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/Topology/Basic.lean
|
mem_closure_of_frequently_of_tendsto
|
[] |
[
1459,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1457,
1
] |
Mathlib/Data/List/AList.lean
|
AList.ext
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl₁ : List (Sigma β)\nh₁ : NodupKeys l₁\nl₂ : List (Sigma β)\nnodupKeys✝ : NodupKeys l₂\nH : { entries := l₁, nodupKeys := h₁ }.entries = { entries := l₂, nodupKeys := nodupKeys✝ }.entries\n⊢ { entries := l₁, nodupKeys := h₁ } = { entries := l₂, nodupKeys := nodupKeys✝ }",
"tactic": "congr"
}
] |
[
68,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithBot.coe_add_eq_bot_iff
|
[] |
[
613,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
612,
1
] |
Mathlib/Order/Disjoint.lean
|
Disjoint.le_of_codisjoint
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : BoundedOrder α\na b c : α\nhab : Disjoint a b\nhbc : Codisjoint b c\n⊢ a ⊓ (b ⊔ c) ≤ (a ⊔ c) ⊓ (b ⊔ c)",
"state_before": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : BoundedOrder α\na b c : α\nhab : Disjoint a b\nhbc : Codisjoint b c\n⊢ a ≤ c",
"tactic": "rw [← @inf_top_eq _ _ _ a, ← @bot_sup_eq _ _ _ c, ← hab.eq_bot, ← hbc.eq_top, sup_inf_right]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : BoundedOrder α\na b c : α\nhab : Disjoint a b\nhbc : Codisjoint b c\n⊢ a ⊓ (b ⊔ c) ≤ (a ⊔ c) ⊓ (b ⊔ c)",
"tactic": "exact inf_le_inf_right _ le_sup_left"
}
] |
[
444,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.PositiveCompacts.ext
|
[] |
[
351,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
11
] |
Mathlib/Topology/Order.lean
|
continuous_empty_function
|
[] |
[
556,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
553,
1
] |
Mathlib/Algebra/Category/GroupCat/EpiMono.lean
|
AddGroupCat.epi_iff_range_eq_top
|
[] |
[
378,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Data/Nat/Interval.lean
|
Nat.Ico_succ_singleton
|
[
{
"state_after": "no goals",
"state_before": "a b c : ℕ\n⊢ Ico a (a + 1) = {a}",
"tactic": "rw [Ico_succ_right, Icc_self]"
}
] |
[
186,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/RingTheory/WittVector/Defs.lean
|
WittVector.wittNeg_vars
|
[] |
[
412,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
411,
1
] |
Mathlib/Topology/Constructions.lean
|
continuous_ofAdd
|
[] |
[
100,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Order/WellFounded.lean
|
WellFounded.min_mem
|
[] |
[
68,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.toDual_map
|
[] |
[
758,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
756,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.map_sup
|
[] |
[
1467,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1466,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
QuadraticForm.map_add_add_add_map
|
[
{
"state_after": "case intro\nS : Type ?u.74856\nR : Type u_1\nR₁ : Type ?u.74862\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx y z : M\nB : BilinForm R M\nh : ∀ (x y : M), ↑Q (x + y) = ↑Q x + ↑Q y + BilinForm.bilin B x y\n⊢ ↑Q (x + y + z) + (↑Q x + ↑Q y + ↑Q z) = ↑Q (x + y) + ↑Q (y + z) + ↑Q (z + x)",
"state_before": "S : Type ?u.74856\nR : Type u_1\nR₁ : Type ?u.74862\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx y z : M\n⊢ ↑Q (x + y + z) + (↑Q x + ↑Q y + ↑Q z) = ↑Q (x + y) + ↑Q (y + z) + ↑Q (z + x)",
"tactic": "obtain ⟨B, h⟩ := Q.exists_companion"
},
{
"state_after": "case intro\nS : Type ?u.74856\nR : Type u_1\nR₁ : Type ?u.74862\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx y z : M\nB : BilinForm R M\nh : ∀ (x y : M), ↑Q (x + y) = ↑Q x + ↑Q y + BilinForm.bilin B x y\n⊢ ↑Q (x + y + z) + (↑Q x + ↑Q y + ↑Q z) = ↑Q (x + y) + ↑Q (y + z) + ↑Q (x + z)",
"state_before": "case intro\nS : Type ?u.74856\nR : Type u_1\nR₁ : Type ?u.74862\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx y z : M\nB : BilinForm R M\nh : ∀ (x y : M), ↑Q (x + y) = ↑Q x + ↑Q y + BilinForm.bilin B x y\n⊢ ↑Q (x + y + z) + (↑Q x + ↑Q y + ↑Q z) = ↑Q (x + y) + ↑Q (y + z) + ↑Q (z + x)",
"tactic": "rw [add_comm z x]"
},
{
"state_after": "case intro\nS : Type ?u.74856\nR : Type u_1\nR₁ : Type ?u.74862\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx y z : M\nB : BilinForm R M\nh : ∀ (x y : M), ↑Q (x + y) = ↑Q x + ↑Q y + BilinForm.bilin B x y\n⊢ ↑Q x + ↑Q y + BilinForm.bilin B x y + ↑Q z + (BilinForm.bilin B x z + BilinForm.bilin B y z) + (↑Q x + ↑Q y + ↑Q z) =\n ↑Q x + ↑Q y + BilinForm.bilin B x y + (↑Q y + ↑Q z + BilinForm.bilin B y z) + (↑Q x + ↑Q z + BilinForm.bilin B x z)",
"state_before": "case intro\nS : Type ?u.74856\nR : Type u_1\nR₁ : Type ?u.74862\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx y z : M\nB : BilinForm R M\nh : ∀ (x y : M), ↑Q (x + y) = ↑Q x + ↑Q y + BilinForm.bilin B x y\n⊢ ↑Q (x + y + z) + (↑Q x + ↑Q y + ↑Q z) = ↑Q (x + y) + ↑Q (y + z) + ↑Q (x + z)",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "case intro\nS : Type ?u.74856\nR : Type u_1\nR₁ : Type ?u.74862\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx y z : M\nB : BilinForm R M\nh : ∀ (x y : M), ↑Q (x + y) = ↑Q x + ↑Q y + BilinForm.bilin B x y\n⊢ ↑Q x + ↑Q y + BilinForm.bilin B x y + ↑Q z + (BilinForm.bilin B x z + BilinForm.bilin B y z) + (↑Q x + ↑Q y + ↑Q z) =\n ↑Q x + ↑Q y + BilinForm.bilin B x y + (↑Q y + ↑Q z + BilinForm.bilin B y z) + (↑Q x + ↑Q z + BilinForm.bilin B x z)",
"tactic": "abel"
}
] |
[
222,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.Red.singleton_iff
|
[] |
[
324,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/Algebra/Associated.lean
|
associated_mul_isUnit_right_iff
|
[] |
[
487,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
485,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.VectorMeasure.MutuallySingular.zero_right
|
[] |
[
1204,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1202,
1
] |
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
|
ProjectiveSpectrum.mem_zeroLocus
|
[] |
[
81,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.rpow_left_surjective
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x ≠ 0\ny : ℝ≥0∞\n⊢ (fun y => y ^ x) (y ^ x⁻¹) = y",
"tactic": "simp_rw [← rpow_mul, _root_.inv_mul_cancel hx, rpow_one]"
}
] |
[
765,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
764,
1
] |
Mathlib/Algebra/Lie/OfAssociative.lean
|
commute_iff_lie_eq
|
[] |
[
61,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean
|
Differentiable.inner
|
[] |
[
143,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisCoinsertion.u_iInf_l
|
[] |
[
795,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
793,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.mod_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.261434\nβ : Type ?u.261437\nγ : Type ?u.261440\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na : Ordinal\n⊢ a % 1 = 0",
"tactic": "simp only [mod_def, div_one, one_mul, sub_self]"
}
] |
[
1062,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1062,
1
] |
Mathlib/Data/Multiset/Powerset.lean
|
Multiset.powersetLen_map
|
[
{
"state_after": "case empty\nα : Type u_2\nβ : Type u_1\nf : α → β\nn✝ n : ℕ\n⊢ powersetLen n (map f 0) = map (map f) (powersetLen n 0)\n\ncase cons\nα : Type u_2\nβ : Type u_1\nf : α → β\nn✝ : ℕ\nt : α\ns : Multiset α\nih : ∀ (n : ℕ), powersetLen n (map f s) = map (map f) (powersetLen n s)\nn : ℕ\n⊢ powersetLen n (map f (t ::ₘ s)) = map (map f) (powersetLen n (t ::ₘ s))",
"state_before": "α : Type u_2\nβ : Type u_1\nf : α → β\nn : ℕ\ns : Multiset α\n⊢ powersetLen n (map f s) = map (map f) (powersetLen n s)",
"tactic": "induction' s using Multiset.induction with t s ih generalizing n"
},
{
"state_after": "no goals",
"state_before": "case empty\nα : Type u_2\nβ : Type u_1\nf : α → β\nn✝ n : ℕ\n⊢ powersetLen n (map f 0) = map (map f) (powersetLen n 0)",
"tactic": "cases n <;> simp [powersetLen_zero_left, powersetLen_zero_right]"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_2\nβ : Type u_1\nf : α → β\nn✝ : ℕ\nt : α\ns : Multiset α\nih : ∀ (n : ℕ), powersetLen n (map f s) = map (map f) (powersetLen n s)\nn : ℕ\n⊢ powersetLen n (map f (t ::ₘ s)) = map (map f) (powersetLen n (t ::ₘ s))",
