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Mathlib/SetTheory/Ordinal/Topology.lean | Ordinal.mem_closure_tfae | [
{
"state_after": "case tfae_1_to_2\ns✝ : Set Ordinal\na✝ a : Ordinal\ns : Set Ordinal\n⊢ a ∈ closure s → a ∈ closure (s ∩ Iic a)\n\ns✝ : Set Ordinal\na✝ a : Ordinal\ns : Set Ordinal\ntfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)\n⊢ TFAE\n [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic... | [
127,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
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Mathlib/Algebra/Lie/Nilpotent.lean | LieModule.nontrivial_max_triv_of_isNilpotent | [] | [
320,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
317,
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Mathlib/Topology/LocalExtr.lean | IsExtrOn.isLocalExtr | [] | [
167,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
166,
1
] |
Mathlib/Data/Set/Prod.lean | Monotone.set_prod | [] | [
436,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
434,
1
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Mathlib/Combinatorics/Quiver/Symmetric.lean | Quiver.Symmetrify.lift_reverse | [
{
"state_after": "U : Type ?u.7670\nV : Type u_2\nW : Type ?u.7676\ninst✝³ : Quiver U\ninst✝² : Quiver V\ninst✝¹ : Quiver W\nV' : Type u_1\ninst✝ : Quiver V'\nh : HasInvolutiveReverse V'\nφ : V ⥤q V'\nX Y : Symmetrify V\nf : X ⟶ Y\n⊢ (match Sum.swap f with\n | Sum.inl g => φ.map g\n | Sum.inr g => reverse... | [
209,
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202,
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Mathlib/Algebra/Quaternion.lean | Quaternion.star_mul_self | [
{
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"state_before": "S : Type ?u.748412\nT : Type ?u.748415\nR : Type u_1\ninst✝ : CommRing R\nr x y z : R\na b c : ℍ[R]\n⊢ star a * a = ↑(↑normSq a)",
"tactic": "rw [star_comm_self, self_mul_star]"
}
] | [
1231,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1231,
1
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Mathlib/Combinatorics/SimpleGraph/Metric.lean | SimpleGraph.dist_comm | [
{
"state_after": "case pos\nV : Type u_1\nG : SimpleGraph V\nu v : V\nh : Reachable G u v\n⊢ dist G u v = dist G v u\n\ncase neg\nV : Type u_1\nG : SimpleGraph V\nu v : V\nh : ¬Reachable G u v\n⊢ dist G u v = dist G v u",
"state_before": "V : Type u_1\nG : SimpleGraph V\nu v : V\n⊢ dist G u v = dist G v u",... | [
125,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
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Mathlib/Analysis/Calculus/MeanValue.lean | concaveOn_univ_of_deriv2_nonpos | [] | [
1265,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1261,
1
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Mathlib/Topology/FiberBundle/Trivialization.lean | Pretrivialization.proj_surjOn_baseSet | [] | [
161,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
158,
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Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tsum_union_disjoint | [] | [
779,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
777,
1
] |
Mathlib/Topology/Support.lean | range_subset_insert_image_mulTSupport | [] | [
79,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
76,
1
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Mathlib/Topology/ContinuousFunction/Bounded.lean | BoundedContinuousFunction.norm_smul_le | [] | [
1421,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1420,
1
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Mathlib/Analysis/Convex/Integral.lean | ConvexOn.average_mem_epigraph | [
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.608928\nm0 : MeasurableSpace α\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\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α... | [
123,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
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Mathlib/LinearAlgebra/FiniteDimensional.lean | FiniteDimensional.finrank_pos_iff | [
{
"state_after": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V\n⊢ 0 < finrank K V ↔ 0 < ↑(finrank K V)",
"state_before": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRin... | [
332,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
327,
1
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Mathlib/Algebra/Tropical/Basic.lean | Tropical.trop_eq_iff_eq_untrop | [] | [
136,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
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Mathlib/Order/Synonym.lean | toLex_inj | [] | [
