file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Nat/Hyperoperation.lean
|
hyperoperation_zero
|
[
{
"state_after": "no goals",
"state_before": "m k : ℕ\n⊢ hyperoperation 0 m k = Nat.succ k",
"tactic": "rw [hyperoperation, Nat.succ_eq_add_one]"
}
] |
[
51,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
src/lean/Init/Core.lean
|
type_eq_of_heq
|
[] |
[
640,
20
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
639,
1
] |
Mathlib/MeasureTheory/Integral/CircleIntegral.lean
|
continuous_circleMap
|
[] |
[
189,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.power_bit0
|
[] |
[
548,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
547,
1
] |
Mathlib/Order/Basic.lean
|
LE.le.ge_iff_eq
|
[] |
[
280,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/Order/Monotone/Monovary.lean
|
MonovaryOn.comp_right
|
[] |
[
156,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/AlgebraicGeometry/PresheafedSpace/HasColimits.lean
|
AlgebraicGeometry.PresheafedSpace.colimit_carrier
|
[] |
[
194,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.rpow_le_rpow_of_exponent_ge
|
[
{
"state_after": "x y z : ℝ\nhx0 : 0 < x\nhx1 : x ≤ 1\nhyz : z ≤ y\n⊢ exp (log x * y) ≤ exp (log x * z)",
"state_before": "x y z : ℝ\nhx0 : 0 < x\nhx1 : x ≤ 1\nhyz : z ≤ y\n⊢ x ^ y ≤ x ^ z",
"tactic": "repeat' rw [rpow_def_of_pos hx0]"
},
{
"state_after": "x y z : ℝ\nhx0 : 0 < x\nhx1 : x ≤ 1\nhyz : z ≤ y\n⊢ log x * y ≤ log x * z",
"state_before": "x y z : ℝ\nhx0 : 0 < x\nhx1 : x ≤ 1\nhyz : z ≤ y\n⊢ exp (log x * y) ≤ exp (log x * z)",
"tactic": "rw [exp_le_exp]"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhx0 : 0 < x\nhx1 : x ≤ 1\nhyz : z ≤ y\n⊢ log x * y ≤ log x * z",
"tactic": "exact mul_le_mul_of_nonpos_left hyz (log_nonpos (le_of_lt hx0) hx1)"
},
{
"state_after": "x y z : ℝ\nhx0 : 0 < x\nhx1 : x ≤ 1\nhyz : z ≤ y\n⊢ exp (log x * y) ≤ exp (log x * z)",
"state_before": "x y z : ℝ\nhx0 : 0 < x\nhx1 : x ≤ 1\nhyz : z ≤ y\n⊢ exp (log x * y) ≤ exp (log x * z)",
"tactic": "rw [rpow_def_of_pos hx0]"
}
] |
[
499,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
HasFDerivWithinAt.add_const
|
[] |
[
189,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
8
] |
Mathlib/LinearAlgebra/Span.lean
|
LinearMap.ext_on
|
[] |
[
965,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
964,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.FTCFilter.finite_at_inner
|
[] |
[
221,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
strictMonoOn_pow
|
[] |
[
492,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
491,
1
] |
Mathlib/Data/Int/Log.lean
|
Int.clog_of_one_le_right
|
[] |
[
193,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.restrict_zero
|
[] |
[
1621,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1620,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.disjiUnion_val
|
[] |
[
3444,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3442,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.uniformEmbedding_iff'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.310325\ninst✝² : PseudoEMetricSpace α\nγ : Type w\ninst✝¹ : EMetricSpace γ\ninst✝ : EMetricSpace β\nf : γ → β\n⊢ UniformEmbedding f ↔\n (∀ (ε : ℝ≥0∞), ε > 0 → ∃ δ, δ > 0 ∧ ∀ {a b : γ}, edist a b < δ → edist (f a) (f b) < ε) ∧\n ∀ (δ : ℝ≥0∞), δ > 0 → ∃ ε, ε > 0 ∧ ∀ {a b : γ}, edist (f a) (f b) < ε → edist a b < δ",
"tactic": "rw [uniformEmbedding_iff_uniformInducing, uniformInducing_iff, uniformContinuous_iff]"
}
] |
[
1040,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1036,
1
] |
Mathlib/Algebra/Star/StarAlgHom.lean
|
NonUnitalStarAlgHom.zero_apply
|
[] |
[
281,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
tsum_empty
|
[] |
[
492,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
491,
1
] |
Mathlib/Computability/Language.lean
|
Language.nil_mem_one
|
[] |
[
109,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Data/Polynomial/Module.lean
|
PolynomialModule.zero_apply
|
[] |
[
78,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Analysis/Calculus/FDerivAnalytic.lean
|
HasFPowerSeriesAt.fderiv_eq
|
[] |
[
66,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzOnWith.of_le_add_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoMetricSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : PseudoMetricSpace γ\nK✝ : ℝ≥0\ns : Set α\nf✝ : α → β\nf : α → ℝ\nK : ℝ≥0\nh : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x ≤ f y + ↑K * dist x y\n⊢ LipschitzOnWith K f s",
"tactic": "simpa only [Real.toNNReal_coe] using LipschitzOnWith.of_le_add_mul' K h"
}
] |
[
558,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
11
] |
Mathlib/CategoryTheory/MorphismProperty.lean
|
CategoryTheory.MorphismProperty.naturalityProperty.stableUnderInverse
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y : C\ne : X ≅ Y\nhe : F₁.map e.hom ≫ app Y = app X ≫ F₂.map e.hom\n⊢ F₁.map e.inv ≫ app X = app Y ≫ F₂.map e.inv",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y : C\ne : X ≅ Y\nhe : naturalityProperty app e.hom\n⊢ naturalityProperty app e.inv",
"tactic": "simp only [naturalityProperty] at he⊢"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y : C\ne : X ≅ Y\nhe : F₁.map e.hom ≫ app Y = app X ≫ F₂.map e.hom\n⊢ F₁.map e.hom ≫ F₁.map e.inv ≫ app X = F₁.map e.hom ≫ app Y ≫ F₂.map e.inv",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y : C\ne : X ≅ Y\nhe : F₁.map e.hom ≫ app Y = app X ≫ F₂.map e.hom\n⊢ F₁.map e.inv ≫ app X = app Y ≫ F₂.map e.inv",
"tactic": "rw [← cancel_epi (F₁.map e.hom)]"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y : C\ne : X ≅ Y\nhe : F₁.map e.hom ≫ app Y = app X ≫ F₂.map e.hom\n⊢ F₁.map e.hom ≫ F₁.map e.inv ≫ app X = (app X ≫ F₂.map e.hom) ≫ F₂.map e.inv",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y : C\ne : X ≅ Y\nhe : F₁.map e.hom ≫ app Y = app X ≫ F₂.map e.hom\n⊢ F₁.map e.hom ≫ F₁.map e.inv ≫ app X = F₁.map e.hom ≫ app Y ≫ F₂.map e.inv",
"tactic": "slice_rhs 1 2 => rw [he]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y : C\ne : X ≅ Y\nhe : F₁.map e.hom ≫ app Y = app X ≫ F₂.map e.hom\n⊢ F₁.map e.hom ≫ F₁.map e.inv ≫ app X = (app X ≫ F₂.map e.hom) ≫ F₂.map e.inv",
"tactic": "simp only [Category.assoc, ← F₁.map_comp_assoc, ← F₂.map_comp, e.hom_inv_id, Functor.map_id,\n Category.id_comp, Category.comp_id]"
}
] |
[
364,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
Mathlib/NumberTheory/LucasLehmer.lean
|
LucasLehmer.X.mul_snd
|
[] |
[
244,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/LinearAlgebra/Trace.lean
|
LinearMap.trace_eq_matrix_trace_of_finset
|
[
{
"state_after": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type ?u.216025\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb✝ : Basis ι R M\nc : Basis κ R M\ns : Finset M\nb : Basis { x // x ∈ s } R M\nf : M →ₗ[R] M\nthis : ∃ s, Nonempty (Basis { x // x ∈ s } R M)\n⊢ ↑(trace R M) f = Matrix.trace (↑(toMatrix b b) f)",
"state_before": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type ?u.216025\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb✝ : Basis ι R M\nc : Basis κ R M\ns : Finset M\nb : Basis { x // x ∈ s } R M\nf : M →ₗ[R] M\n⊢ ↑(trace R M) f = Matrix.trace (↑(toMatrix b b) f)",
"tactic": "have : ∃ s : Finset M, Nonempty (Basis s R M) := ⟨s, ⟨b⟩⟩"
},
{
"state_after": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type ?u.216025\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb✝ : Basis ι R M\nc : Basis κ R M\ns : Finset M\nb : Basis { x // x ∈ s } R M\nf : M →ₗ[R] M\nthis : ∃ s, Nonempty (Basis { x // x ∈ s } R M)\n⊢ ↑(traceAux R (Nonempty.some (_ : Nonempty (Basis { x // x ∈ Exists.choose this } R M)))) f = ↑(traceAux R b) f",
"state_before": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type ?u.216025\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb✝ : Basis ι R M\nc : Basis κ R M\ns : Finset M\nb : Basis { x // x ∈ s } R M\nf : M →ₗ[R] M\nthis : ∃ s, Nonempty (Basis { x // x ∈ s } R M)\n⊢ ↑(trace R M) f = Matrix.trace (↑(toMatrix b b) f)",
"tactic": "rw [trace, dif_pos this, ← traceAux_def]"
},
{