"tactic": "cases n <;> simp [ih, map_comp_cons]"
}
] |
[
300,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
sdiff_inf_self_left
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.138681\nα : Type u_1\nβ : Type ?u.138687\ninst✝ : GeneralizedCoheytingAlgebra α\na✝ b✝ c d a b : α\n⊢ a \\ (a ⊓ b) = a \\ b",
"tactic": "rw [sdiff_inf, sdiff_self, bot_sup_eq]"
}
] |
[
684,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
683,
1
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean
|
Submodule.mem_top
|
[] |
[
155,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
|
ContinuousOn.integrableOn_compact
|
[
{
"state_after": "X : Type u_1\nY : Type ?u.711872\nE : Type u_2\nR : Type ?u.711878\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : MeasureTheory.Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : IsLocallyFiniteMeasure μ\nK : Set X\na b : X\ninst✝ : MetrizableSpace X\nhK : IsCompact K\nhf : ContinuousOn f K\nthis : MetricSpace X := metrizableSpaceMetric X\n⊢ IntegrableOn f K",
"state_before": "X : Type u_1\nY : Type ?u.711872\nE : Type u_2\nR : Type ?u.711878\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : MeasureTheory.Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : IsLocallyFiniteMeasure μ\nK : Set X\na b : X\ninst✝ : MetrizableSpace X\nhK : IsCompact K\nhf : ContinuousOn f K\n⊢ IntegrableOn f K",
"tactic": "letI := metrizableSpaceMetric X"
},
{
"state_after": "X : Type u_1\nY : Type ?u.711872\nE : Type u_2\nR : Type ?u.711878\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : MeasureTheory.Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : IsLocallyFiniteMeasure μ\nK : Set X\na b : X\ninst✝ : MetrizableSpace X\nhK : IsCompact K\nhf : ContinuousOn f K\nthis : MetricSpace X := metrizableSpaceMetric X\nx : X\nhx : x ∈ K\n⊢ IntegrableAtFilter f (𝓝[K] x)",
"state_before": "X : Type u_1\nY : Type ?u.711872\nE : Type u_2\nR : Type ?u.711878\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : MeasureTheory.Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : IsLocallyFiniteMeasure μ\nK : Set X\na b : X\ninst✝ : MetrizableSpace X\nhK : IsCompact K\nhf : ContinuousOn f K\nthis : MetricSpace X := metrizableSpaceMetric X\n⊢ IntegrableOn f K",
"tactic": "refine' LocallyIntegrableOn.integrableOn_isCompact (fun x hx => _) hK"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.711872\nE : Type u_2\nR : Type ?u.711878\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : MeasureTheory.Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : IsLocallyFiniteMeasure μ\nK : Set X\na b : X\ninst✝ : MetrizableSpace X\nhK : IsCompact K\nhf : ContinuousOn f K\nthis : MetricSpace X := metrizableSpaceMetric X\nx : X\nhx : x ∈ K\n⊢ IntegrableAtFilter f (𝓝[K] x)",
"tactic": "exact hf.integrableAt_nhdsWithin_of_isSeparable hK.measurableSet hK.isSeparable hx"
}
] |
[
271,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Topology/CompactOpen.lean
|
ContinuousMap.continuous_eval'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.10621\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : LocallyCompactSpace α\nx✝ : C(α, β) × α\nn : Set β\nf : C(α, β)\nx : α\nhn : n ∈ 𝓝 (↑(f, x).fst (f, x).snd)\nv : Set β\nvn : v ⊆ n\nvo : IsOpen v\nfxv : ↑(f, x).fst (f, x).snd ∈ v\nthis✝² : v ∈ 𝓝 (↑f x)\ns : Set α\nhs : s ∈ 𝓝 x\nsv : s ⊆ ↑f ⁻¹' v\nsc : IsCompact s\nu : Set α\nus : u ⊆ s\nuo : IsOpen u\nxu : x ∈ u\nw : Set (C(α, β) × α) := CompactOpen.gen s v ×ˢ u\nthis✝¹ : w ⊆ (fun p => ↑p.fst p.snd) ⁻¹' n\nthis✝ : IsOpen w\nthis : (f, x) ∈ w\n⊢ w ⊆ (fun p => ↑p.fst p.snd) ⁻¹' n",
"tactic": "assumption"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.10621\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : LocallyCompactSpace α\nx✝ : C(α, β) × α\nn : Set β\nf : C(α, β)\nx : α\nhn : n ∈ 𝓝 (↑(f, x).fst (f, x).snd)\nv : Set β\nvn : v ⊆ n\nvo : IsOpen v\nfxv : ↑(f, x).fst (f, x).snd ∈ v\nthis✝² : v ∈ 𝓝 (↑f x)\ns : Set α\nhs : s ∈ 𝓝 x\nsv : s ⊆ ↑f ⁻¹' v\nsc : IsCompact s\nu : Set α\nus : u ⊆ s\nuo : IsOpen u\nxu : x ∈ u\nw : Set (C(α, β) × α) := CompactOpen.gen s v ×ˢ u\nthis✝¹ : w ⊆ (fun p => ↑p.fst p.snd) ⁻¹' n\nthis✝ : IsOpen w\nthis : (f, x) ∈ w\n⊢ IsOpen w",
"tactic": "assumption"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.10621\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : LocallyCompactSpace α\nx✝ : C(α, β) × α\nn : Set β\nf : C(α, β)\nx : α\nhn : n ∈ 𝓝 (↑(f, x).fst (f, x).snd)\nv : Set β\nvn : v ⊆ n\nvo : IsOpen v\nfxv : ↑(f, x).fst (f, x).snd ∈ v\nthis✝² : v ∈ 𝓝 (↑f x)\ns : Set α\nhs : s ∈ 𝓝 x\nsv : s ⊆ ↑f ⁻¹' v\nsc : IsCompact s\nu : Set α\nus : u ⊆ s\nuo : IsOpen u\nxu : x ∈ u\nw : Set (C(α, β) × α) := CompactOpen.gen s v ×ˢ u\nthis✝¹ : w ⊆ (fun p => ↑p.fst p.snd) ⁻¹' n\nthis✝ : IsOpen w\nthis : (f, x) ∈ w\n⊢ (f, x) ∈ w",
"tactic": "assumption"
}
] |
[
184,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/NumberTheory/Padics/PadicVal.lean
|
padicValNat.eq_zero_of_not_dvd
|
[] |
[
95,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/LinearAlgebra/Pi.lean
|
LinearMap.pi_ext
|
[] |
[
180,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Analysis/Complex/UnitDisc/Basic.lean
|
Complex.UnitDisc.coe_injective
|
[] |
[
45,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/Analysis/Complex/Arg.lean
|
Complex.sameRay_iff
|
[
{
"state_after": "case inl\ny : ℂ\n⊢ SameRay ℝ 0 y ↔ 0 = 0 ∨ y = 0 ∨ arg 0 = arg y\n\ncase inr\nx y : ℂ\nhx : x ≠ 0\n⊢ SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ arg x = arg y",
"state_before": "x y : ℂ\n⊢ SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ arg x = arg y",
"tactic": "rcases eq_or_ne x 0 with (rfl | hx)"
},
{
"state_after": "case inr.inl\nx : ℂ\nhx : x ≠ 0\n⊢ SameRay ℝ x 0 ↔ x = 0 ∨ 0 = 0 ∨ arg x = arg 0\n\ncase inr.inr\nx y : ℂ\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ arg x = arg y",
"state_before": "case inr\nx y : ℂ\nhx : x ≠ 0\n⊢ SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ arg x = arg y",
"tactic": "rcases eq_or_ne y 0 with (rfl | hy)"
},
{
"state_after": "case inr.inr\nx y : ℂ\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ ‖x‖ • y = ‖y‖ • x ↔ ↑(↑abs y) / ↑(↑abs x) * x = y",
"state_before": "case inr.inr\nx y : ℂ\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ arg x = arg y",
"tactic": "simp only [hx, hy, false_or_iff, sameRay_iff_norm_smul_eq, arg_eq_arg_iff hx hy]"
},
{
"state_after": "case inr.inr\nx y : ℂ\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ ↑(↑abs x) * y = ↑(↑abs y) * x ↔ ↑(↑abs y) * x = y * ↑(↑abs x)",
"state_before": "case inr.inr\nx y : ℂ\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ ‖x‖ • y = ‖y‖ • x ↔ ↑(↑abs y) / ↑(↑abs x) * x = y",
"tactic": "field_simp [hx, hy]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nx y : ℂ\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ ↑(↑abs x) * y = ↑(↑abs y) * x ↔ ↑(↑abs y) * x = y * ↑(↑abs x)",
"tactic": "rw [mul_comm, eq_comm]"
},
{
"state_after": "no goals",
"state_before": "case inl\ny : ℂ\n⊢ SameRay ℝ 0 y ↔ 0 = 0 ∨ y = 0 ∨ arg 0 = arg y",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nx : ℂ\nhx : x ≠ 0\n⊢ SameRay ℝ x 0 ↔ x = 0 ∨ 0 = 0 ∨ arg x = arg 0",
"tactic": "simp"
}
] |
[
41,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
34,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Lp.lipschitzWith_pos_part
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.8088288\nE : Type ?u.8088291\nF : Type ?u.8088294\nG : Type ?u.8088297\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\nx y : ℝ\n⊢ dist (max x 0) (max y 0) ≤ ↑1 * dist x y",
"tactic": "simp [Real.dist_eq, abs_max_sub_max_le_abs]"
}
] |
[
1090,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1089,
1
] |
Mathlib/Data/Nat/ModEq.lean
|
Nat.odd_of_mod_four_eq_one
|
[
{
"state_after": "no goals",
"state_before": "m n✝ a b c d n : ℕ\n⊢ n % 4 = 1 → n % 2 = 1",
"tactic": "simpa [ModEq, show 2 * 2 = 4 by norm_num] using @ModEq.of_mul_left 2 n 1 2"
