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/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.mem_incidence_iff_neighbor
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.138365\n𝕜 : Type ?u.138368\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\nv w : V\n⊢ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ incidenceSet G v ↔ w ∈ neighborSet G v",
"tactic": "simp only [mem_incidenceSet, mem_neighborSet]"
}
] |
[
963,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
962,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.homOfLe_apply
|
[] |
[
1063,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1062,
1
] |
Mathlib/Algebra/Homology/ShortExact/Preadditive.lean
|
CategoryTheory.LeftSplit.shortExact
|
[] |
[
64,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Data/Analysis/Filter.lean
|
Filter.Realizer.tendsto_iff
|
[] |
[
347,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
Equiv.Perm.card_cycleType_eq_one
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\n⊢ (∃ a, cycleType σ = {a}) ↔ IsCycle σ",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\n⊢ ↑card (cycleType σ) = 1 ↔ IsCycle σ",
"tactic": "rw [card_eq_one]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\n⊢ (∃ a a_1, (IsCycle σ ∧ σ = a_1) ∧ (Finset.card ∘ support) a_1 = a) ↔ IsCycle σ",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\n⊢ (∃ a, cycleType σ = {a}) ↔ IsCycle σ",
"tactic": "simp_rw [cycleType_def, Multiset.map_eq_singleton, ← Finset.singleton_val, Finset.val_inj,\n cycleFactorsFinset_eq_singleton_iff]"
},
{
"state_after": "case mp\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\n⊢ (∃ a a_1, (IsCycle σ ∧ σ = a_1) ∧ (Finset.card ∘ support) a_1 = a) → IsCycle σ\n\ncase mpr\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\n⊢ IsCycle σ → ∃ a a_1, (IsCycle σ ∧ σ = a_1) ∧ (Finset.card ∘ support) a_1 = a",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\n⊢ (∃ a a_1, (IsCycle σ ∧ σ = a_1) ∧ (Finset.card ∘ support) a_1 = a) ↔ IsCycle σ",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nw✝¹ : ℕ\nw✝ : Perm α\nh : IsCycle σ\n⊢ IsCycle σ",
"state_before": "case mp\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\n⊢ (∃ a a_1, (IsCycle σ ∧ σ = a_1) ∧ (Finset.card ∘ support) a_1 = a) → IsCycle σ",
"tactic": "rintro ⟨_, _, ⟨h, -⟩, -⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nw✝¹ : ℕ\nw✝ : Perm α\nh : IsCycle σ\n⊢ IsCycle σ",
"tactic": "exact h"
},
{
"state_after": "case mpr\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nh : IsCycle σ\n⊢ ∃ a a_1, (IsCycle σ ∧ σ = a_1) ∧ (Finset.card ∘ support) a_1 = a",
"state_before": "case mpr\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\n⊢ IsCycle σ → ∃ a a_1, (IsCycle σ ∧ σ = a_1) ∧ (Finset.card ∘ support) a_1 = a",
"tactic": "intro h"
},
{
"state_after": "case mpr\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nh : IsCycle σ\n⊢ (IsCycle σ ∧ σ = σ) ∧ (Finset.card ∘ support) σ = Finset.card (support σ)",
"state_before": "case mpr\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nh : IsCycle σ\n⊢ ∃ a a_1, (IsCycle σ ∧ σ = a_1) ∧ (Finset.card ∘ support) a_1 = a",
"tactic": "use σ.support.card, σ"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nh : IsCycle σ\n⊢ (IsCycle σ ∧ σ = σ) ∧ (Finset.card ∘ support) σ = Finset.card (support σ)",
"tactic": "simp [h]"
}
] |
[
121,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Algebra/Group/Defs.lean
|
mul_right_cancel
|
[] |
[
215,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigOWith_norm_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.122412\nE : Type ?u.122415\nF : Type u_2\nG : Type ?u.122421\nE' : Type u_3\nF' : Type ?u.122427\nG' : Type ?u.122430\nE'' : Type ?u.122433\nF'' : Type ?u.122436\nG'' : Type ?u.122439\nR : Type ?u.122442\nR' : Type ?u.122445\n𝕜 : Type ?u.122448\n𝕜' : Type ?u.122451\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nu v : α → ℝ\n⊢ IsBigOWith c l (fun x => ‖f' x‖) g ↔ IsBigOWith c l f' g",
"tactic": "simp only [IsBigOWith_def, norm_norm]"
}
] |
[
753,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
752,
1
] |
Mathlib/Data/Prod/Basic.lean
|
Prod.lex_def
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.5817\nδ : Type ?u.5820\nr✝ : α → α → Prop\ns✝ : β → β → Prop\nx y : α × β\nr : α → α → Prop\ns : β → β → Prop\np q : α × β\nh : Prod.Lex r s p q\n⊢ r p.fst q.fst ∨ p.fst = q.fst ∧ s p.snd q.snd",
"tactic": "cases h <;> simp [*]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.5817\nδ : Type ?u.5820\nr✝ : α → α → Prop\ns✝ : β → β → Prop\nx y : α × β\nr : α → α → Prop\ns : β → β → Prop\np q : α × β\nh✝ : r p.fst q.fst ∨ p.fst = q.fst ∧ s p.snd q.snd\na : α\nb d : β\nh : s (a, b).snd ((a, b).fst, d).snd\n⊢ Prod.Lex r s (a, b) ((a, b).fst, d)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.5817\nδ : Type ?u.5820\nr✝ : α → α → Prop\ns✝ : β → β → Prop\nx y : α × β\nr : α → α → Prop\ns : β → β → Prop\np q : α × β\nh✝ : r p.fst q.fst ∨ p.fst = q.fst ∧ s p.snd q.snd\na : α\nb : β\nc : α\nd : β\ne : (a, b).fst = (c, d).fst\nh : s (a, b).snd (c, d).snd\n⊢ Prod.Lex r s (a, b) (c, d)",
"tactic": "subst e"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.5817\nδ : Type ?u.5820\nr✝ : α → α → Prop\ns✝ : β → β → Prop\nx y : α × β\nr : α → α → Prop\ns : β → β → Prop\np q : α × β\nh✝ : r p.fst q.fst ∨ p.fst = q.fst ∧ s p.snd q.snd\na : α\nb d : β\nh : s (a, b).snd ((a, b).fst, d).snd\n⊢ Prod.Lex r s (a, b) ((a, b).fst, d)",
"tactic": "exact Lex.right _ h"
}
] |
[
232,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Tactic/Group.lean
|
Mathlib.Tactic.Group.zpow_trick_one
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝ : Group G\na b : G\nm : ℤ\n⊢ a * b * b ^ m = a * b ^ (m + 1)",
"tactic": "rw [mul_assoc, mul_self_zpow]"
}
] |
[
47,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.norm_two
|
[] |
[
711,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
711,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
Filter.Tendsto.div'
|
[] |
[
1087,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1085,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
IsOpen.div_closure
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : TopologicalGroup α\ns t✝ : Set α\nhs : IsOpen s\nt : Set α\n⊢ s / closure t = s / t",
"tactic": "simp_rw [div_eq_mul_inv, inv_closure, hs.mul_closure]"
}
] |
[
1385,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1384,
1
] |
Mathlib/Topology/Algebra/GroupWithZero.lean
|
Continuous.comp_div_cases
|
[] |
[
229,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.conj_eq_iff_im
|
[] |
[
557,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
555,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measure_biUnion_toMeasurable
