arrows ≫ g",
"state_before": "case a\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : D\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\ng : widePushout B (fun j => objs j) arrows ⟶ X\n⊢ ∀ (j : J), ι arrows j ≫ g = ι arrows j ≫ g\n\ncase a\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : D\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\ng : widePushout B (fun j => objs j) arrows ⟶ X\n⊢ head arrows ≫ g = head arrows ≫ g",
"tactic": "aesop_cat"
},
{
"state_after": "no goals",
"state_before": "case a\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : D\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\ng : widePushout B (fun j => objs j) arrows ⟶ X\n⊢ head arrows ≫ g = head arrows ≫ g",
"tactic": "rfl"
}
] |
[
449,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/GroupTheory/Exponent.lean
|
Monoid.exponent_ne_zero_of_finite
|
[
{
"state_after": "case intro\nG : Type u\ninst✝¹ : LeftCancelMonoid G\ninst✝ : Finite G\nval✝ : Fintype G\n⊢ exponent G ≠ 0",
"state_before": "G : Type u\ninst✝¹ : LeftCancelMonoid G\ninst✝ : Finite G\n⊢ exponent G ≠ 0",
"tactic": "cases nonempty_fintype G"
},
{
"state_after": "no goals",
"state_before": "case intro\nG : Type u\ninst✝¹ : LeftCancelMonoid G\ninst✝ : Finite G\nval✝ : Fintype G\n⊢ exponent G ≠ 0",
"tactic": "simpa [← lcm_order_eq_exponent, Finset.lcm_eq_zero_iff] using fun x => (orderOf_pos x).ne'"
}
] |
[
274,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.image_inter
|
[] |
[
495,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/Algebra/Order/LatticeGroup.lean
|
LatticeOrderedCommGroup.pos_of_one_le
|
[
{
"state_after": "α : Type u\ninst✝¹ : Lattice α\ninst✝ : CommGroup α\na : α\nh : 1 ≤ a\n⊢ a ⊔ 1 = a",
"state_before": "α : Type u\ninst✝¹ : Lattice α\ninst✝ : CommGroup α\na : α\nh : 1 ≤ a\n⊢ a⁺ = a",
"tactic": "rw [m_pos_part_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝¹ : Lattice α\ninst✝ : CommGroup α\na : α\nh : 1 ≤ a\n⊢ a ⊔ 1 = a",
"tactic": "exact sup_of_le_left h"
}
] |
[
468,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
aestronglyMeasurable_union_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.401923\nι : Type ?u.401926\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\nf g : α → β\ninst✝ : PseudoMetrizableSpace β\ns t : Set α\n⊢ AEStronglyMeasurable f (Measure.restrict μ (s ∪ t)) ↔\n AEStronglyMeasurable f (Measure.restrict μ s) ∧ AEStronglyMeasurable f (Measure.restrict μ t)",
"tactic": "simp only [union_eq_iUnion, aestronglyMeasurable_iUnion_iff, Bool.forall_bool, cond, and_comm]"
}
] |
[
1721,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1718,
1
] |
Std/Data/Array/Lemmas.lean
|
Array.ugetElem_eq_getElem
|
[] |
[
45,
27
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
44,
9
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
Real.map_volume_mul_left
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2204078\ninst✝ : Fintype ι\na : ℝ\nh : a ≠ 0\n⊢ Measure.map (fun x => a * x) volume = ofReal (abs a⁻¹) • volume",
"tactic": "conv_rhs =>\n rw [← Real.smul_map_volume_mul_left h, smul_smul, ← ENNReal.ofReal_mul (abs_nonneg _), ←\n abs_mul, inv_mul_cancel h, abs_one, ENNReal.ofReal_one, one_smul]"
}
] |
[
315,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/RingTheory/Adjoin/FG.lean
|
Subalgebra.fg_of_fg_map
|
[
{
"state_after": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nS : Subalgebra R A\nf : A →ₐ[R] B\nhf : Function.Injective ↑f\nhs✝ : FG (map f S)\ns : Finset B\nhs : Algebra.adjoin R ↑s = map f S\n⊢ ↑s ⊆ Set.range ↑f",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nS : Subalgebra R A\nf : A →ₐ[R] B\nhf : Function.Injective ↑f\nhs✝ : FG (map f S)\ns : Finset B\nhs : Algebra.adjoin R ↑s = map f S\n⊢ map f\n (Algebra.adjoin R\n ↑(Finset.preimage s ↑f\n (_ : ∀ (x : A), x ∈ ↑f ⁻¹' ↑s → ∀ (x_2 : A), x_2 ∈ ↑f ⁻¹' ↑s → ↑f x = ↑f x_2 → x = x_2))) =\n map f S",
"tactic": "rw [← Algebra.adjoin_image, Finset.coe_preimage, Set.image_preimage_eq_of_subset, hs]"
},
{
"state_after": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nS : Subalgebra R A\nf : A →ₐ[R] B\nhf : Function.Injective ↑f\nhs✝ : FG (map f S)\ns : Finset B\nhs : Algebra.adjoin R ↑s = map f S\n⊢ map f S ≤ map f ⊤",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nS : Subalgebra R A\nf : A →ₐ[R] B\nhf : Function.Injective ↑f\nhs✝ : FG (map f S)\ns : Finset B\nhs : Algebra.adjoin R ↑s = map f S\n⊢ ↑s ⊆ Set.range ↑f",
"tactic": "rw [← AlgHom.coe_range, ← Algebra.adjoin_le_iff, hs, ← Algebra.map_top]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nS : Subalgebra R A\nf : A →ₐ[R] B\nhf : Function.Injective ↑f\nhs✝ : FG (map f S)\ns : Finset B\nhs : Algebra.adjoin R ↑s = map f S\n⊢ map f S ≤ map f ⊤",
"tactic": "exact map_mono le_top"
}
] |
[
162,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.coe_refl
|
[] |
[
732,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
731,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
|
CategoryTheory.Limits.Cotrident.ofCocone_ι
|
[
{
"state_after": "no goals",
"state_before": "J : Type w\nC : Type u\ninst✝ : Category C\nX Y : C\nf : J → (X ⟶ Y)\nF : WalkingParallelFamily J ⥤ C\nt : Cocone F\nj : WalkingParallelFamily J\n⊢ (parallelFamily fun j => F.map (line j)).obj j = F.obj j",
"tactic": "cases j <;> aesop_cat"
}
] |
[
509,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
507,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
TendstoUniformlyOn.div
|
[] |
[
505,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
|
HomogeneousLocalization.NumDenSameDeg.deg_one
|
[] |
[
129,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Analysis/Calculus/Inverse.lean
|
ApproximatesLinearOn.open_image
|
[
{
"state_after": "case inl\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhs : IsOpen s\nhE : Subsingleton F\n⊢ IsOpen (f '' s)\n\ncase inr\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhs : IsOpen s\nhc : c < f'symm.nnnorm⁻¹\n⊢ IsOpen (f '' s)",
"state_before": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhs : IsOpen s\nhc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹\n⊢ IsOpen (f '' s)",
"tactic": "cases' hc with hE hc"
},
{
"state_after": "case inr\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhc : c < f'symm.nnnorm⁻¹\nhs : ∀ (a : E), a ∈ s → ∃ i, 0 < i ∧ closedBall a i ⊆ s\n⊢ ∀ (x : E), x ∈ s → ∃ i, 0 < i ∧ closedBall (f x) i ⊆ f '' s",
"state_before": "case inr\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhs : IsOpen s\nhc : c < f'symm.nnnorm⁻¹\n⊢ IsOpen (f '' s)",
"tactic": "simp only [isOpen_iff_mem_nhds, nhds_basis_closedBall.mem_iff, ball_image_iff] at hs ⊢"
},
{
"state_after": "case inr\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhc : c < f'symm.nnnorm⁻¹\nhs : ∀ (a : E), a ∈ s → ∃ i, 0 < i ∧ closedBall a i ⊆ s\nx : E\nhx : x ∈ s\n⊢ ∃ i, 0 < i ∧ closedBall (f x) i ⊆ f '' s",
"state_before": "case inr\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhc : c < f'symm.nnnorm⁻¹\nhs : ∀ (a : E), a ∈ s → ∃ i, 0 < i ∧ closedBall a i ⊆ s\n⊢ ∀ (x : E), x ∈ s → ∃ i, 0 < i ∧ closedBall (f x) i ⊆ f '' s",
"tactic": "intro x hx"
},
{
"state_after": "case inr.intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε✝ : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhc : c < f'symm.nnnorm⁻¹\nhs : ∀ (a : E), a ∈ s → ∃ i, 0 < i ∧ closedBall a i ⊆ s\nx : E\nhx : x ∈ s\nε : ℝ\nε0 : 0 < ε\nhε : closedBall x ε ⊆ s\n⊢ ∃ i, 0 < i ∧ closedBall (f x) i ⊆ f '' s",
"state_before": "case inr\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhc : c < f'symm.nnnorm⁻¹\nhs : ∀ (a : E), a ∈ s → ∃ i, 0 < i ∧ closedBall a i ⊆ s\nx : E\nhx : x ∈ s\n⊢ ∃ i, 0 < i ∧ closedBall (f x) i ⊆ f '' s",
"tactic": "rcases hs x hx with ⟨ε, ε0, hε⟩"
},
{
"state_after": "case inr.intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε✝ : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhc : c < f'symm.nnnorm⁻¹\nhs : ∀ (a : E), a ∈ s → ∃ i, 0 < i ∧ closedBall a i ⊆ s\nx : E\nhx : x ∈ s\nε : ℝ\nε0 : 0 < ε\nhε : closedBall x ε ⊆ s\n⊢ closedBall (f x) ((↑f'symm.nnnorm⁻¹ - ↑c) * ε) ⊆ f '' s",
"state_before": "case inr.intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε✝ : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhc : c < f'symm.nnnorm⁻¹\nhs : ∀ (a : E), a ∈ s → ∃ i, 0 < i ∧ closedBall a i ⊆ s\nx : E\nhx : x ∈ s\nε : ℝ\nε0 : 0 < ε\nhε : closedBall x ε ⊆ s\n⊢ ∃ i, 0 < i ∧ closedBall (f x) i ⊆ f '' s",
"tactic": "refine' ⟨(f'symm.nnnorm⁻¹ - c) * ε, mul_pos (sub_pos.2 hc) ε0, _⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε✝ : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhc : c < f'symm.nnnorm⁻¹\nhs : ∀ (a : E), a ∈ s → ∃ i, 0 < i ∧ closedBall a i ⊆ s\nx : E\nhx : x ∈ s\nε : ℝ\nε0 : 0 < ε\nhε : closedBall x ε ⊆ s\n⊢ closedBall (f x) ((↑f'symm.nnnorm⁻¹ - ↑c) * ε) ⊆ f '' s",
"tactic": "exact (hf.surjOn_closedBall_of_nonlinearRightInverse f'symm (le_of_lt ε0) hε).mono hε Subset.rfl"
},
{
"state_after": "case inl\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhs : IsOpen s\nhE : Subsingleton F\n⊢ IsOpen (f '' s)",
"state_before": "case inl\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhs : IsOpen s\nhE : Subsingleton F\n⊢ IsOpen (f '' s)",
"tactic": "skip"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.303710\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.303813\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nhs : IsOpen s\nhE : Subsingleton F\n⊢ IsOpen (f '' s)",
"tactic": "apply isOpen_discrete"
}
] |
[
335,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.subgraphOfAdj_eq_induce
|
[
{
"state_after": "case verts.h\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝ : V\n⊢ x✝ ∈ (subgraphOfAdj G hvw).verts ↔ x✝ ∈ (induce ⊤ {v, w}).verts\n\ncase Adj.h.h.a\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\n⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ ↔ Adj (induce ⊤ {v, w}) x✝¹ x✝",
"state_before": "ι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\n⊢ subgraphOfAdj G hvw = induce ⊤ {v, w}",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case verts.h\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝ : V\n⊢ x✝ ∈ (subgraphOfAdj G hvw).verts ↔ x✝ ∈ (induce ⊤ {v, w}).verts",
"tactic": "simp"
},
{
"state_after": "case Adj.h.h.a.mp\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\n⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ → Adj (induce ⊤ {v, w}) x✝¹ x✝\n\ncase Adj.h.h.a.mpr\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\n⊢ Adj (induce ⊤ {v, w}) x✝¹ x✝ → Adj (subgraphOfAdj G hvw) x✝¹ x✝",
"state_before": "case Adj.h.h.a\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\n⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ ↔ Adj (induce ⊤ {v, w}) x✝¹ x✝",
"tactic": "constructor"
},
{
"state_after": "case Adj.h.h.a.mp\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\nh : Adj (subgraphOfAdj G hvw) x✝¹ x✝\n⊢ Adj (induce ⊤ {v, w}) x✝¹ x✝",
"state_before": "case Adj.h.h.a.mp\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\n⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ → Adj (induce ⊤ {v, w}) x✝¹ x✝",
"tactic": "intro h"
},
{
"state_after": "case Adj.h.h.a.mp\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\nh : v = x✝¹ ∧ w = x✝ ∨ v = x✝ ∧ w = x✝¹\n⊢ Adj (induce ⊤ {v, w}) x✝¹ x✝",
"state_before": "case Adj.h.h.a.mp\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\nh : Adj (subgraphOfAdj G hvw) x✝¹ x✝\n⊢ Adj (induce ⊤ {v, w}) x✝¹ x✝",
"tactic": "simp only [subgraphOfAdj_Adj, Quotient.eq, Sym2.rel_iff] at h"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a.mp\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\nh : v = x✝¹ ∧ w = x✝ ∨ v = x✝ ∧ w = x✝¹\n⊢ Adj (induce ⊤ {v, w}) x✝¹ x✝",
"tactic": "obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]"
},
{
"state_after": "case Adj.h.h.a.mpr\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\nh : Adj (induce ⊤ {v, w}) x✝¹ x✝\n⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝",