},
{
"state_after": "no goals",
"state_before": "m n✝ a b c d n : ℕ\n⊢ 2 * 2 = 4",
"tactic": "norm_num"
}
] |
[
513,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
512,
1
] |
Mathlib/Algebra/Opposites.lean
|
AddOpposite.unop_one
|
[] |
[
378,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.even_mul_succ_self
|
[
{
"state_after": "m n✝ n : ℕ\n⊢ Even n ∨ ¬Even n",
"state_before": "m n✝ n : ℕ\n⊢ Even (n * (n + 1))",
"tactic": "rw [even_mul, even_add_one]"
},
{
"state_after": "no goals",
"state_before": "m n✝ n : ℕ\n⊢ Even n ∨ ¬Even n",
"tactic": "exact em _"
}
] |
[
214,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/RingTheory/Localization/AtPrime.lean
|
IsLocalization.AtPrime.mk'_mem_maximal_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁶ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra R S\nP : Type ?u.75214\ninst✝³ : CommSemiring P\nA : Type ?u.75220\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\nI : Ideal R\nhI : Ideal.IsPrime I\ninst✝ : IsLocalization.AtPrime S I\nx : R\ny : { x // x ∈ Ideal.primeCompl I }\nh : optParam (LocalRing S) (_ : LocalRing S)\n⊢ ¬mk' S x y ∈ LocalRing.maximalIdeal S ↔ ¬x ∈ I",
"tactic": "simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using\n isUnit_mk'_iff S I x y"
}
] |
[
169,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
Antitone.convex_gt
|
[] |
[
409,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
408,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MulHom.comp_id
|
[] |
[
1256,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1255,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.le_def
|
[] |
[
1148,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1147,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
inv_pow
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivisionMonoid α\na✝ b a : α\n⊢ a⁻¹ ^ 0 = (a ^ 0)⁻¹",
"tactic": "rw [pow_zero, pow_zero, inv_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivisionMonoid α\na✝ b a : α\nn : ℕ\n⊢ a⁻¹ ^ (n + 1) = (a ^ (n + 1))⁻¹",
"tactic": "rw [pow_succ', pow_succ, inv_pow _ n, mul_inv_rev]"
}
] |
[
320,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.EqOnSource.symm'
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.56162\nδ : Type ?u.56165\ne✝ : LocalEquiv α β\ne'✝ : LocalEquiv β γ\ne e' : LocalEquiv α β\nh : e ≈ e'\n⊢ RightInvOn (↑(LocalEquiv.symm e')) (↑e) e'.target",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.56162\nδ : Type ?u.56165\ne✝ : LocalEquiv α β\ne'✝ : LocalEquiv β γ\ne e' : LocalEquiv α β\nh : e ≈ e'\n⊢ LocalEquiv.symm e ≈ LocalEquiv.symm e'",
"tactic": "refine' ⟨target_eq h, eqOn_of_leftInvOn_of_rightInvOn e.leftInvOn _ _⟩ <;>\n simp only [symm_source, target_eq h, source_eq h, e'.symm_mapsTo]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.56162\nδ : Type ?u.56165\ne✝ : LocalEquiv α β\ne'✝ : LocalEquiv β γ\ne e' : LocalEquiv α β\nh : e ≈ e'\n⊢ RightInvOn (↑(LocalEquiv.symm e')) (↑e) e'.target",
"tactic": "exact e'.rightInvOn.congr_right e'.symm_mapsTo (source_eq h ▸ h.eqOn.symm)"
}
] |
[
853,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
850,
1
] |
Mathlib/Analysis/Convex/Cone/Dual.lean
|
innerDualCone_univ
|
[
{
"state_after": "𝕜 : Type ?u.5561\nE : Type ?u.5564\nF : Type ?u.5567\nG : Type ?u.5570\nH : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\n⊢ ∀ (x : H), x ∈ innerDualCone univ → x = 0",
"state_before": "𝕜 : Type ?u.5561\nE : Type ?u.5564\nF : Type ?u.5567\nG : Type ?u.5570\nH : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\n⊢ innerDualCone univ = 0",
"tactic": "suffices ∀ x : H, x ∈ (univ : Set H).innerDualCone → x = 0 by\n apply SetLike.coe_injective\n exact eq_singleton_iff_unique_mem.mpr ⟨fun x _ => (inner_zero_right _).ge, this⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.5561\nE : Type ?u.5564\nF : Type ?u.5567\nG : Type ?u.5570\nH : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\n⊢ ∀ (x : H), x ∈ innerDualCone univ → x = 0",
"tactic": "exact fun x hx => by simpa [← real_inner_self_nonpos] using hx (-x) (mem_univ _)"
},
{
"state_after": "case a\n𝕜 : Type ?u.5561\nE : Type ?u.5564\nF : Type ?u.5567\nG : Type ?u.5570\nH : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nthis : ∀ (x : H), x ∈ innerDualCone univ → x = 0\n⊢ ↑(innerDualCone univ) = ↑0",
"state_before": "𝕜 : Type ?u.5561\nE : Type ?u.5564\nF : Type ?u.5567\nG : Type ?u.5570\nH : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nthis : ∀ (x : H), x ∈ innerDualCone univ → x = 0\n⊢ innerDualCone univ = 0",
"tactic": "apply SetLike.coe_injective"
},
{
"state_after": "no goals",
"state_before": "case a\n𝕜 : Type ?u.5561\nE : Type ?u.5564\nF : Type ?u.5567\nG : Type ?u.5570\nH : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nthis : ∀ (x : H), x ∈ innerDualCone univ → x = 0\n⊢ ↑(innerDualCone univ) = ↑0",
"tactic": "exact eq_singleton_iff_unique_mem.mpr ⟨fun x _ => (inner_zero_right _).ge, this⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.5561\nE : Type ?u.5564\nF : Type ?u.5567\nG : Type ?u.5570\nH : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nx : H\nhx : x ∈ innerDualCone univ\n⊢ x = 0",
"tactic": "simpa [← real_inner_self_nonpos] using hx (-x) (mem_univ _)"
}
] |
[
78,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
src/lean/Init/SimpLemmas.lean
|
Bool.or_false
|
[
{
"state_after": "no goals",
"state_before": "b : Bool\n⊢ (b || false) = b",
"tactic": "cases b <;> rfl"
}
] |
[
102,
83
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
102,
9
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.finite_span_isCompactElement
|
[] |
[
679,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
677,
1
] |
Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean
|
Polynomial.IsWeaklyEisensteinAt.pow_natDegree_le_of_aeval_zero_of_monic_mem_map
|
[
{
"state_after": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\n⊢ x ^ natDegree (Polynomial.map (algebraMap R S) f) ∈ Ideal.map (algebraMap R S) 𝓟",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\n⊢ ∀ (i : ℕ), natDegree (Polynomial.map (algebraMap R S) f) ≤ i → x ^ i ∈ Ideal.map (algebraMap R S) 𝓟",
"tactic": "suffices x ^ (f.map (algebraMap R S)).natDegree ∈ 𝓟.map (algebraMap R S) by\n intro i hi\n obtain ⟨k, hk⟩ := exists_add_of_le hi\n rw [hk, pow_add]\n refine' mul_mem_right _ _ this"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : IsRoot (Polynomial.map (algebraMap R S) f) x\nhmo : Monic f\n⊢ x ^ natDegree (Polynomial.map (algebraMap R S) f) ∈ Ideal.map (algebraMap R S) 𝓟",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\n⊢ x ^ natDegree (Polynomial.map (algebraMap R S) f) ∈ Ideal.map (algebraMap R S) 𝓟",
"tactic": "rw [aeval_def, eval₂_eq_eval_map, ← IsRoot.def] at hx"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : IsRoot (Polynomial.map (algebraMap R S) f) x\nhmo : Monic f\n⊢ x ^ natDegree (Polynomial.map (algebraMap R S) f) ∈ Ideal.map (algebraMap R S) 𝓟",
"tactic": "refine' pow_natDegree_le_of_root_of_monic_mem (hf.map _) hx (hmo.map _) _ rfl.le"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nthis : x ^ natDegree (Polynomial.map (algebraMap R S) f) ∈ Ideal.map (algebraMap R S) 𝓟\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\n⊢ x ^ i ∈ Ideal.map (algebraMap R S) 𝓟",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nthis : x ^ natDegree (Polynomial.map (algebraMap R S) f) ∈ Ideal.map (algebraMap R S) 𝓟\n⊢ ∀ (i : ℕ), natDegree (Polynomial.map (algebraMap R S) f) ≤ i → x ^ i ∈ Ideal.map (algebraMap R S) 𝓟",
"tactic": "intro i hi"
},
{
"state_after": "case intro\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nthis : x ^ natDegree (Polynomial.map (algebraMap R S) f) ∈ Ideal.map (algebraMap R S) 𝓟\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\n⊢ x ^ i ∈ Ideal.map (algebraMap R S) 𝓟",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nthis : x ^ natDegree (Polynomial.map (algebraMap R S) f) ∈ Ideal.map (algebraMap R S) 𝓟\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\n⊢ x ^ i ∈ Ideal.map (algebraMap R S) 𝓟",