"state_after": "case e_a\nR : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type ?u.216025\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb✝ : Basis ι R M\nc : Basis κ R M\ns : Finset M\nb : Basis { x // x ∈ s } R M\nf : M →ₗ[R] M\nthis : ∃ s, Nonempty (Basis { x // x ∈ s } R M)\n⊢ traceAux R (Nonempty.some (_ : Nonempty (Basis { x // x ∈ Exists.choose this } R M))) = traceAux R b",
"state_before": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type ?u.216025\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb✝ : Basis ι R M\nc : Basis κ R M\ns : Finset M\nb : Basis { x // x ∈ s } R M\nf : M →ₗ[R] M\nthis : ∃ s, Nonempty (Basis { x // x ∈ s } R M)\n⊢ ↑(traceAux R (Nonempty.some (_ : Nonempty (Basis { x // x ∈ Exists.choose this } R M)))) f = ↑(traceAux R b) f",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case e_a\nR : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type ?u.216025\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb✝ : Basis ι R M\nc : Basis κ R M\ns : Finset M\nb : Basis { x // x ∈ s } R M\nf : M →ₗ[R] M\nthis : ∃ s, Nonempty (Basis { x // x ∈ s } R M)\n⊢ traceAux R (Nonempty.some (_ : Nonempty (Basis { x // x ∈ Exists.choose this } R M))) = traceAux R b",
"tactic": "apply traceAux_eq"
}
] |
[
102,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
sup_sdiff_left_self
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.137716\nα : Type u_1\nβ : Type ?u.137722\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\n⊢ (a ⊔ b) \\ a = b \\ a",
"tactic": "rw [sup_comm, sup_sdiff_right_self]"
}
] |
[
662,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
662,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean
|
NormedSpace.unbounded_univ
|
[] |
[
439,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
11
] |
Mathlib/Analysis/Normed/Group/Pointwise.lean
|
ball_div
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ ball x δ / s = x • thickening δ s⁻¹",
"tactic": "simp [div_eq_mul_inv]"
}
] |
[
248,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.natDegree_of_c_ne_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.591482\nF : Type ?u.591485\nK : Type ?u.591488\nP Q : Cubic R\na b c d a' b' c' d' : R\ninst✝ : Semiring R\nha : P.a = 0\nhb : P.b = 0\nhc : P.c ≠ 0\n⊢ natDegree (toPoly P) = 1",
"tactic": "rw [of_b_eq_zero ha hb, natDegree_linear hc]"
}
] |
[
423,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
EuclideanGeometry.sin_angle_of_angle_eq_pi_div_two
|
[
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₁ ≠ p₂ ∨ p₃ ≠ p₂\n⊢ Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₁ p₃",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ∠ p₁ p₂ p₃ = π / 2\nh0 : p₁ ≠ p₂ ∨ p₃ ≠ p₂\n⊢ Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₁ p₃",
"tactic": "rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←\n inner_neg_left, neg_vsub_eq_vsub_rev] at h"
},
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₁ p₃",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₁ ≠ p₂ ∨ p₃ ≠ p₂\n⊢ Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₁ p₃",
"tactic": "rw [← @vsub_ne_zero V, @ne_comm _ p₃, ← @vsub_ne_zero V _ _ _ p₂, or_comm] at h0"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₁ p₃",
"tactic": "rw [angle, dist_eq_norm_vsub V p₁ p₂, dist_eq_norm_vsub V p₁ p₃, ← vsub_add_vsub_cancel p₁ p₂ p₃,\n add_comm, sin_angle_add_of_inner_eq_zero h h0]"
}
] |
[
449,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measure_union_add_inter'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.6873\nγ : Type ?u.6876\nδ : Type ?u.6879\nι : Type ?u.6882\nR : Type ?u.6885\nR' : Type ?u.6888\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nhs : MeasurableSet s\nt : Set α\n⊢ ↑↑μ (s ∪ t) + ↑↑μ (s ∩ t) = ↑↑μ s + ↑↑μ t",
"tactic": "rw [union_comm, inter_comm, measure_union_add_inter t hs, add_comm]"
}
] |
[
153,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Cardinal.lt_univ'
|
[
{
"state_after": "α : Type ?u.235536\nβ : Type ?u.235539\nγ : Type ?u.235542\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\nh : c < univ\na : Cardinal\ne : lift a = c\nh' : a < (#Ordinal)\n⊢ ∃ c', c = lift c'",
"state_before": "α : Type ?u.235536\nβ : Type ?u.235539\nγ : Type ?u.235542\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\nh : c < univ\n⊢ ∃ c', c = lift c'",
"tactic": "let ⟨a, e, h'⟩ := lt_lift_iff.1 h"
},
{
"state_after": "α : Type ?u.235536\nβ : Type ?u.235539\nγ : Type ?u.235542\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\nh : c < univ\na : Cardinal\ne : lift a = c\nh' : a < univ\n⊢ ∃ c', c = lift c'",
"state_before": "α : Type ?u.235536\nβ : Type ?u.235539\nγ : Type ?u.235542\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\nh : c < univ\na : Cardinal\ne : lift a = c\nh' : a < (#Ordinal)\n⊢ ∃ c', c = lift c'",
"tactic": "rw [← univ_id] at h'"
},
{
"state_after": "case intro\nα : Type ?u.235536\nβ : Type ?u.235539\nγ : Type ?u.235542\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\nh : c < univ\nc' : Cardinal\ne : lift (lift c') = c\nh' : lift c' < univ\n⊢ ∃ c', c = lift c'",
"state_before": "α : Type ?u.235536\nβ : Type ?u.235539\nγ : Type ?u.235542\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\nh : c < univ\na : Cardinal\ne : lift a = c\nh' : a < univ\n⊢ ∃ c', c = lift c'",
"tactic": "rcases lt_univ.{u}.1 h' with ⟨c', rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type ?u.235536\nβ : Type ?u.235539\nγ : Type ?u.235542\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\nh : c < univ\nc' : Cardinal\ne : lift (lift c') = c\nh' : lift c' < univ\n⊢ ∃ c', c = lift c'",
"tactic": "exact ⟨c', by simp only [e.symm, lift_lift]⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.235536\nβ : Type ?u.235539\nγ : Type ?u.235542\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc : Cardinal\nh : c < univ\nc' : Cardinal\ne : lift (lift c') = c\nh' : lift c' < univ\n⊢ c = lift c'",
"tactic": "simp only [e.symm, lift_lift]"
}
] |
[
1537,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1532,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.length_pos_of_pos
|
[
{
"state_after": "case h\nn : ℕ\nc : Composition n\nh : 0 < n\n⊢ 0 < sum c.blocks",
"state_before": "n : ℕ\nc : Composition n\nh : 0 < n\n⊢ 0 < length c",
"tactic": "apply length_pos_of_sum_pos"
},
{
"state_after": "case h.e'_4\nn : ℕ\nc : Composition n\nh : 0 < n\n⊢ sum c.blocks = n",
"state_before": "case h\nn : ℕ\nc : Composition n\nh : 0 < n\n⊢ 0 < sum c.blocks",
"tactic": "convert h"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nn : ℕ\nc : Composition n\nh : 0 < n\n⊢ sum c.blocks = n",
"tactic": "exact c.blocks_sum"
}
] |
[
209,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.hom.injective
|
[] |
[
743,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
742,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.piecewise_empty
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.71934\nγ : Type ?u.71937\nι : Sort ?u.71940\nπ : α → Type ?u.71945\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝ : (i : α) → Decidable (i ∈ ∅)\ni : α\n⊢ piecewise ∅ f g i = g i",
"state_before": "α : Type u_1\nβ : Type ?u.71934\nγ : Type ?u.71937\nι : Sort ?u.71940\nπ : α → Type ?u.71945\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝ : (i : α) → Decidable (i ∈ ∅)\n⊢ piecewise ∅ f g = g",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.71934\nγ : Type ?u.71937\nι : Sort ?u.71940\nπ : α → Type ?u.71945\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝ : (i : α) → Decidable (i ∈ ∅)\ni : α\n⊢ piecewise ∅ f g i = g i",
"tactic": "simp [piecewise]"
}
] |
[
1352,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1350,
1
] |
Mathlib/Computability/PartrecCode.lean
|
Nat.Partrec.Code.eval_const
|
[
{
"state_after": "no goals",
"state_before": "n m : ℕ\n⊢ eval (Code.const (n + 1)) m = Part.some (n + 1)",
"tactic": "simp! [eval_const n m]"
}
] |
[
662,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
660,
1
] |
Mathlib/Logic/Relation.lean
|
Relation.reflexive_join
|
[] |
[
629,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
629,
1
] |
Std/Data/Int/DivMod.lean
|
Int.mul_div_cancel
|
[
{
"state_after": "no goals",
"state_before": "a b : Int\nH✝ : b ≠ 0\na✝ b✝ : Nat\nH : ↑b✝ ≠ 0\n⊢ div (↑a✝ * ↑b✝) ↑b✝ = ↑a✝",
"tactic": "rw [← ofNat_mul, ← ofNat_div,\n Nat.mul_div_cancel _ <| Nat.pos_of_ne_zero <| Int.ofNat_ne_zero.1 H]"
