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/Data/Matrix/Block.lean
|
Matrix.blockDiagonal'_injective
|
[] |
[
859,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
857,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.rotate'_mod
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ rotate' l (n % length l) = rotate' (rotate' l (n % length l)) (length (rotate' l (n % length l)) * (n / length l))",
"tactic": "rw [rotate'_length_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ rotate' (rotate' l (n % length l)) (length (rotate' l (n % length l)) * (n / length l)) = rotate' l n",
"tactic": "rw [rotate'_rotate', length_rotate', Nat.mod_add_div]"
}
] |
[
108,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Algebra/Star/Pointwise.lean
|
Set.star_inv
|
[
{
"state_after": "case h\nα : Type u_1\ns✝ t : Set α\na : α\ninst✝¹ : Group α\ninst✝ : StarSemigroup α\ns : Set α\nx✝ : α\n⊢ x✝ ∈ s⁻¹⋆ ↔ x✝ ∈ s⋆⁻¹",
"state_before": "α : Type u_1\ns✝ t : Set α\na : α\ninst✝¹ : Group α\ninst✝ : StarSemigroup α\ns : Set α\n⊢ s⁻¹⋆ = s⋆⁻¹",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ns✝ t : Set α\na : α\ninst✝¹ : Group α\ninst✝ : StarSemigroup α\ns : Set α\nx✝ : α\n⊢ x✝ ∈ s⁻¹⋆ ↔ x✝ ∈ s⋆⁻¹",
"tactic": "simp only [mem_star, mem_inv, star_inv]"
}
] |
[
142,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
11
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.Integrable.simpleFunc_mul'
|
[
{
"state_after": "α : Type ?u.5138263\nβ : Type u_1\nE : Type ?u.5138269\nF : Type ?u.5138272\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5138278\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\ng : SimpleFunc β ℝ\nhf : Integrable f\n⊢ Integrable (↑(SimpleFunc.toLargerSpace hm g) * f)",
"state_before": "α : Type ?u.5138263\nβ : Type u_1\nE : Type ?u.5138269\nF : Type ?u.5138272\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5138278\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\ng : SimpleFunc β ℝ\nhf : Integrable f\n⊢ Integrable (↑g * f)",
"tactic": "rw [← SimpleFunc.coe_toLargerSpace_eq hm g]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.5138263\nβ : Type u_1\nE : Type ?u.5138269\nF : Type ?u.5138272\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5138278\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\ng : SimpleFunc β ℝ\nhf : Integrable f\n⊢ Integrable (↑(SimpleFunc.toLargerSpace hm g) * f)",
"tactic": "exact hf.simpleFunc_mul (g.toLargerSpace hm)"
}
] |
[
1344,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1342,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.continuous_conj
|
[] |
[
979,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
978,
1
] |
Mathlib/Data/Real/GoldenRatio.lean
|
inv_goldConj
|
[
{
"state_after": "⊢ -ψ = φ⁻¹",
"state_before": "⊢ ψ⁻¹ = -φ",
"tactic": "rw [inv_eq_iff_eq_inv, ← neg_inv, ← neg_eq_iff_eq_neg]"
},
{
"state_after": "no goals",
"state_before": "⊢ -ψ = φ⁻¹",
"tactic": "exact inv_gold.symm"
}
] |
[
59,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEStronglyMeasurable.const_mul
|
[] |
[
1304,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1302,
11
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.frontier_closedBall_subset_sphere
|
[] |
[
1890,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1889,
1
] |
Mathlib/Algebra/Order/Ring/Canonical.lean
|
CanonicallyOrderedCommSemiring.mul_pos
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.21641\ninst✝ : CanonicallyOrderedCommSemiring α\na b : α\n⊢ 0 < a * b ↔ 0 < a ∧ 0 < b",
"tactic": "simp only [pos_iff_ne_zero, ne_eq, mul_eq_zero, not_or]"
}
] |
[
120,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
mul_lt_mul_iff_left
|
[] |
[
110,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/CategoryTheory/EqToHom.lean
|
CategoryTheory.Functor.congr_inv_of_congr_hom
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G : C ⥤ D\nX Y : C\ne : X ≅ Y\nhX : F.obj X = G.obj X\nhY : F.obj Y = G.obj Y\n⊢ F.obj X = G.obj X",
"tactic": "rw [hX]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G : C ⥤ D\nX Y : C\ne : X ≅ Y\nhX : F.obj X = G.obj X\nhY : F.obj Y = G.obj Y\n⊢ G.obj Y = F.obj Y",
"tactic": "rw [hY]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G : C ⥤ D\nX Y : C\ne : X ≅ Y\nhX : F.obj X = G.obj X\nhY : F.obj Y = G.obj Y\nh₂ : F.map e.hom = eqToHom (_ : F.obj X = G.obj X) ≫ G.map e.hom ≫ eqToHom (_ : G.obj Y = F.obj Y)\n⊢ F.obj Y = G.obj Y",
"tactic": "rw [hY]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G : C ⥤ D\nX Y : C\ne : X ≅ Y\nhX : F.obj X = G.obj X\nhY : F.obj Y = G.obj Y\nh₂ : F.map e.hom = eqToHom (_ : F.obj X = G.obj X) ≫ G.map e.hom ≫ eqToHom (_ : G.obj Y = F.obj Y)\n⊢ G.obj X = F.obj X",
"tactic": "rw [hX]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G : C ⥤ D\nX Y : C\ne : X ≅ Y\nhX : F.obj X = G.obj X\nhY : F.obj Y = G.obj Y\nh₂ : F.map e.hom = eqToHom (_ : F.obj X = G.obj X) ≫ G.map e.hom ≫ eqToHom (_ : G.obj Y = F.obj Y)\n⊢ F.map e.inv = eqToHom (_ : F.obj Y = G.obj Y) ≫ G.map e.inv ≫ eqToHom (_ : G.obj X = F.obj X)",
"tactic": "simp only [← IsIso.Iso.inv_hom e, Functor.map_inv, h₂, IsIso.inv_comp, inv_eqToHom,\n Category.assoc]"
}
] |
[
219,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Data/Multiset/Fintype.lean
|
Multiset.prod_toEnumFinset
|
[
{
"state_after": "α : Type u_2\ninst✝¹ : DecidableEq α\nm✝ : Multiset α\nβ : Type u_1\ninst✝ : CommMonoid β\nm : Multiset α\nf : α → ℕ → β\n⊢ ∏ x in toEnumFinset m, f x.fst x.snd = ∏ x : { x // x ∈ toEnumFinset m }, f (↑x).fst (↑x).snd\n\nα : Type u_2\ninst✝¹ : DecidableEq α\nm✝ : Multiset α\nβ : Type u_1\ninst✝ : CommMonoid β\nm : Multiset α\nf : α → ℕ → β\n⊢ ∀ (x : ToType m), f x.fst ↑x.snd = f (↑(↑(coeEquiv m) x)).fst (↑(↑(coeEquiv m) x)).snd",
"state_before": "α : Type u_2\ninst✝¹ : DecidableEq α\nm✝ : Multiset α\nβ : Type u_1\ninst✝ : CommMonoid β\nm : Multiset α\nf : α → ℕ → β\n⊢ ∏ x in toEnumFinset m, f x.fst x.snd = ∏ x : ToType m, f x.fst ↑x.snd",
"tactic": "rw [Fintype.prod_equiv m.coeEquiv (fun x ↦ f x x.2) fun x ↦ f x.1.1 x.1.2]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\ninst✝¹ : DecidableEq α\nm✝ : Multiset α\nβ : Type u_1\ninst✝ : CommMonoid β\nm : Multiset α\nf : α → ℕ → β\n⊢ ∏ x in toEnumFinset m, f x.fst x.snd = ∏ x : { x // x ∈ toEnumFinset m }, f (↑x).fst (↑x).snd",
"tactic": "rw [← m.toEnumFinset.prod_coe_sort fun x ↦ f x.1 x.2]"
},
{
"state_after": "α : Type u_2\ninst✝¹ : DecidableEq α\nm✝ : Multiset α\nβ : Type u_1\ninst✝ : CommMonoid β\nm : Multiset α\nf : α → ℕ → β\nx : ToType m\n⊢ f x.fst ↑x.snd = f (↑(↑(coeEquiv m) x)).fst (↑(↑(coeEquiv m) x)).snd",
"state_before": "α : Type u_2\ninst✝¹ : DecidableEq α\nm✝ : Multiset α\nβ : Type u_1\ninst✝ : CommMonoid β\nm : Multiset α\nf : α → ℕ → β\n⊢ ∀ (x : ToType m), f x.fst ↑x.snd = f (↑(↑(coeEquiv m) x)).fst (↑(↑(coeEquiv m) x)).snd",
"tactic": "intro x"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\ninst✝¹ : DecidableEq α\nm✝ : Multiset α\nβ : Type u_1\ninst✝ : CommMonoid β\nm : Multiset α\nf : α → ℕ → β\nx : ToType m\n⊢ f x.fst ↑x.snd = f (↑(↑(coeEquiv m) x)).fst (↑(↑(coeEquiv m) x)).snd",
"tactic": "rfl"
}
] |
[
284,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.image_uncurry_product
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.67520\nβ : Type u_2\nβ' : Type ?u.67526\nγ : Type u_3\nγ' : Type ?u.67532\nδ : Type ?u.67535\nδ' : Type ?u.67538\nε : Type ?u.67541\nε' : Type ?u.67544\nζ : Type ?u.67547\nζ' : Type ?u.67550\nν : Type ?u.67553\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Finset α\nt✝ t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\nf : α → β → γ\ns : Finset α\nt : Finset β\n⊢ image (uncurry f) (s ×ˢ t) = image₂ f s t",
"tactic": "rw [← image₂_curry, curry_uncurry]"
}
] |
[
347,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
1
] |
Mathlib/Analysis/Calculus/FDerivAnalytic.lean
|
AnalyticOn.iteratedFDeriv
|
[
{
"state_after": "case zero\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\n⊢ AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 Nat.zero f) s\n\ncase succ\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\nn : ℕ\nIH : AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s\n⊢ AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 (Nat.succ n) f) s",
"state_before": "𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\nn : ℕ\n⊢ AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s",
"tactic": "induction' n with n IH"
},
{
"state_after": "case zero\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\n⊢ AnalyticOn 𝕜 (↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) ∘ f) s",
"state_before": "case zero\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\n⊢ AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 Nat.zero f) s",
"tactic": "rw [iteratedFDeriv_zero_eq_comp]"
},
{
"state_after": "no goals",
"state_before": "case zero\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\n⊢ AnalyticOn 𝕜 (↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) ∘ f) s",
"tactic": "exact ((continuousMultilinearCurryFin0 𝕜 E F).symm : F →L[𝕜] E[×0]→L[𝕜] F).comp_analyticOn h"
},
{
"state_after": "case succ\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\nn : ℕ\nIH : AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s\n⊢ AnalyticOn 𝕜 (↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ _root_.fderiv 𝕜 (_root_.iteratedFDeriv 𝕜 n f))\n s",
"state_before": "case succ\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\nn : ℕ\nIH : AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s\n⊢ AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 (Nat.succ n) f) s",
"tactic": "rw [iteratedFDeriv_succ_eq_comp_left]"
},
{
"state_after": "case h.e'_9.h.e'_4\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\nn : ℕ\nIH : AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s\n⊢ ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) = ↑?g\n\ncase g\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\nn : ℕ\nIH : AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s\n⊢ (E →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F) →L[𝕜] ContinuousMultilinearMap 𝕜 (fun x => E) F",
"state_before": "case succ\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\nn : ℕ\nIH : AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s\n⊢ AnalyticOn 𝕜 (↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ _root_.fderiv 𝕜 (_root_.iteratedFDeriv 𝕜 n f))\n s",
"tactic": "convert @ContinuousLinearMap.comp_analyticOn 𝕜 E\n ?_ (ContinuousMultilinearMap 𝕜 (fun _ : Fin (n + 1) ↦ E) F)\n ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_\n s ?g IH.fderiv"
},
{
"state_after": "case h.e'_9.h.e'_4\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\nn : ℕ\nIH : AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s\n⊢ ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) =\n ↑↑(ContinuousLinearEquiv.mk (continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F).toLinearEquiv)",
"state_before": "case h.e'_9.h.e'_4\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\nn : ℕ\nIH : AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s\n⊢ ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) = ↑?g\n\ncase g\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\nn : ℕ\nIH : AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s\n⊢ (E →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F) →L[𝕜] ContinuousMultilinearMap 𝕜 (fun x => E) F",
"tactic": "case g =>\n exact ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_9.h.e'_4\n𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\nn : ℕ\nIH : AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s\n⊢ ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) =\n ↑↑(ContinuousLinearEquiv.mk (continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F).toLinearEquiv)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : AnalyticOn 𝕜 f s\nn : ℕ\nIH : AnalyticOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s\n⊢ (E →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F) →L[𝕜] ContinuousMultilinearMap 𝕜 (fun x => E) F",
"tactic": "exact ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F)"
}
] |
[
143,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
linearIndependent_of_subsingleton
|
[] |
[
251,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.coe_trans_symm
|
[] |
[
809,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
808,
1
] |
Mathlib/LinearAlgebra/Finrank.lean
|
FiniteDimensional.nontrivial_of_finrank_pos
|
[] |
[
103,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_atBot_of_add_const_right
|
[] |
[
710,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
708,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.mem_zeroLocus
|
[] |
[
142,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_X_sub_C
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn m : ℕ\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\na : R\n⊢ degree (X - ↑C a) = 1",
"tactic": "rw [sub_eq_add_neg, ← map_neg C a, degree_X_add_C]"