"state_before": "case Adj.h.h.a.mpr\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\n⊢ Adj (induce ⊤ {v, w}) x✝¹ x✝ → Adj (subgraphOfAdj G hvw) x✝¹ x✝",
"tactic": "intro h"
},
{
"state_after": "case Adj.h.h.a.mpr\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\nh : (x✝¹ = v ∨ x✝¹ = w) ∧ (x✝ = v ∨ x✝ = w) ∧ SimpleGraph.Adj G x✝¹ x✝\n⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝",
"state_before": "case Adj.h.h.a.mpr\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\nh : Adj (induce ⊤ {v, w}) x✝¹ x✝\n⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝",
"tactic": "simp only [induce_Adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a.mpr\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv w : V\nhvw : SimpleGraph.Adj G v w\nx✝¹ x✝ : V\nh : (x✝¹ = v ∨ x✝¹ = w) ∧ (x✝ = v ∨ x✝ = w) ∧ SimpleGraph.Adj G x✝¹ x✝\n⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝",
"tactic": "obtain ⟨rfl | rfl, rfl | rfl, ha⟩ := h <;> first |exact (ha.ne rfl).elim|simp"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a.mpr.intro.inr.intro.inr\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nv x✝ : V\nhvw : SimpleGraph.Adj G v x✝\nha : SimpleGraph.Adj G x✝ x✝\n⊢ Adj (subgraphOfAdj G hvw) x✝ x✝",
"tactic": "exact (ha.ne rfl).elim"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a.mpr.intro.inr.intro.inl\nι : Sort ?u.263704\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' G'' : Subgraph G\ns s' : Set V\nx✝¹ x✝ : V\nha : SimpleGraph.Adj G x✝¹ x✝\nhvw : SimpleGraph.Adj G x✝ x✝¹\n⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝",
"tactic": "simp"
}
] |
[
1196,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1186,
1
] |
Mathlib/Order/Bounded.lean
|
Set.unbounded_ge_inter_not_ge
|
[] |
[
402,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/RingTheory/SimpleModule.lean
|
IsSemisimpleModule.sSup_simples_eq_top
|
[
{
"state_after": "R : Type u_2\ninst✝⁵ : Ring R\nM : Type u_1\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nm : Submodule R M\nN : Type ?u.40765\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : IsSemisimpleModule R M\n⊢ sSup {m | IsAtom m} = ⊤",
"state_before": "R : Type u_2\ninst✝⁵ : Ring R\nM : Type u_1\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nm : Submodule R M\nN : Type ?u.40765\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : IsSemisimpleModule R M\n⊢ sSup {m | IsSimpleModule R { x // x ∈ m }} = ⊤",
"tactic": "simp_rw [isSimpleModule_iff_isAtom]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁵ : Ring R\nM : Type u_1\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nm : Submodule R M\nN : Type ?u.40765\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : IsSemisimpleModule R M\n⊢ sSup {m | IsAtom m} = ⊤",
"tactic": "exact sSup_atoms_eq_top"
}
] |
[
113,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPolynomial.coe_one
|
[] |
[
1094,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1093,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
NNRat.coe_prod
|
[] |
[
287,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.add_le_of_le_sub_left
|
[] |
[
462,
73
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
461,
11
] |
Mathlib/RingTheory/FiniteType.lean
|
Algebra.FiniteType.of_surjective
|
[
{
"state_after": "case h.e'_6\nR : Type u_1\nA : Type u\nB : Type u_2\nM : Type ?u.22919\nN : Type ?u.22922\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nhRA : FiniteType R A\nf : A →ₐ[R] B\nhf : Surjective ↑f\n⊢ ⊤ = Subalgebra.map f ⊤",
"state_before": "R : Type u_1\nA : Type u\nB : Type u_2\nM : Type ?u.22919\nN : Type ?u.22922\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nhRA : FiniteType R A\nf : A →ₐ[R] B\nhf : Surjective ↑f\n⊢ Subalgebra.FG ⊤",
"tactic": "convert hRA.1.map f"
},
{
"state_after": "no goals",
"state_before": "case h.e'_6\nR : Type u_1\nA : Type u\nB : Type u_2\nM : Type ?u.22919\nN : Type ?u.22922\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nhRA : FiniteType R A\nf : A →ₐ[R] B\nhf : Surjective ↑f\n⊢ ⊤ = Subalgebra.map f ⊤",
"tactic": "simpa only [map_top f, @eq_comm _ ⊤, eq_top_iff, AlgHom.mem_range] using hf"
}
] |
[
113,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.pos_of_neg_neg
|
[
{
"state_after": "no goals",
"state_before": "a : Int\nh : -a < 0\n⊢ -a < -0",
"tactic": "rwa [Int.neg_zero]"
}
] |
[
888,
48
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
887,
11
] |
Mathlib/CategoryTheory/Limits/IsLimit.lean
|
CategoryTheory.Limits.IsColimit.OfNatIso.coconeOfHom_fac
|
[
{
"state_after": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cocone F\nX : C\nh : coyoneda.obj X.op ⋙ uliftFunctor ≅ Functor.cocones F\nY : C\nf : X ⟶ Y\n⊢ { pt := Y, ι := h.hom.app Y { down := f } } = { pt := Y, ι := h.hom.app X { down := 𝟙 X } ≫ (const J).map f }",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cocone F\nX : C\nh : coyoneda.obj X.op ⋙ uliftFunctor ≅ Functor.cocones F\nY : C\nf : X ⟶ Y\n⊢ coconeOfHom h f = Cocone.extend (colimitCocone h) f",
"tactic": "dsimp [coconeOfHom, colimitCocone, Cocone.extend]"
},
{
"state_after": "case e_ι.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cocone F\nX : C\nh : coyoneda.obj X.op ⋙ uliftFunctor ≅ Functor.cocones F\nY : C\nf : X ⟶ Y\nj : J\n⊢ (h.hom.app Y { down := f }).app j = (h.hom.app X { down := 𝟙 X } ≫ (const J).map f).app j",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cocone F\nX : C\nh : coyoneda.obj X.op ⋙ uliftFunctor ≅ Functor.cocones F\nY : C\nf : X ⟶ Y\n⊢ { pt := Y, ι := h.hom.app Y { down := f } } = { pt := Y, ι := h.hom.app X { down := 𝟙 X } ≫ (const J).map f }",
"tactic": "congr with j"
},
{
"state_after": "case e_ι.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cocone F\nX : C\nh : coyoneda.obj X.op ⋙ uliftFunctor ≅ Functor.cocones F\nY : C\nf : X ⟶ Y\nj : J\nt :\n ((coyoneda.obj X.op ⋙ uliftFunctor).map f ≫ h.hom.app Y) { down := 𝟙 X } =\n (h.hom.app X ≫ (Functor.cocones F).map f) { down := 𝟙 X }\n⊢ (h.hom.app Y { down := f }).app j = (h.hom.app X { down := 𝟙 X } ≫ (const J).map f).app j",
"state_before": "case e_ι.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cocone F\nX : C\nh : coyoneda.obj X.op ⋙ uliftFunctor ≅ Functor.cocones F\nY : C\nf : X ⟶ Y\nj : J\n⊢ (h.hom.app Y { down := f }).app j = (h.hom.app X { down := 𝟙 X } ≫ (const J).map f).app j",
"tactic": "have t := congrFun (h.hom.naturality f) ⟨𝟙 X⟩"
},
{
"state_after": "case e_ι.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cocone F\nX : C\nh : coyoneda.obj X.op ⋙ uliftFunctor ≅ Functor.cocones F\nY : C\nf : X ⟶ Y\nj : J\nt : h.hom.app Y { down := 𝟙 X ≫ f } = (Functor.cocones F).map f (h.hom.app X { down := 𝟙 X })\n⊢ (h.hom.app Y { down := f }).app j = (h.hom.app X { down := 𝟙 X } ≫ (const J).map f).app j",
"state_before": "case e_ι.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cocone F\nX : C\nh : coyoneda.obj X.op ⋙ uliftFunctor ≅ Functor.cocones F\nY : C\nf : X ⟶ Y\nj : J\nt :\n ((coyoneda.obj X.op ⋙ uliftFunctor).map f ≫ h.hom.app Y) { down := 𝟙 X } =\n (h.hom.app X ≫ (Functor.cocones F).map f) { down := 𝟙 X }\n⊢ (h.hom.app Y { down := f }).app j = (h.hom.app X { down := 𝟙 X } ≫ (const J).map f).app j",
"tactic": "dsimp at t"
},
{
"state_after": "case e_ι.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cocone F\nX : C\nh : coyoneda.obj X.op ⋙ uliftFunctor ≅ Functor.cocones F\nY : C\nf : X ⟶ Y\nj : J\nt : h.hom.app Y { down := f } = (Functor.cocones F).map f (h.hom.app X { down := 𝟙 X })\n⊢ (h.hom.app Y { down := f }).app j = (h.hom.app X { down := 𝟙 X } ≫ (const J).map f).app j",
"state_before": "case e_ι.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cocone F\nX : C\nh : coyoneda.obj X.op ⋙ uliftFunctor ≅ Functor.cocones F\nY : C\nf : X ⟶ Y\nj : J\nt : h.hom.app Y { down := 𝟙 X ≫ f } = (Functor.cocones F).map f (h.hom.app X { down := 𝟙 X })\n⊢ (h.hom.app Y { down := f }).app j = (h.hom.app X { down := 𝟙 X } ≫ (const J).map f).app j",
"tactic": "simp only [id_comp] at t"
},
{
"state_after": "case e_ι.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cocone F\nX : C\nh : coyoneda.obj X.op ⋙ uliftFunctor ≅ Functor.cocones F\nY : C\nf : X ⟶ Y\nj : J\nt : h.hom.app Y { down := f } = (Functor.cocones F).map f (h.hom.app X { down := 𝟙 X })\n⊢ ((Functor.cocones F).map f (h.hom.app X { down := 𝟙 X })).app j =\n (h.hom.app X { down := 𝟙 X } ≫ (const J).map f).app j",
"state_before": "case e_ι.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cocone F\nX : C\nh : coyoneda.obj X.op ⋙ uliftFunctor ≅ Functor.cocones F\nY : C\nf : X ⟶ Y\nj : J\nt : h.hom.app Y { down := f } = (Functor.cocones F).map f (h.hom.app X { down := 𝟙 X })\n⊢ (h.hom.app Y { down := f }).app j = (h.hom.app X { down := 𝟙 X } ≫ (const J).map f).app j",
"tactic": "rw [congrFun (congrArg NatTrans.app t) j]"
},
{
"state_after": "no goals",
"state_before": "case e_ι.w.h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt✝ : Cocone F\nX : C\nh : coyoneda.obj X.op ⋙ uliftFunctor ≅ Functor.cocones F\nY : C\nf : X ⟶ Y\nj : J\nt : h.hom.app Y { down := f } = (Functor.cocones F).map f (h.hom.app X { down := 𝟙 X })\n⊢ ((Functor.cocones F).map f (h.hom.app X { down := 𝟙 X })).app j =\n (h.hom.app X { down := 𝟙 X } ≫ (const J).map f).app j",
"tactic": "rfl"
}
] |
[
1032,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1025,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Finset.isPwo_sup
|
[] |
[
582,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/Deprecated/Group.lean
|
IsMulHom.mul
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ✝ : Type v\ninst✝⁴ : Mul α✝\ninst✝³ : Mul β✝\nγ : Type ?u.931\ninst✝² : Mul γ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Semigroup α\ninst✝ : CommSemigroup β\nf g : α → β\nhf : IsMulHom f\nhg : IsMulHom g\na b : α\n⊢ f (a * b) * g (a * b) = f a * g a * (f b * g b)",
"tactic": "simp only [hf.map_mul, hg.map_mul, mul_comm, mul_assoc, mul_left_comm]"
}
] |
[
82,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
lipschitzWith_min
|
[] |
[
432,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/Data/Real/Irrational.lean
|
Irrational.of_div_nat
|
[] |
[
450,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.Equiv.not_fuzzy'
|
[] |
[
955,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
954,
1
] |
Mathlib/Data/Int/Log.lean
|
Int.clog_zero_right
|
[] |
[
262,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Topology/Algebra/OpenSubgroup.lean
|
Subgroup.isOpen_of_mem_nhds
|
[
{
"state_after": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : ContinuousMul G\nH : Subgroup G\ng : G\nhg : ↑H ∈ 𝓝 g\nx : G\nhx : x ∈ ↑H\n⊢ ↑H ∈ 𝓝 x",
"state_before": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : ContinuousMul G\nH : Subgroup G\ng : G\nhg : ↑H ∈ 𝓝 g\n⊢ IsOpen ↑H",
"tactic": "refine' isOpen_iff_mem_nhds.2 fun x hx => _"
},
{
"state_after": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : ContinuousMul G\nH : Subgroup G\ng : G\nhg : ↑H ∈ 𝓝 g\nx : G\nhx : x ∈ ↑H\nhg' : g ∈ H\n⊢ ↑H ∈ 𝓝 x",
"state_before": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : ContinuousMul G\nH : Subgroup G\ng : G\nhg : ↑H ∈ 𝓝 g\nx : G\nhx : x ∈ ↑H\n⊢ ↑H ∈ 𝓝 x",
"tactic": "have hg' : g ∈ H := SetLike.mem_coe.1 (mem_of_mem_nhds hg)"
},
{
"state_after": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : ContinuousMul G\nH : Subgroup G\ng : G\nhg : ↑H ∈ 𝓝 g\nx : G\nhx : x ∈ ↑H\nhg' : g ∈ H\nthis : Filter.Tendsto (fun y => y * (x⁻¹ * g)) (𝓝 x) (𝓝 g)\n⊢ ↑H ∈ 𝓝 x",
"state_before": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : ContinuousMul G\nH : Subgroup G\ng : G\nhg : ↑H ∈ 𝓝 g\nx : G\nhx : x ∈ ↑H\nhg' : g ∈ H\n⊢ ↑H ∈ 𝓝 x",
