file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Topology/MetricSpace/Basic.lean
|
dist_prod_same_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.245069\nι : Type ?u.245072\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nx : α\ny₁ y₂ : β\n⊢ dist (x, y₁) (x, y₂) = dist y₁ y₂",
"tactic": "simp [Prod.dist_eq, dist_nonneg]"
}
] |
[
1756,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1755,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.val_eq_singleton_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.28858\nγ : Type ?u.28861\ns✝ : Finset α\na✝ b a : α\ns : Finset α\n⊢ s.val = {a} ↔ s.val = {a}.val",
"state_before": "α : Type u_1\nβ : Type ?u.28858\nγ : Type ?u.28861\ns✝ : Finset α\na✝ b a : α\ns : Finset α\n⊢ s.val = {a} ↔ s = {a}",
"tactic": "rw [← val_inj]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.28858\nγ : Type ?u.28861\ns✝ : Finset α\na✝ b a : α\ns : Finset α\n⊢ s.val = {a} ↔ s.val = {a}.val",
"tactic": "rfl"
}
] |
[
701,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
699,
1
] |
Mathlib/Order/Monotone/Monovary.lean
|
Subsingleton.monovary
|
[] |
[
116,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
11
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.equivariantOfLinearOfComm_apply
|
[] |
[
1019,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1018,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.rotate_injective
|
[
{
"state_after": "α : Type u\nn : ℕ\nl l' : List α\nh : rotate l n = rotate l' n\n⊢ l = l'",
"state_before": "α : Type u\nn : ℕ\n⊢ Injective fun l => rotate l n",
"tactic": "rintro l l' (h : l.rotate n = l'.rotate n)"
},
{
"state_after": "α : Type u\nn : ℕ\nl l' : List α\nh : rotate l n = rotate l' n\nhle : length l = length l'\n⊢ l = l'",
"state_before": "α : Type u\nn : ℕ\nl l' : List α\nh : rotate l n = rotate l' n\n⊢ l = l'",
"tactic": "have hle : l.length = l'.length := (l.length_rotate n).symm.trans (h.symm ▸ l'.length_rotate n)"
},
{
"state_after": "α : Type u\nn : ℕ\nl l' : List α\nh : drop (n % length l) l ++ take (n % length l) l = drop (n % length l') l' ++ take (n % length l') l'\nhle : length l = length l'\n⊢ l = l'",
"state_before": "α : Type u\nn : ℕ\nl l' : List α\nh : rotate l n = rotate l' n\nhle : length l = length l'\n⊢ l = l'",
"tactic": "rw [rotate_eq_drop_append_take_mod, rotate_eq_drop_append_take_mod] at h"
},
{
"state_after": "case intro\nα : Type u\nn : ℕ\nl l' : List α\nh : drop (n % length l) l ++ take (n % length l) l = drop (n % length l') l' ++ take (n % length l') l'\nhle : length l = length l'\nhd : drop (n % length l) l = drop (n % length l') l'\nht : take (n % length l) l = take (n % length l') l'\n⊢ l = l'",
"state_before": "α : Type u\nn : ℕ\nl l' : List α\nh : drop (n % length l) l ++ take (n % length l) l = drop (n % length l') l' ++ take (n % length l') l'\nhle : length l = length l'\n⊢ l = l'",
"tactic": "obtain ⟨hd, ht⟩ := append_inj h (by simp_all)"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nn : ℕ\nl l' : List α\nh : drop (n % length l) l ++ take (n % length l) l = drop (n % length l') l' ++ take (n % length l') l'\nhle : length l = length l'\nhd : drop (n % length l) l = drop (n % length l') l'\nht : take (n % length l) l = take (n % length l') l'\n⊢ l = l'",
"tactic": "rw [← take_append_drop _ l, ht, hd, take_append_drop]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nn : ℕ\nl l' : List α\nh : drop (n % length l) l ++ take (n % length l) l = drop (n % length l') l' ++ take (n % length l') l'\nhle : length l = length l'\n⊢ length (drop (n % length l) l) = length (drop (n % length l') l')",
"tactic": "simp_all"
}
] |
[
331,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
326,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieEquiv.lieSubalgebraMap_apply
|
[] |
[
807,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
806,
1
] |
Std/Data/List/Lemmas.lean
|
List.get?_modifyNth_ne
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : α → α\nm n : Nat\nl : List α\nh : m ≠ n\n⊢ get? (modifyNth f m l) n = get? l n",
"tactic": "simp only [get?_modifyNth, if_neg h, id_map']"
}
] |
[
762,
48
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
760,
9
] |
Mathlib/Analysis/Complex/ReImTopology.lean
|
Complex.interior_reProdIm
|
[
{
"state_after": "no goals",
"state_before": "s t : Set ℝ\n⊢ interior (s ×ℂ t) = interior s ×ℂ interior t",
"tactic": "rw [Set.reProdIm, Set.reProdIm, interior_inter, interior_preimage_re, interior_preimage_im]"
}
] |
[
182,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/RingTheory/RootsOfUnity/Complex.lean
|
Complex.isPrimitiveRoot_exp
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\nh0 : n ≠ 0\n⊢ IsPrimitiveRoot (exp (2 * ↑π * I / ↑n)) n",
"tactic": "simpa only [Nat.cast_one, one_div] using\n isPrimitiveRoot_exp_of_coprime 1 n h0 n.coprime_one_left"
}
] |
[
57,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Analysis/InnerProductSpace/Adjoint.lean
|
LinearMap.adjoint_inner_right
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1706942\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 F\ninst✝ : FiniteDimensional 𝕜 G\nA : E →ₗ[𝕜] F\nx : E\ny : F\n⊢ inner x (↑↑(↑ContinuousLinearMap.adjoint (↑toContinuousLinearMap ↑(↑toContinuousLinearMap A))) y) =\n inner (↑↑(↑toContinuousLinearMap A) x) y",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1706942\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 F\ninst✝ : FiniteDimensional 𝕜 G\nA : E →ₗ[𝕜] F\nx : E\ny : F\n⊢ inner x (↑(↑adjoint A) y) = inner (↑A x) y",
"tactic": "rw [← coe_toContinuousLinearMap A, adjoint_eq_toClm_adjoint]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1706942\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 F\ninst✝ : FiniteDimensional 𝕜 G\nA : E →ₗ[𝕜] F\nx : E\ny : F\n⊢ inner x (↑↑(↑ContinuousLinearMap.adjoint (↑toContinuousLinearMap ↑(↑toContinuousLinearMap A))) y) =\n inner (↑↑(↑toContinuousLinearMap A) x) y",
"tactic": "exact ContinuousLinearMap.adjoint_inner_right _ x y"
}
] |
[
400,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
398,
1
] |
Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
|
LiouvilleWith.int_sub_iff
|
[
{
"state_after": "no goals",
"state_before": "p q x y : ℝ\nr : ℚ\nm : ℤ\nn : ℕ\n⊢ LiouvilleWith p (↑m - x) ↔ LiouvilleWith p x",
"tactic": "simp [sub_eq_add_neg]"
}
] |
[
299,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/Data/MvPolynomial/PDeriv.lean
|
MvPolynomial.pderiv_C_mul
|
[] |
[
127,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Data/Sym/Sym2.lean
|
Sym2.lift₂_mk''
|
[] |
[
229,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.aecover_Iio
|
[] |
[
158,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.map_refl
|
[
{
"state_after": "case h.h.a.h\nα : Type u_1\nβ : Type ?u.97471\nγ : Type ?u.97474\nι : Sort ?u.97477\nκ : ι → Sort ?u.97482\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf : α ≃o β\ns t : UpperSet α\na : α\nb : β\nx✝¹ : UpperSet α\nx✝ : α\n⊢ x✝ ∈ ↑(↑(map (OrderIso.refl α)) x✝¹) ↔ x✝ ∈ ↑(↑(OrderIso.refl (UpperSet α)) x✝¹)",
"state_before": "α : Type u_1\nβ : Type ?u.97471\nγ : Type ?u.97474\nι : Sort ?u.97477\nκ : ι → Sort ?u.97482\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf : α ≃o β\ns t : UpperSet α\na : α\nb : β\n⊢ map (OrderIso.refl α) = OrderIso.refl (UpperSet α)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h.a.h\nα : Type u_1\nβ : Type ?u.97471\nγ : Type ?u.97474\nι : Sort ?u.97477\nκ : ι → Sort ?u.97482\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf : α ≃o β\ns t : UpperSet α\na : α\nb : β\nx✝¹ : UpperSet α\nx✝ : α\n⊢ x✝ ∈ ↑(↑(map (OrderIso.refl α)) x✝¹) ↔ x✝ ∈ ↑(↑(OrderIso.refl (UpperSet α)) x✝¹)",
"tactic": "simp"
}
] |
[
983,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
981,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.countp_filter
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.312315\nγ : Type ?u.312318\np : α → Prop\ninst✝¹ : DecidablePred p\nq : α → Prop\ninst✝ : DecidablePred q\ns : Multiset α\n⊢ countp p (filter q s) = countp (fun a => p a ∧ q a) s",
"tactic": "simp [countp_eq_card_filter]"
}
] |
[
2272,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2271,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.C_apply
|
[] |
[
205,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/Analysis/Convex/Cone/Dual.lean
|
innerDualCone_iUnion
|
[
{
"state_after": "𝕜 : Type ?u.15183\nE : Type ?u.15186\nF : Type ?u.15189\nG : Type ?u.15192\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_1\nf : ι → Set H\n⊢ (⨅ (i : ι), innerDualCone (f i)) ≤ innerDualCone (⋃ (i : ι), f i)",
"state_before": "𝕜 : Type ?u.15183\nE : Type ?u.15186\nF : Type ?u.15189\nG : Type ?u.15192\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_1\nf : ι → Set H\n⊢ innerDualCone (⋃ (i : ι), f i) = ⨅ (i : ι), innerDualCone (f i)",
"tactic": "refine' le_antisymm (le_iInf fun i x hx y hy => hx _ <| mem_iUnion_of_mem _ hy) _"
},
{
"state_after": "𝕜 : Type ?u.15183\nE : Type ?u.15186\nF : Type ?u.15189\nG : Type ?u.15192\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_1\nf : ι → Set H\nx : H\nhx : x ∈ ⨅ (i : ι), innerDualCone (f i)\ny : H\nhy : y ∈ ⋃ (i : ι), f i\n⊢ 0 ≤ inner y x",
"state_before": "𝕜 : Type ?u.15183\nE : Type ?u.15186\nF : Type ?u.15189\nG : Type ?u.15192\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_1\nf : ι → Set H\n⊢ (⨅ (i : ι), innerDualCone (f i)) ≤ innerDualCone (⋃ (i : ι), f i)",
"tactic": "intro x hx y hy"
},
{
"state_after": "𝕜 : Type ?u.15183\nE : Type ?u.15186\nF : Type ?u.15189\nG : Type ?u.15192\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_1\nf : ι → Set H\nx : H\nhx : ∀ (i : ι), x ∈ innerDualCone (f i)\ny : H\nhy : y ∈ ⋃ (i : ι), f i\n⊢ 0 ≤ inner y x",
"state_before": "𝕜 : Type ?u.15183\nE : Type ?u.15186\nF : Type ?u.15189\nG : Type ?u.15192\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_1\nf : ι → Set H\nx : H\nhx : x ∈ ⨅ (i : ι), innerDualCone (f i)\ny : H\nhy : y ∈ ⋃ (i : ι), f i\n⊢ 0 ≤ inner y x",
"tactic": "rw [ConvexCone.mem_iInf] at hx"
},
{
"state_after": "case intro\n𝕜 : Type ?u.15183\nE : Type ?u.15186\nF : Type ?u.15189\nG : Type ?u.15192\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_1\nf : ι → Set H\nx : H\nhx : ∀ (i : ι), x ∈ innerDualCone (f i)\ny : H\nhy : y ∈ ⋃ (i : ι), f i\nj : ι\nhj : y ∈ f j\n⊢ 0 ≤ inner y x",
"state_before": "𝕜 : Type ?u.15183\nE : Type ?u.15186\nF : Type ?u.15189\nG : Type ?u.15192\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_1\nf : ι → Set H\nx : H\nhx : ∀ (i : ι), x ∈ innerDualCone (f i)\ny : H\nhy : y ∈ ⋃ (i : ι), f i\n⊢ 0 ≤ inner y x",
"tactic": "obtain ⟨j, hj⟩ := mem_iUnion.mp hy"
},
{
"state_after": "no goals",
"state_before": "case intro\n𝕜 : Type ?u.15183\nE : Type ?u.15186\nF : Type ?u.15189\nG : Type ?u.15192\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_1\nf : ι → Set H\nx : H\nhx : ∀ (i : ι), x ∈ innerDualCone (f i)\ny : H\nhy : y ∈ ⋃ (i : ι), f i\nj : ι\nhj : y ∈ f j\n⊢ 0 ≤ inner y x",
"tactic": "exact hx _ _ hj"
}
] |
[
112,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Monad.lean
|
Pmf.toPmf_dirac
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7626\nγ : Type ?u.7629\na a' : α\ns : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable α\nh : MeasurableSingletonClass α\n⊢ Measure.toPmf (Measure.dirac a) = pure a",
"tactic": "rw [toPmf_eq_iff_toMeasure_eq, toMeasure_pure]"
}
] |
[
101,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/RingTheory/WittVector/IsPoly.lean
|
WittVector.IsPoly₂.compLeft
|
[] |
[
577,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
575,
1
] |
Mathlib/Topology/Algebra/FilterBasis.lean
|
GroupFilterBasis.nhds_one_eq
|
[
{
"state_after": "G : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\n⊢ N B 1 = FilterBasis.filter toFilterBasis",
"state_before": "G : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\n⊢ 𝓝 1 = FilterBasis.filter toFilterBasis",
"tactic": "rw [B.nhds_eq]"
},
{
"state_after": "G : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\n⊢ map (fun y => y) (FilterBasis.filter toFilterBasis) = FilterBasis.filter toFilterBasis",
"state_before": "G : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\n⊢ N B 1 = FilterBasis.filter toFilterBasis",
"tactic": "simp only [N, one_mul]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\n⊢ map (fun y => y) (FilterBasis.filter toFilterBasis) = FilterBasis.filter toFilterBasis",
"tactic": "exact map_id"
}
] |
[
212,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
FormalMultilinearSeries.le_radius_of_isBigO
|
[] |
[
149,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean
|
ContinuousLinearMap.isBoundedLinearMap
|
[] |
[
89,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biprod.inl_map
|
[
{
