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/Fintype/Card.lean
|
Fintype.card_empty
|
[] |
[
345,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.neg_nonneg_of_nonpos
|
[
{
"state_after": "a : Int\nh : a ≤ 0\nthis : -0 ≤ -a\n⊢ 0 ≤ -a",
"state_before": "a : Int\nh : a ≤ 0\n⊢ 0 ≤ -a",
"tactic": "have : -0 ≤ -a := Int.neg_le_neg h"
},
{
"state_after": "no goals",
"state_before": "a : Int\nh : a ≤ 0\nthis : -0 ≤ -a\n⊢ 0 ≤ -a",
"tactic": "rwa [Int.neg_zero] at this"
}
] |
[
877,
29
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
875,
11
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
Convex.norm_image_sub_le_of_norm_fderiv_le
|
[] |
[
543,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
539,
1
] |
Mathlib/ModelTheory/Ultraproducts.lean
|
FirstOrder.Language.Ultraproduct.term_realize_cast
|
[
{
"state_after": "case h.e'_3.h.e'_3\nα : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\nt : Term L β\n⊢ (fun a => Term.realize (fun i => x i a) t) = Term.realize x t",
"state_before": "α : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\nt : Term L β\n⊢ Term.realize (fun i => Quotient.mk' (x i)) t = Quotient.mk' fun a => Term.realize (fun i => x i a) t",
"tactic": "convert @Term.realize_quotient_mk' L _ ((u : Filter α).productSetoid M)\n (Ultraproduct.setoidPrestructure M u) _ t x using 2"
},
{
"state_after": "case h.e'_3.h.e'_3.h\nα : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\nt : Term L β\na : α\n⊢ Term.realize (fun i => x i a) t = Term.realize x t a",
"state_before": "case h.e'_3.h.e'_3\nα : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\nt : Term L β\n⊢ (fun a => Term.realize (fun i => x i a) t) = Term.realize x t",
"tactic": "ext a"
},
{
"state_after": "case h.e'_3.h.e'_3.h.var\nα : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\na : α\n_a✝ : β\n⊢ Term.realize (fun i => x i a) (var _a✝) = Term.realize x (var _a✝) a\n\ncase h.e'_3.h.e'_3.h.func\nα : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\na : α\nl✝ : ℕ\n_f✝ : Functions L l✝\n_ts✝ : Fin l✝ → Term L β\n_ts_ih✝ : ∀ (a_1 : Fin l✝), Term.realize (fun i => x i a) (_ts✝ a_1) = Term.realize x (_ts✝ a_1) a\n⊢ Term.realize (fun i => x i a) (func _f✝ _ts✝) = Term.realize x (func _f✝ _ts✝) a",
"state_before": "case h.e'_3.h.e'_3.h\nα : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\nt : Term L β\na : α\n⊢ Term.realize (fun i => x i a) t = Term.realize x t a",
"tactic": "induction t"
},
{
"state_after": "case h.e'_3.h.e'_3.h.func\nα : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\na : α\nl✝ : ℕ\n_f✝ : Functions L l✝\n_ts✝ : Fin l✝ → Term L β\n_ts_ih✝ : ∀ (a_1 : Fin l✝), Term.realize (fun i => x i a) (_ts✝ a_1) = Term.realize x (_ts✝ a_1) a\n⊢ Term.realize (fun i => x i a) (func _f✝ _ts✝) = Term.realize x (func _f✝ _ts✝) a",
"state_before": "case h.e'_3.h.e'_3.h.var\nα : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\na : α\n_a✝ : β\n⊢ Term.realize (fun i => x i a) (var _a✝) = Term.realize x (var _a✝) a\n\ncase h.e'_3.h.e'_3.h.func\nα : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\na : α\nl✝ : ℕ\n_f✝ : Functions L l✝\n_ts✝ : Fin l✝ → Term L β\n_ts_ih✝ : ∀ (a_1 : Fin l✝), Term.realize (fun i => x i a) (_ts✝ a_1) = Term.realize x (_ts✝ a_1) a\n⊢ Term.realize (fun i => x i a) (func _f✝ _ts✝) = Term.realize x (func _f✝ _ts✝) a",
"tactic": "case var =>\n rfl"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_3.h.func\nα : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\na : α\nl✝ : ℕ\n_f✝ : Functions L l✝\n_ts✝ : Fin l✝ → Term L β\n_ts_ih✝ : ∀ (a_1 : Fin l✝), Term.realize (fun i => x i a) (_ts✝ a_1) = Term.realize x (_ts✝ a_1) a\n⊢ Term.realize (fun i => x i a) (func _f✝ _ts✝) = Term.realize x (func _f✝ _ts✝) a",
"tactic": "case func _ _ _ t_ih =>\n simp only [Term.realize, t_ih]\n rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\na : α\n_a✝ : β\n⊢ Term.realize (fun i => x i a) (var _a✝) = Term.realize x (var _a✝) a",
"tactic": "rfl"
},
{
"state_after": "α : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\na : α\nl✝ : ℕ\n_f✝ : Functions L l✝\n_ts✝ : Fin l✝ → Term L β\nt_ih : ∀ (a_1 : Fin l✝), Term.realize (fun i => x i a) (_ts✝ a_1) = Term.realize x (_ts✝ a_1) a\n⊢ (funMap _f✝ fun i => Term.realize x (_ts✝ i) a) = funMap _f✝ (fun i => Term.realize x (_ts✝ i)) a",
"state_before": "α : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\na : α\nl✝ : ℕ\n_f✝ : Functions L l✝\n_ts✝ : Fin l✝ → Term L β\nt_ih : ∀ (a_1 : Fin l✝), Term.realize (fun i => x i a) (_ts✝ a_1) = Term.realize x (_ts✝ a_1) a\n⊢ Term.realize (fun i => x i a) (func _f✝ _ts✝) = Term.realize x (func _f✝ _ts✝) a",
"tactic": "simp only [Term.realize, t_ih]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nM : α → Type u_3\nu : Ultrafilter α\nL : Language\ninst✝ : (a : α) → Structure L (M a)\nβ : Type u_1\nx : β → (a : α) → M a\na : α\nl✝ : ℕ\n_f✝ : Functions L l✝\n_ts✝ : Fin l✝ → Term L β\nt_ih : ∀ (a_1 : Fin l✝), Term.realize (fun i => x i a) (_ts✝ a_1) = Term.realize x (_ts✝ a_1) a\n⊢ (funMap _f✝ fun i => Term.realize x (_ts✝ i) a) = funMap _f✝ (fun i => Term.realize x (_ts✝ i)) a",
"tactic": "rfl"
}
] |
[
97,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Std/Data/List/Lemmas.lean
|
List.get_map
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nf : α → β\nl : List α\nn : Fin (length (map f l))\n⊢ Option.map f (some (get l { val := n.val, isLt := ?m.72539 })) =\n some (f (get l { val := n.val, isLt := (_ : n.val < length l) }))\n\nα : Type u_1\nβ : Type u_2\nf : α → β\nl : List α\nn : Fin (length (map f l))\n⊢ n.val < length l",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β\nl : List α\nn : Fin (length (map f l))\n⊢ some (get (map f l) n) = some (f (get l { val := n.val, isLt := (_ : n.val < length l) }))",
"tactic": "rw [← get?_eq_get, get?_map, get?_eq_get]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β\nl : List α\nn : Fin (length (map f l))\n⊢ Option.map f (some (get l { val := n.val, isLt := ?m.72539 })) =\n some (f (get l { val := n.val, isLt := (_ : n.val < length l) }))\n\nα : Type u_1\nβ : Type u_2\nf : α → β\nl : List α\nn : Fin (length (map f l))\n⊢ n.val < length l",
"tactic": "rfl"
}
] |
[
574,
71
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
573,
9
] |
Mathlib/Data/Pi/Algebra.lean
|
Pi.one_apply
|
[] |
[
51,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.restrict_preimage
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.812913\nδ : Type ?u.812916\ninst✝¹ : MeasurableSpace α\nK : Type ?u.812922\ninst✝ : Zero β\nf : α →ₛ β\ns : Set α\nhs : MeasurableSet s\nt : Set β\nht : ¬0 ∈ t\n⊢ ↑(restrict f s) ⁻¹' t = s ∩ ↑f ⁻¹' t",
"tactic": "simp [hs, indicator_preimage_of_not_mem _ _ ht, inter_comm]"
}
] |
[
789,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
787,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.map_smul'
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.969460\nR₁ : Type ?u.969463\nR₂ : Type ?u.969466\nR₃ : Type ?u.969469\nR₄ : Type ?u.969472\nS : Type ?u.969475\nK : Type u_1\nK₂ : Type ?u.969481\nM : Type ?u.969484\nM' : Type ?u.969487\nM₁ : Type ?u.969490\nM₂ : Type ?u.969493\nM₃ : Type ?u.969496\nM₄ : Type ?u.969499\nN : Type ?u.969502\nN₂ : Type ?u.969505\nι : Type ?u.969508\nV : Type u_2\nV₂ : Type u_3\ninst✝⁴ : Semifield K\ninst✝³ : AddCommMonoid V\ninst✝² : Module K V\ninst✝¹ : AddCommMonoid V₂\ninst✝ : Module K V₂\nf : V →ₗ[K] V₂\np : Submodule K V\na : K\n⊢ map (a • f) p = ⨆ (_ : a ≠ 0), map f p",
"tactic": "classical by_cases h : a = 0 <;> simp [h, map_smul]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.969460\nR₁ : Type ?u.969463\nR₂ : Type ?u.969466\nR₃ : Type ?u.969469\nR₄ : Type ?u.969472\nS : Type ?u.969475\nK : Type u_1\nK₂ : Type ?u.969481\nM : Type ?u.969484\nM' : Type ?u.969487\nM₁ : Type ?u.969490\nM₂ : Type ?u.969493\nM₃ : Type ?u.969496\nM₄ : Type ?u.969499\nN : Type ?u.969502\nN₂ : Type ?u.969505\nι : Type ?u.969508\nV : Type u_2\nV₂ : Type u_3\ninst✝⁴ : Semifield K\ninst✝³ : AddCommMonoid V\ninst✝² : Module K V\ninst✝¹ : AddCommMonoid V₂\ninst✝ : Module K V₂\nf : V →ₗ[K] V₂\np : Submodule K V\na : K\n⊢ map (a • f) p = ⨆ (_ : a ≠ 0), map f p",
"tactic": "by_cases h : a = 0 <;> simp [h, map_smul]"
}
] |
[
1093,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1091,
1
] |
Mathlib/Order/OrdContinuous.lean
|
LeftOrdContinuous.le_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\nf : α → β\nhf : LeftOrdContinuous f\nh : Injective f\nx y : α\n⊢ f x ≤ f y ↔ x ≤ y",
"tactic": "simp only [← sup_eq_right, ← hf.map_sup, h.eq_iff]"
}
] |
[
102,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.sep_setOf
|
[] |
[
1484,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1483,
1
] |
Mathlib/GroupTheory/GroupAction/Basic.lean
|
MulAction.orbitRel.Quotient.orbit_mk
|
[] |
[
359,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.isEmpty_coe_sort
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.23485\nγ : Type ?u.23488\ns✝ s : Finset α\n⊢ IsEmpty { x // x ∈ s } ↔ s = ∅",
"tactic": "simpa using @Set.isEmpty_coe_sort α s"
}
] |
[
631,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
630,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.add_kroneckerTMul
|
[] |
[
479,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
Std/Data/Option/Lemmas.lean
|
Option.bind_id_eq_join
|
[] |
[
119,
76
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
119,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.SimpleFunc.setToSimpleFunc_smul
|
[
{
"state_after": "α : Type u_4\nE✝ : Type ?u.310418\nF : Type u_3\nF' : Type ?u.310424\nG : Type ?u.310427\n𝕜 : Type u_2\np : ℝ≥0∞\ninst✝¹¹ : NormedAddCommGroup E✝\ninst✝¹⁰ : NormedSpace ℝ E✝\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup F'\ninst✝⁶ : NormedSpace ℝ F'\ninst✝⁵ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 F\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nh_smul : ∀ (c : 𝕜) (s : Set α) (x : E), ↑(T s) (c • x) = c • ↑(T s) x\nc : 𝕜\nf : α →ₛ E\nhf : Integrable ↑f\n⊢ (fun x x_1 => x • x_1) c 0 = 0",
"state_before": "α : Type u_4\nE✝ : Type ?u.310418\nF : Type u_3\nF' : Type ?u.310424\nG : Type ?u.310427\n𝕜 : Type u_2\np : ℝ≥0∞\ninst✝¹¹ : NormedAddCommGroup E✝\ninst✝¹⁰ : NormedSpace ℝ E✝\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup F'\ninst✝⁶ : NormedSpace ℝ F'\ninst✝⁵ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 F\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nh_smul : ∀ (c : 𝕜) (s : Set α) (x : E), ↑(T s) (c • x) = c • ↑(T s) x\nc : 𝕜\nf : α →ₛ E\nhf : Integrable ↑f\n⊢ setToSimpleFunc T (c • f) = ∑ x in SimpleFunc.range f, ↑(T (↑f ⁻¹' {x})) (c • x)",
"tactic": "rw [smul_eq_map c f, map_setToSimpleFunc T h_add hf]"
},
{
"state_after": "α : Type u_4\nE✝ : Type ?u.310418\nF : Type u_3\nF' : Type ?u.310424\nG : Type ?u.310427\n𝕜 : Type u_2\np : ℝ≥0∞\ninst✝¹¹ : NormedAddCommGroup E✝\ninst✝¹⁰ : NormedSpace ℝ E✝\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup F'\ninst✝⁶ : NormedSpace ℝ F'\ninst✝⁵ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 F\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nh_smul : ∀ (c : 𝕜) (s : Set α) (x : E), ↑(T s) (c • x) = c • ↑(T s) x\nc : 𝕜\nf : α →ₛ E\nhf : Integrable ↑f\n⊢ c • 0 = 0",
"state_before": "α : Type u_4\nE✝ : Type ?u.310418\nF : Type u_3\nF' : Type ?u.310424\nG : Type ?u.310427\n𝕜 : Type u_2\np : ℝ≥0∞\ninst✝¹¹ : NormedAddCommGroup E✝\ninst✝¹⁰ : NormedSpace ℝ E✝\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup F'\ninst✝⁶ : NormedSpace ℝ F'\ninst✝⁵ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 F\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nh_smul : ∀ (c : 𝕜) (s : Set α) (x : E), ↑(T s) (c • x) = c • ↑(T s) x\nc : 𝕜\nf : α →ₛ E\nhf : Integrable ↑f\n⊢ (fun x x_1 => x • x_1) c 0 = 0",
"tactic": "dsimp only"
},
{
"state_after": "no goals",
