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/RingTheory/HahnSeries.lean
|
HahnSeries.support_add_subset
|
[
{
"state_after": "Γ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nx y : HahnSeries Γ R\na : Γ\nha : coeff x a + coeff y a ≠ 0\n⊢ a ∈ support x ∪ support y",
"state_before": "Γ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nx y : HahnSeries Γ R\na : Γ\nha : a ∈ support (x + y)\n⊢ a ∈ support x ∪ support y",
"tactic": "rw [mem_support, add_coeff] at ha"
},
{
"state_after": "Γ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nx y : HahnSeries Γ R\na : Γ\nha : coeff x a + coeff y a ≠ 0\n⊢ coeff x a ≠ 0 ∨ coeff y a ≠ 0",
"state_before": "Γ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nx y : HahnSeries Γ R\na : Γ\nha : coeff x a + coeff y a ≠ 0\n⊢ a ∈ support x ∪ support y",
"tactic": "rw [Set.mem_union, mem_support, mem_support]"
},
{
"state_after": "Γ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nx y : HahnSeries Γ R\na : Γ\nha : coeff x a = 0 ∧ coeff y a = 0\n⊢ coeff x a + coeff y a = 0",
"state_before": "Γ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nx y : HahnSeries Γ R\na : Γ\nha : coeff x a + coeff y a ≠ 0\n⊢ coeff x a ≠ 0 ∨ coeff y a ≠ 0",
"tactic": "contrapose! ha"
},
{
"state_after": "no goals",
"state_before": "Γ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nx y : HahnSeries Γ R\na : Γ\nha : coeff x a = 0 ∧ coeff y a = 0\n⊢ coeff x a + coeff y a = 0",
"tactic": "rw [ha.1, ha.2, add_zero]"
}
] |
[
385,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
380,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pushout.hom_ext
|
[] |
[
1313,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1310,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.map_comap_le
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\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 M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂ : LieIdeal R L\nJ : LieIdeal R L'\n⊢ map f (comap f J) ≤ J",
"tactic": "rw [map_le_iff_le_comap]"
}
] |
[
852,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
852,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Polynomial.exists_irreducible_of_natDegree_ne_zero
|
[] |
[
983,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
981,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
QuadraticForm.comp_apply
|
[] |
[
548,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
547,
1
] |
Mathlib/GroupTheory/SemidirectProduct.lean
|
SemidirectProduct.lift_comp_inr
|
[
{
"state_after": "case h\nN : Type u_3\nG : Type u_1\nH : Type u_2\ninst✝² : Group N\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* MulAut N\nf₁ : N →* H\nf₂ : G →* H\nh :\n ∀ (g : G),\n MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑f₂ g))) f₁\nx✝ : G\n⊢ ↑(MonoidHom.comp (lift f₁ f₂ h) inr) x✝ = ↑f₂ x✝",
"state_before": "N : Type u_3\nG : Type u_1\nH : Type u_2\ninst✝² : Group N\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* MulAut N\nf₁ : N →* H\nf₂ : G →* H\nh :\n ∀ (g : G),\n MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑f₂ g))) f₁\n⊢ MonoidHom.comp (lift f₁ f₂ h) inr = f₂",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nN : Type u_3\nG : Type u_1\nH : Type u_2\ninst✝² : Group N\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* MulAut N\nf₁ : N →* H\nf₂ : G →* H\nh :\n ∀ (g : G),\n MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑f₂ g))) f₁\nx✝ : G\n⊢ ↑(MonoidHom.comp (lift f₁ f₂ h) inr) x✝ = ↑f₂ x✝",
"tactic": "simp"
}
] |
[
244,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.ext_iff
|
[] |
[
210,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Algebra/IsPrimePow.lean
|
Nat.Prime.isPrimePow
|
[] |
[
84,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
HasFPowerSeriesAt.add
|
[
{
"state_after": "case intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.522525\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr✝ r' : ℝ≥0∞\nhf : HasFPowerSeriesAt f pf x\nhg : HasFPowerSeriesAt g pg x\nr : ℝ≥0∞\nhr : HasFPowerSeriesOnBall f pf x r ∧ HasFPowerSeriesOnBall g pg x r\n⊢ HasFPowerSeriesAt (f + g) (pf + pg) x",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.522525\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesAt f pf x\nhg : HasFPowerSeriesAt g pg x\n⊢ HasFPowerSeriesAt (f + g) (pf + pg) x",
"tactic": "rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.522525\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr✝ r' : ℝ≥0∞\nhf : HasFPowerSeriesAt f pf x\nhg : HasFPowerSeriesAt g pg x\nr : ℝ≥0∞\nhr : HasFPowerSeriesOnBall f pf x r ∧ HasFPowerSeriesOnBall g pg x r\n⊢ HasFPowerSeriesAt (f + g) (pf + pg) x",
"tactic": "exact ⟨r, hr.1.add hr.2⟩"
}
] |
[
547,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
544,
1
] |
Mathlib/Topology/Constructions.lean
|
frontier_prod_univ_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.61421\nδ : Type ?u.61424\nε : Type ?u.61427\nζ : Type ?u.61430\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\ns : Set α\n⊢ frontier (s ×ˢ univ) = frontier s ×ˢ univ",
"tactic": "simp [frontier_prod_eq]"
}
] |
[
770,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
769,
1
] |
Mathlib/Algebra/DirectLimit.lean
|
Module.DirectLimit.totalize_of_not_le
|
[] |
[
194,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.one_nonempty
|
[] |
[
116,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.isFiniteMeasure_withDensity_ofReal
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.734418\nγ : Type ?u.734421\nδ : Type ?u.734424\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → ℝ\nhfi : HasFiniteIntegral f\nx : α\n⊢ ENNReal.ofReal (f x) ≤ ↑‖f x‖₊",
"state_before": "α : Type u_1\nβ : Type ?u.734418\nγ : Type ?u.734421\nδ : Type ?u.734424\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → ℝ\nhfi : HasFiniteIntegral f\n⊢ IsFiniteMeasure (Measure.withDensity μ fun x => ENNReal.ofReal (f x))",
"tactic": "refine' isFiniteMeasure_withDensity ((lintegral_mono fun x => _).trans_lt hfi).ne"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.734418\nγ : Type ?u.734421\nδ : Type ?u.734424\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → ℝ\nhfi : HasFiniteIntegral f\nx : α\n⊢ ENNReal.ofReal (f x) ≤ ↑‖f x‖₊",
"tactic": "exact Real.ofReal_le_ennnorm (f x)"
}
] |
[
278,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.mul_mul_apply
|
[
{
"state_after": "l : Type ?u.894994\nm : Type ?u.894997\nn : Type u_1\no : Type ?u.895003\nm' : o → Type ?u.895008\nn' : o → Type ?u.895013\nR : Type ?u.895016\nS : Type ?u.895019\nα : Type v\nβ : Type w\nγ : Type ?u.895026\ninst✝¹ : NonUnitalSemiring α\ninst✝ : Fintype n\nA B C : Matrix n n α\ni j : n\n⊢ (A ⬝ (B ⬝ C)) i j = A i ⬝ᵥ mulVec B (Cᵀ j)",
"state_before": "l : Type ?u.894994\nm : Type ?u.894997\nn : Type u_1\no : Type ?u.895003\nm' : o → Type ?u.895008\nn' : o → Type ?u.895013\nR : Type ?u.895016\nS : Type ?u.895019\nα : Type v\nβ : Type w\nγ : Type ?u.895026\ninst✝¹ : NonUnitalSemiring α\ninst✝ : Fintype n\nA B C : Matrix n n α\ni j : n\n⊢ (A ⬝ B ⬝ C) i j = A i ⬝ᵥ mulVec B (Cᵀ j)",
"tactic": "rw [Matrix.mul_assoc]"
},
{
"state_after": "l : Type ?u.894994\nm : Type ?u.894997\nn : Type u_1\no : Type ?u.895003\nm' : o → Type ?u.895008\nn' : o → Type ?u.895013\nR : Type ?u.895016\nS : Type ?u.895019\nα : Type v\nβ : Type w\nγ : Type ?u.895026\ninst✝¹ : NonUnitalSemiring α\ninst✝ : Fintype n\nA B C : Matrix n n α\ni j : n\n⊢ ∑ x : n, A i x * ∑ j_1 : n, B x j_1 * C j_1 j = ∑ x : n, A i x * ∑ x_1 : n, B x x_1 * Cᵀ j x_1",
"state_before": "l : Type ?u.894994\nm : Type ?u.894997\nn : Type u_1\no : Type ?u.895003\nm' : o → Type ?u.895008\nn' : o → Type ?u.895013\nR : Type ?u.895016\nS : Type ?u.895019\nα : Type v\nβ : Type w\nγ : Type ?u.895026\ninst✝¹ : NonUnitalSemiring α\ninst✝ : Fintype n\nA B C : Matrix n n α\ni j : n\n⊢ (A ⬝ (B ⬝ C)) i j = A i ⬝ᵥ mulVec B (Cᵀ j)",
"tactic": "simp only [mul_apply, dotProduct, mulVec]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.894994\nm : Type ?u.894997\nn : Type u_1\no : Type ?u.895003\nm' : o → Type ?u.895008\nn' : o → Type ?u.895013\nR : Type ?u.895016\nS : Type ?u.895019\nα : Type v\nβ : Type w\nγ : Type ?u.895026\ninst✝¹ : NonUnitalSemiring α\ninst✝ : Fintype n\nA B C : Matrix n n α\ni j : n\n⊢ ∑ x : n, A i x * ∑ j_1 : n, B x j_1 * C j_1 j = ∑ x : n, A i x * ∑ x_1 : n, B x x_1 * Cᵀ j x_1",
"tactic": "rfl"
}
] |
[
1859,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1855,
1
] |
Mathlib/Data/Polynomial/EraseLead.lean
|
Polynomial.eraseLead_support_card_lt
|
[
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh : f ≠ 0\n⊢ card (Finset.erase (support f) (natDegree f)) < card (support f)",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh : f ≠ 0\n⊢ card (support (eraseLead f)) < card (support f)",
"tactic": "rw [eraseLead_support]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh : f ≠ 0\n⊢ card (Finset.erase (support f) (natDegree f)) < card (support f)",
"tactic": "exact card_lt_card (erase_ssubset <| natDegree_mem_support_of_nonzero h)"
}
] |
[
114,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Order/Minimal.lean
|
maximals_idem
|
[] |
[
189,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Reflexive.lean
|
CategoryTheory.right_comp_retraction
|
[] |
[
105,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Order/GaloisConnection.lean
|
sSup_image2_eq_sInf_sSup
|
[] |
[
394,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
392,
1
] |
Mathlib/LinearAlgebra/Projection.lean
|
Submodule.prodEquivOfIsCompl_symm_apply_left