}
] |
[
1468,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1467,
1
] |
Mathlib/Data/QPF/Univariate/Basic.lean
|
Qpf.Cofix.bisim_aux
|
[
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : Cofix F\n⊢ ∀ (y : Cofix F), r x y → x = y",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\n⊢ ∀ (x y : Cofix F), r x y → x = y",
"tactic": "intro x"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : Cofix F\n⊢ ∀ (a : PFunctor.M (P F)) (y : Cofix F), r (Quot.mk Mcongr a) y → Quot.mk Mcongr a = y",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : Cofix F\n⊢ ∀ (y : Cofix F), r x y → x = y",
"tactic": "apply Quot.inductionOn (motive := _) x"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\n⊢ ∀ (a : PFunctor.M (P F)) (y : Cofix F), r (Quot.mk Mcongr a) y → Quot.mk Mcongr a = y",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : Cofix F\n⊢ ∀ (a : PFunctor.M (P F)) (y : Cofix F), r (Quot.mk Mcongr a) y → Quot.mk Mcongr a = y",
"tactic": "clear x"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\ny : Cofix F\n⊢ r (Quot.mk Mcongr x) y → Quot.mk Mcongr x = y",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\n⊢ ∀ (a : PFunctor.M (P F)) (y : Cofix F), r (Quot.mk Mcongr a) y → Quot.mk Mcongr a = y",
"tactic": "intro x y"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\ny : Cofix F\n⊢ ∀ (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) → Quot.mk Mcongr x = Quot.mk Mcongr a",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\ny : Cofix F\n⊢ r (Quot.mk Mcongr x) y → Quot.mk Mcongr x = y",
"tactic": "apply Quot.inductionOn (motive := _) y"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\n⊢ ∀ (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) → Quot.mk Mcongr x = Quot.mk Mcongr a",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\ny : Cofix F\n⊢ ∀ (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) → Quot.mk Mcongr x = Quot.mk Mcongr a",
"tactic": "clear y"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n⊢ Quot.mk Mcongr x = Quot.mk Mcongr y",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\n⊢ ∀ (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) → Quot.mk Mcongr x = Quot.mk Mcongr a",
"tactic": "intro y rxy"
},
{
"state_after": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n⊢ Mcongr x y",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n⊢ Quot.mk Mcongr x = Quot.mk Mcongr y",
"tactic": "apply Quot.sound"
},
{
"state_after": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n⊢ Mcongr x y",
"state_before": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n⊢ Mcongr x y",
"tactic": "let r' x y := r (Quot.mk _ x) (Quot.mk _ y)"
},
{
"state_after": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nthis : IsPrecongr r'\n⊢ Mcongr x y",
"state_before": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n⊢ Mcongr x y",
"tactic": "have : IsPrecongr r' := by\n intro a b r'ab\n have h₀ :\n Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =\n Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=\n h _ _ r'ab\n have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by\n intro u v cuv\n apply Quot.sound\n simp only\n rw [Quot.sound cuv]\n apply h'\n let f : Quot r → Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h₁)\n (by\n intro c; apply Quot.inductionOn (motive := _) c; clear c\n intro c d; apply Quot.inductionOn (motive := _) d; clear d\n intro d rcd; apply Quot.sound; apply rcd)\n have : f ∘ Quot.mk r ∘ Quot.mk Mcongr = Quot.mk r' := rfl\n rw [← this, PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map,\n abs_map, h₀]\n rw [PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map, abs_map]"
},
{
"state_after": "no goals",
"state_before": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nthis : IsPrecongr r'\n⊢ Mcongr x y",
"tactic": "refine' ⟨r', this, rxy⟩"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\n⊢ abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n⊢ IsPrecongr r'",
"tactic": "intro a b r'ab"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\n⊢ abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\n⊢ abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)",
"tactic": "have h₀ :\n Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =\n Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=\n h _ _ r'ab"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\n⊢ abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\n⊢ abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)",
"tactic": "have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by\n intro u v cuv\n apply Quot.sound\n simp only\n rw [Quot.sound cuv]\n apply h'"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nf : Quot r → Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h₁)\n (_ : ∀ (c b : Cofix F), r c b → Quot.lift (Quot.mk r') h₁ c = Quot.lift (Quot.mk r') h₁ b)\n⊢ abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\n⊢ abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)",
"tactic": "let f : Quot r → Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h₁)\n (by\n intro c; apply Quot.inductionOn (motive := _) c; clear c\n intro c d; apply Quot.inductionOn (motive := _) d; clear d\n intro d rcd; apply Quot.sound; apply rcd)"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nf : Quot r → Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h₁)\n (_ : ∀ (c b : Cofix F), r c b → Quot.lift (Quot.mk r') h₁ c = Quot.lift (Quot.mk r') h₁ b)\nthis : f ∘ Quot.mk r ∘ Quot.mk Mcongr = Quot.mk r'\n⊢ abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nf : Quot r → Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h₁)\n (_ : ∀ (c b : Cofix F), r c b → Quot.lift (Quot.mk r') h₁ c = Quot.lift (Quot.mk r') h₁ b)\n⊢ abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)",
"tactic": "have : f ∘ Quot.mk r ∘ Quot.mk Mcongr = Quot.mk r' := rfl"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nf : Quot r → Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h₁)\n (_ : ∀ (c b : Cofix F), r c b → Quot.lift (Quot.mk r') h₁ c = Quot.lift (Quot.mk r') h₁ b)\nthis : f ∘ Quot.mk r ∘ Quot.mk Mcongr = Quot.mk r'\n⊢ f <$> Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) =\n abs ((f ∘ Quot.mk r ∘ Quot.mk Mcongr) <$> PFunctor.M.dest b)",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nf : Quot r → Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h₁)\n (_ : ∀ (c b : Cofix F), r c b → Quot.lift (Quot.mk r') h₁ c = Quot.lift (Quot.mk r') h₁ b)\nthis : f ∘ Quot.mk r ∘ Quot.mk Mcongr = Quot.mk r'\n⊢ abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)",
"tactic": "rw [← this, PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map,\n abs_map, h₀]"
},
{
"state_after": "no goals",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nf : Quot r → Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h₁)\n (_ : ∀ (c b : Cofix F), r c b → Quot.lift (Quot.mk r') h₁ c = Quot.lift (Quot.mk r') h₁ b)\nthis : f ∘ Quot.mk r ∘ Quot.mk Mcongr = Quot.mk r'\n⊢ f <$> Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) =\n abs ((f ∘ Quot.mk r ∘ Quot.mk Mcongr) <$> PFunctor.M.dest b)",
"tactic": "rw [PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map, abs_map]"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n⊢ Quot.mk r' u = Quot.mk r' v",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\n⊢ ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v",
"tactic": "intro u v cuv"
},
{
"state_after": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n⊢ r' u v",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n⊢ Quot.mk r' u = Quot.mk r' v",
"tactic": "apply Quot.sound"
},
{
"state_after": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n⊢ r (Quot.mk Mcongr u) (Quot.mk Mcongr v)",
"state_before": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n⊢ r' u v",
"tactic": "simp only"
},
{
"state_after": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n⊢ r (Quot.mk Mcongr v) (Quot.mk Mcongr v)",
"state_before": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n⊢ r (Quot.mk Mcongr u) (Quot.mk Mcongr v)",
"tactic": "rw [Quot.sound cuv]"
},
{
"state_after": "no goals",
"state_before": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n⊢ r (Quot.mk Mcongr v) (Quot.mk Mcongr v)",
"tactic": "apply h'"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc : Cofix F\n⊢ ∀ (b : Cofix F), r c b → Quot.lift (Quot.mk r') h₁ c = Quot.lift (Quot.mk r') h₁ b",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\n⊢ ∀ (a b : Cofix F), r a b → Quot.lift (Quot.mk r') h₁ a = Quot.lift (Quot.mk r') h₁ b",
"tactic": "intro c"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc : Cofix F\n⊢ ∀ (a : PFunctor.M (P F)) (b : Cofix F),\n r (Quot.mk Mcongr a) b → Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h₁ b",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc : Cofix F\n⊢ ∀ (b : Cofix F), r c b → Quot.lift (Quot.mk r') h₁ c = Quot.lift (Quot.mk r') h₁ b",
"tactic": "apply Quot.inductionOn (motive := _) c"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\n⊢ ∀ (a : PFunctor.M (P F)) (b : Cofix F),\n r (Quot.mk Mcongr a) b → Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h₁ b",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc : Cofix F\n⊢ ∀ (a : PFunctor.M (P F)) (b : Cofix F),\n r (Quot.mk Mcongr a) b → Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h₁ b",
"tactic": "clear c"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\nd : Cofix F\n⊢ r (Quot.mk Mcongr c) d → Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ d",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\n⊢ ∀ (a : PFunctor.M (P F)) (b : Cofix F),\n r (Quot.mk Mcongr a) b → Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h₁ b",
"tactic": "intro c d"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\nd : Cofix F\n⊢ ∀ (a : PFunctor.M (P F)),\n r (Quot.mk Mcongr c) (Quot.mk Mcongr a) →\n Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr a)",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\nd : Cofix F\n⊢ r (Quot.mk Mcongr c) d → Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ d",
"tactic": "apply Quot.inductionOn (motive := _) d"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\n⊢ ∀ (a : PFunctor.M (P F)),\n r (Quot.mk Mcongr c) (Quot.mk Mcongr a) →\n Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr a)",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\nd : Cofix F\n⊢ ∀ (a : PFunctor.M (P F)),\n r (Quot.mk Mcongr c) (Quot.mk Mcongr a) →\n Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr a)",
"tactic": "clear d"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc d : PFunctor.M (P F)\nrcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)\n⊢ Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr d)",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\n⊢ ∀ (a : PFunctor.M (P F)),\n r (Quot.mk Mcongr c) (Quot.mk Mcongr a) →\n Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr a)",
"tactic": "intro d rcd"
},
{
"state_after": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc d : PFunctor.M (P F)\nrcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)\n⊢ r' c d",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc d : PFunctor.M (P F)\nrcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)\n⊢ Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr d)",
"tactic": "apply Quot.sound"
},
{
"state_after": "no goals",
"state_before": "case a\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nr : Cofix F → Cofix F → Prop\nh' : ∀ (x : Cofix F), r x x\nh : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v\nc d : PFunctor.M (P F)\nrcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)\n⊢ r' c d",
"tactic": "apply rcd"
}
] |
[
469,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
435,
9
] |
Mathlib/Data/Real/ConjugateExponents.lean
|
Real.IsConjugateExponent.ne_zero
|
[] |
[
56,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPushout.of_hasBinaryCoproduct'
|
[] |
[
401,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
399,
1
] |
Mathlib/Data/Nat/Bitwise.lean
|
Nat.testBit_two_pow_self
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ testBit (2 ^ n) n = true",
"tactic": "rw [testBit, shiftr_eq_div_pow, Nat.div_self (pow_pos (α := ℕ) zero_lt_two n), bodd_one]"
}
] |
[
143,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Analysis/Convex/Exposed.lean
|
exposedPoints_empty
|
[] |
[
218,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Data/Nat/Count.lean
|
Nat.count_le_card
|
[
{
"state_after": "p : ℕ → Prop\ninst✝ : DecidablePred p\nhp : Set.Finite (setOf p)\nn : ℕ\n⊢ card (filter p (range n)) ≤ card (Set.Finite.toFinset hp)",
"state_before": "p : ℕ → Prop\ninst✝ : DecidablePred p\nhp : Set.Finite (setOf p)\nn : ℕ\n⊢ count p n ≤ card (Set.Finite.toFinset hp)",
"tactic": "rw [count_eq_card_filter_range]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ → Prop\ninst✝ : DecidablePred p\nhp : Set.Finite (setOf p)\nn : ℕ\n⊢ card (filter p (range n)) ≤ card (Set.Finite.toFinset hp)",