"state_before": "α : Type u_4\nE✝ : Type ?u.310418\nF : Type u_3\nF' : Type ?u.310424\nG : Type ?u.310427\n𝕜 : Type u_2\np : ℝ≥0∞\ninst✝¹¹ : NormedAddCommGroup E✝\ninst✝¹⁰ : NormedSpace ℝ E✝\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup F'\ninst✝⁶ : NormedSpace ℝ F'\ninst✝⁵ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 F\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nh_smul : ∀ (c : 𝕜) (s : Set α) (x : E), ↑(T s) (c • x) = c • ↑(T s) x\nc : 𝕜\nf : α →ₛ E\nhf : Integrable ↑f\n⊢ c • 0 = 0",
"tactic": "rw [smul_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_4\nE✝ : Type ?u.310418\nF : Type u_3\nF' : Type ?u.310424\nG : Type ?u.310427\n𝕜 : Type u_2\np : ℝ≥0∞\ninst✝¹¹ : NormedAddCommGroup E✝\ninst✝¹⁰ : NormedSpace ℝ E✝\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup F'\ninst✝⁶ : NormedSpace ℝ F'\ninst✝⁵ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 F\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nh_smul : ∀ (c : 𝕜) (s : Set α) (x : E), ↑(T s) (c • x) = c • ↑(T s) x\nc : 𝕜\nf : α →ₛ E\nhf : Integrable ↑f\nb : E\nx✝ : b ∈ SimpleFunc.range f\n⊢ ↑(T (↑f ⁻¹' {b})) (c • b) = c • ↑(T (↑f ⁻¹' {b})) b",
"tactic": "rw [h_smul]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_4\nE✝ : Type ?u.310418\nF : Type u_3\nF' : Type ?u.310424\nG : Type ?u.310427\n𝕜 : Type u_2\np : ℝ≥0∞\ninst✝¹¹ : NormedAddCommGroup E✝\ninst✝¹⁰ : NormedSpace ℝ E✝\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup F'\ninst✝⁶ : NormedSpace ℝ F'\ninst✝⁵ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 F\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nh_smul : ∀ (c : 𝕜) (s : Set α) (x : E), ↑(T s) (c • x) = c • ↑(T s) x\nc : 𝕜\nf : α →ₛ E\nhf : Integrable ↑f\n⊢ ∑ x in SimpleFunc.range f, c • ↑(T (↑f ⁻¹' {x})) x = c • setToSimpleFunc T f",
"tactic": "simp only [setToSimpleFunc, smul_sum, smul_smul, mul_comm]"
}
] |
[
510,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/Data/Sym/Basic.lean
|
SymOptionSuccEquiv.encode_of_not_none_mem
|
[] |
[
624,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
619,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
le_csInf_inter
|
[] |
[
702,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
700,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
Encodable.decode₂_inj
|
[] |
[
226,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
comap_dist_left_atTop_le_cocompact
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.546511\nι : Type ?u.546514\ninst✝ : PseudoMetricSpace α\nx : α\n⊢ comap (dist x) atTop ≤ cocompact α",
"tactic": "simpa only [dist_comm _ x] using comap_dist_right_atTop_le_cocompact x"
}
] |
[
2800,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2799,
1
] |
Mathlib/Data/Quot.lean
|
Trunc.nonempty
|
[] |
[
584,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
11
] |
Mathlib/LinearAlgebra/Matrix/Adjugate.lean
|
Matrix.sum_cramer_apply
|
[
{
"state_after": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\nβ : Type u_1\ns : Finset β\nf : n → β → α\ni : n\n⊢ det (updateColumn A i (∑ x in s, fun j => f j x)) = det (updateColumn A i fun j => ∑ x in s, f j x)",
"state_before": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\nβ : Type u_1\ns : Finset β\nf : n → β → α\ni : n\n⊢ Finset.sum s (fun x => ↑(cramer A) fun j => f j x) i = ↑(cramer A) (fun j => ∑ x in s, f j x) i",
"tactic": "rw [sum_cramer, cramer_apply, cramer_apply]"
},
{
"state_after": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\nβ : Type u_1\ns : Finset β\nf : n → β → α\ni : n\n⊢ det (↑of fun i_1 => Function.update (A i_1) i (Finset.sum s (fun x j => f j x) i_1)) =\n det (↑of fun i_1 => Function.update (A i_1) i (∑ x in s, f i_1 x))",
"state_before": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\nβ : Type u_1\ns : Finset β\nf : n → β → α\ni : n\n⊢ det (updateColumn A i (∑ x in s, fun j => f j x)) = det (updateColumn A i fun j => ∑ x in s, f j x)",
"tactic": "simp only [updateColumn]"
},
{
"state_after": "case e_M.h.e_6.h.h.h\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\nβ : Type u_1\ns : Finset β\nf : n → β → α\ni j x✝ : n\n⊢ Function.update (A j) i (Finset.sum s (fun x j => f j x) j) x✝ = Function.update (A j) i (∑ x in s, f j x) x✝",
"state_before": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\nβ : Type u_1\ns : Finset β\nf : n → β → α\ni : n\n⊢ det (↑of fun i_1 => Function.update (A i_1) i (Finset.sum s (fun x j => f j x) i_1)) =\n det (↑of fun i_1 => Function.update (A i_1) i (∑ x in s, f i_1 x))",
"tactic": "congr with j"
},
{
"state_after": "case e_M.h.e_6.h.h.h.e_v\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\nβ : Type u_1\ns : Finset β\nf : n → β → α\ni j x✝ : n\n⊢ Finset.sum s (fun x j => f j x) j = ∑ x in s, f j x",
"state_before": "case e_M.h.e_6.h.h.h\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\nβ : Type u_1\ns : Finset β\nf : n → β → α\ni j x✝ : n\n⊢ Function.update (A j) i (Finset.sum s (fun x j => f j x) j) x✝ = Function.update (A j) i (∑ x in s, f j x) x✝",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_M.h.e_6.h.h.h.e_v\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\nβ : Type u_1\ns : Finset β\nf : n → β → α\ni j x✝ : n\n⊢ Finset.sum s (fun x j => f j x) j = ∑ x in s, f j x",
"tactic": "apply Finset.sum_apply"
}
] |
[
174,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Analysis/SpecialFunctions/Arsinh.lean
|
Continuous.arsinh
|
[] |
[
233,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.BoundedFormula.realize_bot
|
[] |
[
267,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sSup_apply
|
[] |
[
1760,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1758,
1
] |
Mathlib/LinearAlgebra/Isomorphisms.lean
|
LinearMap.quotientInfEquivSupQuotient_apply_mk
|
[] |
[
124,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/CategoryTheory/Preadditive/Injective.lean
|
CategoryTheory.Equivalence.enoughInjectives_iff
|
[
{
"state_after": "case mp\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\n⊢ EnoughInjectives C → EnoughInjectives D\n\ncase mpr\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\n⊢ EnoughInjectives D → EnoughInjectives C",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\n⊢ EnoughInjectives C ↔ EnoughInjectives D",
"tactic": "constructor"
},
{
"state_after": "case mp.presentation.val\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\nH : EnoughInjectives C\nX : D\n⊢ InjectivePresentation X\n\ncase mpr.presentation.val\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\nH : EnoughInjectives D\nX : C\n⊢ InjectivePresentation X",
"state_before": "case mp\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\n⊢ EnoughInjectives C → EnoughInjectives D\n\ncase mpr\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\n⊢ EnoughInjectives D → EnoughInjectives C",
"tactic": "all_goals intro H; constructor; intro X; constructor"
},
{
"state_after": "case mpr\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\nH : EnoughInjectives D\n⊢ EnoughInjectives C",
"state_before": "case mpr\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\n⊢ EnoughInjectives D → EnoughInjectives C",
"tactic": "intro H"
},
{
"state_after": "case mpr.presentation\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\nH : EnoughInjectives D\n⊢ ∀ (X : C), Nonempty (InjectivePresentation X)",
"state_before": "case mpr\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\nH : EnoughInjectives D\n⊢ EnoughInjectives C",
"tactic": "constructor"
},
{
"state_after": "case mpr.presentation\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\nH : EnoughInjectives D\nX : C\n⊢ Nonempty (InjectivePresentation X)",
"state_before": "case mpr.presentation\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\nH : EnoughInjectives D\n⊢ ∀ (X : C), Nonempty (InjectivePresentation X)",
"tactic": "intro X"
},
{
"state_after": "case mpr.presentation.val\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\nH : EnoughInjectives D\nX : C\n⊢ InjectivePresentation X",
"state_before": "case mpr.presentation\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\nH : EnoughInjectives D\nX : C\n⊢ Nonempty (InjectivePresentation X)",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case mp.presentation.val\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\nH : EnoughInjectives C\nX : D\n⊢ InjectivePresentation X",
"tactic": "exact\n F.symm.injectivePresentationOfMapInjectivePresentation _\n (Nonempty.some (H.presentation (F.inverse.obj X)))"
},
{
"state_after": "no goals",
"state_before": "case mpr.presentation.val\nC : Type u₁\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF✝ F : C ≌ D\nH : EnoughInjectives D\nX : C\n⊢ InjectivePresentation X",
"tactic": "exact\n F.injectivePresentationOfMapInjectivePresentation X\n (Nonempty.some (H.presentation (F.functor.obj X)))"
}
] |
[
375,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Topology/Algebra/Order/ProjIcc.lean
|
Filter.Tendsto.IccExtend
|
[] |
[
30,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
27,
11
] |
Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
|
Finpartition.nonempty_of_not_uniform
|
[] |
[
260,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
mem_closedBall_one_iff
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.94910\n𝕜 : Type ?u.94913\nα : Type ?u.94916\nι : Type ?u.94919\nκ : Type ?u.94922\nE : Type u_1\nF : Type ?u.94928\nG : Type ?u.94931\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\n⊢ a ∈ closedBall 1 r ↔ ‖a‖ ≤ r",
"tactic": "rw [mem_closedBall, dist_one_right]"
}
] |
[
640,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
639,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
Ideal.homogeneousCore'_mono
|
[] |
[
123,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Analysis/InnerProductSpace/Symmetric.lean
|
LinearMap.IsSymmetric.continuous
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\n⊢ y = ↑T x",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\n⊢ Continuous ↑T",
"tactic": "refine' T.continuous_of_seq_closed_graph fun u x y hu hTu => _"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\n⊢ inner (y - ↑T x) (y - ↑T x) = 0",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\n⊢ y = ↑T x",
"tactic": "rw [← sub_eq_zero, ← @inner_self_eq_zero 𝕜]"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nhlhs : ∀ (k : ℕ), inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))\n⊢ inner (y - ↑T x) (y - ↑T x) = 0",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\n⊢ inner (y - ↑T x) (y - ↑T x) = 0",
"tactic": "have hlhs : ∀ k : ℕ, ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫ := by\n intro k\n rw [← T.map_sub, hT]"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nhlhs : ∀ (k : ℕ), inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))\n⊢ Filter.Tendsto (fun t => inner ((↑T ∘ u) t - ↑T x) (y - ↑T x)) Filter.atTop (nhds 0)",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nhlhs : ∀ (k : ℕ), inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))\n⊢ inner (y - ↑T x) (y - ↑T x) = 0",
"tactic": "refine' tendsto_nhds_unique ((hTu.sub_const _).inner tendsto_const_nhds) _"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nhlhs : ∀ (k : ℕ), inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))\n⊢ Filter.Tendsto (fun t => inner (u t - x) (↑T (y - ↑T x))) Filter.atTop (nhds 0)",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nhlhs : ∀ (k : ℕ), inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))\n⊢ Filter.Tendsto (fun t => inner ((↑T ∘ u) t - ↑T x) (y - ↑T x)) Filter.atTop (nhds 0)",
"tactic": "simp_rw [Function.comp_apply, hlhs]"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nhlhs : ∀ (k : ℕ), inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))\n⊢ Filter.Tendsto (fun t => inner (u t - x) (↑T (y - ↑T x))) Filter.atTop (nhds (inner 0 (↑T (y - ↑T x))))",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nhlhs : ∀ (k : ℕ), inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))\n⊢ Filter.Tendsto (fun t => inner (u t - x) (↑T (y - ↑T x))) Filter.atTop (nhds 0)",