"tactic": "obtain ⟨k, hk⟩ := exists_add_of_le hi"
},
{
"state_after": "case intro\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nthis : x ^ natDegree (Polynomial.map (algebraMap R S) f) ∈ Ideal.map (algebraMap R S) 𝓟\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\n⊢ x ^ natDegree (Polynomial.map (algebraMap R S) f) * x ^ k ∈ Ideal.map (algebraMap R S) 𝓟",
"state_before": "case intro\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nthis : x ^ natDegree (Polynomial.map (algebraMap R S) f) ∈ Ideal.map (algebraMap R S) 𝓟\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\n⊢ x ^ i ∈ Ideal.map (algebraMap R S) 𝓟",
"tactic": "rw [hk, pow_add]"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nthis : x ^ natDegree (Polynomial.map (algebraMap R S) f) ∈ Ideal.map (algebraMap R S) 𝓟\ni : ℕ\nhi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i\nk : ℕ\nhk : i = natDegree (Polynomial.map (algebraMap R S) f) + k\n⊢ x ^ natDegree (Polynomial.map (algebraMap R S) f) * x ^ k ∈ Ideal.map (algebraMap R S) 𝓟",
"tactic": "refine' mul_mem_right _ _ this"
}
] |
[
154,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.symm_symm
|
[] |
[
349,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
21
] |
Std/Data/List/Lemmas.lean
|
List.mem_singleton
|
[] |
[
76,
39
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
75,
14
] |
Mathlib/Data/Rat/Floor.lean
|
Nat.coprime_sub_mul_floor_rat_div_of_coprime
|
[
{
"state_after": "n d : ℕ\nn_coprime_d : coprime n d\nthis : ↑n % ↑d = ↑n - ↑d * ⌊↑n / ↑d⌋\n⊢ coprime (natAbs (↑n - ↑d * ⌊↑n / ↑d⌋)) d",
"state_before": "n d : ℕ\nn_coprime_d : coprime n d\n⊢ coprime (natAbs (↑n - ↑d * ⌊↑n / ↑d⌋)) d",
"tactic": "have : (n : ℤ) % d = n - d * ⌊(n : ℚ) / d⌋ := Int.mod_nat_eq_sub_mul_floor_rat_div"
},
{
"state_after": "n d : ℕ\nn_coprime_d : coprime n d\nthis : ↑n % ↑d = ↑n - ↑d * ⌊↑n / ↑d⌋\n⊢ coprime (natAbs (↑n % ↑d)) d",
"state_before": "n d : ℕ\nn_coprime_d : coprime n d\nthis : ↑n % ↑d = ↑n - ↑d * ⌊↑n / ↑d⌋\n⊢ coprime (natAbs (↑n - ↑d * ⌊↑n / ↑d⌋)) d",
"tactic": "rw [← this]"
},
{
"state_after": "n d : ℕ\nn_coprime_d : coprime n d\nthis✝ : ↑n % ↑d = ↑n - ↑d * ⌊↑n / ↑d⌋\nthis : coprime d n\n⊢ coprime (natAbs (↑n % ↑d)) d",
"state_before": "n d : ℕ\nn_coprime_d : coprime n d\nthis : ↑n % ↑d = ↑n - ↑d * ⌊↑n / ↑d⌋\n⊢ coprime (natAbs (↑n % ↑d)) d",
"tactic": "have : d.coprime n := n_coprime_d.symm"
},
{
"state_after": "no goals",
"state_before": "n d : ℕ\nn_coprime_d : coprime n d\nthis✝ : ↑n % ↑d = ↑n - ↑d * ⌊↑n / ↑d⌋\nthis : coprime d n\n⊢ coprime (natAbs (↑n % ↑d)) d",
"tactic": "rwa [Nat.coprime, Nat.gcd_rec] at this"
}
] |
[
106,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Topology/Constructions.lean
|
continuous_pi
|
[] |
[
1194,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1193,
1
] |
Mathlib/GroupTheory/Sylow.lean
|
Sylow.card_coprime_index
|
[] |
[
679,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
676,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.coe_div
|
[] |
[
1337,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1336,
1
] |
Mathlib/Algebra/Order/Group/MinMax.lean
|
min_div_div_right'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedCommGroup α\na✝ b✝ c✝ a b c : α\n⊢ min (a / c) (b / c) = min a b / c",
"tactic": "simpa only [div_eq_mul_inv] using min_mul_mul_right a b c⁻¹"
}
] |
[
56,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.inter_inter_inter_comm
|
[] |
[
1777,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1776,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.restrictScalars_add
|
[] |
[
1700,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1698,
1
] |
Mathlib/Data/Nat/Order/Lemmas.lean
|
Nat.dvd_add_self_right
|
[] |
[
118,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
11
] |
Mathlib/Logic/Nontrivial.lean
|
subsingleton_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.2650\nh : Subsingleton α\n⊢ ∀ (x y : α), x = y",
"state_before": "α : Type u_1\nβ : Type ?u.2650\n⊢ Subsingleton α → ∀ (x y : α), x = y",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.2650\nh : Subsingleton α\n⊢ ∀ (x y : α), x = y",
"tactic": "exact Subsingleton.elim"
}
] |
[
115,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.coe_ennreal_top_mul
|
[
{
"state_after": "case inl\n\n⊢ ↑(⊤ * ↑0) = ⊤ * ↑↑0\n\ncase inr\nx : ℝ≥0\nh0 : x ≠ 0\n⊢ ↑(⊤ * ↑x) = ⊤ * ↑↑x",
"state_before": "x : ℝ≥0\n⊢ ↑(⊤ * ↑x) = ⊤ * ↑↑x",
"tactic": "rcases eq_or_ne x 0 with (rfl | h0)"
},
{
"state_after": "no goals",
"state_before": "case inl\n\n⊢ ↑(⊤ * ↑0) = ⊤ * ↑↑0",
"tactic": "simp"
},
{
"state_after": "case inr\nx : ℝ≥0\nh0 : x ≠ 0\n⊢ ↑⊤ = ⊤ * ↑↑x",
"state_before": "case inr\nx : ℝ≥0\nh0 : x ≠ 0\n⊢ ↑(⊤ * ↑x) = ⊤ * ↑↑x",
"tactic": "rw [ENNReal.top_mul (ENNReal.coe_ne_zero.2 h0)]"
},
{
"state_after": "no goals",
"state_before": "case inr\nx : ℝ≥0\nh0 : x ≠ 0\n⊢ ↑⊤ = ⊤ * ↑↑x",
"tactic": "exact Eq.symm <| if_pos <| NNReal.coe_pos.2 h0.bot_lt"
}
] |
[
575,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
571,
9
] |
Mathlib/Order/Filter/Pi.lean
|
Filter.pi_inf_principal_pi_neBot
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nα : ι → Type u_1\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), NeBot (f i)\nI : Set ι\n⊢ NeBot (pi f ⊓ 𝓟 (Set.pi I s)) ↔ ∀ (i : ι), i ∈ I → NeBot (f i ⊓ 𝓟 (s i))",
"tactic": "simp [neBot_iff]"
}
] |
[
146,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/InformationTheory/Hamming.lean
|
hammingDist_triangle_right
|
[
{
"state_after": "α : Type ?u.6007\nι : Type u_1\nβ : ι → Type u_2\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type ?u.6039\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y z : (i : ι) → β i\n⊢ hammingDist x y ≤ hammingDist x z + hammingDist z y",
"state_before": "α : Type ?u.6007\nι : Type u_1\nβ : ι → Type u_2\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type ?u.6039\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y z : (i : ι) → β i\n⊢ hammingDist x y ≤ hammingDist x z + hammingDist y z",
"tactic": "rw [hammingDist_comm y]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.6007\nι : Type u_1\nβ : ι → Type u_2\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type ?u.6039\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y z : (i : ι) → β i\n⊢ hammingDist x y ≤ hammingDist x z + hammingDist z y",
"tactic": "exact hammingDist_triangle _ _ _"
}
] |
[
85,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.fromEdgeSet_edgeSet
|
[
{
"state_after": "case Adj.h.h.a\nι : Sort ?u.82485\n𝕜 : Type ?u.82488\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w✝ : V\ne : Sym2 V\ns : Set (Sym2 V)\nv w : V\n⊢ Adj (fromEdgeSet (edgeSet G)) v w ↔ Adj G v w",
"state_before": "ι : Sort ?u.82485\n𝕜 : Type ?u.82488\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ns : Set (Sym2 V)\n⊢ fromEdgeSet (edgeSet G) = G",
"tactic": "ext (v w)"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a\nι : Sort ?u.82485\n𝕜 : Type ?u.82488\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w✝ : V\ne : Sym2 V\ns : Set (Sym2 V)\nv w : V\n⊢ Adj (fromEdgeSet (edgeSet G)) v w ↔ Adj G v w",
"tactic": "exact ⟨fun h => h.1, fun h => ⟨h, G.ne_of_adj h⟩⟩"
}
] |
[
633,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
631,
1
] |
Mathlib/Topology/Sets/Closeds.lean
|
TopologicalSpace.Opens.isCoatom_iff
|
[
{
"state_after": "ι : Type ?u.28195\nα : Type u_1\nβ : Type ?u.28201\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : T1Space α\ns : Opens α\n⊢ IsAtom (↑toDual (Closeds.compl (compl s))) ↔ ∃ x, Closeds.compl (compl s) = Closeds.compl (Closeds.singleton x)",
"state_before": "ι : Type ?u.28195\nα : Type u_1\nβ : Type ?u.28201\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : T1Space α\ns : Opens α\n⊢ IsCoatom s ↔ ∃ x, s = Closeds.compl (Closeds.singleton x)",
"tactic": "rw [← s.compl_compl, ← isAtom_dual_iff_isCoatom]"
},
{