"tactic": "exact Finset.card_mono fun x hx ↦ hp.mem_toFinset.2 (mem_filter.1 hx).2"
}
] |
[
145,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Topology/Instances/EReal.lean
|
EReal.tendsto_coe
|
[] |
[
68,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.add_eq_max'
|
[
{
"state_after": "no goals",
"state_before": "a b : Cardinal\nha : ℵ₀ ≤ b\n⊢ a + b = max a b",
"tactic": "rw [add_comm, max_comm, add_eq_max ha]"
}
] |
[
736,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
735,
1
] |
Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
|
Finpartition.IsUniform.mono
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε✝ ε ε' : 𝕜\nhP : IsUniform P G ε\nh : ε ≤ ε'\n⊢ ↑(card P.parts * (card P.parts - 1)) * ε ≤ ↑(card P.parts * (card P.parts - 1)) * ε'",
"tactic": "gcongr"
}
] |
[
252,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
InnerProductSpace.Core.inner_neg_left
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.582097\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ ↑(starRingEnd 𝕜) (-1) * inner x y = -inner x y",
"state_before": "𝕜 : Type u_1\nE : Type ?u.582097\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ inner (-x) y = -inner x y",
"tactic": "rw [← neg_one_smul 𝕜 x, inner_smul_left]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type ?u.582097\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ ↑(starRingEnd 𝕜) (-1) * inner x y = -inner x y",
"tactic": "simp"
}
] |
[
285,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/GroupTheory/GroupAction/SubMulAction.lean
|
SubMulAction.val_smul
|
[] |
[
185,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
ContinuousOn.mono_rng
|
[] |
[
642,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
639,
1
] |
Mathlib/Data/Nat/ModEq.lean
|
Nat.odd_mul_odd
|
[
{
"state_after": "no goals",
"state_before": "m✝ n✝ a b c d n m : ℕ\n⊢ n % 2 = 1 → m % 2 = 1 → n * m % 2 = 1",
"tactic": "simpa [Nat.ModEq] using @ModEq.mul 2 n 1 m 1"
}
] |
[
497,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
496,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.empty_inter
|
[] |
[
922,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
921,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
toTopologicalSpace_mono
|
[] |
[
1333,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1331,
1
] |
Mathlib/Data/List/Basic.lean
|
List.splitOnP_eq_single
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.302293\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nh : ∀ (x : α), x ∈ xs → ¬p x = true\n⊢ splitOnP p xs = [xs]",
"tactic": "induction xs with\n| nil => rfl\n| cons hd tl ih =>\n simp only [splitOnP_cons, h hd (mem_cons_self hd tl), if_neg]\n rw [ih <| forall_mem_of_forall_mem_cons h]\n rfl"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.302293\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nh : ∀ (x : α), x ∈ [] → ¬p x = true\n⊢ splitOnP p [] = [[]]",
"tactic": "rfl"
},
{
"state_after": "case cons\nι : Type ?u.302293\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nhd : α\ntl : List α\nih : (∀ (x : α), x ∈ tl → ¬p x = true) → splitOnP p tl = [tl]\nh : ∀ (x : α), x ∈ hd :: tl → ¬p x = true\n⊢ modifyHead (cons hd) (splitOnP p tl) = [hd :: tl]",
"state_before": "case cons\nι : Type ?u.302293\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nhd : α\ntl : List α\nih : (∀ (x : α), x ∈ tl → ¬p x = true) → splitOnP p tl = [tl]\nh : ∀ (x : α), x ∈ hd :: tl → ¬p x = true\n⊢ splitOnP p (hd :: tl) = [hd :: tl]",
"tactic": "simp only [splitOnP_cons, h hd (mem_cons_self hd tl), if_neg]"
},
{
"state_after": "case cons\nι : Type ?u.302293\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nhd : α\ntl : List α\nih : (∀ (x : α), x ∈ tl → ¬p x = true) → splitOnP p tl = [tl]\nh : ∀ (x : α), x ∈ hd :: tl → ¬p x = true\n⊢ modifyHead (cons hd) [tl] = [hd :: tl]",
"state_before": "case cons\nι : Type ?u.302293\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nhd : α\ntl : List α\nih : (∀ (x : α), x ∈ tl → ¬p x = true) → splitOnP p tl = [tl]\nh : ∀ (x : α), x ∈ hd :: tl → ¬p x = true\n⊢ modifyHead (cons hd) (splitOnP p tl) = [hd :: tl]",
"tactic": "rw [ih <| forall_mem_of_forall_mem_cons h]"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.302293\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nhd : α\ntl : List α\nih : (∀ (x : α), x ∈ tl → ¬p x = true) → splitOnP p tl = [tl]\nh : ∀ (x : α), x ∈ hd :: tl → ¬p x = true\n⊢ modifyHead (cons hd) [tl] = [hd :: tl]",
"tactic": "rfl"
}
] |
[
2972,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2966,
1
] |
Mathlib/RingTheory/Noetherian.lean
|
set_has_maximal_iff_noetherian
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nM : Type u_1\nP : Type ?u.118511\nN : Type w\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\n⊢ (∀ (a : Set (Submodule R M)), Set.Nonempty a → ∃ M', M' ∈ a ∧ ∀ (I : Submodule R M), I ∈ a → ¬M' < I) ↔\n IsNoetherian R M",
"tactic": "rw [isNoetherian_iff_wellFounded, WellFounded.wellFounded_iff_has_min]"
}
] |
[
349,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
347,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
MeasureTheory.IntegrableOn.indicator
|
[] |
[
278,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Algebra/Associated.lean
|
Associates.mk_le_mk_iff_dvd_iff
|
[] |
[
944,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
943,
1
] |
Mathlib/RingTheory/Polynomial/Bernstein.lean
|
bernsteinPolynomial.variance
|
[
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ ν in Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +\n (↑1 - (2 * n) • X) * ∑ ν in Finset.range (n + 1), ν • bernsteinPolynomial R n ν +\n n ^ 2 • X ^ 2 * ∑ ν in Finset.range (n + 1), bernsteinPolynomial R n ν =\n ∑ ν in Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +\n (↑1 - (2 * n) • X) * ∑ ν in Finset.range (n + 1), ν • bernsteinPolynomial R n ν +\n n ^ 2 • X ^ 2 * ∑ ν in Finset.range (n + 1), bernsteinPolynomial R n ν\n⊢ ∑ ν in Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\n⊢ ∑ ν in Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)",
"tactic": "have p : ((((Finset.range (n + 1)).sum fun ν => (ν * (ν - 1)) • bernsteinPolynomial R n ν) +\n (1 - (2 * n) • Polynomial.X) * (Finset.range (n + 1)).sum fun ν =>\n ν • bernsteinPolynomial R n ν) + n ^ 2 • X ^ 2 *\n (Finset.range (n + 1)).sum fun ν => bernsteinPolynomial R n ν) = _ :=\n rfl"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ∑ ν in Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +\n (↑1 - (2 * n) • X) * ∑ ν in Finset.range (n + 1), ν • bernsteinPolynomial R n ν +\n n ^ 2 • X ^ 2 * ∑ ν in Finset.range (n + 1), bernsteinPolynomial R n ν\n⊢ ∑ ν in Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ ν in Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +\n (↑1 - (2 * n) • X) * ∑ ν in Finset.range (n + 1), ν • bernsteinPolynomial R n ν +\n n ^ 2 • X ^ 2 * ∑ ν in Finset.range (n + 1), bernsteinPolynomial R n ν =\n ∑ ν in Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +\n (↑1 - (2 * n) • X) * ∑ ν in Finset.range (n + 1), ν • bernsteinPolynomial R n ν +\n n ^ 2 • X ^ 2 * ∑ ν in Finset.range (n + 1), bernsteinPolynomial R n ν\n⊢ ∑ ν in Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)",
"tactic": "conv at p =>\n lhs\n rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_add_distrib, ← Finset.sum_add_distrib]\n simp only [← nat_cast_mul]\n simp only [← mul_assoc]\n simp only [← add_mul]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\n⊢ ∑ ν in Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ∑ ν in Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +\n (↑1 - (2 * n) • X) * ∑ ν in Finset.range (n + 1), ν • bernsteinPolynomial R n ν +\n n ^ 2 • X ^ 2 * ∑ ν in Finset.range (n + 1), bernsteinPolynomial R n ν\n⊢ ∑ ν in Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)",
"tactic": "conv at p =>\n rhs\n rw [sum, sum_smul, sum_mul_smul, ← nat_cast_mul]"
},
{
"state_after": "case calc_1\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\nk : ℕ\nm : k ∈ Finset.range (n + 1)\n⊢ (n • X - ↑k) ^ 2 * bernsteinPolynomial R n k =\n (↑(k * (k - 1)) + (↑1 - ↑(2 * n) * X) * ↑k + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n k\n\ncase calc_2\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\n⊢ ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1 = n • X * (1 - X)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\n⊢ ∑ ν in Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)",
"tactic": "calc\n _ = _ := Finset.sum_congr rfl fun k m => ?_\n _ = _ := p\n _ = _ := ?_"
},
{
"state_after": "case calc_1.e_a\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\nk : ℕ\nm : k ∈ Finset.range (n + 1)\n⊢ (n • X - ↑k) ^ 2 = ↑(k * (k - 1)) + (↑1 - ↑(2 * n) * X) * ↑k + ↑(n ^ 2) * X ^ 2",
"state_before": "case calc_1\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\nk : ℕ\nm : k ∈ Finset.range (n + 1)\n⊢ (n • X - ↑k) ^ 2 * bernsteinPolynomial R n k =\n (↑(k * (k - 1)) + (↑1 - ↑(2 * n) * X) * ↑k + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n k",
"tactic": "congr 1"
},
{
"state_after": "case calc_1.e_a\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\nk : ℕ\nm : k ∈ Finset.range (n + 1)\n⊢ (↑n * X - ↑k) ^ 2 = ↑k * ↑(k - 1) + (1 - 2 * ↑n * X) * ↑k + ↑n ^ 2 * X ^ 2",
"state_before": "case calc_1.e_a\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\nk : ℕ\nm : k ∈ Finset.range (n + 1)\n⊢ (n • X - ↑k) ^ 2 = ↑(k * (k - 1)) + (↑1 - ↑(2 * n) * X) * ↑k + ↑(n ^ 2) * X ^ 2",
"tactic": "simp only [← nat_cast_mul, push_cast]"
},
{
"state_after": "case calc_1.e_a.succ\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\nn✝ : ℕ\nm : Nat.succ n✝ ∈ Finset.range (n + 1)\n⊢ (↑n * X - (↑n✝ + 1)) ^ 2 = (↑n✝ + 1) * ↑n✝ + (1 - 2 * ↑n * X) * (↑n✝ + 1) + ↑n ^ 2 * X ^ 2",
"state_before": "case calc_1.e_a.succ\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\nn✝ : ℕ\nm : Nat.succ n✝ ∈ Finset.range (n + 1)\n⊢ (↑n * X - ↑(Nat.succ n✝)) ^ 2 =\n ↑(Nat.succ n✝) * ↑(Nat.succ n✝ - 1) + (1 - 2 * ↑n * X) * ↑(Nat.succ n✝) + ↑n ^ 2 * X ^ 2",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case calc_1.e_a.succ\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\nn✝ : ℕ\nm : Nat.succ n✝ ∈ Finset.range (n + 1)\n⊢ (↑n * X - (↑n✝ + 1)) ^ 2 = (↑n✝ + 1) * ↑n✝ + (1 - 2 * ↑n * X) * (↑n✝ + 1) + ↑n ^ 2 * X ^ 2",
"tactic": "ring"
},
{
"state_after": "case calc_2\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\n⊢ ↑n * ↑(n - 1) * X ^ 2 + (1 - 2 * ↑n * X) * (↑n * X) + ↑n ^ 2 * X ^ 2 * 1 = ↑n * X * (1 - X)",
"state_before": "case calc_2\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\n⊢ ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1 = n • X * (1 - X)",
"tactic": "simp only [← nat_cast_mul, push_cast]"
},
{
"state_after": "case calc_2.zero\nR : Type u_1\ninst✝ : CommRing R\np :\n ∑ x in Finset.range (Nat.zero + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * Nat.zero) * X) * ↑x + ↑(Nat.zero ^ 2) * X ^ 2) * bernsteinPolynomial R Nat.zero x =\n ↑(Nat.zero * (Nat.zero - 1)) * X ^ 2 + (↑1 - (2 * Nat.zero) • X) * Nat.zero • X + Nat.zero ^ 2 • X ^ 2 * 1\n⊢ ↑Nat.zero * ↑(Nat.zero - 1) * X ^ 2 + (1 - 2 * ↑Nat.zero * X) * (↑Nat.zero * X) + ↑Nat.zero ^ 2 * X ^ 2 * 1 =\n ↑Nat.zero * X * (1 - X)\n\ncase calc_2.succ\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\np :\n ∑ x in Finset.range (Nat.succ n✝ + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * Nat.succ n✝) * X) * ↑x + ↑(Nat.succ n✝ ^ 2) * X ^ 2) *\n bernsteinPolynomial R (Nat.succ n✝) x =\n ↑(Nat.succ n✝ * (Nat.succ n✝ - 1)) * X ^ 2 + (↑1 - (2 * Nat.succ n✝) • X) * Nat.succ n✝ • X +\n Nat.succ n✝ ^ 2 • X ^ 2 * 1\n⊢ ↑(Nat.succ n✝) * ↑(Nat.succ n✝ - 1) * X ^ 2 + (1 - 2 * ↑(Nat.succ n✝) * X) * (↑(Nat.succ n✝) * X) +\n ↑(Nat.succ n✝) ^ 2 * X ^ 2 * 1 =\n ↑(Nat.succ n✝) * X * (1 - X)",
"state_before": "case calc_2\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x in Finset.range (n + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (↑1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\n⊢ ↑n * ↑(n - 1) * X ^ 2 + (1 - 2 * ↑n * X) * (↑n * X) + ↑n ^ 2 * X ^ 2 * 1 = ↑n * X * (1 - X)",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case calc_2.zero\nR : Type u_1\ninst✝ : CommRing R\np :\n ∑ x in Finset.range (Nat.zero + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * Nat.zero) * X) * ↑x + ↑(Nat.zero ^ 2) * X ^ 2) * bernsteinPolynomial R Nat.zero x =\n ↑(Nat.zero * (Nat.zero - 1)) * X ^ 2 + (↑1 - (2 * Nat.zero) • X) * Nat.zero • X + Nat.zero ^ 2 • X ^ 2 * 1\n⊢ ↑Nat.zero * ↑(Nat.zero - 1) * X ^ 2 + (1 - 2 * ↑Nat.zero * X) * (↑Nat.zero * X) + ↑Nat.zero ^ 2 * X ^ 2 * 1 =\n ↑Nat.zero * X * (1 - X)",
"tactic": "simp"
},
{
"state_after": "case calc_2.succ\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\np :\n ∑ x in Finset.range (Nat.succ n✝ + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * Nat.succ n✝) * X) * ↑x + ↑(Nat.succ n✝ ^ 2) * X ^ 2) *\n bernsteinPolynomial R (Nat.succ n✝) x =\n ↑(Nat.succ n✝ * (Nat.succ n✝ - 1)) * X ^ 2 + (↑1 - (2 * Nat.succ n✝) • X) * Nat.succ n✝ • X +\n Nat.succ n✝ ^ 2 • X ^ 2 * 1\n⊢ (↑n✝ + 1) * ↑n✝ * X ^ 2 + (1 - 2 * (↑n✝ + 1) * X) * ((↑n✝ + 1) * X) + (↑n✝ + 1) ^ 2 * X ^ 2 = (↑n✝ + 1) * X * (1 - X)",