},
{
"state_after": "no goals",
"state_before": "a✝ b✝ : Int\nthis : ∀ {a b : Nat}, ↑b ≠ 0 → div (↑a * ↑b) ↑b = ↑a\na b : Nat\nH : -↑b ≠ 0\n⊢ div (↑a * -↑b) (-↑b) = ↑a",
"tactic": "rw [Int.mul_neg, Int.neg_div, Int.div_neg, Int.neg_neg,\n this (Int.neg_ne_zero.1 H)]"
},
{
"state_after": "no goals",
"state_before": "a✝ b✝ : Int\nthis : ∀ {a b : Nat}, ↑b ≠ 0 → div (↑a * ↑b) ↑b = ↑a\na b : Nat\nH : ↑b ≠ 0\n⊢ div (-↑a * ↑b) ↑b = -↑a",
"tactic": "rw [Int.neg_mul, Int.neg_div, this H]"
},
{
"state_after": "no goals",
"state_before": "a✝ b✝ : Int\nthis : ∀ {a b : Nat}, ↑b ≠ 0 → div (↑a * ↑b) ↑b = ↑a\na b : Nat\nH : -↑b ≠ 0\n⊢ div (-↑a * -↑b) (-↑b) = -↑a",
"tactic": "rw [Int.neg_mul_neg, Int.div_neg, this (Int.neg_ne_zero.1 H)]"
}
] |
[
211,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
200,
19
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.coe_comap
|
[] |
[
1317,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1317,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiffOn_clm_apply
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2800066\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\nn : ℕ∞\nf : E → F →L[𝕜] G\ns : Set E\ninst✝ : FiniteDimensional 𝕜 F\nh : ∀ (y : F), ContDiffOn 𝕜 n (fun x => ↑(f x) y) s\n⊢ ContDiffOn 𝕜 n f s",
"state_before": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2800066\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\nn : ℕ∞\nf : E → F →L[𝕜] G\ns : Set E\ninst✝ : FiniteDimensional 𝕜 F\n⊢ ContDiffOn 𝕜 n f s ↔ ∀ (y : F), ContDiffOn 𝕜 n (fun x => ↑(f x) y) s",
"tactic": "refine' ⟨fun h y => h.clm_apply contDiffOn_const, fun h => _⟩"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2800066\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\nn : ℕ∞\nf : E → F →L[𝕜] G\ns : Set E\ninst✝ : FiniteDimensional 𝕜 F\nh : ∀ (y : F), ContDiffOn 𝕜 n (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 F\n⊢ ContDiffOn 𝕜 n f s",
"state_before": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2800066\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\nn : ℕ∞\nf : E → F →L[𝕜] G\ns : Set E\ninst✝ : FiniteDimensional 𝕜 F\nh : ∀ (y : F), ContDiffOn 𝕜 n (fun x => ↑(f x) y) s\n⊢ ContDiffOn 𝕜 n f s",
"tactic": "let d := finrank 𝕜 F"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2800066\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\nn : ℕ∞\nf : E → F →L[𝕜] G\ns : Set E\ninst✝ : FiniteDimensional 𝕜 F\nh : ∀ (y : F), ContDiffOn 𝕜 n (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 F\nhd : d = finrank 𝕜 (Fin d → 𝕜)\n⊢ ContDiffOn 𝕜 n f s",
"state_before": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2800066\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\nn : ℕ∞\nf : E → F →L[𝕜] G\ns : Set E\ninst✝ : FiniteDimensional 𝕜 F\nh : ∀ (y : F), ContDiffOn 𝕜 n (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 F\n⊢ ContDiffOn 𝕜 n f s",
"tactic": "have hd : d = finrank 𝕜 (Fin d → 𝕜) := (finrank_fin_fun 𝕜).symm"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2800066\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\nn : ℕ∞\nf : E → F →L[𝕜] G\ns : Set E\ninst✝ : FiniteDimensional 𝕜 F\nh : ∀ (y : F), ContDiffOn 𝕜 n (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 F\nhd : d = finrank 𝕜 (Fin d → 𝕜)\ne₁ : F ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd\n⊢ ContDiffOn 𝕜 n f s",
"state_before": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2800066\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\nn : ℕ∞\nf : E → F →L[𝕜] G\ns : Set E\ninst✝ : FiniteDimensional 𝕜 F\nh : ∀ (y : F), ContDiffOn 𝕜 n (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 F\nhd : d = finrank 𝕜 (Fin d → 𝕜)\n⊢ ContDiffOn 𝕜 n f s",
"tactic": "let e₁ := ContinuousLinearEquiv.ofFinrankEq hd"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2800066\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\nn : ℕ∞\nf : E → F →L[𝕜] G\ns : Set E\ninst✝ : FiniteDimensional 𝕜 F\nh : ∀ (y : F), ContDiffOn 𝕜 n (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 F\nhd : d = finrank 𝕜 (Fin d → 𝕜)\ne₁ : F ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd\ne₂ : (F →L[𝕜] G) ≃L[𝕜] Fin d → G :=\n ContinuousLinearEquiv.trans (ContinuousLinearEquiv.arrowCongr e₁ 1) (ContinuousLinearEquiv.piRing (Fin d))\n⊢ ContDiffOn 𝕜 n f s",
"state_before": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2800066\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\nn : ℕ∞\nf : E → F →L[𝕜] G\ns : Set E\ninst✝ : FiniteDimensional 𝕜 F\nh : ∀ (y : F), ContDiffOn 𝕜 n (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 F\nhd : d = finrank 𝕜 (Fin d → 𝕜)\ne₁ : F ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd\n⊢ ContDiffOn 𝕜 n f s",
"tactic": "let e₂ := (e₁.arrowCongr (1 : G ≃L[𝕜] G)).trans (ContinuousLinearEquiv.piRing (Fin d))"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2800066\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\nn : ℕ∞\nf : E → F →L[𝕜] G\ns : Set E\ninst✝ : FiniteDimensional 𝕜 F\nh : ∀ (y : F), ContDiffOn 𝕜 n (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 F\nhd : d = finrank 𝕜 (Fin d → 𝕜)\ne₁ : F ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd\ne₂ : (F →L[𝕜] G) ≃L[𝕜] Fin d → G :=\n ContinuousLinearEquiv.trans (ContinuousLinearEquiv.arrowCongr e₁ 1) (ContinuousLinearEquiv.piRing (Fin d))\n⊢ ContDiffOn 𝕜 n ((↑(ContinuousLinearEquiv.symm e₂) ∘ ↑e₂) ∘ f) s",
"state_before": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2800066\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\nn : ℕ∞\nf : E → F →L[𝕜] G\ns : Set E\ninst✝ : FiniteDimensional 𝕜 F\nh : ∀ (y : F), ContDiffOn 𝕜 n (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 F\nhd : d = finrank 𝕜 (Fin d → 𝕜)\ne₁ : F ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd\ne₂ : (F →L[𝕜] G) ≃L[𝕜] Fin d → G :=\n ContinuousLinearEquiv.trans (ContinuousLinearEquiv.arrowCongr e₁ 1) (ContinuousLinearEquiv.piRing (Fin d))\n⊢ ContDiffOn 𝕜 n f s",
"tactic": "rw [← comp.left_id f, ← e₂.symm_comp_self]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹¹ : NormedAddCommGroup D\ninst✝¹⁰ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\nX : Type ?u.2800066\ninst✝³ : NormedAddCommGroup X\ninst✝² : NormedSpace 𝕜 X\ns✝ s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝¹ : CompleteSpace 𝕜\nn : ℕ∞\nf : E → F →L[𝕜] G\ns : Set E\ninst✝ : FiniteDimensional 𝕜 F\nh : ∀ (y : F), ContDiffOn 𝕜 n (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 F\nhd : d = finrank 𝕜 (Fin d → 𝕜)\ne₁ : F ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd\ne₂ : (F →L[𝕜] G) ≃L[𝕜] Fin d → G :=\n ContinuousLinearEquiv.trans (ContinuousLinearEquiv.arrowCongr e₁ 1) (ContinuousLinearEquiv.piRing (Fin d))\n⊢ ContDiffOn 𝕜 n ((↑(ContinuousLinearEquiv.symm e₂) ∘ ↑e₂) ∘ f) s",
"tactic": "exact e₂.symm.contDiff.comp_contDiffOn (contDiffOn_pi.mpr fun i => h _)"
}
] |
[
1888,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1880,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_image
|
[] |
[
367,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
pi_norm_const_le'
|
[] |
[
2523,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2522,
1
] |
Mathlib/Algebra/Ring/Prod.lean
|
NonUnitalRingHom.coe_fst
|
[] |
[
119,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Data/Set/Intervals/ProjIcc.lean
|
Set.IccExtend_of_le_left
|
[] |
[
115,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/RingTheory/Coprime/Basic.lean
|
isCoprime_zero_left
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nx✝ : IsCoprime 0 x\na b : R\nH : a * 0 + b * x = 1\n⊢ x * b = 1",
"tactic": "rwa [mul_zero, zero_add, mul_comm] at H"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nH : IsUnit x\nb : R\nhb : b * x = 1\n⊢ 1 * 0 + b * x = 1",
"tactic": "rwa [one_mul, zero_add]"
}
] |
[
65,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Data/Dfinsupp/NeLocus.lean
|
Dfinsupp.not_mem_neLocus
|
[] |
[
50,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Data/Bool/Basic.lean
|
Bool.or_inl
|
[
{
"state_after": "no goals",
"state_before": "a b : Bool\nH : a = true\n⊢ (a || b) = true",
"tactic": "simp [H]"
}
] |
[
159,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Std/Data/List/Lemmas.lean
|
List.tailD_eq_tail?