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁹ : Ring R\nE : Type u_1\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.80375\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.80891\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.81854\ninst✝² : Semiring S\nM : Type ?u.81860\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nh : IsCompl p q\nx : { x // x ∈ p }\n⊢ ↑x = ↑(prodEquivOfIsCompl p q h) (x, 0)",
"tactic": "simp"
}
] |
[
124,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Algebra/BigOperators/Ring.lean
|
Finset.sum_boole_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nb : β\nf✝ g : α → β\ninst✝¹ : NonAssocSemiring β\ninst✝ : DecidableEq α\ns : Finset α\nf : α → β\na : α\n⊢ ∑ x in s, (if a = x then 1 else 0) * f x = if a ∈ s then f a else 0",
"tactic": "simp"
}
] |
[
81,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
CochainComplex.mkHom_f_1
|
[] |
[
1103,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1102,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.mem_closedBall
|
[] |
[
542,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
542,
9
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsLittleO.symm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.233556\nE : Type ?u.233559\nF : Type u_3\nG : Type ?u.233565\nE' : Type u_2\nF' : Type ?u.233571\nG' : Type ?u.233574\nE'' : Type ?u.233577\nF'' : Type ?u.233580\nG'' : Type ?u.233583\nR : Type ?u.233586\nR' : Type ?u.233589\n𝕜 : Type ?u.233592\n𝕜' : Type ?u.233595\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf₁ f₂ f₃ : α → E'\nh : (fun x => f₁ x - f₂ x) =o[l] g\n⊢ (fun x => f₂ x - f₁ x) =o[l] g",
"tactic": "simpa only [neg_sub] using h.neg_left"
}
] |
[
1152,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1151,
1
] |
Mathlib/FieldTheory/IntermediateField.lean
|
IntermediateField.smul_mem
|
[] |
[
162,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/CategoryTheory/Subobject/Limits.lean
|
CategoryTheory.Limits.imageSubobject_arrow_comp
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nX Y Z : C\nf : X ⟶ Y\ninst✝ : HasImage f\n⊢ factorThruImageSubobject f ≫ arrow (imageSubobject f) = f",
"tactic": "simp [factorThruImageSubobject, imageSubobject_arrow]"
}
] |
[
332,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/NumberTheory/LucasLehmer.lean
|
LucasLehmer.X.zero_fst
|
[] |
[
207,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
9
] |
Mathlib/Algebra/Regular/Basic.lean
|
IsLeftRegular.right_of_commute
|
[] |
[
87,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.Injective.surjective_comp_right'
|
[] |
[
786,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
784,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
LinearIndependent.ne_zero
|
[
{
"state_after": "ι : Type u'\nι' : Type ?u.83214\nR : Type u_1\nK : Type ?u.83220\nM : Type u_2\nM' : Type ?u.83226\nM'' : Type ?u.83229\nV : Type u\nV' : Type ?u.83234\nv : ι → M\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ni : ι\nhv : LinearIndependent R v\nh : v i = 0\n⊢ ↑(Finsupp.single i 1) i = 0",
"state_before": "ι : Type u'\nι' : Type ?u.83214\nR : Type u_1\nK : Type ?u.83220\nM : Type u_2\nM' : Type ?u.83226\nM'' : Type ?u.83229\nV : Type u\nV' : Type ?u.83234\nv : ι → M\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ni : ι\nhv : LinearIndependent R v\nh : v i = 0\n⊢ 1 = 0",
"tactic": "suffices (Finsupp.single i 1 : ι →₀ R) i = 0 by simpa"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.83214\nR : Type u_1\nK : Type ?u.83220\nM : Type u_2\nM' : Type ?u.83226\nM'' : Type ?u.83229\nV : Type u\nV' : Type ?u.83234\nv : ι → M\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ni : ι\nhv : LinearIndependent R v\nh : v i = 0\n⊢ ↑0 i = 0\n\nι : Type u'\nι' : Type ?u.83214\nR : Type u_1\nK : Type ?u.83220\nM : Type u_2\nM' : Type ?u.83226\nM'' : Type ?u.83229\nV : Type u\nV' : Type ?u.83234\nv : ι → M\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ni : ι\nhv : LinearIndependent R v\nh : v i = 0\n⊢ ↑(Finsupp.total ι M R v) (Finsupp.single i 1) = 0",
"state_before": "ι : Type u'\nι' : Type ?u.83214\nR : Type u_1\nK : Type ?u.83220\nM : Type u_2\nM' : Type ?u.83226\nM'' : Type ?u.83229\nV : Type u\nV' : Type ?u.83234\nv : ι → M\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ni : ι\nhv : LinearIndependent R v\nh : v i = 0\n⊢ ↑(Finsupp.single i 1) i = 0",
"tactic": "rw [linearIndependent_iff.1 hv (Finsupp.single i 1)]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.83214\nR : Type u_1\nK : Type ?u.83220\nM : Type u_2\nM' : Type ?u.83226\nM'' : Type ?u.83229\nV : Type u\nV' : Type ?u.83234\nv : ι → M\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ni : ι\nhv : LinearIndependent R v\nh : v i = 0\nthis : ↑(Finsupp.single i 1) i = 0\n⊢ 1 = 0",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.83214\nR : Type u_1\nK : Type ?u.83220\nM : Type u_2\nM' : Type ?u.83226\nM'' : Type ?u.83229\nV : Type u\nV' : Type ?u.83234\nv : ι → M\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ni : ι\nhv : LinearIndependent R v\nh : v i = 0\n⊢ ↑0 i = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.83214\nR : Type u_1\nK : Type ?u.83220\nM : Type u_2\nM' : Type ?u.83226\nM'' : Type ?u.83229\nV : Type u\nV' : Type ?u.83234\nv : ι → M\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ni : ι\nhv : LinearIndependent R v\nh : v i = 0\n⊢ ↑(Finsupp.total ι M R v) (Finsupp.single i 1) = 0",
"tactic": "simp [h]"
}
] |
[
189,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
BoundedLatticeHom.symm_dual_comp
|
[] |
[
1587,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1584,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
compRel_mono
|
[] |
[
172,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Algebra.coe_sInf
|
[] |
[
856,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
855,
1
] |
Mathlib/CategoryTheory/Monad/Limits.lean
|
CategoryTheory.Monad.hasLimit_of_comp_forget_hasLimit
|
[] |
[
129,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
lt_of_mul_lt_mul_right'
|
[] |
[
149,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.thickening_subset_cthickening
|
[
{
"state_after": "ι : Sort ?u.109089\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns t : Set α\nx✝ : α\nδ : ℝ\nE : Set α\nx : α\nhx : x ∈ thickening δ E\n⊢ x ∈ cthickening δ E",
"state_before": "ι : Sort ?u.109089\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns t : Set α\nx : α\nδ : ℝ\nE : Set α\n⊢ thickening δ E ⊆ cthickening δ E",
"tactic": "intro x hx"
},
{
"state_after": "ι : Sort ?u.109089\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns t : Set α\nx✝ : α\nδ : ℝ\nE : Set α\nx : α\nhx : infEdist x E < ENNReal.ofReal δ\n⊢ x ∈ cthickening δ E",
"state_before": "ι : Sort ?u.109089\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns t : Set α\nx✝ : α\nδ : ℝ\nE : Set α\nx : α\nhx : x ∈ thickening δ E\n⊢ x ∈ cthickening δ E",
"tactic": "rw [thickening, mem_setOf_eq] at hx"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.109089\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns t : Set α\nx✝ : α\nδ : ℝ\nE : Set α\nx : α\nhx : infEdist x E < ENNReal.ofReal δ\n⊢ x ∈ cthickening δ E",
"tactic": "exact hx.le"
}
] |
[
1119,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1116,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.map_add_one
|
[] |
[
163,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Algebra/Group/TypeTags.lean
|
toAdd_mul
|
[] |
[
156,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Topology/Separation.lean
|
nhdsSet_le_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\n⊢ 𝓝ˢ s ≤ 𝓝ˢ t → s ⊆ t",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\n⊢ 𝓝ˢ s ≤ 𝓝ˢ t ↔ s ⊆ t",
"tactic": "refine' ⟨_, fun h => monotone_nhdsSet h⟩"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\n⊢ (∀ (x : Set α), x ∈ 𝓝ˢ t → x ∈ 𝓝ˢ s) → s ⊆ t",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\n⊢ 𝓝ˢ s ≤ 𝓝ˢ t → s ⊆ t",
"tactic": "simp_rw [Filter.le_def]"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\nh : ∀ (x : Set α), x ∈ 𝓝ˢ t → x ∈ 𝓝ˢ s\nx : α\nhx : x ∈ s\n⊢ x ∈ t",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\n⊢ (∀ (x : Set α), x ∈ 𝓝ˢ t → x ∈ 𝓝ˢ s) → s ⊆ t",
"tactic": "intro h x hx"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\nx : α\nhx : x ∈ s\nh : {x}ᶜ ∈ 𝓝ˢ t → {x}ᶜ ∈ 𝓝ˢ s\n⊢ x ∈ t",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\nh : ∀ (x : Set α), x ∈ 𝓝ˢ t → x ∈ 𝓝ˢ s\nx : α\nhx : x ∈ s\n⊢ x ∈ t",
"tactic": "specialize h ({x}ᶜ)"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\nx : α\nhx : x ∈ s\nh : ¬x ∈ t → ¬x ∈ s\n⊢ x ∈ t",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\nx : α\nhx : x ∈ s\nh : {x}ᶜ ∈ 𝓝ˢ t → {x}ᶜ ∈ 𝓝ˢ s\n⊢ x ∈ t",
"tactic": "simp_rw [compl_singleton_mem_nhdsSet_iff] at h"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\nx : α\nhx : x ∈ s\nh : ¬x ∈ t → ¬x ∈ s\nhxt : ¬x ∈ t\n⊢ False",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\nx : α\nhx : x ∈ s\nh : ¬x ∈ t → ¬x ∈ s\n⊢ x ∈ t",
"tactic": "by_contra hxt"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns t : Set α\nx : α\nhx : x ∈ s\nh : ¬x ∈ t → ¬x ∈ s\nhxt : ¬x ∈ t\n⊢ False",
"tactic": "exact h hxt hx"
}
] |
[
687,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
681,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.typein_top
|
[
{
