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/NumberTheory/PellMatiyasevic.lean | Pell.xz_succ_succ | [
{
"state_after": "a : ℕ\na1 : 1 < a\nn : ℕ\n⊢ ↑(xn a1 (n + 2)) + ↑(xn a1 n) = ↑(2 * a) * ↑(xn a1 (n + 1))",
"state_before": "a : ℕ\na1 : 1 < a\nn : ℕ\n⊢ xz a1 (n + 2) + xz a1 n = ↑(2 * a) * xz a1 (n + 1)",
"tactic": "delta xz"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < ... | [
540,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
539,
1
] |
Mathlib/Algebra/Lie/Basic.lean | LieEquiv.bijective | [] | [
662,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
661,
11
] |
Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean | Complex.hasStrictDerivAt_log | [
{
"state_after": "h : 0 < 0.re ∨ 0.im ≠ 0\n⊢ False",
"state_before": "x : ℂ\nh : 0 < x.re ∨ x.im ≠ 0\n⊢ x ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "h : 0 < 0.re ∨ 0.im ≠ 0\n⊢ False",
"tactic": "simp [lt_irrefl] at h"
},
{
"state_after": "no goa... | [
59,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
56,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean | MultilinearMap.comp_linearEquiv_eq_zero_iff | [
{
"state_after": "R : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝¹⁴ : Semiring R\ninst✝¹³ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝¹² : (i : ι) → AddCommMonoid (M₁ i)\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid... | [
446,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
443,
1
] |
Mathlib/LinearAlgebra/Projection.lean | Submodule.linearProjOfIsCompl_apply_right' | [] | [
179,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
177,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.mk_lf_mk | [] | [
459,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
457,
1
] |
Mathlib/Init/Data/Int/Basic.lean | Int.neg_eq_neg | [] | [
73,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
11
] |
Mathlib/RingTheory/WittVector/Basic.lean | WittVector.mapFun.injective | [
{
"state_after": "p : ℕ\nR : Type ?u.19105\nS : Type ?u.19108\nT : Type ?u.19111\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Injective f\na₁✝ a₂✝ : 𝕎 α\nh : mapFun f a₁✝ = mapFun f a₂✝\n⊢ a₁✝ = a₂✝",
"state_before": "p ... | [
83,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
] |
Std/Data/Nat/Lemmas.lean | Nat.le_of_le_of_sub_le_sub_right | [
{
"state_after": "n m k : Nat\nh₀ : succ k ≤ succ m\nh₁ : n - k ≤ m - k\n⊢ succ n ≤ succ m",
"state_before": "n m k : Nat\nh₀ : succ k ≤ succ m\nh₁ : succ n - succ k ≤ succ m - succ k\n⊢ succ n ≤ succ m",
"tactic": "simp [succ_sub_succ] at h₁"
},
{
"state_after": "no goals",
"state_before": ... | [
129,
86
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
122,
11
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.exists_pos_set_lintegral_lt_of_measure_lt | [
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.487463\nγ : Type ?u.487466\nδ : Type ?u.487469\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nε₂ : ℝ≥0∞\nhε₂0 : 0 < ε₂\nhε₂ε : ε₂ < ε\n⊢ ∃ δ, δ > 0 ∧ ∀ (s : Set α), ↑↑μ s < δ → (∫⁻ (x : α) i... | [
510,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
483,
1
] |
Mathlib/CategoryTheory/Bicategory/Basic.lean | CategoryTheory.Bicategory.rightUnitor_comp_inv | [
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\n⊢ (ρ_ (f ≫ g)).inv = f ◁ (ρ_ g).inv ≫ (α_ f g (𝟙 c)).inv",
"tactic": "simp"
}
] | [
480,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
479,
1
] |
Std/Data/Int/Lemmas.lean | Int.neg_mul_eq_neg_mul | [
{
"state_after": "no goals",
"state_before": "a b : Int\n⊢ a * b + -a * b = 0",
"tactic": "rw [← Int.add_mul, Int.add_right_neg, Int.zero_mul]"
}
] | [
462,
86
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
461,
11
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean | Equiv.Perm.parts_partition | [] | [
532,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
