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/CategoryTheory/ConcreteCategory/Basic.lean
|
CategoryTheory.ConcreteCategory.injective_of_mono_of_preservesPullback
|
[] |
[
165,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Algebra/Lie/IdealOperations.lean
|
LieSubmodule.lie_coe_mem_lie
|
[
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nx : { x // x ∈ I }\nm : { x // x ∈ N }\n⊢ ⁅↑x, ↑m⁆ ∈ lieSpan R L {m | ∃ x n, ⁅↑x, ↑n⁆ = m}",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nx : { x // x ∈ I }\nm : { x // x ∈ N }\n⊢ ⁅↑x, ↑m⁆ ∈ ⁅I, N⁆",
"tactic": "rw [lieIdeal_oper_eq_span]"
},
{
"state_after": "case a\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nx : { x // x ∈ I }\nm : { x // x ∈ N }\n⊢ ⁅↑x, ↑m⁆ ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nx : { x // x ∈ I }\nm : { x // x ∈ N }\n⊢ ⁅↑x, ↑m⁆ ∈ lieSpan R L {m | ∃ x n, ⁅↑x, ↑n⁆ = m}",
"tactic": "apply subset_lieSpan"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nx : { x // x ∈ I }\nm : { x // x ∈ N }\n⊢ ⁅↑x, ↑m⁆ ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}",
"tactic": "use x, m"
}
] |
[
111,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.sqrt_normSq_eq_norm
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.5321595\ninst✝ : IsROrC K\nz : K\n⊢ Real.sqrt (↑normSq z) = ‖z‖",
"tactic": "rw [normSq_eq_def', Real.sqrt_sq (norm_nonneg _)]"
}
] |
[
536,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.Monic.normalize_eq_self
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizationMonoid R\np : R[X]\nhp : Monic p\n⊢ ↑normalize p = p",
"tactic": "simp only [Polynomial.coe_normUnit, normalize_apply, hp.leadingCoeff, normUnit_one,\n Units.val_one, Polynomial.C.map_one, mul_one]"
}
] |
[
106,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.ofCauchy_inv
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\nf : Cauchy abs\n⊢ { cauchy := f⁻¹ } = Real.inv' { cauchy := f }",
"tactic": "rw [inv']"
}
] |
[
141,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Algebra/Group/WithOne/Defs.lean
|
WithOne.one_ne_coe
|
[] |
[
161,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
src/lean/Init/Prelude.lean
|
Nat.succ_pos
|
[] |
[
1609,
17
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
1608,
1
] |
Mathlib/Data/MvPolynomial/CommRing.lean
|
MvPolynomial.coeff_neg
|
[] |
[
78,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Order/Filter/Partial.lean
|
Filter.tendsto_iff_rtendsto'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl₁ : Filter α\nl₂ : Filter β\nf : α → β\n⊢ Tendsto f l₁ l₂ ↔ RTendsto' (Function.graph f) l₁ l₂",
"tactic": "simp [tendsto_def, Function.graph, rtendsto'_def, Rel.preimage_def, Set.preimage]"
}
] |
[
205,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Order/BoundedOrder.lean
|
Subtype.coe_top
|
[] |
[
776,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
775,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
Real.differentiableAt_rpow_of_ne
|
[] |
[
307,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/FieldTheory/Finite/Basic.lean
|
FiniteField.even_card_of_char_two
|
[] |
[
517,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
StieltjesFunction.measurableSet_Ioi
|
[
{
"state_after": "f : StieltjesFunction\nc : ℝ\nt : Set ℝ\n⊢ length f (t ∩ Ioi c) + length f (t \\ Ioi c) ≤ length f t",
"state_before": "f : StieltjesFunction\nc : ℝ\n⊢ MeasurableSet (Ioi c)",
"tactic": "refine OuterMeasure.ofFunction_caratheodory fun t => ?_"
},
{
"state_after": "f : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\n⊢ length f (t ∩ Ioi c) + length f (t \\ Ioi c) ≤ ofReal (↑f b - ↑f a)",
"state_before": "f : StieltjesFunction\nc : ℝ\nt : Set ℝ\n⊢ length f (t ∩ Ioi c) + length f (t \\ Ioi c) ≤ length f t",
"tactic": "refine' le_iInf fun a => le_iInf fun b => le_iInf fun h => _"
},
{
"state_after": "f : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\n⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)",
"state_before": "f : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\n⊢ length f (t ∩ Ioi c) + length f (t \\ Ioi c) ≤ ofReal (↑f b - ↑f a)",
"tactic": "refine'\n le_trans\n (add_le_add (f.length_mono <| inter_subset_inter_left _ h)\n (f.length_mono <| diff_subset_diff_left h)) _"
},
{
"state_after": "case inl.inl\nf : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\nhac : a ≤ c\nhbc : b ≤ c\n⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)\n\ncase inl.inr\nf : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\nhac : a ≤ c\nhbc : c ≤ b\n⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)\n\ncase inr.inl\nf : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\nhac : c ≤ a\nhbc : b ≤ c\n⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)\n\ncase inr.inr\nf : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\nhac : c ≤ a\nhbc : c ≤ b\n⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)",
"state_before": "f : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\n⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)",
"tactic": "cases' le_total a c with hac hac <;> cases' le_total b c with hbc hbc"
},
{
"state_after": "no goals",
"state_before": "case inl.inl\nf : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\nhac : a ≤ c\nhbc : b ≤ c\n⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)",
"tactic": "simp only [Ioc_inter_Ioi, f.length_Ioc, hac, _root_.sup_eq_max, hbc, le_refl, Ioc_eq_empty,\n max_eq_right, min_eq_left, Ioc_diff_Ioi, f.length_empty, zero_add, not_lt]"
},
{
"state_after": "no goals",
"state_before": "case inl.inr\nf : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\nhac : a ≤ c\nhbc : c ≤ b\n⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)",
"tactic": "simp only [hac, hbc, Ioc_inter_Ioi, Ioc_diff_Ioi, f.length_Ioc, min_eq_right,\n _root_.sup_eq_max, ← ENNReal.ofReal_add, f.mono hac, f.mono hbc, sub_nonneg,\n sub_add_sub_cancel, le_refl,\n max_eq_right]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nf : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\nhac : c ≤ a\nhbc : b ≤ c\n⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)",
"tactic": "simp only [hbc, le_refl, Ioc_eq_empty, Ioc_inter_Ioi, min_eq_left, Ioc_diff_Ioi, f.length_empty,\n zero_add, or_true_iff, le_sup_iff, f.length_Ioc, not_lt]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nf : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\nhac : c ≤ a\nhbc : c ≤ b\n⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)",
"tactic": "simp only [hac, hbc, Ioc_inter_Ioi, Ioc_diff_Ioi, f.length_Ioc, min_eq_right, _root_.sup_eq_max,\n le_refl, Ioc_eq_empty, add_zero, max_eq_left, f.length_empty, not_lt]"
}
] |
[
467,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
450,
1
] |
Mathlib/Order/WellFounded.lean
|
WellFounded.lt_sup
|
[] |
[
97,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
11
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.Integrable.norm
|
[] |
[
693,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
692,
1
] |
Mathlib/Algebra/Homology/Augment.lean
|
ChainComplex.truncateAugment_inv_f
|
[] |
[
130,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.integral_Icc_eq_integral_Ioo
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.279314\nE : Type u_1\nF : Type ?u.279320\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝³ : CompleteSpace E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : PartialOrder α\na b : α\ninst✝ : NoAtoms μ\n⊢ (∫ (t : α) in Icc a b, f t ∂μ) = ∫ (t : α) in Ico a b, f t ∂μ",
"tactic": "rw [integral_Icc_eq_integral_Ico, integral_Ico_eq_integral_Ioo]"
}
] |
[
664,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
663,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
IsometryEquiv.coe_toEquiv
|
[] |
[
328,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
9
] |
Mathlib/Data/Setoid/Basic.lean
|
Setoid.symm'
|
[] |
[
86,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Data/List/Basic.lean
|
List.surjective_head'
|
[] |
[
867,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
866,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.emetric_ball_top
|
[] |
[
1225,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1224,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
Real.hasStrictDerivAt_const_rpow_of_neg
|
[
{
"state_after": "no goals",
"state_before": "x✝ y z a x : ℝ\nha : a < 0\n⊢ HasStrictDerivAt (fun x => a ^ x) (a ^ x * log a - exp (log a * x) * sin (x * π) * π) x",
"tactic": "simpa using (hasStrictFDerivAt_rpow_of_neg (a, x) ha).comp_hasStrictDerivAt x\n ((hasStrictDerivAt_const _ _).prod (hasStrictDerivAt_id _))"
}
] |
[
338,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Topology/Basic.lean
|
closure_compl
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns : Set α\n⊢ closure (sᶜ) = interior sᶜ",
"tactic": "simp [closure_eq_compl_interior_compl]"
}
] |
[
557,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.ofReal_le_of_le_toReal
|
[] |
[
2089,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2087,
1
] |
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lxor_cancel_right
|
[
{
"state_after": "no goals",
"state_before": "n m : ℕ\n⊢ lxor' (lxor' m n) n = m",
"tactic": "rw [lxor'_assoc, lxor'_self, lxor'_zero]"
}
] |
[
244,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.add_mul
|
[
{
"state_after": "no goals",
"state_before": "a b c : Int\n⊢ (a + b) * c = a * c + b * c",
