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/Order/CompactlyGenerated.lean | DirectedOn.sSup_inf_eq | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.94817\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\n⊢ sSup s ⊓ a = ⨆ (b : α) (_ : b ∈ s), b ⊓ a",
"tactic": "simp_rw ... | [
381,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
380,
11
] |
Mathlib/Data/Set/Lattice.lean | Set.image2_iInter₂_subset_right | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort ?u.254718\nι₂ : Sort ?u.254721\nκ : ι → Sort u_5\nκ₁ : ι → Sort ?u.254731\nκ₂ : ι → Sort ?u.254736\nκ' : ι' → Sort ?u.254741\nf : α → β → γ\ns✝ : Set α\nt✝ : Set β\ns : Set α\nt : (i : ι) → κ i → Set β\n⊢ ∀ (x : α),\n x ∈ s →... | [
1922,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1919,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean | AddLECancellable.le_add_tsub | [
{
"state_after": "α : Type u_1\nβ : Type ?u.23086\ninst✝³ : Preorder α\ninst✝² : AddCommSemigroup α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\n⊢ a ≤ b + a - b",
"state_before": "α : Type u_1\nβ : Type ?u.23086\ninst✝³ : Preorder α\ninst✝² : AddCommSemigroup α\ninst✝¹ : Sub ... | [
208,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
206,
11
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.lintegral_pos_iff_support | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.973547\nγ : Type ?u.973550\nδ : Type ?u.973553\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ (0 < ∫⁻ (a : α), f a ∂μ) ↔ 0 < ↑↑μ (support f)",
"tactic": "simp [pos_iff_ne_zero, hf, Filter.EventuallyEq, a... | [
888,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
886,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean | Ordinal.lift_omega | [] | [
870,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
869,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Images.lean | CategoryTheory.Limits.image.lift_fac | [] | [
353,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
352,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | PowerSeries.evalNegHom_X | [] | [
1975,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1974,
1
] |
Mathlib/Probability/Kernel/Basic.lean | ProbabilityTheory.kernel.set_lintegral_const | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.980848\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nf : β → ℝ≥0∞\nμ : MeasureTheory.Measure β\na : α\ns : Set β\n⊢ (∫⁻ (x : β) in s, f x ∂↑(const α μ) a) = ∫⁻ (x : β) in s, f x ∂μ",
"ta... | [
455,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
454,
1
] |
Mathlib/Data/Nat/Bits.lean | Nat.bit_ne_zero | [
{
"state_after": "no goals",
"state_before": "n✝ : ℕ\nb : Bool\nn : ℕ\nh : n ≠ 0\n⊢ bit b n ≠ 0",
"tactic": "cases b <;> [exact Nat.bit0_ne_zero h; exact Nat.bit1_ne_zero _]"
}
] | [
103,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
] |
Mathlib/Order/BoundedOrder.lean | not_isMin_iff_ne_bot | [] | [
350,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
349,
1
] |
Mathlib/Data/Set/Function.lean | Set.EqOn.surjOn_iff | [] | [
789,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
788,
1
] |
Std/Data/List/Basic.lean | List.dropSlice_eq_dropSliceTR | [
{
"state_after": "case h.h.h.h\nα : Type u_1\nn m : Nat\nl : List α\n⊢ dropSlice n m l = dropSliceTR n m l",
"state_before": "⊢ @dropSlice = @dropSliceTR",
"tactic": "funext α n m l"
},
{
"state_after": "case h.h.h.h\nα : Type u_1\nn m : Nat\nl : List α\n⊢ dropSlice n m l =\n match m with\n ... | [
1272,
30
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1263,
10
] |
Mathlib/AlgebraicTopology/SimplicialObject.lean | CategoryTheory.CosimplicialObject.δ_comp_σ_of_gt | [
{
"state_after": "C : Type u\ninst✝ : Category C\nX : CosimplicialObject C\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : ↑Fin.castSucc j < i\n⊢ X.map (SimplexCategory.δ (Fin.succ i)) ≫ X.map (SimplexCategory.σ (↑Fin.castSucc j)) =\n X.map (SimplexCategory.σ j) ≫ X.map (SimplexCategory.δ i)",