530,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | isAdjointPair_toLinearMap₂' | [
{
"state_after": "R : Type u_1\nR₁ : Type ?u.2034253\nR₂ : Type ?u.2034256\nM : Type ?u.2034259\nM₁ : Type ?u.2034262\nM₂ : Type ?u.2034265\nM₁' : Type ?u.2034268\nM₂' : Type ?u.2034271\nn : Type u_2\nm : Type ?u.2034277\nn' : Type u_3\nm' : Type ?u.2034283\nι : Type ?u.2034286\ninst✝⁸ : CommRing R\ninst✝⁷ : Ad... | [
573,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
559,
1
] |
Mathlib/Topology/Algebra/Order/LeftRightLim.lean | leftLim_eq_of_tendsto | [
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace β\nhα : TopologicalSpace α\nh'α : OrderTopology α\ninst✝ : T2Space β\nf : α → β\na : α\ny : β\nh : 𝓝[Iio a] a ≠ ⊥\nh' : Tendsto f (𝓝[Iio a] a) (𝓝 y)\nh'' : ∃ y, Tendsto f (𝓝[Iio a] a) (𝓝 y)\n⊢ leftLim f a = y",
... | [
75,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
68,
1
] |
Mathlib/Algebra/Category/ModuleCat/Basic.lean | ModuleCat.forget₂_obj_moduleCat_of | [] | [
164,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
162,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean | Ideal.span_pair_comm | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na b x y : α\n⊢ span {x, y} = span {y, x}",
"tactic": "simp only [span_insert, sup_comm]"
}
] | [
362,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
361,
1
] |
Mathlib/Algebra/Lie/NonUnitalNonAssocAlgebra.lean | LieHom.toNonUnitalAlgHom_injective | [] | [
103,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
1
] |
Mathlib/Order/Hom/Lattice.lean | LatticeHom.cancel_right | [] | [
1156,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1154,
1
] |
Mathlib/Algebra/BigOperators/Associated.lean | Associates.rel_associated_iff_map_eq_map | [
{
"state_after": "α : Type u_1\nβ : Type ?u.35843\nγ : Type ?u.35846\nδ : Type ?u.35849\ninst✝ : CommMonoid α\np q : Multiset α\n⊢ Multiset.Rel Associated p q ↔ Multiset.Rel (fun a b => Associates.mk a = Associates.mk b) p q",
"state_before": "α : Type u_1\nβ : Type ?u.35843\nγ : Type ?u.35846\nδ : Type ?u.... | [
127,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
1
] |
Mathlib/LinearAlgebra/Quotient.lean | Submodule.quotientRel_r_def | [
{
"state_after": "R : Type u_2\nM : Type u_1\nr : R\nx✝ y✝ : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' : Submodule R M\nx y : M\n⊢ -x + y ∈ toAddSubgroup p ↔ -x + y ∈ p",
"state_before": "R : Type u_2\nM : Type u_1\nr : R\nx✝ y✝ : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst... | [
46,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
41,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | WithTop.sSup_eq | [] | [
66,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
64,
1
] |
Mathlib/Data/Finset/Prod.lean | Finset.mem_offDiag | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.101302\nγ : Type ?u.101305\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\n⊢ x ∈ offDiag s ↔ x.fst ∈ s ∧ x.snd ∈ s ∧ x.fst ≠ x.snd",
"tactic": "simp [offDiag, and_assoc]"
}
] | [
305,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
304,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean | Ordinal.pos_iff_ne_zero | [] | [
405,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
404,
11
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.Relabelling.mk'_leftMovesEquiv | [] | [
1049,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1047,
1
] |
Mathlib/GroupTheory/GroupAction/Pi.lean | Pi.smul_apply' | [] | [
49,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
47,
1
] |
Mathlib/CategoryTheory/Generator.lean | CategoryTheory.isCodetector_iff_reflectsIsomorphisms_yoneda_obj | [
{
"state_after": "case refine'_1\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nG : C\nhG : IsCodetector G\nA✝ B✝ : Cᵒᵖ\nf : A✝ ⟶ B✝\nhf : IsIso ((yoneda.obj G).map f)\n⊢ IsIso f\n\ncase refine'_2\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nG : C\nh : ReflectsIsomorp... | [