"tactic": "rw [← inner_zero_left (T (y - T x))]"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nhlhs : ∀ (k : ℕ), inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))\n⊢ Filter.Tendsto (fun t => u t - x) Filter.atTop (nhds 0)",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nhlhs : ∀ (k : ℕ), inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))\n⊢ Filter.Tendsto (fun t => inner (u t - x) (↑T (y - ↑T x))) Filter.atTop (nhds (inner 0 (↑T (y - ↑T x))))",
"tactic": "refine' Filter.Tendsto.inner _ tendsto_const_nhds"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nhlhs : ∀ (k : ℕ), inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))\n⊢ Filter.Tendsto (fun t => u t - x) Filter.atTop (nhds (x - x))",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nhlhs : ∀ (k : ℕ), inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))\n⊢ Filter.Tendsto (fun t => u t - x) Filter.atTop (nhds 0)",
"tactic": "rw [← sub_self x]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nhlhs : ∀ (k : ℕ), inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))\n⊢ Filter.Tendsto (fun t => u t - x) Filter.atTop (nhds (x - x))",
"tactic": "exact hu.sub_const _"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nk : ℕ\n⊢ inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\n⊢ ∀ (k : ℕ), inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))",
"tactic": "intro k"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nE' : Type ?u.49605\nF : Type ?u.49608\nG : Type ?u.49611\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℝ E'\ninst✝ : CompleteSpace E\nT : E →ₗ[𝕜] E\nhT : IsSymmetric T\nu : ℕ → E\nx y : E\nhu : Filter.Tendsto u Filter.atTop (nhds x)\nhTu : Filter.Tendsto (↑T ∘ u) Filter.atTop (nhds y)\nk : ℕ\n⊢ inner (↑T (u k) - ↑T x) (y - ↑T x) = inner (u k - x) (↑T (y - ↑T x))",
"tactic": "rw [← T.map_sub, hT]"
}
] |
[
117,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
Measurable.isLUB
|
[
{
"state_after": "α : Type u_3\nβ : Type ?u.1092125\nγ : Type ?u.1092128\nγ₂ : Type ?u.1092131\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)\n⊢ Measurable g",
"state_before": "α : Type u_3\nβ : Type ?u.1092125\nγ : Type ?u.1092128\nγ₂ : Type ?u.1092131\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsLUB {a | ∃ i, f i b = a} (g b)\n⊢ Measurable g",
"tactic": "change ∀ b, IsLUB (range fun i => f i b) (g b) at hg"
},
{
"state_after": "α : Type u_3\nβ : Type ?u.1092125\nγ : Type ?u.1092128\nγ₂ : Type ?u.1092131\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)\n⊢ Measurable g",
"state_before": "α : Type u_3\nβ : Type ?u.1092125\nγ : Type ?u.1092128\nγ₂ : Type ?u.1092131\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)\n⊢ Measurable g",
"tactic": "rw [‹BorelSpace α›.measurable_eq, borel_eq_generateFrom_Ioi α]"
},
{
"state_after": "case h\nα : Type u_3\nβ : Type ?u.1092125\nγ : Type ?u.1092128\nγ₂ : Type ?u.1092131\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)\n⊢ ∀ (t : Set α), t ∈ range Ioi → MeasurableSet (g ⁻¹' t)",
"state_before": "α : Type u_3\nβ : Type ?u.1092125\nγ : Type ?u.1092128\nγ₂ : Type ?u.1092131\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)\n⊢ Measurable g",
"tactic": "apply measurable_generateFrom"
},
{
"state_after": "case h.intro\nα : Type u_3\nβ : Type ?u.1092125\nγ : Type ?u.1092128\nγ₂ : Type ?u.1092131\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)\na : α\n⊢ MeasurableSet (g ⁻¹' Ioi a)",
"state_before": "case h\nα : Type u_3\nβ : Type ?u.1092125\nγ : Type ?u.1092128\nγ₂ : Type ?u.1092131\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)\n⊢ ∀ (t : Set α), t ∈ range Ioi → MeasurableSet (g ⁻¹' t)",
"tactic": "rintro _ ⟨a, rfl⟩"
},
{
"state_after": "case h.intro\nα : Type u_3\nβ : Type ?u.1092125\nγ : Type ?u.1092128\nγ₂ : Type ?u.1092131\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)\na : α\n⊢ MeasurableSet (⋃ (i : ι), {x | a < f i x})",
"state_before": "case h.intro\nα : Type u_3\nβ : Type ?u.1092125\nγ : Type ?u.1092128\nγ₂ : Type ?u.1092131\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)\na : α\n⊢ MeasurableSet (g ⁻¹' Ioi a)",
"tactic": "simp_rw [Set.preimage, mem_Ioi, lt_isLUB_iff (hg _), exists_range_iff, setOf_exists]"
},
{
"state_after": "no goals",
"state_before": "case h.intro\nα : Type u_3\nβ : Type ?u.1092125\nγ : Type ?u.1092128\nγ₂ : Type ?u.1092131\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b)\na : α\n⊢ MeasurableSet (⋃ (i : ι), {x | a < f i x})",
"tactic": "exact MeasurableSet.iUnion fun i => hf i (isOpen_lt' _).measurableSet"
}
] |
[
1082,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1075,
1
] |
Mathlib/Data/Set/Countable.lean
|
Set.countable_isBot
|
[] |
[
253,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/FieldTheory/IntermediateField.lean
|
IntermediateField.inclusion_injective
|
[] |
[
554,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
552,
1
] |
src/lean/Init/Control/Lawful.lean
|
ReaderT.run_mapConst
|
[] |
[
190,
77
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
189,
9
] |
Mathlib/Data/Set/Image.lean
|
Set.disjoint_image_iff
|
[] |
[
1602,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1601,
1
] |
Mathlib/Topology/LocalAtTarget.lean
|
isClosedMap_iff_isClosedMap_of_iSup_eq_top
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\n⊢ (∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)) → IsClosedMap f",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\n⊢ IsClosedMap f ↔ ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)",
"tactic": "refine' ⟨fun h i => Set.restrictPreimage_isClosedMap _ h, _⟩"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\nH : ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set α\nhs : IsClosed s\n⊢ IsClosed (f '' s)",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\n⊢ (∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)) → IsClosedMap f",
"tactic": "rintro H s hs"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\nH : ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set α\nhs : IsClosed s\n⊢ ∀ (i : ι), IsClosed (Subtype.val ⁻¹' (f '' s))",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\nH : ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set α\nhs : IsClosed s\n⊢ IsClosed (f '' s)",
"tactic": "rw [isClosed_iff_coe_preimage_of_iSup_eq_top hU]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\nH : ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set α\nhs : IsClosed s\ni : ι\n⊢ IsClosed (Subtype.val ⁻¹' (f '' s))",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\nH : ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set α\nhs : IsClosed s\n⊢ ∀ (i : ι), IsClosed (Subtype.val ⁻¹' (f '' s))",
"tactic": "intro i"
},
{
"state_after": "case h.e'_3.h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\nH : ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set α\nhs : IsClosed s\ni : ι\ne_1✝ : { x // x ∈ U i } = ↑(U i).carrier\n⊢ Subtype.val ⁻¹' (f '' s) = restrictPreimage (U i).carrier f '' (Subtype.val ⁻¹' sᶜ)ᶜ",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\nH : ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set α\nhs : IsClosed s\ni : ι\n⊢ IsClosed (Subtype.val ⁻¹' (f '' s))",
"tactic": "convert H i _ ⟨⟨_, hs.1, eq_compl_comm.mpr rfl⟩⟩"
},
{
"state_after": "case h.e'_3.h.h.mk\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\nH : ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set α\nhs : IsClosed s\ni : ι\ne_1✝ : { x // x ∈ U i } = ↑(U i).carrier\nx : β\nhx : x ∈ U i\n⊢ { val := x, property := hx } ∈ Subtype.val ⁻¹' (f '' s) ↔\n { val := x, property := hx } ∈ restrictPreimage (U i).carrier f '' (Subtype.val ⁻¹' sᶜ)ᶜ",
"state_before": "case h.e'_3.h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\nH : ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set α\nhs : IsClosed s\ni : ι\ne_1✝ : { x // x ∈ U i } = ↑(U i).carrier\n⊢ Subtype.val ⁻¹' (f '' s) = restrictPreimage (U i).carrier f '' (Subtype.val ⁻¹' sᶜ)ᶜ",
"tactic": "ext ⟨x, hx⟩"
},
{
"state_after": "case h.e'_3.h.h.mk\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\nH : ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set α\nhs : IsClosed s\ni : ι\ne_1✝ : { x // x ∈ U i } = ↑(U i).carrier\nx : β\nhx : x ∈ U i\n⊢ (∃ y, y ∈ s ∧ f y = x) ↔ ∃ y, y ∈ s ∧ f y ∈ U i ∧ f y = x",
"state_before": "case h.e'_3.h.h.mk\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\nH : ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set α\nhs : IsClosed s\ni : ι\ne_1✝ : { x // x ∈ U i } = ↑(U i).carrier\nx : β\nhx : x ∈ U i\n⊢ { val := x, property := hx } ∈ Subtype.val ⁻¹' (f '' s) ↔\n { val := x, property := hx } ∈ restrictPreimage (U i).carrier f '' (Subtype.val ⁻¹' sᶜ)ᶜ",
"tactic": "suffices (∃ y, y ∈ s ∧ f y = x) ↔ ∃ y, y ∈ s ∧ f y ∈ U i ∧ f y = x by\n simpa [Set.restrictPreimage, ← Subtype.coe_inj]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.h.mk\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\nH : ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set α\nhs : IsClosed s\ni : ι\ne_1✝ : { x // x ∈ U i } = ↑(U i).carrier\nx : β\nhx : x ∈ U i\n⊢ (∃ y, y ∈ s ∧ f y = x) ↔ ∃ y, y ∈ s ∧ f y ∈ U i ∧ f y = x",
"tactic": "exact ⟨fun ⟨a, b, c⟩ => ⟨a, b, c.symm ▸ hx, c⟩, fun ⟨a, b, _, c⟩ => ⟨a, b, c⟩⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.21339\nU : ι → Opens β\nhU : iSup U = ⊤\nH : ∀ (i : ι), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set α\nhs : IsClosed s\ni : ι\ne_1✝ : { x // x ∈ U i } = ↑(U i).carrier\nx : β\nhx : x ∈ U i\nthis : (∃ y, y ∈ s ∧ f y = x) ↔ ∃ y, y ∈ s ∧ f y ∈ U i ∧ f y = x\n⊢ { val := x, property := hx } ∈ Subtype.val ⁻¹' (f '' s) ↔\n { val := x, property := hx } ∈ restrictPreimage (U i).carrier f '' (Subtype.val ⁻¹' sᶜ)ᶜ",
"tactic": "simpa [Set.restrictPreimage, ← Subtype.coe_inj]"
}
] |
[
118,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.relIso_enum'
|
[
{
"state_after": "α✝ : Type ?u.97545\nβ✝ : Type ?u.97548\nγ : Type ?u.97551\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≃r s\no : Ordinal\n⊢ ∀ (α_1 : Type u) (r_1 : α_1 → α_1 → Prop) [inst : IsWellOrder α_1 r_1] (hr : type r_1 < type r)\n (hs : type r_1 < type s), ↑f (enum r (type r_1) hr) = enum s (type r_1) hs",
"state_before": "α✝ : Type ?u.97545\nβ✝ : Type ?u.97548\nγ : Type ?u.97551\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≃r s\no : Ordinal\n⊢ ∀ (hr : o < type r) (hs : o < type s), ↑f (enum r o hr) = enum s o hs",
"tactic": "refine' inductionOn o _"
},
{
"state_after": "case intro.intro\nα✝ : Type ?u.97545\nβ✝ : Type ?u.97548\nγ✝ : Type ?u.97551\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt✝ : γ✝ → γ✝ → Prop\nα β : Type u\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≃r s\no : Ordinal\nγ : Type u\nt : γ → γ → Prop\nwo : IsWellOrder γ t\ng : t ≺i r\nh : t ≺i s\n⊢ ↑f (enum r (type t) (_ : Nonempty (t ≺i r))) = enum s (type t) (_ : Nonempty (t ≺i s))",
"state_before": "α✝ : Type ?u.97545\nβ✝ : Type ?u.97548\nγ : Type ?u.97551\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≃r s\no : Ordinal\n⊢ ∀ (α_1 : Type u) (r_1 : α_1 → α_1 → Prop) [inst : IsWellOrder α_1 r_1] (hr : type r_1 < type r)\n (hs : type r_1 < type s), ↑f (enum r (type r_1) hr) = enum s (type r_1) hs",
"tactic": "rintro γ t wo ⟨g⟩ ⟨h⟩"
},
{
"state_after": "case intro.intro\nα✝ : Type ?u.97545\nβ✝ : Type ?u.97548\nγ✝ : Type ?u.97551\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt✝ : γ✝ → γ✝ → Prop\nα β : Type u\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≃r s\no : Ordinal\nγ : Type u\nt : γ → γ → Prop\nwo : IsWellOrder γ t\ng : t ≺i r\nh : t ≺i s\n⊢ ↑f (enum r (type t) (_ : Nonempty (t ≺i r))) = enum s (type t) (_ : Nonempty (t ≺i s))",