"state_before": "case calc_2.succ\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\np :\n ∑ x in Finset.range (Nat.succ n✝ + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * Nat.succ n✝) * X) * ↑x + ↑(Nat.succ n✝ ^ 2) * X ^ 2) *\n bernsteinPolynomial R (Nat.succ n✝) x =\n ↑(Nat.succ n✝ * (Nat.succ n✝ - 1)) * X ^ 2 + (↑1 - (2 * Nat.succ n✝) • X) * Nat.succ n✝ • X +\n Nat.succ n✝ ^ 2 • X ^ 2 * 1\n⊢ ↑(Nat.succ n✝) * ↑(Nat.succ n✝ - 1) * X ^ 2 + (1 - 2 * ↑(Nat.succ n✝) * X) * (↑(Nat.succ n✝) * X) +\n ↑(Nat.succ n✝) ^ 2 * X ^ 2 * 1 =\n ↑(Nat.succ n✝) * X * (1 - X)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case calc_2.succ\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\np :\n ∑ x in Finset.range (Nat.succ n✝ + 1),\n (↑(x * (x - 1)) + (↑1 - ↑(2 * Nat.succ n✝) * X) * ↑x + ↑(Nat.succ n✝ ^ 2) * X ^ 2) *\n bernsteinPolynomial R (Nat.succ n✝) x =\n ↑(Nat.succ n✝ * (Nat.succ n✝ - 1)) * X ^ 2 + (↑1 - (2 * Nat.succ n✝) • X) * Nat.succ n✝ • X +\n Nat.succ n✝ ^ 2 • X ^ 2 * 1\n⊢ (↑n✝ + 1) * ↑n✝ * X ^ 2 + (1 - 2 * (↑n✝ + 1) * X) * ((↑n✝ + 1) * X) + (↑n✝ + 1) ^ 2 * X ^ 2 = (↑n✝ + 1) * X * (1 - X)",
"tactic": "ring"
}
] |
[
417,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
391,
1
] |
Mathlib/LinearAlgebra/Matrix/MvPolynomial.lean
|
Matrix.mvPolynomialX_map_eval₂
|
[] |
[
56,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Topology/PathConnected.lean
|
Path.extend_one
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.157058\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.157073\nγ : Path x y\n⊢ extend γ 1 = y",
"tactic": "simp"
}
] |
[
274,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
integrableOn_Icc_iff_integrableOn_Ico
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.4109768\nE : Type u_2\nF : Type ?u.4109774\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : PartialOrder α\ninst✝¹ : MeasurableSingletonClass α\nf : α → E\nμ : MeasureTheory.Measure α\na b : α\ninst✝ : NoAtoms μ\n⊢ 0 ≠ ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.4109768\nE : Type u_2\nF : Type ?u.4109774\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : PartialOrder α\ninst✝¹ : MeasurableSingletonClass α\nf : α → E\nμ : MeasureTheory.Measure α\na b : α\ninst✝ : NoAtoms μ\n⊢ ↑↑μ {b} ≠ ⊤",
"tactic": "rw [measure_singleton]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4109768\nE : Type u_2\nF : Type ?u.4109774\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : PartialOrder α\ninst✝¹ : MeasurableSingletonClass α\nf : α → E\nμ : MeasureTheory.Measure α\na b : α\ninst✝ : NoAtoms μ\n⊢ 0 ≠ ⊤",
"tactic": "exact ENNReal.zero_ne_top"
}
] |
[
684,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
682,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
Int.ofAdd_mul
|
[] |
[
1228,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1227,
1
] |
Mathlib/Topology/Filter.lean
|
Filter.nhds_eq'
|
[] |
[
84,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.coe_sort_coe
|
[] |
[
289,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIocDiv_zsmul_sub_toIocMod
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIocDiv hp a b • p + toIocMod hp a b = b",
"tactic": "rw [add_comm, toIocMod_add_toIocDiv_zsmul]"
}
] |
[
171,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
pow_sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.95648\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝³ : Group G\ninst✝² : Group H\ninst✝¹ : AddGroup A\ninst✝ : AddGroup B\na : G\nm n : ℕ\nh : n ≤ m\n⊢ a ^ (m - n) * a ^ n = a ^ m",
"tactic": "rw [← pow_add, Nat.sub_add_cancel h]"
}
] |
[
425,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MonoidHom.cancel_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.169579\nβ : Type ?u.169582\nM : Type u_1\nN : Type u_2\nP : Type u_3\nG : Type ?u.169594\nH : Type ?u.169597\nF : Type ?u.169600\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\ng : N →* P\nf₁ f₂ : M →* N\nhg : Function.Injective ↑g\nh : comp g f₁ = comp g f₂\nx : M\n⊢ ↑g (↑f₁ x) = ↑g (↑f₂ x)",
"tactic": "rw [← MonoidHom.comp_apply, h, MonoidHom.comp_apply]"
}
] |
[
1207,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1204,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_le_mk_mul_of_mk_preimage_le
|
[
{
"state_after": "no goals",
"state_before": "α β : Type u\nc : Cardinal\nf : α → β\nhf : ∀ (b : β), (#↑(f ⁻¹' {b})) ≤ c\n⊢ (#α) ≤ (#β) * c",
"tactic": "simpa only [← mk_congr (@Equiv.sigmaFiberEquiv α β f), mk_sigma, ← sum_const'] using\n sum_le_sum _ _ hf"
}
] |
[
915,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
912,
1
] |
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
|
MvPolynomial.isWeightedHomogeneous_C
|
[] |
[
219,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Analysis/Complex/CauchyIntegral.lean
|
Complex.integral_boundary_rect_of_hasFDerivAt_real_off_countable
|
[
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I)) =\n ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • ↑(f' (↑x + ↑y * I)) 1 - ↑(f' (↑x + ↑y * I)) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I)) =\n ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • ↑(f' (↑x + ↑y * I)) 1 - ↑(f' (↑x + ↑y * I)) I",
"tactic": "set e : (ℝ × ℝ) ≃L[ℝ] ℂ := equivRealProdClm.symm"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe : ∀ (x y : ℝ), ↑x + ↑y * I = ↑e (x, y)\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I)) =\n ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • ↑(f' (↑x + ↑y * I)) 1 - ↑(f' (↑x + ↑y * I)) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I)) =\n ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • ↑(f' (↑x + ↑y * I)) 1 - ↑(f' (↑x + ↑y * I)) I",
"tactic": "have he : ∀ x y : ℝ, ↑x + ↑y * I = e (x, y) := fun x y => (mk_eq_add_mul_I x y).symm"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe : ∀ (x y : ℝ), ↑x + ↑y * I = ↑e (x, y)\nhe₁ : ↑e (1, 0) = 1\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I)) =\n ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • ↑(f' (↑x + ↑y * I)) 1 - ↑(f' (↑x + ↑y * I)) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe : ∀ (x y : ℝ), ↑x + ↑y * I = ↑e (x, y)\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I)) =\n ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • ↑(f' (↑x + ↑y * I)) 1 - ↑(f' (↑x + ↑y * I)) I",
"tactic": "have he₁ : e (1, 0) = 1 := rfl"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe : ∀ (x y : ℝ), ↑x + ↑y * I = ↑e (x, y)\nhe₁ : ↑e (1, 0) = 1\nhe₂ : ↑e (0, 1) = I\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I)) =\n ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • ↑(f' (↑x + ↑y * I)) 1 - ↑(f' (↑x + ↑y * I)) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe : ∀ (x y : ℝ), ↑x + ↑y * I = ↑e (x, y)\nhe₁ : ↑e (1, 0) = 1\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I)) =\n ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • ↑(f' (↑x + ↑y * I)) 1 - ↑(f' (↑x + ↑y * I)) I",
"tactic": "have he₂ : e (0, 1) = I := rfl"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe : ∀ (x y : ℝ), ↑x + ↑y * I = ↑e (x, y)\nhe₁ : ↑e (1, 0) = 1\nhe₂ : ↑e (0, 1) = I\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I)) =\n ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • ↑(f' (↑x + ↑y * I)) 1 - ↑(f' (↑x + ↑y * I)) I",
"tactic": "simp only [he] at *"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"tactic": "set F : ℝ × ℝ → E := f ∘ e"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"tactic": "set F' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => (f' (e p)).comp (e : ℝ × ℝ →L[ℝ] ℂ)"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"tactic": "have hF' : ∀ p : ℝ × ℝ, (-(I • F' p)) (1, 0) + F' p (0, 1) = -(I • f' (e p) 1 - f' (e p) I) := by\n rintro ⟨x, y⟩\n simp only [ContinuousLinearMap.neg_apply, ContinuousLinearMap.smul_apply,\n ContinuousLinearMap.comp_apply, ContinuousLinearEquiv.coe_coe, he₁, he₂, neg_add_eq_sub,\n neg_sub]"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"tactic": "set R : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"tactic": "set t : Set (ℝ × ℝ) := e ⁻¹' s"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"tactic": "rw [uIcc_comm z.im] at Hc Hi"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"tactic": "rw [min_comm z.im, max_comm z.im] at Hd"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\nhR : ↑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"tactic": "have hR : e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R := rfl"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\nhR : ↑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\nhR : ↑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"tactic": "have htc : ContinuousOn F R := Hc.comp e.continuousOn hR.ge"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\nhR : ↑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\nhtd :\n ∀ (p : ℝ × ℝ),\n p ∈ Ioo (min z.re w.re) (max z.re w.re) ×ˢ Ioo (min w.im z.im) (max w.im z.im) \\ t → HasFDerivAt F (F' p) p\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\nhR : ↑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"tactic": "have htd :\n ∀ p ∈ Ioo (min z.re w.re) (max z.re w.re) ×ˢ Ioo (min w.im z.im) (max w.im z.im) \\ t,\n HasFDerivAt F (F' p) p :=\n fun p hp => (Hd (e p) hp).comp p e.hasFDerivAt"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\nhR : ↑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\nhtd :\n ∀ (p : ℝ × ℝ),\n p ∈ Ioo (min z.re w.re) (max z.re w.re) ×ˢ Ioo (min w.im z.im) (max w.im z.im) \\ t → HasFDerivAt F (F' p) p\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n ∫ (x : ℝ) in w.im..z.im, -(I • f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, x)))) -\n ∫ (x : ℝ) in w.im..z.im, -(I • f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, x)))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (x_1 : ℝ) in w.im..z.im,\n ↑(-(I •\n ContinuousLinearMap.comp (f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, x_1)))\n ↑(ContinuousLinearEquiv.symm equivRealProdClm)))\n (1, 0) +\n ↑(ContinuousLinearMap.comp (f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, x_1)))\n ↑(ContinuousLinearEquiv.symm equivRealProdClm))\n (0, 1)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\nhR : ↑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\nhtd :\n ∀ (p : ℝ × ℝ),\n p ∈ Ioo (min z.re w.re) (max z.re w.re) ×ˢ Ioo (min w.im z.im) (max w.im z.im) \\ t → HasFDerivAt F (F' p) p\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (y : ℝ) in z.im..w.im,\n I • ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n ↑(f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I",
"tactic": "simp_rw [← intervalIntegral.integral_smul, intervalIntegral.integral_symm w.im z.im, ←\n intervalIntegral.integral_neg, ← hF']"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\nhR : ↑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\nhtd :\n ∀ (p : ℝ × ℝ),\n p ∈ Ioo (min z.re w.re) (max z.re w.re) ×ˢ Ioo (min w.im z.im) (max w.im z.im) \\ t → HasFDerivAt F (F' p) p\n⊢ IntegrableOn (fun x => ↑((fun p => -(I • F' p)) x) (1, 0) + ↑(F' x) (0, 1)) ([[z.re, w.re]] ×ˢ [[w.im, z.im]])",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\nhR : ↑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\nhtd :\n ∀ (p : ℝ × ℝ),\n p ∈ Ioo (min z.re w.re) (max z.re w.re) ×ˢ Ioo (min w.im z.im) (max w.im z.im) \\ t → HasFDerivAt F (F' p) p\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n ∫ (x : ℝ) in z.re..w.re, f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n ∫ (x : ℝ) in w.im..z.im, -(I • f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, x)))) -\n ∫ (x : ℝ) in w.im..z.im, -(I • f (↑(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, x)))) =\n ∫ (x : ℝ) in z.re..w.re,\n ∫ (x_1 : ℝ) in w.im..z.im,\n ↑(-(I •\n ContinuousLinearMap.comp (f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, x_1)))\n ↑(ContinuousLinearEquiv.symm equivRealProdClm)))\n (1, 0) +\n ↑(ContinuousLinearMap.comp (f' (↑(ContinuousLinearEquiv.symm equivRealProdClm) (x, x_1)))\n ↑(ContinuousLinearEquiv.symm equivRealProdClm))\n (0, 1)",
"tactic": "refine' (integral2_divergence_prod_of_hasFDerivWithinAt_off_countable (fun p => -(I • F p)) F\n (fun p => -(I • F' p)) F' z.re w.im w.re z.im t (hs.preimage e.injective)\n (htc.const_smul _).neg htc (fun p hp => ((htd p hp).const_smul I).neg) htd _).symm"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi :\n IntegrableOn ((fun z => I • ↑(f' z) 1 - ↑(f' z) I) ∘ ↑(MeasurableEquiv.symm measurableEquivRealProd))\n (↑(MeasurableEquiv.symm measurableEquivRealProd) ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]))\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\nhR : ↑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\nhtd :\n ∀ (p : ℝ × ℝ),\n p ∈ Ioo (min z.re w.re) (max z.re w.re) ×ˢ Ioo (min w.im z.im) (max w.im z.im) \\ t → HasFDerivAt F (F' p) p\n⊢ IntegrableOn (fun x => ↑((fun p => -(I • F' p)) x) (1, 0) + ↑(F' x) (0, 1)) ([[z.re, w.re]] ×ˢ [[w.im, z.im]])",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\nhR : ↑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\nhtd :\n ∀ (p : ℝ × ℝ),\n p ∈ Ioo (min z.re w.re) (max z.re w.re) ×ˢ Ioo (min w.im z.im) (max w.im z.im) \\ t → HasFDerivAt F (F' p) p\n⊢ IntegrableOn (fun x => ↑((fun p => -(I • F' p)) x) (1, 0) + ↑(F' x) (0, 1)) ([[z.re, w.re]] ×ˢ [[w.im, z.im]])",