"state_after": "J : Type w\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\nP Q W X Y Z : C\ninst✝¹ : HasBinaryBiproduct W X\ninst✝ : HasBinaryBiproduct Y Z\nf : W ⟶ Y\ng : X ⟶ Z\n⊢ inl ≫ map' f g = f ≫ inl",
"state_before": "J : Type w\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\nP Q W X Y Z : C\ninst✝¹ : HasBinaryBiproduct W X\ninst✝ : HasBinaryBiproduct Y Z\nf : W ⟶ Y\ng : X ⟶ Z\n⊢ inl ≫ map f g = f ≫ inl",
"tactic": "rw [biprod.map_eq_map']"
},
{
"state_after": "no goals",
"state_before": "J : Type w\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\nP Q W X Y Z : C\ninst✝¹ : HasBinaryBiproduct W X\ninst✝ : HasBinaryBiproduct Y Z\nf : W ⟶ Y\ng : X ⟶ Z\n⊢ inl ≫ map' f g = f ≫ inl",
"tactic": "exact IsColimit.ι_map (BinaryBiproduct.isColimit W X) _ _ ⟨WalkingPair.left⟩"
}
] |
[
1525,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1522,
1
] |
Mathlib/LinearAlgebra/Finrank.lean
|
Submodule.lt_top_of_finrank_lt_finrank
|
[
{
"state_after": "K : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nV₂ : Type v'\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\ns : Submodule K V\nlt : finrank K { x // x ∈ s } < finrank K { x // x ∈ ⊤ }\n⊢ s < ⊤",
"state_before": "K : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nV₂ : Type v'\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\ns : Submodule K V\nlt : finrank K { x // x ∈ s } < finrank K V\n⊢ s < ⊤",
"tactic": "rw [← finrank_top K V] at lt"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nV₂ : Type v'\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\ns : Submodule K V\nlt : finrank K { x // x ∈ s } < finrank K { x // x ∈ ⊤ }\n⊢ s < ⊤",
"tactic": "exact lt_of_le_of_finrank_lt_finrank le_top lt"
}
] |
[
290,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/Data/Set/Intervals/OrdConnected.lean
|
Set.OrdConnected.uIcc_subset
|
[] |
[
259,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.eqOn_mulIndicator
|
[] |
[
166,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.coe_mul
|
[] |
[
2516,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2515,
1
] |
Mathlib/Algebra/Hom/NonUnitalAlg.lean
|
NonUnitalAlgHom.coe_one
|
[] |
[
263,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.integral_zero_ae
|
[] |
[
1000,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
997,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
ContDiffOn.differentiableOn_iteratedFDerivWithin
|
[] |
[
1078,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1075,
1
] |
Mathlib/Analysis/Convex/Join.lean
|
convexJoin_convexJoin_convexJoin_comm
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.72702\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx y : E\ns t u v : Set E\n⊢ convexJoin 𝕜 (convexJoin 𝕜 s t) (convexJoin 𝕜 u v) = convexJoin 𝕜 (convexJoin 𝕜 s u) (convexJoin 𝕜 t v)",
"tactic": "simp_rw [← convexJoin_assoc, convexJoin_right_comm]"
}
] |
[
177,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/LinearAlgebra/Matrix/ZPow.lean
|
Matrix.zpow_add
|
[
{
"state_after": "no goals",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nha : IsUnit (det A)\nm n : ℤ\n⊢ A ^ (m + n) = A ^ m * A ^ n",
"tactic": "induction n using Int.induction_on with\n| hz => simp\n| hp n ihn => simp only [← add_assoc, zpow_add_one ha, ihn, mul_assoc]\n| hn n ihn => rw [zpow_sub_one ha, ← mul_assoc, ← ihn, ← zpow_sub_one ha, add_sub_assoc]"
},
{
"state_after": "no goals",
"state_before": "case hz\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nha : IsUnit (det A)\nm : ℤ\n⊢ A ^ (m + 0) = A ^ m * A ^ 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case hp\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nha : IsUnit (det A)\nm : ℤ\nn : ℕ\nihn : A ^ (m + ↑n) = A ^ m * A ^ ↑n\n⊢ A ^ (m + (↑n + 1)) = A ^ m * A ^ (↑n + 1)",
"tactic": "simp only [← add_assoc, zpow_add_one ha, ihn, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case hn\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nha : IsUnit (det A)\nm : ℤ\nn : ℕ\nihn : A ^ (m + -↑n) = A ^ m * A ^ (-↑n)\n⊢ A ^ (m + (-↑n - 1)) = A ^ m * A ^ (-↑n - 1)",
"tactic": "rw [zpow_sub_one ha, ← mul_assoc, ← ihn, ← zpow_sub_one ha, add_sub_assoc]"
}
] |
[
174,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/RingTheory/Valuation/Quotient.lean
|
AddValuation.supp_quot_supp
|
[] |
[
138,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Order/SuccPred/Limit.lean
|
Order.isPredLimit_top
|
[] |
[
285,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPushout.paste_horiz
|
[] |
[
731,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
727,
1
] |
Mathlib/Analysis/Normed/Field/Basic.lean
|
NormedField.punctured_nhds_neBot
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.375059\nγ : Type ?u.375062\nι : Type ?u.375065\ninst✝ : NontriviallyNormedField α\nx : α\n⊢ ∀ (ε : ℝ), ε > 0 → ∃ b, b ∈ {x}ᶜ ∧ dist x b < ε",
"state_before": "α : Type u_1\nβ : Type ?u.375059\nγ : Type ?u.375062\nι : Type ?u.375065\ninst✝ : NontriviallyNormedField α\nx : α\n⊢ NeBot (𝓝[{x}ᶜ] x)",
"tactic": "rw [← mem_closure_iff_nhdsWithin_neBot, Metric.mem_closure_iff]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.375059\nγ : Type ?u.375062\nι : Type ?u.375065\ninst✝ : NontriviallyNormedField α\nx : α\nε : ℝ\nε0 : ε > 0\n⊢ ∃ b, b ∈ {x}ᶜ ∧ dist x b < ε",
"state_before": "α : Type u_1\nβ : Type ?u.375059\nγ : Type ?u.375062\nι : Type ?u.375065\ninst✝ : NontriviallyNormedField α\nx : α\n⊢ ∀ (ε : ℝ), ε > 0 → ∃ b, b ∈ {x}ᶜ ∧ dist x b < ε",
"tactic": "rintro ε ε0"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.375059\nγ : Type ?u.375062\nι : Type ?u.375065\ninst✝ : NontriviallyNormedField α\nx : α\nε : ℝ\nε0 : ε > 0\nb : α\nhb0 : 0 < ‖b‖\nhbε : ‖b‖ < ε\n⊢ ∃ b, b ∈ {x}ᶜ ∧ dist x b < ε",
"state_before": "α : Type u_1\nβ : Type ?u.375059\nγ : Type ?u.375062\nι : Type ?u.375065\ninst✝ : NontriviallyNormedField α\nx : α\nε : ℝ\nε0 : ε > 0\n⊢ ∃ b, b ∈ {x}ᶜ ∧ dist x b < ε",
"tactic": "rcases exists_norm_lt α ε0 with ⟨b, hb0, hbε⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.375059\nγ : Type ?u.375062\nι : Type ?u.375065\ninst✝ : NontriviallyNormedField α\nx : α\nε : ℝ\nε0 : ε > 0\nb : α\nhb0 : 0 < ‖b‖\nhbε : ‖b‖ < ε\n⊢ dist x (x + b) < ε",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.375059\nγ : Type ?u.375062\nι : Type ?u.375065\ninst✝ : NontriviallyNormedField α\nx : α\nε : ℝ\nε0 : ε > 0\nb : α\nhb0 : 0 < ‖b‖\nhbε : ‖b‖ < ε\n⊢ ∃ b, b ∈ {x}ᶜ ∧ dist x b < ε",
"tactic": "refine' ⟨x + b, mt (Set.mem_singleton_iff.trans add_right_eq_self).1 <| norm_pos_iff.1 hb0, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.375059\nγ : Type ?u.375062\nι : Type ?u.375065\ninst✝ : NontriviallyNormedField α\nx : α\nε : ℝ\nε0 : ε > 0\nb : α\nhb0 : 0 < ‖b‖\nhbε : ‖b‖ < ε\n⊢ dist x (x + b) < ε",
"tactic": "rwa [dist_comm, dist_eq_norm, add_sub_cancel']"
}
] |
[
752,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
747,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
gcd_eq_left_iff
|
[] |
[
476,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
473,
1
] |
Mathlib/CategoryTheory/Sites/CompatiblePlus.lean
|
CategoryTheory.GrothendieckTopology.plusCompIso_whiskerRight
|
[
{
"state_after": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\n⊢ (whiskerRight (plusMap J η) F ≫ (plusCompIso J F Q).hom).app X =\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerRight η F)).app X",
"state_before": "C : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\n⊢ whiskerRight (plusMap J η) F ≫ (plusCompIso J F Q).hom = (plusCompIso J F P).hom ≫ plusMap J (whiskerRight η F)",
"tactic": "ext X"
},
{
"state_after": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\n⊢ ∀ (j : (Cover J X.unop)ᵒᵖ),\n (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app j ≫\n (whiskerRight (plusMap J η) F ≫ (plusCompIso J F Q).hom).app X =\n (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app j ≫\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerRight η F)).app X",
"state_before": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\n⊢ (whiskerRight (plusMap J η) F ≫ (plusCompIso J F Q).hom).app X =\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerRight η F)).app X",
"tactic": "apply (isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).hom_ext"
},
{
"state_after": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app W ≫\n (whiskerRight (plusMap J η) F ≫ (plusCompIso J F Q).hom).app X =\n (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app W ≫\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerRight η F)).app X",
"state_before": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\n⊢ ∀ (j : (Cover J X.unop)ᵒᵖ),\n (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app j ≫\n (whiskerRight (plusMap J η) F ≫ (plusCompIso J F Q).hom).app X =\n (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app j ≫\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerRight η F)).app X",
"tactic": "intro W"
},
{
"state_after": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map (colimit.ι (diagram J P X.unop) W) ≫\n F.map (colimMap (diagramNatTrans J η X.unop)) ≫ (plusCompIso J F Q).hom.app X =\n F.map (colimit.ι (diagram J P X.unop) W) ≫\n (plusCompIso J F P).hom.app X ≫ colimMap (diagramNatTrans J (whiskerRight η F) X.unop)",
"state_before": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app W ≫\n (whiskerRight (plusMap J η) F ≫ (plusCompIso J F Q).hom).app X =\n (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app W ≫\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerRight η F)).app X",
"tactic": "dsimp [plusObj, plusMap]"
},
{
"state_after": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map (colimit.ι (diagram J P X.unop) W) ≫\n F.map (colimMap (diagramNatTrans J η X.unop)) ≫ (plusCompIso J F Q).hom.app X =\n (diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) b) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"state_before": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map (colimit.ι (diagram J P X.unop) W) ≫\n F.map (colimMap (diagramNatTrans J η X.unop)) ≫ (plusCompIso J F Q).hom.app X =\n F.map (colimit.ι (diagram J P X.unop) W) ≫\n (plusCompIso J F P).hom.app X ≫ colimMap (diagramNatTrans J (whiskerRight η F) X.unop)",
"tactic": "simp only [ι_colimMap, whiskerRight_app, ι_plusCompIso_hom_assoc,\n GrothendieckTopology.diagramNatTrans_app]"
},
{
"state_after": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map (colimit.ι (diagram J P X.unop) W ≫ colimMap (diagramNatTrans J η X.unop)) ≫ (plusCompIso J F Q).hom.app X =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) b)) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"state_before": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map (colimit.ι (diagram J P X.unop) W) ≫\n F.map (colimMap (diagramNatTrans J η X.unop)) ≫ (plusCompIso J F Q).hom.app X =\n (diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) b) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"tactic": "simp only [← Category.assoc, ← F.map_comp]"
},
{
"state_after": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map\n (colimit.ι (diagram J P X.unop) W ≫\n colimit.desc (diagram J P X.unop)\n ((Cocones.precompose (diagramNatTrans J η X.unop)).obj (colimit.cocone (diagram J Q X.unop)))) ≫\n (plusCompIso J F Q).hom.app X =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"state_before": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map (colimit.ι (diagram J P X.unop) W ≫ colimMap (diagramNatTrans J η X.unop)) ≫ (plusCompIso J F Q).hom.app X =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) b)) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"tactic": "dsimp [colimMap, IsColimit.map]"
},
{
"state_after": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map (((Cocones.precompose (diagramNatTrans J η X.unop)).obj (colimit.cocone (diagram J Q X.unop))).ι.app W) ≫\n (plusCompIso J F Q).hom.app X =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"state_before": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map\n (colimit.ι (diagram J P X.unop) W ≫\n colimit.desc (diagram J P X.unop)\n ((Cocones.precompose (diagramNatTrans J η X.unop)).obj (colimit.cocone (diagram J Q X.unop)))) ≫\n (plusCompIso J F Q).hom.app X =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"tactic": "simp only [colimit.ι_desc]"
},
{
"state_after": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i) ≫\n colimit.ι (diagram J Q X.unop) W) ≫\n (plusCompIso J F Q).hom.app X =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"state_before": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map (((Cocones.precompose (diagramNatTrans J η X.unop)).obj (colimit.cocone (diagram J Q X.unop))).ι.app W) ≫\n (plusCompIso J F Q).hom.app X =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"tactic": "dsimp [Cocones.precompose]"
},
{