650,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
642,
1
] |
src/lean/Init/Data/List/Basic.lean | List.map_eq_mapTR | [
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ✝ : Type v\nγ : Type w\nα : Type u_2\nβ : Type u_1\nf : α → β\nas : List α\n⊢ map f as = mapTR f as",
"tactic": "simp [mapTR, mapTR_loop_eq]"
}
] | [
194,
32
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
192,
10
] |
Mathlib/Algebra/Order/Sub/Canonical.lean | add_le_of_le_tsub_left_of_le | [] | [
43,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Slope.lean | sub_div_sub_smul_slope_add_sub_div_sub_smul_slope | [
{
"state_after": "case pos\nk : Type u_2\nE : Type u_1\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → PE\na b c : k\nhab : a = b\n⊢ ((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c\n\ncase neg\nk : Type u_2\nE : ... | [
116,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean | ContinuousLinearMap.op_norm_comp_le' | [] | [
459,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
458,
1
] |
Mathlib/Data/Finset/Lattice.lean | Finset.sup'_mul_le_mul_sup'_of_nonneg | [] | [
1906,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1902,
1
] |
Mathlib/Topology/Sheaves/Presheaf.lean | TopCat.Presheaf.Pushforward.id_eq | [
{
"state_after": "C : Type u\ninst✝ : Category C\nX : TopCat\nℱ : Presheaf C X\n⊢ Functor.op (Opens.map (𝟙 X)) ⋙ ℱ = ℱ",
"state_before": "C : Type u\ninst✝ : Category C\nX : TopCat\nℱ : Presheaf C X\n⊢ 𝟙 X _* ℱ = ℱ",
"tactic": "unfold pushforwardObj"
},
{
"state_after": "C : Type u\ninst✝ : Ca... | [
214,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean | MeasureTheory.Integrable.right_of_add_measure | [
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.840553\nδ : Type ?u.840556\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nh : Memℒp f 1\n⊢ Memℒp f 1",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type... | [
543,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
540,
1
] |
Mathlib/Analysis/NormedSpace/Star/Mul.lean | op_nnnorm_mul_flip | [
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\ninst✝⁶ : DenselyNormedField 𝕜\ninst✝⁵ : NonUnitalNormedRing E\ninst✝⁴ : StarRing E\ninst✝³ : CstarRing E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : IsScalarTower 𝕜 E E\ninst✝ : SMulCommClass 𝕜 E E\na : E\n⊢ sSup ((fun x => ‖↑(↑(ContinuousLinearMap.flip (mul 𝕜 E)) a) ... | [
59,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
52,
1
] |
Mathlib/Topology/MetricSpace/PiNat.lean | PiNat.cylinder_eq_res | [
{
"state_after": "case h\nE : ℕ → Type ?u.22143\nα : Type u_1\nx : ℕ → α\nn : ℕ\ny : ℕ → α\n⊢ y ∈ cylinder x n ↔ y ∈ {y | res y n = res x n}",
"state_before": "E : ℕ → Type ?u.22143\nα : Type u_1\nx : ℕ → α\nn : ℕ\n⊢ cylinder x n = {y | res y n = res x n}",
"tactic": "ext y"
},
{
"state_after": ... | [
253,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
249,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.tendsto_bot | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.307631\nι : Sort x\nf : α → β\nl : Filter β\n⊢ Tendsto f ⊥ l",
"tactic": "simp [Tendsto]"
}
] | [
2854,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2854,
1
] |
Mathlib/Data/Multiset/Lattice.lean | Multiset.inf_mono | [] | [
155,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
154,
1
] |
Mathlib/Data/Bundle.lean | Bundle.coe_snd_map_apply | [] | [
173,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
171,
1
] |
Mathlib/Algebra/Tropical/Basic.lean | Tropical.trop_smul | [] | [
455,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
454,
1
] |
Mathlib/Algebra/Order/Floor.lean | round_le | [
{
"state_after": "F : Type ?u.264762\nα : Type u_1\nβ : Type ?u.264768\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nx : α\nz : ℤ\n⊢ fract x ≤ abs (x - ↑z) ∨ 1 - fract x ≤ abs (x - ↑z)",