"state_after": "ι : Type ?u.28195\nα : Type u_1\nβ : Type ?u.28201\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : T1Space α\ns : Opens α\n⊢ IsAtom (↑(Closeds.complOrderIso α) (compl s)) ↔ ∃ x, Closeds.compl (compl s) = Closeds.compl (Closeds.singleton x)",
"state_before": "ι : Type ?u.28195\nα : Type u_1\nβ : Type ?u.28201\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : T1Space α\ns : Opens α\n⊢ IsAtom (↑toDual (Closeds.compl (compl s))) ↔ ∃ x, Closeds.compl (compl s) = Closeds.compl (Closeds.singleton x)",
"tactic": "change IsAtom (Closeds.complOrderIso α s.compl) ↔ _"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.28195\nα : Type u_1\nβ : Type ?u.28201\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : T1Space α\ns : Opens α\n⊢ IsAtom (↑(Closeds.complOrderIso α) (compl s)) ↔ ∃ x, Closeds.compl (compl s) = Closeds.compl (Closeds.singleton x)",
"tactic": "simp only [(Closeds.complOrderIso α).isAtom_iff, Closeds.isAtom_iff,\n Closeds.compl_bijective.injective.eq_iff]"
}
] |
[
272,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/LinearAlgebra/Matrix/Symmetric.lean
|
Matrix.isSymm_add_transpose_self
|
[] |
[
69,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Order/SymmDiff.lean
|
inf_le_bihimp
|
[] |
[
280,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/Order/InitialSeg.lean
|
PrincipalSeg.equivLT_apply
|
[] |
[
349,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
348,
1
] |
Mathlib/Data/Finset/Interval.lean
|
Finset.Icc_eq_image_powerset
|
[
{
"state_after": "case a\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nu : Finset α\n⊢ u ∈ Icc s t ↔ u ∈ image ((fun x x_1 => x ∪ x_1) s) (powerset (t \\ s))",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\n⊢ Icc s t = image ((fun x x_1 => x ∪ x_1) s) (powerset (t \\ s))",
"tactic": "ext u"
},
{
"state_after": "case a\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nu : Finset α\n⊢ s ≤ u ∧ u ≤ t ↔ ∃ a, a ⊆ t \\ s ∧ s ∪ a = u",
"state_before": "case a\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nu : Finset α\n⊢ u ∈ Icc s t ↔ u ∈ image ((fun x x_1 => x ∪ x_1) s) (powerset (t \\ s))",
"tactic": "simp_rw [mem_Icc, mem_image, mem_powerset]"
},
{
"state_after": "case a.mp\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nu : Finset α\n⊢ s ≤ u ∧ u ≤ t → ∃ a, a ⊆ t \\ s ∧ s ∪ a = u\n\ncase a.mpr\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nu : Finset α\n⊢ (∃ a, a ⊆ t \\ s ∧ s ∪ a = u) → s ≤ u ∧ u ≤ t",
"state_before": "case a\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nu : Finset α\n⊢ s ≤ u ∧ u ≤ t ↔ ∃ a, a ⊆ t \\ s ∧ s ∪ a = u",
"tactic": "constructor"
},
{
"state_after": "case a.mp.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nu : Finset α\nhs : s ≤ u\nht : u ≤ t\n⊢ ∃ a, a ⊆ t \\ s ∧ s ∪ a = u",
"state_before": "case a.mp\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nu : Finset α\n⊢ s ≤ u ∧ u ≤ t → ∃ a, a ⊆ t \\ s ∧ s ∪ a = u",
"tactic": "rintro ⟨hs, ht⟩"
},
{
"state_after": "no goals",
"state_before": "case a.mp.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nu : Finset α\nhs : s ≤ u\nht : u ≤ t\n⊢ ∃ a, a ⊆ t \\ s ∧ s ∪ a = u",
"tactic": "exact ⟨u \\ s, sdiff_le_sdiff_right ht, sup_sdiff_cancel_right hs⟩"
},
{
"state_after": "case a.mpr.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nv : Finset α\nhv : v ⊆ t \\ s\n⊢ s ≤ s ∪ v ∧ s ∪ v ≤ t",
"state_before": "case a.mpr\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nu : Finset α\n⊢ (∃ a, a ⊆ t \\ s ∧ s ∪ a = u) → s ≤ u ∧ u ≤ t",
"tactic": "rintro ⟨v, hv, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case a.mpr.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nv : Finset α\nhv : v ⊆ t \\ s\n⊢ s ≤ s ∪ v ∧ s ∪ v ≤ t",
"tactic": "exact ⟨le_sup_left, union_subset h <| hv.trans <| sdiff_subset _ _⟩"
}
] |
[
85,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.TM0.Machine.map_respects
|
[
{
"state_after": "Γ : Type u_3\ninst✝³ : Inhabited Γ\nΓ' : Type u_4\ninst✝² : Inhabited Γ'\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nΛ' : Type u_2\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁✝ : Λ → Λ'\ng₂✝ : Λ' → Λ\ng₁ : PointedMap Λ Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nss : Supports M S\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂₁ : ∀ (q : Λ), q ∈ S → g₂ (PointedMap.f g₁ q) = q\nc : Cfg Γ Λ\na₂✝ : Cfg Γ' Λ'\ncs : c.q ∈ S\n⊢ match step M c with\n | some b₁ =>\n ∃ b₂,\n (fun a b => a.q ∈ S ∧ Cfg.map f₁ g₁.f a = b) b₁ b₂ ∧ Reaches₁ (step (map M f₁ f₂ g₁.f g₂)) (Cfg.map f₁ g₁.f c) b₂\n | none => step (map M f₁ f₂ g₁.f g₂) (Cfg.map f₁ g₁.f c) = none",
"state_before": "Γ : Type u_3\ninst✝³ : Inhabited Γ\nΓ' : Type u_4\ninst✝² : Inhabited Γ'\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nΛ' : Type u_2\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁✝ : Λ → Λ'\ng₂✝ : Λ' → Λ\ng₁ : PointedMap Λ Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nss : Supports M S\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂₁ : ∀ (q : Λ), q ∈ S → g₂ (PointedMap.f g₁ q) = q\n⊢ Respects (step M) (step (map M f₁ f₂ g₁.f g₂)) fun a b => a.q ∈ S ∧ Cfg.map f₁ g₁.f a = b",
"tactic": "intro c _ ⟨cs, rfl⟩"
},
{
"state_after": "case none\nΓ : Type u_3\ninst✝³ : Inhabited Γ\nΓ' : Type u_4\ninst✝² : Inhabited Γ'\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nΛ' : Type u_2\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁✝ : Λ → Λ'\ng₂✝ : Λ' → Λ\ng₁ : PointedMap Λ Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nss : Supports M S\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂₁ : ∀ (q : Λ), q ∈ S → g₂ (PointedMap.f g₁ q) = q\nc : Cfg Γ Λ\na₂✝ : Cfg Γ' Λ'\ncs : c.q ∈ S\ne : step M c = none\n⊢ match none with\n | some b₁ =>\n ∃ b₂,\n (fun a b => a.q ∈ S ∧ Cfg.map f₁ g₁.f a = b) b₁ b₂ ∧ Reaches₁ (step (map M f₁ f₂ g₁.f g₂)) (Cfg.map f₁ g₁.f c) b₂\n | none => step (map M f₁ f₂ g₁.f g₂) (Cfg.map f₁ g₁.f c) = none\n\ncase some\nΓ : Type u_3\ninst✝³ : Inhabited Γ\nΓ' : Type u_4\ninst✝² : Inhabited Γ'\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nΛ' : Type u_2\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁✝ : Λ → Λ'\ng₂✝ : Λ' → Λ\ng₁ : PointedMap Λ Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nss : Supports M S\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂₁ : ∀ (q : Λ), q ∈ S → g₂ (PointedMap.f g₁ q) = q\nc : Cfg Γ Λ\na₂✝ : Cfg Γ' Λ'\ncs : c.q ∈ S\nval✝ : Cfg Γ Λ\ne : step M c = some val✝\n⊢ match some val✝ with\n | some b₁ =>\n ∃ b₂,\n (fun a b => a.q ∈ S ∧ Cfg.map f₁ g₁.f a = b) b₁ b₂ ∧ Reaches₁ (step (map M f₁ f₂ g₁.f g₂)) (Cfg.map f₁ g₁.f c) b₂\n | none => step (map M f₁ f₂ g₁.f g₂) (Cfg.map f₁ g₁.f c) = none",
"state_before": "Γ : Type u_3\ninst✝³ : Inhabited Γ\nΓ' : Type u_4\ninst✝² : Inhabited Γ'\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nΛ' : Type u_2\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁✝ : Λ → Λ'\ng₂✝ : Λ' → Λ\ng₁ : PointedMap Λ Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nss : Supports M S\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂₁ : ∀ (q : Λ), q ∈ S → g₂ (PointedMap.f g₁ q) = q\nc : Cfg Γ Λ\na₂✝ : Cfg Γ' Λ'\ncs : c.q ∈ S\n⊢ match step M c with\n | some b₁ =>\n ∃ b₂,\n (fun a b => a.q ∈ S ∧ Cfg.map f₁ g₁.f a = b) b₁ b₂ ∧ Reaches₁ (step (map M f₁ f₂ g₁.f g₂)) (Cfg.map f₁ g₁.f c) b₂\n | none => step (map M f₁ f₂ g₁.f g₂) (Cfg.map f₁ g₁.f c) = none",
"tactic": "cases e : step M c"
},
{
"state_after": "case none\nΓ : Type u_3\ninst✝³ : Inhabited Γ\nΓ' : Type u_4\ninst✝² : Inhabited Γ'\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nΛ' : Type u_2\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁✝ : Λ → Λ'\ng₂✝ : Λ' → Λ\ng₁ : PointedMap Λ Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nss : Supports M S\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂₁ : ∀ (q : Λ), q ∈ S → g₂ (PointedMap.f g₁ q) = q\nc : Cfg Γ Λ\na₂✝ : Cfg Γ' Λ'\ncs : c.q ∈ S\ne : step M c = none\n⊢ match none with\n | some b₁ =>\n ∃ b₂,\n (fun a b => a.q ∈ S ∧ Cfg.map f₁ g₁.f a = b) b₁ b₂ ∧ Reaches₁ (step (map M f₁ f₂ g₁.f g₂)) (Cfg.map f₁ g₁.f c) b₂\n | none => Option.map (Cfg.map f₁ g₁.f) none = none",