"state_befor... | [
560,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
557,
1
] |
Mathlib/Algebra/Homology/Exact.lean | CategoryTheory.exact_of_image_eq_kernel | [
{
"state_after": "V : Type u\ninst✝³ : Category V\ninst✝² : HasImages V\ninst✝¹ : HasZeroMorphisms V\ninst✝ : HasKernels V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\np : imageSubobject f = kernelSubobject g\nthis : IsIso (imageToKernel f g (_ : f ≫ g = 0))\n⊢ Epi (imageToKernel f g (_ : f ≫ g = 0))",
"state_before":... | [
162,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
157,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean | Continuous.div' | [] | [
1109,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1108,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.inter_eventuallyEq_left | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.190339\nι : Sort x\ns t : Set α\nl : Filter α\n⊢ s ∩ t =ᶠ[l] s ↔ ∀ᶠ (x : α) in l, x ∈ s → x ∈ t",
"tactic": "simp only [eventuallyEq_set, mem_inter_iff, and_iff_left_iff_imp]"
}
] | [
1592,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1590,
1
] |
Mathlib/Analysis/Convex/StrictConvexSpace.lean | StrictConvexSpace.ofNormComboLtOne | [
{
"state_after": "𝕜 : Type ?u.12558\nE : Type u_1\ninst✝³ : NormedLinearOrderedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, a + b = 1 ∧ ‖a • x + b • y‖ < 1\nx : E\nhx : x ∈ closedBall 0 1 \\ interior (closedBall... | [
112,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.erase_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✝ : (i : ι) → Zero (β i)\ns : Finset ι\nx✝ : (i : ↑↑s) → β ↑i\ni✝ j i : ι\nx : β i\n⊢ erase j (single i x) = if i = j then 0 else single i x",
"tactic": "rw ... | [
795,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
793,
1
] |
Mathlib/Order/Filter/ENNReal.lean | ENNReal.limsup_const_mul | [
{
"state_after": "case pos\nα : Type u_1\nf : Filter α\ninst✝ : CountableInterFilter f\nu : α → ℝ≥0∞\na : ℝ≥0∞\nha_top : a ≠ ⊤\n⊢ limsup (fun x => a * u x) f = a * limsup u f\n\ncase neg\nα : Type u_1\nf : Filter α\ninst✝ : CountableInterFilter f\nu : α → ℝ≥0∞\na : ℝ≥0∞\nha_top : ¬a ≠ ⊤\n⊢ limsup (fun x => a * ... | [
72,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
54,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | MeasureTheory.aecover_Ico | [] | [
170,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
1
] |
Mathlib/Data/List/Sigma.lean | List.NodupKeys.pairwise_ne | [] | [
97,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
95,
1
] |
Mathlib/Algebra/Hom/Aut.lean | MulAut.apply_inv_self | [] | [
103,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.count_erase_of_ne | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.376488\nγ : Type ?u.376491\ninst✝ : DecidableEq α\na b : α\nab : a ≠ b\ns : Multiset α\nl : List α\n⊢ count a (erase (Quotient.mk (isSetoid α) l) b) = count a (Quotient.mk (isSetoid α) l)",
"tactic": "convert List.count_erase_of_ne ... | [
2453,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2450,
1
] |
Mathlib/MeasureTheory/Integral/IntervalAverage.lean | interval_average_symm | [
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\na b : ℝ\n⊢ (⨍ (x : ℝ) in Ι a b, f x) = ⨍ (x : ℝ) in Ι b a, f x",
"tactic": "rw [set_average_eq, set_average_eq, uIoc_comm]"
}
] | [
43,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean | dist_prod_prod_le | [] | [
1508,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1506,
1
] |
Mathlib/LinearAlgebra/Vandermonde.lean | Matrix.eq_zero_of_forall_pow_sum_mul_pow_eq_zero | [] | [
177,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
174,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | dist_toMul | [] | [