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.37295\nδ : Type ?u.37298\nι : Type ?u.37301\nR : Type ?u.37304\nR' : Type ?u.37307\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nI : Set β\nhc : Set.Countable I\ns : β → Set α\nthis : Encodable ↑I\n⊢ ↑↑μ (⋃ (b : β) (_ : b ∈ I), toMeasurable μ (s b)) = ↑↑μ (⋃ (b : β) (_ : b ∈ I), s b)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.37295\nδ : Type ?u.37298\nι : Type ?u.37301\nR : Type ?u.37304\nR' : Type ?u.37307\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nI : Set β\nhc : Set.Countable I\ns : β → Set α\n⊢ ↑↑μ (⋃ (b : β) (_ : b ∈ I), toMeasurable μ (s b)) = ↑↑μ (⋃ (b : β) (_ : b ∈ I), s b)",
"tactic": "haveI := hc.toEncodable"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.37295\nδ : Type ?u.37298\nι : Type ?u.37301\nR : Type ?u.37304\nR' : Type ?u.37307\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nI : Set β\nhc : Set.Countable I\ns : β → Set α\nthis : Encodable ↑I\n⊢ ↑↑μ (⋃ (b : β) (_ : b ∈ I), toMeasurable μ (s b)) = ↑↑μ (⋃ (b : β) (_ : b ∈ I), s b)",
"tactic": "simp only [biUnion_eq_iUnion, measure_iUnion_toMeasurable]"
}
] |
[
370,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
|
AffineEquiv.map_vadd
|
[] |
[
111,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
EuclideanGeometry.angle_lt_pi_div_two_of_angle_eq_pi_div_two
|
[
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₃ ≠ p₂\n⊢ ∠ p₂ p₃ p₁ < π / 2",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ∠ p₁ p₂ p₃ = π / 2\nh0 : p₃ ≠ p₂\n⊢ ∠ p₂ p₃ p₁ < π / 2",
"tactic": "rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←\n inner_neg_left, neg_vsub_eq_vsub_rev] at h"
},
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₂ -ᵥ p₃ ≠ 0\n⊢ ∠ p₂ p₃ p₁ < π / 2",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₃ ≠ p₂\n⊢ ∠ p₂ p₃ p₁ < π / 2",
"tactic": "rw [ne_comm, ← @vsub_ne_zero V] at h0"
},
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₂ -ᵥ p₃ ≠ 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₂ -ᵥ p₃ + (p₁ -ᵥ p₂)) < π / 2",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₂ -ᵥ p₃ ≠ 0\n⊢ ∠ p₂ p₃ p₁ < π / 2",
"tactic": "rw [angle, ← vsub_add_vsub_cancel p₁ p₂ p₃, add_comm]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₂ -ᵥ p₃ ≠ 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₂ -ᵥ p₃ + (p₁ -ᵥ p₂)) < π / 2",
"tactic": "exact angle_add_lt_pi_div_two_of_inner_eq_zero h h0"
}
] |
[
430,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
IsOpen.upperSemicontinuous_indicator
|
[] |
[
740,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
738,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.mod_le
|
[] |
[
1032,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1031,
1
] |
Mathlib/Topology/Constructions.lean
|
IsOpenMap.sum_elim
|
[] |
[
985,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
983,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.ext'
|
[] |
[
1080,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1078,
11
] |
Mathlib/Data/Polynomial/Div.lean
|
Polynomial.rootMultiplicity_zero
|
[] |
[
558,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
557,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
tendsto_comp_of_locally_uniform_limit
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : ContinuousWithinAt f univ x\nhg : Tendsto g p (𝓝 x)\nhunif : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝 x ∧ ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ t → (f y, F n y) ∈ u\n⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : ContinuousAt f x\nhg : Tendsto g p (𝓝 x)\nhunif : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝 x ∧ ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ t → (f y, F n y) ∈ u\n⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))",
"tactic": "rw [← continuousWithinAt_univ] at h"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : ContinuousWithinAt f univ x\nhg : Tendsto g p (𝓝[univ] x)\nhunif : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[univ] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ t → (f y, F n y) ∈ u\n⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : ContinuousWithinAt f univ x\nhg : Tendsto g p (𝓝 x)\nhunif : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝 x ∧ ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ t → (f y, F n y) ∈ u\n⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))",
"tactic": "rw [← nhdsWithin_univ] at hunif hg"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : ContinuousWithinAt f univ x\nhg : Tendsto g p (𝓝[univ] x)\nhunif : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[univ] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ t → (f y, F n y) ∈ u\n⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))",
"tactic": "exact tendsto_comp_of_locally_uniform_limit_within h hg hunif"
}
] |
[
942,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
937,
1
] |
Mathlib/Algebra/Tropical/Basic.lean
|
Tropical.trop_zsmul
|
[] |
[
498,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/LinearAlgebra/Dfinsupp.lean
|
CompleteLattice.Independent.dfinsupp_sumAddHom_injective
|
[
{
"state_after": "ι : Type u_2\nR : Type ?u.683548\nS : Type ?u.683551\nM : ι → Type ?u.683556\nN : Type u_1\ndec_ι : DecidableEq ι\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → AddSubgroup N\nh : Independent (↑AddSubgroup.toIntSubmodule ∘ p)\n⊢ Function.Injective ↑(sumAddHom fun i => AddSubgroup.subtype (p i))",
"state_before": "ι : Type u_2\nR : Type ?u.683548\nS : Type ?u.683551\nM : ι → Type ?u.683556\nN : Type u_1\ndec_ι : DecidableEq ι\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → AddSubgroup N\nh : Independent p\n⊢ Function.Injective ↑(sumAddHom fun i => AddSubgroup.subtype (p i))",
"tactic": "rw [← independent_map_orderIso_iff (AddSubgroup.toIntSubmodule : AddSubgroup N ≃o _)] at h"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nR : Type ?u.683548\nS : Type ?u.683551\nM : ι → Type ?u.683556\nN : Type u_1\ndec_ι : DecidableEq ι\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → AddSubgroup N\nh : Independent (↑AddSubgroup.toIntSubmodule ∘ p)\n⊢ Function.Injective ↑(sumAddHom fun i => AddSubgroup.subtype (p i))",
"tactic": "exact h.dfinsupp_lsum_injective"
}
] |
[
521,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
518,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.le_multiplicity_of_pow_dvd
|
[] |
[
146,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/FieldTheory/Finite/Basic.lean
|
FiniteField.exists_nonsquare
|
[
{
"state_after": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\n⊢ ∃ a, ¬IsSquare a",
"state_before": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\n⊢ ∃ a, ¬IsSquare a",
"tactic": "let sq : F → F := fun x => x ^ 2"
},
{
"state_after": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\nh : ¬Function.Injective sq\n⊢ ∃ a, ¬IsSquare a",
"state_before": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\n⊢ ∃ a, ¬IsSquare a",
"tactic": "have h : ¬Function.Injective sq := by\n simp only [Function.Injective, not_forall, exists_prop]\n refine' ⟨-1, 1, _, Ring.neg_one_ne_one_of_char_ne_two hF⟩\n simp only [one_pow, neg_one_sq]"
},
{
"state_after": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\nh : ¬Function.Surjective sq\n⊢ ∃ a, ¬IsSquare a",
"state_before": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\nh : ¬Function.Injective sq\n⊢ ∃ a, ¬IsSquare a",