"state_after": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n (diagramCompIso J F Q X.unop).hom.app W ≫ colimit.ι (diagram J (Q ⋙ F) X.unop) W =\n (diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"state_before": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i) ≫\n colimit.ι (diagram J Q X.unop) W) ≫\n (plusCompIso J F Q).hom.app X =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"tactic": "simp only [Functor.map_comp, Category.assoc, ι_plusCompIso_hom]"
},
{
"state_after": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ (F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n (diagramCompIso J F Q X.unop).hom.app W) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"state_before": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n (diagramCompIso J F Q X.unop).hom.app W ≫ colimit.ι (diagram J (Q ⋙ F) X.unop) W =\n (diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"tactic": "simp only [← Category.assoc]"
},
{
"state_after": "case w.h.e_a\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n (diagramCompIso J F Q X.unop).hom.app W =\n (diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)",
"state_before": "case w.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ (F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n (diagramCompIso J F Q X.unop).hom.app W) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n colimit.ι (diagram J (Q ⋙ F) X.unop) W",
"tactic": "congr 1"
},
{
"state_after": "case w.h.e_a.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ ∀ (a : (Cover.index W.unop (Q ⋙ F)).L),\n (F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n (diagramCompIso J F Q X.unop).hom.app W) ≫\n Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a",
"state_before": "case w.h.e_a\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n (diagramCompIso J F Q X.unop).hom.app W =\n (diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)",
"tactic": "apply Multiequalizer.hom_ext"
},
{
"state_after": "case w.h.e_a.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\na : (Cover.index W.unop (Q ⋙ F)).L\n⊢ (F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n (diagramCompIso J F Q X.unop).hom.app W) ≫\n Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a",
"state_before": "case w.h.e_a.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ ∀ (a : (Cover.index W.unop (Q ⋙ F)).L),\n (F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n (diagramCompIso J F Q X.unop).hom.app W) ≫\n Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a",
"tactic": "intro a"
},
{
"state_after": "case w.h.e_a.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\na : (Cover.index W.unop (Q ⋙ F)).L\n⊢ (F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n (diagramCompIso J F Q X.unop).hom.app W) ≫\n Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a",
"state_before": "case w.h.e_a.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\na : (Cover.index W.unop (Q ⋙ F)).L\n⊢ (F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n (diagramCompIso J F Q X.unop).hom.app W) ≫\n Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a",
"tactic": "dsimp"
},
{
"state_after": "case w.h.e_a.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\na : (Cover.index W.unop (Q ⋙ F)).L\n⊢ F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n F.map (Multiequalizer.ι (Cover.index W.unop Q) a) =\n F.map (Multiequalizer.ι (Cover.index W.unop P) a) ≫ F.map (η.app a.Y.op)",
"state_before": "case w.h.e_a.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\na : (Cover.index W.unop (Q ⋙ F)).L\n⊢ (F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n (diagramCompIso J F Q X.unop).hom.app W) ≫\n Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫\n Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a",
"tactic": "simp only [diagramCompIso_hom_ι_assoc, Multiequalizer.lift_ι, diagramCompIso_hom_ι,\n Category.assoc]"
},
{
"state_after": "no goals",
"state_before": "case w.h.e_a.h\nC : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP✝ : Cᵒᵖ ⥤ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\na : (Cover.index W.unop (Q ⋙ F)).L\n⊢ F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫\n F.map (Multiequalizer.ι (Cover.index W.unop Q) a) =\n F.map (Multiequalizer.ι (Cover.index W.unop P) a) ≫ F.map (η.app a.Y.op)",
"tactic": "simp only [← F.map_comp, Multiequalizer.lift_ι]"
}
] |
[
192,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.ssubset_iff_subset_ne
|
[] |
[
408,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
11
] |
Mathlib/Algebra/DirectLimit.lean
|
DirectedSystem.map_self
|
[] |
[
60,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.coe_zpow
|
[] |
[
620,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
619,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ico_inter_Ico_consecutive
|
[] |
[
606,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
605,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieSubmodule.sInf_coe_toSubmodule
|
[] |
[
422,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
Bornology.IsVonNBounded.image
|
[
{
"state_after": "𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\n⊢ IsVonNBounded 𝕜₂ (↑f '' s)",
"state_before": "𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\n⊢ IsVonNBounded 𝕜₂ (↑f '' s)",
"tactic": "let σ' := RingEquiv.ofBijective σ ⟨σ.injective, σ.surjective⟩"
},
{
"state_after": "𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\n⊢ IsVonNBounded 𝕜₂ (↑f '' s)",
"state_before": "𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\n⊢ IsVonNBounded 𝕜₂ (↑f '' s)",
"tactic": "have σ_iso : Isometry σ := AddMonoidHomClass.isometry_of_norm σ fun x => RingHomIsometric.is_iso"
},
{
"state_after": "𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\n⊢ IsVonNBounded 𝕜₂ (↑f '' s)",
"state_before": "𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\n⊢ IsVonNBounded 𝕜₂ (↑f '' s)",
"tactic": "have σ'_symm_iso : Isometry σ'.symm := σ_iso.right_inv σ'.right_inv"
},
{
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"tactic": "have f_tendsto_zero := f.continuous.tendsto 0"
},
{
"state_after": "𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\n⊢ IsVonNBounded 𝕜₂ (↑f '' s)",
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"tactic": "rw [map_zero] at f_tendsto_zero"
},
{
"state_after": "𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\n⊢ Absorbs 𝕜₂ V (↑f '' s)",
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"tactic": "intro V hV"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\n⊢ Absorbs 𝕜₂ V (↑f '' s)",
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"tactic": "rcases hs (f_tendsto_zero hV) with ⟨r, hrpos, hr⟩"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\n⊢ ↑f '' s ⊆ a • V",
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"tactic": "refine' ⟨r, hrpos, fun a ha => _⟩"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\n⊢ ↑f '' s ⊆ ↑σ' (↑(RingEquiv.symm σ') a) • V",
"state_before": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\n⊢ ↑f '' s ⊆ a • V",
"tactic": "rw [← σ'.apply_symm_apply a]"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\nhanz : a ≠ 0\n⊢ ↑f '' s ⊆ ↑σ' (↑(RingEquiv.symm σ') a) • V",
"state_before": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\n⊢ ↑f '' s ⊆ ↑σ' (↑(RingEquiv.symm σ') a) • V",
"tactic": "have hanz : a ≠ 0 := norm_pos_iff.mp (hrpos.trans_le ha)"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\nhanz : a ≠ 0\nthis : ↑(RingEquiv.symm σ') a ≠ 0\n⊢ ↑f '' s ⊆ ↑σ' (↑(RingEquiv.symm σ') a) • V",
"state_before": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\nhanz : a ≠ 0\n⊢ ↑f '' s ⊆ ↑σ' (↑(RingEquiv.symm σ') a) • V",
"tactic": "have : σ'.symm a ≠ 0 := (map_ne_zero σ'.symm.toRingHom).mpr hanz"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\nhanz : a ≠ 0\nthis : ↑(RingEquiv.symm σ') a ≠ 0\n⊢ ↑f '' s ⊆ ↑σ (↑(RingEquiv.symm σ') a) • V",
"state_before": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\nhanz : a ≠ 0\nthis : ↑(RingEquiv.symm σ') a ≠ 0\n⊢ ↑f '' s ⊆ ↑σ' (↑(RingEquiv.symm σ') a) • V",
"tactic": "change _ ⊆ σ _ • _"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\nhanz : a ≠ 0\nthis : ↑(RingEquiv.symm σ') a ≠ 0\n⊢ s ⊆ ↑(RingEquiv.symm σ') a • ↑f ⁻¹' V",
"state_before": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\nhanz : a ≠ 0\nthis : ↑(RingEquiv.symm σ') a ≠ 0\n⊢ ↑f '' s ⊆ ↑σ (↑(RingEquiv.symm σ') a) • V",
"tactic": "rw [Set.image_subset_iff, preimage_smul_setₛₗ _ _ _ f this.isUnit]"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\nhanz : a ≠ 0\nthis : ↑(RingEquiv.symm σ') a ≠ 0\n⊢ r ≤ ‖↑(RingEquiv.symm σ') a‖",
"state_before": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\nhanz : a ≠ 0\nthis : ↑(RingEquiv.symm σ') a ≠ 0\n⊢ s ⊆ ↑(RingEquiv.symm σ') a • ↑f ⁻¹' V",
"tactic": "refine' hr (σ'.symm a) _"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\n𝕜 : Type ?u.7414\n𝕜' : Type ?u.7417\nE : Type u_3\nE' : Type ?u.7423\nF : Type u_4\nι : Type ?u.7429\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSurjective σ\ninst✝ : RingHomIsometric σ\ns : Set E\nhs : IsVonNBounded 𝕜₁ s\nf : E →SL[σ] F\nσ' : 𝕜₁ ≃+* 𝕜₂ := RingEquiv.ofBijective σ (_ : Function.Injective ↑σ ∧ Function.Surjective ↑σ)\nσ_iso : Isometry ↑σ\nσ'_symm_iso : Isometry ↑(RingEquiv.symm σ')\nf_tendsto_zero : Tendsto (↑f) (𝓝 0) (𝓝 0)\nV : Set F\nhV : V ∈ 𝓝 0\nr : ℝ\nhrpos : 0 < r\nhr : ∀ (a : 𝕜₁), r ≤ ‖a‖ → s ⊆ a • ↑f ⁻¹' V\na : 𝕜₂\nha : r ≤ ‖a‖\nhanz : a ≠ 0\nthis : ↑(RingEquiv.symm σ') a ≠ 0\n⊢ r ≤ ‖↑(RingEquiv.symm σ') a‖",
"tactic": "rwa [σ'_symm_iso.norm_map_of_map_zero (map_zero _)]"
}
] |
[
140,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
|
CategoryTheory.Limits.lift_comp_kernelIsoOfEq_inv
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasKernel f✝\nZ : C\nf g : X ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasKernel g\nh : f = g\ne : Z ⟶ X\nhe : e ≫ g = 0\n⊢ e ≫ f = 0",
"tactic": "simp [h, he]"
},
{
"state_after": "case refl\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasKernel f✝\nZ : C\nf : X ⟶ Y\ninst✝¹ : HasKernel f\ne : Z ⟶ X\ninst✝ : HasKernel f\nhe : e ≫ f = 0\n⊢ kernel.lift f e he ≫ (kernelIsoOfEq (_ : f = f)).inv = kernel.lift f e (_ : e ≫ f = 0)",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasKernel f✝\nZ : C\nf g : X ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasKernel g\nh : f = g\ne : Z ⟶ X\nhe : e ≫ g = 0\n⊢ kernel.lift g e he ≫ (kernelIsoOfEq h).inv = kernel.lift f e (_ : e ≫ f = 0)",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasKernel f✝\nZ : C\nf : X ⟶ Y\ninst✝¹ : HasKernel f\ne : Z ⟶ X\ninst✝ : HasKernel f\nhe : e ≫ f = 0\n⊢ kernel.lift f e he ≫ (kernelIsoOfEq (_ : f = f)).inv = kernel.lift f e (_ : e ≫ f = 0)",
"tactic": "simp"
}
] |
[
379,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
376,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.sInf_empty
|
[] |
[
793,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
792,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.lt_limit
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.91204\nβ : Type ?u.91207\nγ : Type ?u.91210\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nh : IsLimit o\na : Ordinal\n⊢ a < o ↔ ∃ x, x < o ∧ a < x",
"tactic": "simpa only [not_ball, not_le, bex_def] using not_congr (@limit_le _ h a)"
}
] |
[
275,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/SetTheory/Cardinal/Continuum.lean
|
Cardinal.nat_power_aleph0
|
[] |
[
192,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
QuadraticForm.polar_zero_left
|
[
{
"state_after": "no goals",
"state_before": "S : Type ?u.144337\nR : Type u_1\nR₁ : Type ?u.144343\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : CommRing R₁\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\ny : M\n⊢ polar (↑Q) 0 y = 0",
"tactic": "simp only [polar, zero_add, QuadraticForm.map_zero, sub_zero, sub_self]"
}
] |
[
262,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Order/CompactlyGenerated.lean
|
CompleteLattice.setIndependent_iUnion_of_directed
|
[
{
"state_after": "case pos\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : Nonempty η\n⊢ SetIndependent (⋃ (i : η), s i)\n\ncase neg\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : ¬Nonempty η\n⊢ SetIndependent (⋃ (i : η), s i)",
"state_before": "ι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\n⊢ SetIndependent (⋃ (i : η), s i)",
"tactic": "by_cases hη : Nonempty η"
},
{
"state_after": "case pos\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : Nonempty η\n⊢ ∀ (t : Finset α), (↑t ⊆ ⋃ (i : η), s i) → SetIndependent ↑t",
"state_before": "case pos\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : Nonempty η\n⊢ SetIndependent (⋃ (i : η), s i)",
"tactic": "rw [CompleteLattice.setIndependent_iff_finite]"
},
{
"state_after": "case pos\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : Nonempty η\nt : Finset α\nht : ↑t ⊆ ⋃ (i : η), s i\n⊢ SetIndependent ↑t",