"tactic": "have : Filter.Tendsto (fun y => y * (x⁻¹ * g)) (𝓝 x) (𝓝 g) :=\n (continuous_id.mul continuous_const).tendsto' _ _ (mul_inv_cancel_left _ _)"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : ContinuousMul G\nH : Subgroup G\ng : G\nhg : ↑H ∈ 𝓝 g\nx : G\nhx : x ∈ ↑H\nhg' : g ∈ H\nthis : Filter.Tendsto (fun y => y * (x⁻¹ * g)) (𝓝 x) (𝓝 g)\n⊢ ↑H ∈ 𝓝 x",
"tactic": "simpa only [SetLike.mem_coe, Filter.mem_map',\n H.mul_mem_cancel_right (H.mul_mem (H.inv_mem hx) hg')] using this hg"
}
] |
[
318,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
1
] |
Mathlib/Data/Int/Bitwise.lean
|
Int.testBit_land
|
[
{
"state_after": "no goals",
"state_before": "m n : ℤ\nk : ℕ\n⊢ testBit (land m n) k = (testBit m k && testBit n k)",
"tactic": "rw [← bitwise_and, testBit_bitwise]"
}
] |
[
351,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.limsSup_eq_iInf_sSup
|
[] |
[
676,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
675,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean
|
ContinuousLinearMap.map_zero₂
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_6\ninst✝¹⁶ : NontriviallyNormedField 𝕜\nE : Type ?u.150367\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedSpace 𝕜 E\nF : Type u_5\ninst✝¹³ : NormedAddCommGroup F\ninst✝¹² : NormedSpace 𝕜 F\nG : Type ?u.150552\ninst✝¹¹ : NormedAddCommGroup G\ninst✝¹⁰ : NormedSpace 𝕜 G\nR : Type u_1\n𝕜₂ : Type u_7\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NontriviallyNormedField 𝕜₂\nM : Type u_3\ninst✝⁷ : TopologicalSpace M\nσ₁₂ : 𝕜 →+* 𝕜₂\nG' : Type u_4\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜₂ G'\ninst✝⁴ : NormedSpace 𝕜' G'\ninst✝³ : SMulCommClass 𝕜₂ 𝕜' G'\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nρ₁₂ : R →+* 𝕜'\nf : M →SL[ρ₁₂] F →SL[σ₁₂] G'\ny : F\n⊢ ↑(↑f 0) y = 0",
"tactic": "rw [f.map_zero, zero_apply]"
}
] |
[
301,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.Red.cons_nil_iff_singleton
|
[
{
"state_after": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx : α\nb : Bool\nh : Red ((x, b) :: L) []\nh₁ : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\n⊢ Red L [(x, !b)]",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx : α\nb : Bool\nh : Red ((x, b) :: L) []\n⊢ Red L [(x, !b)]",
"tactic": "have h₁ : Red ((x, not b) :: (x, b) :: L) [(x, not b)] := cons_cons h"
},
{
"state_after": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx : α\nb : Bool\nh : Red ((x, b) :: L) []\nh₁ : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\nh₂ : Red ((x, !b) :: (x, b) :: L) L\n⊢ Red L [(x, !b)]",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx : α\nb : Bool\nh : Red ((x, b) :: L) []\nh₁ : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\n⊢ Red L [(x, !b)]",
"tactic": "have h₂ : Red ((x, not b) :: (x, b) :: L) L := ReflTransGen.single Step.cons_not_rev"
},
{
"state_after": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx : α\nb : Bool\nh : Red ((x, b) :: L) []\nh₁✝ : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\nh₂✝ : Red ((x, !b) :: (x, b) :: L) L\nL' : List (α × Bool)\nh₁ : Red [(x, !b)] L'\nh₂ : Red L L'\n⊢ Red L [(x, !b)]",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx : α\nb : Bool\nh : Red ((x, b) :: L) []\nh₁ : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\nh₂ : Red ((x, !b) :: (x, b) :: L) L\n⊢ Red L [(x, !b)]",
"tactic": "let ⟨L', h₁, h₂⟩ := church_rosser h₁ h₂"
},
{
"state_after": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx : α\nb : Bool\nh : Red ((x, b) :: L) []\nh₁✝ : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\nh₂✝ : Red ((x, !b) :: (x, b) :: L) L\nL' : List (α × Bool)\nh₁ : L' = [(x, !b)]\nh₂ : Red L L'\n⊢ Red L [(x, !b)]",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx : α\nb : Bool\nh : Red ((x, b) :: L) []\nh₁✝ : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\nh₂✝ : Red ((x, !b) :: (x, b) :: L) L\nL' : List (α × Bool)\nh₁ : Red [(x, !b)] L'\nh₂ : Red L L'\n⊢ Red L [(x, !b)]",
"tactic": "rw [singleton_iff] at h₁"
},
{
"state_after": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx : α\nb : Bool\nh : Red ((x, b) :: L) []\nh₁ : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\nh₂✝ : Red ((x, !b) :: (x, b) :: L) L\nh₂ : Red L [(x, !b)]\n⊢ Red L [(x, !b)]",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx : α\nb : Bool\nh : Red ((x, b) :: L) []\nh₁✝ : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\nh₂✝ : Red ((x, !b) :: (x, b) :: L) L\nL' : List (α × Bool)\nh₁ : L' = [(x, !b)]\nh₂ : Red L L'\n⊢ Red L [(x, !b)]",
"tactic": "subst L'"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nx : α\nb : Bool\nh : Red ((x, b) :: L) []\nh₁ : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\nh₂✝ : Red ((x, !b) :: (x, b) :: L) L\nh₂ : Red L [(x, !b)]\n⊢ Red L [(x, !b)]",
"tactic": "assumption"
}
] |
[
342,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/Data/Matrix/Basis.lean
|
Matrix.StdBasisMatrix.apply_of_col_ne
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.23553\nm : Type u_3\nn : Type u_1\nR : Type ?u.23562\nα : Type u_2\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : Semiring α\ni✝ : m\nj✝ : n\nc : α\ni'✝ : m\nj'✝ : n\ni i' : m\nj j' : n\nhj : j ≠ j'\na : α\n⊢ stdBasisMatrix i j a i' j' = 0",
"tactic": "simp [hj]"
}
] |
[
142,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.one
|
[] |
[
394,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
392,
1
] |
Mathlib/CategoryTheory/Sites/Subsheaf.lean
|
CategoryTheory.GrothendieckTopology.Subpresheaf.homOfLe_ι
|
[
{
"state_after": "case w.h.h\nC : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG✝ G'✝ G G' : Subpresheaf F\nh : G ≤ G'\nx✝ : Cᵒᵖ\na✝ : (toPresheaf G).obj x✝\n⊢ (homOfLe h ≫ ι G').app x✝ a✝ = (ι G).app x✝ a✝",
"state_before": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG✝ G'✝ G G' : Subpresheaf F\nh : G ≤ G'\n⊢ homOfLe h ≫ ι G' = ι G",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case w.h.h\nC : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG✝ G'✝ G G' : Subpresheaf F\nh : G ≤ G'\nx✝ : Cᵒᵖ\na✝ : (toPresheaf G).obj x✝\n⊢ (homOfLe h ≫ ι G').app x✝ a✝ = (ι G).app x✝ a✝",
"tactic": "rfl"
}
] |
[
115,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.Infinite.exists_supset_ncard_eq
|
[
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.135186\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : Set.Infinite t\nhst : s ⊆ t\nhs : Set.Finite s\nk : ℕ\nhsk : Set.ncard s ≤ k\ns₁ : Set α\nhs₁ : s₁ ⊆ t \\ s\nhs₁fin : Set.Finite s₁\nhs₁card : Set.ncard s₁ = k - Set.ncard s\n⊢ ∃ s', s ⊆ s' ∧ s' ⊆ t ∧ Set.ncard s' = k",
"state_before": "α : Type u_1\nβ : Type ?u.135186\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : Set.Infinite t\nhst : s ⊆ t\nhs : Set.Finite s\nk : ℕ\nhsk : Set.ncard s ≤ k\n⊢ ∃ s', s ⊆ s' ∧ s' ⊆ t ∧ Set.ncard s' = k",
"tactic": "obtain ⟨s₁, hs₁, hs₁fin, hs₁card⟩ := (ht.diff hs).exists_subset_ncard_eq (k - s.ncard)"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.135186\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : Set.Infinite t\nhst : s ⊆ t\nhs : Set.Finite s\nk : ℕ\nhsk : Set.ncard s ≤ k\ns₁ : Set α\nhs₁ : s₁ ⊆ t \\ s\nhs₁fin : Set.Finite s₁\nhs₁card : Set.ncard s₁ = k - Set.ncard s\n⊢ Set.ncard (s ∪ s₁) = k",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.135186\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : Set.Infinite t\nhst : s ⊆ t\nhs : Set.Finite s\nk : ℕ\nhsk : Set.ncard s ≤ k\ns₁ : Set α\nhs₁ : s₁ ⊆ t \\ s\nhs₁fin : Set.Finite s₁\nhs₁card : Set.ncard s₁ = k - Set.ncard s\n⊢ ∃ s', s ⊆ s' ∧ s' ⊆ t ∧ Set.ncard s' = k",
"tactic": "refine' ⟨s ∪ s₁, subset_union_left _ _, union_subset hst (hs₁.trans (diff_subset _ _)), _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.135186\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : Set.Infinite t\nhst : s ⊆ t\nhs : Set.Finite s\nk : ℕ\nhsk : Set.ncard s ≤ k\ns₁ : Set α\nhs₁ : s₁ ⊆ t \\ s\nhs₁fin : Set.Finite s₁\nhs₁card : Set.ncard s₁ = k - Set.ncard s\n⊢ Set.ncard (s ∪ s₁) = k",
"tactic": "rwa [ncard_union_eq (disjoint_of_subset_right hs₁ disjoint_sdiff_right) hs hs₁fin, hs₁card,\n add_tsub_cancel_of_le]"
}
] |
[
620,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
615,
1
] |
Mathlib/Order/MinMax.lean
|
Monotone.map_max
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\ns : Set α\na b c d : α\nhf : Monotone f\n⊢ f (max a b) = max (f a) (f b)",
"tactic": "cases' le_total a b with h h <;> simp [h, hf h]"
}
] |
[
257,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/NumberTheory/Multiplicity.lean
|
multiplicity.Int.pow_add_pow
|
[
{
"state_after": "R : Type ?u.836891\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℤ\nhxy : ↑p ∣ x - -y\nhx : ¬↑p ∣ x\nn : ℕ\nhn : Odd n\n⊢ multiplicity (↑p) (x ^ n + y ^ n) = multiplicity (↑p) (x + y) + multiplicity p n",
"state_before": "R : Type ?u.836891\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℤ\nhxy : ↑p ∣ x + y\nhx : ¬↑p ∣ x\nn : ℕ\nhn : Odd n\n⊢ multiplicity (↑p) (x ^ n + y ^ n) = multiplicity (↑p) (x + y) + multiplicity p n",
"tactic": "rw [← sub_neg_eq_add] at hxy"
},
{
"state_after": "R : Type ?u.836891\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℤ\nhxy : ↑p ∣ x - -y\nhx : ¬↑p ∣ x\nn : ℕ\nhn : Odd n\n⊢ multiplicity (↑p) (x ^ n - (-y) ^ n) = multiplicity (↑p) (x - -y) + multiplicity p n",
"state_before": "R : Type ?u.836891\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℤ\nhxy : ↑p ∣ x - -y\nhx : ¬↑p ∣ x\nn : ℕ\nhn : Odd n\n⊢ multiplicity (↑p) (x ^ n + y ^ n) = multiplicity (↑p) (x + y) + multiplicity p n",
"tactic": "rw [← sub_neg_eq_add, ← sub_neg_eq_add, ← Odd.neg_pow hn]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.836891\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℤ\nhxy : ↑p ∣ x - -y\nhx : ¬↑p ∣ x\nn : ℕ\nhn : Odd n\n⊢ multiplicity (↑p) (x ^ n - (-y) ^ n) = multiplicity (↑p) (x - -y) + multiplicity p n",
"tactic": "exact Int.pow_sub_pow hp hp1 hxy hx n"
}
] |
[
227,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.map_iSup
|
[] |
[
338,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Analysis/NormedSpace/Banach.lean
|
ContinuousLinearMap.exists_approx_preimage_norm_le
|
[
{
"state_after": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\n⊢ ∃ C, C ≥ 0 ∧ ∀ (y : F), ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\n⊢ ∃ C, C ≥ 0 ∧ ∀ (y : F), ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖",
"tactic": "have A : (⋃ n : ℕ, closure (f '' ball 0 n)) = Set.univ := by\n refine' Subset.antisymm (subset_univ _) fun y _ => _\n rcases surj y with ⟨x, hx⟩\n rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩\n refine' mem_iUnion.2 ⟨n, subset_closure _⟩\n refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩\n rwa [mem_ball, dist_eq_norm, sub_zero]"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nthis : ∃ n x, x ∈ interior (closure (↑f '' ball 0 ↑n))\n⊢ ∃ C, C ≥ 0 ∧ ∀ (y : F), ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\n⊢ ∃ C, C ≥ 0 ∧ ∀ (y : F), ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖",
"tactic": "have : ∃ (n : ℕ)(x : _), x ∈ interior (closure (f '' ball 0 n)) :=\n nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nthis : ∃ n x ε, ε > 0 ∧ ball x ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\n⊢ ∃ C, C ≥ 0 ∧ ∀ (y : F), ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nthis : ∃ n x, x ∈ interior (closure (↑f '' ball 0 ↑n))\n⊢ ∃ C, C ≥ 0 ∧ ∀ (y : F), ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖",
"tactic": "simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\n⊢ ∃ C, C ≥ 0 ∧ ∀ (y : F), ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nthis : ∃ n x ε, ε > 0 ∧ ball x ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\n⊢ ∃ C, C ≥ 0 ∧ ∀ (y : F), ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖",
"tactic": "rcases this with ⟨n, a, ε, ⟨εpos, H⟩⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\n⊢ ∃ C, C ≥ 0 ∧ ∀ (y : F), ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\n⊢ ∃ C, C ≥ 0 ∧ ∀ (y : F), ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖",