"state_before": "F : Type ?u.264762\nα : Type u_1\nβ : Type ?u.264768\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorR... | [
1421,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1411,
1
] |
Mathlib/Order/Filter/Basic.lean | Pairwise.exists_mem_filter_of_disjoint | [
{
"state_after": "case this\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.87127\nι✝ : Sort x\nf g : Filter α\ns t : Set α\nι : Type u_1\ninst✝ : Finite ι\nl : ι → Filter α\nhd : Pairwise (Disjoint on l)\n⊢ ∀ (i j : ι), i ≠ j → ∃ s t, Disjoint ↑s ↑t\n\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.87127\nι✝... | [
738,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
729,
1
] |
Mathlib/LinearAlgebra/Prod.lean | LinearMap.snd_eq_coprod | [
{
"state_after": "case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.85660\nM₆ : Type ?u.85663\nS : Type ?u.85666\ninst✝¹³ : Semiring R\ninst✝¹² : Semiring S\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\... | [
253,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
253,
1
] |
Mathlib/Order/Closure.lean | LowerAdjoint.closed_eq_range_close | [] | [
413,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
412,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean | Finset.prod_ne_zero_iff | [
{
"state_after": "ι : Type ?u.875346\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⊢ (¬∃ a, a ∈ s ∧ f a = 0) ↔ ∀ (a : α), a ∈ s → f a ≠ 0",
"state_before": "ι : Type ?u.875346\nβ : Type u\nα : Type... | [
1933,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1931,
1
] |
Mathlib/NumberTheory/Padics/RingHoms.lean | PadicInt.dvd_appr_sub_appr | [
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝ x : ℤ_[p]\nm k : ℕ\nh : m ≤ m + k\n⊢ p ^ m ∣ appr x (m + k) - appr x m",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝ x : ℤ_[p]\nm n : ℕ\nh : m ≤ n\n⊢ p ^ m ∣ appr x n - appr x m",
"tactic": "obtain ⟨k, rfl... | [
351,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
339,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean | MeasureTheory.Lp.coeFn_posPart | [] | [
1119,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1118,
1
] |
Mathlib/Order/Filter/ENNReal.lean | ENNReal.limsup_const_mul_of_ne_top | [
{
"state_after": "case pos\nα : Type u_1\nf : Filter α\nu : α → ℝ≥0∞\na : ℝ≥0∞\nha_top : a ≠ ⊤\nha_zero : a = 0\n⊢ limsup (fun x => a * u x) f = a * limsup u f\n\ncase neg\nα : Type u_1\nf : Filter α\nu : α → ℝ≥0∞\na : ℝ≥0∞\nha_top : a ≠ ⊤\nha_zero : ¬a = 0\n⊢ limsup (fun x => a * u x) f = a * limsup u f",
... | [
51,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
36,
1
] |
Mathlib/Analysis/Calculus/Deriv/Mul.lean | HasStrictDerivAt.clm_apply | [
{
"state_after": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nG : Type u_1\ninst✝¹ :... | [
387,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
383,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Inner.lean | MeasureTheory.AEStronglyMeasurable.inner | [] | [
57,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
54,
11
] |
Mathlib/LinearAlgebra/Ray.lean | SameRay.exists_pos_right | [] | [
662,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
660,
1
] |
Mathlib/Topology/Algebra/Module/Multilinear.lean | ContinuousMultilinearMap.ext | [] | [
113,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean | AffineIsometryEquiv.ext | [] | [
363,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
362,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | MeasureTheory.AEStronglyMeasurable.const_smul | [] | [
1333,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1331,
11
] |
Mathlib/Order/Monotone/Basic.lean | StrictMono.isMax_of_apply | [] | [
573,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
570,
1
] |
Mathlib/Topology/Homotopy/Basic.lean | ContinuousMap.HomotopyRel.symm_trans | [] | [
655,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
653,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | ContDiff.contDiff_fderiv_apply | [