"tactic": "simp [Int.mul_comm, Int.mul_add]"
}
] |
[
459,
35
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
458,
11
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.natCast_re
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.6224464\ninst✝ : IsROrC K\nn : ℕ\n⊢ ↑re ↑n = ↑n",
"tactic": "rw [← ofReal_natCast, ofReal_re]"
}
] |
[
636,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
636,
1
] |
Mathlib/Topology/Algebra/ContinuousAffineMap.lean
|
ContinuousAffineMap.coe_neg
|
[] |
[
243,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Order/Filter/Extr.lean
|
isMinOn_const
|
[] |
[
193,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Order/Hom/Basic.lean
|
WithTop.toDualBotEquiv_symm_coe
|
[] |
[
1291,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1289,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean
|
CategoryTheory.Limits.map_π_preserves_coequalizer_inv_colimMap
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\nw : f ≫ h = g ≫ h\ninst✝³ : HasCoequalizer f g\ninst✝² : HasCoequalizer (G.map f) (G.map g)\ninst✝¹ : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' ⟶ Y'\ninst✝ : HasCoequalizer f' g'\np : G.obj X ⟶ X'\nq : G.obj Y ⟶ Y'\nwf : G.map f ≫ q = p ≫ f'\nwg : G.map g ≫ q = p ≫ g'\n⊢ G.map (coequalizer.π f g) ≫\n (PreservesCoequalizer.iso G f g).inv ≫ colimMap (parallelPairHom (G.map f) (G.map g) f' g' p q wf wg) =\n q ≫ coequalizer.π f' g'",
"tactic": "rw [← Category.assoc, map_π_preserves_coequalizer_inv, ι_colimMap, parallelPairHom_app_one]"
}
] |
[
227,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
|
ContinuousMap.sup_mem_closed_subalgebra
|
[
{
"state_after": "case h.e'_5\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nh : IsClosed ↑A\nf g : { x // x ∈ A }\n⊢ A = Subalgebra.topologicalClosure A",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nh : IsClosed ↑A\nf g : { x // x ∈ A }\n⊢ ↑f ⊔ ↑g ∈ A",
"tactic": "convert sup_mem_subalgebra_closure A f g"
},
{
"state_after": "case h.e'_5.h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nh : IsClosed ↑A\nf g : { x // x ∈ A }\n⊢ ↑A = ↑(Subalgebra.topologicalClosure A)",
"state_before": "case h.e'_5\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nh : IsClosed ↑A\nf g : { x // x ∈ A }\n⊢ A = Subalgebra.topologicalClosure A",
"tactic": "apply SetLike.ext'"
},
{
"state_after": "case h.e'_5.h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nh : IsClosed ↑A\nf g : { x // x ∈ A }\n⊢ ↑(Subalgebra.topologicalClosure A) = ↑A",
"state_before": "case h.e'_5.h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nh : IsClosed ↑A\nf g : { x // x ∈ A }\n⊢ ↑A = ↑(Subalgebra.topologicalClosure A)",
"tactic": "symm"
},
{
"state_after": "case h.e'_5.h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nh : IsClosed ↑A\nf g : { x // x ∈ A }\n⊢ IsClosed ↑A",
"state_before": "case h.e'_5.h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nh : IsClosed ↑A\nf g : { x // x ∈ A }\n⊢ ↑(Subalgebra.topologicalClosure A) = ↑A",
"tactic": "erw [closure_eq_iff_isClosed]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nh : IsClosed ↑A\nf g : { x // x ∈ A }\n⊢ IsClosed ↑A",
"tactic": "exact h"
}
] |
[
163,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/CategoryTheory/Adjunction/Basic.lean
|
CategoryTheory.Functor.leftAdjoint_of_isEquivalence
|
[] |
[
671,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
670,
1
] |
Mathlib/Algebra/Field/Basic.lean
|
one_div_neg_eq_neg_one_div
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.23152\nβ : Type ?u.23155\nK : Type u_1\ninst✝¹ : DivisionMonoid K\ninst✝ : HasDistribNeg K\na✝ b a : K\n⊢ 1 / -a = 1 / (-1 * a)",
"tactic": "rw [neg_eq_neg_one_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.23152\nβ : Type ?u.23155\nK : Type u_1\ninst✝¹ : DivisionMonoid K\ninst✝ : HasDistribNeg K\na✝ b a : K\n⊢ 1 / (-1 * a) = 1 / a * (1 / -1)",
"tactic": "rw [one_div_mul_one_div_rev]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.23152\nβ : Type ?u.23155\nK : Type u_1\ninst✝¹ : DivisionMonoid K\ninst✝ : HasDistribNeg K\na✝ b a : K\n⊢ 1 / a * (1 / -1) = 1 / a * -1",
"tactic": "rw [one_div_neg_one_eq_neg_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.23152\nβ : Type ?u.23155\nK : Type u_1\ninst✝¹ : DivisionMonoid K\ninst✝ : HasDistribNeg K\na✝ b a : K\n⊢ 1 / a * -1 = -(1 / a)",
"tactic": "rw [mul_neg, mul_one]"
}
] |
[
105,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Order/Monotone/Monovary.lean
|
Subsingleton.antivary
|
[] |
[
120,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
11
] |
Mathlib/Data/Matrix/DMatrix.lean
|
DMatrix.sub_apply
|
[] |
[
141,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Algebra/Algebra/Bilinear.lean
|
LinearMap.mul'_apply
|
[] |
[
97,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
|
Real.arctan_eq_arccos
|
[
{
"state_after": "x : ℝ\nh : 0 ≤ x\n⊢ arcsin (x / sqrt (↑1 + x ^ 2)) = arcsin (sqrt (1 - (sqrt (↑1 + x ^ 2))⁻¹ ^ 2))\n\nx : ℝ\nh : 0 ≤ x\n⊢ 0 ≤ (sqrt (↑1 + x ^ 2))⁻¹",
"state_before": "x : ℝ\nh : 0 ≤ x\n⊢ arctan x = arccos (sqrt (↑1 + x ^ 2))⁻¹",
"tactic": "rw [arctan_eq_arcsin, arccos_eq_arcsin]"
},
{
"state_after": "x : ℝ\nh : 0 ≤ x\n⊢ 0 ≤ (sqrt (↑1 + x ^ 2))⁻¹\n\nx : ℝ\nh : 0 ≤ x\n⊢ arcsin (x / sqrt (↑1 + x ^ 2)) = arcsin (sqrt (1 - (sqrt (↑1 + x ^ 2))⁻¹ ^ 2))",
"state_before": "x : ℝ\nh : 0 ≤ x\n⊢ arcsin (x / sqrt (↑1 + x ^ 2)) = arcsin (sqrt (1 - (sqrt (↑1 + x ^ 2))⁻¹ ^ 2))\n\nx : ℝ\nh : 0 ≤ x\n⊢ 0 ≤ (sqrt (↑1 + x ^ 2))⁻¹",
"tactic": "swap"
},
{
"state_after": "case e_a\nx : ℝ\nh : 0 ≤ x\n⊢ x / sqrt (↑1 + x ^ 2) = sqrt (1 - (sqrt (↑1 + x ^ 2))⁻¹ ^ 2)",
"state_before": "x : ℝ\nh : 0 ≤ x\n⊢ arcsin (x / sqrt (↑1 + x ^ 2)) = arcsin (sqrt (1 - (sqrt (↑1 + x ^ 2))⁻¹ ^ 2))",
"tactic": "congr 1"
},
{
"state_after": "case e_a.hx\nx : ℝ\nh : 0 ≤ x\n⊢ 0 ≤ x ^ 2\n\ncase e_a\nx : ℝ\nh : 0 ≤ x\n⊢ 1 + x ^ 2 ≠ 0\n\ncase e_a\nx : ℝ\nh : 0 ≤ x\n⊢ 0 ≤ (1 + x ^ 2)⁻¹",
"state_before": "case e_a\nx : ℝ\nh : 0 ≤ x\n⊢ x / sqrt (↑1 + x ^ 2) = sqrt (1 - (sqrt (↑1 + x ^ 2))⁻¹ ^ 2)",
"tactic": "rw_mod_cast [← sqrt_inv, sq_sqrt, ← one_div, one_sub_div, add_sub_cancel', sqrt_div, sqrt_sq h]"
},
{
"state_after": "no goals",
"state_before": "case e_a.hx\nx : ℝ\nh : 0 ≤ x\n⊢ 0 ≤ x ^ 2\n\ncase e_a\nx : ℝ\nh : 0 ≤ x\n⊢ 1 + x ^ 2 ≠ 0\n\ncase e_a\nx : ℝ\nh : 0 ≤ x\n⊢ 0 ≤ (1 + x ^ 2)⁻¹",
"tactic": "all_goals positivity"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh : 0 ≤ x\n⊢ 0 ≤ (sqrt (↑1 + x ^ 2))⁻¹",
"tactic": "exact inv_nonneg.2 (sqrt_nonneg _)"
},
{
"state_after": "no goals",
"state_before": "case e_a\nx : ℝ\nh : 0 ≤ x\n⊢ 0 ≤ (1 + x ^ 2)⁻¹",
"tactic": "positivity"
}
] |
[
196,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Data/Quot.lean
|
Quotient.sound'
|
[] |
[
770,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
769,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.foldr_induction
|
[] |
[
1444,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1441,
1
] |
Mathlib/Data/List/Basic.lean
|
List.map_congr
|
[
{
"state_after": "ι : Type ?u.130134\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf g : α → β\na : α\nl : List α\nh : ∀ (x : α), x ∈ a :: l → f x = g x\nh₁ : f a = g a\nh₂ : ∀ (x : α), x ∈ l → f x = g x\n⊢ map f (a :: l) = map g (a :: l)",
"state_before": "ι : Type ?u.130134\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf g : α → β\na : α\nl : List α\nh : ∀ (x : α), x ∈ a :: l → f x = g x\n⊢ map f (a :: l) = map g (a :: l)",
"tactic": "let ⟨h₁, h₂⟩ := forall_mem_cons.1 h"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.130134\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf g : α → β\na : α\nl : List α\nh : ∀ (x : α), x ∈ a :: l → f x = g x\nh₁ : f a = g a\nh₂ : ∀ (x : α), x ∈ l → f x = g x\n⊢ map f (a :: l) = map g (a :: l)",
"tactic": "rw [map, map, h₁, map_congr h₂]"
}
] |
[
1778,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1774,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.one_succAbove_succ
|
[
{
"state_after": "n✝ m n : ℕ\nj : Fin n\nthis : ↑(succAbove (succ 0)) (succ j) = succ (↑(succAbove 0) j)\n⊢ ↑(succAbove 1) (succ j) = succ (succ j)",
"state_before": "n✝ m n : ℕ\nj : Fin n\n⊢ ↑(succAbove 1) (succ j) = succ (succ j)",
"tactic": "have := succ_succAbove_succ 0 j"
},
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\nj : Fin n\nthis : ↑(succAbove (succ 0)) (succ j) = succ (↑(succAbove 0) j)\n⊢ ↑(succAbove 1) (succ j) = succ (succ j)",
"tactic": "rwa [succ_zero_eq_one, zero_succAbove] at this"
}
] |
[
2247,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2244,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.continuousAt_arcsin
|
[] |
[
105,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.infinite_of_not_bddAbove
|
[] |
[
1607,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1606,
1
] |
Mathlib/Algebra/Star/Pi.lean
|
Pi.star_def
|
[] |
[
40,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
one_div_nonneg
|
[] |
[
84,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.comp_eq_of_eq
|
[] |
[
911,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
909,
1
] |
Mathlib/Topology/CompactOpen.lean
|
ContinuousMap.continuous_coev
|
[
{
"state_after": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\n⊢ IsOpen (coev α β ⁻¹' CompactOpen.gen s u)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\n⊢ ∀ (s : Set C(α, β × α)), s ∈ {m | ∃ s x u x, m = CompactOpen.gen s u} → IsOpen (coev α β ⁻¹' s)",