"tactic": "rw [Finite.injective_iff_surjective] at h"
},
{
"state_after": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\nh : ¬Function.Surjective sq\n⊢ ∃ a, ¬∃ r, r ^ 2 = a",
"state_before": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\nh : ¬Function.Surjective sq\n⊢ ∃ a, ¬IsSquare a",
"tactic": "simp_rw [IsSquare, ← pow_two, @eq_comm _ _ (_ ^ 2)]"
},
{
"state_after": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\nh : ¬∀ (b : F), ∃ a, sq a = b\n⊢ ∃ a, ¬∃ r, r ^ 2 = a",
"state_before": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\nh : ¬Function.Surjective sq\n⊢ ∃ a, ¬∃ r, r ^ 2 = a",
"tactic": "unfold Function.Surjective at h"
},
{
"state_after": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\nh : ∃ b, ∀ (a : F), (fun x => x ^ 2) a ≠ b\n⊢ ∃ a, ∀ (r : F), r ^ 2 ≠ a",
"state_before": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\nh : ¬∀ (b : F), ∃ a, sq a = b\n⊢ ∃ a, ¬∃ r, r ^ 2 = a",
"tactic": "push_neg at h⊢"
},
{
"state_after": "no goals",
"state_before": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\nh : ∃ b, ∀ (a : F), (fun x => x ^ 2) a ≠ b\n⊢ ∃ a, ∀ (r : F), r ^ 2 ≠ a",
"tactic": "exact h"
},
{
"state_after": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\n⊢ ∃ x x_1, x ^ 2 = x_1 ^ 2 ∧ ¬x = x_1",
"state_before": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\n⊢ ¬Function.Injective sq",
"tactic": "simp only [Function.Injective, not_forall, exists_prop]"
},
{
"state_after": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\n⊢ (-1) ^ 2 = 1 ^ 2",
"state_before": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\n⊢ ∃ x x_1, x ^ 2 = x_1 ^ 2 ∧ ¬x = x_1",
"tactic": "refine' ⟨-1, 1, _, Ring.neg_one_ne_one_of_char_ne_two hF⟩"
},
{
"state_after": "no goals",
"state_before": "K : Type ?u.1094008\nR : Type ?u.1094011\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nsq : F → F := fun x => x ^ 2\n⊢ (-1) ^ 2 = 1 ^ 2",
"tactic": "simp only [one_pow, neg_one_sq]"
}
] |
[
496,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/Topology/Algebra/Monoid.lean
|
continuousMul_of_comm_of_nhds_one
|
[
{
"state_after": "ι : Type ?u.60957\nα : Type ?u.60960\nX : Type ?u.60963\nM✝ : Type ?u.60966\nN : Type ?u.60969\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace M✝\ninst✝³ : Mul M✝\ninst✝² : ContinuousMul M✝\nM : Type u\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)\nhleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)\n⊢ ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)",
"state_before": "ι : Type ?u.60957\nα : Type ?u.60960\nX : Type ?u.60963\nM✝ : Type ?u.60966\nN : Type ?u.60969\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace M✝\ninst✝³ : Mul M✝\ninst✝² : ContinuousMul M✝\nM : Type u\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)\nhleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)\n⊢ ContinuousMul M",
"tactic": "apply ContinuousMul.of_nhds_one hmul hleft"
},
{
"state_after": "ι : Type ?u.60957\nα : Type ?u.60960\nX : Type ?u.60963\nM✝ : Type ?u.60966\nN : Type ?u.60969\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace M✝\ninst✝³ : Mul M✝\ninst✝² : ContinuousMul M✝\nM : Type u\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)\nhleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)\nx₀ : M\n⊢ 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)",
"state_before": "ι : Type ?u.60957\nα : Type ?u.60960\nX : Type ?u.60963\nM✝ : Type ?u.60966\nN : Type ?u.60969\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace M✝\ninst✝³ : Mul M✝\ninst✝² : ContinuousMul M✝\nM : Type u\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)\nhleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)\n⊢ ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)",
"tactic": "intro x₀"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.60957\nα : Type ?u.60960\nX : Type ?u.60963\nM✝ : Type ?u.60966\nN : Type ?u.60969\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace M✝\ninst✝³ : Mul M✝\ninst✝² : ContinuousMul M✝\nM : Type u\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)\nhleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)\nx₀ : M\n⊢ 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)",
"tactic": "simp_rw [mul_comm, hleft x₀]"
}
] |
[
302,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.coeff_smul
|
[] |
[
533,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
532,
1
] |
Mathlib/Data/Nat/ModEq.lean
|
Nat.ModEq.add_right
|
[] |
[
151,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
11
] |
Mathlib/Computability/Primrec.lean
|
Primrec.list_headI
|
[] |
[
1047,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1046,
1
] |
Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean
|
Multiset.Nat.nodup_antidiagonalTuple
|
[] |
[
207,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.IsPwo.union
|
[] |
[
436,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
435,
18
] |
Mathlib/Data/Polynomial/Inductions.lean
|
Polynomial.divX_add
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ ∀ (n : ℕ), coeff (divX (p + q)) n = coeff (divX p + divX q) n",
"tactic": "simp"
}
] |
[
69,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean
|
rescale_to_shell_zpow
|
[] |
[
408,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
406,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
QuotientAddGroup.equivIcoMod_coe
|
[] |
[
811,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
809,
1
] |
Mathlib/AlgebraicTopology/DoldKan/Projections.lean
|
AlgebraicTopology.DoldKan.map_Q
|
[
{
"state_after": "C : Type u_4\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nX✝ : SimplicialObject C\nD : Type u_1\ninst✝² : Category D\ninst✝¹ : Preadditive D\nG : C ⥤ D\ninst✝ : Functor.Additive G\nX : SimplicialObject C\nq n : ℕ\n⊢ G.map (𝟙 (X.obj [n].op)) = 𝟙 ((((whiskering C D).obj G).obj X).obj [n].op)",
"state_before": "C : Type u_4\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nX✝ : SimplicialObject C\nD : Type u_1\ninst✝² : Category D\ninst✝¹ : Preadditive D\nG : C ⥤ D\ninst✝ : Functor.Additive G\nX : SimplicialObject C\nq n : ℕ\n⊢ G.map (HomologicalComplex.Hom.f (Q q) n) = HomologicalComplex.Hom.f (Q q) n",
"tactic": "rw [← add_right_inj (G.map ((P q : K[X] ⟶ _).f n)), ← G.map_add, map_P G X q n, P_add_Q_f,\n P_add_Q_f]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_4\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nX✝ : SimplicialObject C\nD : Type u_1\ninst✝² : Category D\ninst✝¹ : Preadditive D\nG : C ⥤ D\ninst✝ : Functor.Additive G\nX : SimplicialObject C\nq n : ℕ\n⊢ G.map (𝟙 (X.obj [n].op)) = 𝟙 ((((whiskering C D).obj G).obj X).obj [n].op)",
"tactic": "apply G.map_id"
}
] |
[
238,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snorm_le_snorm_of_exponent_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.4186528\nG : Type ?u.4186531\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq✝ : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\np q : ℝ≥0∞\nhpq : p ≤ q\ninst✝ : IsProbabilityMeasure μ\nf : α → E\nhf : AEStronglyMeasurable f μ\n⊢ snorm f q μ * ↑↑μ Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q) = snorm f q μ",