"state_after": "case intro\nα✝ : Type ?u.72894\nβ✝ : Type ?u.72897\nγ : Type ?u.72900\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u_1\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≺i s\nx✝ : ↑{b | s b f.top}\nb : α\nh : ↑f.toRelEmbedding b ∈ {b | s b f.top}\n⊢ ∃ a,\n ↑(RelEmbedding.codRestrict {b | s b f.top}\n { toRelEmbedding := f.toRelEmbedding,\n init' :=\n (_ :\n ∀ (x : α) (x_1 : β),\n s x_1 (↑f.toRelEmbedding x) → ∃ a', ↑f.toRelEmbedding a' = x_1) }.toRelEmbedding\n (_ : ∀ (a : α), s (↑f.toRelEmbedding a) f.top))\n a =\n { val := ↑f.toRelEmbedding b, property := h }",
"state_before": "α✝ : Type ?u.72894\nβ✝ : Type ?u.72897\nγ : Type ?u.72900\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u_1\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≺i s\nx✝ : ↑{b | s b f.top}\na : β\nh : a ∈ {b | s b f.top}\n⊢ ∃ a_1,\n ↑(RelEmbedding.codRestrict {b | s b f.top}\n { toRelEmbedding := f.toRelEmbedding,\n init' :=\n (_ :\n ∀ (x : α) (x_1 : β),\n s x_1 (↑f.toRelEmbedding x) → ∃ a', ↑f.toRelEmbedding a' = x_1) }.toRelEmbedding\n (_ : ∀ (a : α), s (↑f.toRelEmbedding a) f.top))\n a_1 =\n { val := a, property := h }",
"tactic": "rcases f.down.1 h with ⟨b, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα✝ : Type ?u.72894\nβ✝ : Type ?u.72897\nγ : Type ?u.72900\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u_1\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≺i s\nx✝ : ↑{b | s b f.top}\nb : α\nh : ↑f.toRelEmbedding b ∈ {b | s b f.top}\n⊢ ∃ a,\n ↑(RelEmbedding.codRestrict {b | s b f.top}\n { toRelEmbedding := f.toRelEmbedding,\n init' :=\n (_ :\n ∀ (x : α) (x_1 : β),\n s x_1 (↑f.toRelEmbedding x) → ∃ a', ↑f.toRelEmbedding a' = x_1) }.toRelEmbedding\n (_ : ∀ (a : α), s (↑f.toRelEmbedding a) f.top))\n a =\n { val := ↑f.toRelEmbedding b, property := h }",
"tactic": "exact ⟨b, rfl⟩"
}
] |
[
457,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/Algebra/Ring/Defs.lean
|
ite_mul_zero_left
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\nR : Type x\nα : Type u_1\ninst✝¹ : MulZeroClass α\nP : Prop\ninst✝ : Decidable P\na b : α\n⊢ (if P then a * b else 0) = (if P then a else 0) * b",
"tactic": "by_cases h : P <;> simp [h]"
}
] |
[
230,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/LinearAlgebra/UnitaryGroup.lean
|
Matrix.UnitaryGroup.toLin'_one
|
[] |
[
147,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.trans_of_set'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.71931\nδ : Type ?u.71934\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\ns : Set β\nhs : IsOpen s\n⊢ LocalHomeomorph.trans e (ofSet s hs) = LocalHomeomorph.restr e (e.source ∩ ↑e ⁻¹' s)",
"tactic": "rw [trans_ofSet, restr_source_inter]"
}
] |
[
874,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
873,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.mul_im
|
[] |
[
133,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
RingHom.map_prod
|
[] |
[
265,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
11
] |
Mathlib/Analysis/Convex/Gauge.lean
|
gauge_zero
|
[
{
"state_after": "𝕜 : Type ?u.20500\nE : Type u_1\nF : Type ?u.20506\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • 0 ∈ s} = 0",
"state_before": "𝕜 : Type ?u.20500\nE : Type u_1\nF : Type ?u.20506\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\n⊢ gauge s 0 = 0",
"tactic": "rw [gauge_def']"
},
{
"state_after": "case pos\n𝕜 : Type ?u.20500\nE : Type u_1\nF : Type ?u.20506\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nh : 0 ∈ s\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • 0 ∈ s} = 0\n\ncase neg\n𝕜 : Type ?u.20500\nE : Type u_1\nF : Type ?u.20506\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nh : ¬0 ∈ s\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • 0 ∈ s} = 0",
"state_before": "𝕜 : Type ?u.20500\nE : Type u_1\nF : Type ?u.20506\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • 0 ∈ s} = 0",
"tactic": "by_cases h : (0 : E) ∈ s"
},
{
"state_after": "no goals",
"state_before": "case pos\n𝕜 : Type ?u.20500\nE : Type u_1\nF : Type ?u.20506\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nh : 0 ∈ s\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • 0 ∈ s} = 0",
"tactic": "simp only [smul_zero, sep_true, h, csInf_Ioi]"
},
{
"state_after": "no goals",
"state_before": "case neg\n𝕜 : Type ?u.20500\nE : Type u_1\nF : Type ?u.20506\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nh : ¬0 ∈ s\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • 0 ∈ s} = 0",
"tactic": "simp only [smul_zero, sep_false, h, Real.sInf_empty]"
}
] |
[
106,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Data/Fintype/Lattice.lean
|
Finset.fold_inf_univ
|
[] |
[
47,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Data/List/Basic.lean
|
List.cons_subset_of_subset_of_mem
|
[] |
[
328,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
326,
1
] |
Mathlib/Topology/Algebra/Monoid.lean
|
Submonoid.top_closure_mul_self_eq
|
[] |
[
422,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
419,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.pred_succ_iterate_of_not_isMax
|
[
{
"state_after": "case zero\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn : ℕ\nhin✝ : ¬IsMax ((succ^[n - 1]) i)\nhin : ¬IsMax ((succ^[Nat.zero - 1]) i)\n⊢ (pred^[Nat.zero]) ((succ^[Nat.zero]) i) = i\n\ncase succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n - 1]) i) → (pred^[n]) ((succ^[n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n - 1]) i)\n⊢ (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i",
"state_before": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn : ℕ\nhin : ¬IsMax ((succ^[n - 1]) i)\n⊢ (pred^[n]) ((succ^[n]) i) = i",
"tactic": "induction' n with n hn"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n - 1]) i) → (pred^[n]) ((succ^[n]) i) = i\nhin : ¬IsMax ((succ^[n]) i)\n⊢ (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i",
"state_before": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n - 1]) i) → (pred^[n]) ((succ^[n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n - 1]) i)\n⊢ (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i",
"tactic": "rw [Nat.succ_sub_succ_eq_sub, Nat.sub_zero] at hin"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n - 1]) i) → (pred^[n]) ((succ^[n]) i) = i\nhin : ¬IsMax ((succ^[n]) i)\nh_not_max : ¬IsMax ((succ^[n - 1]) i)\n⊢ (pred^[n] ∘ pred) ((succ ∘ succ^[n]) i) = i",
"state_before": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n - 1]) i) → (pred^[n]) ((succ^[n]) i) = i\nhin : ¬IsMax ((succ^[n]) i)\nh_not_max : ¬IsMax ((succ^[n - 1]) i)\n⊢ (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i",
"tactic": "rw [Function.iterate_succ, Function.iterate_succ']"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n - 1]) i) → (pred^[n]) ((succ^[n]) i) = i\nhin : ¬IsMax ((succ^[n]) i)\nh_not_max : ¬IsMax ((succ^[n - 1]) i)\n⊢ (pred^[n]) (pred (succ ((succ^[n]) i))) = i",
"state_before": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n - 1]) i) → (pred^[n]) ((succ^[n]) i) = i\nhin : ¬IsMax ((succ^[n]) i)\nh_not_max : ¬IsMax ((succ^[n - 1]) i)\n⊢ (pred^[n] ∘ pred) ((succ ∘ succ^[n]) i) = i",
"tactic": "simp only [Function.comp_apply]"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n - 1]) i) → (pred^[n]) ((succ^[n]) i) = i\nhin : ¬IsMax ((succ^[n]) i)\nh_not_max : ¬IsMax ((succ^[n - 1]) i)\n⊢ (pred^[n]) ((succ^[n]) i) = i",
"state_before": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n - 1]) i) → (pred^[n]) ((succ^[n]) i) = i\nhin : ¬IsMax ((succ^[n]) i)\nh_not_max : ¬IsMax ((succ^[n - 1]) i)\n⊢ (pred^[n]) (pred (succ ((succ^[n]) i))) = i",
"tactic": "rw [pred_succ_of_not_isMax hin]"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n - 1]) i) → (pred^[n]) ((succ^[n]) i) = i\nhin : ¬IsMax ((succ^[n]) i)\nh_not_max : ¬IsMax ((succ^[n - 1]) i)\n⊢ (pred^[n]) ((succ^[n]) i) = i",
"tactic": "exact hn h_not_max"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn : ℕ\nhin✝ : ¬IsMax ((succ^[n - 1]) i)\nhin : ¬IsMax ((succ^[Nat.zero - 1]) i)\n⊢ (pred^[Nat.zero]) ((succ^[Nat.zero]) i) = i",
"tactic": "simp only [Nat.zero_eq, Function.iterate_zero, id.def]"
},
{
"state_after": "case zero\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn : ℕ\nhin✝ : ¬IsMax ((succ^[n - 1]) i)\nhn : ¬IsMax ((succ^[Nat.zero - 1]) i) → (pred^[Nat.zero]) ((succ^[Nat.zero]) i) = i\nhin : ¬IsMax ((succ^[Nat.zero]) i)\n⊢ ¬IsMax ((succ^[Nat.zero - 1]) i)\n\ncase succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[Nat.succ n - 1]) i) → (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n]) i)\n⊢ ¬IsMax ((succ^[Nat.succ n - 1]) i)",
"state_before": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n - 1]) i) → (pred^[n]) ((succ^[n]) i) = i\nhin : ¬IsMax ((succ^[n]) i)\n⊢ ¬IsMax ((succ^[n - 1]) i)",
"tactic": "cases' n with n"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n]) i) → (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n]) i)\n⊢ ¬IsMax ((succ^[n]) i)",
"state_before": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[Nat.succ n - 1]) i) → (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n]) i)\n⊢ ¬IsMax ((succ^[Nat.succ n - 1]) i)",
"tactic": "rw [Nat.succ_sub_succ_eq_sub, Nat.sub_zero] at hn⊢"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n]) i) → (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n]) i)\nh_sub_le : (succ^[n]) i ≤ (succ^[Nat.succ n]) i\n⊢ ¬IsMax ((succ^[n]) i)",
"state_before": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n]) i) → (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n]) i)\n⊢ ¬IsMax ((succ^[n]) i)",
"tactic": "have h_sub_le : (succ^[n]) i ≤ (succ^[n.succ]) i := by\n rw [Function.iterate_succ']\n exact le_succ _"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n]) i) → (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n]) i)\nh_sub_le : (succ^[n]) i ≤ (succ^[Nat.succ n]) i\nh_max : IsMax ((succ^[n]) i)\nj : α\nhj : (succ^[Nat.succ n]) i ≤ j\n⊢ j ≤ (succ^[Nat.succ n]) i",