"tactic": "rw [← (volume_preserving_equiv_real_prod.symm _).integrableOn_comp_preimage\n (MeasurableEquiv.measurableEmbedding _)] at Hi"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s → HasFDerivAt f (f' x) x\nHi :\n IntegrableOn ((fun z => I • ↑(f' z) 1 - ↑(f' z) I) ∘ ↑(MeasurableEquiv.symm measurableEquivRealProd))\n (↑(MeasurableEquiv.symm measurableEquivRealProd) ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]))\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nhF' : ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)\nR : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]\nt : Set (ℝ × ℝ) := ↑e ⁻¹' s\nhR : ↑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\nhtd :\n ∀ (p : ℝ × ℝ),\n p ∈ Ioo (min z.re w.re) (max z.re w.re) ×ˢ Ioo (min w.im z.im) (max w.im z.im) \\ t → HasFDerivAt F (F' p) p\n⊢ IntegrableOn (fun x => ↑((fun p => -(I • F' p)) x) (1, 0) + ↑(F' x) (0, 1)) ([[z.re, w.re]] ×ˢ [[w.im, z.im]])",
"tactic": "simpa only [hF'] using Hi.neg"
},
{
"state_after": "case mk\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nx y : ℝ\n⊢ ↑(-(I • F' (x, y))) (1, 0) + ↑(F' (x, y)) (0, 1) = -(I • ↑(f' (↑e (x, y))) 1 - ↑(f' (↑e (x, y))) I)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\n⊢ ∀ (p : ℝ × ℝ), ↑(-(I • F' p)) (1, 0) + ↑(F' p) (0, 1) = -(I • ↑(f' (↑e p)) 1 - ↑(f' (↑e p)) I)",
"tactic": "rintro ⟨x, y⟩"
},
{
"state_after": "no goals",
"state_before": "case mk\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I • ↑(f' z) 1 - ↑(f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\ne : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm\nhe₁ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe₂ : ↑(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : ℝ → ℝ → True\nF : ℝ × ℝ → E := f ∘ ↑e\nF' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (↑e p)) ↑e\nx y : ℝ\n⊢ ↑(-(I • F' (x, y))) (1, 0) + ↑(F' (x, y)) (0, 1) = -(I • ↑(f' (↑e (x, y))) 1 - ↑(f' (↑e (x, y))) I)",
"tactic": "simp only [ContinuousLinearMap.neg_apply, ContinuousLinearMap.smul_apply,\n ContinuousLinearMap.comp_apply, ContinuousLinearEquiv.coe_coe, he₁, he₂, neg_add_eq_sub,\n neg_sub]"
}
] |
[
206,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Order/CompleteLattice.lean
|
Function.Surjective.iSup_comp
|
[
{
"state_after": "α : Type u_3\nβ : Type ?u.60572\nβ₂ : Type ?u.60575\nγ : Type ?u.60578\nι : Sort u_1\nι' : Sort u_2\nκ : ι → Sort ?u.60589\nκ' : ι' → Sort ?u.60594\ninst✝ : SupSet α\nf✝ g✝ : ι → α\nf : ι → ι'\nhf : Surjective f\ng : ι' → α\n⊢ sSup (range fun x => g (f x)) = sSup (range fun y => g y)",
"state_before": "α : Type u_3\nβ : Type ?u.60572\nβ₂ : Type ?u.60575\nγ : Type ?u.60578\nι : Sort u_1\nι' : Sort u_2\nκ : ι → Sort ?u.60589\nκ' : ι' → Sort ?u.60594\ninst✝ : SupSet α\nf✝ g✝ : ι → α\nf : ι → ι'\nhf : Surjective f\ng : ι' → α\n⊢ (⨆ (x : ι), g (f x)) = ⨆ (y : ι'), g y",
"tactic": "simp [iSup]"
},
{
"state_after": "case e_a\nα : Type u_3\nβ : Type ?u.60572\nβ₂ : Type ?u.60575\nγ : Type ?u.60578\nι : Sort u_1\nι' : Sort u_2\nκ : ι → Sort ?u.60589\nκ' : ι' → Sort ?u.60594\ninst✝ : SupSet α\nf✝ g✝ : ι → α\nf : ι → ι'\nhf : Surjective f\ng : ι' → α\n⊢ (range fun x => g (f x)) = range fun y => g y",
"state_before": "α : Type u_3\nβ : Type ?u.60572\nβ₂ : Type ?u.60575\nγ : Type ?u.60578\nι : Sort u_1\nι' : Sort u_2\nκ : ι → Sort ?u.60589\nκ' : ι' → Sort ?u.60594\ninst✝ : SupSet α\nf✝ g✝ : ι → α\nf : ι → ι'\nhf : Surjective f\ng : ι' → α\n⊢ sSup (range fun x => g (f x)) = sSup (range fun y => g y)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_a\nα : Type u_3\nβ : Type ?u.60572\nβ₂ : Type ?u.60575\nγ : Type ?u.60578\nι : Sort u_1\nι' : Sort u_2\nκ : ι → Sort ?u.60589\nκ' : ι' → Sort ?u.60594\ninst✝ : SupSet α\nf✝ g✝ : ι → α\nf : ι → ι'\nhf : Surjective f\ng : ι' → α\n⊢ (range fun x => g (f x)) = range fun y => g y",
"tactic": "exact hf.range_comp g"
}
] |
[
641,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
637,
1
] |
Mathlib/RingTheory/GradedAlgebra/Radical.lean
|
Ideal.IsHomogeneous.radical_eq
|
[
{
"state_after": "ι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\n⊢ InfSet.sInf {J | I ≤ J ∧ IsPrime J} = InfSet.sInf {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ IsPrime J}",
"state_before": "ι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\n⊢ radical I = InfSet.sInf {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ IsPrime J}",
"tactic": "rw [Ideal.radical_eq_sInf]"
},
{
"state_after": "case a\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\n⊢ InfSet.sInf {J | I ≤ J ∧ IsPrime J} ≤ InfSet.sInf {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ IsPrime J}\n\ncase a\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\n⊢ InfSet.sInf {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ IsPrime J} ≤ InfSet.sInf {J | I ≤ J ∧ IsPrime J}",
"state_before": "ι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\n⊢ InfSet.sInf {J | I ≤ J ∧ IsPrime J} = InfSet.sInf {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ IsPrime J}",
"tactic": "apply le_antisymm"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\n⊢ InfSet.sInf {J | I ≤ J ∧ IsPrime J} ≤ InfSet.sInf {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ IsPrime J}",
"tactic": "exact sInf_le_sInf fun J => And.right"
},
{
"state_after": "case a\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\n⊢ ∀ (x : Ideal A), x ∈ {J | I ≤ J ∧ IsPrime J} → ∃ y, y ∈ {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ IsPrime J} ∧ y ≤ x",
"state_before": "case a\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\n⊢ InfSet.sInf {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ IsPrime J} ≤ InfSet.sInf {J | I ≤ J ∧ IsPrime J}",
"tactic": "refine sInf_le_sInf_of_forall_exists_le ?_"
},
{
"state_after": "case a.intro\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\nJ : Ideal A\nHJ₁ : I ≤ J\nHJ₂ : IsPrime J\n⊢ ∃ y, y ∈ {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ IsPrime J} ∧ y ≤ J",
"state_before": "case a\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\n⊢ ∀ (x : Ideal A), x ∈ {J | I ≤ J ∧ IsPrime J} → ∃ y, y ∈ {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ IsPrime J} ∧ y ≤ x",
"tactic": "rintro J ⟨HJ₁, HJ₂⟩"
},
{
"state_after": "case a.intro\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\nJ : Ideal A\nHJ₁ : I ≤ J\nHJ₂ : IsPrime J\n⊢ HomogeneousIdeal.toIdeal (homogeneousCore 𝒜 J) ∈ {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ IsPrime J}",
"state_before": "case a.intro\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\nJ : Ideal A\nHJ₁ : I ≤ J\nHJ₂ : IsPrime J\n⊢ ∃ y, y ∈ {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ IsPrime J} ∧ y ≤ J",
"tactic": "refine ⟨(J.homogeneousCore 𝒜).toIdeal, ?_, J.toIdeal_homogeneousCore_le _⟩"
},
{
"state_after": "case a.intro\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\nJ : Ideal A\nHJ₁ : I ≤ J\nHJ₂ : IsPrime J\n⊢ I ≤ HomogeneousIdeal.toIdeal (homogeneousCore 𝒜 J)",
"state_before": "case a.intro\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\nJ : Ideal A\nHJ₁ : I ≤ J\nHJ₂ : IsPrime J\n⊢ HomogeneousIdeal.toIdeal (homogeneousCore 𝒜 J) ∈ {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ IsPrime J}",
"tactic": "refine ⟨HomogeneousIdeal.isHomogeneous _, ?_, HJ₂.homogeneousCore⟩"
},
{
"state_after": "no goals",
"state_before": "case a.intro\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\nJ : Ideal A\nHJ₁ : I ≤ J\nHJ₂ : IsPrime J\n⊢ I ≤ HomogeneousIdeal.toIdeal (homogeneousCore 𝒜 J)",
"tactic": "exact hI.toIdeal_homogeneousCore_eq_self.symm.trans_le (Ideal.homogeneousCore_mono _ HJ₁)"
}
] |
[
178,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.zero_re
|
[] |
[
71,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Bounded.dual_iff
|
[
{
"state_after": "α : Type u_1\ninst✝ : Preorder α\nt : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nh : Bounded (Ordnode.dual t) o₂ o₁\nthis : Bounded (Ordnode.dual (Ordnode.dual t)) o₁ o₂\n⊢ Bounded t o₁ o₂",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nt : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nh : Bounded (Ordnode.dual t) o₂ o₁\n⊢ Bounded t o₁ o₂",
"tactic": "have := Bounded.dual h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nt : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nh : Bounded (Ordnode.dual t) o₂ o₁\nthis : Bounded (Ordnode.dual (Ordnode.dual t)) o₁ o₂\n⊢ Bounded t o₁ o₂",
"tactic": "rwa [dual_dual, OrderDual.Preorder.dual_dual] at this"
}
] |
[
913,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
910,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.IsNormal.refl
|
[] |
[
472,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
LinearMap.toMatrixAlgEquiv_toLinAlgEquiv
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁹ : CommSemiring R\nl : Type ?u.2299745\nm : Type ?u.2299748\nn : Type u_1\ninst✝⁸ : Fintype n\ninst✝⁷ : Fintype m\ninst✝⁶ : DecidableEq n\nM₁ : Type u_3\nM₂ : Type ?u.2299772\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₁\ninst✝² : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type ?u.2300274\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Module R M₃\nv₃ : Basis l R M₃\nM : Matrix n n R\n⊢ ↑(toMatrixAlgEquiv v₁) (↑(toLinAlgEquiv v₁) M) = M",
"tactic": "rw [← Matrix.toLinAlgEquiv_symm, AlgEquiv.symm_apply_apply]"
}
] |
[
743,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
741,
1
] |
Mathlib/Algebra/Lie/IdealOperations.lean
|
LieSubmodule.mono_lie_right
|
[] |
[
167,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Topology/DenseEmbedding.lean
|
DenseInducing.continuous
|
[] |
[
58,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
11
] |
Mathlib/Algebra/Hom/Iterate.lean
|
AddMonoidHom.iterate_map_smul
|
[] |
[
103,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Order/MinMax.lean
|
max_left_commutative
|
[] |
[
302,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Analysis/Convex/Join.lean
|
subset_convexJoin_left
|
[] |
[
114,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.sub_mem
|
[] |
[
559,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
558,
11
] |
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.BoundedFormula.IsPrenex.relabel
|
[
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.115821\nP : Type ?u.115824\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.115852\nn l : ℕ\nφ✝¹ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nxs : Fin l → M\nm : ℕ\nφ : BoundedFormula L α m\nh✝ : IsPrenex φ\nf : α → β ⊕ Fin n\nn✝ : ℕ\nφ✝ : BoundedFormula L α (n✝ + 1)\nx✝ : IsPrenex φ✝\nh : IsPrenex (BoundedFormula.relabel f φ✝)\n⊢ IsPrenex (BoundedFormula.relabel f (BoundedFormula.all φ✝))",
"tactic": "simp [h.all]"
},
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.115821\nP : Type ?u.115824\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.115852\nn l : ℕ\nφ✝¹ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nxs : Fin l → M\nm : ℕ\nφ : BoundedFormula L α m\nh✝ : IsPrenex φ\nf : α → β ⊕ Fin n\nn✝ : ℕ\nφ✝ : BoundedFormula L α (n✝ + 1)\nx✝ : IsPrenex φ✝\nh : IsPrenex (BoundedFormula.relabel f φ✝)\n⊢ IsPrenex (BoundedFormula.relabel f (BoundedFormula.ex φ✝))",
"tactic": "simp [h.ex]"
}
] |
[
778,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
775,
1
] |
Mathlib/Data/List/Basic.lean
|
List.reverse_surjective
|
[] |
[
620,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
619,
1
] |
Mathlib/RingTheory/Valuation/ValuationRing.lean
|
ValuationRing.le_total
|
[
{
"state_after": "case mk\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na✝ b : ValueGroup A K\na : K\n⊢ Quot.mk Setoid.r a ≤ b ∨ b ≤ Quot.mk Setoid.r a",
"state_before": "A : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na b : ValueGroup A K\n⊢ a ≤ b ∨ b ≤ a",
"tactic": "rcases a with ⟨a⟩"
},
{
"state_after": "case mk.mk\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na✝ b✝ : ValueGroup A K\na b : K\n⊢ Quot.mk Setoid.r a ≤ Quot.mk Setoid.r b ∨ Quot.mk Setoid.r b ≤ Quot.mk Setoid.r a",
"state_before": "case mk\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na✝ b : ValueGroup A K\na : K\n⊢ Quot.mk Setoid.r a ≤ b ∨ b ≤ Quot.mk Setoid.r a",
"tactic": "rcases b with ⟨b⟩"
},
{
"state_after": "case mk.mk.intro.intro.intro\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na b✝ : ValueGroup A K\nb : K\nxa ya : A\nhya : ya ∈ nonZeroDivisors A\n⊢ Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya) ≤ Quot.mk Setoid.r b ∨\n Quot.mk Setoid.r b ≤ Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya)",