199,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
198,
1
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Mathlib/Topology/Constructions.lean | continuous_inf_dom_left₂ | [
{
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446,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
439,
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] |
Mathlib/Order/Filter/NAry.lean | Filter.map₂_map_left_anticomm | [] | [
401,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
398,
1
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Mathlib/Data/Sum/Order.lean | OrderIso.sumLexAssoc_apply_inr | [] | [
672,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
670,
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] |
Mathlib/LinearAlgebra/Basis/Bilinear.lean | LinearMap.sum_repr_mul_repr_mulₛₗ | [
{
"state_after": "ι₁ : Type u_8\nι₂ : Type u_9\nR : Type u_1\nR₂ : Type u_2\nS : Type u_6\nS₂ : Type u_7\nM : Type u_3\nN : Type u_5\nP : Type u_4\nMₗ : Type ?u.19146\nNₗ : Type ?u.19149\nPₗ : Type ?u.19152\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : CommSemiring S\ninst✝¹⁵ : CommSemiring R₂\ninst✝¹⁴ : CommSemiring S₂... | [
59,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
54,
1
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Mathlib/Data/Set/Prod.lean | Set.pi_update_of_not_mem | [
{
"state_after": "case h\nι : Type u_1\nα : ι → Type u_3\nβ : ι → Type u_2\ns s₁ s₂ : Set ι\nt✝ t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nhi : ¬i ∈ s\nf : (j : ι) → α j\na : α i\nt : (j : ι) → α j → Set (β j)\nj : ι\nhj : j ∈ s\n⊢ j ≠ i",
"state_before": "ι : Type u_1\nα : ι → Type u_3\nβ :... | [
774,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
770,
1
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Mathlib/Data/SetLike/Basic.lean | SetLike.coe_subset_coe | [] | [
194,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
193,
1
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Mathlib/Analysis/NormedSpace/OperatorNorm.lean | ContinuousLinearMap.flip_smul | [] | [
820,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
819,
1
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Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.Formula.realize_graph | [
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.246639\nP : Type ?u.246642\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nφ ψ : Formula L α\nv : α → M\nf : Functions L n\nx : Fin n → M\ny : M\n⊢ (y = funMap f fun i => x i) ↔ funMap... | [
696,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
693,
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Mathlib/Algebra/DirectSum/Decomposition.lean | DirectSum.decompose_symm_of | [] | [
108,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
107,
1
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Mathlib/Topology/MetricSpace/CantorScheme.lean | CantorScheme.map_mem | [
{
"state_after": "β : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ↑(inducedMap A).fst\nn : ℕ\nthis : Set.Nonempty.some (_ : ↑x ∈ (inducedMap A).fst) ∈ ⋂ (n : ℕ), A (res (↑x) n)\n⊢ Sigma.snd (inducedMap A) x ∈ A (res (↑x) n)",
"state_before": "β : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ↑(induced... | [
88,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
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Mathlib/Order/Bounded.lean | Set.Unbounded.rel_mono | [] | [
103,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
100,
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Mathlib/Analysis/Convex/StrictConvexSpace.lean | eq_lineMap_of_dist_eq_mul_of_dist_eq_mul | [
{
"state_after": "𝕜 : Type ?u.86035\nE : Type u_2\ninst✝¹⁰ : NormedLinearOrderedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : StrictConvexSpace ℝ E\nx✝ y✝ z✝ : E\na b r : ℝ\nF : Type ?u.86312\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\nPF :... | [
268,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
254,
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Mathlib/GroupTheory/FreeAbelianGroup.lean | FreeAbelianGroup.map_zero | [] | [
213,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
212,
11
] |
Mathlib/GroupTheory/Submonoid/Membership.lean | FreeMonoid.mrange_lift | [
{
"state_after": "no goals",
"state_before": "M : Type u_2\nA : Type ?u.75236\nB : Type ?u.75239\ninst✝ : Monoid M\nα : Type u_1\nf : α → M\n⊢ mrange (↑FreeMonoid.lift f) = closure (Set.range f)",
"tactic": "rw [mrange_eq_map, ← FreeMonoid.closure_range_of, map_mclosure, ← Set.range_comp,\n FreeMonoid.... | [