"state_before": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n]) i) → (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n]) i)\nh_sub_le : (succ^[n]) i ≤ (succ^[Nat.succ n]) i\n⊢ ¬IsMax ((succ^[n]) i)",
"tactic": "refine' fun h_max => hin fun j hj => _"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n]) i) → (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n]) i)\nh_sub_le : (succ^[n]) i ≤ (succ^[Nat.succ n]) i\nh_max : IsMax ((succ^[n]) i)\nj : α\nhj : (succ^[Nat.succ n]) i ≤ j\nhj_le : j ≤ (succ^[n]) i\n⊢ j ≤ (succ^[Nat.succ n]) i",
"state_before": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n]) i) → (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n]) i)\nh_sub_le : (succ^[n]) i ≤ (succ^[Nat.succ n]) i\nh_max : IsMax ((succ^[n]) i)\nj : α\nhj : (succ^[Nat.succ n]) i ≤ j\n⊢ j ≤ (succ^[Nat.succ n]) i",
"tactic": "have hj_le : j ≤ (succ^[n]) i := h_max (h_sub_le.trans hj)"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n]) i) → (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n]) i)\nh_sub_le : (succ^[n]) i ≤ (succ^[Nat.succ n]) i\nh_max : IsMax ((succ^[n]) i)\nj : α\nhj : (succ^[Nat.succ n]) i ≤ j\nhj_le : j ≤ (succ^[n]) i\n⊢ j ≤ (succ^[Nat.succ n]) i",
"tactic": "exact hj_le.trans h_sub_le"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn : ℕ\nhin✝ : ¬IsMax ((succ^[n - 1]) i)\nhn : ¬IsMax ((succ^[Nat.zero - 1]) i) → (pred^[Nat.zero]) ((succ^[Nat.zero]) i) = i\nhin : ¬IsMax ((succ^[Nat.zero]) i)\n⊢ ¬IsMax ((succ^[Nat.zero - 1]) i)",
"tactic": "simpa using hin"
},
{
"state_after": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n]) i) → (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n]) i)\n⊢ (succ^[n]) i ≤ (succ ∘ succ^[n]) i",
"state_before": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n]) i) → (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n]) i)\n⊢ (succ^[n]) i ≤ (succ^[Nat.succ n]) i",
"tactic": "rw [Function.iterate_succ']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\na b i : α\nn✝ : ℕ\nhin✝ : ¬IsMax ((succ^[n✝ - 1]) i)\nn : ℕ\nhn : ¬IsMax ((succ^[n]) i) → (pred^[Nat.succ n]) ((succ^[Nat.succ n]) i) = i\nhin : ¬IsMax ((succ^[Nat.succ n]) i)\n⊢ (succ^[n]) i ≤ (succ ∘ succ^[n]) i",
"tactic": "exact le_succ _"
}
] |
[
979,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
961,
1
] |
Mathlib/Topology/Constructions.lean
|
set_pi_mem_nhds
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.284993\nδ : Type ?u.284996\nε : Type ?u.284999\nζ : Type ?u.285002\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.285013\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\ni : Set ι\ns : (a : ι) → Set (π a)\nx : (a : ι) → π a\nhi : Set.Finite i\nhs : ∀ (a : ι), a ∈ i → s a ∈ 𝓝 (x a)\n⊢ ∀ (i_1 : ι), i_1 ∈ i → eval i_1 ⁻¹' s i_1 ∈ 𝓝 x",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.284993\nδ : Type ?u.284996\nε : Type ?u.284999\nζ : Type ?u.285002\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.285013\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\ni : Set ι\ns : (a : ι) → Set (π a)\nx : (a : ι) → π a\nhi : Set.Finite i\nhs : ∀ (a : ι), a ∈ i → s a ∈ 𝓝 (x a)\n⊢ Set.pi i s ∈ 𝓝 x",
"tactic": "rw [pi_def, biInter_mem hi]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.284993\nδ : Type ?u.284996\nε : Type ?u.284999\nζ : Type ?u.285002\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.285013\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\ni : Set ι\ns : (a : ι) → Set (π a)\nx : (a : ι) → π a\nhi : Set.Finite i\nhs : ∀ (a : ι), a ∈ i → s a ∈ 𝓝 (x a)\n⊢ ∀ (i_1 : ι), i_1 ∈ i → eval i_1 ⁻¹' s i_1 ∈ 𝓝 x",
"tactic": "exact fun a ha => (continuous_apply a).continuousAt (hs a ha)"
}
] |
[
1346,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1343,
1
] |
Mathlib/Order/Atoms.lean
|
isAtomic_of_orderBot_wellFounded_lt
|
[] |
[
320,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/Data/List/Infix.lean
|
List.dropLast_prefix
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1312\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\n⊢ dropLast [] ++ [] = []",
"tactic": "rw [dropLast, List.append_nil]"
}
] |
[
172,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/RingTheory/Ideal/Cotangent.lean
|
Ideal.to_quotient_square_range
|
[
{
"state_after": "R : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\n⊢ LinearMap.range (cotangentToQuotientSquare I) =\n LinearMap.range (LinearMap.comp (cotangentToQuotientSquare I) (toCotangent I))\n\nR : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\n⊢ LinearMap.range (LinearMap.comp (cotangentToQuotientSquare I) (toCotangent I)) =\n Submodule.restrictScalars R (cotangentIdeal I)",
"state_before": "R : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\n⊢ LinearMap.range (cotangentToQuotientSquare I) = Submodule.restrictScalars R (cotangentIdeal I)",
"tactic": "trans LinearMap.range (I.cotangentToQuotientSquare.comp I.toCotangent)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\n⊢ LinearMap.range (cotangentToQuotientSquare I) =\n LinearMap.range (LinearMap.comp (cotangentToQuotientSquare I) (toCotangent I))",
"tactic": "rw [LinearMap.range_comp, I.toCotangent_range, Submodule.map_top]"
},
{
"state_after": "R : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\n⊢ Submodule.map (Submodule.mkQ (I ^ 2)) I = Submodule.restrictScalars R (cotangentIdeal I)",
"state_before": "R : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\n⊢ LinearMap.range (LinearMap.comp (cotangentToQuotientSquare I) (toCotangent I)) =\n Submodule.restrictScalars R (cotangentIdeal I)",
"tactic": "rw [to_quotient_square_comp_toCotangent, LinearMap.range_comp, I.range_subtype]"
},
{
"state_after": "case h\nR : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\nx✝ : R ⧸ I ^ 2\n⊢ x✝ ∈ Submodule.map (Submodule.mkQ (I ^ 2)) I ↔ x✝ ∈ Submodule.restrictScalars R (cotangentIdeal I)",
"state_before": "R : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\n⊢ Submodule.map (Submodule.mkQ (I ^ 2)) I = Submodule.restrictScalars R (cotangentIdeal I)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\nx✝ : R ⧸ I ^ 2\n⊢ x✝ ∈ Submodule.map (Submodule.mkQ (I ^ 2)) I ↔ x✝ ∈ Submodule.restrictScalars R (cotangentIdeal I)",
"tactic": "rfl"
}
] |
[
142,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.nat_iterate
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.109928\nδ : Type ?u.109931\nσ : Type ?u.109934\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nf : α → ℕ\ng : α → β\nh : α → β → β\nhf : Primrec f\nhg : Primrec g\nhh : Primrec₂ h\na : α\n⊢ Nat.rec (g a) (fun n IH => h a (n, IH).snd) (f a) = (h a^[f a]) (g a)",
"tactic": "induction f a <;> simp [*, -Function.iterate_succ, Function.iterate_succ']"
}
] |
[
609,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
606,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/ContinuousLinearMap.lean
|
Measurable.apply_continuousLinearMap
|
[] |
[
89,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.lists_nil
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\n⊢ lists nil = ↑[[]]",
"tactic": "rw [nil, lists_coe, cyclicPermutations_nil]"
}
] |
[
758,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
757,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.mk_bounded_subset_le
|
[
{
"state_after": "α : Type u\ns : Set α\nc : Cardinal\n⊢ (#{ t // t ⊆ s ∧ (#↑t) ≤ c }) ≤ (#{ t // (#↑t) ≤ c })",
"state_before": "α : Type u\ns : Set α\nc : Cardinal\n⊢ (#{ t // t ⊆ s ∧ (#↑t) ≤ c }) ≤ max (#↑s) ℵ₀ ^ c",
"tactic": "refine' le_trans _ (mk_bounded_set_le s c)"
},
{
"state_after": "case refine'_1\nα : Type u\ns : Set α\nc : Cardinal\n⊢ { t // t ⊆ s ∧ (#↑t) ≤ c } ↪ Set ↑s\n\ncase refine'_2\nα : Type u\ns : Set α\nc : Cardinal\n⊢ ∀ (a : { t // t ⊆ s ∧ (#↑t) ≤ c }),\n ↑?refine'_1 a ∈ fun t =>\n Quot.lift ((fun α β => Nonempty (α ↪ β)) ↑t) (_ : ∀ (a b : Type u), a ≈ b → Nonempty (↑t ↪ a) = Nonempty (↑t ↪ b))\n c",
"state_before": "α : Type u\ns : Set α\nc : Cardinal\n⊢ (#{ t // t ⊆ s ∧ (#↑t) ≤ c }) ≤ (#{ t // (#↑t) ≤ c })",
"tactic": "refine' ⟨Embedding.codRestrict _ _ _⟩"
},
{
"state_after": "case refine'_1\nα : Type u\ns : Set α\nc : Cardinal\n⊢ Injective fun t => Subtype.val ⁻¹' ↑t\n\ncase refine'_2\nα : Type u\ns : Set α\nc : Cardinal\n⊢ ∀ (a : { t // t ⊆ s ∧ (#↑t) ≤ c }),\n ↑{ toFun := fun t => Subtype.val ⁻¹' ↑t, inj' := ?refine'_1 } a ∈ fun t =>\n Quot.lift ((fun α β => Nonempty (α ↪ β)) ↑t) (_ : ∀ (a b : Type u), a ≈ b → Nonempty (↑t ↪ a) = Nonempty (↑t ↪ b))\n c",
"state_before": "case refine'_1\nα : Type u\ns : Set α\nc : Cardinal\n⊢ { t // t ⊆ s ∧ (#↑t) ≤ c } ↪ Set ↑s\n\ncase refine'_2\nα : Type u\ns : Set α\nc : Cardinal\n⊢ ∀ (a : { t // t ⊆ s ∧ (#↑t) ≤ c }),\n ↑?refine'_1 a ∈ fun t =>\n Quot.lift ((fun α β => Nonempty (α ↪ β)) ↑t) (_ : ∀ (a b : Type u), a ≈ b → Nonempty (↑t ↪ a) = Nonempty (↑t ↪ b))\n c",
"tactic": "use fun t => (↑) ⁻¹' t.1"
},
{
"state_after": "case refine'_2.mk.intro\nα : Type u\ns : Set α\nc : Cardinal\nt : Set α\nleft✝ : t ⊆ s\nh2t : (#↑t) ≤ c\n⊢ ↑{ toFun := fun t => Subtype.val ⁻¹' ↑t,\n inj' :=\n (_ :\n ∀ ⦃a₁ a₂ : { t // t ⊆ s ∧ (#↑t) ≤ c }⦄,\n (fun t => Subtype.val ⁻¹' ↑t) a₁ = (fun t => Subtype.val ⁻¹' ↑t) a₂ → a₁ = a₂) }\n { val := t, property := (_ : t ⊆ s ∧ (#↑t) ≤ c) } ∈\n fun t =>\n Quot.lift ((fun α β => Nonempty (α ↪ β)) ↑t) (_ : ∀ (a b : Type u), a ≈ b → Nonempty (↑t ↪ a) = Nonempty (↑t ↪ b)) c",