"state_before": "case mk.mk\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na✝ b✝ : ValueGroup A K\na b : K\n⊢ Quot.mk Setoid.r a ≤ Quot.mk Setoid.r b ∨ Quot.mk Setoid.r b ≤ Quot.mk Setoid.r a",
"tactic": "obtain ⟨xa, ya, hya, rfl⟩ : ∃ a b : A, _ := IsFractionRing.div_surjective a"
},
{
"state_after": "case mk.mk.intro.intro.intro.intro.intro.intro\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya ∈ nonZeroDivisors A\nxb yb : A\nhyb : yb ∈ nonZeroDivisors A\n⊢ Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya) ≤\n Quot.mk Setoid.r (↑(algebraMap A K) xb / ↑(algebraMap A K) yb) ∨\n Quot.mk Setoid.r (↑(algebraMap A K) xb / ↑(algebraMap A K) yb) ≤\n Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya)",
"state_before": "case mk.mk.intro.intro.intro\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na b✝ : ValueGroup A K\nb : K\nxa ya : A\nhya : ya ∈ nonZeroDivisors A\n⊢ Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya) ≤ Quot.mk Setoid.r b ∨\n Quot.mk Setoid.r b ≤ Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya)",
"tactic": "obtain ⟨xb, yb, hyb, rfl⟩ : ∃ a b : A, _ := IsFractionRing.div_surjective b"
},
{
"state_after": "case mk.mk.intro.intro.intro.intro.intro.intro\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya ∈ nonZeroDivisors A\nxb yb : A\nhyb : yb ∈ nonZeroDivisors A\nthis : ↑(algebraMap A K) ya ≠ 0\n⊢ Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya) ≤\n Quot.mk Setoid.r (↑(algebraMap A K) xb / ↑(algebraMap A K) yb) ∨\n Quot.mk Setoid.r (↑(algebraMap A K) xb / ↑(algebraMap A K) yb) ≤\n Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya)",
"state_before": "case mk.mk.intro.intro.intro.intro.intro.intro\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya ∈ nonZeroDivisors A\nxb yb : A\nhyb : yb ∈ nonZeroDivisors A\n⊢ Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya) ≤\n Quot.mk Setoid.r (↑(algebraMap A K) xb / ↑(algebraMap A K) yb) ∨\n Quot.mk Setoid.r (↑(algebraMap A K) xb / ↑(algebraMap A K) yb) ≤\n Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya)",
"tactic": "have : (algebraMap A K) ya ≠ 0 := IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors hya"
},
{
"state_after": "case mk.mk.intro.intro.intro.intro.intro.intro\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya ∈ nonZeroDivisors A\nxb yb : A\nhyb : yb ∈ nonZeroDivisors A\nthis✝ : ↑(algebraMap A K) ya ≠ 0\nthis : ↑(algebraMap A K) yb ≠ 0\n⊢ Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya) ≤\n Quot.mk Setoid.r (↑(algebraMap A K) xb / ↑(algebraMap A K) yb) ∨\n Quot.mk Setoid.r (↑(algebraMap A K) xb / ↑(algebraMap A K) yb) ≤\n Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya)",
"state_before": "case mk.mk.intro.intro.intro.intro.intro.intro\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya ∈ nonZeroDivisors A\nxb yb : A\nhyb : yb ∈ nonZeroDivisors A\nthis : ↑(algebraMap A K) ya ≠ 0\n⊢ Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya) ≤\n Quot.mk Setoid.r (↑(algebraMap A K) xb / ↑(algebraMap A K) yb) ∨\n Quot.mk Setoid.r (↑(algebraMap A K) xb / ↑(algebraMap A K) yb) ≤\n Quot.mk Setoid.r (↑(algebraMap A K) xa / ↑(algebraMap A K) ya)",
"tactic": "have : (algebraMap A K) yb ≠ 0 := IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors hyb"
},
{
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"tactic": "obtain ⟨c, h | h⟩ := ValuationRing.cond (xa * yb) (xb * ya)"
},
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"tactic": "ring"
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"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro.intro.intro.intro.intro.intro.inr.h.h.e_6.h\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya ∈ nonZeroDivisors A\nxb yb : A\nhyb : yb ∈ nonZeroDivisors A\nthis✝ : ↑(algebraMap A K) ya ≠ 0\nthis : ↑(algebraMap A K) yb ≠ 0\nc : A\nh : xb * ya * c = xa * yb\n⊢ c * xb * ya = xb * ya * c",
"tactic": "ring"
}
] |
[
127,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
11
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.three_le
|
[
{
"state_after": "case this\nα✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\n⊢ ↑3 ≤ (#α)\n\nα✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\nthis : ↑3 ≤ (#α)\n⊢ ∃ z, z ≠ x ∧ z ≠ y",
"state_before": "α✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\n⊢ ∃ z, z ≠ x ∧ z ≠ y",
"tactic": "have : ↑(3 : ℕ) ≤ (#α)"
},
{
"state_after": "α✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\nthis : ↑3 ≤ (#α)\n⊢ ∃ z, z ≠ x ∧ z ≠ y",
"state_before": "case this\nα✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\n⊢ ↑3 ≤ (#α)\n\nα✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\nthis : ↑3 ≤ (#α)\n⊢ ∃ z, z ≠ x ∧ z ≠ y",
"tactic": "simpa using h"
},
{
"state_after": "case this\nα✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\nthis : ↑3 ≤ (#α)\n⊢ ↑2 < (#α)\n\nα✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\nthis✝ : ↑3 ≤ (#α)\nthis : ↑2 < (#α)\n⊢ ∃ z, z ≠ x ∧ z ≠ y",
"state_before": "α✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\nthis : ↑3 ≤ (#α)\n⊢ ∃ z, z ≠ x ∧ z ≠ y",
"tactic": "have : ↑(2 : ℕ) < (#α)"
},
{
"state_after": "α✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\nthis✝ : ↑3 ≤ (#α)\nthis : ↑2 < (#α)\n⊢ ∃ z, z ≠ x ∧ z ≠ y",
"state_before": "case this\nα✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\nthis : ↑3 ≤ (#α)\n⊢ ↑2 < (#α)\n\nα✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\nthis✝ : ↑3 ≤ (#α)\nthis : ↑2 < (#α)\n⊢ ∃ z, z ≠ x ∧ z ≠ y",
"tactic": "rwa [← succ_le_iff, ← Cardinal.nat_succ]"
},
{
"state_after": "α✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\nthis✝¹ : ↑3 ≤ (#α)\nthis✝ : ↑2 < (#α)\nthis : ∃ z, ¬z ∈ [x, y]\n⊢ ∃ z, z ≠ x ∧ z ≠ y",
"state_before": "α✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\nthis✝ : ↑3 ≤ (#α)\nthis : ↑2 < (#α)\n⊢ ∃ z, z ≠ x ∧ z ≠ y",
"tactic": "have := exists_not_mem_of_length_lt [x, y] this"
},
{
"state_after": "no goals",
"state_before": "α✝ β : Type u\nα : Type u_1\nh : 3 ≤ (#α)\nx y : α\nthis✝¹ : ↑3 ≤ (#α)\nthis✝ : ↑2 < (#α)\nthis : ∃ z, ¬z ∈ [x, y]\n⊢ ∃ z, z ≠ x ∧ z ≠ y",
"tactic": "simpa [not_or] using this"
}
] |
[
2270,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2266,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.single_of_embDomain_single
|
[
{
"state_after": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\n⊢ ∃ x, l = single x b ∧ ↑f x = a",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\n⊢ ∃ x, l = single x b ∧ ↑f x = a",
"tactic": "have h_map_support : Finset.map f l.support = {a} := by\n rw [← support_embDomain, h, support_single_ne_zero _ hb]"
},
{
"state_after": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\n⊢ ∃ x, l = single x b ∧ ↑f x = a",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\n⊢ ∃ x, l = single x b ∧ ↑f x = a",
"tactic": "have ha : a ∈ Finset.map f l.support := by simp only [h_map_support, Finset.mem_singleton]"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\n⊢ ∃ x, l = single x b ∧ ↑f x = a",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\n⊢ ∃ x, l = single x b ∧ ↑f x = a",
"tactic": "rcases Finset.mem_map.1 ha with ⟨c, _hc₁, hc₂⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\n⊢ l = single c b ∧ ↑f c = a",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\n⊢ ∃ x, l = single x b ∧ ↑f x = a",
"tactic": "use c"
},
{
"state_after": "case intro.intro.left\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\n⊢ l = single c b\n\ncase intro.intro.right\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\n⊢ ↑f c = a",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\n⊢ l = single c b ∧ ↑f c = a",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\n⊢ map f l.support = {a}",
"tactic": "rw [← support_embDomain, h, support_single_ne_zero _ hb]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\n⊢ a ∈ map f l.support",
"tactic": "simp only [h_map_support, Finset.mem_singleton]"
},
{
"state_after": "case intro.intro.left.h\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\nd : α\n⊢ ↑l d = ↑(single c b) d",
"state_before": "case intro.intro.left\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\n⊢ l = single c b",
"tactic": "ext d"
},
{
"state_after": "case intro.intro.left.h\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\nd : α\n⊢ ↑(single a b) (↑f d) = ↑(single c b) d",
"state_before": "case intro.intro.left.h\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\nd : α\n⊢ ↑l d = ↑(single c b) d",
"tactic": "rw [← embDomain_apply f l, h]"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\nd : α\nh_cases : c = d\n⊢ ↑(single a b) (↑f d) = ↑(single c b) d\n\ncase neg\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\nd : α\nh_cases : ¬c = d\n⊢ ↑(single a b) (↑f d) = ↑(single c b) d",
"state_before": "case intro.intro.left.h\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\nd : α\n⊢ ↑(single a b) (↑f d) = ↑(single c b) d",
"tactic": "by_cases h_cases : c = d"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\nd : α\nh_cases : c = d\n⊢ ↑(single a b) (↑f d) = ↑(single c b) d",
"tactic": "simp only [Eq.symm h_cases, hc₂, single_eq_same]"
},
{
"state_after": "case neg.hnc\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\nd : α\nh_cases : ¬c = d\n⊢ ¬a = ↑f d",
"state_before": "case neg\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\nd : α\nh_cases : ¬c = d\n⊢ ↑(single a b) (↑f d) = ↑(single c b) d",
"tactic": "rw [single_apply, single_apply, if_neg, if_neg h_cases]"
},
{
"state_after": "case neg.hnc\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\nd : α\nh_cases : ¬c = d\nhfd : a = ↑f d\n⊢ False",
"state_before": "case neg.hnc\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\nd : α\nh_cases : ¬c = d\n⊢ ¬a = ↑f d",
"tactic": "by_contra hfd"
},
{
"state_after": "no goals",
"state_before": "case neg.hnc\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\nd : α\nh_cases : ¬c = d\nhfd : a = ↑f d\n⊢ False",
"tactic": "exact h_cases (f.injective (hc₂.trans hfd))"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.right\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.245637\nι : Type ?u.245640\nM : Type u_2\nM' : Type ?u.245646\nN : Type ?u.245649\nP : Type ?u.245652\nG : Type ?u.245655\nH : Type ?u.245658\nR : Type ?u.245661\nS : Type ?u.245664\ninst✝¹ : Zero M\ninst✝ : Zero N\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : ↑f c = a\n⊢ ↑f c = a",
"tactic": "exact hc₂"
}
] |
[
910,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
894,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.image_perm
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.57571\nγ : Type ?u.57574\nι : Sort ?u.57577\nι' : Sort ?u.57580\nf : α → β\ns✝ t s : Set α\nσ : Equiv.Perm α\nhs : {a | ↑σ a ≠ a} ⊆ s\ni : α\n⊢ i ∈ ↑σ '' s ↔ i ∈ s",
"state_before": "α : Type u_1\nβ : Type ?u.57571\nγ : Type ?u.57574\nι : Sort ?u.57577\nι' : Sort ?u.57580\nf : α → β\ns✝ t s : Set α\nσ : Equiv.Perm α\nhs : {a | ↑σ a ≠ a} ⊆ s\n⊢ ↑σ '' s = s",
"tactic": "ext i"
},
{
"state_after": "case h.inl\nα : Type u_1\nβ : Type ?u.57571\nγ : Type ?u.57574\nι : Sort ?u.57577\nι' : Sort ?u.57580\nf : α → β\ns✝ t s : Set α\nσ : Equiv.Perm α\nhs : {a | ↑σ a ≠ a} ⊆ s\ni : α\nhi : ↑σ i = i\n⊢ i ∈ ↑σ '' s ↔ i ∈ s\n\ncase h.inr\nα : Type u_1\nβ : Type ?u.57571\nγ : Type ?u.57574\nι : Sort ?u.57577\nι' : Sort ?u.57580\nf : α → β\ns✝ t s : Set α\nσ : Equiv.Perm α\nhs : {a | ↑σ a ≠ a} ⊆ s\ni : α\nhi : ↑σ i ≠ i\n⊢ i ∈ ↑σ '' s ↔ i ∈ s",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.57571\nγ : Type ?u.57574\nι : Sort ?u.57577\nι' : Sort ?u.57580\nf : α → β\ns✝ t s : Set α\nσ : Equiv.Perm α\nhs : {a | ↑σ a ≠ a} ⊆ s\ni : α\n⊢ i ∈ ↑σ '' s ↔ i ∈ s",
"tactic": "obtain hi | hi := eq_or_ne (σ i) i"
},
{
"state_after": "case h.inl\nα : Type u_1\nβ : Type ?u.57571\nγ : Type ?u.57574\nι : Sort ?u.57577\nι' : Sort ?u.57580\nf : α → β\ns✝ t s : Set α\nσ : Equiv.Perm α\nhs : {a | ↑σ a ≠ a} ⊆ s\ni : α\nhi : ↑σ i = i\n⊢ i ∈ ↑σ '' s → i ∈ s",
"state_before": "case h.inl\nα : Type u_1\nβ : Type ?u.57571\nγ : Type ?u.57574\nι : Sort ?u.57577\nι' : Sort ?u.57580\nf : α → β\ns✝ t s : Set α\nσ : Equiv.Perm α\nhs : {a | ↑σ a ≠ a} ⊆ s\ni : α\nhi : ↑σ i = i\n⊢ i ∈ ↑σ '' s ↔ i ∈ s",
"tactic": "refine' ⟨_, fun h => ⟨i, h, hi⟩⟩"
},
{
"state_after": "case h.inl.intro.intro\nα : Type u_1\nβ : Type ?u.57571\nγ : Type ?u.57574\nι : Sort ?u.57577\nι' : Sort ?u.57580\nf : α → β\ns✝ t s : Set α\nσ : Equiv.Perm α\nhs : {a | ↑σ a ≠ a} ⊆ s\ni : α\nhi : ↑σ i = i\nj : α\nhj : j ∈ s\nh : ↑σ j = i\n⊢ i ∈ s",
"state_before": "case h.inl\nα : Type u_1\nβ : Type ?u.57571\nγ : Type ?u.57574\nι : Sort ?u.57577\nι' : Sort ?u.57580\nf : α → β\ns✝ t s : Set α\nσ : Equiv.Perm α\nhs : {a | ↑σ a ≠ a} ⊆ s\ni : α\nhi : ↑σ i = i\n⊢ i ∈ ↑σ '' s → i ∈ s",
"tactic": "rintro ⟨j, hj, h⟩"
},
{
"state_after": "no goals",