"state_before": "case intro.intro\nα✝ : Type ?u.97545\nβ✝ : Type ?u.97548\nγ✝ : Type ?u.97551\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt✝ : γ✝ → γ✝ → Prop\nα β : Type u\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≃r s\no : Ordinal\nγ : Type u\nt : γ → γ → Prop\nwo : IsWellOrder γ t\ng : t ≺i r\nh : t ≺i s\n⊢ ↑f (enum r (type t) (_ : Nonempty (t ≺i r))) = enum s (type t) (_ : Nonempty (t ≺i s))",
"tactic": "skip"
},
{
"state_after": "case intro.intro\nα✝ : Type ?u.97545\nβ✝ : Type ?u.97548\nγ✝ : Type ?u.97551\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt✝ : γ✝ → γ✝ → Prop\nα β : Type u\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≃r s\no : Ordinal\nγ : Type u\nt : γ → γ → Prop\nwo : IsWellOrder γ t\ng : t ≺i r\nh : t ≺i s\n⊢ ↑f g.top = (PrincipalSeg.ltEquiv g f).top",
"state_before": "case intro.intro\nα✝ : Type ?u.97545\nβ✝ : Type ?u.97548\nγ✝ : Type ?u.97551\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt✝ : γ✝ → γ✝ → Prop\nα β : Type u\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≃r s\no : Ordinal\nγ : Type u\nt : γ → γ → Prop\nwo : IsWellOrder γ t\ng : t ≺i r\nh : t ≺i s\n⊢ ↑f (enum r (type t) (_ : Nonempty (t ≺i r))) = enum s (type t) (_ : Nonempty (t ≺i s))",
"tactic": "rw [enum_type g, enum_type (PrincipalSeg.ltEquiv g f)]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα✝ : Type ?u.97545\nβ✝ : Type ?u.97548\nγ✝ : Type ?u.97551\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt✝ : γ✝ → γ✝ → Prop\nα β : Type u\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≃r s\no : Ordinal\nγ : Type u\nt : γ → γ → Prop\nwo : IsWellOrder γ t\ng : t ≺i r\nh : t ≺i s\n⊢ ↑f g.top = (PrincipalSeg.ltEquiv g f).top",
"tactic": "rfl"
}
] |
[
546,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
542,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.limit.conePointUniqueUpToIso_hom_comp
|
[] |
[
233,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Analysis/Calculus/Deriv/Inv.lean
|
fderiv_inv
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\n⊢ fderiv 𝕜 (fun x => x⁻¹) x = smulRight 1 (-(x ^ 2)⁻¹)",
"tactic": "rw [← deriv_fderiv, deriv_inv]"
}
] |
[
124,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.map_apply_of_aemeasurable
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.212187\nδ : Type ?u.212190\nι : Type ?u.212193\nR : Type ?u.212196\nR' : Type ?u.212199\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nf : α → β\nhf : AEMeasurable f\ns : Set β\nhs : MeasurableSet s\n⊢ ↑↑(map f μ) s = ↑↑μ (f ⁻¹' s)",
"tactic": "simpa only [mapₗ, hf.measurable_mk, hs, dif_pos, liftLinear_apply, OuterMeasure.map_apply,\n ← mapₗ_mk_apply_of_aemeasurable hf] using\n measure_congr (hf.ae_eq_mk.symm.preimage s)"
}
] |
[
1225,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1221,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
LinearMap.toMatrix'_toLinearMap₂'
|
[] |
[
265,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/MeasureTheory/Integral/CircleIntegral.lean
|
circleMap_mem_closedBall
|
[] |
[
131,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biprod.inr_desc
|
[] |
[
1383,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1381,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.pos
|
[] |
[
150,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
11
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.comap_map_eq
|
[
{
"state_after": "case Adj.h.h.a\nι : Sort ?u.216907\n𝕜 : Type ?u.216910\nV : Type u\nW : Type v\nX : Type w\nG✝ : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\nf : V ↪ W\nG : SimpleGraph V\nx✝¹ x✝ : V\n⊢ Adj (SimpleGraph.comap (↑f) (SimpleGraph.map f G)) x✝¹ x✝ ↔ Adj G x✝¹ x✝",
"state_before": "ι : Sort ?u.216907\n𝕜 : Type ?u.216910\nV : Type u\nW : Type v\nX : Type w\nG✝ : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\nf : V ↪ W\nG : SimpleGraph V\n⊢ SimpleGraph.comap (↑f) (SimpleGraph.map f G) = G",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a\nι : Sort ?u.216907\n𝕜 : Type ?u.216910\nV : Type u\nW : Type v\nX : Type w\nG✝ : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\nf : V ↪ W\nG : SimpleGraph V\nx✝¹ x✝ : V\n⊢ Adj (SimpleGraph.comap (↑f) (SimpleGraph.map f G)) x✝¹ x✝ ↔ Adj G x✝¹ x✝",
"tactic": "simp"
}
] |
[
1272,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1270,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.comp₂_mk_mk
|
[] |
[
305,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.SigmaFinite.out
|
[] |
[
3446,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3445,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
uniformGroup_inf
|
[
{
"state_after": "α : Type ?u.119171\nβ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : Group α\ninst✝¹ : UniformGroup α\ninst✝ : Group β\nu₁ u₂ : UniformSpace β\nh₁ : UniformGroup β\nh₂ : UniformGroup β\n⊢ UniformGroup β",
"state_before": "α : Type ?u.119171\nβ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : Group α\ninst✝¹ : UniformGroup α\ninst✝ : Group β\nu₁ u₂ : UniformSpace β\nh₁ : UniformGroup β\nh₂ : UniformGroup β\n⊢ UniformGroup β",
"tactic": "rw [inf_eq_iInf]"
},
{
"state_after": "α : Type ?u.119171\nβ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : Group α\ninst✝¹ : UniformGroup α\ninst✝ : Group β\nu₁ u₂ : UniformSpace β\nh₁ : UniformGroup β\nh₂ : UniformGroup β\nb : Bool\n⊢ UniformGroup β",
"state_before": "α : Type ?u.119171\nβ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : Group α\ninst✝¹ : UniformGroup α\ninst✝ : Group β\nu₁ u₂ : UniformSpace β\nh₁ : UniformGroup β\nh₂ : UniformGroup β\n⊢ UniformGroup β",
"tactic": "refine' uniformGroup_iInf fun b => _"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.119171\nβ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : Group α\ninst✝¹ : UniformGroup α\ninst✝ : Group β\nu₁ u₂ : UniformSpace β\nh₁ : UniformGroup β\nh₂ : UniformGroup β\nb : Bool\n⊢ UniformGroup β",
"tactic": "cases b <;> assumption"
}
] |
[
242,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_bool
|
[
{
"state_after": "no goals",
"state_before": "α β : Type u\n⊢ (#Bool) = 2",
"tactic": "simp"
}
] |
[
565,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
565,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.prod_univ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.7813\nδ : Type ?u.7816\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\ns : Set α\n⊢ s ×ˢ univ = Prod.fst ⁻¹' s",
"tactic": "simp [prod_eq]"
}
] |
[
131,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
Multiset.measurable_prod
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\nι : Type ?u.4458135\nα : Type u_2\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf : ι → α → M\ns : Multiset (α → M)\nhs : ∀ (f : α → M), f ∈ s → Measurable f\n⊢ Measurable fun x => prod (map (fun f => f x) s)",
"tactic": "simpa only [← Pi.multiset_prod_apply] using s.measurable_prod' hs"
}
] |
[
920,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
918,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.ncard_subtype
|
[
{
"state_after": "case h.e'_3.h.e'_2\nα : Type u_1\nβ : Type ?u.42313\ns✝ t : Set α\na b x y : α\nf : α → β\nP : α → Prop\ns : Set α\n⊢ s ∩ setOf P = (fun a => ↑a) '' {x | ↑x ∈ s}",
"state_before": "α : Type u_1\nβ : Type ?u.42313\ns✝ t : Set α\na b x y : α\nf : α → β\nP : α → Prop\ns : Set α\n⊢ ncard {x | ↑x ∈ s} = ncard (s ∩ setOf P)",
"tactic": "convert (ncard_image_ofInjective _ (@Subtype.coe_injective _ P)).symm"
},
{
"state_after": "case h.e'_3.h.e'_2.h\nα : Type u_1\nβ : Type ?u.42313\ns✝ t : Set α\na b x✝ y : α\nf : α → β\nP : α → Prop\ns : Set α\nx : α\n⊢ x ∈ s ∩ setOf P ↔ x ∈ (fun a => ↑a) '' {x | ↑x ∈ s}",
"state_before": "case h.e'_3.h.e'_2\nα : Type u_1\nβ : Type ?u.42313\ns✝ t : Set α\na b x y : α\nf : α → β\nP : α → Prop\ns : Set α\n⊢ s ∩ setOf P = (fun a => ↑a) '' {x | ↑x ∈ s}",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_2.h\nα : Type u_1\nβ : Type ?u.42313\ns✝ t : Set α\na b x✝ y : α\nf : α → β\nP : α → Prop\ns : Set α\nx : α\n⊢ x ∈ s ∩ setOf P ↔ x ∈ (fun a => ↑a) '' {x | ↑x ∈ s}",
"tactic": "simp [←and_assoc, exists_eq_right]"
}
] |
[
295,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
9
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
EuclideanGeometry.angle_eq_right
|
[
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 : P\n⊢ ∠ p1 p2 p2 = π / 2",
"tactic": "rw [angle_comm, angle_eq_left]"
}
] |
[
151,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.isPairSelfAdjoint_equiv
|
[
{
"state_after": "R : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\nhₗ : compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F₂ f) ↑e ↑e\n⊢ IsPairSelfAdjoint B₂ F₂ f ↔\n IsPairSelfAdjoint (comp B₂ ↑e ↑e) (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)",
"state_before": "R : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\n⊢ IsPairSelfAdjoint B₂ F₂ f ↔\n IsPairSelfAdjoint (comp B₂ ↑e ↑e) (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)",
"tactic": "have hₗ : (F₂.comp ↑e ↑e).compLeft (e.symm.conj f) = (F₂.compLeft f).comp ↑e ↑e := by\n ext\n simp [LinearEquiv.symm_conj_apply]"
},
{
"state_after": "R : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\nhₗ : compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F₂ f) ↑e ↑e\nhᵣ : compRight (comp B₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compRight B₂ f) ↑e ↑e\n⊢ IsPairSelfAdjoint B₂ F₂ f ↔\n IsPairSelfAdjoint (comp B₂ ↑e ↑e) (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)",
"state_before": "R : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\nhₗ : compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F₂ f) ↑e ↑e\n⊢ IsPairSelfAdjoint B₂ F₂ f ↔\n IsPairSelfAdjoint (comp B₂ ↑e ↑e) (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)",
"tactic": "have hᵣ : (B₂.comp ↑e ↑e).compRight (e.symm.conj f) = (B₂.compRight f).comp ↑e ↑e := by\n ext\n simp [LinearEquiv.conj_apply]"
},
{
"state_after": "R : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\nhₗ : compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F₂ f) ↑e ↑e\nhᵣ : compRight (comp B₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compRight B₂ f) ↑e ↑e\nhe : Function.Surjective ↑↑e\n⊢ IsPairSelfAdjoint B₂ F₂ f ↔\n IsPairSelfAdjoint (comp B₂ ↑e ↑e) (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)",
"state_before": "R : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\nhₗ : compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F₂ f) ↑e ↑e\nhᵣ : compRight (comp B₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compRight B₂ f) ↑e ↑e\n⊢ IsPairSelfAdjoint B₂ F₂ f ↔\n IsPairSelfAdjoint (comp B₂ ↑e ↑e) (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)",
"tactic": "have he : Function.Surjective (⇑(↑e : M₂' →ₗ[R₂] M₂) : M₂' → M₂) := e.surjective"
},
{
"state_after": "R : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\nhₗ : compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F₂ f) ↑e ↑e\nhᵣ : compRight (comp B₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compRight B₂ f) ↑e ↑e\nhe : Function.Surjective ↑↑e\n⊢ IsAdjointPair B₂ F₂ f f ↔\n IsAdjointPair (comp B₂ ↑e ↑e) (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)\n (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)",
"state_before": "R : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\nhₗ : compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F₂ f) ↑e ↑e\nhᵣ : compRight (comp B₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compRight B₂ f) ↑e ↑e\nhe : Function.Surjective ↑↑e\n⊢ IsPairSelfAdjoint B₂ F₂ f ↔\n IsPairSelfAdjoint (comp B₂ ↑e ↑e) (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)",
"tactic": "show BilinForm.IsAdjointPair _ _ _ _ ↔ BilinForm.IsAdjointPair _ _ _ _"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\nhₗ : compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F₂ f) ↑e ↑e\nhᵣ : compRight (comp B₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compRight B₂ f) ↑e ↑e\nhe : Function.Surjective ↑↑e\n⊢ IsAdjointPair B₂ F₂ f f ↔\n IsAdjointPair (comp B₂ ↑e ↑e) (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)\n (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)",