"tactic": "simp [measure_univ]"
}
] |
[
1110,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1108,
1
] |
Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean
|
tendsto_atTop_iInf
|
[] |
[
178,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/CategoryTheory/Abelian/Transfer.lean
|
CategoryTheory.AbelianOfAdjunction.hasCokernels
|
[
{
"state_after": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis : PreservesColimits G\n⊢ HasCokernel f",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\n⊢ HasCokernel f",
"tactic": "have : PreservesColimits G := adj.leftAdjointPreservesColimits"
},
{
"state_after": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis✝ : PreservesColimits G\nthis : i.inv.app X✝ ≫ (F ⋙ G).map f ≫ i.hom.app Y✝ = (𝟭 C).map f\n⊢ HasCokernel f",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis : PreservesColimits G\n⊢ HasCokernel f",
"tactic": "have := NatIso.naturality_1 i f"
},
{
"state_after": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis✝ : PreservesColimits G\nthis : i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝ = f\n⊢ HasCokernel f",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis✝ : PreservesColimits G\nthis : i.inv.app X✝ ≫ (F ⋙ G).map f ≫ i.hom.app Y✝ = (𝟭 C).map f\n⊢ HasCokernel f",
"tactic": "simp at this"
},
{
"state_after": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis✝ : PreservesColimits G\nthis : i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝ = f\n⊢ HasCokernel (i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝)",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis✝ : PreservesColimits G\nthis : i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝ = f\n⊢ HasCokernel f",
"tactic": "rw [← this]"
},
{
"state_after": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis✝¹ : PreservesColimits G\nthis✝ : i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝ = f\nthis : HasCokernel (G.map (F.map f) ≫ i.hom.app Y✝)\n⊢ HasCokernel (i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝)",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis✝ : PreservesColimits G\nthis : i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝ = f\n⊢ HasCokernel (i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝)",
"tactic": "haveI : HasCokernel (G.map (F.map f) ≫ i.hom.app _) := Limits.hasCokernel_comp_iso _ _"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\ninst✝³ : Preadditive C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝ : Functor.PreservesZeroMorphisms G\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nthis✝¹ : PreservesColimits G\nthis✝ : i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝ = f\nthis : HasCokernel (G.map (F.map f) ≫ i.hom.app Y✝)\n⊢ HasCokernel (i.inv.app X✝ ≫ G.map (F.map f) ≫ i.hom.app Y✝)",
"tactic": "apply Limits.hasCokernel_epi_comp"
}
] |
[
77,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_eq_self_of_not_mem
|
[] |
[
411,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/MeasureTheory/Integral/CircleTransform.lean
|
Complex.abs_circleTransformBoundingFunction_le
|
[
{
"state_after": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ univ)\n⊢ ∃ x,\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),\n ↑abs (circleTransformBoundingFunction R z ↑y) ≤ ↑abs (circleTransformBoundingFunction R z ↑x)",
"state_before": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\n⊢ ∃ x,\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),\n ↑abs (circleTransformBoundingFunction R z ↑y) ≤ ↑abs (circleTransformBoundingFunction R z ↑x)",
"tactic": "have cts := continuousOn_abs_circleTransformBoundingFunction hr z"
},
{
"state_after": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\n⊢ ∃ x,\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),\n ↑abs (circleTransformBoundingFunction R z ↑y) ≤ ↑abs (circleTransformBoundingFunction R z ↑x)",
"state_before": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ univ)\n⊢ ∃ x,\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),\n ↑abs (circleTransformBoundingFunction R z ↑y) ≤ ↑abs (circleTransformBoundingFunction R z ↑x)",
"tactic": "have comp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]]) := by\n apply_rules [IsCompact.prod, ProperSpace.isCompact_closedBall z r, isCompact_uIcc]"
},
{
"state_after": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\nnone : Set.Nonempty (closedBall z r ×ˢ [[0, 2 * π]])\n⊢ ∃ x,\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),\n ↑abs (circleTransformBoundingFunction R z ↑y) ≤ ↑abs (circleTransformBoundingFunction R z ↑x)",
"state_before": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\n⊢ ∃ x,\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),\n ↑abs (circleTransformBoundingFunction R z ↑y) ≤ ↑abs (circleTransformBoundingFunction R z ↑x)",
"tactic": "have none : (closedBall z r ×ˢ [[0, 2 * π]]).Nonempty :=\n (nonempty_closedBall.2 hr').prod nonempty_uIcc"
},
{
"state_after": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\nnone : Set.Nonempty (closedBall z r ×ˢ [[0, 2 * π]])\nthis :\n ∃ x,\n x ∈ closedBall z r ×ˢ [[0, 2 * π]] ∧\n IsMaxOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ [[0, 2 * π]]) x\n⊢ ∃ x,\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),\n ↑abs (circleTransformBoundingFunction R z ↑y) ≤ ↑abs (circleTransformBoundingFunction R z ↑x)",
"state_before": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\nnone : Set.Nonempty (closedBall z r ×ˢ [[0, 2 * π]])\n⊢ ∃ x,\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),\n ↑abs (circleTransformBoundingFunction R z ↑y) ≤ ↑abs (circleTransformBoundingFunction R z ↑x)",
"tactic": "have := IsCompact.exists_isMaxOn comp none (cts.mono\n (by intro z; simp only [mem_prod, mem_closedBall, mem_univ, and_true_iff, and_imp]; tauto))"
},
{
"state_after": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\nnone : Set.Nonempty (closedBall z r ×ˢ [[0, 2 * π]])\nthis :\n ∃ x,\n x ∈ closedBall z r ×ˢ [[0, 2 * π]] ∧\n ∀ᶠ (x_1 : ℂ × ℝ) in 𝓟 (closedBall z r ×ˢ [[0, 2 * π]]),\n (↑abs ∘ fun t => circleTransformBoundingFunction R z t) x_1 ≤\n (↑abs ∘ fun t => circleTransformBoundingFunction R z t) x\n⊢ ∃ x,\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),\n ↑abs (circleTransformBoundingFunction R z ↑y) ≤ ↑abs (circleTransformBoundingFunction R z ↑x)",
"state_before": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\nnone : Set.Nonempty (closedBall z r ×ˢ [[0, 2 * π]])\nthis :\n ∃ x,\n x ∈ closedBall z r ×ˢ [[0, 2 * π]] ∧\n IsMaxOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ [[0, 2 * π]]) x\n⊢ ∃ x,\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),\n ↑abs (circleTransformBoundingFunction R z ↑y) ≤ ↑abs (circleTransformBoundingFunction R z ↑x)",