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl l' : List α\n⊢ tailD l l' = Option.getD (tail? l) l'",
"tactic": "cases l <;> rfl"
}
] |
[
448,
18
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
447,
9
] |
Mathlib/Logic/Basic.lean
|
ExistsUnique.elim₂
|
[
{
"state_after": "ι : Sort ?u.24682\nα✝ : Sort ?u.24687\nκ : ι → Sort ?u.24684\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\nb : Prop\nh₁ : ∀ (x : α) (h : p x), q x h → (∀ (y : α) (hy : p y), q y hy → y = x) → b\nh₂ : ∃! x, ∃ h, q x h\n⊢ b",
"state_before": "ι : Sort ?u.24682\nα✝ : Sort ?u.24687\nκ : ι → Sort ?u.24684\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\nb : Prop\nh₂ : ∃! x h, q x h\nh₁ : ∀ (x : α) (h : p x), q x h → (∀ (y : α) (hy : p y), q y hy → y = x) → b\n⊢ b",
"tactic": "simp only [exists_unique_iff_exists] at h₂"
},
{
"state_after": "ι : Sort ?u.24682\nα✝ : Sort ?u.24687\nκ : ι → Sort ?u.24684\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\nb : Prop\nh₁ : ∀ (x : α) (h : p x), q x h → (∀ (y : α) (hy : p y), q y hy → y = x) → b\nh₂ : ∃! x, ∃ h, q x h\n⊢ ∀ (x : α), (∃ h, q x h) → (∀ (y : α), (∃ h, q y h) → y = x) → b",
"state_before": "ι : Sort ?u.24682\nα✝ : Sort ?u.24687\nκ : ι → Sort ?u.24684\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\nb : Prop\nh₁ : ∀ (x : α) (h : p x), q x h → (∀ (y : α) (hy : p y), q y hy → y = x) → b\nh₂ : ∃! x, ∃ h, q x h\n⊢ b",
"tactic": "apply h₂.elim"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.24682\nα✝ : Sort ?u.24687\nκ : ι → Sort ?u.24684\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\nb : Prop\nh₁ : ∀ (x : α) (h : p x), q x h → (∀ (y : α) (hy : p y), q y hy → y = x) → b\nh₂ : ∃! x, ∃ h, q x h\n⊢ ∀ (x : α), (∃ h, q x h) → (∀ (y : α), (∃ h, q y h) → y = x) → b",
"tactic": "exact fun x ⟨hxp, hxq⟩ H ↦ h₁ x hxp hxq fun y hyp hyq ↦ H y ⟨hyp, hyq⟩"
}
] |
[
935,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
930,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.isBridge_iff_mem_and_forall_cycle_not_mem
|
[] |
[
2536,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2534,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.eq_top_iff
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Monoid α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\n⊢ (∀ (n : ℕ), a ^ (n + 1) ∣ b) ↔ ∀ (n : ℕ), a ^ n ∣ b",
"state_before": "α : Type u_1\ninst✝¹ : Monoid α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\n⊢ (∀ (n : ℕ), ¬¬a ^ (n + 1) ∣ b) ↔ ∀ (n : ℕ), a ^ n ∣ b",
"tactic": "simp only [Classical.not_not]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Monoid α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\n⊢ (∀ (n : ℕ), a ^ (n + 1) ∣ b) ↔ ∀ (n : ℕ), a ^ n ∣ b",
"tactic": "exact\n ⟨fun h n =>\n Nat.casesOn n\n (by\n rw [_root_.pow_zero]\n exact one_dvd _)\n fun n => h _,\n fun h n => h _⟩"
},
{
"state_after": "α : Type u_1\ninst✝¹ : Monoid α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nh : ∀ (n : ℕ), a ^ (n + 1) ∣ b\nn : ℕ\n⊢ 1 ∣ b",
"state_before": "α : Type u_1\ninst✝¹ : Monoid α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nh : ∀ (n : ℕ), a ^ (n + 1) ∣ b\nn : ℕ\n⊢ a ^ zero ∣ b",
"tactic": "rw [_root_.pow_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Monoid α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nh : ∀ (n : ℕ), a ^ (n + 1) ∣ b\nn : ℕ\n⊢ 1 ∣ b",
"tactic": "exact one_dvd _"
}
] |
[
180,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.join_principal_eq_sSup
|
[] |
[
665,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
665,
9
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.ppow_apply
|
[
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nf : ArithmeticFunction R\nk x : ℕ\nkpos : 0 < k\n⊢ ↑{ toFun := fun x => ↑f x ^ k, map_zero' := (_ : (fun x => ↑f x ^ k) 0 = 0) } x = ↑f x ^ k",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf : ArithmeticFunction R\nk x : ℕ\nkpos : 0 < k\n⊢ ↑(ppow f k) x = ↑f x ^ k",
"tactic": "rw [ppow, dif_neg (ne_of_gt kpos)]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf : ArithmeticFunction R\nk x : ℕ\nkpos : 0 < k\n⊢ ↑{ toFun := fun x => ↑f x ^ k, map_zero' := (_ : (fun x => ↑f x ^ k) 0 = 0) } x = ↑f x ^ k",
"tactic": "rfl"
}
] |
[
557,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
555,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.constr_def
|
[] |
[
626,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
624,
1
] |
Mathlib/Data/Set/Intervals/Instances.lean
|
Set.Icc.coe_one
|
[] |
[
66,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.SimpleFunc.norm_approxOn_zero_le
|
[
{
"state_after": "α : Type ?u.14817\nβ : Type u_2\nι : Type ?u.14823\nE : Type u_1\nF : Type ?u.14829\n𝕜 : Type ?u.14832\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\nq : ℝ\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\nh₀ : 0 ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\nthis : edist 0 (↑(approxOn f hf s 0 h₀ n) x) ≤ edist 0 (f x) + edist 0 (f x)\n⊢ ‖↑(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖",
"state_before": "α : Type ?u.14817\nβ : Type u_2\nι : Type ?u.14823\nE : Type u_1\nF : Type ?u.14829\n𝕜 : Type ?u.14832\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\nq : ℝ\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\nh₀ : 0 ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\n⊢ ‖↑(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖",
"tactic": "have := edist_approxOn_y0_le hf h₀ x n"
},
{
"state_after": "α : Type ?u.14817\nβ : Type u_2\nι : Type ?u.14823\nE : Type u_1\nF : Type ?u.14829\n𝕜 : Type ?u.14832\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\nq : ℝ\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\nh₀ : 0 ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\nthis : ↑‖↑(approxOn f hf s 0 h₀ n) x‖₊ ≤ ↑‖f x‖₊ + ↑‖f x‖₊\n⊢ ‖↑(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖",
"state_before": "α : Type ?u.14817\nβ : Type u_2\nι : Type ?u.14823\nE : Type u_1\nF : Type ?u.14829\n𝕜 : Type ?u.14832\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\nq : ℝ\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\nh₀ : 0 ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\nthis : edist 0 (↑(approxOn f hf s 0 h₀ n) x) ≤ edist 0 (f x) + edist 0 (f x)\n⊢ ‖↑(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖",
"tactic": "simp [edist_comm (0 : E), edist_eq_coe_nnnorm] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.14817\nβ : Type u_2\nι : Type ?u.14823\nE : Type u_1\nF : Type ?u.14829\n𝕜 : Type ?u.14832\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\nq : ℝ\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\nh₀ : 0 ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\nthis : ↑‖↑(approxOn f hf s 0 h₀ n) x‖₊ ≤ ↑‖f x‖₊ + ↑‖f x‖₊\n⊢ ‖↑(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖",
"tactic": "exact_mod_cast this"
}
] |
[
95,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/MeasureTheory/MeasurableSpaceDef.lean
|
MeasurableSpace.generateFrom_singleton_empty
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.20176\nγ : Type ?u.20179\nδ : Type ?u.20182\nδ' : Type ?u.20185\nι : Sort ?u.20188\ns t u : Set α\n⊢ ∀ (t : Set α), t ∈ {∅} → MeasurableSet t",
"tactic": "simp [@MeasurableSet.empty α ⊥]"
}
] |
[
443,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/ModelTheory/LanguageMap.lean
|
FirstOrder.Language.withConstants_funMap_sum_inr
|
[
{
"state_after": "L : Language\nL' : Language\nM : Type w\ninst✝¹ : Structure L M\nα : Type u_1\ninst✝ : Structure (constantsOn α) M\na : α\nx : Fin 0 → M\n⊢ funMap (Sum.inr a) default = ↑(Language.con L a)",
"state_before": "L : Language\nL' : Language\nM : Type w\ninst✝¹ : Structure L M\nα : Type u_1\ninst✝ : Structure (constantsOn α) M\na : α\nx : Fin 0 → M\n⊢ funMap (Sum.inr a) x = ↑(Language.con L a)",
"tactic": "rw [Unique.eq_default x]"
},
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\nM : Type w\ninst✝¹ : Structure L M\nα : Type u_1\ninst✝ : Structure (constantsOn α) M\na : α\nx : Fin 0 → M\n⊢ funMap (Sum.inr a) default = ↑(Language.con L a)",
"tactic": "exact (LHom.sumInr : constantsOn α →ᴸ L.sum _).map_onFunction _ _"
}
] |
[
576,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
573,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.erase_add_right_neg
|
[] |
[
1061,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1059,
1
] |
Std/Data/List/Lemmas.lean
|
List.disjoint_iff_ne
|
[] |
[
1359,
83
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1358,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.map_mono
|
[] |
[
471,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
470,
1
] |
Mathlib/Computability/Primrec.lean
|
Nat.Primrec.of_eq
|
[] |
[