"tactic": "rw [isAdjointPair_iff_compLeft_eq_compRight, isAdjointPair_iff_compLeft_eq_compRight, hᵣ,\n hₗ, comp_inj _ _ he he]"
},
{
"state_after": "case H\nR : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\nx✝ y✝ : M₂'\n⊢ bilin (compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)) x✝ y✝ =\n bilin (comp (compLeft F₂ f) ↑e ↑e) x✝ y✝",
"state_before": "R : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\n⊢ compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F₂ f) ↑e ↑e",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H\nR : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\nx✝ y✝ : M₂'\n⊢ bilin (compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)) x✝ y✝ =\n bilin (comp (compLeft F₂ f) ↑e ↑e) x✝ y✝",
"tactic": "simp [LinearEquiv.symm_conj_apply]"
},
{
"state_after": "case H\nR : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\nhₗ : compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F₂ f) ↑e ↑e\nx✝ y✝ : M₂'\n⊢ bilin (compRight (comp B₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)) x✝ y✝ =\n bilin (comp (compRight B₂ f) ↑e ↑e) x✝ y✝",
"state_before": "R : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\nhₗ : compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F₂ f) ↑e ↑e\n⊢ compRight (comp B₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compRight B₂ f) ↑e ↑e",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H\nR : Type ?u.1139594\nM : Type ?u.1139597\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1139633\nM₁ : Type ?u.1139636\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_3\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type ?u.1140435\nM₃ : Type ?u.1140438\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1141026\nK : Type ?u.1141029\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type u_2\nM₂'' : Type ?u.1142246\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1142557\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1142994\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1145473\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\ne : M₂' ≃ₗ[R₂] M₂\nf : Module.End R₂ M₂\nhₗ : compLeft (comp F₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F₂ f) ↑e ↑e\nx✝ y✝ : M₂'\n⊢ bilin (compRight (comp B₂ ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)) x✝ y✝ =\n bilin (comp (compRight B₂ f) ↑e ↑e) x✝ y✝",
"tactic": "simp [LinearEquiv.conj_apply]"
}
] |
[
1119,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1107,
1
] |
Mathlib/Computability/Reduce.lean
|
ManyOneReducible.mk
|
[] |
[
52,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Order/Filter/Partial.lean
|
Filter.rmap_sets
|
[] |
[
72,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.adjoinIntegral_coe
|
[] |
[
1625,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1622,
1
] |
Mathlib/Data/Pi/Lex.lean
|
Pi.lex_lt_of_lt
|
[
{
"state_after": "ι : Type u_2\nβ : ι → Type u_1\nr✝ : ι → ι → Prop\ns : {i : ι} → β i → β i → Prop\ninst✝ : (i : ι) → PartialOrder (β i)\nr : ι → ι → Prop\nhwf : WellFounded r\nx y : (i : ι) → β i\nhlt : x < y\n⊢ ∃ i, (∀ (j : ι), r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i",
"state_before": "ι : Type u_2\nβ : ι → Type u_1\nr✝ : ι → ι → Prop\ns : {i : ι} → β i → β i → Prop\ninst✝ : (i : ι) → PartialOrder (β i)\nr : ι → ι → Prop\nhwf : WellFounded r\nx y : (i : ι) → β i\nhlt : x < y\n⊢ Pi.Lex r (fun i x x_1 => x < x_1) x y",
"tactic": "simp_rw [Pi.Lex, le_antisymm_iff]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nβ : ι → Type u_1\nr✝ : ι → ι → Prop\ns : {i : ι} → β i → β i → Prop\ninst✝ : (i : ι) → PartialOrder (β i)\nr : ι → ι → Prop\nhwf : WellFounded r\nx y : (i : ι) → β i\nhlt : x < y\n⊢ ∃ i, (∀ (j : ι), r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i",
"tactic": "exact lex_lt_of_lt_of_preorder hwf hlt"
}
] |
[
76,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Data/Nat/Bits.lean
|
Nat.bit_eq_zero_iff
|
[
{
"state_after": "case mp\nn✝ n : ℕ\nb : Bool\n⊢ bit b n = 0 → n = 0 ∧ b = false\n\ncase mpr\nn✝ n : ℕ\nb : Bool\n⊢ n = 0 ∧ b = false → bit b n = 0",
"state_before": "n✝ n : ℕ\nb : Bool\n⊢ bit b n = 0 ↔ n = 0 ∧ b = false",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case mp\nn✝ n : ℕ\nb : Bool\n⊢ bit b n = 0 → n = 0 ∧ b = false",
"tactic": "cases b <;> simp [Nat.bit, Nat.bit0_eq_zero, Nat.bit1_ne_zero]"
},
{
"state_after": "case mpr.intro\nn : ℕ\n⊢ bit false 0 = 0",
"state_before": "case mpr\nn✝ n : ℕ\nb : Bool\n⊢ n = 0 ∧ b = false → bit b n = 0",
"tactic": "rintro ⟨rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nn : ℕ\n⊢ bit false 0 = 0",
"tactic": "rfl"
}
] |
[
161,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/MeasureTheory/Measure/Portmanteau.lean
|
MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto
|
[
{
"state_after": "case pos\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : L = ⊥\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F\n\ncase neg\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F",
"state_before": "Ω✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F",
"tactic": "by_cases L = ⊥"
},
{
"state_after": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\n⊢ ∀ (ε : ℝ≥0), 0 < ε → ↑↑↑μ F < ⊤ → limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F",
"tactic": "apply ENNReal.le_of_forall_pos_le_add"
},
{
"state_after": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\n⊢ ∀ (ε : ℝ≥0), 0 < ε → ↑↑↑μ F < ⊤ → limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "intro ε ε_pos _"
},
{
"state_after": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "let δs := fun n : ℕ => (1 : ℝ) / (n + 1)"
},
{
"state_after": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "have δs_pos : ∀ n, 0 < δs n := fun n => Nat.one_div_pos_of_nat"
},
{
"state_after": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "have δs_lim : Tendsto δs atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_0_nat"
},
{
"state_after": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "have key₁ :=\n tendsto_lintegral_thickenedIndicator_of_isClosed (μ : Measure Ω) F_closed δs_pos δs_lim"
},
{
"state_after": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "have room₁ : (μ : Measure Ω) F < (μ : Measure Ω) F + ε / 2 := by\n apply\n ENNReal.lt_add_right (measure_lt_top (μ : Measure Ω) F).ne\n (ENNReal.div_pos_iff.mpr ⟨(ENNReal.coe_pos.mpr ε_pos).ne.symm, ENNReal.two_ne_top⟩).ne.symm"
},
{
"state_after": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "rcases eventually_atTop.mp (eventually_lt_of_tendsto_lt room₁ key₁) with ⟨M, hM⟩"
},
{
"state_after": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "have key₂ :=\n FiniteMeasure.tendsto_iff_forall_lintegral_tendsto.mp μs_lim (thickenedIndicator (δs_pos M) F)"
},
{
"state_after": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "have room₂ :\n (lintegral (μ : Measure Ω) fun a => thickenedIndicator (δs_pos M) F a) <\n (lintegral (μ : Measure Ω) fun a => thickenedIndicator (δs_pos M) F a) + ε / 2 := by\n apply\n ENNReal.lt_add_right (lintegral_lt_top_of_boundedContinuous_to_nnreal (μ : Measure Ω) _).ne\n (ENNReal.div_pos_iff.mpr ⟨(ENNReal.coe_pos.mpr ε_pos).ne.symm, ENNReal.two_ne_top⟩).ne.symm"
},
{
"state_after": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near :\n ∀ᶠ (x : ι) in L,\n (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs x)) ≤\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "have ev_near := Eventually.mono (eventually_lt_of_tendsto_lt room₂ key₂) fun n => le_of_lt"
},
{
"state_after": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near :\n ∀ᶠ (x : ι) in L,\n (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs x)) ≤\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near' : ∀ᶠ (x : ι) in L, ↑↑↑(μs x) F ≤ (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near :\n ∀ᶠ (x : ι) in L,\n (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs x)) ≤\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "have ev_near' := Eventually.mono ev_near fun n => le_trans\n (measure_le_lintegral_thickenedIndicator (μs n : Measure Ω) F_closed.measurableSet (δs_pos M))"
},
{
"state_after": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near :\n ∀ᶠ (x : ι) in L,\n (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs x)) ≤\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near' : ∀ᶠ (x : ι) in L, ↑↑↑(μs x) F ≤ (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\n⊢ limsup (fun x => (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near :\n ∀ᶠ (x : ι) in L,\n (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs x)) ≤\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near' : ∀ᶠ (x : ι) in L, ↑↑↑(μs x) F ≤ (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "apply (Filter.limsup_le_limsup ev_near').trans"
},
{
"state_after": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near :\n ∀ᶠ (x : ι) in L,\n (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs x)) ≤\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near' : ∀ᶠ (x : ι) in L, ↑↑↑(μs x) F ≤ (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nthis : NeBot L\n⊢ limsup (fun x => (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2) L ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near :\n ∀ᶠ (x : ι) in L,\n (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs x)) ≤\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near' : ∀ᶠ (x : ι) in L, ↑↑↑(μs x) F ≤ (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\n⊢ limsup (fun x => (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "have : NeBot L := ⟨h⟩"
},
{
"state_after": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near :\n ∀ᶠ (x : ι) in L,\n (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs x)) ≤\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near' : ∀ᶠ (x : ι) in L, ↑↑↑(μs x) F ≤ (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nthis : NeBot L\n⊢ (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2 ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near :\n ∀ᶠ (x : ι) in L,\n (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs x)) ≤\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near' : ∀ᶠ (x : ι) in L, ↑↑↑(μs x) F ≤ (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nthis : NeBot L\n⊢ limsup (fun x => (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2) L ≤ ↑↑↑μ F + ↑ε",
"tactic": "rw [limsup_const]"
},
{
"state_after": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near :\n ∀ᶠ (x : ι) in L,\n (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs x)) ≤\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near' : ∀ᶠ (x : ι) in L, ↑↑↑(μs x) F ≤ (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nthis : NeBot L\n⊢ ↑↑↑μ F + ↑ε / 2 + ↑ε / 2 ≤ ↑↑↑μ F + ↑ε",
"state_before": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near :\n ∀ᶠ (x : ι) in L,\n (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs x)) ≤\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near' : ∀ᶠ (x : ι) in L, ↑↑↑(μs x) F ≤ (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nthis : NeBot L\n⊢ (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2 ≤ ↑↑↑μ F + ↑ε",
"tactic": "apply le_trans (add_le_add (hM M rfl.le).le (le_refl (ε / 2 : ℝ≥0∞)))"
},
{
"state_after": "no goals",
"state_before": "case neg.h.intro\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\nroom₂ :\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near :\n ∀ᶠ (x : ι) in L,\n (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs x)) ≤\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nev_near' : ∀ᶠ (x : ι) in L, ↑↑↑(μs x) F ≤ (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2\nthis : NeBot L\n⊢ ↑↑↑μ F + ↑ε / 2 + ↑ε / 2 ≤ ↑↑↑μ F + ↑ε",