"state_before": "case pos\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : Nonempty η\n⊢ ∀ (t : Finset α), (↑t ⊆ ⋃ (i : η), s i) → SetIndependent ↑t",
"tactic": "intro t ht"
},
{
"state_after": "case pos.intro.intro\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : Nonempty η\nt : Finset α\nht : ↑t ⊆ ⋃ (i : η), s i\nI : Set η\nfi : Set.Finite I\nhI : ↑t ⊆ ⋃ (i : η) (_ : i ∈ I), s i\n⊢ SetIndependent ↑t",
"state_before": "case pos\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : Nonempty η\nt : Finset α\nht : ↑t ⊆ ⋃ (i : η), s i\n⊢ SetIndependent ↑t",
"tactic": "obtain ⟨I, fi, hI⟩ := Set.finite_subset_iUnion t.finite_toSet ht"
},
{
"state_after": "case pos.intro.intro.intro\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : Nonempty η\nt : Finset α\nht : ↑t ⊆ ⋃ (i : η), s i\nI : Set η\nfi : Set.Finite I\nhI : ↑t ⊆ ⋃ (i : η) (_ : i ∈ I), s i\ni : η\nhi : ∀ (i_1 : η), i_1 ∈ Set.Finite.toFinset fi → s i_1 ⊆ s i\n⊢ SetIndependent ↑t",
"state_before": "case pos.intro.intro\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : Nonempty η\nt : Finset α\nht : ↑t ⊆ ⋃ (i : η), s i\nI : Set η\nfi : Set.Finite I\nhI : ↑t ⊆ ⋃ (i : η) (_ : i ∈ I), s i\n⊢ SetIndependent ↑t",
"tactic": "obtain ⟨i, hi⟩ := hs.finset_le fi.toFinset"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro.intro\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : Nonempty η\nt : Finset α\nht : ↑t ⊆ ⋃ (i : η), s i\nI : Set η\nfi : Set.Finite I\nhI : ↑t ⊆ ⋃ (i : η) (_ : i ∈ I), s i\ni : η\nhi : ∀ (i_1 : η), i_1 ∈ Set.Finite.toFinset fi → s i_1 ⊆ s i\n⊢ SetIndependent ↑t",
"tactic": "exact (h i).mono\n (Set.Subset.trans hI <| Set.iUnion₂_subset fun j hj => hi j (fi.mem_toFinset.2 hj))"
},
{
"state_after": "case neg.intro.intro.intro\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na✝ b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : ¬Nonempty η\na : α\nw✝ : Set α\nright✝ : a ∈ w✝\ni : η\nh✝ : (fun i => s i) i = w✝\n⊢ Disjoint a (sSup ((⋃ (i : η), s i) \\ {a}))",
"state_before": "case neg\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : ¬Nonempty η\n⊢ SetIndependent (⋃ (i : η), s i)",
"tactic": "rintro a ⟨_, ⟨i, _⟩, _⟩"
},
{
"state_after": "case neg.intro.intro.intro.h\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na✝ b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : ¬Nonempty η\na : α\nw✝ : Set α\nright✝ : a ∈ w✝\ni : η\nh✝ : (fun i => s i) i = w✝\n⊢ False",
"state_before": "case neg.intro.intro.intro\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na✝ b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : ¬Nonempty η\na : α\nw✝ : Set α\nright✝ : a ∈ w✝\ni : η\nh✝ : (fun i => s i) i = w✝\n⊢ Disjoint a (sSup ((⋃ (i : η), s i) \\ {a}))",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.h\nι : Sort ?u.107199\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na✝ b : α\ns✝ : Set α\nη : Type u_1\ns : η → Set α\nhs : Directed (fun x x_1 => x ⊆ x_1) s\nh : ∀ (i : η), SetIndependent (s i)\nhη : ¬Nonempty η\na : α\nw✝ : Set α\nright✝ : a ∈ w✝\ni : η\nh✝ : (fun i => s i) i = w✝\n⊢ False",
"tactic": "exact hη ⟨i⟩"
}
] |
[
455,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.add_le_add
|
[] |
[
792,
70
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
791,
11
] |
Mathlib/Analysis/LocallyConvex/Basic.lean
|
balanced_univ
|
[] |
[
179,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.set_support_inv_eq
|
[
{
"state_after": "case h\nα : Type u_1\np q : Perm α\nx : α\n⊢ x ∈ {x | ↑p⁻¹ x ≠ x} ↔ x ∈ {x | ↑p x ≠ x}",
"state_before": "α : Type u_1\np q : Perm α\n⊢ {x | ↑p⁻¹ x ≠ x} = {x | ↑p x ≠ x}",
"tactic": "ext x"
},
{
"state_after": "case h\nα : Type u_1\np q : Perm α\nx : α\n⊢ ¬↑p⁻¹ x = x ↔ ¬↑p x = x",
"state_before": "case h\nα : Type u_1\np q : Perm α\nx : α\n⊢ x ∈ {x | ↑p⁻¹ x ≠ x} ↔ x ∈ {x | ↑p x ≠ x}",
"tactic": "simp only [Set.mem_setOf_eq, Ne.def]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\np q : Perm α\nx : α\n⊢ ¬↑p⁻¹ x = x ↔ ¬↑p x = x",
"tactic": "rw [inv_def, symm_apply_eq, eq_comm]"
}
] |
[
263,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.IsCycle.exists_pow_eq
|
[
{
"state_after": "ι : Type ?u.377965\nα : Type u_1\nβ : Type ?u.377971\nf g : Perm α\nx y : α\ninst✝ : Finite α\nhf : IsCycle f\nhx : ↑f x ≠ x\nhy : ↑f y ≠ y\nn : ℤ\nhn : ↑(f ^ n) x = y\n⊢ ∃ i, ↑(f ^ i) x = y",
"state_before": "ι : Type ?u.377965\nα : Type u_1\nβ : Type ?u.377971\nf g : Perm α\nx y : α\ninst✝ : Finite α\nhf : IsCycle f\nhx : ↑f x ≠ x\nhy : ↑f y ≠ y\n⊢ ∃ i, ↑(f ^ i) x = y",
"tactic": "let ⟨n, hn⟩ := hf.exists_zpow_eq hx hy"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.377965\nα : Type u_1\nβ : Type ?u.377971\nf g : Perm α\nx y : α\ninst✝ : Finite α\nhf : IsCycle f\nhx : ↑f x ≠ x\nhy : ↑f y ≠ y\nn : ℤ\nhn : ↑(f ^ n) x = y\n⊢ ∃ i, ↑(f ^ i) x = y",
"tactic": "classical exact\n ⟨(n % orderOf f).toNat, by\n {have := n.emod_nonneg (Int.coe_nat_ne_zero.mpr (ne_of_gt (orderOf_pos f)))\n rwa [← zpow_ofNat, Int.toNat_of_nonneg this, ← zpow_eq_mod_orderOf]}⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.377965\nα : Type u_1\nβ : Type ?u.377971\nf g : Perm α\nx y : α\ninst✝ : Finite α\nhf : IsCycle f\nhx : ↑f x ≠ x\nhy : ↑f y ≠ y\nn : ℤ\nhn : ↑(f ^ n) x = y\n⊢ ∃ i, ↑(f ^ i) x = y",
"tactic": "exact\n⟨(n % orderOf f).toNat, by\n{have := n.emod_nonneg (Int.coe_nat_ne_zero.mpr (ne_of_gt (orderOf_pos f)))\nrwa [← zpow_ofNat, Int.toNat_of_nonneg this, ← zpow_eq_mod_orderOf]}⟩"
}
] |
[
360,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iSup_of_empty'
|
[] |
[
1499,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1498,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.isCaratheodory_compl_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nm : OuterMeasure α\ns s₁ s₂ : Set α\nh : IsCaratheodory m (sᶜ)\n⊢ IsCaratheodory m s",
"tactic": "simpa using isCaratheodory_compl m h"
}
] |
[
963,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
962,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.EqOn.image_eq
|
[] |
[
214,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
AddLECancellable.tsub_add_tsub_comm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nhd : AddLECancellable d\nhba : b ≤ a\nhdc : d ≤ c\n⊢ a - b + (c - d) = a + c - (b + d)",
"tactic": "rw [hb.tsub_add_eq_add_tsub hba, ← hd.add_tsub_assoc_of_le hdc, tsub_tsub, add_comm d]"
}
] |
[
106,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
11
] |
Mathlib/Algebra/Lie/BaseChange.lean
|
LieAlgebra.ExtendScalars.bracket_tmul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type w\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns t : A\nx y : L\n⊢ ⁅s ⊗ₜ[R] x, t ⊗ₜ[R] y⁆ = (s * t) ⊗ₜ[R] ⁅x, y⁆",
"tactic": "rw [bracket_def, bracket'_tmul]"
}
] |
[
63,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegrable_const_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.927792\n𝕜 : Type ?u.927795\nE : Type u_1\nF : Type ?u.927801\nA : Type ?u.927804\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nc : E\n⊢ IntervalIntegrable (fun x => c) μ a b ↔ c = 0 ∨ ↑↑μ (Ι a b) < ⊤",
"tactic": "simp only [intervalIntegrable_iff, integrableOn_const]"
}
] |
[
124,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Set.iUnion_eq_iUnion_finset'
|
[] |
[
1863,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1861,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
nhds_eq_uniformity
|
[] |
[
789,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
788,
1
] |
Mathlib/Order/Partition/Finpartition.lean
|
Finpartition.card_extend
|
[] |
[
426,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/RingTheory/Subsemiring/Pointwise.lean
|
Subsemiring.pointwise_smul_toAddSubmonoid
|
[] |
[
68,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieModuleEquiv.symm_apply_apply
|
[] |
[
1075,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1074,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.natAbs_of_nonneg
|
[] |
[
1245,
23
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1243,
1
] |
Mathlib/Data/Int/ModEq.lean
|
Int.modEq_neg
|
[
{
"state_after": "no goals",
"state_before": "m n a b c d : ℤ\n⊢ a ≡ b [ZMOD -n] ↔ a ≡ b [ZMOD n]",
"tactic": "simp [modEq_iff_dvd]"
}
] |
[
122,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Data/Fintype/Prod.lean
|
infinite_prod
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : Infinite (α × β)\n⊢ Infinite α ∧ Nonempty β ∨ Nonempty α ∧ Infinite β",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\n⊢ Infinite (α × β) ↔ Infinite α ∧ Nonempty β ∨ Nonempty α ∧ Infinite β",
"tactic": "refine'\n ⟨fun H => _, fun H =>\n H.elim (and_imp.2 <| @Prod.infinite_of_left α β) (and_imp.2 <| @Prod.infinite_of_right α β)⟩"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : Infinite (α × β)\n⊢ Nonempty β ∧ Infinite α ∨ Nonempty α ∧ Infinite β",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : Infinite (α × β)\n⊢ Infinite α ∧ Nonempty β ∨ Nonempty α ∧ Infinite β",
"tactic": "rw [and_comm]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : (Nonempty β → ¬Infinite α) ∧ (Nonempty α → ¬Infinite β)\n⊢ ¬Infinite (α × β)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : Infinite (α × β)\n⊢ Nonempty β ∧ Infinite α ∨ Nonempty α ∧ Infinite β",
"tactic": "contrapose! H"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : (Nonempty β → ¬Infinite α) ∧ (Nonempty α → ¬Infinite β)\nH' : Infinite (α × β)\n⊢ False",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : (Nonempty β → ¬Infinite α) ∧ (Nonempty α → ¬Infinite β)\n⊢ ¬Infinite (α × β)",
"tactic": "intro H'"
},
{
"state_after": "case intro.mk\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : (Nonempty β → ¬Infinite α) ∧ (Nonempty α → ¬Infinite β)\nH' : Infinite (α × β)\na : α\nb : β\n⊢ False",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : (Nonempty β → ¬Infinite α) ∧ (Nonempty α → ¬Infinite β)\nH' : Infinite (α × β)\n⊢ False",
"tactic": "rcases Infinite.nonempty (α × β) with ⟨a, b⟩"
},
{
"state_after": "case intro.mk\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : (Nonempty β → ¬Infinite α) ∧ (Nonempty α → ¬Infinite β)\nH' : Infinite (α × β)\na : α\nb : β\nthis : Fintype α\n⊢ False",
"state_before": "case intro.mk\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : (Nonempty β → ¬Infinite α) ∧ (Nonempty α → ¬Infinite β)\nH' : Infinite (α × β)\na : α\nb : β\n⊢ False",
"tactic": "haveI := fintypeOfNotInfinite (H.1 ⟨b⟩)"
},
{
"state_after": "case intro.mk\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : (Nonempty β → ¬Infinite α) ∧ (Nonempty α → ¬Infinite β)\nH' : Infinite (α × β)\na : α\nb : β\nthis✝ : Fintype α\nthis : Fintype β\n⊢ False",
"state_before": "case intro.mk\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : (Nonempty β → ¬Infinite α) ∧ (Nonempty α → ¬Infinite β)\nH' : Infinite (α × β)\na : α\nb : β\nthis : Fintype α\n⊢ False",
"tactic": "haveI := fintypeOfNotInfinite (H.2 ⟨a⟩)"
},
{
"state_after": "no goals",
"state_before": "case intro.mk\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.4496\nH : (Nonempty β → ¬Infinite α) ∧ (Nonempty α → ¬Infinite β)\nH' : Infinite (α × β)\na : α\nb : β\nthis✝ : Fintype α\nthis : Fintype β\n⊢ False",
"tactic": "exact H'.false"
}
] |
[
74,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.continuous_integral_integral
|
[
{
"state_after": "α : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\n⊢ ∀ (x : { x // x ∈ Lp E 1 }), ContinuousAt (fun f => ∫ (x : α), ∫ (y : β), ↑↑f (x, y) ∂ν ∂μ) x",
"state_before": "α : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\n⊢ Continuous fun f => ∫ (x : α), ∫ (y : β), ↑↑f (x, y) ∂ν ∂μ",
"tactic": "rw [continuous_iff_continuousAt]"
},
{
"state_after": "α : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ ContinuousAt (fun f => ∫ (x : α), ∫ (y : β), ↑↑f (x, y) ∂ν ∂μ) g",
"state_before": "α : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\n⊢ ∀ (x : { x // x ∈ Lp E 1 }), ContinuousAt (fun f => ∫ (x : α), ∫ (y : β), ↑↑f (x, y) ∂ν ∂μ) x",
"tactic": "intro g"
},
{
"state_after": "α : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ Tendsto (fun i => ∫⁻ (x : α), ↑‖(∫ (y : β), ↑↑i (x, y) ∂ν) - ∫ (y : β), ↑↑g (x, y) ∂ν‖₊ ∂μ) (𝓝 g) (𝓝 0)",
"state_before": "α : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ ContinuousAt (fun f => ∫ (x : α), ∫ (y : β), ↑↑f (x, y) ∂ν ∂μ) g",
"tactic": "refine'\n tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_prod_left\n (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_prod_left) _"
},
{
"state_after": "α : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ Tendsto (fun i => ∫⁻ (x : α), ↑‖∫ (y : β), ↑↑i (x, y) - ↑↑g (x, y) ∂ν‖₊ ∂μ) (𝓝 g) (𝓝 0)",
"state_before": "α : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ Tendsto (fun i => ∫⁻ (x : α), ↑‖(∫ (y : β), ↑↑i (x, y) ∂ν) - ∫ (y : β), ↑↑g (x, y) ∂ν‖₊ ∂μ) (𝓝 g) (𝓝 0)",