"state_before": "case h.inl.intro.intro\nα : Type u_1\nβ : Type ?u.57571\nγ : Type ?u.57574\nι : Sort ?u.57577\nι' : Sort ?u.57580\nf : α → β\ns✝ t s : Set α\nσ : Equiv.Perm α\nhs : {a | ↑σ a ≠ a} ⊆ s\ni : α\nhi : ↑σ i = i\nj : α\nhj : j ∈ s\nh : ↑σ j = i\n⊢ i ∈ s",
"tactic": "rwa [σ.injective (hi.trans h.symm)]"
},
{
"state_after": "case h.inr\nα : Type u_1\nβ : Type ?u.57571\nγ : Type ?u.57574\nι : Sort ?u.57577\nι' : Sort ?u.57580\nf : α → β\ns✝ t s : Set α\nσ : Equiv.Perm α\nhs : {a | ↑σ a ≠ a} ⊆ s\ni : α\nhi : ↑σ i ≠ i\nh : ↑σ (↑σ.symm i) = ↑σ.symm i\n⊢ ↑σ i = i",
"state_before": "case h.inr\nα : Type u_1\nβ : Type ?u.57571\nγ : Type ?u.57574\nι : Sort ?u.57577\nι' : Sort ?u.57580\nf : α → β\ns✝ t s : Set α\nσ : Equiv.Perm α\nhs : {a | ↑σ a ≠ a} ⊆ s\ni : α\nhi : ↑σ i ≠ i\n⊢ i ∈ ↑σ '' s ↔ i ∈ s",
"tactic": "refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)"
},
{
"state_after": "no goals",
"state_before": "case h.inr\nα : Type u_1\nβ : Type ?u.57571\nγ : Type ?u.57574\nι : Sort ?u.57577\nι' : Sort ?u.57580\nf : α → β\ns✝ t s : Set α\nσ : Equiv.Perm α\nhs : {a | ↑σ a ≠ a} ⊆ s\ni : α\nhi : ↑σ i ≠ i\nh : ↑σ (↑σ.symm i) = ↑σ.symm i\n⊢ ↑σ i = i",
"tactic": "convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm"
}
] |
[
608,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
601,
1
] |
Mathlib/Data/Finset/PImage.lean
|
Part.mem_toFinset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.68\no : Part α\ninst✝ : Decidable o.Dom\nx : α\n⊢ x ∈ toFinset o ↔ x ∈ o",
"tactic": "simp [toFinset]"
}
] |
[
38,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.deterministic_apply
|
[] |
[
356,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/Data/Fin/VecNotation.lean
|
Matrix.add_cons
|
[
{
"state_after": "case h\nα : Type u\nm n o : ℕ\nm' : Type ?u.61403\nn' : Type ?u.61406\no' : Type ?u.61409\ninst✝ : Add α\nv : Fin (Nat.succ n) → α\ny : α\nw : Fin n → α\ni : Fin (Nat.succ n)\n⊢ (v + vecCons y w) i = vecCons (vecHead v + y) (vecTail v + w) i",
"state_before": "α : Type u\nm n o : ℕ\nm' : Type ?u.61403\nn' : Type ?u.61406\no' : Type ?u.61409\ninst✝ : Add α\nv : Fin (Nat.succ n) → α\ny : α\nw : Fin n → α\n⊢ v + vecCons y w = vecCons (vecHead v + y) (vecTail v + w)",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nm n o : ℕ\nm' : Type ?u.61403\nn' : Type ?u.61406\no' : Type ?u.61409\ninst✝ : Add α\nv : Fin (Nat.succ n) → α\ny : α\nw : Fin n → α\ni : Fin (Nat.succ n)\n⊢ (v + vecCons y w) i = vecCons (vecHead v + y) (vecTail v + w) i",
"tactic": "refine' Fin.cases _ _ i <;> simp [vecHead, vecTail]"
}
] |
[
467,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
464,
1
] |
Mathlib/Data/Multiset/Fold.lean
|
Multiset.max_le_of_forall_le
|
[
{
"state_after": "case h\nα✝ : Type ?u.20522\nβ : Type ?u.20525\nα : Type u_1\ninst✝ : CanonicallyLinearOrderedAddMonoid α\nn : α\na✝ : List α\nh : ∀ (x : α), x ∈ Quotient.mk (List.isSetoid α) a✝ → x ≤ n\n⊢ fold max ⊥ (Quotient.mk (List.isSetoid α) a✝) ≤ n",
"state_before": "α✝ : Type ?u.20522\nβ : Type ?u.20525\nα : Type u_1\ninst✝ : CanonicallyLinearOrderedAddMonoid α\nl : Multiset α\nn : α\nh : ∀ (x : α), x ∈ l → x ≤ n\n⊢ fold max ⊥ l ≤ n",
"tactic": "induction l using Quotient.inductionOn"
},
{
"state_after": "no goals",
"state_before": "case h\nα✝ : Type ?u.20522\nβ : Type ?u.20525\nα : Type u_1\ninst✝ : CanonicallyLinearOrderedAddMonoid α\nn : α\na✝ : List α\nh : ∀ (x : α), x ∈ Quotient.mk (List.isSetoid α) a✝ → x ≤ n\n⊢ fold max ⊥ (Quotient.mk (List.isSetoid α) a✝) ≤ n",
"tactic": "simpa using List.max_le_of_forall_le _ _ h"
}
] |
[
133,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/LinearAlgebra/StdBasis.lean
|
LinearMap.stdBasis_eq_pi_diag
|
[
{
"state_after": "case h.h\nR : Type u_3\nι : Type u_1\ninst✝³ : Semiring R\nφ : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\ni : ι\nx : φ i\nj : ι\n⊢ ↑(stdBasis R φ i) x j = ↑(pi (diag i)) x j",
"state_before": "R : Type u_3\nι : Type u_1\ninst✝³ : Semiring R\nφ : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\ni : ι\n⊢ stdBasis R φ i = pi (diag i)",
"tactic": "ext (x j)"
},
{
"state_after": "case h.e'_2.h.e'_1\nR : Type u_3\nι : Type u_1\ninst✝³ : Semiring R\nφ : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\ni : ι\nx : φ i\nj : ι\n⊢ x = ↑id x",
"state_before": "case h.h\nR : Type u_3\nι : Type u_1\ninst✝³ : Semiring R\nφ : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\ni : ι\nx : φ i\nj : ι\n⊢ ↑(stdBasis R φ i) x j = ↑(pi (diag i)) x j",
"tactic": "convert (update_apply (R := R) (φ := φ) (ι := ι) 0 x i j _).symm"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_1\nR : Type u_3\nι : Type u_1\ninst✝³ : Semiring R\nφ : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\ni : ι\nx : φ i\nj : ι\n⊢ x = ↑id x",
"tactic": "rfl"
}
] |
[
82,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieHom.mem_range
|
[] |
[
318,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.tan_eq_sin_div_cos
|
[] |
[
780,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
779,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
|
InnerProductGeometry.angle_smul_left_of_pos
|
[
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nr : ℝ\nhr : 0 < r\n⊢ angle (r • x) y = angle x y",
"tactic": "rw [angle_comm, angle_smul_right_of_pos y x hr, angle_comm]"
}
] |
[
164,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
AlternatingMap.domDomCongr_refl
|
[] |
[
740,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
739,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.L1.SimpleFunc.integral_L1_eq_integral
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.423076\n𝕜 : Type ?u.423079\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : SMulCommClass ℝ 𝕜 E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace E\nf : { x // x ∈ simpleFunc E 1 μ }\n⊢ ↑L1.integralCLM ↑f = MeasureTheory.SimpleFunc.integral μ (toSimpleFunc f)",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.423076\n𝕜 : Type ?u.423079\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : SMulCommClass ℝ 𝕜 E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace E\nf : { x // x ∈ simpleFunc E 1 μ }\n⊢ L1.integral ↑f = integral f",
"tactic": "simp only [integral, L1.integral]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.423076\n𝕜 : Type ?u.423079\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : SMulCommClass ℝ 𝕜 E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace E\nf : { x // x ∈ simpleFunc E 1 μ }\n⊢ ↑L1.integralCLM ↑f = MeasureTheory.SimpleFunc.integral μ (toSimpleFunc f)",
"tactic": "exact setToL1_eq_setToL1SCLM (dominatedFinMeasAdditive_weightedSMul μ) f"
}
] |
[
692,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
689,
1
] |
Mathlib/Data/Polynomial/Div.lean
|
Polynomial.degree_modByMonic_le
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Ring R\np✝ q✝ p q : R[X]\nhq : Monic q\n✝ : Nontrivial R\n⊢ degree (p %ₘ q) ≤ degree q",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Ring R\np✝ q✝ p q : R[X]\nhq : Monic q\n⊢ degree (p %ₘ q) ≤ degree q",
"tactic": "nontriviality R"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Ring R\np✝ q✝ p q : R[X]\nhq : Monic q\n✝ : Nontrivial R\n⊢ degree (p %ₘ q) ≤ degree q",
"tactic": "exact (degree_modByMonic_lt _ hq).le"
}
] |
[
222,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
smul_vsub_vadd_mem_affineSpan_pair
|
[] |
[
1326,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1324,
1
] |
Mathlib/RingTheory/AlgebraicIndependent.lean
|
AlgebraicIndependent.repr_ker
|
[] |
[
416,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
real_inner_smul_self_right
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.3142239\nE : Type ?u.3142242\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : F\nr : ℝ\n⊢ inner x (r • x) = r * (‖x‖ * ‖x‖)",
"tactic": "rw [inner_smul_right, ← real_inner_self_eq_norm_mul_norm]"
}
] |
[
1548,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1547,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.eq_zero_of_comapDomain_eq_zero
|
[
{
"state_after": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.301425\nι : Type ?u.301428\nM : Type u_1\nM' : Type ?u.301434\nN : Type ?u.301437\nP : Type ?u.301440\nG : Type ?u.301443\nH : Type ?u.301446\nR : Type ?u.301449\nS : Type ?u.301452\ninst✝ : AddCommMonoid M\nf : α → β\nl : β →₀ M\nhf : Set.BijOn f (f ⁻¹' ↑l.support) ↑l.support\n⊢ { support := preimage l.support f (_ : Set.InjOn f (f ⁻¹' ↑l.support)), toFun := fun a => ↑l (f a),\n mem_support_toFun :=\n (_ :\n ∀ (a : α),\n a ∈ preimage l.support f (_ : Set.InjOn f (f ⁻¹' ↑l.support)) ↔ (fun a => ↑l (f a)) a ≠ 0) }.support =\n ∅ →\n l.support = ∅",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.301425\nι : Type ?u.301428\nM : Type u_1\nM' : Type ?u.301434\nN : Type ?u.301437\nP : Type ?u.301440\nG : Type ?u.301443\nH : Type ?u.301446\nR : Type ?u.301449\nS : Type ?u.301452\ninst✝ : AddCommMonoid M\nf : α → β\nl : β →₀ M\nhf : Set.BijOn f (f ⁻¹' ↑l.support) ↑l.support\n⊢ comapDomain f l (_ : Set.InjOn f (f ⁻¹' ↑l.support)) = 0 → l = 0",
"tactic": "rw [← support_eq_empty, ← support_eq_empty, comapDomain]"
},
{
"state_after": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.301425\nι : Type ?u.301428\nM : Type u_1\nM' : Type ?u.301434\nN : Type ?u.301437\nP : Type ?u.301440\nG : Type ?u.301443\nH : Type ?u.301446\nR : Type ?u.301449\nS : Type ?u.301452\ninst✝ : AddCommMonoid M\nf : α → β\nl : β →₀ M\nhf : Set.BijOn f (f ⁻¹' ↑l.support) ↑l.support\n⊢ (∀ (a : α), ¬f a ∈ l.support) → ∀ (a : β), ¬a ∈ l.support",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.301425\nι : Type ?u.301428\nM : Type u_1\nM' : Type ?u.301434\nN : Type ?u.301437\nP : Type ?u.301440\nG : Type ?u.301443\nH : Type ?u.301446\nR : Type ?u.301449\nS : Type ?u.301452\ninst✝ : AddCommMonoid M\nf : α → β\nl : β →₀ M\nhf : Set.BijOn f (f ⁻¹' ↑l.support) ↑l.support\n⊢ { support := preimage l.support f (_ : Set.InjOn f (f ⁻¹' ↑l.support)), toFun := fun a => ↑l (f a),\n mem_support_toFun :=\n (_ :\n ∀ (a : α),\n a ∈ preimage l.support f (_ : Set.InjOn f (f ⁻¹' ↑l.support)) ↔ (fun a => ↑l (f a)) a ≠ 0) }.support =\n ∅ →\n l.support = ∅",
"tactic": "simp only [Finset.ext_iff, Finset.not_mem_empty, iff_false_iff, mem_preimage]"
},
{
"state_after": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.301425\nι : Type ?u.301428\nM : Type u_1\nM' : Type ?u.301434\nN : Type ?u.301437\nP : Type ?u.301440\nG : Type ?u.301443\nH : Type ?u.301446\nR : Type ?u.301449\nS : Type ?u.301452\ninst✝ : AddCommMonoid M\nf : α → β\nl : β →₀ M\nhf : Set.BijOn f (f ⁻¹' ↑l.support) ↑l.support\nh : ∀ (a : α), ¬f a ∈ l.support\na : β\nha : a ∈ l.support\n⊢ False",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.301425\nι : Type ?u.301428\nM : Type u_1\nM' : Type ?u.301434\nN : Type ?u.301437\nP : Type ?u.301440\nG : Type ?u.301443\nH : Type ?u.301446\nR : Type ?u.301449\nS : Type ?u.301452\ninst✝ : AddCommMonoid M\nf : α → β\nl : β →₀ M\nhf : Set.BijOn f (f ⁻¹' ↑l.support) ↑l.support\n⊢ (∀ (a : α), ¬f a ∈ l.support) → ∀ (a : β), ¬a ∈ l.support",
"tactic": "intro h a ha"
},
{
"state_after": "case intro\nα : Type u_3\nβ : Type u_2\nγ : Type ?u.301425\nι : Type ?u.301428\nM : Type u_1\nM' : Type ?u.301434\nN : Type ?u.301437\nP : Type ?u.301440\nG : Type ?u.301443\nH : Type ?u.301446\nR : Type ?u.301449\nS : Type ?u.301452\ninst✝ : AddCommMonoid M\nf : α → β\nl : β →₀ M\nhf : Set.BijOn f (f ⁻¹' ↑l.support) ↑l.support\nh : ∀ (a : α), ¬f a ∈ l.support\na : β\nha : a ∈ l.support\nb : α\nhb : b ∈ f ⁻¹' ↑l.support ∧ f b = a\n⊢ False",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.301425\nι : Type ?u.301428\nM : Type u_1\nM' : Type ?u.301434\nN : Type ?u.301437\nP : Type ?u.301440\nG : Type ?u.301443\nH : Type ?u.301446\nR : Type ?u.301449\nS : Type ?u.301452\ninst✝ : AddCommMonoid M\nf : α → β\nl : β →₀ M\nhf : Set.BijOn f (f ⁻¹' ↑l.support) ↑l.support\nh : ∀ (a : α), ¬f a ∈ l.support\na : β\nha : a ∈ l.support\n⊢ False",
"tactic": "cases' hf.2.2 ha with b hb"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_3\nβ : Type u_2\nγ : Type ?u.301425\nι : Type ?u.301428\nM : Type u_1\nM' : Type ?u.301434\nN : Type ?u.301437\nP : Type ?u.301440\nG : Type ?u.301443\nH : Type ?u.301446\nR : Type ?u.301449\nS : Type ?u.301452\ninst✝ : AddCommMonoid M\nf : α → β\nl : β →₀ M\nhf : Set.BijOn f (f ⁻¹' ↑l.support) ↑l.support\nh : ∀ (a : α), ¬f a ∈ l.support\na : β\nha : a ∈ l.support\nb : α\nhb : b ∈ f ⁻¹' ↑l.support ∧ f b = a\n⊢ False",
"tactic": "exact h b (hb.2.symm ▸ ha)"
}
] |
[
723,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
717,
1
] |
Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
|
FreeAbelianGroup.support_zero
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\n⊢ support 0 = ∅",
"tactic": "simp only [support, Finsupp.support_zero, AddMonoidHom.map_zero]"
}
] |
[
168,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigO.of_pow