"tactic": "simp only [add_assoc, ENNReal.add_halves, le_refl]"
},
{
"state_after": "no goals",
"state_before": "case pos\nΩ✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : L = ⊥\n⊢ limsup (fun i => ↑↑↑(μs i) F) L ≤ ↑↑↑μ F",
"tactic": "simp only [h, limsup, Filter.map_bot, limsSup_bot, ENNReal.bot_eq_zero, zero_le]"
},
{
"state_after": "no goals",
"state_before": "Ω✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\n⊢ ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2",
"tactic": "apply\n ENNReal.lt_add_right (measure_lt_top (μ : Measure Ω) F).ne\n (ENNReal.div_pos_iff.mpr ⟨(ENNReal.coe_pos.mpr ε_pos).ne.symm, ENNReal.two_ne_top⟩).ne.symm"
},
{
"state_after": "no goals",
"state_before": "Ω✝ : Type ?u.26248\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nμs : ι → FiniteMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nF : Set Ω\nF_closed : IsClosed F\nh : ¬L = ⊥\nε : ℝ≥0\nε_pos : 0 < ε\na✝ : ↑↑↑μ F < ⊤\nδs : ℕ → ℝ := fun n => 1 / (↑n + 1)\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey₁ : Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂↑μ) atTop (𝓝 (↑↑↑μ F))\nroom₁ : ↑↑↑μ F < ↑↑↑μ F + ↑ε / 2\nM : ℕ\nhM : ∀ (b : ℕ), b ≥ M → (∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs b) F) ω) ∂↑μ) < ↑↑↑μ F + ↑ε / 2\nkey₂ :\n Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑(μs i)) L\n (𝓝 (∫⁻ (x : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) x) ∂↑μ))\n⊢ (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) <\n (∫⁻ (a : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs M) F) a) ∂↑μ) + ↑ε / 2",
"tactic": "apply\n ENNReal.lt_add_right (lintegral_lt_top_of_boundedContinuous_to_nnreal (μ : Measure Ω) _).ne\n (ENNReal.div_pos_iff.mpr ⟨(ENNReal.coe_pos.mpr ε_pos).ne.symm, ENNReal.two_ne_top⟩).ne.symm"
}
] |
[
373,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Ico_ofDual
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.114883\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\n⊢ Ico (↑ofDual a) (↑ofDual b) = Ioc b a",
"state_before": "α : Type u_1\nβ : Type ?u.114883\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\n⊢ Ico (↑ofDual a) (↑ofDual b) = map (Equiv.toEmbedding ofDual) (Ioc b a)",
"tactic": "refine' Eq.trans _ map_refl.symm"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.114883\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\nc : (fun x => α) a\n⊢ c ∈ Ico (↑ofDual a) (↑ofDual b) ↔ c ∈ Ioc b a",
"state_before": "α : Type u_1\nβ : Type ?u.114883\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\n⊢ Ico (↑ofDual a) (↑ofDual b) = Ioc b a",
"tactic": "ext c"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.114883\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\nc : (fun x => α) a\n⊢ ↑ofDual a ≤ c ∧ c < ↑ofDual b ↔ b < c ∧ c ≤ a",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.114883\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\nc : (fun x => α) a\n⊢ c ∈ Ico (↑ofDual a) (↑ofDual b) ↔ c ∈ Ioc b a",
"tactic": "rw [mem_Ico, mem_Ioc (α := αᵒᵈ)]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.114883\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na✝ b✝ : α\na b : αᵒᵈ\nc : (fun x => α) a\n⊢ ↑ofDual a ≤ c ∧ c < ↑ofDual b ↔ b < c ∧ c ≤ a",
"tactic": "exact and_comm"
}
] |
[
866,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
862,
1
] |
Mathlib/Algebra/Order/Rearrangement.lean
|
Monovary.sum_mul_comp_perm_eq_sum_mul_iff
|
[] |
[
465,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
463,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.lsum_apply
|
[] |
[
376,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.map_const
|
[
{
"state_after": "case a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.244483\nι : Sort x\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm : α → β\nm' : β → γ\ns✝ : Set α\nt : Set β\ninst✝ : NeBot f\nc : β\ns : Set β\n⊢ s ∈ map (fun x => c) f ↔ s ∈ pure c",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.244483\nι : Sort x\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\ninst✝ : NeBot f\nc : β\n⊢ map (fun x => c) f = pure c",
"tactic": "ext s"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.244483\nι : Sort x\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm : α → β\nm' : β → γ\ns✝ : Set α\nt : Set β\ninst✝ : NeBot f\nc : β\ns : Set β\n⊢ s ∈ map (fun x => c) f ↔ s ∈ pure c",
"tactic": "by_cases h : c ∈ s <;> simp [h]"
}
] |
[
2083,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2081,
1
] |
Mathlib/Topology/Bornology/Hom.lean
|
LocallyBoundedMap.cancel_left
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.9512\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.9524\ninst✝³ : Bornology α\ninst✝² : Bornology β\ninst✝¹ : Bornology γ\ninst✝ : Bornology δ\ng : LocallyBoundedMap β γ\nf₁ f₂ : LocallyBoundedMap α β\nhg : Injective ↑g\nh : comp g f₁ = comp g f₂\na : α\n⊢ ↑g (↑f₁ a) = ↑g (↑f₂ a)",
"tactic": "rw [← comp_apply, h, comp_apply]"
}
] |
[
202,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.prod_mul
|
[] |
[
1795,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1792,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.disjoint_def'
|
[] |
[
3645,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3643,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.Tape.move_right_n_head
|
[
{
"state_after": "case zero\nΓ : Type u_1\ninst✝ : Inhabited Γ\nT : Tape Γ\n⊢ ((move Dir.right^[Nat.zero]) T).head = nth T ↑Nat.zero\n\ncase succ\nΓ : Type u_1\ninst✝ : Inhabited Γ\nn✝ : ℕ\nn_ih✝ : ∀ (T : Tape Γ), ((move Dir.right^[n✝]) T).head = nth T ↑n✝\nT : Tape Γ\n⊢ ((move Dir.right^[Nat.succ n✝]) T).head = nth T ↑(Nat.succ n✝)",
"state_before": "Γ : Type u_1\ninst✝ : Inhabited Γ\nT : Tape Γ\ni : ℕ\n⊢ ((move Dir.right^[i]) T).head = nth T ↑i",
"tactic": "induction i generalizing T"
},
{
"state_after": "no goals",
"state_before": "case zero\nΓ : Type u_1\ninst✝ : Inhabited Γ\nT : Tape Γ\n⊢ ((move Dir.right^[Nat.zero]) T).head = nth T ↑Nat.zero",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case succ\nΓ : Type u_1\ninst✝ : Inhabited Γ\nn✝ : ℕ\nn_ih✝ : ∀ (T : Tape Γ), ((move Dir.right^[n✝]) T).head = nth T ↑n✝\nT : Tape Γ\n⊢ ((move Dir.right^[Nat.succ n✝]) T).head = nth T ↑(Nat.succ n✝)",
"tactic": "simp only [*, Tape.move_right_nth, Int.ofNat_succ, iterate_succ, Function.comp_apply]"
}
] |
[
656,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
652,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
LowerSemicontinuous.add'
|
[] |
[
476,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
473,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.liftAddHom_apply_single
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → AddZeroClass (β i)\ninst✝ : AddCommMonoid γ\nf : (i : ι) → β i →+ γ\ni : ι\nx : β i\n⊢ ↑(↑liftAddHom f) (single i x) = ↑(f i) x",
"tactic": "simp"
}
] |
[
2036,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2035,
1
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
TrivSqZeroExt.snd_pow
|
[] |
[
634,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
632,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.bot_vsub
|
[] |
[
1098,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1097,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
iteratedFDerivWithin_of_isOpen
|
[
{
"state_after": "case zero\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\n⊢ EqOn (iteratedFDerivWithin 𝕜 Nat.zero f s) (iteratedFDeriv 𝕜 Nat.zero f) s\n\ncase succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\n⊢ EqOn (iteratedFDerivWithin 𝕜 (Nat.succ n) f s) (iteratedFDeriv 𝕜 (Nat.succ n) f) s",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhs : IsOpen s\n⊢ EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s",
"tactic": "induction' n with n IH"
},
{
"state_after": "case zero\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nx : E\na✝ : x ∈ s\n⊢ iteratedFDerivWithin 𝕜 Nat.zero f s x = iteratedFDeriv 𝕜 Nat.zero f x",
"state_before": "case zero\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\n⊢ EqOn (iteratedFDerivWithin 𝕜 Nat.zero f s) (iteratedFDeriv 𝕜 Nat.zero f) s",
"tactic": "intro x _"
},
{
"state_after": "case zero.H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝¹ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nx : E\na✝ : x ∈ s\nx✝ : Fin Nat.zero → E\n⊢ ↑(iteratedFDerivWithin 𝕜 Nat.zero f s x) x✝ = ↑(iteratedFDeriv 𝕜 Nat.zero f x) x✝",
"state_before": "case zero\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nx : E\na✝ : x ∈ s\n⊢ iteratedFDerivWithin 𝕜 Nat.zero f s x = iteratedFDeriv 𝕜 Nat.zero f x",
"tactic": "ext1"
},
{
"state_after": "no goals",
"state_before": "case zero.H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝¹ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nx : E\na✝ : x ∈ s\nx✝ : Fin Nat.zero → E\n⊢ ↑(iteratedFDerivWithin 𝕜 Nat.zero f s x) x✝ = ↑(iteratedFDeriv 𝕜 Nat.zero f x) x✝",
"tactic": "simp only [Nat.zero_eq, iteratedFDerivWithin_zero_apply, iteratedFDeriv_zero_apply]"
},
{
"state_after": "case succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ iteratedFDerivWithin 𝕜 (Nat.succ n) f s x = iteratedFDeriv 𝕜 (Nat.succ n) f x",
"state_before": "case succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\n⊢ EqOn (iteratedFDerivWithin 𝕜 (Nat.succ n) f s) (iteratedFDeriv 𝕜 (Nat.succ n) f) s",
"tactic": "intro x hx"
},
{
"state_after": "case succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ (↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s) x =\n (↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ fderiv 𝕜 (iteratedFDeriv 𝕜 n f)) x",
"state_before": "case succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ iteratedFDerivWithin 𝕜 (Nat.succ n) f s x = iteratedFDeriv 𝕜 (Nat.succ n) f x",
"tactic": "rw [iteratedFDeriv_succ_eq_comp_left, iteratedFDerivWithin_succ_eq_comp_left]"
},
{
"state_after": "case succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x) =\n ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) (fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x)",
"state_before": "case succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ (↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s) x =\n (↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ fderiv 𝕜 (iteratedFDeriv 𝕜 n f)) x",
"tactic": "dsimp"
},
{
"state_after": "case succ.h.e_6.h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x = fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x",
"state_before": "case succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x) =\n ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) (fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x)",
"tactic": "congr 1"
},
{
"state_after": "case succ.h.e_6.h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ fderiv 𝕜 (iteratedFDerivWithin 𝕜 n f s) x = fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x",
"state_before": "case succ.h.e_6.h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x = fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x",
"tactic": "rw [fderivWithin_of_open hs hx]"
},
{
"state_after": "case succ.h.e_6.h.h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDeriv 𝕜 n f",
"state_before": "case succ.h.e_6.h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ fderiv 𝕜 (iteratedFDerivWithin 𝕜 n f s) x = fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x",
"tactic": "apply Filter.EventuallyEq.fderiv_eq"
},
{
"state_after": "case h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ ∀ (a : E), a ∈ s → iteratedFDerivWithin 𝕜 n f s a = iteratedFDeriv 𝕜 n f a",
"state_before": "case succ.h.e_6.h.h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDeriv 𝕜 n f",