"tactic": "simp_rw [←\n lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)\n (L1.integrable_coeFn g)]"
},
{
"state_after": "case refine'_1\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ { x // x ∈ Lp E 1 } → ℝ≥0∞\n\ncase refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ Tendsto ?refine'_1 (𝓝 g) (𝓝 0)\n\ncase refine'_3\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ (fun i => ∫⁻ (x : α), ↑‖∫ (y : β), ↑↑i (x, y) - ↑↑g (x, y) ∂ν‖₊ ∂μ) ≤ ?refine'_1",
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"tactic": "refine' tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _"
},
{
"state_after": "case refine'_3\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ (fun i => ∫⁻ (x : α), ↑‖∫ (y : β), ↑↑i (x, y) - ↑↑g (x, y) ∂ν‖₊ ∂μ) ≤ fun i =>\n ∫⁻ (x : α), ∫⁻ (y : β), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂ν ∂μ\n\ncase refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ Tendsto (fun i => ∫⁻ (x : α), ∫⁻ (y : β), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂ν ∂μ) (𝓝 g) (𝓝 0)",
"state_before": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ Tendsto (fun i => ∫⁻ (x : α), ∫⁻ (y : β), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂ν ∂μ) (𝓝 g) (𝓝 0)\n\ncase refine'_3\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ (fun i => ∫⁻ (x : α), ↑‖∫ (y : β), ↑↑i (x, y) - ↑↑g (x, y) ∂ν‖₊ ∂μ) ≤ fun i =>\n ∫⁻ (x : α), ∫⁻ (y : β), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂ν ∂μ",
"tactic": "swap"
},
{
"state_after": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ Tendsto (fun i => ∫⁻ (x : α), ∫⁻ (y : β), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂ν ∂μ) (𝓝 g) (𝓝 0)",
"state_before": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ Tendsto (fun i => ∫⁻ (x : α), ∫⁻ (y : β), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂ν ∂μ) (𝓝 g) (𝓝 0)",
"tactic": "show\n Tendsto (fun i : α × β →₁[μ.prod ν] E => ∫⁻ x, ∫⁻ y : β, ‖i (x, y) - g (x, y)‖₊ ∂ν ∂μ) (𝓝 g)\n (𝓝 0)"
},
{
"state_after": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\nthis : ∀ (i : { x // x ∈ Lp E 1 }), Measurable fun z => ↑‖↑↑i z - ↑↑g z‖₊\n⊢ Tendsto (fun i => ∫⁻ (x : α), ∫⁻ (y : β), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂ν ∂μ) (𝓝 g) (𝓝 0)",
"state_before": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ Tendsto (fun i => ∫⁻ (x : α), ∫⁻ (y : β), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂ν ∂μ) (𝓝 g) (𝓝 0)",
"tactic": "have : ∀ i : α × β →₁[μ.prod ν] E, Measurable fun z => (‖i z - g z‖₊ : ℝ≥0∞) := fun i =>\n ((Lp.stronglyMeasurable i).sub (Lp.stronglyMeasurable g)).ennnorm"
},
{
"state_after": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\nthis : ∀ (i : { x // x ∈ Lp E 1 }), Measurable fun z => ↑‖↑↑i z - ↑↑g z‖₊\n⊢ Tendsto (fun x => ENNReal.ofReal ‖x - g‖) (𝓝 g) (𝓝 0)",
"state_before": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\nthis : ∀ (i : { x // x ∈ Lp E 1 }), Measurable fun z => ↑‖↑↑i z - ↑↑g z‖₊\n⊢ Tendsto (fun i => ∫⁻ (x : α), ∫⁻ (y : β), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂ν ∂μ) (𝓝 g) (𝓝 0)",
"tactic": "conv =>\n congr\n ext\n rw [← lintegral_prod_of_measurable _ (this _), ← L1.ofReal_norm_sub_eq_lintegral]"
},
{
"state_after": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\nthis : ∀ (i : { x // x ∈ Lp E 1 }), Measurable fun z => ↑‖↑↑i z - ↑↑g z‖₊\n⊢ Tendsto (fun x => ENNReal.ofReal ‖x - g‖) (𝓝 g) (𝓝 (ENNReal.ofReal 0))",
"state_before": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\nthis : ∀ (i : { x // x ∈ Lp E 1 }), Measurable fun z => ↑‖↑↑i z - ↑↑g z‖₊\n⊢ Tendsto (fun x => ENNReal.ofReal ‖x - g‖) (𝓝 g) (𝓝 0)",
"tactic": "rw [← ofReal_zero]"
},
{
"state_after": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\nthis : ∀ (i : { x // x ∈ Lp E 1 }), Measurable fun z => ↑‖↑↑i z - ↑↑g z‖₊\n⊢ Tendsto (fun x => ‖x - g‖) (𝓝 g) (𝓝 0)",
"state_before": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\nthis : ∀ (i : { x // x ∈ Lp E 1 }), Measurable fun z => ↑‖↑↑i z - ↑↑g z‖₊\n⊢ Tendsto (fun x => ENNReal.ofReal ‖x - g‖) (𝓝 g) (𝓝 (ENNReal.ofReal 0))",
"tactic": "refine' (continuous_ofReal.tendsto 0).comp _"
},
{
"state_after": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\nthis : ∀ (i : { x // x ∈ Lp E 1 }), Measurable fun z => ↑‖↑↑i z - ↑↑g z‖₊\n⊢ Tendsto (fun x => x) (𝓝 g) (𝓝 g)",
"state_before": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\nthis : ∀ (i : { x // x ∈ Lp E 1 }), Measurable fun z => ↑‖↑↑i z - ↑↑g z‖₊\n⊢ Tendsto (fun x => ‖x - g‖) (𝓝 g) (𝓝 0)",
"tactic": "rw [← tendsto_iff_norm_tendsto_zero]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\nthis : ∀ (i : { x // x ∈ Lp E 1 }), Measurable fun z => ↑‖↑↑i z - ↑↑g z‖₊\n⊢ Tendsto (fun x => x) (𝓝 g) (𝓝 g)",
"tactic": "exact tendsto_id"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ { x // x ∈ Lp E 1 } → ℝ≥0∞",
"tactic": "exact fun i => ∫⁻ x, ∫⁻ y, ‖i (x, y) - g (x, y)‖₊ ∂ν ∂μ"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα : Type u_1\nα' : Type ?u.2437594\nβ : Type u_2\nβ' : Type ?u.2437600\nγ : Type ?u.2437603\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2437868\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\ng : { x // x ∈ Lp E 1 }\n⊢ (fun i => ∫⁻ (x : α), ↑‖∫ (y : β), ↑↑i (x, y) - ↑↑g (x, y) ∂ν‖₊ ∂μ) ≤ fun i =>\n ∫⁻ (x : α), ∫⁻ (y : β), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂ν ∂μ",
"tactic": "exact fun i => lintegral_mono fun x => ennnorm_integral_le_lintegral_ennnorm _"
}
] |
[
448,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
422,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
AlternatingMap.compLinearMap_injective
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹⁴ : Semiring R\nM : Type u_3\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Module R M\nN : Type u_4\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : Module R N\nP : Type ?u.370920\ninst✝⁹ : AddCommMonoid P\ninst✝⁸ : Module R P\nM' : Type ?u.370950\ninst✝⁷ : AddCommGroup M'\ninst✝⁶ : Module R M'\nN' : Type ?u.371338\ninst✝⁵ : AddCommGroup N'\ninst✝⁴ : Module R N'\nι : Type u_5\nι' : Type ?u.371729\nι'' : Type ?u.371732\nM₂ : Type u_2\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₂\nM₃ : Type ?u.371765\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Module R M₃\nf : M₂ →ₗ[R] M\nhf : Function.Surjective ↑f\ng₁ g₂ : AlternatingMap R M N ι\nh : (fun g => compLinearMap g f) g₁ = (fun g => compLinearMap g f) g₂\nx : ι → M\n⊢ ↑g₁ x = ↑g₂ x",
"tactic": "simpa [Function.surjInv_eq hf] using ext_iff.mp h (Function.surjInv hf ∘ x)"
}
] |
[
594,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
592,
1
] |
Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
|
ModuleCat.MonoidalCategory.rightUnitor_inv_apply
|
[] |
[
243,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
LinearMap.compAlternatingMap_codRestrict
|
[] |
[
527,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/MeasureTheory/Function/Jacobian.lean
|
exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt
|
[
{
"state_after": "case inl\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\nf' : E → E →L[ℝ] F\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhf' : ∀ (x : E), x ∈ ∅ → HasFDerivWithinAt f (f' x) ∅ x\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (∅ ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (∅ ∩ t n) (r (A n))) ∧\n (Set.Nonempty ∅ → ∀ (n : ℕ), ∃ y, y ∈ ∅ ∧ A n = f' y)\n\ncase inr\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"tactic": "rcases eq_empty_or_nonempty s with (rfl | hs)"
},
{
"state_after": "case inr.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"state_before": "case inr\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"tactic": "obtain ⟨T, T_count, hT⟩ :\n ∃ T : Set s,\n T.Countable ∧ (⋃ x ∈ T, ball (f' (x : E)) (r (f' x))) = ⋃ x : s, ball (f' x) (r (f' x)) :=\n TopologicalSpace.isOpen_iUnion_countable _ fun x => isOpen_ball"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"state_before": "case inr.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"tactic": "obtain ⟨u, _, u_pos, u_lim⟩ :\n ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=\n exists_seq_strictAnti_tendsto (0 : ℝ)"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"state_before": "case inr.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"tactic": "let M : ℕ → T → Set E := fun n z =>\n {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - f' z (y - x)‖ ≤ r (f' z) * ‖y - x‖}"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"state_before": "case inr.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"tactic": "have s_subset : ∀ x ∈ s, ∃ (n : ℕ) (z : T), x ∈ M n z := by\n intro x xs\n obtain ⟨z, zT, hz⟩ : ∃ z ∈ T, f' x ∈ ball (f' (z : E)) (r (f' z)) := by\n have : f' x ∈ ⋃ z ∈ T, ball (f' (z : E)) (r (f' z)) := by\n rw [hT]\n refine' mem_iUnion.2 ⟨⟨x, xs⟩, _⟩\n simpa only [mem_ball, Subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt\n rwa [mem_iUnion₂, bex_def] at this\n obtain ⟨ε, εpos, hε⟩ : ∃ ε : ℝ, 0 < ε ∧ ‖f' x - f' z‖ + ε ≤ r (f' z) := by\n refine' ⟨r (f' z) - ‖f' x - f' z‖, _, le_of_eq (by abel)⟩\n simpa only [sub_pos] using mem_ball_iff_norm.mp hz\n obtain ⟨δ, δpos, hδ⟩ :\n ∃ (δ : ℝ), 0 < δ ∧ ball x δ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖} :=\n Metric.mem_nhdsWithin_iff.1 (IsLittleO.def (hf' x xs) εpos)\n obtain ⟨n, hn⟩ : ∃ n, u n < δ := ((tendsto_order.1 u_lim).2 _ δpos).exists\n refine' ⟨n, ⟨z, zT⟩, ⟨xs, _⟩⟩\n intro y hy\n calc\n ‖f y - f x - (f' z) (y - x)‖ = ‖f y - f x - (f' x) (y - x) + (f' x - f' z) (y - x)‖ := by\n congr 1\n simp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply]\n abel\n _ ≤ ‖f y - f x - (f' x) (y - x)‖ + ‖(f' x - f' z) (y - x)‖ := (norm_add_le _ _)\n _ ≤ ε * ‖y - x‖ + ‖f' x - f' z‖ * ‖y - x‖ := by\n refine' add_le_add (hδ _) (ContinuousLinearMap.le_op_norm _ _)\n rw [inter_comm]\n exact inter_subset_inter_right _ (ball_subset_ball hn.le) hy\n _ ≤ r (f' z) * ‖y - x‖ := by\n rw [← add_mul, add_comm]\n exact mul_le_mul_of_nonneg_right hε (norm_nonneg _)"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"state_before": "case inr.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"tactic": "have closure_M_subset : ∀ n z, s ∩ closure (M n z) ⊆ M n z := by\n rintro n z x ⟨xs, hx⟩\n refine' ⟨xs, fun y hy => _⟩\n obtain ⟨a, aM, a_lim⟩ : ∃ a : ℕ → E, (∀ k, a k ∈ M n z) ∧ Tendsto a atTop (𝓝 x) :=\n mem_closure_iff_seq_limit.1 hx\n have L1 :\n Tendsto (fun k : ℕ => ‖f y - f (a k) - (f' z) (y - a k)‖) atTop\n (𝓝 ‖f y - f x - (f' z) (y - x)‖) := by\n apply Tendsto.norm\n have L : Tendsto (fun k => f (a k)) atTop (𝓝 (f x)) := by\n apply (hf' x xs).continuousWithinAt.tendsto.comp\n apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ a_lim\n exact eventually_of_forall fun k => (aM k).1\n apply Tendsto.sub (tendsto_const_nhds.sub L)\n exact ((f' z).continuous.tendsto _).comp (tendsto_const_nhds.sub a_lim)\n have L2 : Tendsto (fun k : ℕ => (r (f' z) : ℝ) * ‖y - a k‖) atTop (𝓝 (r (f' z) * ‖y - x‖)) :=\n (tendsto_const_nhds.sub a_lim).norm.const_mul _\n have I : ∀ᶠ k in atTop, ‖f y - f (a k) - (f' z) (y - a k)‖ ≤ r (f' z) * ‖y - a k‖ := by\n have L : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x)) :=\n tendsto_const_nhds.dist a_lim\n filter_upwards [(tendsto_order.1 L).2 _ hy.2]\n intro k hk\n exact (aM k).2 y ⟨hy.1, hk⟩\n exact le_of_tendsto_of_tendsto L1 L2 I"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"state_before": "case inr.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"tactic": "rcases TopologicalSpace.exists_dense_seq E with ⟨d, hd⟩"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"state_before": "case inr.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"tactic": "let K : ℕ → T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"state_before": "case inr.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"tactic": "have K_approx : ∀ (n) (z : T) (p), ApproximatesLinearOn f (f' z) (s ∩ K n z p) (r (f' z)) := by\n intro n z p x hx y hy\n have yM : y ∈ M n z := closure_M_subset _ _ ⟨hy.1, hy.2.1⟩\n refine' yM.2 _ ⟨hx.1, _⟩\n calc\n dist x y ≤ dist x (d p) + dist y (d p) := dist_triangle_right _ _ _\n _ ≤ u n / 3 + u n / 3 := (add_le_add hx.2.2 hy.2.2)\n _ < u n := by linarith [u_pos n]"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"state_before": "case inr.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"tactic": "have K_closed : ∀ (n) (z : T) (p), IsClosed (K n z p) := fun n z p =>\n isClosed_closure.inter isClosed_ball"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\n⊢ x ∈ ⋃ (n : ℕ), (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) n",
"state_before": "case inr.intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (s ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧\n (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y)",