3193,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3193,
9
] |
Mathlib/Data/Matrix/Notation.lean | Matrix.cons_vecMul_cons | [
{
"state_after": "no goals",
"state_before": "α : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\ninst✝ : Semiring α\nx : α\nv : Fin n → α\nw : o' → α\nB : Fin n → o' → α\n⊢ vecMul (vecCons x v) (↑of (vecCons w B)) = x • w + vecMul v (↑of B)",
"tactic": "simp"
}
] | [
291,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
290,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean | CategoryTheory.Limits.parallelFamily_map_left | [] | [
152,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
151,
1
] |
Mathlib/RingTheory/Polynomial/Content.lean | Polynomial.content_zero | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\n⊢ content 0 = 0",
"tactic": "rw [← C_0, content_C, normalize_zero]"
}
] | [
105,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
105,
1
] |
Mathlib/Data/Bool/Basic.lean | Bool.decide_not | [
{
"state_after": "no goals",
"state_before": "p : Prop\ninst✝ : Decidable p\n⊢ (decide ¬p) = !decide p",
"tactic": "by_cases p <;> simp [*]"
}
] | [
64,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
1
] |
Mathlib/Data/Option/Basic.lean | Option.Mem.leftUnique | [] | [
62,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
61,
1
] |
Mathlib/CategoryTheory/Category/ULift.lean | CategoryTheory.objDown_objUp | [] | [
124,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
] |
Mathlib/CategoryTheory/Sites/Sheafification.lean | CategoryTheory.Meq.pullback_apply | [] | [
99,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
97,
1
] |
Mathlib/Tactic/Abel.lean | Mathlib.Tactic.Abel.term_atom | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddCommMonoid α\nx : α\n⊢ x = term 1 x 0",
"tactic": "simp [term]"
}
] | [
235,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
235,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean | LinearPMap.snd_apply | [] | [
211,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
209,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | Subalgebra.map_toSubmodule | [] | [
495,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
493,
1
] |
Mathlib/GroupTheory/Perm/Support.lean | Equiv.Perm.disjoint_refl_iff | [
{
"state_after": "α : Type u_1\nf g h✝ : Perm α\nh : Disjoint f f\n⊢ f = 1",
"state_before": "α : Type u_1\nf g h : Perm α\n⊢ Disjoint f f ↔ f = 1",
"tactic": "refine' ⟨fun h => _, fun h => h.symm ▸ disjoint_one_left 1⟩"
},
{
"state_after": "case H\nα : Type u_1\nf g h✝ : Perm α\nh : Disjoint f ... | [
87,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/Analysis/Calculus/Deriv/Pow.lean | differentiable_pow | [] | [
83,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
] |
Mathlib/Data/Finset/PImage.lean | Part.toFinset_none | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.311\ninst✝ : Decidable none.Dom\n⊢ toFinset none = ∅",
"tactic": "simp [toFinset]"
}
] | [
43,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean | dfinsupp_prod_mem | [] | [
1844,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1840,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean | LinearIsometryEquiv.coe_trans | [] | [
801,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
800,
1
] |
Mathlib/Data/Finsupp/Defs.lean | Finsupp.coe_zero | [] | [
172,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
171,
1
] |
Mathlib/Dynamics/OmegaLimit.lean | Flow.isInvariant_omegaLimit | [
{
"state_after": "τ : Type u_1\ninst✝³ : TopologicalSpace τ\ninst✝² : AddMonoid τ\ninst✝¹ : ContinuousAdd τ\nα : Type u_2\ninst✝ : TopologicalSpace α\nf : Filter τ\nϕ : Flow τ α\ns : Set α\nhf : ∀ (t : τ), Tendsto ((fun x x_1 => x + x_1) t) f f\nt : τ\n⊢ MapsTo (toFun ϕ t) (ω f ϕ.toFun s) (ω f (fun t_1 x => toF... | [