94,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
indicator_meas_zero
|
[] |
[
4695,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4694,
1
] |
Mathlib/Data/Set/Intervals/Monotone.lean
|
AntitoneOn.Ici
|
[] |
[
53,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
11
] |
Mathlib/Data/Finset/NoncommProd.lean
|
Finset.noncommProd_congr
|
[
{
"state_after": "F : Type ?u.139635\nι : Type ?u.139638\nα : Type ?u.139641\nβ : Type ?u.139644\nγ : Type ?u.139647\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns₁ s₂ : Finset α\nf g : α → β\nh₁ : s₁ = s₂\nh₂ : ∀ (x : α), x ∈ s₂ → f x = g x\ncomm : Set.Pairwise ↑s₁ fun a b => Commute (f a) (f b)\nx : α\nhx : x ∈ ↑s₂\ny : α\nhy : y ∈ ↑s₂\nh : x ≠ y\n⊢ Commute (g x) (g y)",
"state_before": "F : Type ?u.139635\nι : Type ?u.139638\nα : Type ?u.139641\nβ : Type ?u.139644\nγ : Type ?u.139647\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns₁ s₂ : Finset α\nf g : α → β\nh₁ : s₁ = s₂\nh₂ : ∀ (x : α), x ∈ s₂ → f x = g x\ncomm : Set.Pairwise ↑s₁ fun a b => Commute (f a) (f b)\nx : α\nhx : x ∈ ↑s₂\ny : α\nhy : y ∈ ↑s₂\nh : x ≠ y\n⊢ (fun a b => Commute (g a) (g b)) x y",
"tactic": "dsimp only"
},
{
"state_after": "F : Type ?u.139635\nι : Type ?u.139638\nα : Type ?u.139641\nβ : Type ?u.139644\nγ : Type ?u.139647\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns₁ s₂ : Finset α\nf g : α → β\nh₁ : s₁ = s₂\nh₂ : ∀ (x : α), x ∈ s₂ → f x = g x\ncomm : Set.Pairwise ↑s₁ fun a b => Commute (f a) (f b)\nx : α\nhx : x ∈ ↑s₂\ny : α\nhy : y ∈ ↑s₂\nh : x ≠ y\n⊢ Commute (f x) (f y)",
"state_before": "F : Type ?u.139635\nι : Type ?u.139638\nα : Type ?u.139641\nβ : Type ?u.139644\nγ : Type ?u.139647\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns₁ s₂ : Finset α\nf g : α → β\nh₁ : s₁ = s₂\nh₂ : ∀ (x : α), x ∈ s₂ → f x = g x\ncomm : Set.Pairwise ↑s₁ fun a b => Commute (f a) (f b)\nx : α\nhx : x ∈ ↑s₂\ny : α\nhy : y ∈ ↑s₂\nh : x ≠ y\n⊢ Commute (g x) (g y)",
"tactic": "rw [← h₂ _ hx, ← h₂ _ hy]"
},
{
"state_after": "F : Type ?u.139635\nι : Type ?u.139638\nα : Type ?u.139641\nβ : Type ?u.139644\nγ : Type ?u.139647\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns₁ : Finset α\nf g : α → β\ncomm : Set.Pairwise ↑s₁ fun a b => Commute (f a) (f b)\nx y : α\nh : x ≠ y\nh₂ : ∀ (x : α), x ∈ s₁ → f x = g x\nhx : x ∈ ↑s₁\nhy : y ∈ ↑s₁\n⊢ Commute (f x) (f y)",
"state_before": "F : Type ?u.139635\nι : Type ?u.139638\nα : Type ?u.139641\nβ : Type ?u.139644\nγ : Type ?u.139647\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns₁ s₂ : Finset α\nf g : α → β\nh₁ : s₁ = s₂\nh₂ : ∀ (x : α), x ∈ s₂ → f x = g x\ncomm : Set.Pairwise ↑s₁ fun a b => Commute (f a) (f b)\nx : α\nhx : x ∈ ↑s₂\ny : α\nhy : y ∈ ↑s₂\nh : x ≠ y\n⊢ Commute (f x) (f y)",
"tactic": "subst h₁"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.139635\nι : Type ?u.139638\nα : Type ?u.139641\nβ : Type ?u.139644\nγ : Type ?u.139647\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns₁ : Finset α\nf g : α → β\ncomm : Set.Pairwise ↑s₁ fun a b => Commute (f a) (f b)\nx y : α\nh : x ≠ y\nh₂ : ∀ (x : α), x ∈ s₁ → f x = g x\nhx : x ∈ ↑s₁\nhy : y ∈ ↑s₁\n⊢ Commute (f x) (f y)",
"tactic": "exact comm hx hy h"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.139635\nι : Type ?u.139638\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.139647\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns₁ s₂ : Finset α\nf g : α → β\nh₁ : s₁ = s₂\nh₂ : ∀ (x : α), x ∈ s₂ → f x = g x\ncomm : Set.Pairwise ↑s₁ fun a b => Commute (f a) (f b)\n⊢ noncommProd s₁ f comm =\n noncommProd s₂ g (_ : ∀ (x : α), x ∈ ↑s₂ → ∀ (y : α), y ∈ ↑s₂ → x ≠ y → (fun a b => Commute (g a) (g b)) x y)",
"tactic": "simp_rw [noncommProd, Multiset.map_congr (congr_arg _ h₁) h₂]"
}
] |
[
260,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Std/Data/List/Lemmas.lean
|
List.mem_eraseP_of_neg
|
[
{
"state_after": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nal : a ∈ l\n⊢ a ∈ eraseP p l",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\n⊢ a ∈ eraseP p l ↔ a ∈ l",
"tactic": "refine ⟨mem_of_mem_eraseP, fun al => ?_⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nal : a ∈ l\n⊢ a ∈ eraseP p l",
"tactic": "match exists_or_eq_self_of_eraseP p l with\n| .inl h => rw [h]; assumption\n| .inr ⟨c, l₁, l₂, h₁, h₂, h₃, h₄⟩ =>\n rw [h₄]; rw [h₃] at al\n have : a ≠ c := fun h => (h ▸ pa).elim h₂\n simp [this] at al; simp [al]"
},
{
"state_after": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nal : a ∈ l\nh : eraseP p l = l\n⊢ a ∈ l",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nal : a ∈ l\nh : eraseP p l = l\n⊢ a ∈ eraseP p l",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nal : a ∈ l\nh : eraseP p l = l\n⊢ a ∈ l",
"tactic": "assumption"
},
{
"state_after": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nal : a ∈ l\nc : α\nl₁ l₂ : List α\nh₁ : ∀ (b : α), b ∈ l₁ → ¬p b = true\nh₂ : p c = true\nh₃ : l = l₁ ++ c :: l₂\nh₄ : eraseP p l = l₁ ++ l₂\n⊢ a ∈ l₁ ++ l₂",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nal : a ∈ l\nc : α\nl₁ l₂ : List α\nh₁ : ∀ (b : α), b ∈ l₁ → ¬p b = true\nh₂ : p c = true\nh₃ : l = l₁ ++ c :: l₂\nh₄ : eraseP p l = l₁ ++ l₂\n⊢ a ∈ eraseP p l",
"tactic": "rw [h₄]"
},
{
"state_after": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nc : α\nl₁ l₂ : List α\nal : a ∈ l₁ ++ c :: l₂\nh₁ : ∀ (b : α), b ∈ l₁ → ¬p b = true\nh₂ : p c = true\nh₃ : l = l₁ ++ c :: l₂\nh₄ : eraseP p l = l₁ ++ l₂\n⊢ a ∈ l₁ ++ l₂",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nal : a ∈ l\nc : α\nl₁ l₂ : List α\nh₁ : ∀ (b : α), b ∈ l₁ → ¬p b = true\nh₂ : p c = true\nh₃ : l = l₁ ++ c :: l₂\nh₄ : eraseP p l = l₁ ++ l₂\n⊢ a ∈ l₁ ++ l₂",
"tactic": "rw [h₃] at al"
},
{
"state_after": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nc : α\nl₁ l₂ : List α\nal : a ∈ l₁ ++ c :: l₂\nh₁ : ∀ (b : α), b ∈ l₁ → ¬p b = true\nh₂ : p c = true\nh₃ : l = l₁ ++ c :: l₂\nh₄ : eraseP p l = l₁ ++ l₂\nthis : a ≠ c\n⊢ a ∈ l₁ ++ l₂",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nc : α\nl₁ l₂ : List α\nal : a ∈ l₁ ++ c :: l₂\nh₁ : ∀ (b : α), b ∈ l₁ → ¬p b = true\nh₂ : p c = true\nh₃ : l = l₁ ++ c :: l₂\nh₄ : eraseP p l = l₁ ++ l₂\n⊢ a ∈ l₁ ++ l₂",
"tactic": "have : a ≠ c := fun h => (h ▸ pa).elim h₂"
},
{
"state_after": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nc : α\nl₁ l₂ : List α\nh₁ : ∀ (b : α), b ∈ l₁ → ¬p b = true\nh₂ : p c = true\nh₃ : l = l₁ ++ c :: l₂\nh₄ : eraseP p l = l₁ ++ l₂\nthis : a ≠ c\nal : a ∈ l₁ ∨ a ∈ l₂\n⊢ a ∈ l₁ ++ l₂",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nc : α\nl₁ l₂ : List α\nal : a ∈ l₁ ++ c :: l₂\nh₁ : ∀ (b : α), b ∈ l₁ → ¬p b = true\nh₂ : p c = true\nh₃ : l = l₁ ++ c :: l₂\nh₄ : eraseP p l = l₁ ++ l₂\nthis : a ≠ c\n⊢ a ∈ l₁ ++ l₂",
"tactic": "simp [this] at al"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl : List α\npa : ¬p a = true\nc : α\nl₁ l₂ : List α\nh₁ : ∀ (b : α), b ∈ l₁ → ¬p b = true\nh₂ : p c = true\nh₃ : l = l₁ ++ c :: l₂\nh₄ : eraseP p l = l₁ ++ l₂\nthis : a ≠ c\nal : a ∈ l₁ ∨ a ∈ l₂\n⊢ a ∈ l₁ ++ l₂",
"tactic": "simp [al]"
}
] |
[
1007,
33
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1000,
9
] |
Mathlib/CategoryTheory/Action.lean
|
CategoryTheory.ActionCategory.id_val
|
[] |
[
138,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
11
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
IntervalIntegrable.smul
|
[] |
[
244,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.Cofork.IsColimit.existsUnique
|
[] |
[
487,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Std/Data/List/Lemmas.lean
|
List.leftpad_prefix
|
[
{
"state_after": "α : Type u_1\nn : Nat\na : α\nl : List α\n⊢ ∃ t, replicate (n - length l) a ++ t = replicate (n - length l) a ++ l",
"state_before": "α : Type u_1\nn : Nat\na : α\nl : List α\n⊢ replicate (n - length l) a <+: leftpad n a l",
"tactic": "simp only [isPrefix, leftpad]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nn : Nat\na : α\nl : List α\n⊢ ∃ t, replicate (n - length l) a ++ t = replicate (n - length l) a ++ l",
"tactic": "exact Exists.intro l rfl"
}
] |
[
1466,
27
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1463,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Set.infinite_iff_exists_lt
|
[] |
[
857,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
856,
1
] |
Mathlib/Data/Finset/Prod.lean
|
Finset.offDiag_card
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.112877\nγ : Type ?u.112880\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\nthis : card (diag s) + card (offDiag s) = card s * card s\n⊢ card (offDiag s) = card (diag s) * card (diag s) - card (diag s)",
"state_before": "α : Type u_1\nβ : Type ?u.112877\nγ : Type ?u.112880\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\nthis : card (diag s) + card (offDiag s) = card s * card s\n⊢ card (offDiag s) = card s * card s - card s",