"tactic": "refine'\n ⟨fun q => K (F q).1 (F q).2.1 (F q).2.2, fun q => f' (F q).2.1, fun n => K_closed _ _ _,\n fun x xs => _, fun q => K_approx _ _ _, fun _ q => ⟨(F q).2.1, (F q).2.1.1.2, rfl⟩⟩"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\n⊢ x ∈ ⋃ (n : ℕ), (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) n",
"state_before": "case inr.intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\n⊢ x ∈ ⋃ (n : ℕ), (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) n",
"tactic": "obtain ⟨n, z, hnz⟩ : ∃ (n : ℕ) (z : T), x ∈ M n z := s_subset x xs"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\np : ℕ\nhp : x ∈ closedBall (d p) (u n / 3)\n⊢ x ∈ ⋃ (n : ℕ), (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) n",
"state_before": "case inr.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\n⊢ x ∈ ⋃ (n : ℕ), (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) n",
"tactic": "obtain ⟨p, hp⟩ : ∃ p : ℕ, x ∈ closedBall (d p) (u n / 3) := by\n have : Set.Nonempty (ball x (u n / 3)) := by simp only [nonempty_ball]; linarith [u_pos n]\n obtain ⟨p, hp⟩ : ∃ p : ℕ, d p ∈ ball x (u n / 3) := hd.exists_mem_open isOpen_ball this\n exact ⟨p, (mem_ball'.1 hp).le⟩"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\np : ℕ\nhp : x ∈ closedBall (d p) (u n / 3)\nq : ℕ\nhq : F q = (n, z, p)\n⊢ x ∈ ⋃ (n : ℕ), (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) n",
"state_before": "case inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\np : ℕ\nhp : x ∈ closedBall (d p) (u n / 3)\n⊢ x ∈ ⋃ (n : ℕ), (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) n",
"tactic": "obtain ⟨q, hq⟩ : ∃ q, F q = (n, z, p) := hF _"
},
{
"state_after": "E : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\np : ℕ\nhp : x ∈ closedBall (d p) (u n / 3)\nq : ℕ\nhq : F q = (n, z, p)\n⊢ x ∈ (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) q",
"state_before": "case inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\np : ℕ\nhp : x ∈ closedBall (d p) (u n / 3)\nq : ℕ\nhq : F q = (n, z, p)\n⊢ x ∈ ⋃ (n : ℕ), (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) n",
"tactic": "apply mem_iUnion.2 ⟨q, _⟩"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\np : ℕ\nhp : x ∈ closedBall (d p) (u n / 3)\nq : ℕ\nhq : F q = (n, z, p)\n⊢ x ∈ (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) q",
"tactic": "simp (config := { zeta := false }) only [hq, subset_closure hnz, hp, mem_inter_iff, and_true, hnz]"
},
{
"state_after": "no goals",
"state_before": "case inl\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\nf' : E → E →L[ℝ] F\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhf' : ∀ (x : E), x ∈ ∅ → HasFDerivWithinAt f (f' x) ∅ x\n⊢ ∃ t A,\n (∀ (n : ℕ), IsClosed (t n)) ∧\n (∅ ⊆ ⋃ (n : ℕ), t n) ∧\n (∀ (n : ℕ), ApproximatesLinearOn f (A n) (∅ ∩ t n) (r (A n))) ∧\n (Set.Nonempty ∅ → ∀ (n : ℕ), ∃ y, y ∈ ∅ ∧ A n = f' y)",
"tactic": "refine' ⟨fun _ => ∅, fun _ => 0, _, _, _, _⟩ <;> simp"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\n⊢ ∃ n z, x ∈ M n z",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\n⊢ ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z",
"tactic": "intro x xs"
},
{
"state_after": "case intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\n⊢ ∃ n z, x ∈ M n z",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\n⊢ ∃ n z, x ∈ M n z",
"tactic": "obtain ⟨z, zT, hz⟩ : ∃ z ∈ T, f' x ∈ ball (f' (z : E)) (r (f' z)) := by\n have : f' x ∈ ⋃ z ∈ T, ball (f' (z : E)) (r (f' z)) := by\n rw [hT]\n refine' mem_iUnion.2 ⟨⟨x, xs⟩, _⟩\n simpa only [mem_ball, Subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt\n rwa [mem_iUnion₂, bex_def] at this"
},
{
"state_after": "case intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\n⊢ ∃ n z, x ∈ M n z",
"state_before": "case intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\n⊢ ∃ n z, x ∈ M n z",
"tactic": "obtain ⟨ε, εpos, hε⟩ : ∃ ε : ℝ, 0 < ε ∧ ‖f' x - f' z‖ + ε ≤ r (f' z) := by\n refine' ⟨r (f' z) - ‖f' x - f' z‖, _, le_of_eq (by abel)⟩\n simpa only [sub_pos] using mem_ball_iff_norm.mp hz"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\n⊢ ∃ n z, x ∈ M n z",
"state_before": "case intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\n⊢ ∃ n z, x ∈ M n z",
"tactic": "obtain ⟨δ, δpos, hδ⟩ :\n ∃ (δ : ℝ), 0 < δ ∧ ball x δ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖} :=\n Metric.mem_nhdsWithin_iff.1 (IsLittleO.def (hf' x xs) εpos)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\n⊢ ∃ n z, x ∈ M n z",
"state_before": "case intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\n⊢ ∃ n z, x ∈ M n z",
"tactic": "obtain ⟨n, hn⟩ : ∃ n, u n < δ := ((tendsto_order.1 u_lim).2 _ δpos).exists"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\n⊢ ∀ (y : E),\n y ∈ s ∩ ball x (u n) →\n ‖f y - f x - ↑(f' ↑↑{ val := z, property := zT }) (y - x)‖ ≤ ↑(r (f' ↑↑{ val := z, property := zT })) * ‖y - x‖",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\n⊢ ∃ n z, x ∈ M n z",
"tactic": "refine' ⟨n, ⟨z, zT⟩, ⟨xs, _⟩⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ ‖f y - f x - ↑(f' ↑↑{ val := z, property := zT }) (y - x)‖ ≤ ↑(r (f' ↑↑{ val := z, property := zT })) * ‖y - x‖",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\n⊢ ∀ (y : E),\n y ∈ s ∩ ball x (u n) →\n ‖f y - f x - ↑(f' ↑↑{ val := z, property := zT }) (y - x)‖ ≤ ↑(r (f' ↑↑{ val := z, property := zT })) * ‖y - x‖",
"tactic": "intro y hy"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ ‖f y - f x - ↑(f' ↑↑{ val := z, property := zT }) (y - x)‖ ≤ ↑(r (f' ↑↑{ val := z, property := zT })) * ‖y - x‖",
"tactic": "calc\n ‖f y - f x - (f' z) (y - x)‖ = ‖f y - f x - (f' x) (y - x) + (f' x - f' z) (y - x)‖ := by\n congr 1\n simp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply]\n abel\n _ ≤ ‖f y - f x - (f' x) (y - x)‖ + ‖(f' x - f' z) (y - x)‖ := (norm_add_le _ _)\n _ ≤ ε * ‖y - x‖ + ‖f' x - f' z‖ * ‖y - x‖ := by\n refine' add_le_add (hδ _) (ContinuousLinearMap.le_op_norm _ _)\n rw [inter_comm]\n exact inter_subset_inter_right _ (ball_subset_ball hn.le) hy\n _ ≤ r (f' z) * ‖y - x‖ := by\n rw [← add_mul, add_comm]\n exact mul_le_mul_of_nonneg_right hε (norm_nonneg _)"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nthis : f' x ∈ ⋃ (z : ↑s) (_ : z ∈ T), ball (f' ↑z) ↑(r (f' ↑z))\n⊢ ∃ z, z ∈ T ∧ f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\n⊢ ∃ z, z ∈ T ∧ f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))",
"tactic": "have : f' x ∈ ⋃ z ∈ T, ball (f' (z : E)) (r (f' z)) := by\n rw [hT]\n refine' mem_iUnion.2 ⟨⟨x, xs⟩, _⟩\n simpa only [mem_ball, Subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nthis : f' x ∈ ⋃ (z : ↑s) (_ : z ∈ T), ball (f' ↑z) ↑(r (f' ↑z))\n⊢ ∃ z, z ∈ T ∧ f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))",
"tactic": "rwa [mem_iUnion₂, bex_def] at this"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\n⊢ f' x ∈ ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\n⊢ f' x ∈ ⋃ (z : ↑s) (_ : z ∈ T), ball (f' ↑z) ↑(r (f' ↑z))",
"tactic": "rw [hT]"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\n⊢ f' x ∈ ball (f' ↑{ val := x, property := xs }) ↑(r (f' ↑{ val := x, property := xs }))",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\n⊢ f' x ∈ ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))",
"tactic": "refine' mem_iUnion.2 ⟨⟨x, xs⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\n⊢ f' x ∈ ball (f' ↑{ val := x, property := xs }) ↑(r (f' ↑{ val := x, property := xs }))",
"tactic": "simpa only [mem_ball, Subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\n⊢ 0 < ↑(r (f' ↑z)) - ‖f' x - f' ↑z‖",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\n⊢ ∃ ε, 0 < ε ∧ ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))",
"tactic": "refine' ⟨r (f' z) - ‖f' x - f' z‖, _, le_of_eq (by abel)⟩"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\n⊢ 0 < ↑(r (f' ↑z)) - ‖f' x - f' ↑z‖",
"tactic": "simpa only [sub_pos] using mem_ball_iff_norm.mp hz"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\n⊢ ‖f' x - f' ↑z‖ + (↑(r (f' ↑z)) - ‖f' x - f' ↑z‖) = ↑(r (f' ↑z))",
"tactic": "abel"
},
{
"state_after": "case e_a\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ f y - f x - ↑(f' ↑z) (y - x) = f y - f x - ↑(f' x) (y - x) + ↑(f' x - f' ↑z) (y - x)",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ ‖f y - f x - ↑(f' ↑z) (y - x)‖ = ‖f y - f x - ↑(f' x) (y - x) + ↑(f' x - f' ↑z) (y - x)‖",
"tactic": "congr 1"
},
{
"state_after": "case e_a\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ f y - f x - (↑(f' ↑z) y - ↑(f' ↑z) x) =\n f y - f x - (↑(f' x) y - ↑(f' x) x) + (↑(f' x) y - ↑(f' ↑z) y - (↑(f' x) x - ↑(f' ↑z) x))",
"state_before": "case e_a\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ f y - f x - ↑(f' ↑z) (y - x) = f y - f x - ↑(f' x) (y - x) + ↑(f' x - f' ↑z) (y - x)",
"tactic": "simp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply]"
},
{
"state_after": "no goals",
"state_before": "case e_a\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ f y - f x - (↑(f' ↑z) y - ↑(f' ↑z) x) =\n f y - f x - (↑(f' x) y - ↑(f' x) x) + (↑(f' x) y - ↑(f' ↑z) y - (↑(f' x) x - ↑(f' ↑z) x))",
"tactic": "abel"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ y ∈ ball x δ ∩ s",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ ‖f y - f x - ↑(f' x) (y - x)‖ + ‖↑(f' x - f' ↑z) (y - x)‖ ≤ ε * ‖y - x‖ + ‖f' x - f' ↑z‖ * ‖y - x‖",
"tactic": "refine' add_le_add (hδ _) (ContinuousLinearMap.le_op_norm _ _)"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ y ∈ s ∩ ball x δ",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ y ∈ ball x δ ∩ s",
"tactic": "rw [inter_comm]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ y ∈ s ∩ ball x δ",
"tactic": "exact inter_subset_inter_right _ (ball_subset_ball hn.le) hy"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ (‖f' x - f' ↑z‖ + ε) * ‖y - x‖ ≤ ↑(r (f' ↑z)) * ‖y - x‖",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ ε * ‖y - x‖ + ‖f' x - f' ↑z‖ * ‖y - x‖ ≤ ↑(r (f' ↑z)) * ‖y - x‖",
"tactic": "rw [← add_mul, add_comm]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\nx : E\nxs : x ∈ s\nz : ↑s\nzT : z ∈ T\nhz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))\nε : ℝ\nεpos : 0 < ε\nhε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))\nδ : ℝ\nδpos : 0 < δ\nhδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖}\nn : ℕ\nhn : u n < δ\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ (‖f' x - f' ↑z‖ + ε) * ‖y - x‖ ≤ ↑(r (f' ↑z)) * ‖y - x‖",
"tactic": "exact mul_le_mul_of_nonneg_right hε (norm_nonneg _)"
},
{
"state_after": "case intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\n⊢ x ∈ M n z",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\n⊢ ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z",
"tactic": "rintro n z x ⟨xs, hx⟩"
},
{
"state_after": "case intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖",
"state_before": "case intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\n⊢ x ∈ M n z",
"tactic": "refine' ⟨xs, fun y hy => _⟩"
},
{
"state_after": "case intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\n⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖",
"state_before": "case intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\n⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖",
"tactic": "obtain ⟨a, aM, a_lim⟩ : ∃ a : ℕ → E, (∀ k, a k ∈ M n z) ∧ Tendsto a atTop (𝓝 x) :=\n mem_closure_iff_seq_limit.1 hx"
},
{
"state_after": "case intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)\n⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖",
"state_before": "case intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\n⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖",
"tactic": "have L1 :\n Tendsto (fun k : ℕ => ‖f y - f (a k) - (f' z) (y - a k)‖) atTop\n (𝓝 ‖f y - f x - (f' z) (y - x)‖) := by\n apply Tendsto.norm\n have L : Tendsto (fun k => f (a k)) atTop (𝓝 (f x)) := by\n apply (hf' x xs).continuousWithinAt.tendsto.comp\n apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ a_lim\n exact eventually_of_forall fun k => (aM k).1\n apply Tendsto.sub (tendsto_const_nhds.sub L)\n exact ((f' z).continuous.tendsto _).comp (tendsto_const_nhds.sub a_lim)"
},
{
"state_after": "case intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)\nL2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖))\n⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖",
"state_before": "case intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)\n⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖",
"tactic": "have L2 : Tendsto (fun k : ℕ => (r (f' z) : ℝ) * ‖y - a k‖) atTop (𝓝 (r (f' z) * ‖y - x‖)) :=\n (tendsto_const_nhds.sub a_lim).norm.const_mul _"
},
{