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.518903\nE : Type ?u.518906\nF : Type ?u.518909\nG : Type ?u.518912\nE' : Type ?u.518915\nF' : Type ?u.518918\nG' : Type ?u.518921\nE'' : Type ?u.518924\nF'' : Type ?u.518927\nG'' : Type ?u.518930\nR : Type u_3\nR' : Type ?u.518936\n𝕜 : Type u_2\n𝕜' : Type ?u.518942\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf : α → 𝕜\ng : α → R\nn : ℕ\nhn : n ≠ 0\nh : (f ^ n) =O[l] (g ^ n)\nC : ℝ\n_hC₀ : 0 < C\nhC : IsBigOWith C l (f ^ n) (g ^ n)\n⊢ f =O[l] g",
"state_before": "α : Type u_1\nβ : Type ?u.518903\nE : Type ?u.518906\nF : Type ?u.518909\nG : Type ?u.518912\nE' : Type ?u.518915\nF' : Type ?u.518918\nG' : Type ?u.518921\nE'' : Type ?u.518924\nF'' : Type ?u.518927\nG'' : Type ?u.518930\nR : Type u_3\nR' : Type ?u.518936\n𝕜 : Type u_2\n𝕜' : Type ?u.518942\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf : α → 𝕜\ng : α → R\nn : ℕ\nhn : n ≠ 0\nh : (f ^ n) =O[l] (g ^ n)\n⊢ f =O[l] g",
"tactic": "rcases h.exists_pos with ⟨C, _hC₀, hC⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.518903\nE : Type ?u.518906\nF : Type ?u.518909\nG : Type ?u.518912\nE' : Type ?u.518915\nF' : Type ?u.518918\nG' : Type ?u.518921\nE'' : Type ?u.518924\nF'' : Type ?u.518927\nG'' : Type ?u.518930\nR : Type u_3\nR' : Type ?u.518936\n𝕜 : Type u_2\n𝕜' : Type ?u.518942\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf : α → 𝕜\ng : α → R\nn : ℕ\nhn : n ≠ 0\nh : (f ^ n) =O[l] (g ^ n)\nC : ℝ\n_hC₀ : 0 < C\nhC : IsBigOWith C l (f ^ n) (g ^ n)\n⊢ ∃ c, 0 ≤ c ∧ C ≤ c ^ n\n\ncase intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.518903\nE : Type ?u.518906\nF : Type ?u.518909\nG : Type ?u.518912\nE' : Type ?u.518915\nF' : Type ?u.518918\nG' : Type ?u.518921\nE'' : Type ?u.518924\nF'' : Type ?u.518927\nG'' : Type ?u.518930\nR : Type u_3\nR' : Type ?u.518936\n𝕜 : Type u_2\n𝕜' : Type ?u.518942\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf : α → 𝕜\ng : α → R\nn : ℕ\nhn : n ≠ 0\nh : (f ^ n) =O[l] (g ^ n)\nC : ℝ\n_hC₀ : 0 < C\nhC : IsBigOWith C l (f ^ n) (g ^ n)\nc : ℝ\nhc₀ : 0 ≤ c\nhc : C ≤ c ^ n\n⊢ f =O[l] g",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.518903\nE : Type ?u.518906\nF : Type ?u.518909\nG : Type ?u.518912\nE' : Type ?u.518915\nF' : Type ?u.518918\nG' : Type ?u.518921\nE'' : Type ?u.518924\nF'' : Type ?u.518927\nG'' : Type ?u.518930\nR : Type u_3\nR' : Type ?u.518936\n𝕜 : Type u_2\n𝕜' : Type ?u.518942\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf : α → 𝕜\ng : α → R\nn : ℕ\nhn : n ≠ 0\nh : (f ^ n) =O[l] (g ^ n)\nC : ℝ\n_hC₀ : 0 < C\nhC : IsBigOWith C l (f ^ n) (g ^ n)\n⊢ f =O[l] g",
"tactic": "obtain ⟨c, hc₀, hc⟩ : ∃ c : ℝ, 0 ≤ c ∧ C ≤ c ^ n"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.518903\nE : Type ?u.518906\nF : Type ?u.518909\nG : Type ?u.518912\nE' : Type ?u.518915\nF' : Type ?u.518918\nG' : Type ?u.518921\nE'' : Type ?u.518924\nF'' : Type ?u.518927\nG'' : Type ?u.518930\nR : Type u_3\nR' : Type ?u.518936\n𝕜 : Type u_2\n𝕜' : Type ?u.518942\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf : α → 𝕜\ng : α → R\nn : ℕ\nhn : n ≠ 0\nh : (f ^ n) =O[l] (g ^ n)\nC : ℝ\n_hC₀ : 0 < C\nhC : IsBigOWith C l (f ^ n) (g ^ n)\nc : ℝ\nhc₀ : 0 ≤ c\nhc : C ≤ c ^ n\n⊢ f =O[l] g",
"state_before": "α : Type u_1\nβ : Type ?u.518903\nE : Type ?u.518906\nF : Type ?u.518909\nG : Type ?u.518912\nE' : Type ?u.518915\nF' : Type ?u.518918\nG' : Type ?u.518921\nE'' : Type ?u.518924\nF'' : Type ?u.518927\nG'' : Type ?u.518930\nR : Type u_3\nR' : Type ?u.518936\n𝕜 : Type u_2\n𝕜' : Type ?u.518942\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf : α → 𝕜\ng : α → R\nn : ℕ\nhn : n ≠ 0\nh : (f ^ n) =O[l] (g ^ n)\nC : ℝ\n_hC₀ : 0 < C\nhC : IsBigOWith C l (f ^ n) (g ^ n)\n⊢ ∃ c, 0 ≤ c ∧ C ≤ c ^ n\n\ncase intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.518903\nE : Type ?u.518906\nF : Type ?u.518909\nG : Type ?u.518912\nE' : Type ?u.518915\nF' : Type ?u.518918\nG' : Type ?u.518921\nE'' : Type ?u.518924\nF'' : Type ?u.518927\nG'' : Type ?u.518930\nR : Type u_3\nR' : Type ?u.518936\n𝕜 : Type u_2\n𝕜' : Type ?u.518942\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf : α → 𝕜\ng : α → R\nn : ℕ\nhn : n ≠ 0\nh : (f ^ n) =O[l] (g ^ n)\nC : ℝ\n_hC₀ : 0 < C\nhC : IsBigOWith C l (f ^ n) (g ^ n)\nc : ℝ\nhc₀ : 0 ≤ c\nhc : C ≤ c ^ n\n⊢ f =O[l] g",
"tactic": "exact ((eventually_ge_atTop _).and <| (tendsto_pow_atTop hn).eventually_ge_atTop C).exists"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.518903\nE : Type ?u.518906\nF : Type ?u.518909\nG : Type ?u.518912\nE' : Type ?u.518915\nF' : Type ?u.518918\nG' : Type ?u.518921\nE'' : Type ?u.518924\nF'' : Type ?u.518927\nG'' : Type ?u.518930\nR : Type u_3\nR' : Type ?u.518936\n𝕜 : Type u_2\n𝕜' : Type ?u.518942\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf : α → 𝕜\ng : α → R\nn : ℕ\nhn : n ≠ 0\nh : (f ^ n) =O[l] (g ^ n)\nC : ℝ\n_hC₀ : 0 < C\nhC : IsBigOWith C l (f ^ n) (g ^ n)\nc : ℝ\nhc₀ : 0 ≤ c\nhc : C ≤ c ^ n\n⊢ f =O[l] g",
"tactic": "exact (hC.of_pow hn hc hc₀).isBigO"
}
] |
[
1649,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1644,
1
] |
Mathlib/Data/Sign.lean
|
sign_eq_zero_iff
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na : α\nh : ↑sign a = 0\n⊢ a = 0",
"state_before": "α : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na : α\n⊢ ↑sign a = 0 ↔ a = 0",
"tactic": "refine' ⟨fun h => _, fun h => h.symm ▸ sign_zero⟩"
},
{
"state_after": "α : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na : α\nh : (if 0 < a then 1 else if a < 0 then -1 else 0) = 0\n⊢ a = 0",
"state_before": "α : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na : α\nh : ↑sign a = 0\n⊢ a = 0",
"tactic": "rw [sign_apply] at h"
},
{
"state_after": "case inr.inr\nα : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na : α\nh_1 : ¬0 < a\nh_2 : ¬a < 0\nh : 0 = 0\n⊢ a = 0",
"state_before": "α : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na : α\nh : (if 0 < a then 1 else if a < 0 then -1 else 0) = 0\n⊢ a = 0",
"tactic": "split_ifs at h with h_1 h_2"
},
{
"state_after": "case inr.inr.refl\nα : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na : α\nh_1 : ¬0 < a\nh_2 : ¬a < 0\n⊢ a = 0",
"state_before": "case inr.inr\nα : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na : α\nh_1 : ¬0 < a\nh_2 : ¬a < 0\nh : 0 = 0\n⊢ a = 0",
"tactic": "cases' h"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.refl\nα : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na : α\nh_1 : ¬0 < a\nh_2 : ¬a < 0\n⊢ a = 0",
"tactic": "exact (le_of_not_lt h_1).eq_of_not_lt h_2"
}
] |
[
357,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.union_subset_union
|
[] |
[
834,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
833,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Iso.connectedComponentEquiv_refl
|
[
{
"state_after": "case H.mk\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nx✝ : ConnectedComponent G\nv : V\n⊢ ↑(connectedComponentEquiv refl) (Quot.mk (Reachable G) v) =\n ↑(Equiv.refl (ConnectedComponent G)) (Quot.mk (Reachable G) v)",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\n⊢ connectedComponentEquiv refl = Equiv.refl (ConnectedComponent G)",
"tactic": "ext ⟨v⟩"
},
{
"state_after": "no goals",
"state_before": "case H.mk\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nx✝ : ConnectedComponent G\nv : V\n⊢ ↑(connectedComponentEquiv refl) (Quot.mk (Reachable G) v) =\n ↑(Equiv.refl (ConnectedComponent G)) (Quot.mk (Reachable G) v)",
"tactic": "rfl"
}
] |
[
2110,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2107,
1
] |
Mathlib/Order/GameAdd.lean
|
Sym2.gameAdd_mk'_iff
|
[] |
[
174,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.mod_eq_of_lt
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.257723\nβ : Type ?u.257726\nγ : Type ?u.257729\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nh : a < b\n⊢ a % b = a",
"tactic": "simp only [mod_def, div_eq_zero_of_lt h, mul_zero, sub_zero]"
}
] |
[
1040,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1039,
1
] |
Mathlib/Topology/Order/LowerTopology.lean
|
LowerTopology.closure_singleton
|
[] |
[
208,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Topology/UniformSpace/Completion.lean
|
CauchyFilter.uniformInducing_pureCauchy
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\nx✝ : α × α\na₁ a₂ : α\n⊢ (a₁, a₂) ∈ ((preimage fun x => (pureCauchy x.fst, pureCauchy x.snd)) ∘ gen) s ↔ (a₁, a₂) ∈ id s",
"tactic": "simp [preimage, gen, pureCauchy, prod_principal_principal]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nthis : (preimage fun x => (pureCauchy x.fst, pureCauchy x.snd)) ∘ gen = id\n⊢ Filter.lift' (𝓤 α) ((preimage fun x => (pureCauchy x.fst, pureCauchy x.snd)) ∘ gen) = 𝓤 α",
"tactic": "simp [this]"
}
] |
[
169,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.graph_inj
|
[] |
[
114,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tendsto_atTop_zero_of_tsum_ne_top
|
[
{
"state_after": "α : Type ?u.262359\nβ : Type ?u.262362\nγ : Type ?u.262365\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : ℕ → ℝ≥0∞\nhf : (∑' (x : ℕ), f x) ≠ ⊤\n⊢ Tendsto f cofinite (𝓝 0)",
"state_before": "α : Type ?u.262359\nβ : Type ?u.262362\nγ : Type ?u.262365\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : ℕ → ℝ≥0∞\nhf : (∑' (x : ℕ), f x) ≠ ⊤\n⊢ Tendsto f atTop (𝓝 0)",
"tactic": "rw [← Nat.cofinite_eq_atTop]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.262359\nβ : Type ?u.262362\nγ : Type ?u.262365\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : ℕ → ℝ≥0∞\nhf : (∑' (x : ℕ), f x) ≠ ⊤\n⊢ Tendsto f cofinite (𝓝 0)",
"tactic": "exact tendsto_cofinite_zero_of_tsum_ne_top hf"
}
] |
[
960,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
957,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
AlternatingMap.coe_neg
|
[] |
[
382,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/Algebra/Homology/Augment.lean
|
ChainComplex.augmentTruncate_hom_f_zero
|
[] |
[
171,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.weightedSMul_smul_measure
|
[
{
"state_after": "case h\nα : Type u_1\nE : Type ?u.30268\nF : Type u_2\n𝕜 : Type ?u.30274\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nμ : Measure α\nc : ℝ≥0∞\ns : Set α\nx : F\n⊢ ↑(weightedSMul (c • μ) s) x = ↑(ENNReal.toReal c • weightedSMul μ s) x",
"state_before": "α : Type u_1\nE : Type ?u.30268\nF : Type u_2\n𝕜 : Type ?u.30274\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nμ : Measure α\nc : ℝ≥0∞\ns : Set α\n⊢ weightedSMul (c • μ) s = ENNReal.toReal c • weightedSMul μ s",
"tactic": "ext1 x"
},
{
"state_after": "case h\nα : Type u_1\nE : Type ?u.30268\nF : Type u_2\n𝕜 : Type ?u.30274\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nμ : Measure α\nc : ℝ≥0∞\ns : Set α\nx : F\n⊢ ↑(weightedSMul (c • μ) s) x = (ENNReal.toReal c • ↑(weightedSMul μ s)) x",
"state_before": "case h\nα : Type u_1\nE : Type ?u.30268\nF : Type u_2\n𝕜 : Type ?u.30274\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nμ : Measure α\nc : ℝ≥0∞\ns : Set α\nx : F\n⊢ ↑(weightedSMul (c • μ) s) x = ↑(ENNReal.toReal c • weightedSMul μ s) x",
"tactic": "push_cast"
},
{
"state_after": "case h\nα : Type u_1\nE : Type ?u.30268\nF : Type u_2\n𝕜 : Type ?u.30274\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nμ : Measure α\nc : ℝ≥0∞\ns : Set α\nx : F\n⊢ ENNReal.toReal (↑↑(c • μ) s) • x = ENNReal.toReal c • ENNReal.toReal (↑↑μ s) • x",
"state_before": "case h\nα : Type u_1\nE : Type ?u.30268\nF : Type u_2\n𝕜 : Type ?u.30274\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nμ : Measure α\nc : ℝ≥0∞\ns : Set α\nx : F\n⊢ ↑(weightedSMul (c • μ) s) x = (ENNReal.toReal c • ↑(weightedSMul μ s)) x",
"tactic": "simp_rw [Pi.smul_apply, weightedSMul_apply]"
},
{
"state_after": "case h\nα : Type u_1\nE : Type ?u.30268\nF : Type u_2\n𝕜 : Type ?u.30274\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nμ : Measure α\nc : ℝ≥0∞\ns : Set α\nx : F\n⊢ ENNReal.toReal ((c • ↑↑μ) s) • x = ENNReal.toReal c • ENNReal.toReal (↑↑μ s) • x",
"state_before": "case h\nα : Type u_1\nE : Type ?u.30268\nF : Type u_2\n𝕜 : Type ?u.30274\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nμ : Measure α\nc : ℝ≥0∞\ns : Set α\nx : F\n⊢ ENNReal.toReal (↑↑(c • μ) s) • x = ENNReal.toReal c • ENNReal.toReal (↑↑μ s) • x",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nE : Type ?u.30268\nF : Type u_2\n𝕜 : Type ?u.30274\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nμ : Measure α\nc : ℝ≥0∞\ns : Set α\nx : F\n⊢ ENNReal.toReal ((c • ↑↑μ) s) • x = ENNReal.toReal c • ENNReal.toReal (↑↑μ s) • x",
"tactic": "simp_rw [Pi.smul_apply, smul_eq_mul, toReal_mul, smul_smul]"
}
] |
[
203,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Topology/Bornology/Basic.lean
|
Bornology.isBounded_union
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.4828\nα : Type u_1\nβ : Type ?u.4834\ninst✝ : Bornology α\ns t : Set α\nx : α\n⊢ IsBounded (s ∪ t) ↔ IsBounded s ∧ IsBounded t",
"tactic": "simp only [← isCobounded_compl_iff, compl_union, isCobounded_inter]"
}
] |
[
201,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
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