"tactic": "rintro _ ⟨s, sc, u, uo, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\n⊢ ∀ (x : β), x ∈ coev α β ⁻¹' CompactOpen.gen s u → ∃ t, t ⊆ coev α β ⁻¹' CompactOpen.gen s u ∧ IsOpen t ∧ x ∈ t",
"state_before": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\n⊢ IsOpen (coev α β ⁻¹' CompactOpen.gen s u)",
"tactic": "rw [isOpen_iff_forall_mem_open]"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\n⊢ ∃ t, t ⊆ coev α β ⁻¹' CompactOpen.gen s u ∧ IsOpen t ∧ y ∈ t",
"state_before": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\n⊢ ∀ (x : β), x ∈ coev α β ⁻¹' CompactOpen.gen s u → ∃ t, t ⊆ coev α β ⁻¹' CompactOpen.gen s u ∧ IsOpen t ∧ x ∈ t",
"tactic": "intro y hy"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy' : ↑(coev α β y) '' s ⊆ u\n⊢ ∃ t, t ⊆ coev α β ⁻¹' CompactOpen.gen s u ∧ IsOpen t ∧ y ∈ t",
"state_before": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\n⊢ ∃ t, t ⊆ coev α β ⁻¹' CompactOpen.gen s u ∧ IsOpen t ∧ y ∈ t",
"tactic": "have hy' : (↑(coev α β y) '' s ⊆ u) := hy"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy' : {y} ×ˢ s ⊆ u\n⊢ ∃ t, t ⊆ coev α β ⁻¹' CompactOpen.gen s u ∧ IsOpen t ∧ y ∈ t",
"state_before": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy' : ↑(coev α β y) '' s ⊆ u\n⊢ ∃ t, t ⊆ coev α β ⁻¹' CompactOpen.gen s u ∧ IsOpen t ∧ y ∈ t",
"tactic": "rw [image_coev s] at hy'"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy' : {y} ×ˢ s ⊆ u\nv : Set β\nw : Set α\nvo : IsOpen v\nleft✝ : IsOpen w\nyv : {y} ⊆ v\nsw : s ⊆ w\nvwu : v ×ˢ w ⊆ u\n⊢ ∃ t, t ⊆ coev α β ⁻¹' CompactOpen.gen s u ∧ IsOpen t ∧ y ∈ t",
"state_before": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy' : {y} ×ˢ s ⊆ u\n⊢ ∃ t, t ⊆ coev α β ⁻¹' CompactOpen.gen s u ∧ IsOpen t ∧ y ∈ t",
"tactic": "rcases generalized_tube_lemma isCompact_singleton sc uo hy' with ⟨v, w, vo, _, yv, sw, vwu⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy' : {y} ×ˢ s ⊆ u\nv : Set β\nw : Set α\nvo : IsOpen v\nleft✝ : IsOpen w\nyv : {y} ⊆ v\nsw : s ⊆ w\nvwu : v ×ˢ w ⊆ u\n⊢ v ⊆ coev α β ⁻¹' CompactOpen.gen s u",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy' : {y} ×ˢ s ⊆ u\nv : Set β\nw : Set α\nvo : IsOpen v\nleft✝ : IsOpen w\nyv : {y} ⊆ v\nsw : s ⊆ w\nvwu : v ×ˢ w ⊆ u\n⊢ ∃ t, t ⊆ coev α β ⁻¹' CompactOpen.gen s u ∧ IsOpen t ∧ y ∈ t",
"tactic": "refine' ⟨v, _, vo, singleton_subset_iff.mp yv⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy'✝ : {y} ×ˢ s ⊆ u\nv : Set β\nw : Set α\nvo : IsOpen v\nleft✝ : IsOpen w\nyv : {y} ⊆ v\nsw : s ⊆ w\nvwu : v ×ˢ w ⊆ u\ny' : β\nhy' : y' ∈ v\n⊢ y' ∈ coev α β ⁻¹' CompactOpen.gen s u",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy' : {y} ×ˢ s ⊆ u\nv : Set β\nw : Set α\nvo : IsOpen v\nleft✝ : IsOpen w\nyv : {y} ⊆ v\nsw : s ⊆ w\nvwu : v ×ˢ w ⊆ u\n⊢ v ⊆ coev α β ⁻¹' CompactOpen.gen s u",
"tactic": "intro y' hy'"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy'✝ : {y} ×ˢ s ⊆ u\nv : Set β\nw : Set α\nvo : IsOpen v\nleft✝ : IsOpen w\nyv : {y} ⊆ v\nsw : s ⊆ w\nvwu : v ×ˢ w ⊆ u\ny' : β\nhy' : y' ∈ v\n⊢ ↑(coev α β y') '' s ⊆ u",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy'✝ : {y} ×ˢ s ⊆ u\nv : Set β\nw : Set α\nvo : IsOpen v\nleft✝ : IsOpen w\nyv : {y} ⊆ v\nsw : s ⊆ w\nvwu : v ×ˢ w ⊆ u\ny' : β\nhy' : y' ∈ v\n⊢ y' ∈ coev α β ⁻¹' CompactOpen.gen s u",
"tactic": "change coev α β y' '' s ⊆ u"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy'✝ : {y} ×ˢ s ⊆ u\nv : Set β\nw : Set α\nvo : IsOpen v\nleft✝ : IsOpen w\nyv : {y} ⊆ v\nsw : s ⊆ w\nvwu : v ×ˢ w ⊆ u\ny' : β\nhy' : y' ∈ v\n⊢ {y'} ×ˢ s ⊆ u",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy'✝ : {y} ×ˢ s ⊆ u\nv : Set β\nw : Set α\nvo : IsOpen v\nleft✝ : IsOpen w\nyv : {y} ⊆ v\nsw : s ⊆ w\nvwu : v ×ˢ w ⊆ u\ny' : β\nhy' : y' ∈ v\n⊢ ↑(coev α β y') '' s ⊆ u",
"tactic": "rw [image_coev s]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41809\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nsc : IsCompact s\nu : Set (β × α)\nuo : IsOpen u\ny : β\nhy : y ∈ coev α β ⁻¹' CompactOpen.gen s u\nhy'✝ : {y} ×ˢ s ⊆ u\nv : Set β\nw : Set α\nvo : IsOpen v\nleft✝ : IsOpen w\nyv : {y} ⊆ v\nsw : s ⊆ w\nvwu : v ×ˢ w ⊆ u\ny' : β\nhy' : y' ∈ v\n⊢ {y'} ×ˢ s ⊆ u",
"tactic": "exact (prod_mono (singleton_subset_iff.mpr hy') sw).trans vwu"
}
] |
[
343,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
1
] |
Mathlib/Control/LawfulFix.lean
|
Pi.uncurry_curry_continuous
|
[] |
[
283,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/Topology/Order/Basic.lean
|
TFAE_mem_nhdsWithin_Iic
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na b : α\nh : a < b\ns : Set α\n⊢ TFAE\n [s ∈ 𝓝[Iic b] b, s ∈ 𝓝[Icc a b] b, s ∈ 𝓝[Ioc a b] b, ∃ l, l ∈ Ico a b ∧ Ioc l b ⊆ s, ∃ l, l ∈ Iio b ∧ Ioc l b ⊆ s]",
"tactic": "simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)"
}
] |
[
1809,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1802,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
uniformGroup_sInf
|
[] |
[
225,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.Memℒp.induction_dense
|
[
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\nμ : Measure α\nP : (α → E) → Prop\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_ne_top : 0 ≠ ⊤\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) 0 μ ≤ ε ∧ P g\nhf : Memℒp f 0\n⊢ ∃ g, snorm (f - g) 0 μ ≤ ε ∧ P g\n\ncase inr\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\n⊢ ∃ g, snorm (f - g) p μ ≤ ε ∧ P g",
"state_before": "α : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∃ g, snorm (f - g) p μ ≤ ε ∧ P g",
"tactic": "rcases eq_or_ne p 0 with (rfl | hp_pos)"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nH : ∀ (f' : α →ₛ E) (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\n⊢ ∃ g, snorm (f - g) p μ ≤ ε ∧ P g\n\ncase H\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\n⊢ ∀ (f' : α →ₛ E) (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\n⊢ ∃ g, snorm (f - g) p μ ≤ ε ∧ P g",
"tactic": "suffices H :\n ∀ (f' : α →ₛ E) (δ : ℝ≥0∞) (hδ : δ ≠ 0), Memℒp f' p μ → ∃ g, snorm (⇑f' - g) p μ ≤ δ ∧ P g"
},
{
"state_after": "case H.h_ind\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\n⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (δ : ℝ≥0∞),\n δ ≠ 0 →\n Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p →\n ∃ g, snorm (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) - g) p μ ≤ δ ∧ P g\n\ncase H.h_add\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\n⊢ ∀ ⦃f g : α →ₛ E⦄,\n Disjoint (support ↑f) (support ↑g) →\n (∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g) →\n (∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑g) p → ∃ g_1, snorm (↑g - g_1) p μ ≤ δ ∧ P g_1) →\n ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑(f + g)) p → ∃ g_1, snorm (↑(f + g) - g_1) p μ ≤ δ ∧ P g_1",
"state_before": "case H\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\n⊢ ∀ (f' : α →ₛ E) (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g",
"tactic": "apply SimpleFunc.induction"
},
{
"state_after": "case inl.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\nμ : Measure α\nP : (α → E) → Prop\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_ne_top : 0 ≠ ⊤\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) 0 μ ≤ ε ∧ P g\nhf : Memℒp f 0\ng : α → E\nleft✝ : snorm (g - Set.indicator ∅ fun x => 0) 0 μ ≤ ε\nPg : P g\n⊢ ∃ g, snorm (f - g) 0 μ ≤ ε ∧ P g",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\nμ : Measure α\nP : (α → E) → Prop\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_ne_top : 0 ≠ ⊤\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) 0 μ ≤ ε ∧ P g\nhf : Memℒp f 0\n⊢ ∃ g, snorm (f - g) 0 μ ≤ ε ∧ P g",
"tactic": "rcases h0P (0 : E) MeasurableSet.empty (by simp only [measure_empty, WithTop.zero_lt_top])\n hε with ⟨g, _, Pg⟩"
},
{
"state_after": "no goals",
"state_before": "case inl.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\nμ : Measure α\nP : (α → E) → Prop\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_ne_top : 0 ≠ ⊤\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) 0 μ ≤ ε ∧ P g\nhf : Memℒp f 0\ng : α → E\nleft✝ : snorm (g - Set.indicator ∅ fun x => 0) 0 μ ≤ ε\nPg : P g\n⊢ ∃ g, snorm (f - g) 0 μ ≤ ε ∧ P g",
"tactic": "exact ⟨g, by simp only [snorm_exponent_zero, zero_le'], Pg⟩"
},
{
"state_after": "no goals",
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{
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{
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"tactic": "obtain ⟨η, ηpos, hη⟩ := exists_Lp_half E μ p hε"
},
{
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"tactic": "rcases hf.exists_simpleFunc_snorm_sub_lt hp_ne_top ηpos.ne' with ⟨f', hf', f'_mem⟩"
},
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"tactic": "rcases H f' η ηpos.ne' f'_mem with ⟨g, hg, Pg⟩"
},
{
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"tactic": "refine' ⟨g, _, Pg⟩"
},
{
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"tactic": "convert (hη _ _ (hf.aestronglyMeasurable.sub f'.aestronglyMeasurable)\n (f'.aestronglyMeasurable.sub (h2P g Pg)) hf'.le hg).le using 2"
},
{
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"tactic": "simp only [sub_add_sub_cancel]"
},
{
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"tactic": "intro c s hs ε εpos Hs"
},
{
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"tactic": "rcases eq_or_ne c 0 with (rfl | hc)"