"state_after": "case intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)\nL2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖))\nI : ∀ᶠ (k : ℕ) in atTop, ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a k‖\n⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖",
"state_before": "case intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)\nL2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖))\n⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖",
"tactic": "have I : ∀ᶠ k in atTop, ‖f y - f (a k) - (f' z) (y - a k)‖ ≤ r (f' z) * ‖y - a k‖ := by\n have L : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x)) :=\n tendsto_const_nhds.dist a_lim\n filter_upwards [(tendsto_order.1 L).2 _ hy.2]\n intro k hk\n exact (aM k).2 y ⟨hy.1, hk⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)\nL2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖))\nI : ∀ᶠ (k : ℕ) in atTop, ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a k‖\n⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖",
"tactic": "exact le_of_tendsto_of_tendsto L1 L2 I"
},
{
"state_after": "case h\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\n⊢ Tendsto (fun x => f y - f (a x) - ↑(f' ↑↑z) (y - a x)) atTop (𝓝 (f y - f x - ↑(f' ↑↑z) (y - x)))",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\n⊢ Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)",
"tactic": "apply Tendsto.norm"
},
{
"state_after": "case h\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL : Tendsto (fun k => f (a k)) atTop (𝓝 (f x))\n⊢ Tendsto (fun x => f y - f (a x) - ↑(f' ↑↑z) (y - a x)) atTop (𝓝 (f y - f x - ↑(f' ↑↑z) (y - x)))",
"state_before": "case h\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\n⊢ Tendsto (fun x => f y - f (a x) - ↑(f' ↑↑z) (y - a x)) atTop (𝓝 (f y - f x - ↑(f' ↑↑z) (y - x)))",
"tactic": "have L : Tendsto (fun k => f (a k)) atTop (𝓝 (f x)) := by\n apply (hf' x xs).continuousWithinAt.tendsto.comp\n apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ a_lim\n exact eventually_of_forall fun k => (aM k).1"
},
{
"state_after": "case h\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL : Tendsto (fun k => f (a k)) atTop (𝓝 (f x))\n⊢ Tendsto (fun x => ↑(f' ↑↑z) (y - a x)) atTop (𝓝 (↑(f' ↑↑z) (y - x)))",
"state_before": "case h\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL : Tendsto (fun k => f (a k)) atTop (𝓝 (f x))\n⊢ Tendsto (fun x => f y - f (a x) - ↑(f' ↑↑z) (y - a x)) atTop (𝓝 (f y - f x - ↑(f' ↑↑z) (y - x)))",
"tactic": "apply Tendsto.sub (tendsto_const_nhds.sub L)"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL : Tendsto (fun k => f (a k)) atTop (𝓝 (f x))\n⊢ Tendsto (fun x => ↑(f' ↑↑z) (y - a x)) atTop (𝓝 (↑(f' ↑↑z) (y - x)))",
"tactic": "exact ((f' z).continuous.tendsto _).comp (tendsto_const_nhds.sub a_lim)"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\n⊢ Tendsto (fun k => a k) atTop (𝓝[s] x)",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\n⊢ Tendsto (fun k => f (a k)) atTop (𝓝 (f x))",
"tactic": "apply (hf' x xs).continuousWithinAt.tendsto.comp"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\n⊢ ∀ᶠ (x : ℕ) in atTop, a x ∈ s",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\n⊢ Tendsto (fun k => a k) atTop (𝓝[s] x)",
"tactic": "apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ a_lim"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\n⊢ ∀ᶠ (x : ℕ) in atTop, a x ∈ s",
"tactic": "exact eventually_of_forall fun k => (aM k).1"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)\nL2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖))\nL : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x))\n⊢ ∀ᶠ (k : ℕ) in atTop, ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a k‖",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)\nL2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖))\n⊢ ∀ᶠ (k : ℕ) in atTop, ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a k‖",
"tactic": "have L : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x)) :=\n tendsto_const_nhds.dist a_lim"
},
{
"state_after": "case h\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)\nL2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖))\nL : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x))\n⊢ ∀ (a_1 : ℕ), dist y (a a_1) < u n → ‖f y - f (a a_1) - ↑(f' ↑↑z) (y - a a_1)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a a_1‖",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)\nL2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖))\nL : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x))\n⊢ ∀ᶠ (k : ℕ) in atTop, ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a k‖",
"tactic": "filter_upwards [(tendsto_order.1 L).2 _ hy.2]"
},
{
"state_after": "case h\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)\nL2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖))\nL : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x))\nk : ℕ\nhk : dist y (a k) < u n\n⊢ ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a k‖",
"state_before": "case h\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)\nL2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖))\nL : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x))\n⊢ ∀ (a_1 : ℕ), dist y (a a_1) < u n → ‖f y - f (a a_1) - ↑(f' ↑↑z) (y - a a_1)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a a_1‖",
"tactic": "intro k hk"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nn : ℕ\nz : ↑T\nx : E\nxs : x ∈ s\nhx : x ∈ closure (M n z)\ny : E\nhy : y ∈ s ∩ ball x (u n)\na : ℕ → E\naM : ∀ (k : ℕ), a k ∈ M n z\na_lim : Tendsto a atTop (𝓝 x)\nL1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖)\nL2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖))\nL : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x))\nk : ℕ\nhk : dist y (a k) < u n\n⊢ ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a k‖",
"tactic": "exact (aM k).2 y ⟨hy.1, hk⟩"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nn : ℕ\nz : ↑T\np : ℕ\nx : E\nhx : x ∈ s ∩ K n z p\ny : E\nhy : y ∈ s ∩ K n z p\n⊢ ‖f x - f y - ↑(f' ↑↑z) (x - y)‖ ≤ ↑(r (f' ↑↑z)) * ‖x - y‖",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\n⊢ ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))",
"tactic": "intro n z p x hx y hy"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nn : ℕ\nz : ↑T\np : ℕ\nx : E\nhx : x ∈ s ∩ K n z p\ny : E\nhy : y ∈ s ∩ K n z p\nyM : y ∈ M n z\n⊢ ‖f x - f y - ↑(f' ↑↑z) (x - y)‖ ≤ ↑(r (f' ↑↑z)) * ‖x - y‖",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nn : ℕ\nz : ↑T\np : ℕ\nx : E\nhx : x ∈ s ∩ K n z p\ny : E\nhy : y ∈ s ∩ K n z p\n⊢ ‖f x - f y - ↑(f' ↑↑z) (x - y)‖ ≤ ↑(r (f' ↑↑z)) * ‖x - y‖",
"tactic": "have yM : y ∈ M n z := closure_M_subset _ _ ⟨hy.1, hy.2.1⟩"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nn : ℕ\nz : ↑T\np : ℕ\nx : E\nhx : x ∈ s ∩ K n z p\ny : E\nhy : y ∈ s ∩ K n z p\nyM : y ∈ M n z\n⊢ x ∈ ball y (u n)",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nn : ℕ\nz : ↑T\np : ℕ\nx : E\nhx : x ∈ s ∩ K n z p\ny : E\nhy : y ∈ s ∩ K n z p\nyM : y ∈ M n z\n⊢ ‖f x - f y - ↑(f' ↑↑z) (x - y)‖ ≤ ↑(r (f' ↑↑z)) * ‖x - y‖",
"tactic": "refine' yM.2 _ ⟨hx.1, _⟩"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nn : ℕ\nz : ↑T\np : ℕ\nx : E\nhx : x ∈ s ∩ K n z p\ny : E\nhy : y ∈ s ∩ K n z p\nyM : y ∈ M n z\n⊢ x ∈ ball y (u n)",
"tactic": "calc\n dist x y ≤ dist x (d p) + dist y (d p) := dist_triangle_right _ _ _\n _ ≤ u n / 3 + u n / 3 := (add_le_add hx.2.2 hy.2.2)\n _ < u n := by linarith [u_pos n]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nn : ℕ\nz : ↑T\np : ℕ\nx : E\nhx : x ∈ s ∩ K n z p\ny : E\nhy : y ∈ s ∩ K n z p\nyM : y ∈ M n z\n⊢ u n / 3 + u n / 3 < u n",
"tactic": "linarith [u_pos n]"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nthis : Encodable ↑T\n⊢ ∃ F, Function.Surjective F",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\n⊢ ∃ F, Function.Surjective F",
"tactic": "haveI : Encodable T := T_count.toEncodable"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nthis✝ : Encodable ↑T\nthis : Nonempty ↑T\ninhabited_h : Inhabited ↑T\n⊢ ∃ F, Function.Surjective F",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nthis✝ : Encodable ↑T\nthis : Nonempty ↑T\n⊢ ∃ F, Function.Surjective F",
"tactic": "inhabit ↥T"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nthis✝ : Encodable ↑T\nthis : Nonempty ↑T\ninhabited_h : Inhabited ↑T\n⊢ ∃ F, Function.Surjective F",
"tactic": "exact ⟨_, Encodable.surjective_decode_iget (ℕ × T × ℕ)⟩"
},
{
"state_after": "case inl\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nd : ℕ → E\nhd : DenseRange d\nT_count : Set.Countable ∅\nhT : (⋃ (x : ↑s) (_ : x ∈ ∅), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nM : ℕ → ↑∅ → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑∅), s ∩ closure (M n z) ⊆ M n z\nK : ℕ → ↑∅ → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), IsClosed (K n z p)\nthis : Encodable ↑∅\n⊢ Nonempty ↑∅\n\ncase inr\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT✝ : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nthis : Encodable ↑T\nhT : Set.Nonempty T\n⊢ Nonempty ↑T",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nthis : Encodable ↑T\n⊢ Nonempty ↑T",
"tactic": "rcases eq_empty_or_nonempty T with (rfl | hT)"
},
{
"state_after": "case inl.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nd : ℕ → E\nhd : DenseRange d\nT_count : Set.Countable ∅\nhT : (⋃ (x : ↑s) (_ : x ∈ ∅), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nM : ℕ → ↑∅ → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑∅), s ∩ closure (M n z) ⊆ M n z\nK : ℕ → ↑∅ → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), IsClosed (K n z p)\nthis : Encodable ↑∅\nx : E\nxs : x ∈ s\n⊢ Nonempty ↑∅",
"state_before": "case inl\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nd : ℕ → E\nhd : DenseRange d\nT_count : Set.Countable ∅\nhT : (⋃ (x : ↑s) (_ : x ∈ ∅), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nM : ℕ → ↑∅ → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑∅), s ∩ closure (M n z) ⊆ M n z\nK : ℕ → ↑∅ → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), IsClosed (K n z p)\nthis : Encodable ↑∅\n⊢ Nonempty ↑∅",
"tactic": "rcases hs with ⟨x, xs⟩"
},
{
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"state_before": "case inl.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nd : ℕ → E\nhd : DenseRange d\nT_count : Set.Countable ∅\nhT : (⋃ (x : ↑s) (_ : x ∈ ∅), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nM : ℕ → ↑∅ → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑∅), s ∩ closure (M n z) ⊆ M n z\nK : ℕ → ↑∅ → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), IsClosed (K n z p)\nthis : Encodable ↑∅\nx : E\nxs : x ∈ s\n⊢ Nonempty ↑∅",
"tactic": "rcases s_subset x xs with ⟨n, z, _⟩"
},
{
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"state_before": "case inl.intro.intro.intro\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nd : ℕ → E\nhd : DenseRange d\nT_count : Set.Countable ∅\nhT : (⋃ (x : ↑s) (_ : x ∈ ∅), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nM : ℕ → ↑∅ → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑∅), s ∩ closure (M n z) ⊆ M n z\nK : ℕ → ↑∅ → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), IsClosed (K n z p)\nthis : Encodable ↑∅\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑∅\nh✝ : x ∈ M n z\n⊢ Nonempty ↑∅",
"tactic": "exact False.elim z.2"
},
{
"state_after": "no goals",
"state_before": "case inr\nE : Type u_2\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT✝ : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nthis : Encodable ↑T\nhT : Set.Nonempty T\n⊢ Nonempty ↑T",
"tactic": "exact hT.coe_sort"
},
{
"state_after": "E : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\nthis : Set.Nonempty (ball x (u n / 3))\n⊢ ∃ p, x ∈ closedBall (d p) (u n / 3)",
"state_before": "E : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\n⊢ ∃ p, x ∈ closedBall (d p) (u n / 3)",
"tactic": "have : Set.Nonempty (ball x (u n / 3)) := by simp only [nonempty_ball]; linarith [u_pos n]"
},
{
"state_after": "case intro\nE : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\nthis : Set.Nonempty (ball x (u n / 3))\np : ℕ\nhp : d p ∈ ball x (u n / 3)\n⊢ ∃ p, x ∈ closedBall (d p) (u n / 3)",
"state_before": "E : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\nthis : Set.Nonempty (ball x (u n / 3))\n⊢ ∃ p, x ∈ closedBall (d p) (u n / 3)",
"tactic": "obtain ⟨p, hp⟩ : ∃ p : ℕ, d p ∈ ball x (u n / 3) := hd.exists_mem_open isOpen_ball this"
},
{
"state_after": "no goals",
"state_before": "case intro\nE : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\nthis : Set.Nonempty (ball x (u n / 3))\np : ℕ\nhp : d p ∈ ball x (u n / 3)\n⊢ ∃ p, x ∈ closedBall (d p) (u n / 3)",
"tactic": "exact ⟨p, (mem_ball'.1 hp).le⟩"
},
{
"state_after": "E : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\n⊢ 0 < u n / 3",