352,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
349,
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Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.zero_div | [] | [
934,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
933,
1
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Mathlib/Order/Monotone/Union.lean | StrictMonoOn.Iic_union_Ici | [
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\nh₁ : StrictMonoOn f (Iic a)\nh₂ : StrictMonoOn f (Ici a)\n⊢ StrictMonoOn f (Iic a ∪ Ici a)",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\n... | [
62,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
59,
11
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | MeasureTheory.aecover_Ioo | [] | [
162,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
1
] |
Mathlib/Topology/Connected.lean | isConnected_connectedComponentIn_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.54113\nπ : ι → Type ?u.54118\ninst✝ : TopologicalSpace α\ns t u v : Set α\nx : α\nF : Set α\n⊢ IsConnected (connectedComponentIn F x) ↔ x ∈ F",
"tactic": "simp_rw [← connectedComponentIn_nonempty_iff, IsConnected, isPrecon... | [
637,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
634,
1
] |
Mathlib/Topology/Bases.lean | TopologicalSpace.isSeparable_iUnion | [
{
"state_after": "α : Type u\nt : TopologicalSpace α\nι : Type u_1\ninst✝ : Countable ι\ns c : ι → Set α\nhc : ∀ (i : ι), Set.Countable (c i)\nh'c : ∀ (i : ι), s i ⊆ closure (c i)\n⊢ IsSeparable (⋃ (i : ι), s i)",
"state_before": "α : Type u\nt : TopologicalSpace α\nι : Type u_1\ninst✝ : Countable ι\ns : ι ... | [
409,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
405,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.zero_ne_top | [] | [
315,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
9
] |
Mathlib/LinearAlgebra/FreeModule/Finite/Rank.lean | FiniteDimensional.finrank_matrix | [
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\nN : Type w\ninst✝¹¹ : Ring R\ninst✝¹⁰ : StrongRankCondition R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : Module.Free R M\ninst✝⁶ : Module.Finite R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Module.Free R N\nins... | [
136,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
1
] |
Mathlib/Data/Int/Log.lean | Int.le_zpow_iff_clog_le | [] | [
305,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
303,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean | reflection_involutive | [] | [
677,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
676,
1
] |
Mathlib/Topology/Sequences.lean | QuotientMap.sequentialSpace | [] | [
237,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
235,
1
] |
Mathlib/Topology/ContinuousFunction/Compact.lean | BoundedContinuousFunction.dist_mkOfCompact | [] | [
117,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
115,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.union_distrib_left | [] | [
1749,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1748,
1
] |
Std/Data/RBMap/Lemmas.lean | Std.RBNode.Balanced.depth_bound | [] | [
85,
76
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
83,
1
] |
Mathlib/Topology/Order/Basic.lean | dense_iff_exists_between | [] | [
1305,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1303,
1
] |
Mathlib/Order/Compare.lean | cmpLE_ofDual | [] | [
196,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
194,
1
] |
Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | EuclideanGeometry.Sphere.secondInter_zero | [
{
"state_after": "no goals",
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"tactic": "simp [Sphere.secondInter]"
}
] | [
66,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
65,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | CategoryTheory.BicartesianSq.of_isPullback_isPushout | [] | [
917,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
915,
1
] |
Mathlib/Topology/Algebra/Order/IntermediateValue.lean | Continuous.surjective' | [] | [
627,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
625,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean | BoundedContinuousFunction.uniformContinuous_coe | [] | [
316,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
1
] |
Mathlib/Order/Bounded.lean | Set.bounded_ge_Ico | [] | [
270,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
269,
1
] |
Mathlib/Algebra/Lie/Basic.lean | LieModuleEquiv.self_trans_symm | [] | [
1103,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1102,