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"tactic": "rcases h0P (0 : E) MeasurableSet.empty (by simp only [measure_empty, WithTop.zero_lt_top])\n εpos with ⟨g, hg, Pg⟩"
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"tactic": "rw [← snorm_neg, neg_sub] at hg"
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{
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"tactic": "refine' ⟨g, _, Pg⟩"
},
{
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"tactic": "convert hg"
},
{
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"tactic": "ext x"
},
{
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"tactic": "simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_zero,\n piecewise_eq_indicator, indicator_zero', Pi.zero_apply, indicator_zero]"
},
{
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"tactic": "simp only [measure_empty, WithTop.zero_lt_top]"
},
{
"state_after": "case H.h_ind.inr\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε : ε✝ ≠ 0\nhp_pos : p ≠ 0\nc : E\ns : Set α\nhs : MeasurableSet s\nε : ℝ≥0∞\nεpos : ε ≠ 0\nHs : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\nhc : c ≠ 0\nthis : ↑↑μ s < ⊤\n⊢ ∃ g, snorm (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) - g) p μ ≤ ε ∧ P g",
"state_before": "case H.h_ind.inr\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε : ε✝ ≠ 0\nhp_pos : p ≠ 0\nc : E\ns : Set α\nhs : MeasurableSet s\nε : ℝ≥0∞\nεpos : ε ≠ 0\nHs : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\nhc : c ≠ 0\n⊢ ∃ g, snorm (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) - g) p μ ≤ ε ∧ P g",
"tactic": "have : μ s < ∞ := SimpleFunc.measure_lt_top_of_memℒp_indicator hp_pos hp_ne_top hc hs Hs"
},
{
"state_after": "case H.h_ind.inr.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε : ε✝ ≠ 0\nhp_pos : p ≠ 0\nc : E\ns : Set α\nhs : MeasurableSet s\nε : ℝ≥0∞\nεpos : ε ≠ 0\nHs : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\nhc : c ≠ 0\nthis : ↑↑μ s < ⊤\ng : α → E\nhg : snorm (g - Set.indicator s fun x => c) p μ ≤ ε\nPg : P g\n⊢ ∃ g, snorm (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) - g) p μ ≤ ε ∧ P g",
"state_before": "case H.h_ind.inr\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε : ε✝ ≠ 0\nhp_pos : p ≠ 0\nc : E\ns : Set α\nhs : MeasurableSet s\nε : ℝ≥0∞\nεpos : ε ≠ 0\nHs : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\nhc : c ≠ 0\nthis : ↑↑μ s < ⊤\n⊢ ∃ g, snorm (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) - g) p μ ≤ ε ∧ P g",
"tactic": "rcases h0P c hs this εpos with ⟨g, hg, Pg⟩"
},
{
"state_after": "case H.h_ind.inr.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε : ε✝ ≠ 0\nhp_pos : p ≠ 0\nc : E\ns : Set α\nhs : MeasurableSet s\nε : ℝ≥0∞\nεpos : ε ≠ 0\nHs : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\nhc : c ≠ 0\nthis : ↑↑μ s < ⊤\ng : α → E\nhg : snorm ((Set.indicator s fun x => c) - g) p μ ≤ ε\nPg : P g\n⊢ ∃ g, snorm (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) - g) p μ ≤ ε ∧ P g",
"state_before": "case H.h_ind.inr.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε : ε✝ ≠ 0\nhp_pos : p ≠ 0\nc : E\ns : Set α\nhs : MeasurableSet s\nε : ℝ≥0∞\nεpos : ε ≠ 0\nHs : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\nhc : c ≠ 0\nthis : ↑↑μ s < ⊤\ng : α → E\nhg : snorm (g - Set.indicator s fun x => c) p μ ≤ ε\nPg : P g\n⊢ ∃ g, snorm (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) - g) p μ ≤ ε ∧ P g",
"tactic": "rw [← snorm_neg, neg_sub] at hg"
},
{
"state_after": "no goals",
"state_before": "case H.h_ind.inr.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε : ε✝ ≠ 0\nhp_pos : p ≠ 0\nc : E\ns : Set α\nhs : MeasurableSet s\nε : ℝ≥0∞\nεpos : ε ≠ 0\nHs : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\nhc : c ≠ 0\nthis : ↑↑μ s < ⊤\ng : α → E\nhg : snorm ((Set.indicator s fun x => c) - g) p μ ≤ ε\nPg : P g\n⊢ ∃ g, snorm (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) - g) p μ ≤ ε ∧ P g",
"tactic": "exact ⟨g, hg, Pg⟩"
},
{
"state_after": "case H.h_add\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑(f + f')) p\n⊢ ∃ g, snorm (↑(f + f') - g) p μ ≤ δ ∧ P g",
"state_before": "case H.h_add\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\n⊢ ∀ ⦃f g : α →ₛ E⦄,\n Disjoint (support ↑f) (support ↑g) →\n (∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g) →\n (∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑g) p → ∃ g_1, snorm (↑g - g_1) p μ ≤ δ ∧ P g_1) →\n ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑(f + g)) p → ∃ g_1, snorm (↑(f + g) - g_1) p μ ≤ δ ∧ P g_1",
"tactic": "intro f f' hff' hf hf' δ δpos int_ff'"
},
{
"state_after": "case H.h_add.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑(f + f')) p\nη : ℝ≥0∞\nηpos : 0 < η\nhη :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ\n⊢ ∃ g, snorm (↑(f + f') - g) p μ ≤ δ ∧ P g",
"state_before": "case H.h_add\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑(f + f')) p\n⊢ ∃ g, snorm (↑(f + f') - g) p μ ≤ δ ∧ P g",
"tactic": "obtain ⟨η, ηpos, hη⟩ := exists_Lp_half E μ p δpos"
},
{
"state_after": "case H.h_add.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑f) p ∧ Memℒp (↑f') p\nη : ℝ≥0∞\nηpos : 0 < η\nhη :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ\n⊢ ∃ g, snorm (↑(f + f') - g) p μ ≤ δ ∧ P g",
"state_before": "case H.h_add.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑(f + f')) p\nη : ℝ≥0∞\nηpos : 0 < η\nhη :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ\n⊢ ∃ g, snorm (↑(f + f') - g) p μ ≤ δ ∧ P g",
"tactic": "rw [SimpleFunc.coe_add,\n memℒp_add_of_disjoint hff' f.stronglyMeasurable f'.stronglyMeasurable] at int_ff'"
},
{
"state_after": "case H.h_add.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑f) p ∧ Memℒp (↑f') p\nη : ℝ≥0∞\nηpos : 0 < η\nhη :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ\ng : α → E\nhg : snorm (↑f - g) p μ ≤ η\nPg : P g\n⊢ ∃ g, snorm (↑(f + f') - g) p μ ≤ δ ∧ P g",
"state_before": "case H.h_add.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑f) p ∧ Memℒp (↑f') p\nη : ℝ≥0∞\nηpos : 0 < η\nhη :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ\n⊢ ∃ g, snorm (↑(f + f') - g) p μ ≤ δ ∧ P g",
"tactic": "rcases hf η ηpos.ne' int_ff'.1 with ⟨g, hg, Pg⟩"
},
{
"state_after": "case H.h_add.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑f) p ∧ Memℒp (↑f') p\nη : ℝ≥0∞\nηpos : 0 < η\nhη :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ\ng : α → E\nhg : snorm (↑f - g) p μ ≤ η\nPg : P g\ng' : α → E\nhg' : snorm (↑f' - g') p μ ≤ η\nPg' : P g'\n⊢ ∃ g, snorm (↑(f + f') - g) p μ ≤ δ ∧ P g",
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"tactic": "rcases hf' η ηpos.ne' int_ff'.2 with ⟨g', hg', Pg'⟩"
},
{
"state_after": "case H.h_add.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑f) p ∧ Memℒp (↑f') p\nη : ℝ≥0∞\nηpos : 0 < η\nhη :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ\ng : α → E\nhg : snorm (↑f - g) p μ ≤ η\nPg : P g\ng' : α → E\nhg' : snorm (↑f' - g') p μ ≤ η\nPg' : P g'\n⊢ snorm (↑(f + f') - (g + g')) p μ ≤ δ",
"state_before": "case H.h_add.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑f) p ∧ Memℒp (↑f') p\nη : ℝ≥0∞\nηpos : 0 < η\nhη :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ\ng : α → E\nhg : snorm (↑f - g) p μ ≤ η\nPg : P g\ng' : α → E\nhg' : snorm (↑f' - g') p μ ≤ η\nPg' : P g'\n⊢ ∃ g, snorm (↑(f + f') - g) p μ ≤ δ ∧ P g",
"tactic": "refine' ⟨g + g', _, h1P g g' Pg Pg'⟩"
},
{
"state_after": "case h.e'_3.h.e'_5\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑f) p ∧ Memℒp (↑f') p\nη : ℝ≥0∞\nηpos : 0 < η\nhη :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ\ng : α → E\nhg : snorm (↑f - g) p μ ≤ η\nPg : P g\ng' : α → E\nhg' : snorm (↑f' - g') p μ ≤ η\nPg' : P g'\n⊢ ↑(f + f') - (g + g') = ↑f - g + (↑f' - g')",
"state_before": "case H.h_add.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑f) p ∧ Memℒp (↑f') p\nη : ℝ≥0∞\nηpos : 0 < η\nhη :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ\ng : α → E\nhg : snorm (↑f - g) p μ ≤ η\nPg : P g\ng' : α → E\nhg' : snorm (↑f' - g') p μ ≤ η\nPg' : P g'\n⊢ snorm (↑(f + f') - (g + g')) p μ ≤ δ",
"tactic": "convert (hη _ _ (f.aestronglyMeasurable.sub (h2P g Pg))\n (f'.aestronglyMeasurable.sub (h2P g' Pg')) hg hg').le using 2"
},
{
"state_after": "case h.e'_3.h.e'_5\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑f) p ∧ Memℒp (↑f') p\nη : ℝ≥0∞\nηpos : 0 < η\nhη :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ\ng : α → E\nhg : snorm (↑f - g) p μ ≤ η\nPg : P g\ng' : α → E\nhg' : snorm (↑f' - g') p μ ≤ η\nPg' : P g'\n⊢ ↑f + ↑f' - (g + g') = ↑f - g + (↑f' - g')",
"state_before": "case h.e'_3.h.e'_5\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑f) p ∧ Memℒp (↑f') p\nη : ℝ≥0∞\nηpos : 0 < η\nhη :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ\ng : α → E\nhg : snorm (↑f - g) p μ ≤ η\nPg : P g\ng' : α → E\nhg' : snorm (↑f' - g') p μ ≤ η\nPg' : P g'\n⊢ ↑(f + f') - (g + g') = ↑f - g + (↑f' - g')",