"state_before": "E : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\n⊢ Set.Nonempty (ball x (u n / 3))",
"tactic": "simp only [nonempty_ball]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nF✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F✝\ninst✝¹ : NormedSpace ℝ F✝\ns✝ : Set E\nf✝ : E → E\nf'✝ : E → E →L[ℝ] E\ninst✝ : SecondCountableTopology F✝\nf : E → F✝\ns : Set E\nf' : E → E →L[ℝ] F✝\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nr : (E →L[ℝ] F✝) → ℝ≥0\nrpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0\nhs : Set.Nonempty s\nT : Set ↑s\nT_count : Set.Countable T\nhT : (⋃ (x : ↑s) (_ : x ∈ T), ball (f' ↑x) ↑(r (f' ↑x))) = ⋃ (x : ↑s), ball (f' ↑x) ↑(r (f' ↑x))\nu : ℕ → ℝ\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nM : ℕ → ↑T → Set E :=\n fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}\ns_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z\nclosure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z\nd : ℕ → E\nhd : DenseRange d\nK : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)\nK_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))\nK_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)\nF : ℕ → ℕ × ↑T × ℕ\nhF : Function.Surjective F\nx : E\nxs : x ∈ s\nn : ℕ\nz : ↑T\nhnz : x ∈ M n z\n⊢ 0 < u n / 3",
"tactic": "linarith [u_pos n]"
}
] |
[
249,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Data/Finsupp/Multiset.lean
|
Finsupp.toMultiset_single
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.10567\nι : Type ?u.10570\na : α\nn : ℕ\n⊢ 0 • {a} = 0",
"state_before": "α : Type u_1\nβ : Type ?u.10567\nι : Type ?u.10570\na : α\nn : ℕ\n⊢ ↑toMultiset (single a n) = n • {a}",
"tactic": "rw [toMultiset_apply, sum_single_index]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.10567\nι : Type ?u.10570\na : α\nn : ℕ\n⊢ 0 • {a} = 0",
"tactic": "apply zero_nsmul"
}
] |
[
71,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Logic/Equiv/Fin.lean
|
finAddFlip_apply_natAdd
|
[
{
"state_after": "no goals",
"state_before": "m✝ n : ℕ\nk : Fin n\nm : ℕ\n⊢ ↑finAddFlip (↑(Fin.natAdd m) k) = ↑(Fin.castAdd m) k",
"tactic": "simp [finAddFlip]"
}
] |
[
370,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iSup_iInf_le_iInf_iSup
|
[] |
[
889,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
888,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.mem_of_subset_of_mem
|
[] |
[
385,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
384,
1
] |
Mathlib/Combinatorics/Configuration.lean
|
Configuration.ProjectivePlane.pointCount_eq_pointCount
|
[
{
"state_after": "no goals",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : ProjectivePlane P L\ninst✝¹ : Finite P\ninst✝ : Finite L\nl m : L\n⊢ pointCount P l = pointCount P m",
"tactic": "apply lineCount_eq_lineCount (Dual P)"
}
] |
[
425,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
RingHom.codomain_trivial
|
[] |
[
628,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
626,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.mem_graph_snd_inj'
|
[
{
"state_after": "case mk\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.558597\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.559113\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : E × F\nhy : y ∈ graph f\nfst✝ : E\nsnd✝ : F\nhx : (fst✝, snd✝) ∈ graph f\nhxy : (fst✝, snd✝).fst = y.fst\n⊢ (fst✝, snd✝).snd = y.snd",
"state_before": "R : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.558597\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.559113\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nx y : E × F\nhx : x ∈ graph f\nhy : y ∈ graph f\nhxy : x.fst = y.fst\n⊢ x.snd = y.snd",
"tactic": "cases x"
},
{
"state_after": "case mk.mk\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.558597\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.559113\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nfst✝¹ : E\nsnd✝¹ : F\nhx : (fst✝¹, snd✝¹) ∈ graph f\nfst✝ : E\nsnd✝ : F\nhy : (fst✝, snd✝) ∈ graph f\nhxy : (fst✝¹, snd✝¹).fst = (fst✝, snd✝).fst\n⊢ (fst✝¹, snd✝¹).snd = (fst✝, snd✝).snd",
"state_before": "case mk\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.558597\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.559113\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : E × F\nhy : y ∈ graph f\nfst✝ : E\nsnd✝ : F\nhx : (fst✝, snd✝) ∈ graph f\nhxy : (fst✝, snd✝).fst = y.fst\n⊢ (fst✝, snd✝).snd = y.snd",
"tactic": "cases y"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.558597\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.559113\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nfst✝¹ : E\nsnd✝¹ : F\nhx : (fst✝¹, snd✝¹) ∈ graph f\nfst✝ : E\nsnd✝ : F\nhy : (fst✝, snd✝) ∈ graph f\nhxy : (fst✝¹, snd✝¹).fst = (fst✝, snd✝).fst\n⊢ (fst✝¹, snd✝¹).snd = (fst✝, snd✝).snd",
"tactic": "exact f.mem_graph_snd_inj hx hy hxy"
}
] |
[
822,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
818,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.single_vecMul_diagonal
|
[
{
"state_after": "case h\nl : Type ?u.881533\nm : Type ?u.881536\nn : Type u_1\no : Type ?u.881542\nm' : o → Type ?u.881547\nn' : o → Type ?u.881552\nR : Type u_2\nS : Type ?u.881558\nα : Type v\nβ : Type w\nγ : Type ?u.881565\ninst✝³ : NonUnitalNonAssocSemiring α\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : NonUnitalNonAssocSemiring R\nv : n → R\nj : n\nx : R\ni : n\n⊢ vecMul (Pi.single j x) (diagonal v) i = Pi.single j (x * v j) i",
"state_before": "l : Type ?u.881533\nm : Type ?u.881536\nn : Type u_1\no : Type ?u.881542\nm' : o → Type ?u.881547\nn' : o → Type ?u.881552\nR : Type u_2\nS : Type ?u.881558\nα : Type v\nβ : Type w\nγ : Type ?u.881565\ninst✝³ : NonUnitalNonAssocSemiring α\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : NonUnitalNonAssocSemiring R\nv : n → R\nj : n\nx : R\n⊢ vecMul (Pi.single j x) (diagonal v) = Pi.single j (x * v j)",
"tactic": "ext i"
},
{
"state_after": "case h\nl : Type ?u.881533\nm : Type ?u.881536\nn : Type u_1\no : Type ?u.881542\nm' : o → Type ?u.881547\nn' : o → Type ?u.881552\nR : Type u_2\nS : Type ?u.881558\nα : Type v\nβ : Type w\nγ : Type ?u.881565\ninst✝³ : NonUnitalNonAssocSemiring α\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : NonUnitalNonAssocSemiring R\nv : n → R\nj : n\nx : R\ni : n\n⊢ Pi.single j x i * v i = Pi.single j (x * v j) i",
"state_before": "case h\nl : Type ?u.881533\nm : Type ?u.881536\nn : Type u_1\no : Type ?u.881542\nm' : o → Type ?u.881547\nn' : o → Type ?u.881552\nR : Type u_2\nS : Type ?u.881558\nα : Type v\nβ : Type w\nγ : Type ?u.881565\ninst✝³ : NonUnitalNonAssocSemiring α\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : NonUnitalNonAssocSemiring R\nv : n → R\nj : n\nx : R\ni : n\n⊢ vecMul (Pi.single j x) (diagonal v) i = Pi.single j (x * v j) i",
"tactic": "rw [vecMul_diagonal]"
},
{
"state_after": "no goals",
"state_before": "case h\nl : Type ?u.881533\nm : Type ?u.881536\nn : Type u_1\no : Type ?u.881542\nm' : o → Type ?u.881547\nn' : o → Type ?u.881552\nR : Type u_2\nS : Type ?u.881558\nα : Type v\nβ : Type w\nγ : Type ?u.881565\ninst✝³ : NonUnitalNonAssocSemiring α\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : NonUnitalNonAssocSemiring R\nv : n → R\nj : n\nx : R\ni : n\n⊢ Pi.single j x i * v i = Pi.single j (x * v j) i",
"tactic": "exact Pi.apply_single (fun i x => x * v i) (fun i => MulZeroClass.zero_mul _) j x i"
}
] |
[
1811,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1807,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.coe_int_mul_eq_zsmul
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nn : ℤ\n⊢ ↑(↑n * x) = n • ↑x",
"tactic": "simpa only [zsmul_eq_mul] using coeHom.map_zsmul x n"
}
] |
[
116,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Topology/Connected.lean
|
IsConnected.preimage_of_closedMap
|
[] |
[
414,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
411,
1
] |
src/lean/Init/Core.lean
|
proofIrrel
|
[] |
[
525,
59
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
525,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiff_succ_iff_fderiv_apply
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2834577\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nf : E → F\n⊢ ContDiff 𝕜 (↑(n + 1)) f ↔ Differentiable 𝕜 f ∧ ∀ (y : E), ContDiff 𝕜 ↑n fun x => ↑(fderiv 𝕜 f x) y",
"tactic": "rw [contDiff_succ_iff_fderiv, contDiff_clm_apply_iff]"
}
] |
[
1907,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1905,
1
] |
Std/Logic.lean
|
ite_id
|
[
{
"state_after": "no goals",
"state_before": "c : Prop\ninst✝ : Decidable c\nα : Sort u_1\nt : α\n⊢ (if c then t else t) = t",
"tactic": "split <;> rfl"
}
] |
[
689,
88
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
689,
1
] |
Mathlib/RingTheory/Polynomial/Content.lean
|
Polynomial.content_one
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\n⊢ content 1 = 1",
"tactic": "rw [← C_1, content_C, normalize_one]"
}
] |
[
109,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/Deprecated/Subgroup.lean
|
IsGroupHom.inv_iff_ker
|
[
{
"state_after": "G : Type u_1\nH : Type u_2\nA : Type ?u.83924\na✝ a₁ a₂ b✝ c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\na b : G\n⊢ f a = f b ↔ f (a * b⁻¹) = 1",
"state_before": "G : Type u_1\nH : Type u_2\nA : Type ?u.83924\na✝ a₁ a₂ b✝ c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\na b : G\n⊢ f a = f b ↔ a * b⁻¹ ∈ ker f",
"tactic": "rw [mem_ker]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nH : Type u_2\nA : Type ?u.83924\na✝ a₁ a₂ b✝ c : G\ninst✝¹ : Group G\ninst✝ : Group H\nf : G → H\nhf : IsGroupHom f\na b : G\n⊢ f a = f b ↔ f (a * b⁻¹) = 1",
"tactic": "exact one_iff_ker_inv hf _ _"
}
] |
[
392,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
391,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.not_dom_iff_eq_top
|
[] |
[
374,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/LinearAlgebra/Matrix/Symmetric.lean
|
Matrix.IsSymm.conjTranspose
|
[] |
[
98,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
ball_pi'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.401932\nι : Type ?u.401935\ninst✝³ : PseudoMetricSpace α\nπ : β → Type u_1\ninst✝² : Fintype β\ninst✝¹ : (b : β) → PseudoMetricSpace (π b)\ninst✝ : Nonempty β\nx : (b : β) → π b\nr : ℝ\nhr : r ≤ 0\n⊢ ball x r = Set.pi Set.univ fun b => ball (x b) r",
"tactic": "simp [ball_eq_empty.2 hr]"
}
] |
[
2059,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2057,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Pretrivialization.symm_apply_apply
|
[] |
[
174,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/GroupTheory/Subgroup/ZPowers.lean
|
Subgroup.exists_zpowers
|
[] |
[
69,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/CategoryTheory/Limits/ExactFunctor.lean
|
CategoryTheory.LeftExactFunctor.ofExact_map
|
[] |
[
156,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Data/Polynomial/DenomsClearable.lean
|
denomsClearable_natDegree
|
[] |
[
82,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Ioc_mem_nhdsWithin_Iic
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderClosedTopology α\na✝ b✝ : α\ninst✝ : TopologicalSpace γ\na b c : α\nH : b ∈ Ioc a c\n⊢ Ioc a c ∈ 𝓝[Iic b] b",
"tactic": "simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)"
}
] |
[
579,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Mathlib/Topology/Order.lean
|
coinduced_id
|
[] |
[
482,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/GroupTheory/Nilpotent.lean
|
of_quotient_center_nilpotent
|
[
{
"state_after": "case intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nh : Group.IsNilpotent (G ⧸ center G)\nn : ℕ\nhn : upperCentralSeries (G ⧸ center G) n = ⊤\n⊢ Group.IsNilpotent G",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nh : Group.IsNilpotent (G ⧸ center G)\n⊢ Group.IsNilpotent G",
"tactic": "obtain ⟨n, hn⟩ := h.nilpotent"
},
{
"state_after": "case intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nh : Group.IsNilpotent (G ⧸ center G)\nn : ℕ\nhn : upperCentralSeries (G ⧸ center G) n = ⊤\n⊢ upperCentralSeries G (Nat.succ n) = ⊤",
"state_before": "case intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nh : Group.IsNilpotent (G ⧸ center G)\nn : ℕ\nhn : upperCentralSeries (G ⧸ center G) n = ⊤\n⊢ Group.IsNilpotent G",
"tactic": "use n.succ"
},
{
"state_after": "no goals",
"state_before": "case intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nh : Group.IsNilpotent (G ⧸ center G)\nn : ℕ\nhn : upperCentralSeries (G ⧸ center G) n = ⊤\n⊢ upperCentralSeries G (Nat.succ n) = ⊤",
"tactic": "simp [← comap_upperCentralSeries_quotient_center, hn]"
}
] |
[
658,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
655,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.Ici_prod_Ici
|
[] |
[
1668,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1667,
1
] |
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