350,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
346,
1
] |
Mathlib/ModelTheory/Types.lean | FirstOrder.Language.Theory.CompleteType.toList_foldr_inf_mem | [
{
"state_after": "L : Language\nT : Theory L\nα : Type w\np : CompleteType T α\nt : Finset (Sentence (L[[α]]))\n⊢ (∀ (M : ModelType ↑p) (φ : BoundedFormula (L[[α]]) Empty 0), φ ∈ t → BoundedFormula.Realize φ default default) ↔\n ∀ (x : Sentence (L[[α]])), x ∈ ↑t → ∀ (M : ModelType ↑p), BoundedFormula.Realize... | [
171,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.tendsto_atTop_principal | [
{
"state_after": "ι : Type ?u.256503\nι' : Type ?u.256506\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.256515\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nf : β → α\ns : Set α\n⊢ (∃ a, ∀ (b : β), b ≥ a → b ∈ f ⁻¹' s) ↔ ∃ N, ∀ (n : β), n ≥ N → f n ∈ s",
"state_before": "ι : Type ?u.256503\nι' : Type ?u.256506... | [
1265,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1263,
1
] |
Mathlib/Control/Applicative.lean | Comp.seq_mk | [] | [
160,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
157,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean | nullMeasurableSet_eq_fun | [
{
"state_after": "G : Type ?u.2047564\nα : Type u_2\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Div G\nm : MeasurableSpace α\nf✝ g✝ : α → G\nμ : MeasureTheory.Measure α\nE : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : AddGroup E\ninst✝¹ : MeasurableSingletonClass E\ninst✝ : MeasurableSub₂ E\nf g : α → E\nhf : AEMe... | [
396,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
390,
1
] |
Mathlib/Data/Polynomial/Taylor.lean | Polynomial.taylor_one | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nr : R\nf : R[X]\n⊢ ↑(taylor r) 1 = ↑C 1",
"tactic": "rw [← C_1, taylor_C]"
}
] | [
69,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
69,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | ContDiffOn.const_smul | [] | [
1565,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1564,
1
] |
Mathlib/Topology/ContinuousFunction/CocompactMap.lean | CocompactMap.coe_comp | [] | [
151,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
150,
1
] |
Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | strictConvexOn_zpow | [
{
"state_after": "case hf\nm : ℤ\nhm₀ : m ≠ 0\nhm₁ : m ≠ 1\n⊢ ContinuousOn (fun x => x ^ m) (Ioi 0)\n\ncase hf''\nm : ℤ\nhm₀ : m ≠ 0\nhm₁ : m ≠ 1\n⊢ ∀ (x : ℝ), x ∈ Ioi 0 → 0 < (deriv^[2]) (fun x => x ^ m) x",
"state_before": "m : ℤ\nhm₀ : m ≠ 0\nhm₁ : m ≠ 1\n⊢ StrictConvexOn ℝ (Ioi 0) fun x => x ^ m",
"... | [
114,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean | toIcoMod_eq_sub | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIcoMod hp a b = toIcoMod hp 0 (b - a) + a",
"tactic": "rw [toIcoMod_sub_eq_sub, zero_add, sub_add_cancel]"
}
] | [
777,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
776,
1
] |
Mathlib/Order/CompleteLattice.lean | iSup_false | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.104937\nβ₂ : Type ?u.104940\nγ : Type ?u.104943\nι : Sort ?u.104946\nι' : Sort ?u.104949\nκ : ι → Sort ?u.104954\nκ' : ι' → Sort ?u.104959\ninst✝ : CompleteLattice α\nf g s✝ t : ι → α\na b : α\ns : False → α\n⊢ iSup s = ⊥",
"tactic"... | [
1292,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1291,
1
] |
Mathlib/Data/Finset/Sigma.lean | Finset.sigma_nonempty | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : ι → Type u_2\nβ : Type ?u.1273\ns s₁ s₂ : Finset ι\nt t₁ t₂ : (i : ι) → Finset (α i)\n⊢ Finset.Nonempty (Finset.sigma s t) ↔ ∃ i, i ∈ s ∧ Finset.Nonempty (t i)",
"tactic": "simp [Finset.Nonempty]"
}
] | [
63,
101
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
1
] |
Mathlib/CategoryTheory/Monoidal/Braided.lean | CategoryTheory.leftUnitor_inv_braiding | [