"state_before": "case none\nΓ : Type u_3\ninst✝³ : Inhabited Γ\nΓ' : Type u_4\ninst✝² : Inhabited Γ'\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nΛ' : Type u_2\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁✝ : Λ → Λ'\ng₂✝ : Λ' → Λ\ng₁ : PointedMap Λ Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nss : Supports M S\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂₁ : ∀ (q : Λ), q ∈ S → g₂ (PointedMap.f g₁ q) = q\nc : Cfg Γ Λ\na₂✝ : Cfg Γ' Λ'\ncs : c.q ∈ S\ne : step M c = none\n⊢ match none with\n | some b₁ =>\n ∃ b₂,\n (fun a b => a.q ∈ S ∧ Cfg.map f₁ g₁.f a = b) b₁ b₂ ∧ Reaches₁ (step (map M f₁ f₂ g₁.f g₂)) (Cfg.map f₁ g₁.f c) b₂\n | none => step (map M f₁ f₂ g₁.f g₂) (Cfg.map f₁ g₁.f c) = none",
"tactic": "rw [← M.map_step f₁ f₂ g₁ g₂ f₂₁ g₂₁ _ cs, e]"
},
{
"state_after": "no goals",
"state_before": "case none\nΓ : Type u_3\ninst✝³ : Inhabited Γ\nΓ' : Type u_4\ninst✝² : Inhabited Γ'\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nΛ' : Type u_2\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁✝ : Λ → Λ'\ng₂✝ : Λ' → Λ\ng₁ : PointedMap Λ Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nss : Supports M S\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂₁ : ∀ (q : Λ), q ∈ S → g₂ (PointedMap.f g₁ q) = q\nc : Cfg Γ Λ\na₂✝ : Cfg Γ' Λ'\ncs : c.q ∈ S\ne : step M c = none\n⊢ match none with\n | some b₁ =>\n ∃ b₂,\n (fun a b => a.q ∈ S ∧ Cfg.map f₁ g₁.f a = b) b₁ b₂ ∧ Reaches₁ (step (map M f₁ f₂ g₁.f g₂)) (Cfg.map f₁ g₁.f c) b₂\n | none => Option.map (Cfg.map f₁ g₁.f) none = none",
"tactic": "rfl"
},
{
"state_after": "case some\nΓ : Type u_3\ninst✝³ : Inhabited Γ\nΓ' : Type u_4\ninst✝² : Inhabited Γ'\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nΛ' : Type u_2\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁✝ : Λ → Λ'\ng₂✝ : Λ' → Λ\ng₁ : PointedMap Λ Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nss : Supports M S\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂₁ : ∀ (q : Λ), q ∈ S → g₂ (PointedMap.f g₁ q) = q\nc : Cfg Γ Λ\na₂✝ : Cfg Γ' Λ'\ncs : c.q ∈ S\nval✝ : Cfg Γ Λ\ne : step M c = some val✝\n⊢ Cfg.map f₁ g₁.f val✝ ∈ step (map M f₁ f₂ g₁.f g₂) (Cfg.map f₁ g₁.f c)",
"state_before": "case some\nΓ : Type u_3\ninst✝³ : Inhabited Γ\nΓ' : Type u_4\ninst✝² : Inhabited Γ'\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nΛ' : Type u_2\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁✝ : Λ → Λ'\ng₂✝ : Λ' → Λ\ng₁ : PointedMap Λ Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nss : Supports M S\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂₁ : ∀ (q : Λ), q ∈ S → g₂ (PointedMap.f g₁ q) = q\nc : Cfg Γ Λ\na₂✝ : Cfg Γ' Λ'\ncs : c.q ∈ S\nval✝ : Cfg Γ Λ\ne : step M c = some val✝\n⊢ match some val✝ with\n | some b₁ =>\n ∃ b₂,\n (fun a b => a.q ∈ S ∧ Cfg.map f₁ g₁.f a = b) b₁ b₂ ∧ Reaches₁ (step (map M f₁ f₂ g₁.f g₂)) (Cfg.map f₁ g₁.f c) b₂\n | none => step (map M f₁ f₂ g₁.f g₂) (Cfg.map f₁ g₁.f c) = none",
"tactic": "refine' ⟨_, ⟨step_supports M ss e cs, rfl⟩, TransGen.single _⟩"
},
{
"state_after": "case some\nΓ : Type u_3\ninst✝³ : Inhabited Γ\nΓ' : Type u_4\ninst✝² : Inhabited Γ'\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nΛ' : Type u_2\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁✝ : Λ → Λ'\ng₂✝ : Λ' → Λ\ng₁ : PointedMap Λ Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nss : Supports M S\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂₁ : ∀ (q : Λ), q ∈ S → g₂ (PointedMap.f g₁ q) = q\nc : Cfg Γ Λ\na₂✝ : Cfg Γ' Λ'\ncs : c.q ∈ S\nval✝ : Cfg Γ Λ\ne : step M c = some val✝\n⊢ Cfg.map f₁ g₁.f val✝ ∈ Option.map (Cfg.map f₁ g₁.f) (some val✝)",
"state_before": "case some\nΓ : Type u_3\ninst✝³ : Inhabited Γ\nΓ' : Type u_4\ninst✝² : Inhabited Γ'\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nΛ' : Type u_2\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁✝ : Λ → Λ'\ng₂✝ : Λ' → Λ\ng₁ : PointedMap Λ Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nss : Supports M S\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂₁ : ∀ (q : Λ), q ∈ S → g₂ (PointedMap.f g₁ q) = q\nc : Cfg Γ Λ\na₂✝ : Cfg Γ' Λ'\ncs : c.q ∈ S\nval✝ : Cfg Γ Λ\ne : step M c = some val✝\n⊢ Cfg.map f₁ g₁.f val✝ ∈ step (map M f₁ f₂ g₁.f g₂) (Cfg.map f₁ g₁.f c)",
"tactic": "rw [← M.map_step f₁ f₂ g₁ g₂ f₂₁ g₂₁ _ cs, e]"
},
{
"state_after": "no goals",
"state_before": "case some\nΓ : Type u_3\ninst✝³ : Inhabited Γ\nΓ' : Type u_4\ninst✝² : Inhabited Γ'\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nΛ' : Type u_2\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁✝ : Λ → Λ'\ng₂✝ : Λ' → Λ\ng₁ : PointedMap Λ Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nss : Supports M S\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂₁ : ∀ (q : Λ), q ∈ S → g₂ (PointedMap.f g₁ q) = q\nc : Cfg Γ Λ\na₂✝ : Cfg Γ' Λ'\ncs : c.q ∈ S\nval✝ : Cfg Γ Λ\ne : step M c = some val✝\n⊢ Cfg.map f₁ g₁.f val✝ ∈ Option.map (Cfg.map f₁ g₁.f) (some val✝)",
"tactic": "rfl"
}
] |
[
1185,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1176,
1
] |
Mathlib/LinearAlgebra/Matrix/Circulant.lean
|
Matrix.circulant_sub
|
[] |
[
123,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
|
Pmf.pure_map
|
[] |
[
75,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Analysis/Convex/StrictConvexSpace.lean
|
eq_midpoint_of_dist_eq_half
|
[
{
"state_after": "case hxy\n𝕜 : Type ?u.100748\nE : Type u_2\ninst✝¹⁰ : NormedLinearOrderedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : StrictConvexSpace ℝ E\nx✝ y✝ z✝ : E\na b r : ℝ\nF : Type ?u.101025\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\nPF : Type u\nPE : Type u_1\ninst✝³ : MetricSpace PF\ninst✝² : MetricSpace PE\ninst✝¹ : NormedAddTorsor F PF\ninst✝ : NormedAddTorsor E PE\nx y z : PE\nhx : dist x y = dist x z / 2\nhy : dist y z = dist x z / 2\n⊢ dist x y = ⅟2 * dist x z\n\ncase hyz\n𝕜 : Type ?u.100748\nE : Type u_2\ninst✝¹⁰ : NormedLinearOrderedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : StrictConvexSpace ℝ E\nx✝ y✝ z✝ : E\na b r : ℝ\nF : Type ?u.101025\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\nPF : Type u\nPE : Type u_1\ninst✝³ : MetricSpace PF\ninst✝² : MetricSpace PE\ninst✝¹ : NormedAddTorsor F PF\ninst✝ : NormedAddTorsor E PE\nx y z : PE\nhx : dist x y = dist x z / 2\nhy : dist y z = dist x z / 2\n⊢ dist y z = (1 - ⅟2) * dist x z",
"state_before": "𝕜 : Type ?u.100748\nE : Type u_2\ninst✝¹⁰ : NormedLinearOrderedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : StrictConvexSpace ℝ E\nx✝ y✝ z✝ : E\na b r : ℝ\nF : Type ?u.101025\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\nPF : Type u\nPE : Type u_1\ninst✝³ : MetricSpace PF\ninst✝² : MetricSpace PE\ninst✝¹ : NormedAddTorsor F PF\ninst✝ : NormedAddTorsor E PE\nx y z : PE\nhx : dist x y = dist x z / 2\nhy : dist y z = dist x z / 2\n⊢ y = midpoint ℝ x z",
"tactic": "apply eq_lineMap_of_dist_eq_mul_of_dist_eq_mul"
},
{
"state_after": "no goals",
"state_before": "case hxy\n𝕜 : Type ?u.100748\nE : Type u_2\ninst✝¹⁰ : NormedLinearOrderedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : StrictConvexSpace ℝ E\nx✝ y✝ z✝ : E\na b r : ℝ\nF : Type ?u.101025\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\nPF : Type u\nPE : Type u_1\ninst✝³ : MetricSpace PF\ninst✝² : MetricSpace PE\ninst✝¹ : NormedAddTorsor F PF\ninst✝ : NormedAddTorsor E PE\nx y z : PE\nhx : dist x y = dist x z / 2\nhy : dist y z = dist x z / 2\n⊢ dist x y = ⅟2 * dist x z",
"tactic": "rwa [invOf_eq_inv, ← div_eq_inv_mul]"
},
{
"state_after": "no goals",
"state_before": "case hyz\n𝕜 : Type ?u.100748\nE : Type u_2\ninst✝¹⁰ : NormedLinearOrderedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : StrictConvexSpace ℝ E\nx✝ y✝ z✝ : E\na b r : ℝ\nF : Type ?u.101025\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\nPF : Type u\nPE : Type u_1\ninst✝³ : MetricSpace PF\ninst✝² : MetricSpace PE\ninst✝¹ : NormedAddTorsor F PF\ninst✝ : NormedAddTorsor E PE\nx y z : PE\nhx : dist x y = dist x z / 2\nhy : dist y z = dist x z / 2\n⊢ dist y z = (1 - ⅟2) * dist x z",