"state_before": "case refine'_2\nα : Type u\ns : Set α\nc : Cardinal\n⊢ ∀ (a : { t // t ⊆ s ∧ (#↑t) ≤ c }),\n ↑{ toFun := fun t => Subtype.val ⁻¹' ↑t,\n inj' :=\n (_ :\n ∀ ⦃a₁ a₂ : { t // t ⊆ s ∧ (#↑t) ≤ c }⦄,\n (fun t => Subtype.val ⁻¹' ↑t) a₁ = (fun t => Subtype.val ⁻¹' ↑t) a₂ → a₁ = a₂) }\n a ∈\n fun t =>\n Quot.lift ((fun α β => Nonempty (α ↪ β)) ↑t) (_ : ∀ (a b : Type u), a ≈ b → Nonempty (↑t ↪ a) = Nonempty (↑t ↪ b))\n c",
"tactic": "rintro ⟨t, _, h2t⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.mk.intro\nα : Type u\ns : Set α\nc : Cardinal\nt : Set α\nleft✝ : t ⊆ s\nh2t : (#↑t) ≤ c\n⊢ ↑{ toFun := fun t => Subtype.val ⁻¹' ↑t,\n inj' :=\n (_ :\n ∀ ⦃a₁ a₂ : { t // t ⊆ s ∧ (#↑t) ≤ c }⦄,\n (fun t => Subtype.val ⁻¹' ↑t) a₁ = (fun t => Subtype.val ⁻¹' ↑t) a₂ → a₁ = a₂) }\n { val := t, property := (_ : t ⊆ s ∧ (#↑t) ≤ c) } ∈\n fun t =>\n Quot.lift ((fun α β => Nonempty (α ↪ β)) ↑t) (_ : ∀ (a b : Type u), a ≈ b → Nonempty (↑t ↪ a) = Nonempty (↑t ↪ b)) c",
"tactic": "exact (mk_preimage_of_injective _ _ Subtype.val_injective).trans h2t"
},
{
"state_after": "case refine'_1.mk.intro.mk.intro\nα : Type u\ns : Set α\nc : Cardinal\nt : Set α\nht1 : t ⊆ s\nht2 : (#↑t) ≤ c\nt' : Set α\nh1t' : t' ⊆ s\nh2t' : (#↑t') ≤ c\nh :\n (fun t => Subtype.val ⁻¹' ↑t) { val := t, property := (_ : t ⊆ s ∧ (#↑t) ≤ c) } =\n (fun t => Subtype.val ⁻¹' ↑t) { val := t', property := (_ : t' ⊆ s ∧ (#↑t') ≤ c) }\n⊢ { val := t, property := (_ : t ⊆ s ∧ (#↑t) ≤ c) } = { val := t', property := (_ : t' ⊆ s ∧ (#↑t') ≤ c) }",
"state_before": "case refine'_1\nα : Type u\ns : Set α\nc : Cardinal\n⊢ Injective fun t => Subtype.val ⁻¹' ↑t",
"tactic": "rintro ⟨t, ht1, ht2⟩ ⟨t', h1t', h2t'⟩ h"
},
{
"state_after": "case refine'_1.mk.intro.mk.intro.a\nα : Type u\ns : Set α\nc : Cardinal\nt : Set α\nht1 : t ⊆ s\nht2 : (#↑t) ≤ c\nt' : Set α\nh1t' : t' ⊆ s\nh2t' : (#↑t') ≤ c\nh :\n (fun t => Subtype.val ⁻¹' ↑t) { val := t, property := (_ : t ⊆ s ∧ (#↑t) ≤ c) } =\n (fun t => Subtype.val ⁻¹' ↑t) { val := t', property := (_ : t' ⊆ s ∧ (#↑t') ≤ c) }\n⊢ ↑{ val := t, property := (_ : t ⊆ s ∧ (#↑t) ≤ c) } = ↑{ val := t', property := (_ : t' ⊆ s ∧ (#↑t') ≤ c) }",
"state_before": "case refine'_1.mk.intro.mk.intro\nα : Type u\ns : Set α\nc : Cardinal\nt : Set α\nht1 : t ⊆ s\nht2 : (#↑t) ≤ c\nt' : Set α\nh1t' : t' ⊆ s\nh2t' : (#↑t') ≤ c\nh :\n (fun t => Subtype.val ⁻¹' ↑t) { val := t, property := (_ : t ⊆ s ∧ (#↑t) ≤ c) } =\n (fun t => Subtype.val ⁻¹' ↑t) { val := t', property := (_ : t' ⊆ s ∧ (#↑t') ≤ c) }\n⊢ { val := t, property := (_ : t ⊆ s ∧ (#↑t) ≤ c) } = { val := t', property := (_ : t' ⊆ s ∧ (#↑t') ≤ c) }",
"tactic": "apply Subtype.eq"
},
{
"state_after": "case refine'_1.mk.intro.mk.intro.a\nα : Type u\ns : Set α\nc : Cardinal\nt : Set α\nht1 : t ⊆ s\nht2 : (#↑t) ≤ c\nt' : Set α\nh1t' : t' ⊆ s\nh2t' : (#↑t') ≤ c\nh : Subtype.val ⁻¹' t = Subtype.val ⁻¹' t'\n⊢ t = t'",
"state_before": "case refine'_1.mk.intro.mk.intro.a\nα : Type u\ns : Set α\nc : Cardinal\nt : Set α\nht1 : t ⊆ s\nht2 : (#↑t) ≤ c\nt' : Set α\nh1t' : t' ⊆ s\nh2t' : (#↑t') ≤ c\nh :\n (fun t => Subtype.val ⁻¹' ↑t) { val := t, property := (_ : t ⊆ s ∧ (#↑t) ≤ c) } =\n (fun t => Subtype.val ⁻¹' ↑t) { val := t', property := (_ : t' ⊆ s ∧ (#↑t') ≤ c) }\n⊢ ↑{ val := t, property := (_ : t ⊆ s ∧ (#↑t) ≤ c) } = ↑{ val := t', property := (_ : t' ⊆ s ∧ (#↑t') ≤ c) }",
"tactic": "dsimp only at h⊢"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.mk.intro.mk.intro.a\nα : Type u\ns : Set α\nc : Cardinal\nt : Set α\nht1 : t ⊆ s\nht2 : (#↑t) ≤ c\nt' : Set α\nh1t' : t' ⊆ s\nh2t' : (#↑t') ≤ c\nh : Subtype.val ⁻¹' t = Subtype.val ⁻¹' t'\n⊢ t = t'",
"tactic": "refine' (preimage_eq_preimage' _ _).1 h <;> rw [Subtype.range_coe] <;> assumption"
}
] |
[
1143,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1134,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
Monotone.compRel
|
[] |
[
167,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Algebra/Regular/Basic.lean
|
not_isRightRegular_zero_iff
|
[
{
"state_after": "R : Type u_1\ninst✝ : MulZeroClass R\na b : R\n⊢ (¬∃ x y, x ≠ y) ↔ ∀ (x y : R), x = y",
"state_before": "R : Type u_1\ninst✝ : MulZeroClass R\na b : R\n⊢ ¬IsRightRegular 0 ↔ Nontrivial R",
"tactic": "rw [nontrivial_iff, not_iff_comm, isRightRegular_zero_iff_subsingleton, subsingleton_iff]"
},
{
"state_after": "R : Type u_1\ninst✝ : MulZeroClass R\na b : R\n⊢ (∀ (x y : R), x = y) ↔ ∀ (x y : R), x = y",
"state_before": "R : Type u_1\ninst✝ : MulZeroClass R\na b : R\n⊢ (¬∃ x y, x ≠ y) ↔ ∀ (x y : R), x = y",
"tactic": "push_neg"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : MulZeroClass R\na b : R\n⊢ (∀ (x y : R), x = y) ↔ ∀ (x y : R), x = y",
"tactic": "exact Iff.rfl"
}
] |
[
228,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.Fork.ι_ofι
|
[] |
[
383,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
382,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.cos_neg_iff_pi_div_two_lt_abs_toReal
|
[
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ cos θ < 0 ↔ π / 2 < abs (toReal θ)",
"tactic": "rw [← not_le, ← not_le, not_iff_not, cos_nonneg_iff_abs_toReal_le_pi_div_two]"
}
] |
[
759,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
758,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
IsCompact.elim_nhds_subcover'
|
[] |
[
204,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Data/Set/Sups.lean
|
Set.sups_empty
|
[] |
[
138,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Data/Int/ModEq.lean
|
Int.ModEq.sub_left
|
[] |
[
191,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
21
] |
Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
|
MeasureTheory.FiniteMeasure.testAgainstNN_eq_mass_mul
|
[
{
"state_after": "Ω : Type u_1\ninst✝¹ : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ninst✝ : TopologicalSpace Ω\nf : Ω →ᵇ ℝ≥0\n⊢ testAgainstNN (mass μ • ProbabilityMeasure.toFiniteMeasure (normalize μ)) f =\n mass μ * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize μ)) f",
"state_before": "Ω : Type u_1\ninst✝¹ : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ninst✝ : TopologicalSpace Ω\nf : Ω →ᵇ ℝ≥0\n⊢ testAgainstNN μ f = mass μ * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize μ)) f",
"tactic": "nth_rw 1 [μ.self_eq_mass_smul_normalize]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\ninst✝¹ : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ninst✝ : TopologicalSpace Ω\nf : Ω →ᵇ ℝ≥0\n⊢ testAgainstNN (mass μ • ProbabilityMeasure.toFiniteMeasure (normalize μ)) f =\n mass μ * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize μ)) f",
"tactic": "rw [μ.normalize.toFiniteMeasure.smul_testAgainstNN_apply μ.mass f, smul_eq_mul]"
}
] |
[
403,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/Data/Finite/Card.lean
|
Finite.card_le_of_injective'
|
[] |
[
129,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.ball_mono
|
[] |
[
738,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
737,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.prod_eq_bot
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.188893\nι : Sort ?u.188896\nκ : ι → Sort ?u.188901\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns s₁ s₂ : LowerSet α\nt t₁ t₂ : LowerSet β\nx : α × β\n⊢ ↑(s ×ˢ t) = ↑⊥ ↔ ↑s = ↑⊥ ∨ ↑t = ↑⊥",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.188893\nι : Sort ?u.188896\nκ : ι → Sort ?u.188901\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns s₁ s₂ : LowerSet α\nt t₁ t₂ : LowerSet β\nx : α × β\n⊢ s ×ˢ t = ⊥ ↔ s = ⊥ ∨ t = ⊥",
"tactic": "simp_rw [SetLike.ext'_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.188893\nι : Sort ?u.188896\nκ : ι → Sort ?u.188901\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns s₁ s₂ : LowerSet α\nt t₁ t₂ : LowerSet β\nx : α × β\n⊢ ↑(s ×ˢ t) = ↑⊥ ↔ ↑s = ↑⊥ ∨ ↑t = ↑⊥",
"tactic": "exact prod_eq_empty_iff"
}
] |
[
1738,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1736,
1
] |
Mathlib/Data/Nat/Sqrt.lean
|
Nat.sqrt_eq_zero
|
[
{
"state_after": "n : ℕ\nh : sqrt n = 0\n⊢ 0 < 1",
"state_before": "n : ℕ\nh : sqrt n = 0\n⊢ sqrt n < 1",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nh : sqrt n = 0\n⊢ 0 < 1",
"tactic": "decide"
},
{
"state_after": "⊢ sqrt 0 = 0",
"state_before": "n : ℕ\n⊢ n = 0 → sqrt n = 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "⊢ sqrt 0 = 0",
"tactic": "simp"
}
] |
[
118,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Algebra/Order/Interval.lean
|
NonemptyInterval.snd_inv
|
[] |
[
495,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
494,
1
] |
Mathlib/GroupTheory/Subgroup/ZPowers.lean
|
Subgroup.zpowers_eq_closure
|
[
{
"state_after": "case h\nG : Type u_1\ninst✝² : Group G\nA : Type ?u.4059\ninst✝¹ : AddGroup A\nN : Type ?u.4065\ninst✝ : Group N\ng x✝ : G\n⊢ x✝ ∈ zpowers g ↔ x✝ ∈ closure {g}",
"state_before": "G : Type u_1\ninst✝² : Group G\nA : Type ?u.4059\ninst✝¹ : AddGroup A\nN : Type ?u.4065\ninst✝ : Group N\ng : G\n⊢ zpowers g = closure {g}",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nG : Type u_1\ninst✝² : Group G\nA : Type ?u.4059\ninst✝¹ : AddGroup A\nN : Type ?u.4065\ninst✝ : Group N\ng x✝ : G\n⊢ x✝ ∈ zpowers g ↔ x✝ ∈ closure {g}",
"tactic": "exact mem_closure_singleton.symm"
}
] |
[
45,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
LinearMap.isOrtho_zero_left
|
[
{
"state_after": "R : Type u_2\nR₁ : Type u_1\nR₂ : Type u_5\nR₃ : Type ?u.6685\nM : Type ?u.6688\nM₁ : Type u_3\nM₂ : Type u_4\nMₗ₁ : Type ?u.6697\nMₗ₁' : Type ?u.6700\nMₗ₂ : Type ?u.6703\nMₗ₂' : Type ?u.6706\nK : Type ?u.6709\nK₁ : Type ?u.6712\nK₂ : Type ?u.6715\nV : Type ?u.6718\nV₁ : Type ?u.6721\nV₂ : Type ?u.6724\nn : Type ?u.6727\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : Module R₁ M₁\ninst✝² : CommSemiring R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\nI₁ : R₁ →+* R\nI₂ : R₂ →+* R\nI₁' : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R\nx : M₂\n⊢ ↑(↑B 0) x = 0",