"tactic": "rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.refine'_1\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\n⊢ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n ≥ 0\n\ncase intro.intro.intro.intro.intro.refine'_2\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
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"tactic": "refine' ⟨(ε / 2)⁻¹ * ‖c‖ * 2 * n, _, fun y => _⟩"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\ny : F\nx✝ : y ∈ Set.univ\n⊢ y ∈ ⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)",
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"tactic": "refine' Subset.antisymm (subset_univ _) fun y _ => _"
},
{
"state_after": "case intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\ny : F\nx✝ : y ∈ Set.univ\nx : E\nhx : ↑f x = y\n⊢ y ∈ ⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)",
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"tactic": "rcases surj y with ⟨x, hx⟩"
},
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"tactic": "rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩"
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{
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"tactic": "refine' mem_iUnion.2 ⟨n, subset_closure _⟩"
},
{
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"tactic": "refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩"
},
{
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"tactic": "rwa [mem_ball, dist_eq_norm, sub_zero]"
},
{
"state_after": "case intro.intro.intro.intro.intro.refine'_1.refine'_1\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\n⊢ 0 ≤ (ε / 2)⁻¹\n\ncase intro.intro.intro.intro.intro.refine'_1.refine'_2\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\n⊢ 0 ≤ ↑n",
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"tactic": "refine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.refine'_1.refine'_1\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\n⊢ 0 ≤ (ε / 2)⁻¹\n\ncase intro.intro.intro.intro.intro.refine'_1.refine'_2\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\n⊢ 0 ≤ ↑n",
"tactic": "exacts [inv_nonneg.2 (div_nonneg (le_of_lt εpos) (by norm_num)), n.cast_nonneg]"
},
{
"state_after": "no goals",
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"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\n⊢ 0 ≤ 2",
"tactic": "norm_num"
},
{
"state_after": "case pos\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : y = 0\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖\n\ncase neg\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
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"tactic": "by_cases hy : y = 0"
},
{
"state_after": "case pos\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : y = 0\n⊢ dist (↑f 0) y ≤ 1 / 2 * ‖y‖ ∧ ‖0‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
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"tactic": "use 0"
},
{
"state_after": "no goals",
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"tactic": "simp [hy]"
},
{
"state_after": "case neg.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "rcases rescale_to_shell hc (half_pos εpos) hy with ⟨d, hd, ydlt, -, dinv⟩"
},
{
"state_after": "case neg.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "let δ := ‖d‖ * ‖y‖ / 4"
},
{
"state_after": "case neg.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "have δpos : 0 < δ := div_pos (mul_pos (norm_pos_iff.2 hd) (norm_pos_iff.2 hy)) (by norm_num)"
},
{
"state_after": "case neg.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis : a + d • y ∈ ball a ε\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "have : a + d • y ∈ ball a ε := by\n simp [dist_eq_norm, lt_of_le_of_lt ydlt.le (half_lt_self εpos)]"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nh₁ : dist (a + d • y) z₁ < δ\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis : a + d • y ∈ ball a ε\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "rcases Metric.mem_closure_iff.1 (H this) _ δpos with ⟨z₁, z₁im, h₁⟩"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nh₁ : dist (a + d • y) z₁ < δ\nx₁ : E\nhx₁ : x₁ ∈ ball 0 ↑n\nxz₁ : ↑f x₁ = z₁\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nh₁ : dist (a + d • y) z₁ < δ\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "rcases(mem_image _ _ _).1 z₁im with ⟨x₁, hx₁, xz₁⟩"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : x₁ ∈ ball 0 ↑n\nxz₁ : ↑f x₁ = z₁\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nh₁ : dist (a + d • y) z₁ < δ\nx₁ : E\nhx₁ : x₁ ∈ ball 0 ↑n\nxz₁ : ↑f x₁ = z₁\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "rw [← xz₁] at h₁"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : x₁ ∈ ball 0 ↑n\nxz₁ : ↑f x₁ = z₁\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "rw [mem_ball, dist_eq_norm, sub_zero] at hx₁"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "have : a ∈ ball a ε := by\n simp\n exact εpos"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nh₂ : dist a z₂ < δ\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "rcases Metric.mem_closure_iff.1 (H this) _ δpos with ⟨z₂, z₂im, h₂⟩"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nh₂ : dist a z₂ < δ\nx₂ : E\nhx₂ : x₂ ∈ ball 0 ↑n\nxz₂ : ↑f x₂ = z₂\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nh₂ : dist a z₂ < δ\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "rcases(mem_image _ _ _).1 z₂im with ⟨x₂, hx₂, xz₂⟩"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : x₂ ∈ ball 0 ↑n\nxz₂ : ↑f x₂ = z₂\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nh₂ : dist a z₂ < δ\nx₂ : E\nhx₂ : x₂ ∈ ball 0 ↑n\nxz₂ : ↑f x₂ = z₂\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "rw [← xz₂] at h₂"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : x₂ ∈ ball 0 ↑n\nxz₂ : ↑f x₂ = z₂\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "rw [mem_ball, dist_eq_norm, sub_zero] at hx₂"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "let x := x₁ - x₂"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\nJ : ‖↑f (d⁻¹ • x) - y‖ ≤ 1 / 2 * ‖y‖\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "have J : ‖f (d⁻¹ • x) - y‖ ≤ 1 / 2 * ‖y‖ :=\n calc\n ‖f (d⁻¹ • x) - y‖ = ‖d⁻¹ • f x - (d⁻¹ * d) • y‖ := by\n rwa [f.map_smul _, inv_mul_cancel, one_smul]\n _ = ‖d⁻¹ • (f x - d • y)‖ := by rw [mul_smul, smul_sub]\n _ = ‖d‖⁻¹ * ‖f x - d • y‖ := by rw [norm_smul, norm_inv]\n _ ≤ ‖d‖⁻¹ * (2 * δ) := by\n apply mul_le_mul_of_nonneg_left I\n rw [inv_nonneg]\n exact norm_nonneg _\n _ = ‖d‖⁻¹ * ‖d‖ * ‖y‖ / 2 := by\n simp only\n ring\n _ = ‖y‖ / 2 := by\n rw [inv_mul_cancel, one_mul]\n simp [norm_eq_zero, hd]\n _ = 1 / 2 * ‖y‖ := by ring"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\nJ : dist (↑f (d⁻¹ • x)) y ≤ 1 / 2 * ‖y‖\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\nJ : ‖↑f (d⁻¹ • x) - y‖ ≤ 1 / 2 * ‖y‖\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "rw [← dist_eq_norm] at J"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\nJ : dist (↑f (d⁻¹ • x)) y ≤ 1 / 2 * ‖y‖\nK : ‖d⁻¹ • x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖\n⊢ ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "exact ⟨d⁻¹ • x, J, K⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\n⊢ 0 < 4",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\n⊢ a + d • y ∈ ball a ε",
"tactic": "simp [dist_eq_norm, lt_of_le_of_lt ydlt.le (half_lt_self εpos)]"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\n⊢ 0 < ε",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\n⊢ a ∈ ball a ε",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\n⊢ 0 < ε",
"tactic": "exact εpos"
},
{
"state_after": "case e_a\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ ↑f x - d • y = ↑f x₁ - (a + d • y) - (↑f x₂ - a)",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ ‖↑f x - d • y‖ = ‖↑f x₁ - (a + d • y) - (↑f x₂ - a)‖",
"tactic": "congr 1"
},
{
"state_after": "case e_a\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ ↑f x₁ - ↑f x₂ - d • y = ↑f x₁ - (a + d • y) - (↑f x₂ - a)",
"state_before": "case e_a\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ ↑f x - d • y = ↑f x₁ - (a + d • y) - (↑f x₂ - a)",
"tactic": "simp only [f.map_sub]"
},
{
"state_after": "no goals",
"state_before": "case e_a\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ ↑f x₁ - ↑f x₂ - d • y = ↑f x₁ - (a + d • y) - (↑f x₂ - a)",
"tactic": "abel"
},
{
"state_after": "case h₁\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ ‖↑f x₁ - (a + d • y)‖ ≤ δ\n\ncase h₂\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ ‖↑f x₂ - a‖ ≤ δ",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ ‖↑f x₁ - (a + d • y)‖ + ‖↑f x₂ - a‖ ≤ δ + δ",
"tactic": "apply add_le_add"
},
{
"state_after": "case h₁\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ dist (a + d • y) (↑f x₁) ≤ δ",
"state_before": "case h₁\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ ‖↑f x₁ - (a + d • y)‖ ≤ δ",
"tactic": "rw [← dist_eq_norm, dist_comm]"
},
{
"state_after": "no goals",
"state_before": "case h₁\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ dist (a + d • y) (↑f x₁) ≤ δ",
"tactic": "exact le_of_lt h₁"
},
{
"state_after": "case h₂\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ dist a (↑f x₂) ≤ δ",
"state_before": "case h₂\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ ‖↑f x₂ - a‖ ≤ δ",
"tactic": "rw [← dist_eq_norm, dist_comm]"
},
{
"state_after": "no goals",
"state_before": "case h₂\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\n⊢ dist a (↑f x₂) ≤ δ",
"tactic": "exact le_of_lt h₂"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ ‖↑f (d⁻¹ • x) - y‖ = ‖d⁻¹ • ↑f x - (d⁻¹ * d) • y‖",
"tactic": "rwa [f.map_smul _, inv_mul_cancel, one_smul]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ ‖d⁻¹ • ↑f x - (d⁻¹ * d) • y‖ = ‖d⁻¹ • (↑f x - d • y)‖",
"tactic": "rw [mul_smul, smul_sub]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ ‖d⁻¹ • (↑f x - d • y)‖ = ‖d‖⁻¹ * ‖↑f x - d • y‖",
"tactic": "rw [norm_smul, norm_inv]"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ 0 ≤ ‖d‖⁻¹",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ ‖d‖⁻¹ * ‖↑f x - d • y‖ ≤ ‖d‖⁻¹ * (2 * δ)",
"tactic": "apply mul_le_mul_of_nonneg_left I"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ 0 ≤ ‖d‖",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ 0 ≤ ‖d‖⁻¹",
"tactic": "rw [inv_nonneg]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ 0 ≤ ‖d‖",
"tactic": "exact norm_nonneg _"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ ‖d‖⁻¹ * (2 * (‖d‖ * ‖y‖ / 4)) = ‖d‖⁻¹ * ‖d‖ * ‖y‖ / 2",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ ‖d‖⁻¹ * (2 * δ) = ‖d‖⁻¹ * ‖d‖ * ‖y‖ / 2",
"tactic": "simp only"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ ‖d‖⁻¹ * (2 * (‖d‖ * ‖y‖ / 4)) = ‖d‖⁻¹ * ‖d‖ * ‖y‖ / 2",
"tactic": "ring"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ ‖d‖ ≠ 0",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ ‖d‖⁻¹ * ‖d‖ * ‖y‖ / 2 = ‖y‖ / 2",
"tactic": "rw [inv_mul_cancel, one_mul]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ ‖d‖ ≠ 0",
"tactic": "simp [norm_eq_zero, hd]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\n⊢ ‖y‖ / 2 = 1 / 2 * ‖y‖",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\nJ : dist (↑f (d⁻¹ • x)) y ≤ 1 / 2 * ‖y‖\n⊢ ‖d⁻¹ • x‖ = ‖d‖⁻¹ * ‖x₁ - x₂‖",
"tactic": "rw [norm_smul, norm_inv]"
},
{
"state_after": "case refine'_1\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\nJ : dist (↑f (d⁻¹ • x)) y ≤ 1 / 2 * ‖y‖\n⊢ ‖x₁ - x₂‖ ≤ ↑n + ↑n\n\ncase refine'_2\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\nJ : dist (↑f (d⁻¹ • x)) y ≤ 1 / 2 * ‖y‖\n⊢ 0 ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\nJ : dist (↑f (d⁻¹ • x)) y ≤ 1 / 2 * ‖y‖\n⊢ ‖d‖⁻¹ * ‖x₁ - x₂‖ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖ * (↑n + ↑n)",