{
"state_after": "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 : Normed... | [
1104,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1100,
1
] |
Mathlib/MeasureTheory/Function/Egorov.lean | MeasureTheory.tendstoUniformlyOn_of_ae_tendsto' | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\ninst✝⁵ : MetricSpace β\nμ : Measure α\ninst✝⁴ : SemilatticeSup ι\ninst✝³ : Nonempty ι\ninst✝² : Countable ι\nγ : Type ?u.46709\ninst✝¹ : TopologicalSpace γ\nf : ι → α → β\ng : α → β\ns : Set α\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (... | [
223,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
216,
1
] |
Mathlib/RingTheory/Polynomial/Vieta.lean | Multiset.esymm_neg | [
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\ns : Multiset R\nk : ℕ\n⊢ ∀ (x : Multiset R), x ∈ powersetLen k s → (prod ∘ map Neg.neg) x = (-1) ^ k * prod x",
"state_before": "R : Type u_1\ninst✝ : CommRing R\ns : Multiset R\nk : ℕ\n⊢ esymm (map Neg.neg s) k = (-1) ^ k * esymm s k",
"tactic": "rw [... | [
104,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
97,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | rel_iSup_sum | [
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.376229\nδ : Type u_3\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Countable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\nR : α → α → Prop\nm_iSup : ∀ (s : ℕ → β), R (m (⨆ (i : ℕ)... | [
756,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
752,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean | norm_add_sq_real | [
{
"state_after": "𝕜 : Type ?u.2366731\nE : Type ?u.2366734\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\nh : ‖x + y‖ ^ 2 = ‖x‖ ^ 2 + 2 * ↑re (inner x y) + ‖y‖ ^ 2\n... | [
1043,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1041,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean | MeasureTheory.snorm'_lt_top_of_snorm'_lt_top_of_exponent_le | [
{
"state_after": "case inl\nα : Type u_1\nE : Type u_2\nF : Type ?u.4187857\nG : Type ?u.4187860\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq✝ : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\np q : ℝ\ninst✝ : IsFiniteMeasure μ\nf : α → E\nhf : AEStron... | [
1126,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1113,
1
] |
Mathlib/Order/GaloisConnection.lean | GaloisConnection.l_comm_of_u_comm | [] | [
353,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
347,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | Subalgebra.map_injective | [] | [
476,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
474,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | tendsto_iff_dist_tendsto_zero | [
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.289039\nι : Type ?u.289042\ninst✝ : PseudoMetricSpace α\nf : β → α\nx : Filter β\na : α\n⊢ Tendsto ((fun x => dist x a) ∘ f) x (𝓝 0) ↔ Tendsto (fun b => dist (f b) a) x (𝓝 0)",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.289039\nι : Type ?u.28... | [
1825,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1823,
1
] |
Mathlib/Topology/Instances/ENNReal.lean | ENNReal.biSup_add' | [
{
"state_after": "α : Type ?u.130531\nβ : Type ?u.130534\nγ : Type ?u.130537\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\np : ι → Prop\nh : ∃ i, p i\nf : ι → ℝ≥0∞\nthis : Nonempty { i // p i }\n⊢ (⨆ (i : ι) (_ : p i), f i) + a = ⨆ (i : ι) (_ : p i), f i + a",
"state_befor... | [
571,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
568,
1
] |
Mathlib/Data/Complex/Basic.lean | Complex.ofReal_eq_zero | [] | [
156,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
] |
Mathlib/Topology/ContinuousOn.lean | ContinuousOn.restrict_mapsTo | [] | [
615,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
613,
1
] |
Mathlib/FieldTheory/Subfield.lean | Subfield.mem_carrier | [] | [
182,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