"tactic": "rw [SimpleFunc.coe_add]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_5\nα : Type u_1\nβ : Type ?u.3350178\nι : Type ?u.3350181\nE : Type u_2\nF : Type ?u.3350187\n𝕜 : Type ?u.3350190\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝¹ : α → E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → ↑↑μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E), P f → P g → P (f + g)\nh2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ\nf✝ : α → E\nhf✝ : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nhp_pos : p ≠ 0\nf f' : α →ₛ E\nhff' : Disjoint (support ↑f) (support ↑f')\nhf : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f) p → ∃ g, snorm (↑f - g) p μ ≤ δ ∧ P g\nhf' : ∀ (δ : ℝ≥0∞), δ ≠ 0 → Memℒp (↑f') p → ∃ g, snorm (↑f' - g) p μ ≤ δ ∧ P g\nδ : ℝ≥0∞\nδpos : δ ≠ 0\nint_ff' : Memℒp (↑f) p ∧ Memℒp (↑f') p\nη : ℝ≥0∞\nηpos : 0 < η\nhη :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ\ng : α → E\nhg : snorm (↑f - g) p μ ≤ η\nPg : P g\ng' : α → E\nhg' : snorm (↑f' - g') p μ ≤ η\nPg' : P g'\n⊢ ↑f + ↑f' - (g + g') = ↑f - g + (↑f' - g')",
"tactic": "abel"
}
] |
[
1071,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1025,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.cons_inter_of_pos
|
[] |
[
1764,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1763,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
top_himp
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.29794\nα : Type u_1\nβ : Type ?u.29800\ninst✝ : GeneralizedHeytingAlgebra α\na b✝ c d b : α\n⊢ b ≤ ⊤ ⇨ a ↔ b ≤ a",
"tactic": "rw [le_himp_iff, inf_top_eq]"
}
] |
[
378,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
ContinuousAt.upperSemicontinuousAt
|
[] |
[
816,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
815,
1
] |
Mathlib/Data/Multiset/Sum.lean
|
Multiset.mem_disjSum
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na : α\nb : β\nx : α ⊕ β\n⊢ x ∈ disjSum s t ↔ (∃ a, a ∈ s ∧ inl a = x) ∨ ∃ b, b ∈ t ∧ inr b = x",
"tactic": "simp_rw [disjSum, mem_add, mem_map]"
}
] |
[
54,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.le_def'
|
[
{
"state_after": "no goals",
"state_before": "x✝ y✝ x y : ℝ\n⊢ Real.le x y ↔ x < y ∨ x = y",
"tactic": "rw [le_def]"
}
] |
[
356,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
9
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
|
CategoryTheory.Limits.IsZero.iff_id_eq_zero
|
[
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type u'\ninst✝¹ : Category D\ninst✝ : HasZeroMorphisms C\nX : C\nh : 𝟙 X = 0\nY : C\nf : X ⟶ Y\n⊢ 0 = default",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type u'\ninst✝¹ : Category D\ninst✝ : HasZeroMorphisms C\nX : C\nh : 𝟙 X = 0\nY : C\nf : X ⟶ Y\n⊢ f = default",
"tactic": "rw [← id_comp f, ← id_comp (0: X ⟶ Y), h, zero_comp, zero_comp]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type u'\ninst✝¹ : Category D\ninst✝ : HasZeroMorphisms C\nX : C\nh : 𝟙 X = 0\nY : C\nf : X ⟶ Y\n⊢ 0 = default",
"tactic": "simp only"
},
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type u'\ninst✝¹ : Category D\ninst✝ : HasZeroMorphisms C\nX : C\nh : 𝟙 X = 0\nY : C\nf : Y ⟶ X\n⊢ 0 = default",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type u'\ninst✝¹ : Category D\ninst✝ : HasZeroMorphisms C\nX : C\nh : 𝟙 X = 0\nY : C\nf : Y ⟶ X\n⊢ f = default",
"tactic": "rw [← comp_id f, ← comp_id (0 : Y ⟶ X), h, comp_zero, comp_zero]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type u'\ninst✝¹ : Category D\ninst✝ : HasZeroMorphisms C\nX : C\nh : 𝟙 X = 0\nY : C\nf : Y ⟶ X\n⊢ 0 = default",
"tactic": "simp only"
}
] |
[
194,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.get?_zipWith
|
[] |
[
592,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
590,
1
] |
Mathlib/Topology/Algebra/Polynomial.lean
|
Polynomial.continuousOn_aeval
|
[] |
[
97,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
11
] |
Mathlib/RingTheory/Localization/LocalizationLocalization.lean
|
IsFractionRing.isFractionRing_of_isLocalization
|
[
{
"state_after": "R : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\n⊢ IsFractionRing S T",
"state_before": "R : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\n⊢ IsFractionRing S T",
"tactic": "have := isLocalization_of_submonoid_le S T M (nonZeroDivisors R) hM"
},
{
"state_after": "case refine_1\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\n⊢ Submonoid.map (algebraMap R S) (nonZeroDivisors R) ≤ nonZeroDivisors S\n\ncase refine_2\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\n⊢ ∀ (x : { x // x ∈ nonZeroDivisors S }), ∃ m, m * ↑x ∈ Submonoid.map (algebraMap R S) (nonZeroDivisors R)",
"state_before": "R : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\n⊢ IsFractionRing S T",
"tactic": "refine @isLocalization_of_is_exists_mul_mem _ _ _ _ _ _ _ this ?_ ?_"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\n⊢ Submonoid.map (algebraMap R S) (nonZeroDivisors R) ≤ nonZeroDivisors S",
"tactic": "exact map_nonZeroDivisors_le M S"
},
{
"state_after": "case refine_2.mk\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\n⊢ ∃ m, m * ↑{ val := x, property := hx } ∈ Submonoid.map (algebraMap R S) (nonZeroDivisors R)",
"state_before": "case refine_2\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\n⊢ ∀ (x : { x // x ∈ nonZeroDivisors S }), ∃ m, m * ↑x ∈ Submonoid.map (algebraMap R S) (nonZeroDivisors R)",
"tactic": "rintro ⟨x, hx⟩"
},
{
"state_after": "case refine_2.mk.intro.mk\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\n⊢ ∃ m, m * ↑{ val := x, property := hx } ∈ Submonoid.map (algebraMap R S) (nonZeroDivisors R)",
"state_before": "case refine_2.mk\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\n⊢ ∃ m, m * ↑{ val := x, property := hx } ∈ Submonoid.map (algebraMap R S) (nonZeroDivisors R)",
"tactic": "obtain ⟨⟨y, s⟩, e⟩ := IsLocalization.surj M x"
},
{
"state_after": "case refine_2.mk.intro.mk\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\n⊢ ↑(algebraMap R S) ↑s * ↑{ val := x, property := hx } ∈ Submonoid.map (algebraMap R S) (nonZeroDivisors R)",
"state_before": "case refine_2.mk.intro.mk\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\n⊢ ∃ m, m * ↑{ val := x, property := hx } ∈ Submonoid.map (algebraMap R S) (nonZeroDivisors R)",
"tactic": "use algebraMap R S s"
},
{
"state_after": "case refine_2.mk.intro.mk\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\n⊢ ↑(algebraMap R S) (y, s).fst ∈ Submonoid.map (algebraMap R S) (nonZeroDivisors R)",
"state_before": "case refine_2.mk.intro.mk\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\n⊢ ↑(algebraMap R S) ↑s * ↑{ val := x, property := hx } ∈ Submonoid.map (algebraMap R S) (nonZeroDivisors R)",
"tactic": "rw [mul_comm, Subtype.coe_mk, e]"
},
{
"state_after": "case refine_2.mk.intro.mk\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\n⊢ (y, s).fst ∈ ↑(nonZeroDivisors R)",
"state_before": "case refine_2.mk.intro.mk\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\n⊢ ↑(algebraMap R S) (y, s).fst ∈ Submonoid.map (algebraMap R S) (nonZeroDivisors R)",
"tactic": "refine' Set.mem_image_of_mem (algebraMap R S) _"
},
{
"state_after": "case refine_2.mk.intro.mk\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\nz : R\nhz : z * (y, s).fst = 0\n⊢ z = 0",
"state_before": "case refine_2.mk.intro.mk\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\n⊢ (y, s).fst ∈ ↑(nonZeroDivisors R)",
"tactic": "intro z hz"
},
{
"state_after": "case refine_2.mk.intro.mk.a\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\nz : R\nhz : z * (y, s).fst = 0\n⊢ ↑(algebraMap R S) z = ↑(algebraMap R S) 0",
"state_before": "case refine_2.mk.intro.mk\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\nz : R\nhz : z * (y, s).fst = 0\n⊢ z = 0",
"tactic": "apply IsLocalization.injective S hM"
},
{
"state_after": "case refine_2.mk.intro.mk.a\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\nz : R\nhz : z * (y, s).fst = 0\n⊢ ↑(algebraMap R S) z = 0",
"state_before": "case refine_2.mk.intro.mk.a\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\nz : R\nhz : z * (y, s).fst = 0\n⊢ ↑(algebraMap R S) z = ↑(algebraMap R S) 0",
"tactic": "rw [map_zero]"
},
{
"state_after": "case refine_2.mk.intro.mk.a.a\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\nz : R\nhz : z * (y, s).fst = 0\n⊢ ↑(algebraMap R S) z * x = 0",
"state_before": "case refine_2.mk.intro.mk.a\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\nz : R\nhz : z * (y, s).fst = 0\n⊢ ↑(algebraMap R S) z = 0",
"tactic": "apply hx"
},
{
"state_after": "no goals",
"state_before": "case refine_2.mk.intro.mk.a.a\nR : Type u_3\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS✝ : Type ?u.661559\ninst✝¹⁰ : CommRing S✝\ninst✝⁹ : Algebra R S✝\nP : Type ?u.661815\ninst✝⁸ : CommRing P\nS : Type u_1\nT : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T\nx : S\nhx : x ∈ nonZeroDivisors S\ny : R\ns : { x // x ∈ M }\ne : x * ↑(algebraMap R S) ↑(y, s).snd = ↑(algebraMap R S) (y, s).fst\nz : R\nhz : z * (y, s).fst = 0\n⊢ ↑(algebraMap R S) z * x = 0",
"tactic": "rw [← (map_units S s).mul_left_inj, mul_assoc, e, ← map_mul, hz, map_zero,\n MulZeroClass.zero_mul]"
}
] |
[
293,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.le_of_le_of_eq
|
[] |
[
753,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
752,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.comp_toGerm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.444612\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ng : β → γ\nhg : Continuous g\nf✝ : α →ₘ[μ] β\nf : α → β\nx✝ : AEStronglyMeasurable f μ\n⊢ toGerm (comp g hg (mk f x✝)) = Germ.map g (toGerm (mk f x✝))",
"tactic": "simp"
}
] |
[
392,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
HasDerivWithinAt.sub_const
|
[] |
[