"tactic": "refine' mul_le_mul dinv _ (norm_nonneg _) _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\nJ : dist (↑f (d⁻¹ • x)) y ≤ 1 / 2 * ‖y‖\n⊢ ‖x₁ - x₂‖ ≤ ↑n + ↑n",
"tactic": "exact le_trans (norm_sub_le _ _) (add_le_add (le_of_lt hx₁) (le_of_lt hx₂))"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\nJ : dist (↑f (d⁻¹ • x)) y ≤ 1 / 2 * ‖y‖\n⊢ 0 ≤ (ε / 2)⁻¹",
"state_before": "case refine'_2\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\nJ : dist (↑f (d⁻¹ • x)) y ≤ 1 / 2 * ‖y‖\n⊢ 0 ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖",
"tactic": "apply mul_nonneg (mul_nonneg _ (norm_nonneg _)) (norm_nonneg _)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\nJ : dist (↑f (d⁻¹ • x)) y ≤ 1 / 2 * ‖y‖\n⊢ 0 ≤ (ε / 2)⁻¹",
"tactic": "exact inv_nonneg.2 (le_of_lt (half_pos εpos))"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝ : CompleteSpace F\nsurj : Surjective ↑f\nA : (⋃ (n : ℕ), closure (↑f '' ball 0 ↑n)) = Set.univ\nn : ℕ\na : F\nε : ℝ\nεpos : ε > 0\nH : ball a ε ⊆ closure ((fun a => ↑f a) '' ball 0 ↑n)\nc : 𝕜\nhc : 1 < ‖c‖\ny : F\nhy : ¬y = 0\nd : 𝕜\nhd : d ≠ 0\nydlt : ‖d • y‖ < ε / 2\ndinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖\nδ : ℝ := ‖d‖ * ‖y‖ / 4\nδpos : 0 < δ\nthis✝ : a + d • y ∈ ball a ε\nz₁ : F\nz₁im : z₁ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₁ : E\nh₁ : dist (a + d • y) (↑f x₁) < δ\nhx₁ : ‖x₁‖ < ↑n\nxz₁ : ↑f x₁ = z₁\nthis : a ∈ ball a ε\nz₂ : F\nz₂im : z₂ ∈ (fun a => ↑f a) '' ball 0 ↑n\nx₂ : E\nh₂ : dist a (↑f x₂) < δ\nhx₂ : ‖x₂‖ < ↑n\nxz₂ : ↑f x₂ = z₂\nx : E := x₁ - x₂\nI : ‖↑f x - d • y‖ ≤ 2 * δ\nJ : dist (↑f (d⁻¹ • x)) y ≤ 1 / 2 * ‖y‖\n⊢ (ε / 2)⁻¹ * ‖c‖ * ‖y‖ * (↑n + ↑n) = (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖",
"tactic": "ring"
}
] |
[
165,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_eq_zero
|
[] |
[
374,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasurableEmbedding.lintegral_map
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\n⊢ (⨆ (g_1 : β →ₛ ℝ≥0∞) (_ : ↑g_1 ≤ fun a => f a), SimpleFunc.lintegral g_1 (Measure.map g μ)) =\n ⨆ (g_1 : α →ₛ ℝ≥0∞) (_ : ↑g_1 ≤ fun a => f (g a)), SimpleFunc.lintegral g_1 μ",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\n⊢ (∫⁻ (a : β), f a ∂Measure.map g μ) = ∫⁻ (a : α), f (g a) ∂μ",
"tactic": "rw [lintegral, lintegral]"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\nf₀ : β →ₛ ℝ≥0∞\nhf₀ : ↑f₀ ≤ fun a => f a\n⊢ SimpleFunc.lintegral f₀ (Measure.map g μ) ≤\n ⨆ (g_1 : α →ₛ ℝ≥0∞) (_ : ↑g_1 ≤ fun a => f (g a)), SimpleFunc.lintegral g_1 μ\n\ncase refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\nf₀ : α →ₛ ℝ≥0∞\nhf₀ : ↑f₀ ≤ fun a => f (g a)\n⊢ SimpleFunc.lintegral f₀ μ ≤ ⨆ (g_1 : β →ₛ ℝ≥0∞) (_ : ↑g_1 ≤ fun a => f a), SimpleFunc.lintegral g_1 (Measure.map g μ)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\n⊢ (⨆ (g_1 : β →ₛ ℝ≥0∞) (_ : ↑g_1 ≤ fun a => f a), SimpleFunc.lintegral g_1 (Measure.map g μ)) =\n ⨆ (g_1 : α →ₛ ℝ≥0∞) (_ : ↑g_1 ≤ fun a => f (g a)), SimpleFunc.lintegral g_1 μ",
"tactic": "refine' le_antisymm (iSup₂_le fun f₀ hf₀ => _) (iSup₂_le fun f₀ hf₀ => _)"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\nf₀ : β →ₛ ℝ≥0∞\nhf₀ : ↑f₀ ≤ fun a => f a\n⊢ SimpleFunc.lintegral (comp f₀ g (_ : Measurable g)) μ ≤\n ⨆ (g_1 : α →ₛ ℝ≥0∞) (_ : ↑g_1 ≤ fun a => f (g a)), SimpleFunc.lintegral g_1 μ",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\nf₀ : β →ₛ ℝ≥0∞\nhf₀ : ↑f₀ ≤ fun a => f a\n⊢ SimpleFunc.lintegral f₀ (Measure.map g μ) ≤\n ⨆ (g_1 : α →ₛ ℝ≥0∞) (_ : ↑g_1 ≤ fun a => f (g a)), SimpleFunc.lintegral g_1 μ",
"tactic": "rw [SimpleFunc.lintegral_map _ hg.measurable]"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\nf₀ : β →ₛ ℝ≥0∞\nhf₀ : ↑f₀ ≤ fun a => f a\nthis : ↑(comp f₀ g (_ : Measurable g)) ≤ f ∘ g\n⊢ SimpleFunc.lintegral (comp f₀ g (_ : Measurable g)) μ ≤\n ⨆ (g_1 : α →ₛ ℝ≥0∞) (_ : ↑g_1 ≤ fun a => f (g a)), SimpleFunc.lintegral g_1 μ",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\nf₀ : β →ₛ ℝ≥0∞\nhf₀ : ↑f₀ ≤ fun a => f a\n⊢ SimpleFunc.lintegral (comp f₀ g (_ : Measurable g)) μ ≤\n ⨆ (g_1 : α →ₛ ℝ≥0∞) (_ : ↑g_1 ≤ fun a => f (g a)), SimpleFunc.lintegral g_1 μ",
"tactic": "have : (f₀.comp g hg.measurable : α → ℝ≥0∞) ≤ f ∘ g := fun x => hf₀ (g x)"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\nf₀ : β →ₛ ℝ≥0∞\nhf₀ : ↑f₀ ≤ fun a => f a\nthis : ↑(comp f₀ g (_ : Measurable g)) ≤ f ∘ g\n⊢ SimpleFunc.lintegral (comp f₀ g (_ : Measurable g)) μ ≤\n ⨆ (g_1 : α →ₛ ℝ≥0∞) (_ : ↑g_1 ≤ fun a => f (g a)), SimpleFunc.lintegral g_1 μ",
"tactic": "exact le_iSup_of_le (comp f₀ g hg.measurable) (by exact le_iSup (α := ℝ≥0∞) _ this)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\nf₀ : β →ₛ ℝ≥0∞\nhf₀ : ↑f₀ ≤ fun a => f a\nthis : ↑(comp f₀ g (_ : Measurable g)) ≤ f ∘ g\n⊢ SimpleFunc.lintegral (comp f₀ g (_ : Measurable g)) μ ≤\n ⨆ (_ : ↑(comp f₀ g (_ : Measurable g)) ≤ fun a => f (g a)), SimpleFunc.lintegral (comp f₀ g (_ : Measurable g)) μ",
"tactic": "exact le_iSup (α := ℝ≥0∞) _ this"
},
{
"state_after": "case refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\nf₀ : α →ₛ ℝ≥0∞\nhf₀ : ↑f₀ ≤ fun a => f (g a)\n⊢ (∫⁻ (a : β), ↑(SimpleFunc.extend f₀ g hg (const β 0)) a ∂Measure.map g μ) ≤ ∫⁻ (a : β), f a ∂Measure.map g μ",
"state_before": "case refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\nf₀ : α →ₛ ℝ≥0∞\nhf₀ : ↑f₀ ≤ fun a => f (g a)\n⊢ SimpleFunc.lintegral f₀ μ ≤ ⨆ (g_1 : β →ₛ ℝ≥0∞) (_ : ↑g_1 ≤ fun a => f a), SimpleFunc.lintegral g_1 (Measure.map g μ)",
"tactic": "rw [← f₀.extend_comp_eq hg (const _ 0), ← SimpleFunc.lintegral_map, ←\n SimpleFunc.lintegral_eq_lintegral, ← lintegral]"
},
{
"state_after": "case refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\nf₀ : α →ₛ ℝ≥0∞\nhf₀ : ↑f₀ ≤ fun a => f (g a)\nx : α\n⊢ ↑(SimpleFunc.extend f₀ g hg (const β 0)) (g x) ≤ f (g x)",
"state_before": "case refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\nf₀ : α →ₛ ℝ≥0∞\nhf₀ : ↑f₀ ≤ fun a => f (g a)\n⊢ (∫⁻ (a : β), ↑(SimpleFunc.extend f₀ g hg (const β 0)) a ∂Measure.map g μ) ≤ ∫⁻ (a : β), f a ∂Measure.map g μ",
"tactic": "refine' lintegral_mono_ae (hg.ae_map_iff.2 <| eventually_of_forall fun x => _)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1366326\nδ : Type ?u.1366329\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace β\ng : α → β\nhg : MeasurableEmbedding g\nf : β → ℝ≥0∞\nf₀ : α →ₛ ℝ≥0∞\nhf₀ : ↑f₀ ≤ fun a => f (g a)\nx : α\n⊢ ↑(SimpleFunc.extend f₀ g hg (const β 0)) (g x) ≤ f (g x)",
"tactic": "exact (extend_apply _ _ _ _).trans_le (hf₀ _)"
}
] |
[
1321,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1311,
1
] |
Mathlib/Algebra/DirectSum/Module.lean
|
DirectSum.mk_smul
|
[] |
[
85,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.PartrecToTM2.supports_biUnion
|
[
{
"state_after": "K : Option Γ' → Finset Λ'\nS : Finset Λ'\n⊢ (∀ (q : Λ') (x : Option Γ'), q ∈ K x → TM2.SupportsStmt S (tr q)) ↔\n ∀ (a : Option Γ') (q : Λ'), q ∈ K a → TM2.SupportsStmt S (tr q)",
"state_before": "K : Option Γ' → Finset Λ'\nS : Finset Λ'\n⊢ Supports (Finset.biUnion Finset.univ K) S ↔ ∀ (a : Option Γ'), Supports (K a) S",
"tactic": "simp [Supports]"
},
{
"state_after": "no goals",
"state_before": "K : Option Γ' → Finset Λ'\nS : Finset Λ'\n⊢ (∀ (q : Λ') (x : Option Γ'), q ∈ K x → TM2.SupportsStmt S (tr q)) ↔\n ∀ (a : Option Γ') (q : Λ'), q ∈ K a → TM2.SupportsStmt S (tr q)",
"tactic": "apply forall_swap"
}
] |
[
1931,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1929,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.sin_pi_div_six
|
[
{
"state_after": "x : ℝ\n⊢ cos (π / 2 - π / 6) = cos (π / 3)",
"state_before": "x : ℝ\n⊢ sin (π / 6) = 1 / 2",
"tactic": "rw [← cos_pi_div_two_sub, ← cos_pi_div_three]"
},
{
"state_after": "case e_x\nx : ℝ\n⊢ π / 2 - π / 6 = π / 3",
"state_before": "x : ℝ\n⊢ cos (π / 2 - π / 6) = cos (π / 3)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_x\nx : ℝ\n⊢ π / 2 - π / 6 = π / 3",
"tactic": "ring"
}
] |
[
893,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
890,
1
] |
Mathlib/RingTheory/DedekindDomain/PID.lean
|
FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_top
|
[
{
"state_after": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nh : Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v}) = ⊤\nhinv : ↑I * ↑I⁻¹ = 1\n⊢ Submodule.IsPrincipal ↑↑I",
"state_before": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nh : Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v}) = ⊤\n⊢ Submodule.IsPrincipal ↑↑I",
"tactic": "have hinv := I.mul_inv"
},
{
"state_after": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\n⊢ Submodule.IsPrincipal ↑↑I",
"state_before": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nh : Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v}) = ⊤\nhinv : ↑I * ↑I⁻¹ = 1\n⊢ Submodule.IsPrincipal ↑↑I",
"tactic": "set J := Submodule.comap (Algebra.linearMap R A) ((I : Submodule R A) * Submodule.span R {v})"
},
{
"state_after": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\n⊢ Submodule.IsPrincipal ↑↑I",
"state_before": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\n⊢ Submodule.IsPrincipal ↑↑I",
"tactic": "have hJ : IsLocalization.coeSubmodule A J = ↑I * Submodule.span R {v} := by\n rw [Subtype.ext_iff] at hinv\n simp only [coe_mul, val_eq_coe, coe_one] at hinv\n apply Submodule.map_comap_eq_self\n rw [← Submodule.one_eq_range, ← hinv]\n exact Submodule.mul_le_mul_right ((Submodule.span_singleton_le_iff_mem _ _).2 hv)"
},
{
"state_after": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\n⊢ Submodule.IsPrincipal ↑↑I",
"state_before": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\n⊢ Submodule.IsPrincipal ↑↑I",
"tactic": "have : (1 : A) ∈ ↑I * Submodule.span R {v} := by\n rw [← hJ, h, IsLocalization.coeSubmodule_top, Submodule.mem_one]\n exact ⟨1, (algebraMap R _).map_one⟩"
},
{
"state_after": "case intro.intro\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ Submodule.IsPrincipal ↑↑I",
"state_before": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\n⊢ Submodule.IsPrincipal ↑↑I",
"tactic": "obtain ⟨w, hw, hvw⟩ := Submodule.mem_mul_span_singleton.1 this"
},
{
"state_after": "case intro.intro\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ ↑↑I = Submodule.span R {w}",
"state_before": "case intro.intro\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ Submodule.IsPrincipal ↑↑I",
"tactic": "refine' ⟨⟨w, _⟩⟩"
},
{
"state_after": "case intro.intro\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ ↑(spanSingleton S w) = ↑↑I⁻¹⁻¹",
"state_before": "case intro.intro\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ ↑↑I = Submodule.span R {w}",
"tactic": "rw [← FractionalIdeal.coe_spanSingleton S, ← inv_inv I, eq_comm]"
},
{
"state_after": "case intro.intro.refine'_1\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ ↑I⁻¹ * spanSingleton S w ≤ 1\n\ncase intro.intro.refine'_2\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ 1 ≤ ↑I⁻¹ * spanSingleton S w",
"state_before": "case intro.intro\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ ↑(spanSingleton S w) = ↑↑I⁻¹⁻¹",
"tactic": "refine' congr_arg (↑) (Units.eq_inv_of_mul_eq_one_left (le_antisymm _ _))"
},
{
"state_after": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑(↑I * ↑I⁻¹) = ↑1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\n⊢ IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}",
"state_before": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\n⊢ IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}",
"tactic": "rw [Subtype.ext_iff] at hinv"
},
{
"state_after": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhinv : ↑↑I * ↑↑I⁻¹ = 1\n⊢ IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}",
"state_before": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑(↑I * ↑I⁻¹) = ↑1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\n⊢ IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}",