"state_before": "R : Type u_2\nR₁ : Type u_1\nR₂ : Type u_5\nR₃ : Type ?u.6685\nM : Type ?u.6688\nM₁ : Type u_3\nM₂ : Type u_4\nMₗ₁ : Type ?u.6697\nMₗ₁' : Type ?u.6700\nMₗ₂ : Type ?u.6703\nMₗ₂' : Type ?u.6706\nK : Type ?u.6709\nK₁ : Type ?u.6712\nK₂ : Type ?u.6715\nV : Type ?u.6718\nV₁ : Type ?u.6721\nV₂ : Type ?u.6724\nn : Type ?u.6727\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : Module R₁ M₁\ninst✝² : CommSemiring R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\nI₁ : R₁ →+* R\nI₂ : R₂ →+* R\nI₁' : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R\nx : M₂\n⊢ IsOrtho B 0 x",
"tactic": "dsimp only [IsOrtho]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₁ : Type u_1\nR₂ : Type u_5\nR₃ : Type ?u.6685\nM : Type ?u.6688\nM₁ : Type u_3\nM₂ : Type u_4\nMₗ₁ : Type ?u.6697\nMₗ₁' : Type ?u.6700\nMₗ₂ : Type ?u.6703\nMₗ₂' : Type ?u.6706\nK : Type ?u.6709\nK₁ : Type ?u.6712\nK₂ : Type ?u.6715\nV : Type ?u.6718\nV₁ : Type ?u.6721\nV₂ : Type ?u.6724\nn : Type ?u.6727\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : Module R₁ M₁\ninst✝² : CommSemiring R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\nI₁ : R₁ →+* R\nI₂ : R₂ →+* R\nI₁' : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R\nx : M₂\n⊢ ↑(↑B 0) x = 0",
"tactic": "rw [map_zero B, zero_apply]"
}
] |
[
71,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.uniformity_basis_edist
|
[] |
[
1187,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1185,
11
] |
Mathlib/Order/Hom/Basic.lean
|
WithBot.toDualTopEquiv_symm_coe
|
[] |
[
1254,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1252,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.neg_lt_of_tendsto_zero_of_pos
|
[
{
"state_after": "f : ℕ → ℝ\nhf : Tendsto f atTop (𝓝 0)\nr✝ : ℝ\nhr : 0 < r✝\nhg : Tendsto (fun x => -f x) atTop (𝓝 0)\n⊢ -ofSeq f < ↑r✝",
"state_before": "f : ℕ → ℝ\nhf : Tendsto f atTop (𝓝 0)\nr✝ : ℝ\nhr : 0 < r✝\nhg : Tendsto (fun x => -f x) atTop (𝓝 (-0))\n⊢ -ofSeq f < ↑r✝",
"tactic": "rw [neg_zero] at hg"
},
{
"state_after": "no goals",
"state_before": "f : ℕ → ℝ\nhf : Tendsto f atTop (𝓝 0)\nr✝ : ℝ\nhr : 0 < r✝\nhg : Tendsto (fun x => -f x) atTop (𝓝 0)\n⊢ -ofSeq f < ↑r✝",
"tactic": "exact lt_of_tendsto_zero_of_pos hg hr"
}
] |
[
209,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.strictAntiOn_arccos
|
[] |
[
370,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
ClosedEmbedding.isCompact_preimage
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.93816\nπ : ι → Type ?u.93821\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nf : α → β\nhf : ClosedEmbedding f\nK : Set β\nhK : IsCompact (K ∩ range f)\n⊢ IsCompact (f ⁻¹' K)",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.93816\nπ : ι → Type ?u.93821\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nf : α → β\nhf : ClosedEmbedding f\nK : Set β\nhK : IsCompact K\n⊢ IsCompact (f ⁻¹' K)",
"tactic": "replace hK := hK.inter_right hf.closed_range"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.93816\nπ : ι → Type ?u.93821\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nf : α → β\nhf : ClosedEmbedding f\nK : Set β\nhK : IsCompact (K ∩ range f)\n⊢ IsCompact (f ⁻¹' K)",
"tactic": "rwa [← hf.toInducing.isCompact_iff, image_preimage_eq_inter_range]"
}
] |
[
898,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
895,
1
] |
Mathlib/CategoryTheory/Equivalence.lean
|
CategoryTheory.Equivalence.cancel_counitInv_right_assoc'
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\ne : C ≌ D\nW X X' Y Y' Z : D\nf : W ⟶ X\ng : X ⟶ Y\nh : Y ⟶ Z\nf' : W ⟶ X'\ng' : X' ⟶ Y'\nh' : Y' ⟶ Z\n⊢ f ≫ g ≫ h ≫ (counitInv e).app Z = f' ≫ g' ≫ h' ≫ (counitInv e).app Z ↔ f ≫ g ≫ h = f' ≫ g' ≫ h'",
"tactic": "simp only [← Category.assoc, cancel_mono]"
}
] |
[
428,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
Submodule.ker_inr
|
[
{
"state_after": "no goals",
"state_before": "R : 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.318044\nM₆ : Type ?u.318047\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\n⊢ ker (inr R M M₂) = ⊥",
"tactic": "rw [ker, ← prod_bot, prod_comap_inr]"
}
] |
[
593,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
593,
1
] |
Mathlib/RingTheory/GradedAlgebra/Radical.lean
|
Ideal.IsHomogeneous.isPrime_iff
|
[] |
[
155,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.single_dotProduct
|
[
{
"state_after": "l : Type ?u.160745\nm : Type u_1\nn : Type ?u.160751\no : Type ?u.160754\nm' : o → Type ?u.160759\nn' : o → Type ?u.160764\nR : Type ?u.160767\nS : Type ?u.160770\nα : Type v\nβ : Type w\nγ : Type ?u.160777\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : NonUnitalNonAssocSemiring α\nu v w : m → α\nx : α\ni : m\nthis : ∀ (j : m), j ≠ i → Pi.single i x j * v j = 0\n⊢ Pi.single i x ⬝ᵥ v = x * v i",
"state_before": "l : Type ?u.160745\nm : Type u_1\nn : Type ?u.160751\no : Type ?u.160754\nm' : o → Type ?u.160759\nn' : o → Type ?u.160764\nR : Type ?u.160767\nS : Type ?u.160770\nα : Type v\nβ : Type w\nγ : Type ?u.160777\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : NonUnitalNonAssocSemiring α\nu v w : m → α\nx : α\ni : m\n⊢ Pi.single i x ⬝ᵥ v = x * v i",
"tactic": "have : ∀ (j) (_ : j ≠ i), Pi.single (f := fun _ => α) i x j * v j = 0 := fun j hij => by\n simp [Pi.single_eq_of_ne hij]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.160745\nm : Type u_1\nn : Type ?u.160751\no : Type ?u.160754\nm' : o → Type ?u.160759\nn' : o → Type ?u.160764\nR : Type ?u.160767\nS : Type ?u.160770\nα : Type v\nβ : Type w\nγ : Type ?u.160777\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : NonUnitalNonAssocSemiring α\nu v w : m → α\nx : α\ni : m\nthis : ∀ (j : m), j ≠ i → Pi.single i x j * v j = 0\n⊢ Pi.single i x ⬝ᵥ v = x * v i",
"tactic": "convert Finset.sum_eq_single i (fun j _ => this j) _ using 1 <;> simp"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.160745\nm : Type u_1\nn : Type ?u.160751\no : Type ?u.160754\nm' : o → Type ?u.160759\nn' : o → Type ?u.160764\nR : Type ?u.160767\nS : Type ?u.160770\nα : Type v\nβ : Type w\nγ : Type ?u.160777\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : NonUnitalNonAssocSemiring α\nu v w : m → α\nx : α\ni j : m\nhij : j ≠ i\n⊢ Pi.single i x j * v j = 0",
"tactic": "simp [Pi.single_eq_of_ne hij]"
}
] |
[
833,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
829,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.numeratorHom_inj
|
[
{
"state_after": "R : Type u_1\ninst✝¹ : Ring R\nS : Submonoid R\ninst✝ : OreSet S\nhS : S ≤ R⁰\nr₁ r₂ : R\nh : ∃ u v, r₂ * ↑u = r₁ * v ∧ ↑1 * ↑u = ↑1 * v\n⊢ r₁ = r₂",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\nS : Submonoid R\ninst✝ : OreSet S\nhS : S ≤ R⁰\nr₁ r₂ : R\nh : ↑numeratorHom r₁ = ↑numeratorHom r₂\n⊢ r₁ = r₂",
"tactic": "rw [numeratorHom_apply, numeratorHom_apply, oreDiv_eq_iff] at h"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\ninst✝¹ : Ring R\nS : Submonoid R\ninst✝ : OreSet S\nhS : S ≤ R⁰\nr₁ r₂ : R\nu : { x // x ∈ S }\nv : R\nh₁ : r₂ * ↑u = r₁ * v\nh₂ : ↑1 * ↑u = ↑1 * v\n⊢ r₁ = r₂",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\nS : Submonoid R\ninst✝ : OreSet S\nhS : S ≤ R⁰\nr₁ r₂ : R\nh : ∃ u v, r₂ * ↑u = r₁ * v ∧ ↑1 * ↑u = ↑1 * v\n⊢ r₁ = r₂",
"tactic": "rcases h with ⟨u, v, h₁, h₂⟩"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\ninst✝¹ : Ring R\nS : Submonoid R\ninst✝ : OreSet S\nhS : S ≤ R⁰\nr₁ r₂ : R\nu : { x // x ∈ S }\nv : R\nh₁ : r₂ * ↑u = r₁ * v\nh₂ : ↑u = v\n⊢ r₁ = r₂",
"state_before": "case intro.intro.intro\nR : Type u_1\ninst✝¹ : Ring R\nS : Submonoid R\ninst✝ : OreSet S\nhS : S ≤ R⁰\nr₁ r₂ : R\nu : { x // x ∈ S }\nv : R\nh₁ : r₂ * ↑u = r₁ * v\nh₂ : ↑1 * ↑u = ↑1 * v\n⊢ r₁ = r₂",
"tactic": "simp only [S.coe_one, one_mul] at h₂"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nR : Type u_1\ninst✝¹ : Ring R\nS : Submonoid R\ninst✝ : OreSet S\nhS : S ≤ R⁰\nr₁ r₂ : R\nu : { x // x ∈ S }\nv : R\nh₁ : r₂ * ↑u = r₁ * v\nh₂ : ↑u = v\n⊢ r₁ = r₂",
"tactic": "rwa [← h₂, mul_cancel_right_mem_nonZeroDivisors (hS (SetLike.coe_mem u)), eq_comm] at h₁"
}
] |
[
867,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
862,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean
|
DoubleCentralizer.algebraMap_toProd
|
[] |
[
406,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Topology/MetricSpace/IsometricSMul.lean
|
dist_mul_left
|
[] |
[
360,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
MeasureTheory.FiniteMeasure.testAgainstNN_mono
|
[
{
"state_after": "Ω : Type u_1\ninst✝⁵ : MeasurableSpace Ω\nR : Type ?u.49329\ninst✝⁴ : SMul R ℝ≥0\ninst✝³ : SMul R ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : TopologicalSpace Ω\nμ : FiniteMeasure Ω\nf g : Ω →ᵇ ℝ≥0\nf_le_g : ↑f ≤ ↑g\n⊢ (∫⁻ (ω : Ω), ↑(↑f ω) ∂↑μ) ≤ ∫⁻ (ω : Ω), ↑(↑g ω) ∂↑μ",
"state_before": "Ω : Type u_1\ninst✝⁵ : MeasurableSpace Ω\nR : Type ?u.49329\ninst✝⁴ : SMul R ℝ≥0\ninst✝³ : SMul R ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : TopologicalSpace Ω\nμ : FiniteMeasure Ω\nf g : Ω →ᵇ ℝ≥0\nf_le_g : ↑f ≤ ↑g\n⊢ testAgainstNN μ f ≤ testAgainstNN μ g",
"tactic": "simp only [← ENNReal.coe_le_coe, testAgainstNN_coe_eq]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\ninst✝⁵ : MeasurableSpace Ω\nR : Type ?u.49329\ninst✝⁴ : SMul R ℝ≥0\ninst✝³ : SMul R ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : TopologicalSpace Ω\nμ : FiniteMeasure Ω\nf g : Ω →ᵇ ℝ≥0\nf_le_g : ↑f ≤ ↑g\n⊢ (∫⁻ (ω : Ω), ↑(↑f ω) ∂↑μ) ≤ ∫⁻ (ω : Ω), ↑(↑g ω) ∂↑μ",