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((λ_ X).inv ≫ (β_ (𝟙_ C) X).hom) ≫ (ρ_ X).hom = (ρ_ X).inv ≫ (ρ_ X).hom",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ... | [
212,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
210,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | CircleDeg1Lift.sup_apply | [] | [
428,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
427,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | mul_lt_mul_left | [] | [
202,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
200,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean | MeasureTheory.integrable_withDensity_iff_integrable_smul' | [
{
"state_after": "α : Type u_1\nβ : Type ?u.1021419\nγ : Type ?u.1021422\nδ : Type ?u.1021425\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0∞\nhf ... | [
878,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
872,
1
] |
Mathlib/Data/Nat/Log.lean | Nat.log_lt_of_lt_pow | [] | [
127,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
126,
1
] |
Mathlib/LinearAlgebra/Pi.lean | LinearEquiv.sumArrowLequivProdArrow_symm_apply_inr | [] | [
470,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
468,
1
] |
Mathlib/Data/Sym/Sym2.lean | Sym2.fromRel_prop | [] | [
500,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
499,
1
] |
Mathlib/Algebra/Lie/IdealOperations.lean | LieSubmodule.lie_le_right | [
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\n... | [
128,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
126,
1
] |
Mathlib/Topology/Category/TopCat/Limits/Products.lean | TopCat.inducing_prod_map | [
{
"state_after": "case induced\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ X\ng : Y ⟶ Z\nhf : Inducing ((forget TopCat).map f)\nhg : Inducing ((forget TopCat).map g)\n⊢ topologicalSpace_forget (W ⨯ Y) = induced ((forget TopCat).map (prod.map f g)) (topologicalSpace_forget (X ⨯ Z))",
"sta... | [
283,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
] |
Mathlib/Analysis/Complex/ReImTopology.lean | Complex.frontier_setOf_re_lt | [
{
"state_after": "no goals",
"state_before": "a : ℝ\n⊢ frontier {z | z.re < a} = {z | z.re = a}",
"tactic": "simpa only [frontier_Iio] using frontier_preimage_re (Iio a)"
}
] | [
158,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
157,
1
] |
Mathlib/Algebra/Hom/Centroid.lean | CentroidHom.toEnd_nsmul | [] | [
355,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
354,
1
] |
Mathlib/Topology/Algebra/OpenSubgroup.lean | Subgroup.isOpen_mono | [] | [
325,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
323,
1
] |
Mathlib/Topology/Instances/ENNReal.lean | NNReal.tendsto_sum_nat_add | [
{
"state_after": "α : Type ?u.319791\nβ : Type ?u.319794\nγ : Type ?u.319797\nf : ℕ → ℝ≥0\n⊢ Tendsto (fun a => ↑(∑' (k : ℕ), f (k + a))) atTop (𝓝 ↑0)",
"state_before": "α : Type ?u.319791\nβ : Type ?u.319794\nγ : Type ?u.319797\nf : ℕ → ℝ≥0\n⊢ Tendsto (fun i => ∑' (k : ℕ), f (k + i)) atTop (𝓝 0)",
"ta... | [
1197,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1194,
1
] |
Std/Data/RBMap/WF.lean | Std.RBNode.setBlack_idem | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nt : RBNode α\n⊢ setBlack (setBlack t) = setBlack t",
"tactic": "cases t <;> rfl"
}
] | [
86,
94
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
86,
1
] |
Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | MeasureTheory.withDensityᵥ_sub' | [] | [
112,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
] |
Mathlib/RingTheory/Localization/Basic.lean | IsLocalization.lift_eq | [] | [
512,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
511,
1
] |
Mathlib/GroupTheory/Sylow.lean | IsPGroup.inf_normalizer_sylow | [] | [
288,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
282,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean | AddMonoidAlgebra.single_zero_mul_apply | [