"tactic": "simp only [IsMaxOn, IsMaxFilter] at this"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\nnone : Set.Nonempty (closedBall z r ×ˢ [[0, 2 * π]])\nthis :\n ∃ x,\n x ∈ closedBall z r ×ˢ [[0, 2 * π]] ∧\n ∀ᶠ (x_1 : ℂ × ℝ) in 𝓟 (closedBall z r ×ˢ [[0, 2 * π]]),\n (↑abs ∘ fun t => circleTransformBoundingFunction R z t) x_1 ≤\n (↑abs ∘ fun t => circleTransformBoundingFunction R z t) x\n⊢ ∃ x,\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),\n ↑abs (circleTransformBoundingFunction R z ↑y) ≤ ↑abs (circleTransformBoundingFunction R z ↑x)",
"tactic": "simpa [SetCoe.forall, Subtype.coe_mk, SetCoe.exists]"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ univ)\n⊢ IsCompact (closedBall z r ×ˢ [[0, 2 * π]])",
"tactic": "apply_rules [IsCompact.prod, ProperSpace.isCompact_closedBall z r, isCompact_uIcc]"
},
{
"state_after": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝¹ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz✝ : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z✝ t) (closedBall z✝ r ×ˢ univ)\ncomp : IsCompact (closedBall z✝ r ×ˢ [[0, 2 * π]])\nnone : Set.Nonempty (closedBall z✝ r ×ˢ [[0, 2 * π]])\nz : ℂ × ℝ\n⊢ z ∈ closedBall z✝ r ×ˢ [[0, 2 * π]] → z ∈ closedBall z✝ r ×ˢ univ",
"state_before": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z t) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\nnone : Set.Nonempty (closedBall z r ×ˢ [[0, 2 * π]])\n⊢ closedBall z r ×ˢ [[0, 2 * π]] ⊆ closedBall z r ×ˢ univ",
"tactic": "intro z"
},
{
"state_after": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝¹ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz✝ : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z✝ t) (closedBall z✝ r ×ˢ univ)\ncomp : IsCompact (closedBall z✝ r ×ˢ [[0, 2 * π]])\nnone : Set.Nonempty (closedBall z✝ r ×ˢ [[0, 2 * π]])\nz : ℂ × ℝ\n⊢ dist z.fst z✝ ≤ r → z.snd ∈ [[0, 2 * π]] → dist z.fst z✝ ≤ r",
"state_before": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝¹ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz✝ : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z✝ t) (closedBall z✝ r ×ˢ univ)\ncomp : IsCompact (closedBall z✝ r ×ˢ [[0, 2 * π]])\nnone : Set.Nonempty (closedBall z✝ r ×ˢ [[0, 2 * π]])\nz : ℂ × ℝ\n⊢ z ∈ closedBall z✝ r ×ˢ [[0, 2 * π]] → z ∈ closedBall z✝ r ×ˢ univ",
"tactic": "simp only [mem_prod, mem_closedBall, mem_univ, and_true_iff, and_imp]"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.139500\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR✝ : ℝ\nz✝¹ w : ℂ\nR r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz✝ : ℂ\ncts : ContinuousOn (↑abs ∘ fun t => circleTransformBoundingFunction R z✝ t) (closedBall z✝ r ×ˢ univ)\ncomp : IsCompact (closedBall z✝ r ×ˢ [[0, 2 * π]])\nnone : Set.Nonempty (closedBall z✝ r ×ˢ [[0, 2 * π]])\nz : ℂ × ℝ\n⊢ dist z.fst z✝ ≤ r → z.snd ∈ [[0, 2 * π]] → dist z.fst z✝ ≤ r",
"tactic": "tauto"
}
] |
[
142,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.diagonal_kroneckerTMul_diagonal
|
[] |
[
499,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/GroupTheory/Submonoid/Basic.lean
|
Submonoid.coe_inf
|
[] |
[
297,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Monotone.tendsto_nhdsWithin_Ioi
|
[] |
[
2873,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2870,
1
] |
Mathlib/Topology/Order.lean
|
generateFrom_iInter_of_generateFrom_eq_self
|
[] |
[
991,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
988,
1
] |
src/lean/Init/Prelude.lean
|
Char.eq_of_val_eq
|
[] |
[
2083,
31
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
2082,
1
] |
Mathlib/Topology/Order.lean
|
nhdsAdjoint_nhds
|
[
{
"state_after": "α✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\n⊢ 𝓝 a = pure a ⊔ f",
"state_before": "α✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\n⊢ 𝓝 a = pure a ⊔ f",
"tactic": "letI := nhdsAdjoint a f"
},
{
"state_after": "case a\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\nU : Set α\n⊢ U ∈ 𝓝 a ↔ U ∈ pure a ⊔ f",
"state_before": "α✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\n⊢ 𝓝 a = pure a ⊔ f",
"tactic": "ext U"
},
{
"state_after": "case a\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\nU : Set α\n⊢ (∃ t, t ⊆ U ∧ IsOpen t ∧ a ∈ t) ↔ U ∈ pure a ⊔ f",
"state_before": "case a\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\nU : Set α\n⊢ U ∈ 𝓝 a ↔ U ∈ pure a ⊔ f",
"tactic": "rw [mem_nhds_iff]"
},
{
"state_after": "case a.mp\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\nU : Set α\n⊢ (∃ t, t ⊆ U ∧ IsOpen t ∧ a ∈ t) → U ∈ pure a ⊔ f\n\ncase a.mpr\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\nU : Set α\n⊢ U ∈ pure a ⊔ f → ∃ t, t ⊆ U ∧ IsOpen t ∧ a ∈ t",
"state_before": "case a\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\nU : Set α\n⊢ (∃ t, t ⊆ U ∧ IsOpen t ∧ a ∈ t) ↔ U ∈ pure a ⊔ f",
"tactic": "constructor"
},
{
"state_after": "case a.mp.intro.intro.intro\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\nU t : Set α\nhtU : t ⊆ U\nht : IsOpen t\nhat : a ∈ t\n⊢ U ∈ pure a ⊔ f",
"state_before": "case a.mp\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\nU : Set α\n⊢ (∃ t, t ⊆ U ∧ IsOpen t ∧ a ∈ t) → U ∈ pure a ⊔ f",
"tactic": "rintro ⟨t, htU, ht, hat⟩"
},
{
"state_after": "no goals",
"state_before": "case a.mp.intro.intro.intro\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\nU t : Set α\nhtU : t ⊆ U\nht : IsOpen t\nhat : a ∈ t\n⊢ U ∈ pure a ⊔ f",
"tactic": "exact ⟨htU hat, mem_of_superset (ht hat) htU⟩"
},
{
"state_after": "case a.mpr.intro\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\nU : Set α\nhaU : U ∈ (pure a).sets\nhU : U ∈ f.sets\n⊢ ∃ t, t ⊆ U ∧ IsOpen t ∧ a ∈ t",
"state_before": "case a.mpr\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\nU : Set α\n⊢ U ∈ pure a ⊔ f → ∃ t, t ⊆ U ∧ IsOpen t ∧ a ∈ t",
"tactic": "rintro ⟨haU, hU⟩"
},
{
"state_after": "no goals",
"state_before": "case a.mpr.intro\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nthis : TopologicalSpace α := nhdsAdjoint a f\nU : Set α\nhaU : U ∈ (pure a).sets\nhU : U ∈ f.sets\n⊢ ∃ t, t ⊆ U ∧ IsOpen t ∧ a ∈ t",
"tactic": "exact ⟨U, Subset.rfl, fun _ => hU, haU⟩"
}
] |
[
611,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
602,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.domRestrict'_apply
|
[] |
[
531,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pullbackRightPullbackFstIso_inv_fst
|
[] |
[
2178,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2176,
1
] |
Mathlib/Data/List/Nodup.lean
|
List.nodup_map_iff_inj_on
|
[] |
[
264,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.coe_ennreal_le_coe_ennreal_iff
|
[] |
[
502,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
501,
1
] |
Mathlib/Data/List/Basic.lean
|
List.mem_takeWhile_imp
|
[
{
"state_after": "case cons.false\nι : Type ?u.414998\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\nx hd : α\ntl : List α\nIH : x ∈ takeWhile p tl → p x = true\nhx :\n x ∈\n match p hd with\n | true => hd :: takeWhile p tl\n | false => []\nhp : p hd = false\n⊢ p x = true\n\ncase cons.true\nι : Type ?u.414998\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\nx hd : α\ntl : List α\nIH : x ∈ takeWhile p tl → p x = true\nhx :\n x ∈\n match p hd with\n | true => hd :: takeWhile p tl\n | false => []\nhp : p hd = true\n⊢ p x = true",