"tactic": "conv_rhs => { rw [← s.diag_card] }"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.112877\nγ : Type ?u.112880\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\nthis : card s + card (offDiag s) = card s * card s\n⊢ card (offDiag s) = card s * card s - card s",
"state_before": "α : Type u_1\nβ : Type ?u.112877\nγ : Type ?u.112880\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\nthis : card (diag s) + card (offDiag s) = card s * card s\n⊢ card (offDiag s) = card (diag s) * card (diag s) - card (diag s)",
"tactic": "simp only [diag_card] at *"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.112877\nγ : Type ?u.112880\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\nthis : card s + card (offDiag s) = card s * card s\n⊢ card s * card s = card s + card (offDiag s)",
"state_before": "α : Type u_1\nβ : Type ?u.112877\nγ : Type ?u.112880\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\nthis : card s + card (offDiag s) = card s * card s\n⊢ card (offDiag s) = card s * card s - card s",
"tactic": "rw [tsub_eq_of_eq_add_rev]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.112877\nγ : Type ?u.112880\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\nthis : card s + card (offDiag s) = card s * card s\n⊢ card s * card s = card s + card (offDiag s)",
"tactic": "rw [this]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.112877\nγ : Type ?u.112880\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\n⊢ card (filter (fun a => a.fst = a.snd) (s ×ˢ s)) + card (filter (fun a => a.fst ≠ a.snd) (s ×ˢ s)) = card (s ×ˢ s)",
"state_before": "α : Type u_1\nβ : Type ?u.112877\nγ : Type ?u.112880\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\n⊢ card (diag s) + card (offDiag s) = card s * card s",
"tactic": "rw [← card_product, diag, offDiag]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.112877\nγ : Type ?u.112880\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\n⊢ card (filter (fun a => a.fst = a.snd) (s ×ˢ s)) + card (filter (fun a => a.fst ≠ a.snd) (s ×ˢ s)) = card (s ×ˢ s)",
"tactic": "conv_rhs => rw [← filter_card_add_filter_neg_card_eq_card (fun a => a.1 = a.2)]"
}
] |
[
340,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_insert
|
[
{
"state_after": "α✝ β α : Type u\ns : Set α\na : α\nh : ¬a ∈ s\n⊢ Disjoint s {a}",
"state_before": "α✝ β α : Type u\ns : Set α\na : α\nh : ¬a ∈ s\n⊢ (#↑(insert a s)) = (#↑s) + 1",
"tactic": "rw [← union_singleton, mk_union_of_disjoint, mk_singleton]"
},
{
"state_after": "no goals",
"state_before": "α✝ β α : Type u\ns : Set α\na : α\nh : ¬a ∈ s\n⊢ Disjoint s {a}",
"tactic": "simpa"
}
] |
[
2109,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2106,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.mul_add_div
|
[
{
"state_after": "case a\nα : Type ?u.240963\nβ : Type ?u.240966\nγ : Type ?u.240969\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nb0 : b ≠ 0\nc : Ordinal\n⊢ (b * a + c) / b ≤ a + c / b\n\ncase a\nα : Type ?u.240963\nβ : Type ?u.240966\nγ : Type ?u.240969\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nb0 : b ≠ 0\nc : Ordinal\n⊢ a + c / b ≤ (b * a + c) / b",
"state_before": "α : Type ?u.240963\nβ : Type ?u.240966\nγ : Type ?u.240969\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nb0 : b ≠ 0\nc : Ordinal\n⊢ (b * a + c) / b = a + c / b",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\nα : Type ?u.240963\nβ : Type ?u.240966\nγ : Type ?u.240969\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nb0 : b ≠ 0\nc : Ordinal\n⊢ b * a + c < b * succ (a + c / b)",
"state_before": "case a\nα : Type ?u.240963\nβ : Type ?u.240966\nγ : Type ?u.240969\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nb0 : b ≠ 0\nc : Ordinal\n⊢ (b * a + c) / b ≤ a + c / b",
"tactic": "apply (div_le b0).2"
},
{
"state_after": "case a\nα : Type ?u.240963\nβ : Type ?u.240966\nγ : Type ?u.240969\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nb0 : b ≠ 0\nc : Ordinal\n⊢ c < b * (c / b) + b",
"state_before": "case a\nα : Type ?u.240963\nβ : Type ?u.240966\nγ : Type ?u.240969\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nb0 : b ≠ 0\nc : Ordinal\n⊢ b * a + c < b * succ (a + c / b)",
"tactic": "rw [mul_succ, mul_add, add_assoc, add_lt_add_iff_left]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type ?u.240963\nβ : Type ?u.240966\nγ : Type ?u.240969\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nb0 : b ≠ 0\nc : Ordinal\n⊢ c < b * (c / b) + b",
"tactic": "apply lt_mul_div_add _ b0"
},
{
"state_after": "case a\nα : Type ?u.240963\nβ : Type ?u.240966\nγ : Type ?u.240969\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nb0 : b ≠ 0\nc : Ordinal\n⊢ b * (c / b) ≤ c",
"state_before": "case a\nα : Type ?u.240963\nβ : Type ?u.240966\nγ : Type ?u.240969\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nb0 : b ≠ 0\nc : Ordinal\n⊢ a + c / b ≤ (b * a + c) / b",
"tactic": "rw [le_div b0, mul_add, add_le_add_iff_left]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type ?u.240963\nβ : Type ?u.240966\nγ : Type ?u.240969\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nb0 : b ≠ 0\nc : Ordinal\n⊢ b * (c / b) ≤ c",
"tactic": "apply mul_div_le"
}
] |
[
947,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
941,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.cast_nat_cast'
|
[] |
[
399,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
398,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.aecover_Ioc_of_Ioo
|
[] |
[
234,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
LinearMap.toMatrix_toLin
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\ninst✝⁷ : CommSemiring R\nl : Type ?u.1454587\nm : Type u_1\nn : Type u_2\ninst✝⁶ : Fintype n\ninst✝⁵ : Fintype m\ninst✝⁴ : DecidableEq n\nM₁ : Type u_4\nM₂ : Type u_5\ninst✝³ : AddCommMonoid M₁\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM : Matrix m n R\n⊢ ↑(toMatrix v₁ v₂) (↑(toLin v₁ v₂) M) = M",
"tactic": "rw [← Matrix.toLin_symm, LinearEquiv.symm_apply_apply]"
}
] |
[
572,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
570,
1
] |
Std/Data/Range/Lemmas.lean
|
Std.Range.numElems_stop_le_start
|
[
{
"state_after": "start stop step : Nat\nh : { start := start, stop := stop, step := step }.stop ≤ { start := start, stop := stop, step := step }.start\n⊢ (if step = 0 then if stop ≤ start then 0 else stop else (stop - start + step - 1) / step) = 0",
"state_before": "start stop step : Nat\nh : { start := start, stop := stop, step := step }.stop ≤ { start := start, stop := stop, step := step }.start\n⊢ numElems { start := start, stop := stop, step := step } = 0",
"tactic": "simp [numElems]"
},
{
"state_after": "case inr\nstart stop step : Nat\nh : stop ≤ start\nh✝ : ¬step = 0\n⊢ (stop - start + step - 1) / step = 0",
"state_before": "start stop step : Nat\nh : { start := start, stop := stop, step := step }.stop ≤ { start := start, stop := stop, step := step }.start\n⊢ (if step = 0 then if stop ≤ start then 0 else stop else (stop - start + step - 1) / step) = 0",
"tactic": "split <;> simp_all"
},
{
"state_after": "case inr.h₀\nstart stop step : Nat\nh : stop ≤ start\nh✝ : ¬step = 0\n⊢ stop - start + step - 1 < step",
"state_before": "case inr\nstart stop step : Nat\nh : stop ≤ start\nh✝ : ¬step = 0\n⊢ (stop - start + step - 1) / step = 0",
"tactic": "apply Nat.div_eq_of_lt"
},
{
"state_after": "case inr.h₀\nstart stop step : Nat\nh : stop ≤ start\nh✝ : ¬step = 0\n⊢ step - 1 < step",
"state_before": "case inr.h₀\nstart stop step : Nat\nh : stop ≤ start\nh✝ : ¬step = 0\n⊢ stop - start + step - 1 < step",
"tactic": "simp [Nat.sub_eq_zero_of_le h]"
},
{
"state_after": "no goals",
"state_before": "case inr.h₀\nstart stop step : Nat\nh : stop ≤ start\nh✝ : ¬step = 0\n⊢ step - 1 < step",
"tactic": "exact Nat.pred_lt ‹_›"
}
] |
[
18,
26
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
14,
1
] |
Std/Logic.lean
|
Decidable.or_iff_not_imp_right
|
[] |
[
547,
34
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
546,
1
] |
Mathlib/Data/Set/Intervals/OrdConnected.lean
|
Set.ordConnected_Icc
|
[] |
[
156,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/GroupTheory/Index.lean
|
Subgroup.finiteIndex_of_le
|
[] |
[
584,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.fp_family_unbounded
|
[
{
"state_after": "ι : Type u\nf : ι → Ordinal → Ordinal\nH : ∀ (i : ι), IsNormal (f i)\na : Ordinal\ns : Set Ordinal\nx✝ : s ∈ Set.range fun i => fixedPoints (f i)\ni : ι\nhi : (fun i => fixedPoints (f i)) i = s\n⊢ f i (nfpFamily f a) = nfpFamily f a",