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.2059773\ninst✝¹ : Semiring k\ninst✝ : AddZeroClass G\nf : AddMonoidAlgebra k G\nr : k\nx a : G\n⊢ 0 + a = x ↔ a = x",
"tactic": "rw [zero_add]"
}
] | [
1647,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1645,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.compl_eq_deleteEdges | [
{
"state_after": "case Adj.h.h.a\nι : Sort ?u.154998\n𝕜 : Type ?u.155001\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\nx✝¹ x✝ : V\n⊢ Adj (Gᶜ) x✝¹ x✝ ↔ Adj (deleteEdges ⊤ (edgeSet G)) x✝¹ x✝",
"state_before": "ι : Sort ?u.154998\n𝕜 : Type ?u.155001... | [
1128,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1126,
1
] |
Mathlib/Analysis/PSeries.lean | ENNReal.tsum_condensed_le | [
{
"state_after": "f : ℕ → ℝ≥0∞\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\n⊢ (⨆ (i : ℕ), ∑ a in Finset.range (Nat.succ i), 2 ^ a * f (2 ^ a)) ≤ f 1 + 2 • ∑' (k : ℕ), f k",
"state_before": "f : ℕ → ℝ≥0∞\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\n⊢ (∑' (k : ℕ), 2 ^ k * f (2 ^ k)) ≤ f 1 + 2 * ∑' (k : ℕ), ... | [
116,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
108,
1
] |
Mathlib/Algebra/Order/Field/Power.lean | one_le_zpow_of_nonneg | [] | [
47,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
46,
1
] |
Mathlib/Order/Filter/Lift.lean | Filter.lift_lift_same_eq_lift | [] | [
167,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.Ioc_union_Ioc_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.211189\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\n⊢ Ioc a b ∪ Ioc a c = Ioc a (max b c)",
"tactic": "rw [Ioc_union_Ioc, min_self] <;> exact (min_le_left _ _).trans (le_max_left _ _)"
}
] | [
1842,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1841,
1
] |
Mathlib/Algebra/FreeAlgebra.lean | FreeAlgebra.ι_ne_one | [] | [
471,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
470,
1
] |
Mathlib/Topology/Order/Basic.lean | interior_Ioc | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\ninst✝ : NoMaxOrder α\na b : α\n⊢ interior (Ioc a b) = Ioo a b",
"tactic": "rw [← Ioi_inter_Iic... | [
2301,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2300,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.eventually_atBot_prod_self' | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.301355\nι' : Type ?u.301358\nα : Type u_1\nβ : Type ?u.301364\nγ : Type ?u.301367\ninst✝¹ : SemilatticeInf α\ninst✝ : Nonempty α\np : α × α → Prop\n⊢ (∀ᶠ (x : α × α) in atBot, p x) ↔ ∃ a, ∀ (k : α), k ≤ a → ∀ (l : α), l ≤ a → p (k, l)",
"tactic":... | [
1480,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1478,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Images.lean | CategoryTheory.Limits.HasImageMap.transport | [] | [
728,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
725,
1
] |
Mathlib/Data/Num/Lemmas.lean | Num.add_succ | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.173867\nn : Num\n⊢ 0 + succ n = succ (0 + n)",
"tactic": "simp [zero_add]"
},
{
"state_after": "α : Type ?u.173867\np : PosNum\n⊢ pos (PosNum.succ p) = succ (pos p)",
"state_before": "α : Type ?u.173867\np : PosNum\n⊢ pos (p + 1) = su... | [
224,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
221,
1
] |
Mathlib/Order/SuccPred/IntervalSucc.lean | Antitone.pairwise_disjoint_on_Ioo_succ | [] | [
121,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
119,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean | zpow_lt_zpow' | [] | [
397,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
396,
1
] |
Mathlib/Order/Basic.lean | LE.le.le_or_lt | [] | [
290,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
289,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | Polynomial.prod_cyclotomic_eq_X_pow_sub_one | [
{
"state_after": "n : ℕ\nhpos : 0 < n\nR : Type u_1\ninst✝ : CommRing R\ninteger : ∏ i in Nat.divisors n, cyclotomic i ℤ = X ^ n - 1\n⊢ ∏ i in Nat.divisors n, cyclotomic i R = X ^ n - 1",