333,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
8
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measure_eq_zero_of_trim_eq_zero
|
[] |
[
4381,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4380,
1
] |
Mathlib/Algebra/GroupWithZero/Power.lean
|
pow_sub_of_lt
|
[
{
"state_after": "case inl\nG₀ : Type u_1\ninst✝ : GroupWithZero G₀\na : G₀\nm✝ n✝ m n : ℕ\nh : n < m\n⊢ 0 ^ (m - n) = 0 ^ m * (0 ^ n)⁻¹\n\ncase inr\nG₀ : Type u_1\ninst✝ : GroupWithZero G₀\na✝ : G₀\nm✝ n✝ : ℕ\na : G₀\nm n : ℕ\nh : n < m\nha : a ≠ 0\n⊢ a ^ (m - n) = a ^ m * (a ^ n)⁻¹",
"state_before": "G₀ : Type u_1\ninst✝ : GroupWithZero G₀\na✝ : G₀\nm✝ n✝ : ℕ\na : G₀\nm n : ℕ\nh : n < m\n⊢ a ^ (m - n) = a ^ m * (a ^ n)⁻¹",
"tactic": "obtain rfl | ha := eq_or_ne a 0"
},
{
"state_after": "no goals",
"state_before": "case inl\nG₀ : Type u_1\ninst✝ : GroupWithZero G₀\na : G₀\nm✝ n✝ m n : ℕ\nh : n < m\n⊢ 0 ^ (m - n) = 0 ^ m * (0 ^ n)⁻¹",
"tactic": "rw [zero_pow (tsub_pos_of_lt h), zero_pow (n.zero_le.trans_lt h), zero_mul]"
},
{
"state_after": "no goals",
"state_before": "case inr\nG₀ : Type u_1\ninst✝ : GroupWithZero G₀\na✝ : G₀\nm✝ n✝ : ℕ\na : G₀\nm n : ℕ\nh : n < m\nha : a ≠ 0\n⊢ a ^ (m - n) = a ^ m * (a ^ n)⁻¹",
"tactic": "exact pow_sub₀ _ ha h.le"
}
] |
[
37,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
34,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean
|
ContDiffBump.continuous
|
[] |
[
477,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
476,
11
] |
Mathlib/Analysis/Calculus/LocalExtr.lean
|
posTangentConeAt_univ
|
[] |
[
111,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.infiniteNeg_add_infiniteNeg
|
[] |
[
569,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
567,
1
] |
Mathlib/MeasureTheory/Measure/OpenPos.lean
|
MeasureTheory.Measure.eqOn_Ico_of_ae_eq
|
[] |
[
165,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.optionEquivSumPUnit_none
|
[] |
[
425,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Topology/Algebra/Order/Compact.lean
|
IsCompact.exists_isLUB
|
[] |
[
204,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
Filter.HasBasis.uniformity_of_nhds_one
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.173356\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\nι : Sort u_1\np : ι → Prop\nU : ι → Set α\nh : HasBasis (𝓝 1) p U\n⊢ HasBasis (Filter.comap (fun x => x.snd / x.fst) (𝓝 1)) p fun i => {x | x.snd / x.fst ∈ U i}",
"state_before": "α : Type u_2\nβ : Type ?u.173356\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\nι : Sort u_1\np : ι → Prop\nU : ι → Set α\nh : HasBasis (𝓝 1) p U\n⊢ HasBasis (𝓤 α) p fun i => {x | x.snd / x.fst ∈ U i}",
"tactic": "rw [uniformity_eq_comap_nhds_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.173356\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\nι : Sort u_1\np : ι → Prop\nU : ι → Set α\nh : HasBasis (𝓝 1) p U\n⊢ HasBasis (Filter.comap (fun x => x.snd / x.fst) (𝓝 1)) p fun i => {x | x.snd / x.fst ∈ U i}",
"tactic": "exact h.comap _"
}
] |
[
343,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/Algebra/DirectSum/Internal.lean
|
SetLike.homogeneous_zero_submodule
|
[] |
[
347,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.diagonal_smul
|
[
{
"state_after": "case a.h\nl : Type ?u.54869\nm : Type ?u.54872\nn : Type u_2\no : Type ?u.54878\nm' : o → Type ?u.54883\nn' : o → Type ?u.54888\nR : Type u_1\nS : Type ?u.54894\nα : Type v\nβ : Type w\nγ : Type ?u.54901\ninst✝³ : DecidableEq n\ninst✝² : Monoid R\ninst✝¹ : AddMonoid α\ninst✝ : DistribMulAction R α\nr : R\nd : n → α\ni j : n\n⊢ diagonal (r • d) i j = (r • diagonal d) i j",
"state_before": "l : Type ?u.54869\nm : Type ?u.54872\nn : Type u_2\no : Type ?u.54878\nm' : o → Type ?u.54883\nn' : o → Type ?u.54888\nR : Type u_1\nS : Type ?u.54894\nα : Type v\nβ : Type w\nγ : Type ?u.54901\ninst✝³ : DecidableEq n\ninst✝² : Monoid R\ninst✝¹ : AddMonoid α\ninst✝ : DistribMulAction R α\nr : R\nd : n → α\n⊢ diagonal (r • d) = r • diagonal d",
"tactic": "ext i j"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type ?u.54869\nm : Type ?u.54872\nn : Type u_2\no : Type ?u.54878\nm' : o → Type ?u.54883\nn' : o → Type ?u.54888\nR : Type u_1\nS : Type ?u.54894\nα : Type v\nβ : Type w\nγ : Type ?u.54901\ninst✝³ : DecidableEq n\ninst✝² : Monoid R\ninst✝¹ : AddMonoid α\ninst✝ : DistribMulAction R α\nr : R\nd : n → α\ni j : n\n⊢ diagonal (r • d) i j = (r • diagonal d) i j",
"tactic": "by_cases h : i = j <;>\nsimp [h]"
}
] |
[
485,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.diag_map
|
[] |
[
676,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
675,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
OneHom.toFun_eq_coe
|
[] |
[
591,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
591,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.log_b_ne_zero
|
[
{
"state_after": "case b_ne_zero\nb x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\n⊢ b ≠ 0\n\nb x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nb_ne_zero : b ≠ 0\n⊢ log b ≠ 0",
"state_before": "b x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\n⊢ log b ≠ 0",
"tactic": "have b_ne_zero : b ≠ 0"
},
{
"state_after": "b x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nb_ne_zero : b ≠ 0\n⊢ log b ≠ 0",
"state_before": "case b_ne_zero\nb x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\n⊢ b ≠ 0\n\nb x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nb_ne_zero : b ≠ 0\n⊢ log b ≠ 0",
"tactic": "linarith"
},
{
"state_after": "case b_ne_minus_one\nb x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nb_ne_zero : b ≠ 0\n⊢ b ≠ -1\n\nb x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nb_ne_zero : b ≠ 0\nb_ne_minus_one : b ≠ -1\n⊢ log b ≠ 0",
"state_before": "b x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nb_ne_zero : b ≠ 0\n⊢ log b ≠ 0",
"tactic": "have b_ne_minus_one : b ≠ -1"
},
{
"state_after": "b x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nb_ne_zero : b ≠ 0\nb_ne_minus_one : b ≠ -1\n⊢ log b ≠ 0",
"state_before": "case b_ne_minus_one\nb x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nb_ne_zero : b ≠ 0\n⊢ b ≠ -1\n\nb x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nb_ne_zero : b ≠ 0\nb_ne_minus_one : b ≠ -1\n⊢ log b ≠ 0",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "b x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nb_ne_zero : b ≠ 0\nb_ne_minus_one : b ≠ -1\n⊢ log b ≠ 0",
"tactic": "simp [b_ne_one, b_ne_zero, b_ne_minus_one]"
}
] |
[
87,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
9
] |
Mathlib/Data/Dfinsupp/NeLocus.lean
|
Dfinsupp.neLocus_comm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → Zero (N a)\nf g : Π₀ (a : α), N a\n⊢ neLocus f g = neLocus g f",
"tactic": "simp_rw [neLocus, Finset.union_comm, ne_comm]"
}
] |
[
71,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean
|
spectrum.map_polynomial_aeval_of_nonempty
|
[
{
"state_after": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhnon : Set.Nonempty (σ a)\n✝ : Nontrivial A\n⊢ σ (↑(aeval a) p) = (fun k => eval k p) '' σ a",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhnon : Set.Nonempty (σ a)\n⊢ σ (↑(aeval a) p) = (fun k => eval k p) '' σ a",
"tactic": "nontriviality A"
},
{
"state_after": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhnon : Set.Nonempty (σ a)\n✝ : Nontrivial A\nh : degree p ≤ 0\n⊢ σ (↑(aeval a) p) = (fun k => eval k p) '' σ a",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhnon : Set.Nonempty (σ a)\n✝ : Nontrivial A\n⊢ σ (↑(aeval a) p) = (fun k => eval k p) '' σ a",
"tactic": "refine' Or.elim (le_or_gt (degree p) 0) (fun h => _) (map_polynomial_aeval_of_degree_pos a p)"
},
{
"state_after": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhnon : Set.Nonempty (σ a)\n✝ : Nontrivial A\nh : degree p ≤ 0\n⊢ σ (↑(aeval a) (↑C (coeff p 0))) = (fun k => eval k (↑C (coeff p 0))) '' σ a",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhnon : Set.Nonempty (σ a)\n✝ : Nontrivial A\nh : degree p ≤ 0\n⊢ σ (↑(aeval a) p) = (fun k => eval k p) '' σ a",
"tactic": "rw [eq_C_of_degree_le_zero h]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhnon : Set.Nonempty (σ a)\n✝ : Nontrivial A\nh : degree p ≤ 0\n⊢ σ (↑(aeval a) (↑C (coeff p 0))) = (fun k => eval k (↑C (coeff p 0))) '' σ a",
"tactic": "simp only [Set.image_congr, eval_C, aeval_C, scalar_eq, Set.Nonempty.image_const hnon]"
}
] |
[
132,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Analysis/NormedSpace/Pointwise.lean
|
exists_dist_le_lt
|
[
{
"state_after": "𝕜 : Type ?u.210210\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y✝ z : E\nδ ε : ℝ\nhδ : 0 ≤ δ\nhε : 0 < ε\nh : dist x z < ε + δ\ny : E\nhy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z\n⊢ dist x y ≤ δ ∧ dist y z < ε",
"state_before": "𝕜 : Type ?u.210210\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y z : E\nδ ε : ℝ\nhδ : 0 ≤ δ\nhε : 0 < ε\nh : dist x z < ε + δ\n⊢ ∃ y, dist x y ≤ δ ∧ dist y z < ε",
"tactic": "refine' (exists_dist_eq x z (div_nonneg hε.le <| add_nonneg hε.le hδ)\n (div_nonneg hδ <| add_nonneg hε.le hδ) <| by\n rw [← add_div, div_self (add_pos_of_pos_of_nonneg hε hδ).ne']).imp\n fun y hy => _"
},
{
"state_after": "𝕜 : Type ?u.210210\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y✝ z : E\nδ ε : ℝ\nhδ : 0 ≤ δ\nhε : 0 < ε\nh : dist x z < ε + δ\ny : E\nhy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z\n⊢ dist x z / (ε + δ) * δ ≤ δ ∧ dist x z / (ε + δ) * ε < ε",