"tactic": "simp only [coe_mul, val_eq_coe, coe_one] at hinv"
},
{
"state_after": "case h\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhinv : ↑↑I * ↑↑I⁻¹ = 1\n⊢ ↑↑I * Submodule.span R {v} ≤ LinearMap.range (Algebra.linearMap R A)",
"state_before": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhinv : ↑↑I * ↑↑I⁻¹ = 1\n⊢ IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}",
"tactic": "apply Submodule.map_comap_eq_self"
},
{
"state_after": "case h\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhinv : ↑↑I * ↑↑I⁻¹ = 1\n⊢ ↑↑I * Submodule.span R {v} ≤ ↑↑I * ↑↑I⁻¹",
"state_before": "case h\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhinv : ↑↑I * ↑↑I⁻¹ = 1\n⊢ ↑↑I * Submodule.span R {v} ≤ LinearMap.range (Algebra.linearMap R A)",
"tactic": "rw [← Submodule.one_eq_range, ← hinv]"
},
{
"state_after": "no goals",
"state_before": "case h\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhinv : ↑↑I * ↑↑I⁻¹ = 1\n⊢ ↑↑I * Submodule.span R {v} ≤ ↑↑I * ↑↑I⁻¹",
"tactic": "exact Submodule.mul_le_mul_right ((Submodule.span_singleton_le_iff_mem _ _).2 hv)"
},
{
"state_after": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\n⊢ ∃ y, ↑(algebraMap R A) y = 1",
"state_before": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\n⊢ 1 ∈ ↑↑I * Submodule.span R {v}",
"tactic": "rw [← hJ, h, IsLocalization.coeSubmodule_top, Submodule.mem_one]"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\n⊢ ∃ y, ↑(algebraMap R A) y = 1",
"tactic": "exact ⟨1, (algebraMap R _).map_one⟩"
},
{
"state_after": "case intro.intro.refine'_1\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ ↑I⁻¹ * spanSingleton S w ≤ ↑I⁻¹ * ↑I",
"state_before": "case intro.intro.refine'_1\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ ↑I⁻¹ * spanSingleton S w ≤ 1",
"tactic": "conv_rhs => rw [← hinv, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ ↑I⁻¹ * spanSingleton S w ≤ ↑I⁻¹ * ↑I",
"tactic": "apply FractionalIdeal.mul_le_mul_left (FractionalIdeal.spanSingleton_le_iff_mem.mpr hw)"
},
{
"state_after": "case intro.intro.refine'_2\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ v * w ∈ ↑I⁻¹ * spanSingleton S w",
"state_before": "case intro.intro.refine'_2\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ 1 ≤ ↑I⁻¹ * spanSingleton S w",
"tactic": "rw [FractionalIdeal.one_le, ← hvw, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_2\nR✝ : Type ?u.9292\ninst✝⁴ : CommRing R✝\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v})\nh : J = ⊤\nhJ : IsLocalization.coeSubmodule A J = ↑↑I * Submodule.span R {v}\nthis : 1 ∈ ↑↑I * Submodule.span R {v}\nw : A\nhw : w ∈ ↑↑I\nhvw : w * v = 1\n⊢ v * w ∈ ↑I⁻¹ * spanSingleton S w",
"tactic": "exact FractionalIdeal.mul_mem_mul hv (FractionalIdeal.mem_spanSingleton_self _ _)"
}
] |
[
108,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.card_filter_mem_Icc_le
|
[
{
"state_after": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx✝ : ι → ℝ\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ card (filter (fun J => x ∈ ↑Box.Icc J) π.boxes) ≤ Fintype.card (Set ι)",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx✝ : ι → ℝ\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ card (filter (fun J => x ∈ ↑Box.Icc J) π.boxes) ≤ 2 ^ Fintype.card ι",
"tactic": "rw [← Fintype.card_set]"
},
{
"state_after": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx✝ : ι → ℝ\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ ∀ (a₁ : Box ι),\n a₁ ∈ filter (fun J => x ∈ ↑Box.Icc J) π.boxes →\n ∀ (a₂ : Box ι),\n a₂ ∈ filter (fun J => x ∈ ↑Box.Icc J) π.boxes →\n (fun J => {i | Box.lower J i = x i}) a₁ = (fun J => {i | Box.lower J i = x i}) a₂ → a₁ = a₂",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx✝ : ι → ℝ\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ card (filter (fun J => x ∈ ↑Box.Icc J) π.boxes) ≤ Fintype.card (Set ι)",
"tactic": "refine' Finset.card_le_card_of_inj_on (fun J : Box ι => { i | J.lower i = x i })\n (fun _ _ => Finset.mem_univ _) _"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx✝ : ι → ℝ\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ ∀ (a₁ : Box ι),\n a₁ ∈ filter (fun J => x ∈ ↑Box.Icc J) π.boxes →\n ∀ (a₂ : Box ι),\n a₂ ∈ filter (fun J => x ∈ ↑Box.Icc J) π.boxes →\n (fun J => {i | Box.lower J i = x i}) a₁ = (fun J => {i | Box.lower J i = x i}) a₂ → a₁ = a₂",
"tactic": "simpa only [Finset.mem_filter] using π.injOn_setOf_mem_Icc_setOf_lower_eq x"
}
] |
[
207,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Probability/CondCount.lean
|
ProbabilityTheory.condCount_self
|
[
{
"state_after": "case h0\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\n⊢ ↑↑Measure.count s ≠ 0\n\ncase ht\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\n⊢ ↑↑Measure.count s ≠ ⊤",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\n⊢ ↑↑(condCount s) s = 1",
"tactic": "rw [condCount, cond_apply _ hs.measurableSet, Set.inter_self, ENNReal.inv_mul_cancel]"
},
{
"state_after": "no goals",
"state_before": "case h0\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\n⊢ ↑↑Measure.count s ≠ 0",
"tactic": "exact fun h => hs'.ne_empty <| Measure.empty_of_count_eq_zero h"
},
{
"state_after": "no goals",
"state_before": "case ht\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\n⊢ ↑↑Measure.count s ≠ ⊤",
"tactic": "exact (Measure.count_apply_lt_top.2 hs).ne"
}
] |
[
109,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.ofReal_sinh_ofReal_re
|
[
{
"state_after": "no goals",
"state_before": "x✝ y : ℂ\nx : ℝ\n⊢ ↑(starRingEnd ℂ) (sinh ↑x) = sinh ↑x",
"tactic": "rw [← sinh_conj, conj_ofReal]"
}
] |
[
652,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
651,
1
] |
Mathlib/Data/Sign.lean
|
sign_sum
|
[
{
"state_after": "case zero\nα : Type u_2\ninst✝ : LinearOrderedAddCommGroup α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : Finset.Nonempty s\nh : ∀ (i : ι), i ∈ s → ↑sign (f i) = zero\n⊢ ↑sign (∑ i in s, f i) = zero\n\ncase neg\nα : Type u_2\ninst✝ : LinearOrderedAddCommGroup α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : Finset.Nonempty s\nh : ∀ (i : ι), i ∈ s → ↑sign (f i) = neg\n⊢ ↑sign (∑ i in s, f i) = neg\n\ncase pos\nα : Type u_2\ninst✝ : LinearOrderedAddCommGroup α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : Finset.Nonempty s\nh : ∀ (i : ι), i ∈ s → ↑sign (f i) = pos\n⊢ ↑sign (∑ i in s, f i) = pos",
"state_before": "α : Type u_2\ninst✝ : LinearOrderedAddCommGroup α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : Finset.Nonempty s\nt : SignType\nh : ∀ (i : ι), i ∈ s → ↑sign (f i) = t\n⊢ ↑sign (∑ i in s, f i) = t",
"tactic": "cases t"
},
{
"state_after": "case zero\nα : Type u_2\ninst✝ : LinearOrderedAddCommGroup α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : Finset.Nonempty s\nh : ∀ (i : ι), i ∈ s → f i = 0\n⊢ ∑ i in s, f i = 0",
"state_before": "case zero\nα : Type u_2\ninst✝ : LinearOrderedAddCommGroup α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : Finset.Nonempty s\nh : ∀ (i : ι), i ∈ s → ↑sign (f i) = zero\n⊢ ↑sign (∑ i in s, f i) = zero",
"tactic": "simp_rw [zero_eq_zero, sign_eq_zero_iff] at h⊢"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_2\ninst✝ : LinearOrderedAddCommGroup α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : Finset.Nonempty s\nh : ∀ (i : ι), i ∈ s → f i = 0\n⊢ ∑ i in s, f i = 0",
"tactic": "exact Finset.sum_eq_zero h"
},
{
"state_after": "case neg\nα : Type u_2\ninst✝ : LinearOrderedAddCommGroup α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : Finset.Nonempty s\nh : ∀ (i : ι), i ∈ s → f i < 0\n⊢ ∑ i in s, f i < 0",
"state_before": "case neg\nα : Type u_2\ninst✝ : LinearOrderedAddCommGroup α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : Finset.Nonempty s\nh : ∀ (i : ι), i ∈ s → ↑sign (f i) = neg\n⊢ ↑sign (∑ i in s, f i) = neg",
"tactic": "simp_rw [neg_eq_neg_one, sign_eq_neg_one_iff] at h⊢"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_2\ninst✝ : LinearOrderedAddCommGroup α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : Finset.Nonempty s\nh : ∀ (i : ι), i ∈ s → f i < 0\n⊢ ∑ i in s, f i < 0",
"tactic": "exact Finset.sum_neg h hs"
},
{
"state_after": "case pos\nα : Type u_2\ninst✝ : LinearOrderedAddCommGroup α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : Finset.Nonempty s\nh : ∀ (i : ι), i ∈ s → 0 < f i\n⊢ 0 < ∑ i in s, f i",
"state_before": "case pos\nα : Type u_2\ninst✝ : LinearOrderedAddCommGroup α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : Finset.Nonempty s\nh : ∀ (i : ι), i ∈ s → ↑sign (f i) = pos\n⊢ ↑sign (∑ i in s, f i) = pos",
"tactic": "simp_rw [pos_eq_one, sign_eq_one_iff] at h⊢"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_2\ninst✝ : LinearOrderedAddCommGroup α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : Finset.Nonempty s\nh : ∀ (i : ι), i ∈ s → 0 < f i\n⊢ 0 < ∑ i in s, f i",
"tactic": "exact Finset.sum_pos h hs"
}
] |
[
478,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
470,
1
] |
Mathlib/Data/Nat/GCD/Basic.lean
|
Nat.lcm_dvd_mul
|
[] |
[
97,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.lift_apply
|
[
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.1218573\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\nA : Type u₃\ninst✝³ : Semiring A\ninst✝² : Algebra k A\nB : Type ?u.1218610\ninst✝¹ : Semiring B\ninst✝ : Algebra k B\nF : G →* A\nf : MonoidAlgebra k G\n⊢ ↑(↑(lift k G A) F) f = sum f fun a b => b • ↑F a",
"tactic": "simp only [lift_apply', Algebra.smul_def]"
}
] |
[
903,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
902,
1
] |
Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
|
MeasureTheory.FiniteMeasure.normalize_eq_inv_mass_smul_of_nonzero
|
[
{
"state_after": "Ω : Type u_1\ninst✝ : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nnonzero : μ ≠ 0\n⊢ ProbabilityMeasure.toFiniteMeasure (normalize μ) =\n (mass μ)⁻¹ • mass μ • ProbabilityMeasure.toFiniteMeasure (normalize μ)",
"state_before": "Ω : Type u_1\ninst✝ : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nnonzero : μ ≠ 0\n⊢ ProbabilityMeasure.toFiniteMeasure (normalize μ) = (mass μ)⁻¹ • μ",
"tactic": "nth_rw 3 [μ.self_eq_mass_smul_normalize]"
},
{
"state_after": "Ω : Type u_1\ninst✝ : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nnonzero : μ ≠ 0\n⊢ ProbabilityMeasure.toFiniteMeasure (normalize μ) =\n ((mass μ)⁻¹ • mass μ) • ProbabilityMeasure.toFiniteMeasure (normalize μ)",
"state_before": "Ω : Type u_1\ninst✝ : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nnonzero : μ ≠ 0\n⊢ ProbabilityMeasure.toFiniteMeasure (normalize μ) =\n (mass μ)⁻¹ • mass μ • ProbabilityMeasure.toFiniteMeasure (normalize μ)",
"tactic": "rw [← smul_assoc]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\ninst✝ : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nnonzero : μ ≠ 0\n⊢ ProbabilityMeasure.toFiniteMeasure (normalize μ) =\n ((mass μ)⁻¹ • mass μ) • ProbabilityMeasure.toFiniteMeasure (normalize μ)",
"tactic": "simp only [μ.mass_nonzero_iff.mpr nonzero, Algebra.id.smul_eq_mul, inv_mul_cancel, Ne.def,\n not_false_iff, one_smul]"
}
] |
[
366,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.le_sup
|
[] |
[
1265,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1264,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_singleton_eq_empty
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\n⊢ s ∩ {a} = ∅ ↔ ¬a ∈ s",
"tactic": "rw [inter_comm, singleton_inter_eq_empty]"
}
] |
[
1351,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1350,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.measurableSet_preimage
|
[] |
[
210,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.support_single_ne_zero
|
[
{
"state_after": "case a\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ni : ι\nb : β i\nhb : b ≠ 0\nj : ι\n⊢ j ∈ support (single i b) ↔ j ∈ {i}",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ni : ι\nb : β i\nhb : b ≠ 0\n⊢ support (single i b) = {i}",