1
] |
Mathlib/Data/Complex/Basic.lean | Complex.inv_def | [] | [
735,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
734,
1
] |
Mathlib/Combinatorics/HalesJewett.lean | Combinatorics.Line.exists_mono_in_high_dimension | [
{
"state_after": "no goals",
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342,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
337,
1
] |
Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | TopCat.fst_iso_of_right_embedding_range_subset | [
{
"state_after": "J : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nhg : Embedding ((forget TopCat).map g)\nH : Set.range ((forget TopCat).map f) ⊆ Set.range ((forget TopCat).map g)\nesto : ↑(pullback f g) ≃ₜ ↑X :=\n Homeomorph.trans\n (Homeomorph.ofEmbedding ((for... | [
356,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
344,
1
] |
Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_left_pure | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7567\nγ : Type ?u.7570\ninst✝³ : LinearOrderedRing α\ninst✝² : FloorRing α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ Tendsto ceil (𝓝[Iic ↑n] ↑n) (pure n)",
"tactic": "simpa only [ceil_intCast] using tend... | [
87,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean | MeasureTheory.snorm'_sum_le | [] | [
1213,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1208,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | MvPowerSeries.ext_iff | [] | [
143,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
142,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | HomogeneousLocalization.den_mem_deg | [] | [
523,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
522,
1
] |
Mathlib/Algebra/Associated.lean | Associates.mk_dvdNotUnit_mk_iff | [
{
"state_after": "α : Type u_1\nβ : Type ?u.321829\nγ : Type ?u.321832\nδ : Type ?u.321835\ninst✝ : CommMonoidWithZero α\na b : α\n⊢ (a ≠ 0 ∧ ∃ x, ¬IsUnit x ∧ Associates.mk b = Associates.mk a * x) ↔ a ≠ 0 ∧ ∃ x, ¬IsUnit x ∧ b = a * x",
"state_before": "α : Type u_1\nβ : Type ?u.321829\nγ : Type ?u.321832\n... | [
1063,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1044,
1
] |
Mathlib/CategoryTheory/CofilteredSystem.lean | CategoryTheory.Functor.eval_section_injective_of_eventually_injective | [
{
"state_after": "J : Type u\ninst✝¹ : Category J\nF : J ⥤ Type v\ni✝ j✝ k✝ : J\ns : Set (F.obj i✝)\ninst✝ : IsCofilteredOrEmpty J\nj : J\nFinj : ∀ (i : J) (f : i ⟶ j), Function.Injective (F.map f)\ni : J\nf : i ⟶ j\ns₀ s₁ : ↑(sections F)\nh : (fun s => ↑s j) s₀ = (fun s => ↑s j) s₁\nk : J\n⊢ ↑s₀ k = ↑s₁ k",
... | [
345,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
337,
1
] |
Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean | aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet | [
{
"state_after": "no goals",
"state_before": "ι : Sort u_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.259052\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\nhf : ∀ (i : ι), AEMeasurable (f i)\nx : α\nhx : x ∈ aeSeqSet hf p\ni : ι\n⊢ a... | [
69,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
67,
1
] |
Mathlib/LinearAlgebra/Basic.lean | LinearMap.ker_toAddSubgroup | [] | [
1481,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1480,
1
] |
Mathlib/GroupTheory/Index.lean | Subgroup.index_bot_eq_card | [] | [
238,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
237,
1
] |
Mathlib/Topology/LocalHomeomorph.lean | LocalHomeomorph.leftInvOn | [] | [
178,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
178,
11
] |
Mathlib/Data/List/NodupEquivFin.lean | List.duplicate_iff_exists_distinct_get | [
{
"state_after": "α : Type u_1\nl : List α\nx : α\n⊢ (∃ f, ∀ (ix : Fin (length (replicate 2 x))), get (replicate 2 x) ix = get l (↑f ix)) ↔\n ∃ n m x_1, x = get l n ∧ x = get l m",
"state_before": "α : Type u_1\nl : List α\nx : α\n⊢ Duplicate x l ↔ ∃ n m x_1, x = get l n ∧ x = get l m",
"tactic": "rw... | [
238,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
215,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean | Ultrafilter.eventually_not | [] | [
195,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
194,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean | intervalIntegral.intervalIntegrable_pow | [] | [
58,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
57,
1
] |
Mathlib/Order/Lattice.lean | Antitone.le_map_inf | [] | [
1195,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1193,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean | PadicInt.norm_int_lt_one_iff_dvd | [