"tactic": "exact lintegral_mono fun ω => ENNReal.coe_mono (f_le_g ω)"
}
] |
[
350,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
347,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.support_prod_of_pairwise_disjoint
|
[
{
"state_after": "case nil\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : List.Pairwise Disjoint l\nh : List.Pairwise Disjoint []\n⊢ support (List.prod []) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support [])\n\ncase cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nhl : List.Pairwise Disjoint tl → support (List.prod tl) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)\nh : List.Pairwise Disjoint (hd :: tl)\n⊢ support (List.prod (hd :: tl)) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support (hd :: tl))",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh : List.Pairwise Disjoint l\n⊢ support (List.prod l) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support l)",
"tactic": "induction' l with hd tl hl"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : List.Pairwise Disjoint l\nh : List.Pairwise Disjoint []\n⊢ support (List.prod []) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support [])",
"tactic": "simp"
},
{
"state_after": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nhl : List.Pairwise Disjoint tl → support (List.prod tl) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)\nh : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\n⊢ support (List.prod (hd :: tl)) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support (hd :: tl))",
"state_before": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nhl : List.Pairwise Disjoint tl → support (List.prod tl) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)\nh : List.Pairwise Disjoint (hd :: tl)\n⊢ support (List.prod (hd :: tl)) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support (hd :: tl))",
"tactic": "rw [List.pairwise_cons] at h"
},
{
"state_after": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nhl : List.Pairwise Disjoint tl → support (List.prod tl) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)\nh : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\nthis : Disjoint hd (List.prod tl)\n⊢ support (List.prod (hd :: tl)) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support (hd :: tl))",
"state_before": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nhl : List.Pairwise Disjoint tl → support (List.prod tl) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)\nh : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\n⊢ support (List.prod (hd :: tl)) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support (hd :: tl))",
"tactic": "have : Disjoint hd tl.prod := disjoint_prod_right _ h.left"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\nh✝ : List.Pairwise Disjoint l\nhd : Perm α\ntl : List (Perm α)\nhl : List.Pairwise Disjoint tl → support (List.prod tl) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)\nh : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl\nthis : Disjoint hd (List.prod tl)\n⊢ support (List.prod (hd :: tl)) = List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support (hd :: tl))",
"tactic": "simp [this.support_mul, hl h.right]"
}
] |
[
417,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
411,
1
] |
Mathlib/Logic/Basic.lean
|
ExistsUnique.intro₂
|
[
{
"state_after": "ι : Sort ?u.25148\nα✝ : Sort ?u.25153\nκ : ι → Sort ?u.25150\np✝ q✝ : α✝ → Prop\nα : Sort u_1\np : α → Sort u_2\ninst✝ : ∀ (x : α), Subsingleton (p x)\nq : (x : α) → p x → Prop\nw : α\nhp : p w\nhq : q w hp\nH : ∀ (y : α) (hy : p y), q y hy → y = w\n⊢ ∃! x, ∃ hx, q x hx",
"state_before": "ι : Sort ?u.25148\nα✝ : Sort ?u.25153\nκ : ι → Sort ?u.25150\np✝ q✝ : α✝ → Prop\nα : Sort u_1\np : α → Sort u_2\ninst✝ : ∀ (x : α), Subsingleton (p x)\nq : (x : α) → p x → Prop\nw : α\nhp : p w\nhq : q w hp\nH : ∀ (y : α) (hy : p y), q y hy → y = w\n⊢ ∃! x hx, q x hx",
"tactic": "simp only [exists_unique_iff_exists]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.25148\nα✝ : Sort ?u.25153\nκ : ι → Sort ?u.25150\np✝ q✝ : α✝ → Prop\nα : Sort u_1\np : α → Sort u_2\ninst✝ : ∀ (x : α), Subsingleton (p x)\nq : (x : α) → p x → Prop\nw : α\nhp : p w\nhq : q w hp\nH : ∀ (y : α) (hy : p y), q y hy → y = w\n⊢ ∃! x, ∃ hx, q x hx",
"tactic": "exact ExistsUnique.intro w ⟨hp, hq⟩ fun y ⟨hyp, hyq⟩ ↦ H y hyp hyq"
}
] |
[
942,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
938,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffAt.exists_lipschitzOnWith
|
[] |
[
2046,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2044,
1
] |
Mathlib/Data/List/Intervals.lean
|
List.Ico.eq_empty_iff
|
[
{
"state_after": "no goals",
"state_before": "n m : ℕ\nh : Ico n m = []\n⊢ m - n = 0",
"tactic": "rw [← length, h, List.length]"
}
] |
[
95,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Analysis/Calculus/TangentCone.lean
|
tangentCone_mono_nhds
|
[
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.28939\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.29029\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y✝ : E\ns t : Set E\nh : 𝓝[s] x ≤ 𝓝[t] x\ny : E\nc : ℕ → 𝕜\nd : ℕ → E\nds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nctop : Tendsto (fun n => ‖c n‖) atTop atTop\nclim : Tendsto (fun n => c n • d n) atTop (𝓝 y)\n⊢ y ∈ tangentConeAt 𝕜 t x",
"state_before": "𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.28939\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.29029\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y : E\ns t : Set E\nh : 𝓝[s] x ≤ 𝓝[t] x\n⊢ tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 t x",
"tactic": "rintro y ⟨c, d, ds, ctop, clim⟩"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.28939\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.29029\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y✝ : E\ns t : Set E\nh : 𝓝[s] x ≤ 𝓝[t] x\ny : E\nc : ℕ → 𝕜\nd : ℕ → E\nds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nctop : Tendsto (fun n => ‖c n‖) atTop atTop\nclim : Tendsto (fun n => c n • d n) atTop (𝓝 y)\n⊢ ∀ᶠ (n : ℕ) in atTop, x + d n ∈ t",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.28939\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.29029\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y✝ : E\ns t : Set E\nh : 𝓝[s] x ≤ 𝓝[t] x\ny : E\nc : ℕ → 𝕜\nd : ℕ → E\nds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nctop : Tendsto (fun n => ‖c n‖) atTop atTop\nclim : Tendsto (fun n => c n • d n) atTop (𝓝 y)\n⊢ y ∈ tangentConeAt 𝕜 t x",
"tactic": "refine' ⟨c, d, _, ctop, clim⟩"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.28939\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.29029\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y✝ : E\ns t : Set E\nh : 𝓝[s] x ≤ 𝓝[t] x\ny : E\nc : ℕ → 𝕜\nd : ℕ → E\nds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nctop : Tendsto (fun n => ‖c n‖) atTop atTop\nclim : Tendsto (fun n => c n • d n) atTop (𝓝 y)\nthis : Tendsto (fun n => x + d n) atTop (𝓝[t] x)\n⊢ ∀ᶠ (n : ℕ) in atTop, x + d n ∈ t\n\ncase this\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.28939\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.29029\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y✝ : E\ns t : Set E\nh : 𝓝[s] x ≤ 𝓝[t] x\ny : E\nc : ℕ → 𝕜\nd : ℕ → E\nds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nctop : Tendsto (fun n => ‖c n‖) atTop atTop\nclim : Tendsto (fun n => c n • d n) atTop (𝓝 y)\n⊢ Tendsto (fun n => x + d n) atTop (𝓝[t] x)",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.28939\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.29029\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y✝ : E\ns t : Set E\nh : 𝓝[s] x ≤ 𝓝[t] x\ny : E\nc : ℕ → 𝕜\nd : ℕ → E\nds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nctop : Tendsto (fun n => ‖c n‖) atTop atTop\nclim : Tendsto (fun n => c n • d n) atTop (𝓝 y)\n⊢ ∀ᶠ (n : ℕ) in atTop, x + d n ∈ t",
"tactic": "suffices : Tendsto (fun n => x + d n) atTop (𝓝[t] x)"
},
{
"state_after": "case this\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.28939\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.29029\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y✝ : E\ns t : Set E\nh : 𝓝[s] x ≤ 𝓝[t] x\ny : E\nc : ℕ → 𝕜\nd : ℕ → E\nds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nctop : Tendsto (fun n => ‖c n‖) atTop atTop\nclim : Tendsto (fun n => c n • d n) atTop (𝓝 y)\n⊢ Tendsto (fun n => x + d n) atTop (𝓝[t] x)",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.28939\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.29029\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y✝ : E\ns t : Set E\nh : 𝓝[s] x ≤ 𝓝[t] x\ny : E\nc : ℕ → 𝕜\nd : ℕ → E\nds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nctop : Tendsto (fun n => ‖c n‖) atTop atTop\nclim : Tendsto (fun n => c n • d n) atTop (𝓝 y)\nthis : Tendsto (fun n => x + d n) atTop (𝓝[t] x)\n⊢ ∀ᶠ (n : ℕ) in atTop, x + d n ∈ t\n\ncase this\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.28939\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.29029\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y✝ : E\ns t : Set E\nh : 𝓝[s] x ≤ 𝓝[t] x\ny : E\nc : ℕ → 𝕜\nd : ℕ → E\nds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nctop : Tendsto (fun n => ‖c n‖) atTop atTop\nclim : Tendsto (fun n => c n • d n) atTop (𝓝 y)\n⊢ Tendsto (fun n => x + d n) atTop (𝓝[t] x)",