"state_before": "𝕜 : Type ?u.210210\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y✝ z : E\nδ ε : ℝ\nhδ : 0 ≤ δ\nhε : 0 < ε\nh : dist x z < ε + δ\ny : E\nhy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z\n⊢ dist x y ≤ δ ∧ dist y z < ε",
"tactic": "rw [hy.1, hy.2, div_mul_comm, div_mul_comm ε]"
},
{
"state_after": "𝕜 : Type ?u.210210\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y✝ z : E\nδ ε : ℝ\nhδ : 0 ≤ δ\nhε : 0 < ε\nh : dist x z / (ε + δ) < 1\ny : E\nhy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z\n⊢ dist x z / (ε + δ) * δ ≤ δ ∧ dist x z / (ε + δ) * ε < ε",
"state_before": "𝕜 : Type ?u.210210\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y✝ z : E\nδ ε : ℝ\nhδ : 0 ≤ δ\nhε : 0 < ε\nh : dist x z < ε + δ\ny : E\nhy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z\n⊢ dist x z / (ε + δ) * δ ≤ δ ∧ dist x z / (ε + δ) * ε < ε",
"tactic": "rw [← div_lt_one (add_pos_of_pos_of_nonneg hε hδ)] at h"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.210210\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y✝ z : E\nδ ε : ℝ\nhδ : 0 ≤ δ\nhε : 0 < ε\nh : dist x z / (ε + δ) < 1\ny : E\nhy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z\n⊢ dist x z / (ε + δ) * δ ≤ δ ∧ dist x z / (ε + δ) * ε < ε",
"tactic": "exact ⟨mul_le_of_le_one_left hδ h.le, mul_lt_of_lt_one_left hε h⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.210210\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y z : E\nδ ε : ℝ\nhδ : 0 ≤ δ\nhε : 0 < ε\nh : dist x z < ε + δ\n⊢ ε / (ε + δ) + δ / (ε + δ) = 1",
"tactic": "rw [← add_div, div_self (add_pos_of_pos_of_nonneg hε hδ).ne']"
}
] |
[
187,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean
|
ContinuousAffineMap.norm_comp_le
|
[
{
"state_after": "𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ ‖↑(comp f g) 0‖ ≤ ‖f‖ * ‖g‖ + ‖↑f 0‖ ∧ ‖contLinear (comp f g)‖ ≤ ‖f‖ * ‖g‖ + ‖↑f 0‖",
"state_before": "𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ ‖comp f g‖ ≤ ‖f‖ * ‖g‖ + ‖↑f 0‖",
"tactic": "rw [norm_def, max_le_iff]"
},
{
"state_after": "case left\n𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ ‖↑(comp f g) 0‖ ≤ ‖f‖ * ‖g‖ + ‖↑f 0‖\n\ncase right\n𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ ‖contLinear (comp f g)‖ ≤ ‖f‖ * ‖g‖ + ‖↑f 0‖",
"state_before": "𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ ‖↑(comp f g) 0‖ ≤ ‖f‖ * ‖g‖ + ‖↑f 0‖ ∧ ‖contLinear (comp f g)‖ ≤ ‖f‖ * ‖g‖ + ‖↑f 0‖",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case left\n𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ ‖↑(comp f g) 0‖ ≤ ‖f‖ * ‖g‖ + ‖↑f 0‖",
"tactic": "calc\n ‖f.comp g 0‖ = ‖f (g 0)‖ := by simp\n _ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp\n _ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=\n ((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))\n _ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=\n add_le_add_right\n (mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ ‖↑(comp f g) 0‖ = ‖↑f (↑g 0)‖",
"tactic": "simp"
},
{
"state_after": "𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ ‖(↑(contLinear f) + Function.const V (↑f 0)) (↑g 0)‖ =\n ‖↑(contLinear f) (↑g 0) + (↑(contLinear f) + Function.const V (↑f 0)) 0‖",
"state_before": "𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ ‖↑f (↑g 0)‖ = ‖↑(contLinear f) (↑g 0) + ↑f 0‖",
"tactic": "rw [f.decomp]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ ‖(↑(contLinear f) + Function.const V (↑f 0)) (↑g 0)‖ =\n ‖↑(contLinear f) (↑g 0) + (↑(contLinear f) + Function.const V (↑f 0)) 0‖",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case right\n𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ ‖contLinear (comp f g)‖ ≤ ‖f‖ * ‖g‖ + ‖↑f 0‖",
"tactic": "calc\n ‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=\n (g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _\n _ ≤ ‖f‖ * ‖g‖ :=\n (mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))\n _ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by rw [le_add_iff_nonneg_right]; apply norm_nonneg"
},
{
"state_after": "𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ 0 ≤ ‖↑f 0‖",
"state_before": "𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ ‖f‖ * ‖g‖ ≤ ‖f‖ * ‖g‖ + ‖↑f 0‖",
"tactic": "rw [le_add_iff_nonneg_right]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nR : Type ?u.276372\nV : Type u_3\nW : Type u_4\nW₂ : Type u_2\nP : Type ?u.276384\nQ : Type ?u.276387\nQ₂ : Type ?u.276390\ninst✝¹⁶ : NormedAddCommGroup V\ninst✝¹⁵ : MetricSpace P\ninst✝¹⁴ : NormedAddTorsor V P\ninst✝¹³ : NormedAddCommGroup W\ninst✝¹² : MetricSpace Q\ninst✝¹¹ : NormedAddTorsor W Q\ninst✝¹⁰ : NormedAddCommGroup W₂\ninst✝⁹ : MetricSpace Q₂\ninst✝⁸ : NormedAddTorsor W₂ Q₂\ninst✝⁷ : NormedField R\ninst✝⁶ : NormedSpace R V\ninst✝⁵ : NormedSpace R W\ninst✝⁴ : NormedSpace R W₂\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : NormedSpace 𝕜 W₂\nf : V →A[𝕜] W\ng : W₂ →A[𝕜] V\n⊢ 0 ≤ ‖↑f 0‖",
"tactic": "apply norm_nonneg"
}
] |
[
250,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.add_one_le_ceil_iff
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.209172\nα : Type u_1\nβ : Type ?u.209178\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\n⊢ z + 1 ≤ ⌈a⌉ ↔ ↑z < a",
"tactic": "rw [← lt_ceil, add_one_le_iff]"
}
] |
[
1112,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1112,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
ZFSet.mk_eq
|
[] |
[
684,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
683,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Ordered.lean
|
lineMap_lt_left_iff_lt
|
[] |
[
96,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
HasFDerivAt.sub_const
|
[] |
[
544,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
542,
8
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
inv_nonneg
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.5703\nα : Type u_1\nβ : Type ?u.5709\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\n⊢ 0 ≤ a⁻¹ ↔ 0 ≤ a",
"tactic": "simp only [le_iff_eq_or_lt, inv_pos, zero_eq_inv]"
}
] |
[
61,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/RingTheory/MvPolynomial/Symmetric.lean
|
MvPolynomial.IsSymmetric.one
|
[] |
[
120,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Analysis/SpecialFunctions/Stirling.lean
|
Stirling.log_stirlingSeq_sub_log_stirlingSeq_succ
|
[
{
"state_after": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\n⊢ Real.log (stirlingSeq (succ n)) - Real.log (stirlingSeq (succ (succ n))) ≤ 1 / (4 * ↑(succ n) ^ 2)",
"state_before": "n : ℕ\n⊢ Real.log (stirlingSeq (succ n)) - Real.log (stirlingSeq (succ (succ n))) ≤ 1 / (4 * ↑(succ n) ^ 2)",
"tactic": "have h₁ : ↑0 < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity"
},
{
"state_after": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\n⊢ Real.log (stirlingSeq (succ n)) - Real.log (stirlingSeq (succ (succ n))) ≤ 1 / (4 * ↑(succ n) ^ 2)",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\n⊢ Real.log (stirlingSeq (succ n)) - Real.log (stirlingSeq (succ (succ n))) ≤ 1 / (4 * ↑(succ n) ^ 2)",
"tactic": "have h₃ : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity"
},
{
"state_after": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ Real.log (stirlingSeq (succ n)) - Real.log (stirlingSeq (succ (succ n))) ≤ 1 / (4 * ↑(succ n) ^ 2)",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\n⊢ Real.log (stirlingSeq (succ n)) - Real.log (stirlingSeq (succ (succ n))) ≤ 1 / (4 * ↑(succ n) ^ 2)",
"tactic": "have h₂ : ↑0 < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by\n rw [← mul_lt_mul_right h₃]\n have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]\n convert H using 1 <;> field_simp [h₃.ne']"
},
{
"state_after": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ (1 / (2 * ↑(succ n) + 1)) ^ 2 / (1 - (1 / (2 * ↑(succ n) + 1)) ^ 2) ≤ 1 / (4 * ↑(succ n) ^ 2)",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ Real.log (stirlingSeq (succ n)) - Real.log (stirlingSeq (succ (succ n))) ≤ 1 / (4 * ↑(succ n) ^ 2)",
"tactic": "refine' (log_stirlingSeq_diff_le_geo_sum n).trans _"
},
{
"state_after": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ (1 / (2 * (↑n + 1) + 1)) ^ 2 / (1 - (1 / (2 * (↑n + 1) + 1)) ^ 2) ≤ 1 / (4 * (↑n + 1) ^ 2)",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ (1 / (2 * ↑(succ n) + 1)) ^ 2 / (1 - (1 / (2 * ↑(succ n) + 1)) ^ 2) ≤ 1 / (4 * ↑(succ n) ^ 2)",
"tactic": "push_cast"
},
{
"state_after": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ (1 / (2 * (↑n + 1) + 1)) ^ 2 * (4 * (↑n + 1) ^ 2) ≤ 1 * (1 - (1 / (2 * (↑n + 1) + 1)) ^ 2)",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ (1 / (2 * (↑n + 1) + 1)) ^ 2 / (1 - (1 / (2 * (↑n + 1) + 1)) ^ 2) ≤ 1 / (4 * (↑n + 1) ^ 2)",
"tactic": "rw [div_le_div_iff h₂ h₁]"
},
{
"state_after": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ 4 * (↑n + 1) ^ 2 / (2 * (↑n + 1) + 1) ^ 2 ≤ ((2 * (↑n + 1) + 1) ^ 2 - 1) / (2 * (↑n + 1) + 1) ^ 2",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ (1 / (2 * (↑n + 1) + 1)) ^ 2 * (4 * (↑n + 1) ^ 2) ≤ 1 * (1 - (1 / (2 * (↑n + 1) + 1)) ^ 2)",
"tactic": "field_simp [h₃.ne']"
},
{
"state_after": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ 4 * (↑n + 1) ^ 2 ≤ (2 * (↑n + 1) + 1) ^ 2 - 1",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ 4 * (↑n + 1) ^ 2 / (2 * (↑n + 1) + 1) ^ 2 ≤ ((2 * (↑n + 1) + 1) ^ 2 - 1) / (2 * (↑n + 1) + 1) ^ 2",