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : ℤ\nthis : ‖↑k‖ < 1 ↔ ↑p ∣ k\n⊢ ‖↑k‖ < 1 ↔ ↑p ∣ k",
"tactic": "rwa [norm_int_cast_eq_padic_norm]"
}
] | [
362,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
360,
1
] |
Mathlib/Algebra/CharP/Quotient.lean | CharP.quotient' | [
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\np : ℕ\ninst✝ : CharP R p\nI : Ideal R\nh : ∀ (x : ℕ), ↑x ∈ I → ↑x = 0\nx : ℕ\n⊢ ↑(Ideal.Quotient.mk I) ↑x = 0 ↔ ↑x = 0",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\np : ℕ\ninst✝ : CharP R p\nI : Ideal R\nh : ∀ (x : ℕ), ↑x ∈ I → ↑x = 0\nx : ℕ\n⊢ ↑x... | [
45,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
39,
1
] |
Mathlib/GroupTheory/SpecificGroups/Quaternion.lean | QuaternionGroup.a_one_pow | [
{
"state_after": "case zero\nn : ℕ\n⊢ a 1 ^ Nat.zero = a ↑Nat.zero\n\ncase succ\nn k : ℕ\nIH : a 1 ^ k = a ↑k\n⊢ a 1 ^ Nat.succ k = a ↑(Nat.succ k)",
"state_before": "n k : ℕ\n⊢ a 1 ^ k = a ↑k",
"tactic": "induction' k with k IH"
},
{
"state_after": "case zero\nn : ℕ\n⊢ a 1 ^ Nat.zero = a 0",
... | [
189,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
183,
1
] |
Mathlib/Combinatorics/Hindman.lean | Hindman.FP.mul_two | [
{
"state_after": "M : Type u_1\ninst✝ : Semigroup M\na : Stream' M\ni j : ℕ\nij : i < j\n⊢ Stream'.nth a i * Stream'.nth a j ∈ FP (Stream'.drop i a)",
"state_before": "M : Type u_1\ninst✝ : Semigroup M\na : Stream' M\ni j : ℕ\nij : i < j\n⊢ Stream'.nth a i * Stream'.nth a j ∈ FP a",
"tactic": "refine' F... | [
280,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
269,
1
] |
Mathlib/Analysis/Complex/Schwarz.lean | Complex.abs_deriv_le_div_of_mapsTo_ball | [] | [
172,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
170,
1
] |
Mathlib/Data/Nat/Parity.lean | Nat.even_mul | [
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ Even (m * n) ↔ Even m ∨ Even n",
"tactic": "cases' mod_two_eq_zero_or_one m with h₁ h₁ <;> cases' mod_two_eq_zero_or_one n with h₂ h₂ <;>\n simp [even_iff, h₁, h₂, Nat.mul_mod]"
}
] | [
158,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
156,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean | AffineEquiv.coe_homothetyUnitsMulHom_eq_homothetyHom_coe | [] | [
547,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
544,
1
] |
Mathlib/Data/Finset/Lattice.lean | Finset.sup_insert | [] | [
61,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
60,
1
] |
Mathlib/Order/Lattice.lean | MonotoneOn.max | [] | [
1149,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1147,
11
] |
Mathlib/RingTheory/Polynomial/Pochhammer.lean | pochhammer_pos | [
{
"state_after": "case zero\nS : Type u_1\ninst✝ : StrictOrderedSemiring S\ns : S\nh : 0 < s\n⊢ 0 < eval s (pochhammer S Nat.zero)\n\ncase succ\nS : Type u_1\ninst✝ : StrictOrderedSemiring S\ns : S\nh : 0 < s\nn : ℕ\nih : 0 < eval s (pochhammer S n)\n⊢ 0 < eval s (pochhammer S (Nat.succ n))",
"state_before"... | [
179,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
173,
1
] |
Mathlib/Logic/Function/Basic.lean | Function.apply_update | [
{
"state_after": "case pos\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g✝ : (a : α✝) → β✝ a\na : α✝\nb : β✝ a\nι : Sort u_1\ninst✝ : DecidableEq ι\nα : ι → Sort u_2\nβ : ι → Sort u_3\nf : (i : ι) → α i → β i\ng : (i : ι) → α i\ni : ι\nv : α i\nj : ι\nh : j =... | [
648,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
642,
1
] |
Mathlib/Algebra/GradedMonoid.lean | GradedMonoid.mk_mul_mk | [] | [
146,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean | Multiset.nodup_Ioo | [] | [
44,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
43,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean | MeasureTheory.OuterMeasure.ofFunction_apply | [] | [
698,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
696,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean | contDiffAt_one_iff | [
{
"state_after": "no goals",
"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\... | [
1456,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1451,
1
] |
Mathlib/Data/Set/Basic.lean | Set.inter_inter_distrib_right | [] | [
1086,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1085,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean | hasFDerivAt_pi' | [] | [
410,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
408,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.filter_single | [
{