"state_before": "ι : Type u\nf : ι → Ordinal → Ordinal\nH : ∀ (i : ι), IsNormal (f i)\na : Ordinal\ns : Set Ordinal\nx✝ : s ∈ Set.range fun i => fixedPoints (f i)\ni : ι\nhi : (fun i => fixedPoints (f i)) i = s\n⊢ nfpFamily f a ∈ s",
"tactic": "rw [← hi, mem_fixedPoints_iff]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nf : ι → Ordinal → Ordinal\nH : ∀ (i : ι), IsNormal (f i)\na : Ordinal\ns : Set Ordinal\nx✝ : s ∈ Set.range fun i => fixedPoints (f i)\ni : ι\nhi : (fun i => fixedPoints (f i)) i = s\n⊢ f i (nfpFamily f a) = nfpFamily f a",
"tactic": "exact nfpFamily_fp.{u, v} (H i) a"
}
] |
[
152,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
StrictMonoOn.exists_deriv_lt_slope
|
[
{
"state_after": "case pos\nE : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)\n\ncase neg\nE : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ¬∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"state_before": "E : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"tactic": "by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nE : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"tactic": "apply StrictMonoOn.exists_deriv_lt_slope_aux hf hxy hf'_mono h"
},
{
"state_after": "case neg\nE : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∃ w, w ∈ Ioo x y ∧ deriv f w = 0\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"state_before": "case neg\nE : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ¬∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"tactic": "push_neg at h"
},
{
"state_after": "case neg.intro.intro.intro\nE : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : ℝ\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"state_before": "case neg\nE : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∃ w, w ∈ Ioo x y ∧ deriv f w = 0\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"tactic": "rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : ℝ\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : ℝ\nha : deriv f a < (f w - f x) / (w - x)\nhxa : x < a\nhaw : a < w\nb : ℝ\nhb : deriv f b < (f y - f w) / (y - w)\nhwb : w < b\nhby : b < y\n⊢ deriv f a < (f y - f x) / (y - x)",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : ℝ\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : ℝ\nha : deriv f a < (f w - f x) / (w - x)\nhxa : x < a\nhaw : a < w\nb : ℝ\nhb : deriv f b < (f y - f w) / (y - w)\nhwb : w < b\nhby : b < y\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"tactic": "refine' ⟨a, ⟨hxa, haw.trans hwy⟩, _⟩"
},
{
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{
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"state_before": "E : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : ℝ\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : ℝ\nhxa : x < a\nhaw : a < w\nb : ℝ\nhwb : w < b\nhby : b < y\nha : deriv f a * (w - x) < f w - f x\nhb : deriv f b * (y - w) < f y - f w\n⊢ deriv f a < deriv f b",
"tactic": "exact hf'_mono ⟨hxa, haw.trans hwy⟩ ⟨hxw.trans hwb, hby⟩ (haw.trans hwb)"
},
{
"state_after": "E : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : ℝ\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : ℝ\nhxa : x < a\nhaw : a < w\nb : ℝ\nhwb : w < b\nhby : b < y\nha : deriv f a * (w - x) < f w - f x\nhb : deriv f b * (y - w) < f y - f w\n⊢ deriv f w ≤ deriv f b",
"state_before": "E : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : ℝ\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : ℝ\nhxa : x < a\nhaw : a < w\nb : ℝ\nhwb : w < b\nhby : b < y\nha : deriv f a * (w - x) < f w - f x\nhb : deriv f b * (y - w) < f y - f w\n⊢ 0 ≤ deriv f b",
"tactic": "rw [← hw]"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.439267\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.439363\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : ℝ\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : ℝ\nhxa : x < a\nhaw : a < w\nb : ℝ\nhwb : w < b\nhby : b < y\nha : deriv f a * (w - x) < f w - f x\nhb : deriv f b * (y - w) < f y - f w\n⊢ deriv f w ≤ deriv f b",
"tactic": "exact (hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hwb, hby⟩ hwb).le"
}
] |
[
1097,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1067,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
inner_mul_symm_re_eq_norm
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1772395\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ↑re (inner y x * ↑(starRingEnd 𝕜) (inner y x)) = ‖inner y x * ↑(starRingEnd 𝕜) (inner y x)‖",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1772395\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ↑re (inner x y * inner y x) = ‖inner x y * inner y x‖",
"tactic": "rw [← inner_conj_symm, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1772395\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ↑re (inner y x * ↑(starRingEnd 𝕜) (inner y x)) = ‖inner y x * ↑(starRingEnd 𝕜) (inner y x)‖",
"tactic": "exact re_eq_norm_of_mul_conj (inner y x)"
}
] |
[
661,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
659,
1
] |
Mathlib/Data/Multiset/Fintype.lean
|
Multiset.card_coe
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nm✝ m : Multiset α\n⊢ Fintype.card { x // x ∈ toEnumFinset m } = ↑card m",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nm✝ m : Multiset α\n⊢ Fintype.card (ToType m) = ↑card m",
"tactic": "rw [Fintype.card_congr m.coeEquiv]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nm✝ m : Multiset α\n⊢ Fintype.card { x // x ∈ toEnumFinset m } = ↑card m",
"tactic": "simp only [Fintype.card_coe, card_toEnumFinset]"
}
] |
[
259,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.bound_of_isBigO_cofinite
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : 0 < C\nhC : IsBigOWith C cofinite f g''\n⊢ ∃ C, C > 0 ∧ ∀ ⦃x : α⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖",
"state_before": "α : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\n⊢ ∃ C, C > 0 ∧ ∀ ⦃x : α⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖",
"tactic": "rcases h.exists_pos with ⟨C, C₀, hC⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : 0 < C\nhC : Set.Finite {x | ¬‖f x‖ ≤ C * ‖g'' x‖}\n⊢ ∃ C, C > 0 ∧ ∀ ⦃x : α⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : 0 < C\nhC : IsBigOWith C cofinite f g''\n⊢ ∃ C, C > 0 ∧ ∀ ⦃x : α⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖",
"tactic": "rw [IsBigOWith_def, eventually_cofinite] at hC"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : 0 < C\nhC : Set.Finite {x | ¬‖f x‖ ≤ C * ‖g'' x‖}\nC' : ℝ\nhC' : ∀ (i : ℝ), i ∈ Finset.image (fun x => ‖f x‖ / ‖g'' x‖) (Finite.toFinset hC) → i ≤ C'\n⊢ ∃ C, C > 0 ∧ ∀ ⦃x : α⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : 0 < C\nhC : Set.Finite {x | ¬‖f x‖ ≤ C * ‖g'' x‖}\n⊢ ∃ C, C > 0 ∧ ∀ ⦃x : α⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖",
"tactic": "rcases (hC.toFinset.image fun x => ‖f x‖ / ‖g'' x‖).exists_le with ⟨C', hC'⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : 0 < C\nhC : Set.Finite {x | ¬‖f x‖ ≤ C * ‖g'' x‖}\nC' : ℝ\nhC' : ∀ (i : ℝ), i ∈ Finset.image (fun x => ‖f x‖ / ‖g'' x‖) (Finite.toFinset hC) → i ≤ C'\nthis : ∀ (x : α), C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ / ‖g'' x‖ ≤ C'\n⊢ ∃ C, C > 0 ∧ ∀ ⦃x : α⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : 0 < C\nhC : Set.Finite {x | ¬‖f x‖ ≤ C * ‖g'' x‖}\nC' : ℝ\nhC' : ∀ (i : ℝ), i ∈ Finset.image (fun x => ‖f x‖ / ‖g'' x‖) (Finite.toFinset hC) → i ≤ C'\n⊢ ∃ C, C > 0 ∧ ∀ ⦃x : α⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖",
"tactic": "have : ∀ x, C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ / ‖g'' x‖ ≤ C' := by simpa using hC'"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : 0 < C\nhC : Set.Finite {x | ¬‖f x‖ ≤ C * ‖g'' x‖}\nC' : ℝ\nhC' : ∀ (i : ℝ), i ∈ Finset.image (fun x => ‖f x‖ / ‖g'' x‖) (Finite.toFinset hC) → i ≤ C'\nthis : ∀ (x : α), C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ / ‖g'' x‖ ≤ C'\nx : α\nh₀ : g'' x ≠ 0\n⊢ ‖f x‖ ≤ max C C' * ‖g'' x‖",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : 0 < C\nhC : Set.Finite {x | ¬‖f x‖ ≤ C * ‖g'' x‖}\nC' : ℝ\nhC' : ∀ (i : ℝ), i ∈ Finset.image (fun x => ‖f x‖ / ‖g'' x‖) (Finite.toFinset hC) → i ≤ C'\nthis : ∀ (x : α), C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ / ‖g'' x‖ ≤ C'\n⊢ ∃ C, C > 0 ∧ ∀ ⦃x : α⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖",