"tactic": "exact tendsto_principal.1 (tendsto_inf.1 this).2"
},
{
"state_after": "case this\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.28939\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.29029\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y✝ : E\ns t : Set E\nh : 𝓝[s] x ≤ 𝓝[t] x\ny : E\nc : ℕ → 𝕜\nd : ℕ → E\nds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nctop : Tendsto (fun n => ‖c n‖) atTop atTop\nclim : Tendsto (fun n => c n • d n) atTop (𝓝 y)\n⊢ Tendsto (fun a => x + d a) atTop (𝓝 x)",
"state_before": "case this\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.28939\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.29029\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y✝ : E\ns t : Set E\nh : 𝓝[s] x ≤ 𝓝[t] x\ny : E\nc : ℕ → 𝕜\nd : ℕ → E\nds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nctop : Tendsto (fun n => ‖c n‖) atTop atTop\nclim : Tendsto (fun n => c n • d n) atTop (𝓝 y)\n⊢ Tendsto (fun n => x + d n) atTop (𝓝[t] x)",
"tactic": "refine' (tendsto_inf.2 ⟨_, tendsto_principal.2 ds⟩).mono_right h"
},
{
"state_after": "no goals",
"state_before": "case this\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.28939\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.29029\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y✝ : E\ns t : Set E\nh : 𝓝[s] x ≤ 𝓝[t] x\ny : E\nc : ℕ → 𝕜\nd : ℕ → E\nds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nctop : Tendsto (fun n => ‖c n‖) atTop atTop\nclim : Tendsto (fun n => c n • d n) atTop (𝓝 y)\n⊢ Tendsto (fun a => x + d a) atTop (𝓝 x)",
"tactic": "simpa only [add_zero] using tendsto_const_nhds.add (tangentConeAt.lim_zero atTop ctop clim)"
}
] |
[
135,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.IsImage.symm_mapsTo
|
[] |
[
384,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.card_Ioi_eq_card_Ici_sub_one
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.116791\nα : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrderTop α\na : α\n⊢ card (Ioi a) = card (Ici a) - 1",
"tactic": "rw [Ici_eq_cons_Ioi, card_cons, add_tsub_cancel_right]"
}
] |
[
703,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
702,
1
] |
Mathlib/LinearAlgebra/Determinant.lean
|
LinearMap.detAux_def''
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.143873\ninst✝¹¹ : CommRing R\nM : Type u_2\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.144464\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι : Type u_4\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ne : Basis ι R M\nA : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : Module A M\nκ : Type ?u.145982\ninst✝² : Fintype κ\nι' : Type u_1\ninst✝¹ : Fintype ι'\ninst✝ : DecidableEq ι'\ntb : Trunc (Basis ι A M)\nb' : Basis ι' A M\nf : M →ₗ[A] M\n⊢ ↑(detAux tb) f = det (↑(toMatrix b' b') f)",
"tactic": "induction tb using Trunc.induction_on with\n| h b => rw [detAux_def', det_toMatrix_eq_det_toMatrix b b']"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type ?u.143873\ninst✝¹¹ : CommRing R\nM : Type u_2\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.144464\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι : Type u_4\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ne : Basis ι R M\nA : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : Module A M\nκ : Type ?u.145982\ninst✝² : Fintype κ\nι' : Type u_1\ninst✝¹ : Fintype ι'\ninst✝ : DecidableEq ι'\nb' : Basis ι' A M\nf : M →ₗ[A] M\nb : Basis ι A M\n⊢ ↑(detAux (Trunc.mk b)) f = det (↑(toMatrix b' b') f)",
"tactic": "rw [detAux_def', det_toMatrix_eq_det_toMatrix b b']"
}
] |
[
158,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.ofReal_sub
|
[
{
"state_after": "case inl\nα : Type ?u.808531\nβ : Type ?u.808534\na b c d : ℝ≥0∞\nr p✝ q✝ : ℝ≥0\np q : ℝ\nhq : 0 ≤ q\nh : p ≤ q\n⊢ ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q\n\ncase inr\nα : Type ?u.808531\nβ : Type ?u.808534\na b c d : ℝ≥0∞\nr p✝ q✝ : ℝ≥0\np q : ℝ\nhq : 0 ≤ q\nh : q ≤ p\n⊢ ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q",
"state_before": "α : Type ?u.808531\nβ : Type ?u.808534\na b c d : ℝ≥0∞\nr p✝ q✝ : ℝ≥0\np q : ℝ\nhq : 0 ≤ q\n⊢ ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q",
"tactic": "obtain h | h := le_total p q"
},
{
"state_after": "case inr\nα : Type ?u.808531\nβ : Type ?u.808534\na b c d : ℝ≥0∞\nr p✝ q✝ : ℝ≥0\np q : ℝ\nhq : 0 ≤ q\nh : q ≤ p\n⊢ ENNReal.ofReal (p - q) + ENNReal.ofReal q = ENNReal.ofReal p",
"state_before": "case inr\nα : Type ?u.808531\nβ : Type ?u.808534\na b c d : ℝ≥0∞\nr p✝ q✝ : ℝ≥0\np q : ℝ\nhq : 0 ≤ q\nh : q ≤ p\n⊢ ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q",
"tactic": "refine' ENNReal.eq_sub_of_add_eq ofReal_ne_top _"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type ?u.808531\nβ : Type ?u.808534\na b c d : ℝ≥0∞\nr p✝ q✝ : ℝ≥0\np q : ℝ\nhq : 0 ≤ q\nh : q ≤ p\n⊢ ENNReal.ofReal (p - q) + ENNReal.ofReal q = ENNReal.ofReal p",
"tactic": "rw [← ofReal_add (sub_nonneg_of_le h) hq, sub_add_cancel]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type ?u.808531\nβ : Type ?u.808534\na b c d : ℝ≥0∞\nr p✝ q✝ : ℝ≥0\np q : ℝ\nhq : 0 ≤ q\nh : p ≤ q\n⊢ ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q",
"tactic": "rw [ofReal_of_nonpos (sub_nonpos_of_le h), tsub_eq_zero_of_le (ofReal_le_ofReal h)]"
}
] |
[
2134,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2129,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.cast_id
|
[] |
[
420,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.SimpleFunc.measure_lt_top_of_memℒp_indicator
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.1445144\nι : Type ?u.1445147\nE : Type u_1\nF : Type ?u.1445153\n𝕜 : Type ?u.1445156\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nμ : Measure α\np : ℝ≥0∞\nhp_pos : p ≠ 0\nhp_ne_top : p ≠ ⊤\nc : E\nhc : c ≠ 0\ns : Set α\nhs : MeasurableSet s\nhcs : Memℒp (↑(piecewise s hs (const α c) (const α 0))) p\nthis : support ↑(const α c) = Set.univ\n⊢ ↑↑μ s < ⊤",
"state_before": "α : Type u_2\nβ : Type ?u.1445144\nι : Type ?u.1445147\nE : Type u_1\nF : Type ?u.1445153\n𝕜 : Type ?u.1445156\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nμ : Measure α\np : ℝ≥0∞\nhp_pos : p ≠ 0\nhp_ne_top : p ≠ ⊤\nc : E\nhc : c ≠ 0\ns : Set α\nhs : MeasurableSet s\nhcs : Memℒp (↑(piecewise s hs (const α c) (const α 0))) p\n⊢ ↑↑μ s < ⊤",
"tactic": "have : Function.support (const α c) = Set.univ := Function.support_const hc"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.1445144\nι : Type ?u.1445147\nE : Type u_1\nF : Type ?u.1445153\n𝕜 : Type ?u.1445156\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nμ : Measure α\np : ℝ≥0∞\nhp_pos : p ≠ 0\nhp_ne_top : p ≠ ⊤\nc : E\nhc : c ≠ 0\ns : Set α\nhs : MeasurableSet s\nhcs : Memℒp (↑(piecewise s hs (const α c) (const α 0))) p\nthis : support ↑(const α c) = Set.univ\n⊢ ↑↑μ s < ⊤",
"tactic": "simpa only [memℒp_iff_finMeasSupp hp_pos hp_ne_top, finMeasSupp_iff_support,\n support_indicator, Set.inter_univ, this] using hcs"
}
] |
[
428,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.isClosed_zeroLocus
|
[
{
"state_after": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ns : Set R\n⊢ ∃ s_1, zeroLocus s = zeroLocus s_1",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ns : Set R\n⊢ IsClosed (zeroLocus s)",
"tactic": "rw [isClosed_iff_zeroLocus]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ns : Set R\n⊢ ∃ s_1, zeroLocus s = zeroLocus s_1",
"tactic": "exact ⟨s, rfl⟩"
}
] |
[
440,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
438,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
div_inv_eq_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.28599\nG : Type ?u.28602\ninst✝ : DivisionMonoid α\na b c : α\n⊢ a / b⁻¹ = a * b",
"tactic": "simp"
}
] |
[
467,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
467,
1
] |
Mathlib/Algebra/Group/Conj.lean
|
conj_inv
|
[] |
[
102,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Algebra/Algebra/Tower.lean
|
Submodule.map_mem_span_algebraMap_image
|
[
{
"state_after": "R : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring S✝\ninst✝¹⁰ : AddCommMonoid A\ninst✝⁹ : Algebra R S✝\ninst✝⁸ : Module S✝ A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R S✝ A\nS : Type u_1\nT : Type u_2\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Semiring T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx : S\na : Set S\nhx : x ∈ span R a\n⊢ ∃ y, y ∈ span R a ∧ ↑(↑R (Algebra.linearMap S T)) y = ↑(algebraMap S T) x",
"state_before": "R : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring S✝\ninst✝¹⁰ : AddCommMonoid A\ninst✝⁹ : Algebra R S✝\ninst✝⁸ : Module S✝ A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R S✝ A\nS : Type u_1\nT : Type u_2\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Semiring T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx : S\na : Set S\nhx : x ∈ span R a\n⊢ ↑(algebraMap S T) x ∈ span R (↑(algebraMap S T) '' a)",
"tactic": "rw [span_algebraMap_image_of_tower, mem_map]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring S✝\ninst✝¹⁰ : AddCommMonoid A\ninst✝⁹ : Algebra R S✝\ninst✝⁸ : Module S✝ A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R S✝ A\nS : Type u_1\nT : Type u_2\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Semiring T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx : S\na : Set S\nhx : x ∈ span R a\n⊢ ∃ y, y ∈ span R a ∧ ↑(↑R (Algebra.linearMap S T)) y = ↑(algebraMap S T) x",
"tactic": "exact ⟨x, hx, rfl⟩"
}
] |
[
347,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
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