"state_after": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ns : Finset ι\nx✝ : (i : ↑↑s) → β ↑i\ni✝ : ι\np : ι → Prop\ninst✝ : DecidablePred p\ni : ι\nx : β i\nj : ι\n⊢ ↑(filter p (single i x)) j = ↑(if p i then single... | [
706,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
698,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.eventuallyLE_antisymm_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.206109\nι : Sort x\ninst✝ : PartialOrder β\nl : Filter α\nf g : α → β\n⊢ f =ᶠ[l] g ↔ f ≤ᶠ[l] g ∧ g ≤ᶠ[l] f",
"tactic": "simp only [EventuallyEq, EventuallyLE, le_antisymm_iff, eventually_and]"
}
] | [
1684,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1682,
1
] |
Mathlib/Algebra/Ring/Equiv.lean | RingEquiv.ext_iff | [] | [
188,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
187,
11
] |
Mathlib/Topology/FiberBundle/Basic.lean | FiberPrebundle.mem_pretrivializationAt_source | [
{
"state_after": "ι : Type ?u.61867\nB : Type u_2\nF : Type u_3\nX : Type ?u.61876\ninst✝³ : TopologicalSpace X\nE : B → Type u_1\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\ninst✝ : (x : B) → TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nb : B\nx : E b\n... | [
858,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
855,
1
] |
Mathlib/CategoryTheory/Sites/Closed.lean | CategoryTheory.GrothendieckTopology.le_close_of_isClosed | [] | [
102,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
100,
1
] |
Mathlib/Data/Set/Image.lean | Set.image_preimage_inl_union_image_preimage_inr | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.86447\nι : Sort ?u.86450\nι' : Sort ?u.86453\nf : ι → α\ns✝ t : Set α\ns : Set (α ⊕ β)\n⊢ Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s",
"tactic": "rw [image_preimage_eq_inter_range, image_preimage_eq_in... | [
952,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
949,
1
] |
Mathlib/Data/Real/EReal.lean | EReal.zero_ne_top | [] | [
337,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
336,
1
] |
Mathlib/Algebra/Quaternion.lean | QuaternionAlgebra.star_add_self | [
{
"state_after": "no goals",
"state_before": "S : Type ?u.312006\nT : Type ?u.312009\nR : Type u_1\ninst✝ : CommRing R\nc₁ c₂ r x y z : R\na b c : ℍ[R,c₁,c₂]\n⊢ star a + a = 2 * ↑a.re",
"tactic": "rw [add_comm, self_add_star]"
}
] | [
658,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
658,
1
] |
Mathlib/Data/Set/Image.lean | Set.preimage_subset_iff | [] | [
552,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
550,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | midpoint_eq_iff' | [] | [
189,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
188,
1
] |
Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.Theory.Model.union | [
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.419567\nP : Type ?u.419570\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT T' : Theory L\nh : ∀ (φ : Sentence L), φ ∈ T → M ⊨ φ\nh' : ∀ (φ : Sentence L), φ ∈ T' → M ⊨ φ\n⊢ ∀ (φ : Sen... | [
847,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
845,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | EuclideanGeometry.angle_eq_arccos_of_angle_eq_pi_div_two | [
{
"state_after": "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 p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\n⊢ ∠ p₂ p₃ p₁ = Real.arccos (dist p₃ p₂ / dist p₁ p₃)",
"state_before": "V : Type u_1\nP : Type ... | [
381,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
376,
1
] |
Std/Data/Nat/Gcd.lean | Nat.gcd_gcd_self_right_right | [
{
"state_after": "no goals",
"state_before": "m n : Nat\n⊢ gcd m (gcd n m) = gcd n m",
"tactic": "rw [gcd_comm n m, gcd_gcd_self_right_left]"
}
] | [
165,
45
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
164,
9
] |
Mathlib/LinearAlgebra/LinearPMap.lean | LinearPMap.le_of_le_graph | [
{
"state_after": "case left\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.595684\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.596200\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M... | [
893,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
880,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean | Nat.descFactorial_le_pow | [
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ descFactorial n 0 ≤ n ^ 0",
"tactic": "rw [descFactorial_zero, pow_zero]"
},
{
"state_after": "n k : ℕ\n⊢ (n - k) * descFactorial n k ≤ n * n ^ k",
"state_before": "n k : ℕ\n⊢ descFactorial n (k + 1) ≤ n ^ (k + 1)",
"tactic": "rw [de... | [
473,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
469,
1
] |
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