"tactic": "ext j"
},
{
"state_after": "case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ni : ι\nb : β i\nhb : b ≠ 0\nj : ι\nh : i = j\n⊢ j ∈ support (single i b) ↔ j ∈ {i}\n\ncase neg\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ni : ι\nb : β i\nhb : b ≠ 0\nj : ι\nh : ¬i = j\n⊢ j ∈ support (single i b) ↔ j ∈ {i}",
"state_before": "case a\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ni : ι\nb : β i\nhb : b ≠ 0\nj : ι\n⊢ j ∈ support (single i b) ↔ j ∈ {i}",
"tactic": "by_cases h : i = j"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ni : ι\nb : β i\nhb : b ≠ 0\nj : ι\nh : ¬i = j\n⊢ j ∈ support (single i b) ↔ j ∈ {i}",
"tactic": "simp [Ne.symm h, h]"
},
{
"state_after": "case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ni : ι\nb : β i\nhb : b ≠ 0\n⊢ i ∈ support (single i b) ↔ i ∈ {i}",
"state_before": "case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ni : ι\nb : β i\nhb : b ≠ 0\nj : ι\nh : i = j\n⊢ j ∈ support (single i b) ↔ j ∈ {i}",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ni : ι\nb : β i\nhb : b ≠ 0\n⊢ i ∈ support (single i b) ↔ i ∈ {i}",
"tactic": "simp [hb]"
}
] |
[
1165,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1161,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.measure_inter_lt_top_of_right_ne_top
|
[] |
[
340,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/Topology/GDelta.lean
|
Set.Finite.isGδ
|
[] |
[
170,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Function.Injective.wbtw_map_iff
|
[
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z : P\nf : P →ᵃ[R] P'\nhf : Injective ↑f\nh : Wbtw R (↑f x) (↑f y) (↑f z)\n⊢ Wbtw R x y z",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z : P\nf : P →ᵃ[R] P'\nhf : Injective ↑f\n⊢ Wbtw R (↑f x) (↑f y) (↑f z) ↔ Wbtw R x y z",
"tactic": "refine' ⟨fun h => _, fun h => h.map _⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z : P\nf : P →ᵃ[R] P'\nhf : Injective ↑f\nh : Wbtw R (↑f x) (↑f y) (↑f z)\n⊢ Wbtw R x y z",
"tactic": "rwa [Wbtw, ← affineSegment_image, hf.mem_set_image] at h"
}
] |
[
164,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.pow_sub_one_eq
|
[
{
"state_after": "no goals",
"state_before": "M : Type ?u.2780127\nN : Type ?u.2780130\nG : Type ?u.2780133\nR : Type u_1\nS : Type ?u.2780139\nF : Type ?u.2780142\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh : IsPrimitiveRoot ζ✝ k\ninst✝ : IsDomain R\nζ : R\nhζ : IsPrimitiveRoot ζ k\nhk : 1 < k\n⊢ ζ ^ Nat.pred k = -∑ i in range (Nat.pred k), ζ ^ i",
"tactic": "rw [eq_neg_iff_add_eq_zero, add_comm, ← sum_range_succ, ← Nat.succ_eq_add_one,\n Nat.succ_pred_eq_of_pos (pos_of_gt hk), hζ.geom_sum_eq_zero hk]"
}
] |
[
675,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
672,
1
] |
Mathlib/Topology/DiscreteQuotient.lean
|
DiscreteQuotient.map_comp
|
[
{
"state_after": "case h.mk\nα : Type ?u.31962\nX : Type u_3\nY : Type u_1\nZ : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\nh1 : LEComap g B C\nh2 : LEComap f A B\nx✝ : Quotient A.toSetoid\na✝ : X\n⊢ map (ContinuousMap.comp g f) (_ : LEComap (ContinuousMap.comp g f) A C) (Quot.mk Setoid.r a✝) =\n (map g h1 ∘ map f h2) (Quot.mk Setoid.r a✝)",
"state_before": "α : Type ?u.31962\nX : Type u_3\nY : Type u_1\nZ : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\nh1 : LEComap g B C\nh2 : LEComap f A B\n⊢ map (ContinuousMap.comp g f) (_ : LEComap (ContinuousMap.comp g f) A C) = map g h1 ∘ map f h2",
"tactic": "ext ⟨⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nα : Type ?u.31962\nX : Type u_3\nY : Type u_1\nZ : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\nh1 : LEComap g B C\nh2 : LEComap f A B\nx✝ : Quotient A.toSetoid\na✝ : X\n⊢ map (ContinuousMap.comp g f) (_ : LEComap (ContinuousMap.comp g f) A C) (Quot.mk Setoid.r a✝) =\n (map g h1 ∘ map f h2) (Quot.mk Setoid.r a✝)",
"tactic": "rfl"
}
] |
[
336,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.tendsto_fst
|
[] |
[
140,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.keys_kreplace
|
[
{
"state_after": "case mk.mk\nα : Type u\nβ : α → Type v\nl l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\na₁ : α\nb₂✝ : β a₁\na₂ : α\nb₂ : β a₂\n⊢ ({ fst := a₂, snd := b₂ } ∈ if a = { fst := a₁, snd := b₂✝ }.fst then some { fst := a, snd := b } else none) →\n { fst := a₁, snd := b₂✝ }.fst = { fst := a₂, snd := b₂ }.fst",
"state_before": "α : Type u\nβ : α → Type v\nl l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\n⊢ ∀ (a_1 b_1 : Sigma β), (b_1 ∈ if a = a_1.fst then some { fst := a, snd := b } else none) → a_1.fst = b_1.fst",
"tactic": "rintro ⟨a₁, b₂⟩ ⟨a₂, b₂⟩"
},
{
"state_after": "case mk.mk\nα : Type u\nβ : α → Type v\nl l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\na₁ : α\nb₂✝ : β a₁\na₂ : α\nb₂ : β a₂\n⊢ ({ fst := a₂, snd := b₂ } ∈ if a = a₁ then some { fst := a, snd := b } else none) → a₁ = a₂",
"state_before": "case mk.mk\nα : Type u\nβ : α → Type v\nl l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\na₁ : α\nb₂✝ : β a₁\na₂ : α\nb₂ : β a₂\n⊢ ({ fst := a₂, snd := b₂ } ∈ if a = { fst := a₁, snd := b₂✝ }.fst then some { fst := a, snd := b } else none) →\n { fst := a₁, snd := b₂✝ }.fst = { fst := a₂, snd := b₂ }.fst",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nα : Type u\nβ : α → Type v\nl l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\na₁ : α\nb₂✝ : β a₁\na₂ : α\nb₂ : β a₂\n⊢ ({ fst := a₂, snd := b₂ } ∈ if a = a₁ then some { fst := a, snd := b } else none) → a₁ = a₂",
"tactic": "split_ifs with h <;> simp (config := { contextual := true }) [h]"
}
] |
[
374,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
370,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.cos_pi
|
[
{
"state_after": "⊢ ↑(-1) = -1",
"state_before": "⊢ cos ↑π = -1",
"tactic": "rw [← ofReal_cos, Real.cos_pi]"
},
{
"state_after": "no goals",
"state_before": "⊢ ↑(-1) = -1",
"tactic": "simp"
}
] |
[
1126,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1126,
1
] |
Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean
|
GeneralizedContinuedFraction.denominators_stable_of_terminated
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn m : ℕ\ninst✝ : DivisionRing K\nn_le_m : n ≤ m\nterminated_at_n : TerminatedAt g n\n⊢ denominators g m = denominators g n",
"tactic": "simp only [denom_eq_conts_b, continuants_stable_of_terminated n_le_m terminated_at_n]"
}
] |
[
85,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.coe_ennreal_strictMono
|
[] |
[
481,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
480,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.neighborSet_sSup
|
[
{
"state_after": "case h\nι : Sort ?u.90110\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\ns : Set (Subgraph G)\nv x✝ : V\n⊢ x✝ ∈ neighborSet (sSup s) v ↔ x✝ ∈ ⋃ (G' : Subgraph G) (_ : G' ∈ s), neighborSet G' v",
"state_before": "ι : Sort ?u.90110\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\ns : Set (Subgraph G)\nv : V\n⊢ neighborSet (sSup s) v = ⋃ (G' : Subgraph G) (_ : G' ∈ s), neighborSet G' v",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Sort ?u.90110\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\ns : Set (Subgraph G)\nv x✝ : V\n⊢ x✝ ∈ neighborSet (sSup s) v ↔ x✝ ∈ ⋃ (G' : Subgraph G) (_ : G' ∈ s), neighborSet G' v",
"tactic": "simp"
}
] |
[
518,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Subperm.cons_left
|
[
{
"state_after": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\nx : α\nhx : count x l₁ < count x l₂\n⊢ ∀ (x_1 : α), x_1 ∈ x :: l₁ → count x_1 (x :: l₁) ≤ count x_1 l₂",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh : l₁ <+~ l₂\nx : α\nhx : count x l₁ < count x l₂\n⊢ x :: l₁ <+~ l₂",
"tactic": "rw [subperm_ext_iff] at h⊢"
},
{
"state_after": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\nx : α\nhx : count x l₁ < count x l₂\ny : α\nhy : y ∈ x :: l₁\n⊢ count y (x :: l₁) ≤ count y l₂",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\nx : α\nhx : count x l₁ < count x l₂\n⊢ ∀ (x_1 : α), x_1 ∈ x :: l₁ → count x_1 (x :: l₁) ≤ count x_1 l₂",
"tactic": "intro y hy"
},
{
"state_after": "case pos\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\nx : α\nhx : count x l₁ < count x l₂\ny : α\nhy : y ∈ x :: l₁\nhy' : y = x\n⊢ count y (x :: l₁) ≤ count y l₂\n\ncase neg\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\nx : α\nhx : count x l₁ < count x l₂\ny : α\nhy : y ∈ x :: l₁\nhy' : ¬y = x\n⊢ count y (x :: l₁) ≤ count y l₂",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\nx : α\nhx : count x l₁ < count x l₂\ny : α\nhy : y ∈ x :: l₁\n⊢ count y (x :: l₁) ≤ count y l₂",
"tactic": "by_cases hy' : y = x"
},
{
"state_after": "case pos\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\ny : α\nhx : count y l₁ < count y l₂\nhy : y ∈ y :: l₁\n⊢ count y (y :: l₁) ≤ count y l₂",
"state_before": "case pos\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\nx : α\nhx : count x l₁ < count x l₂\ny : α\nhy : y ∈ x :: l₁\nhy' : y = x\n⊢ count y (x :: l₁) ≤ count y l₂",
"tactic": "subst x"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\ny : α\nhx : count y l₁ < count y l₂\nhy : y ∈ y :: l₁\n⊢ count y (y :: l₁) ≤ count y l₂",
"tactic": "simpa using Nat.succ_le_of_lt hx"
},
{
"state_after": "case neg\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\nx : α\nhx : count x l₁ < count x l₂\ny : α\nhy : y ∈ x :: l₁\nhy' : ¬y = x\n⊢ count y l₁ ≤ count y l₂",
"state_before": "case neg\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\nx : α\nhx : count x l₁ < count x l₂\ny : α\nhy : y ∈ x :: l₁\nhy' : ¬y = x\n⊢ count y (x :: l₁) ≤ count y l₂",
"tactic": "rw [count_cons_of_ne hy']"
},
{
"state_after": "case neg\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\nx : α\nhx : count x l₁ < count x l₂\ny : α\nhy : y ∈ x :: l₁\nhy' : ¬y = x\n⊢ y ∈ l₁",
"state_before": "case neg\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\nx : α\nhx : count x l₁ < count x l₂\ny : α\nhy : y ∈ x :: l₁\nhy' : ¬y = x\n⊢ count y l₁ ≤ count y l₂",
"tactic": "refine' h y _"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ : List α\nh✝ : l₁ <+~ l₂\nh : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂\nx : α\nhx : count x l₁ < count x l₂\ny : α\nhy : y ∈ x :: l₁\nhy' : ¬y = x\n⊢ y ∈ l₁",
"tactic": "simpa [hy'] using hy"
}
] |
[
945,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
936,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean
|
Multiset.Ico_sub_Ico_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ c✝ d a b c : α\n⊢ Ico a b - Ico a c = Ico (max a c) b",
"tactic": "rw [Ico, Ico, Ico, ← Finset.sdiff_val, Finset.Ico_diff_Ico_left]"
}
] |
[
275,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.closedBall_finset_sup
|
[
{
"state_after": "R : Type ?u.1100644\nR' : Type ?u.1100647\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1100653\n𝕜₃ : Type ?u.1100656\n𝕝 : Type ?u.1100659\nE : Type u_2\nE₂ : Type ?u.1100665\nE₃ : Type ?u.1100668\nF : Type ?u.1100671\nG : Type ?u.1100674\nι : Type u_3\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ : Seminorm 𝕜 E\np : ι → Seminorm 𝕜 E\ns : Finset ι\nx : E\nr : ℝ\nhr : 0 ≤ r\n⊢ closedBall (Finset.sup s p) x r = ⨅ (a : ι) (_ : a ∈ s), closedBall (p a) x r",
"state_before": "R : Type ?u.1100644\nR' : Type ?u.1100647\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1100653\n𝕜₃ : Type ?u.1100656\n𝕝 : Type ?u.1100659\nE : Type u_2\nE₂ : Type ?u.1100665\nE₃ : Type ?u.1100668\nF : Type ?u.1100671\nG : Type ?u.1100674\nι : Type u_3\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ : Seminorm 𝕜 E\np : ι → Seminorm 𝕜 E\ns : Finset ι\nx : E\nr : ℝ\nhr : 0 ≤ r\n⊢ closedBall (Finset.sup s p) x r = Finset.inf s fun i => closedBall (p i) x r",