181,
1
] |
Mathlib/CategoryTheory/Whiskering.lean | CategoryTheory.isoWhiskerRight_hom | [] | [
183,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
181,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean | toIocDiv_neg' | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIocDiv hp (-a) b = -(toIcoDiv hp a (-b) + 1)",
"tactic": "simpa only [neg_neg] using toIocDiv_neg hp (-a) (-b)"
}
] | [
401,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
400,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | EuclideanGeometry.angle_eq_pi_iff_sbtw | [
{
"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\n⊢ ∠ p₁ p₂ p₃ = π → Sbtw ℝ p₁ p₂ p₃",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProdu... | [
323,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
309,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Subgroup.comap_iInf | [] | [
1544,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1542,
1
] |
Mathlib/LinearAlgebra/Basic.lean | LinearMap.mem_range | [] | [
1212,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1211,
1
] |
Mathlib/Algebra/GCDMonoid/Div.lean | Finset.Int.gcd_div_id_eq_one | [] | [
69,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
67,
1
] |
Mathlib/LinearAlgebra/Orientation.lean | Basis.orientation_ne_iff_eq_neg | [] | [
244,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
241,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | NonUnitalSubsemiring.subset_closure | [] | [
569,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
569,
1
] |
Mathlib/Algebra/Polynomial/BigOperators.lean | Polynomial.coeff_multiset_prod_of_natDegree_le | [
{
"state_after": "case h\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nn : ℕ\na✝ : List R[X]\nhl : ∀ (p : R[X]), p ∈ Quotient.mk (List.isSetoid R[X]) a✝ → natDegree p ≤ n\n⊢ coeff (prod (Quotient.mk (List.isSetoid R[X]) a✝)) (↑Multiset.card (Quotient.mk (List.is... | [
217,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
214,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite | [] | [
1139,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1132,
1
] |
Mathlib/Algebra/Homology/Exact.lean | CategoryTheory.exact_epi_comp | [
{
"state_after": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh : C ⟶ D\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasEqualizers V\nhgh : Exact g h\ninst✝ : Epi f\n⊢ Epi (imageToKernel (f ≫ g) h (_ : (f ≫ g) ≫ h = 0))",
"state_before": "V : Type u\ninst✝⁴ : Category... | [
194,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
190,
1
] |
Mathlib/CategoryTheory/Functor/Flat.lean | CategoryTheory.PreservesFiniteLimitsOfFlat.uniq | [
{
"state_after": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nJ : Type v₁\ninst✝² : SmallCategory J\ninst✝¹ : FinCategory J\nK✝ : J ⥤ C\nF : C ⥤ D\ninst✝ : RepresentablyFlat F\nc✝ : Cone K✝\nhc✝ : IsLimit c✝\ns✝ : Cone (K✝ ⋙ F)\nK : J ⥤ C\nc : Cone K\nhc : IsLimit c\ns : Cone (K ⋙ F)\nf₁... | [
309,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
231,
1
] |
Mathlib/Data/List/Basic.lean | List.takeI_left' | [
{
"state_after": "ι : Type ?u.226663\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁✝ l₂✝ : List α\ninst✝ : Inhabited α\nl₁ l₂ : List α\nn : ℕ\nh : length l₁ = n\n⊢ takeI (length l₁) (l₁ ++ l₂) = l₁",
"state_before": "ι : Type ?u.226663\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁✝ l₂✝ : List α\n... | [
2346,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2345,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean | lp.single_apply | [] | [
1003,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1001,
11
] |
Mathlib/Topology/MetricSpace/Basic.lean | Metric.emetric_ball | [
{
"state_after": "case h\nα : Type u\nβ : Type v\nX : Type ?u.164248\nι : Type ?u.164251\ninst✝ : PseudoMetricSpace α\nx : α\nε : ℝ\ny : α\n⊢ y ∈ EMetric.ball x (ENNReal.ofReal ε) ↔ y ∈ ball x ε",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.164248\nι : Type ?u.164251\ninst✝ : PseudoMetricSpace α\nx... | [