"tactic": "filter_upwards [hs.mem_nhds hx]"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhs : IsOpen s\nn : ℕ\nIH : EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s\nx : E\nhx : x ∈ s\n⊢ ∀ (a : E), a ∈ s → iteratedFDerivWithin 𝕜 n f s a = iteratedFDeriv 𝕜 n f a",
"tactic": "exact IH"
}
] |
[
1596,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1583,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
iUnion_Ico_int_cast
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedRing α\ninst✝ : Archimedean α\na : α\n⊢ (⋃ (n : ℤ), Ico (↑n) (↑n + 1)) = univ",
"tactic": "simpa only [zero_add] using iUnion_Ico_add_int_cast (0 : α)"
}
] |
[
1114,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1113,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.tail_nil
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\n⊢ tail nil = nil",
"tactic": "simp [tail]"
}
] |
[
706,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
706,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffWithinAt.sub
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1842691\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ns : Set E\nf g : E → F\nhf : ContDiffWithinAt 𝕜 n f s x\nhg : ContDiffWithinAt 𝕜 n g s x\n⊢ ContDiffWithinAt 𝕜 n (fun x => f x - g x) s x",
"tactic": "simpa only [sub_eq_add_neg] using hf.add hg.neg"
}
] |
[
1322,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1320,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MonoidHom.coe_of_map_mul_inv
|
[] |
[
1626,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1624,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.minFac_le_of_dvd
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ∀ {m : ℕ}, 2 ≤ m → m ∣ n → minFac n ≤ m",
"tactic": "by_cases n1 : n = 1 <;> [exact fun m2 _ => n1.symm ▸ le_trans (by decide) m2;\n apply (minFac_has_prop n1).2.2]"
},
{
"state_after": "no goals",
"state_before": "n m✝ : ℕ\nn1 : n = 1\nm2 : 2 ≤ m✝\nx✝ : m✝ ∣ n\n⊢ minFac 1 ≤ 2",
"tactic": "decide"
}
] |
[
340,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Analysis/Convex/Gauge.lean
|
gauge_add_le
|
[
{
"state_after": "𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\n⊢ gauge s (x + y) < gauge s x + gauge s y + ε",
"state_before": "𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\n⊢ gauge s (x + y) ≤ gauge s x + gauge s y",
"tactic": "refine' le_of_forall_pos_lt_add fun ε hε => _"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : x ∈ a • s\n⊢ gauge s (x + y) < gauge s x + gauge s y + ε",
"state_before": "𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\n⊢ gauge s (x + y) < gauge s x + gauge s y + ε",
"tactic": "obtain ⟨a, ha, ha', hx⟩ :=\n exists_lt_of_gauge_lt absorbs (lt_add_of_pos_right (gauge s x) (half_pos hε))"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : x ∈ a • s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : y ∈ b • s\n⊢ gauge s (x + y) < gauge s x + gauge s y + ε",
"state_before": "case intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : x ∈ a • s\n⊢ gauge s (x + y) < gauge s x + gauge s y + ε",
"tactic": "obtain ⟨b, hb, hb', hy⟩ :=\n exists_lt_of_gauge_lt absorbs (lt_add_of_pos_right (gauge s y) (half_pos hε))"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : a⁻¹ • x ∈ s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : y ∈ b • s\n⊢ gauge s (x + y) < gauge s x + gauge s y + ε",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : x ∈ a • s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : y ∈ b • s\n⊢ gauge s (x + y) < gauge s x + gauge s y + ε",
"tactic": "rw [mem_smul_set_iff_inv_smul_mem₀ ha.ne'] at hx"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : a⁻¹ • x ∈ s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : b⁻¹ • y ∈ s\n⊢ gauge s (x + y) < gauge s x + gauge s y + ε",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : a⁻¹ • x ∈ s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : y ∈ b • s\n⊢ gauge s (x + y) < gauge s x + gauge s y + ε",
"tactic": "rw [mem_smul_set_iff_inv_smul_mem₀ hb.ne'] at hy"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : a⁻¹ • x ∈ s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : b⁻¹ • y ∈ s\n⊢ gauge s (x + y) ≤ a + b",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : a⁻¹ • x ∈ s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : b⁻¹ • y ∈ s\n⊢ gauge s (x + y) < gauge s x + gauge s y + ε",
"tactic": "suffices gauge s (x + y) ≤ a + b by linarith"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : a⁻¹ • x ∈ s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : b⁻¹ • y ∈ s\nhab : 0 < a + b\n⊢ gauge s (x + y) ≤ a + b",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : a⁻¹ • x ∈ s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : b⁻¹ • y ∈ s\n⊢ gauge s (x + y) ≤ a + b",
"tactic": "have hab : 0 < a + b := add_pos ha hb"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : a⁻¹ • x ∈ s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : b⁻¹ • y ∈ s\nhab : 0 < a + b\n⊢ x + y ∈ (a + b) • s",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : a⁻¹ • x ∈ s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : b⁻¹ • y ∈ s\nhab : 0 < a + b\n⊢ gauge s (x + y) ≤ a + b",
"tactic": "apply gauge_le_of_mem hab.le"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : a⁻¹ • x ∈ s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : b⁻¹ • y ∈ s\nhab : 0 < a + b\nthis : (a / (a + b)) • a⁻¹ • x + (b / (a + b)) • b⁻¹ • y ∈ s\n⊢ x + y ∈ (a + b) • s",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : a⁻¹ • x ∈ s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : b⁻¹ • y ∈ s\nhab : 0 < a + b\n⊢ x + y ∈ (a + b) • s",
"tactic": "have := convex_iff_div.1 hs hx hy ha.le hb.le hab"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : a⁻¹ • x ∈ s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : b⁻¹ • y ∈ s\nhab : 0 < a + b\nthis : (a / (a + b)) • a⁻¹ • x + (b / (a + b)) • b⁻¹ • y ∈ s\n⊢ x + y ∈ (a + b) • s",
"tactic": "rwa [smul_smul, smul_smul, ← mul_div_right_comm, ← mul_div_right_comm, mul_inv_cancel ha.ne',\n mul_inv_cancel hb.ne', ← smul_add, one_div, ← mem_smul_set_iff_inv_smul_mem₀ hab.ne'] at this"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.211297\nE : Type u_1\nF : Type ?u.211303\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nx y : E\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nha' : a < gauge s x + ε / 2\nhx : a⁻¹ • x ∈ s\nb : ℝ\nhb : 0 < b\nhb' : b < gauge s y + ε / 2\nhy : b⁻¹ • y ∈ s\nthis : gauge s (x + y) ≤ a + b\n⊢ gauge s (x + y) < gauge s x + gauge s y + ε",
"tactic": "linarith"
}
] |
[
389,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Data/Complex/Module.lean
|
Complex.liftAux_apply_I
|
[
{
"state_after": "no goals",
"state_before": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\nI' : A\nhI' : I' * I' = -1\n⊢ ↑(liftAux I' hI') I = I'",
"tactic": "simp"
}
] |
[
366,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
IsOpen.mul_right
|
[
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul αᵐᵒᵖ α\ns t : Set α\nhs : IsOpen s\n⊢ IsOpen (⋃ (a : α) (_ : a ∈ t), MulOpposite.op a • s)",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul αᵐᵒᵖ α\ns t : Set α\nhs : IsOpen s\n⊢ IsOpen (s * t)",
"tactic": "rw [← iUnion_op_smul_set]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul αᵐᵒᵖ α\ns t : Set α\nhs : IsOpen s\n⊢ IsOpen (⋃ (a : α) (_ : a ∈ t), MulOpposite.op a • s)",
"tactic": "exact isOpen_biUnion fun a _ => hs.smul _"
}
] |
[
1299,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1297,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
Orthonormal.sum_inner_products_le
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\n⊢ ∑ i in s, ‖inner (v i) x‖ ^ 2 ≤ ‖x‖ ^ 2",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\n⊢ ∑ i in s, ‖inner (v i) x‖ ^ 2 ≤ ‖x‖ ^ 2",
"tactic": "have h₂ :\n (∑ i in s, ∑ j in s, ⟪v i, x⟫ * ⟪x, v j⟫ * ⟪v j, v i⟫) = (∑ k in s, ⟪v k, x⟫ * ⟪x, v k⟫ : 𝕜) :=\n by classical exact hv.inner_left_right_finset"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\n⊢ ∑ i in s, ‖inner (v i) x‖ ^ 2 ≤ ‖x‖ ^ 2",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\n⊢ ∑ i in s, ‖inner (v i) x‖ ^ 2 ≤ ‖x‖ ^ 2",
"tactic": "have h₃ : ∀ z : 𝕜, re (z * conj z) = ‖z‖ ^ 2 := by\n intro z\n simp only [mul_conj, normSq_eq_def']\n norm_cast"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\nhbf : ‖x - ∑ i in s, inner (v i) x • v i‖ ^ 2 = ‖x‖ ^ 2 - ∑ i in s, ‖inner (v i) x‖ ^ 2\n⊢ ∑ i in s, ‖inner (v i) x‖ ^ 2 ≤ ‖x‖ ^ 2\n\ncase hbf\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\n⊢ ‖x - ∑ i in s, inner (v i) x • v i‖ ^ 2 = ‖x‖ ^ 2 - ∑ i in s, ‖inner (v i) x‖ ^ 2",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\n⊢ ∑ i in s, ‖inner (v i) x‖ ^ 2 ≤ ‖x‖ ^ 2",
"tactic": "suffices hbf : ‖x - ∑ i in s, ⟪v i, x⟫ • v i‖ ^ 2 = ‖x‖ ^ 2 - ∑ i in s, ‖⟪v i, x⟫‖ ^ 2"
},
{
"state_after": "case hbf\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\n⊢ ‖x‖ ^ 2 - (2 * ↑re (inner x (∑ i in s, inner (v i) x • v i)) - ‖∑ i in s, inner (v i) x • v i‖ ^ 2) =\n ‖x‖ ^ 2 - ∑ i in s, ‖inner (v i) x‖ ^ 2",
"state_before": "case hbf\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\n⊢ ‖x - ∑ i in s, inner (v i) x • v i‖ ^ 2 = ‖x‖ ^ 2 - ∑ i in s, ‖inner (v i) x‖ ^ 2",
"tactic": "rw [@norm_sub_sq 𝕜, sub_add]"
},
{
"state_after": "no goals",
"state_before": "case hbf\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\n⊢ ‖x‖ ^ 2 - (2 * ↑re (inner x (∑ i in s, inner (v i) x • v i)) - ‖∑ i in s, inner (v i) x • v i‖ ^ 2) =\n ‖x‖ ^ 2 - ∑ i in s, ‖inner (v i) x‖ ^ 2",
"tactic": "classical\n simp only [@InnerProductSpace.norm_sq_eq_inner 𝕜, _root_.inner_sum, _root_.sum_inner]\n simp only [inner_smul_right, two_mul, inner_smul_left, inner_conj_symm, ←mul_assoc, h₂,\n add_sub_cancel, sub_right_inj]\n simp only [map_sum, ← inner_conj_symm x, ← h₃]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\n⊢ ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)",
"tactic": "classical exact hv.inner_left_right_finset"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\n⊢ ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)",
"tactic": "exact hv.inner_left_right_finset"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nz : 𝕜\n⊢ ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\n⊢ ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2",
"tactic": "intro z"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nz : 𝕜\n⊢ ↑re ↑(‖z‖ ^ 2) = ‖z‖ ^ 2",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nz : 𝕜\n⊢ ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2",
"tactic": "simp only [mul_conj, normSq_eq_def']"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nz : 𝕜\n⊢ ↑re ↑(‖z‖ ^ 2) = ‖z‖ ^ 2",
"tactic": "norm_cast"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\nhbf : ‖x - ∑ i in s, inner (v i) x • v i‖ ^ 2 = ‖x‖ ^ 2 - ∑ i in s, ‖inner (v i) x‖ ^ 2\n⊢ 0 ≤ ‖x - ∑ i in s, inner (v i) x • v i‖ ^ 2",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\nhbf : ‖x - ∑ i in s, inner (v i) x • v i‖ ^ 2 = ‖x‖ ^ 2 - ∑ i in s, ‖inner (v i) x‖ ^ 2\n⊢ ∑ i in s, ‖inner (v i) x‖ ^ 2 ≤ ‖x‖ ^ 2",
"tactic": "rw [← sub_nonneg, ← hbf]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\nhbf : ‖x - ∑ i in s, inner (v i) x • v i‖ ^ 2 = ‖x‖ ^ 2 - ∑ i in s, ‖inner (v i) x‖ ^ 2\n⊢ 0 ≤ ‖x - ∑ i in s, inner (v i) x • v i‖ ^ 2",
"tactic": "simp only [norm_nonneg, pow_nonneg]"
},
{
"state_after": "case hbf\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\n⊢ ↑re (inner x x) -\n (2 * ↑re (∑ i in s, inner x (inner (v i) x • v i)) -\n ↑re (∑ x_1 in s, ∑ i in s, inner (inner (v i) x • v i) (inner (v x_1) x • v x_1))) =\n ↑re (inner x x) - ∑ i in s, ‖inner (v i) x‖ ^ 2",
"state_before": "case hbf\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\n⊢ ‖x‖ ^ 2 - (2 * ↑re (inner x (∑ i in s, inner (v i) x • v i)) - ‖∑ i in s, inner (v i) x • v i‖ ^ 2) =\n ‖x‖ ^ 2 - ∑ i in s, ‖inner (v i) x‖ ^ 2",
"tactic": "simp only [@InnerProductSpace.norm_sq_eq_inner 𝕜, _root_.inner_sum, _root_.sum_inner]"