"state_before": "n : ℕ\nhpos : 0 < n\nR : Type u_1\ninst✝ : CommRing R\n⊢ ∏ i in Nat.divisors n, cyclotomic i R = X ^ n ... | [
373,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
365,
1
] |
Mathlib/Algebra/EuclideanDomain/Basic.lean | EuclideanDomain.gcd_self | [] | [
192,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
191,
1
] |
Mathlib/GroupTheory/FreeProduct.lean | FreeProduct.lift_word_prod_nontrivial_of_head_eq_last | [
{
"state_after": "case intro\nι : Type u_1\nM : ι → Type ?u.773601\ninst✝⁴ : (i : ι) → Monoid (M i)\nN : Type ?u.773612\ninst✝³ : Monoid N\nhnontriv : Nontrivial ι\nG : Type u_3\ninst✝² : Group G\nH : ι → Type u_2\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nhcard : 3 ≤ (#ι) ∨ ∃ i, 3 ≤ (#H i)\nα : T... | [
752,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
750,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | MvPowerSeries.coeff_one | [] | [
193,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean | BoundedContinuousFunction.continuous_coe | [] | [
320,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
319,
1
] |
Mathlib/Algebra/Order/Pointwise.lean | LinearOrderedField.smul_Ioo | [
{
"state_after": "case h\nα : Type ?u.52465\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ x ∈ r • Ioo a b ↔ x ∈ Ioo (r • a) (r • b)",
"state_before": "α : Type ?u.52465\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\n⊢ r • Ioo a b = Ioo (r • a) (r • b)",
... | [
199,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
188,
1
] |
Mathlib/Algebra/CharP/Basic.lean | CharP.cast_eq_mod | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : NonAssocRing R\np : ℕ\ninst✝ : CharP R p\nk : ℕ\n⊢ ↑k = ↑(k % p + p * (k / p))",
"tactic": "rw [Nat.mod_add_div]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : NonAssocRing R\np : ℕ\ninst✝ : CharP ... | [
475,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
472,
1
] |
Mathlib/Order/Filter/Pointwise.lean | Filter.vsub_eq_bot_iff | [] | [
1108,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1107,
1
] |
Mathlib/Order/WellFoundedSet.lean | Set.IsWf.not_lt_min | [] | [
627,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
626,
8
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean | MeasureTheory.IntegrableOn.mono_set | [] | [
131,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
130,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.val_succ | [
{
"state_after": "case mk\nn m val✝ : ℕ\nisLt✝ : val✝ < n\n⊢ ↑(succ { val := val✝, isLt := isLt✝ }) = ↑{ val := val✝, isLt := isLt✝ } + 1",
"state_before": "n m : ℕ\nj : Fin n\n⊢ ↑(succ j) = ↑j + 1",
"tactic": "cases j"
},
{
"state_after": "no goals",
"state_before": "case mk\nn m val✝ : ℕ\n... | [
870,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
870,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean | Finset.prod_eq_zero_iff | [
{
"state_after": "case empty\nι : Type ?u.872555\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoidWithZero β\ninst✝¹ : Nontrivial β\ninst✝ : NoZeroDivisors β\n⊢ ∏ x in ∅, f x = 0 ↔ ∃ a, a ∈ ∅ ∧ f a = 0\n\ncase insert\nι : Type ?u.872555\nβ : Type u\nα : Type v\nγ :... | [
1928,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1924,
1
] |
Mathlib/Analysis/Convex/Combination.lean | Finset.centerMass_insert | [
{
"state_after": "R : Type u_2\nE : Type u_3\nF : Type ?u.18158\nι : Type u_1\nι' : Type ?u.18164\nα : Type ?u.18167\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : Orde... | [
68,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
62,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean | LinearIsometryEquiv.trans_apply | [] | [
807,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
805,
1
] |
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