"state_before": "case cons\nι : Type ?u.414998\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\nx hd : α\ntl : List α\nIH : x ∈ takeWhile p tl → p x = true\nhx :\n x ∈\n match p hd with\n | true => hd :: takeWhile p tl\n | false => []\n⊢ p x = true",
"tactic": "cases hp : p hd"
},
{
"state_after": "no goals",
"state_before": "case cons.false\nι : Type ?u.414998\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\nx hd : α\ntl : List α\nIH : x ∈ takeWhile p tl → p x = true\nhx :\n x ∈\n match p hd with\n | true => hd :: takeWhile p tl\n | false => []\nhp : p hd = false\n⊢ p x = true",
"tactic": "simp [hp] at hx"
},
{
"state_after": "case cons.true\nι : Type ?u.414998\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\nx hd : α\ntl : List α\nIH : x ∈ takeWhile p tl → p x = true\nhx : x = hd ∨ x ∈ takeWhile p tl\nhp : p hd = true\n⊢ p x = true",
"state_before": "case cons.true\nι : Type ?u.414998\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\nx hd : α\ntl : List α\nIH : x ∈ takeWhile p tl → p x = true\nhx :\n x ∈\n match p hd with\n | true => hd :: takeWhile p tl\n | false => []\nhp : p hd = true\n⊢ p x = true",
"tactic": "rw [hp, mem_cons] at hx"
},
{
"state_after": "case cons.true.inl\nι : Type ?u.414998\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\nx : α\ntl : List α\nIH : x ∈ takeWhile p tl → p x = true\nhp : p x = true\n⊢ p x = true\n\ncase cons.true.inr\nι : Type ?u.414998\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\nx hd : α\ntl : List α\nIH : x ∈ takeWhile p tl → p x = true\nhp : p hd = true\nhx : x ∈ takeWhile p tl\n⊢ p x = true",
"state_before": "case cons.true\nι : Type ?u.414998\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\nx hd : α\ntl : List α\nIH : x ∈ takeWhile p tl → p x = true\nhx : x = hd ∨ x ∈ takeWhile p tl\nhp : p hd = true\n⊢ p x = true",
"tactic": "rcases hx with (rfl | hx)"
},
{
"state_after": "no goals",
"state_before": "case cons.true.inl\nι : Type ?u.414998\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\nx : α\ntl : List α\nIH : x ∈ takeWhile p tl → p x = true\nhp : p x = true\n⊢ p x = true",
"tactic": "exact hp"
},
{
"state_after": "no goals",
"state_before": "case cons.true.inr\nι : Type ?u.414998\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\nx hd : α\ntl : List α\nIH : x ∈ takeWhile p tl → p x = true\nhp : p hd = true\nhx : x ∈ takeWhile p tl\n⊢ p x = true",
"tactic": "exact IH hx"
}
] |
[
3659,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3651,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.neg_fuzzy_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "x : PGame\n⊢ -x ‖ 0 ↔ x ‖ 0",
"tactic": "rw [neg_fuzzy_iff, neg_zero]"
}
] |
[
1400,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1400,
1
] |
Mathlib/LinearAlgebra/Projection.lean
|
Submodule.linearProjOfIsCompl_idempotent
|
[] |
[
200,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
Subspace.dualPairing_eq
|
[
{
"state_after": "case h.h.h\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW : Subspace K V₁\nx✝¹ : Dual K V₁\nx✝ : { x // x ∈ W }\n⊢ ↑(↑(LinearMap.comp (Submodule.dualPairing W) (mkQ (dualAnnihilator W))) x✝¹) x✝ =\n ↑(↑(LinearMap.comp (↑(quotAnnihilatorEquiv W)) (mkQ (dualAnnihilator W))) x✝¹) x✝",
"state_before": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW : Subspace K V₁\n⊢ Submodule.dualPairing W = ↑(quotAnnihilatorEquiv W)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW : Subspace K V₁\nx✝¹ : Dual K V₁\nx✝ : { x // x ∈ W }\n⊢ ↑(↑(LinearMap.comp (Submodule.dualPairing W) (mkQ (dualAnnihilator W))) x✝¹) x✝ =\n ↑(↑(LinearMap.comp (↑(quotAnnihilatorEquiv W)) (mkQ (dualAnnihilator W))) x✝¹) x✝",
"tactic": "rfl"
}
] |
[
1388,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1385,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
finrank_vectorSpan_image_finset_le
|
[
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\nhn : Finset.Nonempty (Finset.image p s)\n⊢ finrank k { x // x ∈ vectorSpan k ↑(Finset.image p s) } ≤ n",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\n⊢ finrank k { x // x ∈ vectorSpan k ↑(Finset.image p s) } ≤ n",
"tactic": "have hn : (s.image p).Nonempty := by\n rw [Finset.Nonempty.image_iff, ← Finset.card_pos, hc]\n apply Nat.succ_pos"
},
{
"state_after": "case intro\nk : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\np₁ : P\nhp₁ : p₁ ∈ Finset.image p s\n⊢ finrank k { x // x ∈ vectorSpan k ↑(Finset.image p s) } ≤ n",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\nhn : Finset.Nonempty (Finset.image p s)\n⊢ finrank k { x // x ∈ vectorSpan k ↑(Finset.image p s) } ≤ n",
"tactic": "rcases hn with ⟨p₁, hp₁⟩"
},
{
"state_after": "case intro\nk : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\np₁ : P\nhp₁ : p₁ ∈ Finset.image p s\n⊢ finrank k { x // x ∈ Submodule.span k ↑(Finset.image (fun x => x -ᵥ p₁) (Finset.erase (Finset.image p s) p₁)) } ≤ n",
"state_before": "case intro\nk : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\np₁ : P\nhp₁ : p₁ ∈ Finset.image p s\n⊢ finrank k { x // x ∈ vectorSpan k ↑(Finset.image p s) } ≤ n",
"tactic": "rw [vectorSpan_eq_span_vsub_finset_right_ne k hp₁]"
},
{
"state_after": "case intro\nk : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\np₁ : P\nhp₁ : p₁ ∈ Finset.image p s\n⊢ Finset.card (Finset.image (fun p => p -ᵥ p₁) (Finset.erase (Finset.image p s) p₁)) ≤ n",
"state_before": "case intro\nk : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\np₁ : P\nhp₁ : p₁ ∈ Finset.image p s\n⊢ finrank k { x // x ∈ Submodule.span k ↑(Finset.image (fun x => x -ᵥ p₁) (Finset.erase (Finset.image p s) p₁)) } ≤ n",
"tactic": "refine' le_trans (finrank_span_finset_le_card (((s.image p).erase p₁).image fun p => p -ᵥ p₁)) _"
},
{
"state_after": "case intro\nk : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\np₁ : P\nhp₁ : p₁ ∈ Finset.image p s\n⊢ Finset.card (Finset.image p s) ≤ Finset.card s",
"state_before": "case intro\nk : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\np₁ : P\nhp₁ : p₁ ∈ Finset.image p s\n⊢ Finset.card (Finset.image (fun p => p -ᵥ p₁) (Finset.erase (Finset.image p s) p₁)) ≤ n",
"tactic": "rw [Finset.card_image_of_injective _ (vsub_left_injective p₁), Finset.card_erase_of_mem hp₁,\n tsub_le_iff_right, ← hc]"
},
{
"state_after": "no goals",
"state_before": "case intro\nk : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\np₁ : P\nhp₁ : p₁ ∈ Finset.image p s\n⊢ Finset.card (Finset.image p s) ≤ Finset.card s",
"tactic": "apply Finset.card_image_le"
},
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\n⊢ 0 < n + 1",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\n⊢ Finset.Nonempty (Finset.image p s)",
"tactic": "rw [Finset.Nonempty.image_iff, ← Finset.card_pos, hc]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\ns : Finset ι\nn : ℕ\nhc : Finset.card s = n + 1\n⊢ 0 < n + 1",
"tactic": "apply Nat.succ_pos"
}
] |
[
154,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