1200,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1197,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean | edist_toDual | [] | [
1234,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1233,
1
] |
Mathlib/Algebra/Algebra/Spectrum.lean | spectrum.add_mem_add_iff | [
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr s : R\n⊢ r + s ∈ σ (↑↑ₐ s + a) ↔ r ∈ σ a",
"tactic": "rw [add_mem_iff, neg_add_cancel_left]"
}
] | [
209,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
208,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean | tendsto_one_iff_norm_tendsto_one | [
{
"state_after": "𝓕 : Type ?u.450626\n𝕜 : Type ?u.450629\nα : Type u_1\nι : Type ?u.450635\nκ : Type ?u.450638\nE : Type u_2\nF : Type ?u.450644\nG : Type ?u.450647\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\nf : α → E\na : ... | [
1035,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1032,
1
] |
Mathlib/Analysis/NormedSpace/MazurUlam.lean | IsometryEquiv.coe_to_real_linear_equiv_of_map_zero | [] | [
120,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
118,
1
] |
Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | one_le | [] | [
220,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
219,
1
] |
Mathlib/AlgebraicTopology/MooreComplex.lean | AlgebraicTopology.normalizedMooreComplex_objD | [] | [
167,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
1
] |
Mathlib/LinearAlgebra/Pi.lean | LinearEquiv.piCongrRight_refl | [] | [
363,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
362,
1
] |
Mathlib/Analysis/Convex/Basic.lean | Convex.smul | [] | [
425,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
424,
1
] |
Mathlib/Data/Matrix/Basic.lean | Matrix.add_apply | [] | [
303,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
302,
1
] |
Mathlib/Data/Sigma/Interval.lean | Sigma.Icc_mk_mk | [] | [
85,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/Algebra/Order/Interval.lean | NonemptyInterval.length_neg | [] | [
657,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
656,
1
] |
Std/Data/String/Lemmas.lean | String.foldr_eq | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : Char → α → α\ns : String\na : α\n⊢ foldr f a s = List.foldr f a s.data",
"tactic": "simpa using foldrAux_of_valid f [] s.1 [] a"
}
] | [
704,
46
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
703,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean | InnerProductSpace.Core.inner_neg_right | [
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.598333\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ ↑(starRingEnd 𝕜) (-inner y x) = -inner x y",
"state_before": "𝕜 : Type u_1\nE : Type ?u.598333\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommG... | [
289,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
288,
1
] |
Mathlib/Data/Part.lean | Part.mem_bind_iff | [] | [
499,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
496,
1
] |
Mathlib/RingTheory/PrincipalIdealDomain.lean | Submodule.IsPrincipal.of_comap | [
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\nS✝ : Type ?u.178108\nN : Type u_1\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Ring S✝\ninst✝¹ : Module R M\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Surjective ↑f\nS : Submodule R N\nhI : IsPrincipal (comap... | [
332,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
328,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean | LinearMap.isUnit_iff_ker_eq_bot | [
{
"state_after": "case mp\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : V →ₗ[K] V\n⊢ IsUnit f → ker f = ⊥\n\ncase mpr\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Finit... | [
1011,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1002,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.