"tactic": "rwa [invOf_eq_inv, ← one_div, sub_half, one_div, ← div_eq_inv_mul]"
}
] |
[
275,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/Order/FixedPoints.lean
|
OrderHom.isGreatest_gfp_le
|
[] |
[
140,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.inducedOuterMeasure_eq_extend
|
[] |
[
1586,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1584,
1
] |
Mathlib/Data/Seq/Parallel.lean
|
Computation.mem_parallel
|
[
{
"state_after": "α : Type u\nβ : Type v\nS : WSeq (Computation α)\na : α\nH : ∀ (s : Computation α), s ∈ S → s ~> a\nc : Computation α\ncs : c ∈ S\nac : a ∈ c\nthis : Terminates c\n⊢ a ∈ parallel S",
"state_before": "α : Type u\nβ : Type v\nS : WSeq (Computation α)\na : α\nH : ∀ (s : Computation α), s ∈ S → s ~> a\nc : Computation α\ncs : c ∈ S\nac : a ∈ c\n⊢ a ∈ parallel S",
"tactic": "haveI := terminates_of_mem ac"
},
{
"state_after": "α : Type u\nβ : Type v\nS : WSeq (Computation α)\na : α\nH : ∀ (s : Computation α), s ∈ S → s ~> a\nc : Computation α\ncs : c ∈ S\nac : a ∈ c\nthis✝ : Terminates c\nthis : Terminates (parallel S)\n⊢ a ∈ parallel S",
"state_before": "α : Type u\nβ : Type v\nS : WSeq (Computation α)\na : α\nH : ∀ (s : Computation α), s ∈ S → s ~> a\nc : Computation α\ncs : c ∈ S\nac : a ∈ c\nthis : Terminates c\n⊢ a ∈ parallel S",
"tactic": "haveI := terminates_parallel cs"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nS : WSeq (Computation α)\na : α\nH : ∀ (s : Computation α), s ∈ S → s ~> a\nc : Computation α\ncs : c ∈ S\nac : a ∈ c\nthis✝ : Terminates c\nthis : Terminates (parallel S)\n⊢ a ∈ parallel S",
"tactic": "exact mem_of_promises _ (parallel_promises H)"
}
] |
[
370,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/Topology/UniformSpace/AbstractCompletion.lean
|
AbstractCompletion.uniformContinuous_compareEquiv
|
[] |
[
282,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.map_map
|
[] |
[
1553,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1551,
1
] |
Std/Data/List/Lemmas.lean
|
List.filter_filterMap
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nf : α → Option β\np : β → Bool\nl : List α\n⊢ filterMap (fun x => Option.bind (f x) (Option.guard fun x => p x = true)) l =\n filterMap (fun x => Option.filter p (f x)) l",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → Option β\np : β → Bool\nl : List α\n⊢ filter p (filterMap f l) = filterMap (fun x => Option.filter p (f x)) l",
"tactic": "rw [← filterMap_eq_filter, filterMap_filterMap]"
},
{
"state_after": "case e_f\nα : Type u_1\nβ : Type u_2\nf : α → Option β\np : β → Bool\nl : List α\n⊢ (fun x => Option.bind (f x) (Option.guard fun x => p x = true)) = fun x => Option.filter p (f x)",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → Option β\np : β → Bool\nl : List α\n⊢ filterMap (fun x => Option.bind (f x) (Option.guard fun x => p x = true)) l =\n filterMap (fun x => Option.filter p (f x)) l",
"tactic": "congr"
},
{
"state_after": "case e_f.h\nα : Type u_1\nβ : Type u_2\nf : α → Option β\np : β → Bool\nl : List α\nx : α\n⊢ Option.bind (f x) (Option.guard fun x => p x = true) = Option.filter p (f x)",
"state_before": "case e_f\nα : Type u_1\nβ : Type u_2\nf : α → Option β\np : β → Bool\nl : List α\n⊢ (fun x => Option.bind (f x) (Option.guard fun x => p x = true)) = fun x => Option.filter p (f x)",
"tactic": "funext x"
},
{
"state_after": "no goals",
"state_before": "case e_f.h\nα : Type u_1\nβ : Type u_2\nf : α → Option β\np : β → Bool\nl : List α\nx : α\n⊢ Option.bind (f x) (Option.guard fun x => p x = true) = Option.filter p (f x)",
"tactic": "cases f x <;> simp [Option.filter, Option.guard]"
}
] |
[
1193,
68
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1190,
1
] |
Mathlib/Data/Finmap.lean
|
Finmap.lookup_toFinmap
|
[] |
[
271,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
norm_expSeries_summable
|
[] |
[
413,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
412,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.neg_eq_comm
|
[
{
"state_after": "no goals",
"state_before": "a b : Int\n⊢ -a = b ↔ -b = a",
"tactic": "rw [eq_comm, Int.eq_neg_comm, eq_comm]"
}
] |
[
315,
41
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
314,
11
] |
Mathlib/Data/Polynomial/Coeff.lean
|
Polynomial.int_cast_coeff_zero
|
[
{
"state_after": "no goals",
"state_before": "R✝ : Type u\nS : Type v\na b : R✝\nn m : ℕ\ninst✝¹ : Semiring R✝\np q r : R✝[X]\ni : ℤ\nR : Type u_1\ninst✝ : Ring R\n⊢ coeff (↑i) 0 = ↑i",
"tactic": "cases i <;> simp"
}
] |
[
396,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
395,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.measure_univ_pos
|
[] |
[
1106,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1105,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.half_le_of_sub_le_half
|
[
{
"state_after": "m n k l a b : ℕ\nh✝ : a * 2 - b * 2 ≤ a\nh : a ≤ b * 2\n⊢ a / 2 ≤ b",
"state_before": "m n k l a b : ℕ\nh : a - b ≤ a / 2\n⊢ a / 2 ≤ b",
"tactic": "rw [Nat.le_div_iff_mul_le two_pos, Nat.mul_sub_right_distrib, tsub_le_iff_right, mul_two,\n add_le_add_iff_left] at h"
},
{
"state_after": "m n k l a b : ℕ\nh✝ : a * 2 - b * 2 ≤ a\nh : a ≤ b * 2\n⊢ a / 2 ≤ b * 2 / 2",
"state_before": "m n k l a b : ℕ\nh✝ : a * 2 - b * 2 ≤ a\nh : a ≤ b * 2\n⊢ a / 2 ≤ b",
"tactic": "rw [← Nat.mul_div_left b two_pos]"
},
{
"state_after": "no goals",
"state_before": "m n k l a b : ℕ\nh✝ : a * 2 - b * 2 ≤ a\nh : a ≤ b * 2\n⊢ a / 2 ≤ b * 2 / 2",
"tactic": "exact Nat.div_le_div_right h"
}
] |
[
447,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_eq_integral_pos_part_sub_integral_neg_part
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.1057071\nF : Type ?u.1057074\n𝕜 : Type ?u.1057077\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1059768\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : Integrable f\n⊢ (∫ (a : α), f a ∂μ) = ∫ (a : α), ↑(Real.toNNReal (f a)) - ↑(Real.toNNReal (-f a)) ∂μ\n\nα : Type u_1\nE : Type ?u.1057071\nF : Type ?u.1057074\n𝕜 : Type ?u.1057077\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1059768\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : Integrable f\n⊢ Integrable fun a => ↑(Real.toNNReal (-f a))",
"state_before": "α : Type u_1\nE : Type ?u.1057071\nF : Type ?u.1057074\n𝕜 : Type ?u.1057077\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1059768\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : Integrable f\n⊢ (∫ (a : α), f a ∂μ) = (∫ (a : α), ↑(Real.toNNReal (f a)) ∂μ) - ∫ (a : α), ↑(Real.toNNReal (-f a)) ∂μ",
"tactic": "rw [← integral_sub hf.real_toNNReal]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.1057071\nF : Type ?u.1057074\n𝕜 : Type ?u.1057077\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1059768\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : Integrable f\n⊢ (∫ (a : α), f a ∂μ) = ∫ (a : α), ↑(Real.toNNReal (f a)) - ↑(Real.toNNReal (-f a)) ∂μ",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.1057071\nF : Type ?u.1057074\n𝕜 : Type ?u.1057077\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1059768\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : Integrable f\n⊢ Integrable fun a => ↑(Real.toNNReal (-f a))",
"tactic": "exact hf.neg.real_toNNReal"
}
] |
[
1156,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1152,
1
] |
Mathlib/Order/Compare.lean
|
cmp_eq_lt_iff
|
[] |
[
230,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.comp_elim
|
[] |
[
203,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.quot_mk_eq_mk
|
[] |
[
498,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Logic/Equiv/Fin.lean
|
finRotate_succ_apply
|
[
{
"state_after": "case zero\nm : ℕ\ni : Fin (Nat.zero + 1)\n⊢ ↑(finRotate (Nat.zero + 1)) i = i + 1\n\ncase succ\nm n✝ : ℕ\ni : Fin (Nat.succ n✝ + 1)\n⊢ ↑(finRotate (Nat.succ n✝ + 1)) i = i + 1",
"state_before": "m n : ℕ\ni : Fin (n + 1)\n⊢ ↑(finRotate (n + 1)) i = i + 1",
"tactic": "cases n"
},
{