Memℓp.const_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type ?u.91570\np q : ℝ≥0∞\ninst✝³ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_2\ninst✝² : NormedRing 𝕜\ninst✝¹ : (i : α) → Module 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜 (E i)\nf : α → 𝕜\nhf : Memℓp f p\nc : 𝕜\ni : α\n⊢ BoundedSMul 𝕜 ((fun x => 𝕜) i)",
"tactic": "infer_instance"
}
] |
[
288,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean
|
TopCat.Presheaf.SheafConditionEqualizerProducts.w
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\n⊢ ((Pi.lift fun i => F.map (leSupr U i).op) ≫\n Pi.lift fun p => Pi.π (fun i => F.obj (U i).op) p.fst ≫ F.map (infLELeft (U p.fst) (U p.snd)).op) =\n (Pi.lift fun i => F.map (leSupr U i).op) ≫\n Pi.lift fun p => Pi.π (fun i => F.obj (U i).op) p.snd ≫ F.map (infLERight (U p.fst) (U p.snd)).op",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\n⊢ res F U ≫ leftRes F U = res F U ≫ rightRes F U",
"tactic": "dsimp [res, leftRes, rightRes]"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nx✝ : Discrete (ι × ι)\n⊢ ((Pi.lift fun i => F.map (leSupr U i).op) ≫\n Pi.lift fun p => Pi.π (fun i => F.obj (U i).op) p.fst ≫ F.map (infLELeft (U p.fst) (U p.snd)).op) ≫\n limit.π (Discrete.functor fun p => F.obj (U p.fst ⊓ U p.snd).op) x✝ =\n ((Pi.lift fun i => F.map (leSupr U i).op) ≫\n Pi.lift fun p => Pi.π (fun i => F.obj (U i).op) p.snd ≫ F.map (infLERight (U p.fst) (U p.snd)).op) ≫\n limit.π (Discrete.functor fun p => F.obj (U p.fst ⊓ U p.snd).op) x✝",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\n⊢ ((Pi.lift fun i => F.map (leSupr U i).op) ≫\n Pi.lift fun p => Pi.π (fun i => F.obj (U i).op) p.fst ≫ F.map (infLELeft (U p.fst) (U p.snd)).op) =\n (Pi.lift fun i => F.map (leSupr U i).op) ≫\n Pi.lift fun p => Pi.π (fun i => F.obj (U i).op) p.snd ≫ F.map (infLERight (U p.fst) (U p.snd)).op",
"tactic": "refine limit.hom_ext (fun _ => ?_)"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nx✝ : Discrete (ι × ι)\n⊢ F.map (leSupr U x✝.as.fst).op ≫ F.map (infLELeft (U x✝.as.fst) (U x✝.as.snd)).op =\n F.map (leSupr U x✝.as.snd).op ≫ F.map (infLERight (U x✝.as.fst) (U x✝.as.snd)).op",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nx✝ : Discrete (ι × ι)\n⊢ ((Pi.lift fun i => F.map (leSupr U i).op) ≫\n Pi.lift fun p => Pi.π (fun i => F.obj (U i).op) p.fst ≫ F.map (infLELeft (U p.fst) (U p.snd)).op) ≫\n limit.π (Discrete.functor fun p => F.obj (U p.fst ⊓ U p.snd).op) x✝ =\n ((Pi.lift fun i => F.map (leSupr U i).op) ≫\n Pi.lift fun p => Pi.π (fun i => F.obj (U i).op) p.snd ≫ F.map (infLERight (U p.fst) (U p.snd)).op) ≫\n limit.π (Discrete.functor fun p => F.obj (U p.fst ⊓ U p.snd).op) x✝",
"tactic": "simp only [limit.lift_π, limit.lift_π_assoc, Fan.mk_π_app, Category.assoc]"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nx✝ : Discrete (ι × ι)\n⊢ F.map ((leSupr U x✝.as.fst).op ≫ (infLELeft (U x✝.as.fst) (U x✝.as.snd)).op) =\n F.map (leSupr U x✝.as.snd).op ≫ F.map (infLERight (U x✝.as.fst) (U x✝.as.snd)).op",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nx✝ : Discrete (ι × ι)\n⊢ F.map (leSupr U x✝.as.fst).op ≫ F.map (infLELeft (U x✝.as.fst) (U x✝.as.snd)).op =\n F.map (leSupr U x✝.as.snd).op ≫ F.map (infLERight (U x✝.as.fst) (U x✝.as.snd)).op",
"tactic": "rw [← F.map_comp]"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nx✝ : Discrete (ι × ι)\n⊢ F.map ((leSupr U x✝.as.fst).op ≫ (infLELeft (U x✝.as.fst) (U x✝.as.snd)).op) =\n F.map ((leSupr U x✝.as.snd).op ≫ (infLERight (U x✝.as.fst) (U x✝.as.snd)).op)",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nx✝ : Discrete (ι × ι)\n⊢ F.map ((leSupr U x✝.as.fst).op ≫ (infLELeft (U x✝.as.fst) (U x✝.as.snd)).op) =\n F.map (leSupr U x✝.as.snd).op ≫ F.map (infLERight (U x✝.as.fst) (U x✝.as.snd)).op",
"tactic": "rw [← F.map_comp]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nx✝ : Discrete (ι × ι)\n⊢ F.map ((leSupr U x✝.as.fst).op ≫ (infLELeft (U x✝.as.fst) (U x✝.as.snd)).op) =\n F.map ((leSupr U x✝.as.snd).op ≫ (infLERight (U x✝.as.fst) (U x✝.as.snd)).op)",
"tactic": "congr 1"
}
] |
[
97,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Std/Data/AssocList.lean
|
Std.AssocList.forM_eq
|
[
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝ : Monad m\nf : α → β → m PUnit\nl : AssocList α β\n⊢ forM f l =\n List.forM (toList l) fun x =>\n match x with\n | (a, b) => f a b",
"tactic": "induction l <;> simp [*, forM]"
}
] |
[
78,
33
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
76,
9
] |
Mathlib/Order/RelClasses.lean
|
transitive_ge
|
[] |
[
909,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
908,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.nat_mod
|
[
{
"state_after": "case h\nα : Type ?u.174271\nβ : Type ?u.174274\nγ : Type ?u.174277\nδ : Type ?u.174280\nσ : Type ?u.174283\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nm n : ℕ\n⊢ (m, n).fst = m % n + (m, n).snd * (fun x x_1 => x / x_1) (m, n).fst (m, n).snd",
"state_before": "α : Type ?u.174271\nβ : Type ?u.174274\nγ : Type ?u.174277\nδ : Type ?u.174280\nσ : Type ?u.174283\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nm n : ℕ\n⊢ (m, n).fst - (m, n).snd * (fun x x_1 => x / x_1) (m, n).fst (m, n).snd = m % n",
"tactic": "apply Nat.sub_eq_of_eq_add"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type ?u.174271\nβ : Type ?u.174274\nγ : Type ?u.174277\nδ : Type ?u.174280\nσ : Type ?u.174283\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nm n : ℕ\n⊢ (m, n).fst = m % n + (m, n).snd * (fun x x_1 => x / x_1) (m, n).fst (m, n).snd",
"tactic": "simp [add_comm (m % n), Nat.div_add_mod]"
}
] |
[
837,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
834,
1
] |
Mathlib/Analysis/Normed/Field/Basic.lean
|
nnnorm_pow
|
[] |
[
553,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
552,
1
] |
Std/Data/Int/DivMod.lean
|
Int.emod_sub_cancel_right
|
[] |
[
448,
26
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
447,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
LinearIsometry.map_orthogonalProjection'
|
[
{
"state_after": "case refine'_1\n𝕜 : Type u_3\nE✝ : Type ?u.533832\nF : Type ?u.533835\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : InnerProductSpace 𝕜 E✝\ninst✝⁶ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E✝\ninst✝⁵ : CompleteSpace { x // x ∈ K }\nE : Type u_1\nE' : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace 𝕜 E'\nf : E →ₗᵢ[𝕜] E'\np : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ p }\nx : E\n⊢ ↑f ↑(↑(orthogonalProjection p) x) ∈ Submodule.map f p\n\ncase refine'_2\n𝕜 : Type u_3\nE✝ : Type ?u.533832\nF : Type ?u.533835\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : InnerProductSpace 𝕜 E✝\ninst✝⁶ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E✝\ninst✝⁵ : CompleteSpace { x // x ∈ K }\nE : Type u_1\nE' : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace 𝕜 E'\nf : E →ₗᵢ[𝕜] E'\np : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ p }\nx : E\ny : (fun x => E') ↑(↑(orthogonalProjection p) x)\nhy : y ∈ Submodule.map f p\n⊢ inner (↑f x - ↑f ↑(↑(orthogonalProjection p) x)) y = 0",