"tactic": "rw [Finset.inf_eq_iInf]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1100644\nR' : Type ?u.1100647\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1100653\n𝕜₃ : Type ?u.1100656\n𝕝 : Type ?u.1100659\nE : Type u_2\nE₂ : Type ?u.1100665\nE₃ : Type ?u.1100668\nF : Type ?u.1100671\nG : Type ?u.1100674\nι : Type u_3\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ : Seminorm 𝕜 E\np : ι → Seminorm 𝕜 E\ns : Finset ι\nx : E\nr : ℝ\nhr : 0 ≤ r\n⊢ closedBall (Finset.sup s p) x r = ⨅ (a : ι) (_ : a ∈ s), closedBall (p a) x r",
"tactic": "exact closedBall_finset_sup_eq_iInter _ _ _ hr"
}
] |
[
879,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
876,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.domCongr_refl
|
[] |
[
1755,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1753,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.map.of
|
[] |
[
827,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
826,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.PartrecToTM2.contSupp_halt
|
[] |
[
1891,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1890,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.ball_image_of_ball
|
[] |
[
235,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Data/Multiset/Nodup.lean
|
Multiset.Nodup.inter_right
|
[] |
[
215,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
ZFSet.mem_powerset
|
[
{
"state_after": "no goals",
"state_before": "x y : ZFSet\nx✝¹ x✝ : PSet\nα : Type u\nA : α → PSet\nβ : Type u\nB : β → PSet\n⊢ PSet.mk β B ∈ PSet.powerset (PSet.mk α A) ↔ Quotient.mk setoid (PSet.mk β B) ⊆ Quotient.mk setoid (PSet.mk α A)",
"tactic": "simp [mem_powerset, subset_iff]"
}
] |
[
1013,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1011,
1
] |
Mathlib/LinearAlgebra/Projection.lean
|
Submodule.linearProjOfIsCompl_apply_right
|
[] |
[
185,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieSubalgebra.coe_injective
|
[] |
[
207,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Data/Vector/Basic.lean
|
Vector.mOfFn_pure
|
[
{
"state_after": "n✝ : ℕ\nα✝ : Type ?u.39247\nm : Type u_1 → Type u_2\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nα : Type u_1\nn : ℕ\nf : Fin (n + 1) → α\n⊢ (do\n let a ← pure (f 0)\n let v ← pure (ofFn fun i => f (Fin.succ i))\n pure (a ::ᵥ v)) =\n pure (f 0 ::ᵥ ofFn fun i => f (Fin.succ i))",
"state_before": "n✝ : ℕ\nα✝ : Type ?u.39247\nm : Type u_1 → Type u_2\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nα : Type u_1\nn : ℕ\nf : Fin (n + 1) → α\n⊢ (mOfFn fun i => pure (f i)) = pure (ofFn f)",
"tactic": "rw [mOfFn, @mOfFn_pure m _ _ _ n _, ofFn]"
},
{
"state_after": "no goals",
"state_before": "n✝ : ℕ\nα✝ : Type ?u.39247\nm : Type u_1 → Type u_2\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nα : Type u_1\nn : ℕ\nf : Fin (n + 1) → α\n⊢ (do\n let a ← pure (f 0)\n let v ← pure (ofFn fun i => f (Fin.succ i))\n pure (a ::ᵥ v)) =\n pure (f 0 ::ᵥ ofFn fun i => f (Fin.succ i))",
"tactic": "simp"
}
] |
[
413,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
408,
1
] |
Mathlib/Algebra/Order/Hom/Monoid.lean
|
OrderMonoidHom.copy_eq
|
[] |
[
383,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
382,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Finset.sum_smul_sum_eq_sum_perm
|
[
{
"state_after": "ι : Type u_1\nα : Type u_3\nβ : Type u_2\ninst✝³ : DecidableEq α\ninst✝² : Semiring α\ninst✝¹ : AddCommMonoid β\ninst✝ : Module α β\ns : Finset ι\nσ : Perm ι\nhσ : IsCycleOn σ ↑s\nf : ι → α\ng : ι → β\n⊢ ∑ x in range (card s),\n ∑ x_1 in s,\n f\n (↑{ toFun := fun i => (i, ↑(σ ^ x) i),\n inj' :=\n (_ :\n ∀ (i j : ι),\n (fun i => (i, ↑(σ ^ x) i)) i = (fun i => (i, ↑(σ ^ x) i)) j →\n ((fun i => (i, ↑(σ ^ x) i)) i).fst = ((fun i => (i, ↑(σ ^ x) i)) j).fst) }\n x_1).fst •\n g\n (↑{ toFun := fun i => (i, ↑(σ ^ x) i),\n inj' :=\n (_ :\n ∀ (i j : ι),\n (fun i => (i, ↑(σ ^ x) i)) i = (fun i => (i, ↑(σ ^ x) i)) j →\n ((fun i => (i, ↑(σ ^ x) i)) i).fst = ((fun i => (i, ↑(σ ^ x) i)) j).fst) }\n x_1).snd =\n ∑ x in range (card s), ∑ x_1 in s, f x_1 • g (↑(σ ^ x) x_1)",
"state_before": "ι : Type u_1\nα : Type u_3\nβ : Type u_2\ninst✝³ : DecidableEq α\ninst✝² : Semiring α\ninst✝¹ : AddCommMonoid β\ninst✝ : Module α β\ns : Finset ι\nσ : Perm ι\nhσ : IsCycleOn σ ↑s\nf : ι → α\ng : ι → β\n⊢ (∑ i in s, f i) • ∑ i in s, g i = ∑ k in range (card s), ∑ i in s, f i • g (↑(σ ^ k) i)",
"tactic": "simp_rw [sum_smul_sum, product_self_eq_disjUnion_perm hσ, sum_disjiUnion, sum_map]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_3\nβ : Type u_2\ninst✝³ : DecidableEq α\ninst✝² : Semiring α\ninst✝¹ : AddCommMonoid β\ninst✝ : Module α β\ns : Finset ι\nσ : Perm ι\nhσ : IsCycleOn σ ↑s\nf : ι → α\ng : ι → β\n⊢ ∑ x in range (card s),\n ∑ x_1 in s,\n f\n (↑{ toFun := fun i => (i, ↑(σ ^ x) i),\n inj' :=\n (_ :\n ∀ (i j : ι),\n (fun i => (i, ↑(σ ^ x) i)) i = (fun i => (i, ↑(σ ^ x) i)) j →\n ((fun i => (i, ↑(σ ^ x) i)) i).fst = ((fun i => (i, ↑(σ ^ x) i)) j).fst) }\n x_1).fst •\n g\n (↑{ toFun := fun i => (i, ↑(σ ^ x) i),\n inj' :=\n (_ :\n ∀ (i j : ι),\n (fun i => (i, ↑(σ ^ x) i)) i = (fun i => (i, ↑(σ ^ x) i)) j →\n ((fun i => (i, ↑(σ ^ x) i)) i).fst = ((fun i => (i, ↑(σ ^ x) i)) j).fst) }\n x_1).snd =\n ∑ x in range (card s), ∑ x_1 in s, f x_1 • g (↑(σ ^ x) x_1)",
"tactic": "rfl"
}
] |
[
1978,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1975,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.fixed_point_card_lt_of_ne_one
|
[
{
"state_after": "ι : Type ?u.3061128\nα : Type u_1\nβ : Type ?u.3061134\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nσ✝ τ : Perm α\ninst✝ : Fintype α\nσ : Perm α\nh : σ ≠ 1\n⊢ 1 < card (filter (fun x => ¬↑σ x = x) univ)",
"state_before": "ι : Type ?u.3061128\nα : Type u_1\nβ : Type ?u.3061134\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nσ✝ τ : Perm α\ninst✝ : Fintype α\nσ : Perm α\nh : σ ≠ 1\n⊢ card (filter (fun x => ↑σ x = x) univ) < Fintype.card α - 1",
"tactic": "rw [lt_tsub_iff_left, ← lt_tsub_iff_right, ← Finset.card_compl, Finset.compl_filter]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.3061128\nα : Type u_1\nβ : Type ?u.3061134\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nσ✝ τ : Perm α\ninst✝ : Fintype α\nσ : Perm α\nh : σ ≠ 1\n⊢ 1 < card (filter (fun x => ¬↑σ x = x) univ)",
"tactic": "exact one_lt_card_support_of_ne_one h"
}
] |
[
1828,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1825,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
FiniteDimensional.finset_card_le_finrank_of_linearIndependent
|
[
{
"state_after": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V\nb : Finset V\nh : LinearIndependent K fun x => ↑x\n⊢ Fintype.card { x // x ∈ b } ≤ finrank K V",
"state_before": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V\nb : Finset V\nh : LinearIndependent K fun x => ↑x\n⊢ Finset.card b ≤ finrank K V",
"tactic": "rw [← Fintype.card_coe]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V\nb : Finset V\nh : LinearIndependent K fun x => ↑x\n⊢ Fintype.card { x // x ∈ b } ≤ finrank K V",
"tactic": "exact fintype_card_le_finrank_of_linearIndependent h"
}
] |
[
292,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/CategoryTheory/GlueData.lean
|
CategoryTheory.GlueData.diagram_r
|
[] |
[
160,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Std/Data/PairingHeap.lean
|
Std.PairingHeapImp.Heap.WF.deleteMin
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nle : α → α → Bool\na : α\ns' s : Heap α\nh : WF le s\neq : Heap.deleteMin le s = some (a, s')\n⊢ WF le s'",
"tactic": "cases h with cases eq | node h => exact Heap.WF.combine h"
},
{
"state_after": "no goals",
"state_before": "case node.refl\nα : Type u_1\nle : α → α → Bool\na : α\nc✝ : Heap α\nh : NodeWF le a c✝\n⊢ WF le (Heap.combine le c✝)",
"tactic": "exact Heap.WF.combine h"
}
] |
[
255,
60
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
253,
1
] |
Mathlib/CategoryTheory/Filtered.lean
|
CategoryTheory.IsFiltered.of_right_adjoint
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : IsFiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nJ : Type v\ninst✝² : SmallCategory J\ninst✝¹ : FinCategory J\nD : Type u₁\ninst✝ : Category D\nL : D ⥤ C\nR : C ⥤ D\nh : L ⊣ R\nX Y : D\nf g : X ⟶ Y\n⊢ f ≫\n ↑(Adjunction.homEquiv h Y (coeq (?m.33962 h X Y f g) (?m.33963 h X Y f g)))\n (coeqHom (?m.33962 h X Y f g) (?m.33963 h X Y f g)) =\n g ≫\n ↑(Adjunction.homEquiv h Y (coeq (?m.33962 h X Y f g) (?m.33963 h X Y f g)))\n (coeqHom (?m.33962 h X Y f g) (?m.33963 h X Y f g))",
"tactic": "rw [← h.homEquiv_naturality_left, ← h.homEquiv_naturality_left, coeq_condition]"
}
] |
[
319,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.tanh_zero
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ tanh 0 = 0",
"tactic": "simp [tanh]"
}
] |
[
1395,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1395,
1
] |
Mathlib/CategoryTheory/Category/Grpd.lean
|
CategoryTheory.Grpd.coe_of
|
[] |
[
67,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Algebra/Algebra/Operations.lean
|
Submodule.mem_span_mul_finite_of_mem_mul
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P✝ Q✝ : Submodule R A\nm n : A\nP Q : Submodule R A\nx : A\nhx : x ∈ P * Q\n⊢ x ∈ span R (↑P * ↑Q)",
"tactic": "rwa [← Submodule.span_eq P, ← Submodule.span_eq Q, Submodule.span_mul_span] at hx"
}
] |
[
368,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
map_nhdsWithin
|
[] |
[
363,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.sinh_sub
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ sinh (x - y) = sinh x * cosh y - cosh x * sinh y",
"tactic": "simp [sub_eq_add_neg, sinh_add, sinh_neg, cosh_neg]"
}
] |
[
1383,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1382,
1
] |
Mathlib/CategoryTheory/Limits/Opposites.lean
|
CategoryTheory.Limits.hasLimit_of_hasColimit_leftOp
|
[] |
[
245,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/LinearAlgebra/PiTensorProduct.lean
|
PiTensorProduct.lift.tprod
|
[] |
[
419,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
418,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
Encodable.decode_prod_val
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Encodable α\ninst✝ : Encodable β\ni : Encodable α\nn : ℕ\n⊢ Option.map (↑(Equiv.sigmaEquivProd α β))\n (Option.bind (decode (unpair n).fst) fun a => Option.map (Sigma.mk a) (decode (unpair n).snd)) =\n Option.bind (decode (unpair n).fst) fun a => Option.map (Prod.mk a) (decode (unpair n).snd)",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Encodable α\ninst✝ : Encodable β\ni : Encodable α\nn : ℕ\n⊢ decode n = Option.bind (decode (unpair n).fst) fun a => Option.map (Prod.mk a) (decode (unpair n).snd)",
"tactic": "simp only [decode_ofEquiv, Equiv.symm_symm, decode_sigma_val]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Encodable α\ninst✝ : Encodable β\ni : Encodable α\nn : ℕ\n⊢ Option.map (↑(Equiv.sigmaEquivProd α β))\n (Option.bind (decode (unpair n).fst) fun a => Option.map (Sigma.mk a) (decode (unpair n).snd)) =\n Option.bind (decode (unpair n).fst) fun a => Option.map (Prod.mk a) (decode (unpair n).snd)",
"tactic": "cases (decode n.unpair.1 : Option α) <;> cases (decode n.unpair.2 : Option β)\n<;> rfl"
}
] |
[
393,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
388,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.le_normalizer_of_normal
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.385870\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.385879\ninst✝ : AddGroup A\nH K : Subgroup G\nhK : Normal (subgroupOf H K)\nHK : H ≤ K\nx : G\nhx : x ∈ K\ny : G\nyH : x * y * x⁻¹ ∈ H\n⊢ y ∈ H",
"tactic": "simpa [mem_subgroupOf, mul_assoc] using\n hK.conj_mem ⟨x * y * x⁻¹, HK yH⟩ yH ⟨x⁻¹, K.inv_mem hx⟩"
}
] |
[
2204,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2200,
1
] |
Mathlib/RingTheory/WittVector/Defs.lean
|
WittVector.pow_coeff
|
[] |
[
386,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
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