"state_after": "case succ.inl\nm n✝ : ℕ\n⊢ ↑(finRotate (Nat.succ n✝ + 1)) (Fin.last (n✝ + 1)) = Fin.last (n✝ + 1) + 1\n\ncase succ.inr\nm n✝ : ℕ\ni : Fin (Nat.succ n✝ + 1)\nh : i < Fin.last (n✝ + 1)\n⊢ ↑(finRotate (Nat.succ n✝ + 1)) i = i + 1",
"state_before": "case succ\nm n✝ : ℕ\ni : Fin (Nat.succ n✝ + 1)\n⊢ ↑(finRotate (Nat.succ n✝ + 1)) i = i + 1",
"tactic": "rcases i.le_last.eq_or_lt with (rfl | h)"
},
{
"state_after": "no goals",
"state_before": "case zero\nm : ℕ\ni : Fin (Nat.zero + 1)\n⊢ ↑(finRotate (Nat.zero + 1)) i = i + 1",
"tactic": "exact @Subsingleton.elim (Fin 1) _ _ _"
},
{
"state_after": "no goals",
"state_before": "case succ.inl\nm n✝ : ℕ\n⊢ ↑(finRotate (Nat.succ n✝ + 1)) (Fin.last (n✝ + 1)) = Fin.last (n✝ + 1) + 1",
"tactic": "simp [finRotate_last]"
},
{
"state_after": "case succ.inr.mk\nm n✝ val✝ : ℕ\nisLt✝ : val✝ < Nat.succ n✝ + 1\nh : { val := val✝, isLt := isLt✝ } < Fin.last (n✝ + 1)\n⊢ ↑(finRotate (Nat.succ n✝ + 1)) { val := val✝, isLt := isLt✝ } = { val := val✝, isLt := isLt✝ } + 1",
"state_before": "case succ.inr\nm n✝ : ℕ\ni : Fin (Nat.succ n✝ + 1)\nh : i < Fin.last (n✝ + 1)\n⊢ ↑(finRotate (Nat.succ n✝ + 1)) i = i + 1",
"tactic": "cases i"
},
{
"state_after": "case succ.inr.mk\nm n✝ val✝ : ℕ\nisLt✝ : val✝ < Nat.succ n✝ + 1\nh : val✝ < n✝ + 1\n⊢ ↑(finRotate (Nat.succ n✝ + 1)) { val := val✝, isLt := isLt✝ } = { val := val✝, isLt := isLt✝ } + 1",
"state_before": "case succ.inr.mk\nm n✝ val✝ : ℕ\nisLt✝ : val✝ < Nat.succ n✝ + 1\nh : { val := val✝, isLt := isLt✝ } < Fin.last (n✝ + 1)\n⊢ ↑(finRotate (Nat.succ n✝ + 1)) { val := val✝, isLt := isLt✝ } = { val := val✝, isLt := isLt✝ } + 1",
"tactic": "simp only [Fin.lt_iff_val_lt_val, Fin.val_last, Fin.val_mk] at h"
},
{
"state_after": "no goals",
"state_before": "case succ.inr.mk\nm n✝ val✝ : ℕ\nisLt✝ : val✝ < Nat.succ n✝ + 1\nh : val✝ < n✝ + 1\n⊢ ↑(finRotate (Nat.succ n✝ + 1)) { val := val✝, isLt := isLt✝ } = { val := val✝, isLt := isLt✝ } + 1",
"tactic": "simp [finRotate_of_lt h, Fin.eq_iff_veq, Fin.add_def, Nat.mod_eq_of_lt (Nat.succ_lt_succ h)]"
}
] |
[
442,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
435,
9
] |
Mathlib/Order/Filter/Germ.lean
|
Filter.Germ.const_compTendsto
|
[] |
[
282,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.map_comp
|
[] |
[
620,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
619,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Slope.lean
|
slope_vadd_const
|
[
{
"state_after": "case h.h\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → E\nc : PE\na b : k\n⊢ slope (fun x => f x +ᵥ c) a b = slope f a b",
"state_before": "k : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → E\nc : PE\n⊢ (slope fun x => f x +ᵥ c) = slope f",
"tactic": "ext (a b)"
},
{
"state_after": "no goals",
"state_before": "case h.h\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → E\nc : PE\na b : k\n⊢ slope (fun x => f x +ᵥ c) a b = slope f a b",
"tactic": "simp only [slope, vadd_vsub_vadd_cancel_right, vsub_eq_sub]"
}
] |
[
72,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Tactic/NormNum/Core.lean
|
Mathlib.Meta.NormNum.IsNat.raw_refl
|
[] |
[
39,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Topology/Order.lean
|
discreteTopology_iff_singleton_mem_nhds
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nt t₁ t₂ : TopologicalSpace α\ns : Set α\ninst✝ : TopologicalSpace α\n⊢ DiscreteTopology α ↔ ∀ (x : α), {x} ∈ 𝓝 x",
"tactic": "simp only [← singletons_open_iff_discrete, isOpen_iff_mem_nhds, mem_singleton_iff, forall_eq]"
}
] |
[
330,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/RingTheory/EuclideanDomain.lean
|
EuclideanDomain.gcd_isUnit_iff
|
[] |
[
89,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
sdiff_inf
|
[] |
[
679,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
678,
1
] |
Mathlib/Analysis/NormedSpace/Ray.lean
|
not_sameRay_iff_of_norm_eq
|
[] |
[
110,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Std/Classes/LawfulMonad.lean
|
SatisfiesM.of_true
|
[
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nα : Type u_1\np : α → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx : m α\nh : ∀ (a : α), p a\n⊢ Subtype.val <$> (fun a => { val := a, property := (_ : p a) }) <$> x = x",
"tactic": "simp [← comp_map, Function.comp]"
}
] |
[
77,
67
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
75,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.mem_rightTransversals_iff_existsUnique_mul_inv_mem
|
[
{
"state_after": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\n⊢ (∀ (g : G), ∃! x, ↑x.fst * ↑x.snd = g) ↔ ∀ (g : G), ∃! s, g * (↑s)⁻¹ ∈ T",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\n⊢ S ∈ rightTransversals T ↔ ∀ (g : G), ∃! s, g * (↑s)⁻¹ ∈ T",
"tactic": "rw [rightTransversals, Set.mem_setOf_eq, isComplement_iff_existsUnique]"
},
{
"state_after": "case refine'_1\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : ∀ (g : G), ∃! x, ↑x.fst * ↑x.snd = g\ng : G\n⊢ ∃! s, g * (↑s)⁻¹ ∈ T\n\ncase refine'_2\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : ∀ (g : G), ∃! s, g * (↑s)⁻¹ ∈ T\ng : G\n⊢ ∃! x, ↑x.fst * ↑x.snd = g",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\n⊢ (∀ (g : G), ∃! x, ↑x.fst * ↑x.snd = g) ↔ ∀ (g : G), ∃! s, g * (↑s)⁻¹ ∈ T",
"tactic": "refine' ⟨fun h g => _, fun h g => _⟩"
},
{
"state_after": "case refine'_1.intro.intro\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : ∀ (g : G), ∃! x, ↑x.fst * ↑x.snd = g\ng : G\nx : ↑T × ↑S\nh1 : ↑x.fst * ↑x.snd = g\nh2 : ∀ (y : ↑T × ↑S), (fun x => ↑x.fst * ↑x.snd = g) y → y = x\n⊢ ∃! s, g * (↑s)⁻¹ ∈ T",
"state_before": "case refine'_1\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : ∀ (g : G), ∃! x, ↑x.fst * ↑x.snd = g\ng : G\n⊢ ∃! s, g * (↑s)⁻¹ ∈ T",
"tactic": "obtain ⟨x, h1, h2⟩ := h g"
},
{
"state_after": "case refine'_2.intro.intro\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : ∀ (g : G), ∃! s, g * (↑s)⁻¹ ∈ T\ng : G\nx : ↑S\nh1 : g * (↑x)⁻¹ ∈ T\nh2 : ∀ (y : ↑S), (fun s => g * (↑s)⁻¹ ∈ T) y → y = x\n⊢ ∃! x, ↑x.fst * ↑x.snd = g",
"state_before": "case refine'_2\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : ∀ (g : G), ∃! s, g * (↑s)⁻¹ ∈ T\ng : G\n⊢ ∃! x, ↑x.fst * ↑x.snd = g",
"tactic": "obtain ⟨x, h1, h2⟩ := h g"
},
{
"state_after": "case refine'_2.intro.intro\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : ∀ (g : G), ∃! s, g * (↑s)⁻¹ ∈ T\ng : G\nx : ↑S\nh1 : g * (↑x)⁻¹ ∈ T\nh2 : ∀ (y : ↑S), (fun s => g * (↑s)⁻¹ ∈ T) y → y = x\ny : ↑T × ↑S\nhy : (fun x => ↑x.fst * ↑x.snd = g) y\n⊢ y = ({ val := g * (↑x)⁻¹, property := h1 }, x)",
"state_before": "case refine'_2.intro.intro\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : ∀ (g : G), ∃! s, g * (↑s)⁻¹ ∈ T\ng : G\nx : ↑S\nh1 : g * (↑x)⁻¹ ∈ T\nh2 : ∀ (y : ↑S), (fun s => g * (↑s)⁻¹ ∈ T) y → y = x\n⊢ ∃! x, ↑x.fst * ↑x.snd = g",
"tactic": "refine' ⟨⟨⟨g * (↑x)⁻¹, h1⟩, x⟩, inv_mul_cancel_right g x, fun y hy => _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nh : ∀ (g : G), ∃! s, g * (↑s)⁻¹ ∈ T\ng : G\nx : ↑S\nh1 : g * (↑x)⁻¹ ∈ T\nh2 : ∀ (y : ↑S), (fun s => g * (↑s)⁻¹ ∈ T) y → y = x\ny : ↑T × ↑S\nhy : (fun x => ↑x.fst * ↑x.snd = g) y\nhf : y.snd = x\n⊢ y = ({ val := g * (↑x)⁻¹, property := h1 }, x)",
"tactic": "exact Prod.ext (Subtype.ext (eq_mul_inv_of_mul_eq (hf ▸ hy))) hf"
}
] |
[
238,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
Finset.prod_le_prod'
|
[] |
[
116,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderIso.withTopCongr_trans
|
[] |
[
1336,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1334,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.memℒp_map_measure_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nE : Type u_2\nF : Type ?u.3098262\nG : Type ?u.3098265\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nβ : Type u_1\nmβ : MeasurableSpace β\nf : α → β\ng : β → E\nhg : AEStronglyMeasurable g (Measure.map f μ)\nhf : AEMeasurable f\n⊢ Memℒp g p ↔ Memℒp (g ∘ f) p",
"tactic": "simp [Memℒp, snorm_map_measure hg hf, hg.comp_aemeasurable hf, hg]"
}
] |
[
921,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
919,
1
] |
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