"tactic": "rw [div_le_div_right h₃]"
},
{
"state_after": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ 4 + ↑n * 8 + ↑n ^ 2 * 4 ≤ 8 + ↑n * 12 + ↑n ^ 2 * 4",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ 4 * (↑n + 1) ^ 2 ≤ (2 * (↑n + 1) + 1) ^ 2 - 1",
"tactic": "ring_nf"
},
{
"state_after": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ 4 + n * 8 + n ^ 2 * 4 ≤ 8 + n * 12 + n ^ 2 * 4",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ 4 + ↑n * 8 + ↑n ^ 2 * 4 ≤ 8 + ↑n * 12 + ↑n ^ 2 * 4",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nh₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2\n⊢ 4 + n * 8 + n ^ 2 * 4 ≤ 8 + n * 12 + n ^ 2 * 4",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ 0 < 4 * (↑n + 1) ^ 2",
"tactic": "positivity"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\n⊢ 0 < (2 * (↑n + 1) + 1) ^ 2",
"tactic": "positivity"
},
{
"state_after": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\n⊢ 0 * (2 * (↑n + 1) + 1) ^ 2 < (1 - (1 / (2 * (↑n + 1) + 1)) ^ 2) * (2 * (↑n + 1) + 1) ^ 2",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\n⊢ 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2",
"tactic": "rw [← mul_lt_mul_right h₃]"
},
{
"state_after": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nH : 0 < (2 * (↑n + 1) + 1) ^ 2 - 1\n⊢ 0 * (2 * (↑n + 1) + 1) ^ 2 < (1 - (1 / (2 * (↑n + 1) + 1)) ^ 2) * (2 * (↑n + 1) + 1) ^ 2",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\n⊢ 0 * (2 * (↑n + 1) + 1) ^ 2 < (1 - (1 / (2 * (↑n + 1) + 1)) ^ 2) * (2 * (↑n + 1) + 1) ^ 2",
"tactic": "have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\nH : 0 < (2 * (↑n + 1) + 1) ^ 2 - 1\n⊢ 0 * (2 * (↑n + 1) + 1) ^ 2 < (1 - (1 / (2 * (↑n + 1) + 1)) ^ 2) * (2 * (↑n + 1) + 1) ^ 2",
"tactic": "convert H using 1 <;> field_simp [h₃.ne']"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nh₁ : 0 < 4 * (↑n + 1) ^ 2\nh₃ : 0 < (2 * (↑n + 1) + 1) ^ 2\n⊢ 0 < (2 * (↑n + 1) + 1) ^ 2 - 1",
"tactic": "nlinarith [@cast_nonneg ℝ _ n]"
}
] |
[
153,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.eq_of_eq_on_source_univ
|
[] |
[
1005,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1003,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.pow_eq_on_of_mem_support
|
[
{
"state_after": "case zero\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nh : ∀ (x : α), x ∈ support f ∩ support g → ↑f x = ↑g x\n⊢ ∀ (x : α), x ∈ support f ∩ support g → ↑(f ^ Nat.zero) x = ↑(g ^ Nat.zero) x\n\ncase succ\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nh : ∀ (x : α), x ∈ support f ∩ support g → ↑f x = ↑g x\nk : ℕ\nhk : ∀ (x : α), x ∈ support f ∩ support g → ↑(f ^ k) x = ↑(g ^ k) x\n⊢ ∀ (x : α), x ∈ support f ∩ support g → ↑(f ^ Nat.succ k) x = ↑(g ^ Nat.succ k) x",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nh : ∀ (x : α), x ∈ support f ∩ support g → ↑f x = ↑g x\nk : ℕ\n⊢ ∀ (x : α), x ∈ support f ∩ support g → ↑(f ^ k) x = ↑(g ^ k) x",
"tactic": "induction' k with k hk"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nh : ∀ (x : α), x ∈ support f ∩ support g → ↑f x = ↑g x\n⊢ ∀ (x : α), x ∈ support f ∩ support g → ↑(f ^ Nat.zero) x = ↑(g ^ Nat.zero) x",
"tactic": "simp"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nh : ∀ (x : α), x ∈ support f ∩ support g → ↑f x = ↑g x\nk : ℕ\nhk : ∀ (x : α), x ∈ support f ∩ support g → ↑(f ^ k) x = ↑(g ^ k) x\nx : α\nhx : x ∈ support f ∩ support g\n⊢ ↑(f ^ Nat.succ k) x = ↑(g ^ Nat.succ k) x",
"state_before": "case succ\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nh : ∀ (x : α), x ∈ support f ∩ support g → ↑f x = ↑g x\nk : ℕ\nhk : ∀ (x : α), x ∈ support f ∩ support g → ↑(f ^ k) x = ↑(g ^ k) x\n⊢ ∀ (x : α), x ∈ support f ∩ support g → ↑(f ^ Nat.succ k) x = ↑(g ^ Nat.succ k) x",
"tactic": "intro x hx"
},
{
"state_after": "case succ.a\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nh : ∀ (x : α), x ∈ support f ∩ support g → ↑f x = ↑g x\nk : ℕ\nhk : ∀ (x : α), x ∈ support f ∩ support g → ↑(f ^ k) x = ↑(g ^ k) x\nx : α\nhx : x ∈ support f ∩ support g\n⊢ ↑g x ∈ support f ∩ support g",
"state_before": "case succ\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nh : ∀ (x : α), x ∈ support f ∩ support g → ↑f x = ↑g x\nk : ℕ\nhk : ∀ (x : α), x ∈ support f ∩ support g → ↑(f ^ k) x = ↑(g ^ k) x\nx : α\nhx : x ∈ support f ∩ support g\n⊢ ↑(f ^ Nat.succ k) x = ↑(g ^ Nat.succ k) x",
"tactic": "rw [pow_succ', mul_apply, pow_succ', mul_apply, h _ hx, hk]"
},
{
"state_after": "no goals",
"state_before": "case succ.a\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nh : ∀ (x : α), x ∈ support f ∩ support g → ↑f x = ↑g x\nk : ℕ\nhk : ∀ (x : α), x ∈ support f ∩ support g → ↑(f ^ k) x = ↑(g ^ k) x\nx : α\nhx : x ∈ support f ∩ support g\n⊢ ↑g x ∈ support f ∩ support g",
"tactic": "rwa [mem_inter, apply_mem_support, ← h _ hx, apply_mem_support, ← mem_inter]"
}
] |
[
391,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/Algebra/BigOperators/Associated.lean
|
Associates.prod_eq_one_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.37103\nγ : Type ?u.37106\nδ : Type ?u.37109\ninst✝ : CommMonoid α\np : Multiset (Associates α)\n⊢ Multiset.prod 0 = 1 ↔ ∀ (a : Associates α), a ∈ 0 → a = 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.37103\nγ : Type ?u.37106\nδ : Type ?u.37109\ninst✝ : CommMonoid α\np : Multiset (Associates α)\n⊢ ∀ ⦃a : Associates α⦄ {s : Multiset (Associates α)},\n (Multiset.prod s = 1 ↔ ∀ (a : Associates α), a ∈ s → a = 1) →\n (Multiset.prod (a ::ₘ s) = 1 ↔ ∀ (a_2 : Associates α), a_2 ∈ a ::ₘ s → a_2 = 1)",
"tactic": "simp (config := { contextual := true }) [mul_eq_one_iff, or_imp, forall_and]"
}
] |
[
133,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.comap_injective_of_surjective
|
[] |
[
628,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
624,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
Submodule.comap_fst
|
[
{
"state_after": "case h.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.298370\nM₆ : Type ?u.298373\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nx : M\ny : M₂\n⊢ (x, y) ∈ comap (fst R M M₂) p ↔ (x, y) ∈ prod p ⊤",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.298370\nM₆ : Type ?u.298373\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\n⊢ comap (fst R M M₂) p = prod p ⊤",
"tactic": "ext ⟨x, y⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.298370\nM₆ : Type ?u.298373\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nx : M\ny : M₂\n⊢ (x, y) ∈ comap (fst R M M₂) p ↔ (x, y) ∈ prod p ⊤",
"tactic": "simp"
}
] |
[
563,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
563,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.EventuallyEq.prod_map
|
[] |
[
175,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
|
BoxIntegral.TaggedPrepartition.iUnion_filter_not
|
[] |
[
118,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
List.prod_toFinset
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.887074\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\nM : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid M\nf : α → M\nx✝ : Nodup []\n⊢ Finset.prod (toFinset []) f = prod (map f [])",
"tactic": "simp"
},
{
"state_after": "ι : Type ?u.887074\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\nM : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid M\nf : α → M\na : α\nl : List α\nhl✝ : Nodup (a :: l)\nnot_mem : ¬a ∈ l\nhl : Nodup l\n⊢ Finset.prod (toFinset (a :: l)) f = prod (map f (a :: l))",
"state_before": "ι : Type ?u.887074\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\nM : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid M\nf : α → M\na : α\nl : List α\nhl : Nodup (a :: l)\n⊢ Finset.prod (toFinset (a :: l)) f = prod (map f (a :: l))",
"tactic": "let ⟨not_mem, hl⟩ := List.nodup_cons.mp hl"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.887074\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\nM : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid M\nf : α → M\na : α\nl : List α\nhl✝ : Nodup (a :: l)\nnot_mem : ¬a ∈ l\nhl : Nodup l\n⊢ Finset.prod (toFinset (a :: l)) f = prod (map f (a :: l))",
"tactic": "simp [Finset.prod_insert (mt List.mem_toFinset.mp not_mem), prod_toFinset _ hl]"
}
] |
[
2046,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2041,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigO.exists_mem_basis
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.44090\nE : Type u_3\nF : Type ?u.44096\nG : Type ?u.44099\nE' : Type ?u.44102\nF' : Type u_4\nG' : Type ?u.44108\nE'' : Type ?u.44111\nF'' : Type ?u.44114\nG'' : Type ?u.44117\nR : Type ?u.44120\nR' : Type ?u.44123\n𝕜 : Type ?u.44126\n𝕜' : Type ?u.44129\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\nι : Sort u_1\np : ι → Prop\ns : ι → Set α\nh✝ : f =O[l] g'\nhb : HasBasis l p s\nc : ℝ\n_hc : 0 < c\nh : IsBigOWith c l f g'\n⊢ ∃ i _hi, ∀ (x : α), x ∈ s i → ‖f x‖ ≤ c * ‖g' x‖",
"tactic": "simpa only [isBigOWith_iff, hb.eventually_iff, exists_prop] using h"
}
] |
[
238,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Topology/UrysohnsLemma.lean
|
Urysohns.CU.approx_mono
|
[] |
[
209,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
IsUpperSet.prod
|
[] |
[
1520,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1519,
1
] |
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