file_path
stringlengths
11
79
full_name
stringlengths
2
100
traced_tactics
list
end
sequence
commit
stringclasses
4 values
url
stringclasses
4 values
start
sequence
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 ]