},
{
"state_after": "case hbf\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\n⊢ ↑re (∑ x_1 in s, inner (v x_1) x * inner x (v x_1)) = ∑ i in s, ‖inner (v i) x‖ ^ 2",
"state_before": "case hbf\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\n⊢ ↑re (inner x x) -\n (2 * ↑re (∑ i in s, inner x (inner (v i) x • v i)) -\n ↑re (∑ x_1 in s, ∑ i in s, inner (inner (v i) x • v i) (inner (v x_1) x • v x_1))) =\n ↑re (inner x x) - ∑ i in s, ‖inner (v i) x‖ ^ 2",
"tactic": "simp only [inner_smul_right, two_mul, inner_smul_left, inner_conj_symm, ←mul_assoc, h₂,\n add_sub_cancel, sub_right_inj]"
},
{
"state_after": "no goals",
"state_before": "case hbf\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3426813\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\nι : Type u_1\nx : E\nv : ι → E\ns : Finset ι\nhv : Orthonormal 𝕜 v\nh₂ : ∑ i in s, ∑ j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k in s, inner (v k) x * inner x (v k)\nh₃ : ∀ (z : 𝕜), ↑re (z * ↑(starRingEnd 𝕜) z) = ‖z‖ ^ 2\n⊢ ↑re (∑ x_1 in s, inner (v x_1) x * inner x (v x_1)) = ∑ i in s, ‖inner (v i) x‖ ^ 2",
"tactic": "simp only [map_sum, ← inner_conj_symm x, ← h₃]"
}
] |
[
1897,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1880,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.set_integral_eq_of_subset_of_forall_diff_eq_zero
|
[] |
[
389,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.deriv_zero
|
[] |
[
518,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
517,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Polynomial.frange_zero
|
[] |
[
192,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.diag_add
|
[] |
[
632,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
631,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
UniformSpace.core_eq
|
[] |
[
293,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.inf_neBot_iff
|
[] |
[
638,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
636,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.smul_iSup
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.156923\nβ : Type ?u.156926\nγ : Type ?u.156929\na b c✝ d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nR : Type u_2\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nf : ι → ℝ≥0∞\nc : R\n⊢ (c • ⨆ (i : ι), f i) = ⨆ (i : ι), c • f i",
"tactic": "simp only [← smul_one_mul c (f _), ← smul_one_mul c (iSup f), ENNReal.mul_iSup]"
}
] |
[
662,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
659,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
lp.star_apply
|
[] |
[
751,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
750,
11
] |
Mathlib/Data/MvPolynomial/Monad.lean
|
MvPolynomial.map_bind₁
|
[
{
"state_after": "σ : Type u_4\nτ : Type u_3\nR : Type u_1\nS : Type u_2\nT : Type ?u.836499\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring T\nf✝ : σ → MvPolynomial τ R\nf : R →+* S\ng : σ → MvPolynomial τ R\nφ : MvPolynomial σ R\n⊢ ↑(eval₂Hom C fun i => ↑(map f) (g i)) (↑(map f) φ) = ↑(bind₁ fun i => ↑(map f) (g i)) (↑(map f) φ)",
"state_before": "σ : Type u_4\nτ : Type u_3\nR : Type u_1\nS : Type u_2\nT : Type ?u.836499\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring T\nf✝ : σ → MvPolynomial τ R\nf : R →+* S\ng : σ → MvPolynomial τ R\nφ : MvPolynomial σ R\n⊢ ↑(map f) (↑(bind₁ g) φ) = ↑(bind₁ fun i => ↑(map f) (g i)) (↑(map f) φ)",
"tactic": "rw [hom_bind₁, map_comp_C, ← eval₂Hom_map_hom]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_4\nτ : Type u_3\nR : Type u_1\nS : Type u_2\nT : Type ?u.836499\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring T\nf✝ : σ → MvPolynomial τ R\nf : R →+* S\ng : σ → MvPolynomial τ R\nφ : MvPolynomial σ R\n⊢ ↑(eval₂Hom C fun i => ↑(map f) (g i)) (↑(map f) φ) = ↑(bind₁ fun i => ↑(map f) (g i)) (↑(map f) φ)",
"tactic": "rfl"
}
] |
[
296,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/Order/WellFounded.lean
|
WellFounded.isIrrefl
|
[] |
[
40,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
11
] |
Mathlib/Order/Bounds/Basic.lean
|
bddBelow_iff_exists_le
|
[] |
[
491,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/Order/ModularLattice.lean
|
inf_lt_inf_of_lt_of_sup_le_sup
|
[] |
[
255,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.cons_add
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.44964\nγ : Type ?u.44967\na : α\ns t : Multiset α\n⊢ a ::ₘ s + t = a ::ₘ (s + t)",
"tactic": "rw [← singleton_add, ← singleton_add, add_assoc]"
}
] |
[
674,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
673,
1
] |
Mathlib/Analysis/Calculus/LagrangeMultipliers.lean
|
IsLocalExtrOn.linear_dependent_of_hasStrictFDerivAt
|
[
{
"state_after": "case intro\nE : Type u_2\nF : Type ?u.129783\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf✝ : E → F\nφ : E → ℝ\nx₀ : E\nf'✝ : E →L[ℝ] F\nφ' : E →L[ℝ] ℝ\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → E →L[ℝ] ℝ\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nval✝ : Fintype ι\n⊢ ¬LinearIndependent ℝ (Option.elim' φ' f')",
"state_before": "E : Type u_2\nF : Type ?u.129783\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf✝ : E → F\nφ : E → ℝ\nx₀ : E\nf'✝ : E →L[ℝ] F\nφ' : E →L[ℝ] ℝ\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → E →L[ℝ] ℝ\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\n⊢ ¬LinearIndependent ℝ (Option.elim' φ' f')",
"tactic": "cases nonempty_fintype ι"
},
{
"state_after": "case intro\nE : Type u_2\nF : Type ?u.129783\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf✝ : E → F\nφ : E → ℝ\nx₀ : E\nf'✝ : E →L[ℝ] F\nφ' : E →L[ℝ] ℝ\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → E →L[ℝ] ℝ\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nval✝ : Fintype ι\n⊢ ¬∀ (g : Option ι → ℝ), ∑ i : Option ι, g i • Option.elim' φ' f' i = 0 → ∀ (i : Option ι), g i = 0",
"state_before": "case intro\nE : Type u_2\nF : Type ?u.129783\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf✝ : E → F\nφ : E → ℝ\nx₀ : E\nf'✝ : E →L[ℝ] F\nφ' : E →L[ℝ] ℝ\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → E →L[ℝ] ℝ\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nval✝ : Fintype ι\n⊢ ¬LinearIndependent ℝ (Option.elim' φ' f')",
"tactic": "rw [Fintype.linearIndependent_iff]"
},
{
"state_after": "case intro\nE : Type u_2\nF : Type ?u.129783\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf✝ : E → F\nφ : E → ℝ\nx₀ : E\nf'✝ : E →L[ℝ] F\nφ' : E →L[ℝ] ℝ\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → E →L[ℝ] ℝ\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nval✝ : Fintype ι\n⊢ ∃ g, ∑ i : Option ι, g i • Option.elim' φ' f' i = 0 ∧ ∃ i, g i ≠ 0",
"state_before": "case intro\nE : Type u_2\nF : Type ?u.129783\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf✝ : E → F\nφ : E → ℝ\nx₀ : E\nf'✝ : E →L[ℝ] F\nφ' : E →L[ℝ] ℝ\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → E →L[ℝ] ℝ\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nval✝ : Fintype ι\n⊢ ¬∀ (g : Option ι → ℝ), ∑ i : Option ι, g i • Option.elim' φ' f' i = 0 → ∀ (i : Option ι), g i = 0",
"tactic": "push_neg"
},
{
"state_after": "case intro.intro.intro.intro\nE : Type u_2\nF : Type ?u.129783\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf✝ : E → F\nφ : E → ℝ\nx₀ : E\nf'✝ : E →L[ℝ] F\nφ' : E →L[ℝ] ℝ\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → E →L[ℝ] ℝ\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nval✝ : Fintype ι\nΛ : ι → ℝ\nΛ₀ : ℝ\nhΛ : (Λ, Λ₀) ≠ 0\nhΛf : ∑ i : ι, Λ i • f' i + Λ₀ • φ' = 0\n⊢ ∃ g, ∑ i : Option ι, g i • Option.elim' φ' f' i = 0 ∧ ∃ i, g i ≠ 0",
"state_before": "case intro\nE : Type u_2\nF : Type ?u.129783\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf✝ : E → F\nφ : E → ℝ\nx₀ : E\nf'✝ : E →L[ℝ] F\nφ' : E →L[ℝ] ℝ\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → E →L[ℝ] ℝ\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nval✝ : Fintype ι\n⊢ ∃ g, ∑ i : Option ι, g i • Option.elim' φ' f' i = 0 ∧ ∃ i, g i ≠ 0",
"tactic": "rcases hextr.exists_multipliers_of_hasStrictFDerivAt hf' hφ' with ⟨Λ, Λ₀, hΛ, hΛf⟩"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1\nE : Type u_2\nF : Type ?u.129783\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf✝ : E → F\nφ : E → ℝ\nx₀ : E\nf'✝ : E →L[ℝ] F\nφ' : E →L[ℝ] ℝ\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → E →L[ℝ] ℝ\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nval✝ : Fintype ι\nΛ : ι → ℝ\nΛ₀ : ℝ\nhΛ : (Λ, Λ₀) ≠ 0\nhΛf : ∑ i : ι, Λ i • f' i + Λ₀ • φ' = 0\n⊢ ∑ i : Option ι, Option.elim' Λ₀ Λ i • Option.elim' φ' f' i = 0\n\ncase intro.intro.intro.intro.refine'_2\nE : Type u_2\nF : Type ?u.129783\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf✝ : E → F\nφ : E → ℝ\nx₀ : E\nf'✝ : E →L[ℝ] F\nφ' : E →L[ℝ] ℝ\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → E →L[ℝ] ℝ\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nval✝ : Fintype ι\nΛ : ι → ℝ\nΛ₀ : ℝ\nhΛ : (Λ, Λ₀) ≠ 0\nhΛf : ∑ i : ι, Λ i • f' i + Λ₀ • φ' = 0\n⊢ ∃ i, Option.elim' Λ₀ Λ i ≠ 0",
"state_before": "case intro.intro.intro.intro\nE : Type u_2\nF : Type ?u.129783\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf✝ : E → F\nφ : E → ℝ\nx₀ : E\nf'✝ : E →L[ℝ] F\nφ' : E →L[ℝ] ℝ\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → E →L[ℝ] ℝ\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nval✝ : Fintype ι\nΛ : ι → ℝ\nΛ₀ : ℝ\nhΛ : (Λ, Λ₀) ≠ 0\nhΛf : ∑ i : ι, Λ i • f' i + Λ₀ • φ' = 0\n⊢ ∃ g, ∑ i : Option ι, g i • Option.elim' φ' f' i = 0 ∧ ∃ i, g i ≠ 0",
"tactic": "refine' ⟨Option.elim' Λ₀ Λ, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_1\nE : Type u_2\nF : Type ?u.129783\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf✝ : E → F\nφ : E → ℝ\nx₀ : E\nf'✝ : E →L[ℝ] F\nφ' : E →L[ℝ] ℝ\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → E →L[ℝ] ℝ\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nval✝ : Fintype ι\nΛ : ι → ℝ\nΛ₀ : ℝ\nhΛ : (Λ, Λ₀) ≠ 0\nhΛf : ∑ i : ι, Λ i • f' i + Λ₀ • φ' = 0\n⊢ ∑ i : Option ι, Option.elim' Λ₀ Λ i • Option.elim' φ' f' i = 0",
"tactic": "simpa [add_comm] using hΛf"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_2\nE : Type u_2\nF : Type ?u.129783\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nf✝ : E → F\nφ : E → ℝ\nx₀ : E\nf'✝ : E →L[ℝ] F\nφ' : E →L[ℝ] ℝ\nι : Type u_1\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → E →L[ℝ] ℝ\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nval✝ : Fintype ι\nΛ : ι → ℝ\nΛ₀ : ℝ\nhΛ : (Λ, Λ₀) ≠ 0\nhΛf : ∑ i : ι, Λ i • f' i + Λ₀ • φ' = 0\n⊢ ∃ i, Option.elim' Λ₀ Λ i ≠ 0",
"tactic": "simpa only [Function.funext_iff, not_and_or, or_comm, Option.exists, Prod.mk_eq_zero, Ne.def,\n not_forall] using hΛ"
}
] |
[
144,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Algebra/EuclideanDomain/Defs.lean
|
EuclideanDomain.xgcdAux_rec
|
[
{
"state_after": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nr s t r' s' t' : R\nh : r ≠ 0\n⊢ (fun s t r' s' t' =>\n if _hr : r = 0 then (r', s', t')\n else\n let q := r' / r;\n xgcdAux (r' % r) (s' - q * s) (t' - q * t) r s t)\n s t r' s' t' =\n xgcdAux (r' % r) (s' - r' / r * s) (t' - r' / r * t) r s t",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nr s t r' s' t' : R\nh : r ≠ 0\n⊢ xgcdAux r s t r' s' t' = xgcdAux (r' % r) (s' - r' / r * s) (t' - r' / r * t) r s t",
"tactic": "conv =>\n lhs\n rw [xgcdAux]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nr s t r' s' t' : R\nh : r ≠ 0\n⊢ (fun s t r' s' t' =>\n if _hr : r = 0 then (r', s', t')\n else\n let q := r' / r;\n xgcdAux (r' % r) (s' - q * s) (t' - q * t) r s t)\n s t r' s' t' =\n xgcdAux (r' % r) (s' - r' / r * s) (t' - r' / r * t) r s t",
"tactic": "exact if_neg h"
}
] |
[
247,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieSubmodule.homOfLe_apply
|
[] |
[
591,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
590,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
lie_zero
|
[] |
[
141,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.