"tactic": "refine' ⟨max C C', lt_max_iff.2 (Or.inl C₀), fun x h₀ => _⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : 0 < C\nhC : Set.Finite {x | ¬‖f x‖ ≤ C * ‖g'' x‖}\nC' : ℝ\nhC' : ∀ (i : ℝ), i ∈ Finset.image (fun x => ‖f x‖ / ‖g'' x‖) (Finite.toFinset hC) → i ≤ C'\nthis : ∀ (x : α), C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ / ‖g'' x‖ ≤ C'\nx : α\nh₀ : g'' x ≠ 0\n⊢ C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ ≤ C' * ‖g'' x‖",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : 0 < C\nhC : Set.Finite {x | ¬‖f x‖ ≤ C * ‖g'' x‖}\nC' : ℝ\nhC' : ∀ (i : ℝ), i ∈ Finset.image (fun x => ‖f x‖ / ‖g'' x‖) (Finite.toFinset hC) → i ≤ C'\nthis : ∀ (x : α), C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ / ‖g'' x‖ ≤ C'\nx : α\nh₀ : g'' x ≠ 0\n⊢ ‖f x‖ ≤ max C C' * ‖g'' x‖",
"tactic": "rw [max_mul_of_nonneg _ _ (norm_nonneg _), le_max_iff, or_iff_not_imp_left, not_le]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : 0 < C\nhC : Set.Finite {x | ¬‖f x‖ ≤ C * ‖g'' x‖}\nC' : ℝ\nhC' : ∀ (i : ℝ), i ∈ Finset.image (fun x => ‖f x‖ / ‖g'' x‖) (Finite.toFinset hC) → i ≤ C'\nthis : ∀ (x : α), C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ / ‖g'' x‖ ≤ C'\nx : α\nh₀ : g'' x ≠ 0\n⊢ C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ ≤ C' * ‖g'' x‖",
"tactic": "exact fun hx => (div_le_iff (norm_pos_iff.2 h₀)).1 (this _ hx)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.669259\nE : Type u_2\nF : Type ?u.669265\nG : Type ?u.669268\nE' : Type ?u.669271\nF' : Type ?u.669274\nG' : Type ?u.669277\nE'' : Type ?u.669280\nF'' : Type u_3\nG'' : Type ?u.669286\nR : Type ?u.669289\nR' : Type ?u.669292\n𝕜 : Type ?u.669295\n𝕜' : Type ?u.669298\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\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : 0 < C\nhC : Set.Finite {x | ¬‖f x‖ ≤ C * ‖g'' x‖}\nC' : ℝ\nhC' : ∀ (i : ℝ), i ∈ Finset.image (fun x => ‖f x‖ / ‖g'' x‖) (Finite.toFinset hC) → i ≤ C'\n⊢ ∀ (x : α), C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ / ‖g'' x‖ ≤ C'",
"tactic": "simpa using hC'"
}
] |
[
2107,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2099,
1
] |
Mathlib/Data/Matrix/Notation.lean
|
Matrix.mulVec_cons
|
[
{
"state_after": "case h\nα✝ : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\ninst✝¹ : Semiring α✝\nα : Type u_1\ninst✝ : CommSemiring α\nA : m' → Fin (Nat.succ n) → α\nx : α\nv : Fin n → α\ni : m'\n⊢ mulVec (↑of A) (vecCons x v) i = (x • vecHead ∘ A + mulVec (↑of (vecTail ∘ A)) v) i",
"state_before": "α✝ : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\ninst✝¹ : Semiring α✝\nα : Type u_1\ninst✝ : CommSemiring α\nA : m' → Fin (Nat.succ n) → α\nx : α\nv : Fin n → α\n⊢ mulVec (↑of A) (vecCons x v) = x • vecHead ∘ A + mulVec (↑of (vecTail ∘ A)) v",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case h\nα✝ : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\ninst✝¹ : Semiring α✝\nα : Type u_1\ninst✝ : CommSemiring α\nA : m' → Fin (Nat.succ n) → α\nx : α\nv : Fin n → α\ni : m'\n⊢ mulVec (↑of A) (vecCons x v) i = (x • vecHead ∘ A + mulVec (↑of (vecTail ∘ A)) v) i",
"tactic": "simp [mulVec, mul_comm]"
}
] |
[
321,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/Topology/UniformSpace/AbstractCompletion.lean
|
AbstractCompletion.uniformContinuous_map
|
[] |
[
194,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Data/List/Chain.lean
|
List.chain_of_chain_pmap
|
[
{
"state_after": "case nil\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl✝ l₁ l₂ : List α\na✝¹ b : α\nS : β → β → Prop\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nhl₁✝ : ∀ (a : α), a ∈ l → p a\na✝ : α\nha✝ : p a✝\nhl₂✝ : Chain S (f a✝ ha✝) (pmap f l hl₁✝)\nH : ∀ (a b : α) (ha : p a) (hb : p b), S (f a ha) (f b hb) → R a b\nhl₁ : ∀ (a : α), a ∈ [] → p a\na : α\nha : p a\nhl₂ : Chain S (f a ha) (pmap f [] hl₁)\n⊢ Chain R a []\n\ncase cons\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl✝ l₁ l₂ : List α\na✝¹ b : α\nS : β → β → Prop\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nhl₁✝ : ∀ (a : α), a ∈ l → p a\na✝ : α\nha✝ : p a✝\nhl₂✝ : Chain S (f a✝ ha✝) (pmap f l hl₁✝)\nH : ∀ (a b : α) (ha : p a) (hb : p b), S (f a ha) (f b hb) → R a b\nlh : α\nlt : List α\nl_ih : ∀ (hl₁ : ∀ (a : α), a ∈ lt → p a) {a : α} (ha : p a), Chain S (f a ha) (pmap f lt hl₁) → Chain R a lt\nhl₁ : ∀ (a : α), a ∈ lh :: lt → p a\na : α\nha : p a\nhl₂ : Chain S (f a ha) (pmap f (lh :: lt) hl₁)\n⊢ Chain R a (lh :: lt)",
"state_before": "α : Type u\nβ : Type v\nR r : α → α → Prop\nl✝ l₁ l₂ : List α\na✝ b : α\nS : β → β → Prop\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nhl₁ : ∀ (a : α), a ∈ l → p a\na : α\nha : p a\nhl₂ : Chain S (f a ha) (pmap f l hl₁)\nH : ∀ (a b : α) (ha : p a) (hb : p b), S (f a ha) (f b hb) → R a b\n⊢ Chain R a l",
"tactic": "induction' l with lh lt l_ih generalizing a"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl✝ l₁ l₂ : List α\na✝¹ b : α\nS : β → β → Prop\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nhl₁✝ : ∀ (a : α), a ∈ l → p a\na✝ : α\nha✝ : p a✝\nhl₂✝ : Chain S (f a✝ ha✝) (pmap f l hl₁✝)\nH : ∀ (a b : α) (ha : p a) (hb : p b), S (f a ha) (f b hb) → R a b\nhl₁ : ∀ (a : α), a ∈ [] → p a\na : α\nha : p a\nhl₂ : Chain S (f a ha) (pmap f [] hl₁)\n⊢ Chain R a []",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nβ : Type v\nR r : α → α → Prop\nl✝ l₁ l₂ : List α\na✝¹ b : α\nS : β → β → Prop\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nhl₁✝ : ∀ (a : α), a ∈ l → p a\na✝ : α\nha✝ : p a✝\nhl₂✝ : Chain S (f a✝ ha✝) (pmap f l hl₁✝)\nH : ∀ (a b : α) (ha : p a) (hb : p b), S (f a ha) (f b hb) → R a b\nlh : α\nlt : List α\nl_ih : ∀ (hl₁ : ∀ (a : α), a ∈ lt → p a) {a : α} (ha : p a), Chain S (f a ha) (pmap f lt hl₁) → Chain R a lt\nhl₁ : ∀ (a : α), a ∈ lh :: lt → p a\na : α\nha : p a\nhl₂ : Chain S (f a ha) (pmap f (lh :: lt) hl₁)\n⊢ Chain R a (lh :: lt)",
"tactic": "simp [H _ _ _ _ (rel_of_chain_cons hl₂), l_ih _ _ (chain_of_chain_cons hl₂)]"
}
] |
[
113,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalRingHom.map_sclosure
|
[] |
[
933,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
929,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.mem_inv_smul_set_iff₀
|
[] |
[
1022,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1021,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingIso.toRingEquiv_eq_coe
|
[] |
[
410,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.L1.SimpleFunc.coe_posPart
|
[] |
[
497,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Submodule.annihilator_bot
|
[] |
[
85,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
Module.Ray.someVector_ne_zero
|
[] |
[
377,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
376,
1
] |
Mathlib/CategoryTheory/Sites/DenseSubsite.lean
|
CategoryTheory.CoverDense.sheafHom_eq
|
[
{
"state_after": "case w.h\nC : Type u_5\ninst✝⁴ : Category C\nD : Type u_1\ninst✝³ : Category D\nE : Type ?u.131136\ninst✝² : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝¹ : Category A\nG : C ⥤ D\nH : CoverDense K G\ninst✝ : Full G\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nα : ℱ ⟶ ℱ'.val\nX : Dᵒᵖ\n⊢ (sheafHom H (whiskerLeft (Functor.op G) α)).app X = α.app X",
"state_before": "C : Type u_5\ninst✝⁴ : Category C\nD : Type u_1\ninst✝³ : Category D\nE : Type ?u.131136\ninst✝² : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝¹ : Category A\nG : C ⥤ D\nH : CoverDense K G\ninst✝ : Full G\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nα : ℱ ⟶ ℱ'.val\n⊢ sheafHom H (whiskerLeft (Functor.op G) α) = α",
"tactic": "ext X"
},
{
"state_after": "case w.h.a\nC : Type u_5\ninst✝⁴ : Category C\nD : Type u_1\ninst✝³ : Category D\nE : Type ?u.131136\ninst✝² : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝¹ : Category A\nG : C ⥤ D\nH : CoverDense K G\ninst✝ : Full G\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nα : ℱ ⟶ ℱ'.val\nX : Dᵒᵖ\n⊢ yoneda.map ((sheafHom H (whiskerLeft (Functor.op G) α)).app X) = yoneda.map (α.app X)",
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},
{
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{
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"tactic": "conv_lhs => rw [← hf.some.fac]"
},
{
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{
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"tactic": "simp"
},
{
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"tactic": "exact (pushforwardFamily_compatible H _ _)"
}
] |
[
448,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Data/Finset/Basic.lean
|
List.disjoint_toFinset_iff_disjoint
|
[] |
[
3804,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3803,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometryEquiv.edist_map
|
[] |
[
635,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
634,
1
] |
Mathlib/Algebra/Order/Field/Power.lean
|
zpow_le_one_of_nonpos
|
[] |
[
43,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
PSet.Mem.congr_left
|
[] |
[
311,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
310,
1
] |
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