"state_before": "𝕜 : Type u_3\nE✝ : Type ?u.533832\nF : Type ?u.533835\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : InnerProductSpace 𝕜 E✝\ninst✝⁶ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E✝\ninst✝⁵ : CompleteSpace { x // x ∈ K }\nE : Type u_1\nE' : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace 𝕜 E'\nf : E →ₗᵢ[𝕜] E'\np : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ p }\nx : E\n⊢ ↑f ↑(↑(orthogonalProjection p) x) = ↑(↑(orthogonalProjection (Submodule.map f p)) (↑f x))",
"tactic": "refine' (eq_orthogonalProjection_of_mem_of_inner_eq_zero _ fun y hy => _).symm"
},
{
"state_after": "case refine'_2\n𝕜 : Type u_3\nE✝ : Type ?u.533832\nF : Type ?u.533835\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : InnerProductSpace 𝕜 E✝\ninst✝⁶ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E✝\ninst✝⁵ : CompleteSpace { x // x ∈ K }\nE : Type u_1\nE' : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace 𝕜 E'\nf : E →ₗᵢ[𝕜] E'\np : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ p }\nx : E\ny : (fun x => E') ↑(↑(orthogonalProjection p) x)\nhy : y ∈ Submodule.map f p\n⊢ inner (↑f x - ↑f ↑(↑(orthogonalProjection p) x)) y = 0",
"state_before": "case refine'_1\n𝕜 : Type u_3\nE✝ : Type ?u.533832\nF : Type ?u.533835\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : InnerProductSpace 𝕜 E✝\ninst✝⁶ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E✝\ninst✝⁵ : CompleteSpace { x // x ∈ K }\nE : Type u_1\nE' : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace 𝕜 E'\nf : E →ₗᵢ[𝕜] E'\np : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ p }\nx : E\n⊢ ↑f ↑(↑(orthogonalProjection p) x) ∈ Submodule.map f p\n\ncase refine'_2\n𝕜 : Type u_3\nE✝ : Type ?u.533832\nF : Type ?u.533835\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : InnerProductSpace 𝕜 E✝\ninst✝⁶ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E✝\ninst✝⁵ : CompleteSpace { x // x ∈ K }\nE : Type u_1\nE' : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace 𝕜 E'\nf : E →ₗᵢ[𝕜] E'\np : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ p }\nx : E\ny : (fun x => E') ↑(↑(orthogonalProjection p) x)\nhy : y ∈ Submodule.map f p\n⊢ inner (↑f x - ↑f ↑(↑(orthogonalProjection p) x)) y = 0",
"tactic": "refine' Submodule.apply_coe_mem_map _ _"
},
{
"state_after": "case refine'_2.intro.intro\n𝕜 : Type u_3\nE✝ : Type ?u.533832\nF : Type ?u.533835\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : InnerProductSpace 𝕜 E✝\ninst✝⁶ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E✝\ninst✝⁵ : CompleteSpace { x // x ∈ K }\nE : Type u_1\nE' : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace 𝕜 E'\nf : E →ₗᵢ[𝕜] E'\np : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ p }\nx x' : E\nhx' : x' ∈ ↑p\n⊢ inner (↑f x - ↑f ↑(↑(orthogonalProjection p) x)) (↑f x') = 0",
"state_before": "case refine'_2\n𝕜 : Type u_3\nE✝ : Type ?u.533832\nF : Type ?u.533835\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : InnerProductSpace 𝕜 E✝\ninst✝⁶ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E✝\ninst✝⁵ : CompleteSpace { x // x ∈ K }\nE : Type u_1\nE' : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace 𝕜 E'\nf : E →ₗᵢ[𝕜] E'\np : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ p }\nx : E\ny : (fun x => E') ↑(↑(orthogonalProjection p) x)\nhy : y ∈ Submodule.map f p\n⊢ inner (↑f x - ↑f ↑(↑(orthogonalProjection p) x)) y = 0",
"tactic": "rcases hy with ⟨x', hx', rfl : f x' = y⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\n𝕜 : Type u_3\nE✝ : Type ?u.533832\nF : Type ?u.533835\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : InnerProductSpace 𝕜 E✝\ninst✝⁶ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E✝\ninst✝⁵ : CompleteSpace { x // x ∈ K }\nE : Type u_1\nE' : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace 𝕜 E'\nf : E →ₗᵢ[𝕜] E'\np : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ p }\nx x' : E\nhx' : x' ∈ ↑p\n⊢ inner (↑f x - ↑f ↑(↑(orthogonalProjection p) x)) (↑f x') = 0",
"tactic": "rw [← f.map_sub, f.inner_map_map, orthogonalProjection_inner_eq_zero x x' hx']"
}
] |
[
547,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
1
] |
Mathlib/Init/Data/Int/Basic.lean
|
Int.natAbs_pos_of_ne_zero
|
[] |
[
91,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean
|
MvQPF.wrepr_wMk
|
[
{
"state_after": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n⊢ (MvPFunctor.wMk' (P F) ∘ repr)\n (abs ((TypeVec.id ::: recF (MvPFunctor.wMk' (P F) ∘ repr)) <$$> { fst := a, snd := splitFun f' f })) =\n MvPFunctor.wMk' (P F)\n (repr\n (abs\n ((TypeVec.id ::: recF (MvPFunctor.wMk' (P F) ∘ repr)) <$$>\n { fst := a, snd := MvPFunctor.appendContents (P F) f' f })))",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n⊢ wrepr (MvPFunctor.wMk (P F) a f' f) =\n MvPFunctor.wMk' (P F)\n (repr (abs ((TypeVec.id ::: wrepr) <$$> { fst := a, snd := MvPFunctor.appendContents (P F) f' f })))",
"tactic": "rw [wrepr, recF_eq', q.P.wDest'_wMk]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n⊢ (MvPFunctor.wMk' (P F) ∘ repr)\n (abs ((TypeVec.id ::: recF (MvPFunctor.wMk' (P F) ∘ repr)) <$$> { fst := a, snd := splitFun f' f })) =\n MvPFunctor.wMk' (P F)\n (repr\n (abs\n ((TypeVec.id ::: recF (MvPFunctor.wMk' (P F) ∘ repr)) <$$>\n { fst := a, snd := MvPFunctor.appendContents (P F) f' f })))",
"tactic": "rfl"
}
] |
[
146,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
AffineEquiv.coe_toHomeomorphOfFiniteDimensional_symm
|
[] |
[
161,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Algebra.map_sup
|
[] |
[
831,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
830,
1
] |
Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
|
TopCat.colimit_isOpen_iff
|
[
{
"state_after": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TopCatMax\nU : Set ↑(colimit F)\n⊢ IsOpen U ↔ ∀ (j : J), IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TopCatMax\nU : Set ↑(colimit F)\n⊢ IsOpen U ↔ ∀ (j : J), IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)",
"tactic": "dsimp [topologicalSpace_coe]"
},
{
"state_after": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TopCatMax\nU : Set ↑(colimit F)\n⊢ IsOpen U ↔ ∀ (j : J), IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TopCatMax\nU : Set ↑(colimit F)\n⊢ IsOpen U ↔ ∀ (j : J), IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)",
"tactic": "conv_lhs => rw [colimit_topology F]"
},
{
"state_after": "no goals",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TopCatMax\nU : Set ↑(colimit F)\n⊢ IsOpen U ↔ ∀ (j : J), IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)",
"tactic": "exact isOpen_iSup_iff"
}
] |
[
418,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/CategoryTheory/Monoidal/Category.lean
|
CategoryTheory.MonoidalCategory.inv_hom_id_tensor
|
[
{
"state_after": "no goals",
"state_before": "C✝ : Type u\n𝒞 : Category C✝\ninst✝² : MonoidalCategory C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nU V✝ W✝ X✝ Y✝ Z✝ V W X Y Z : C\nf : V ≅ W\ng : X ⟶ Y\nh : Y ⟶ Z\n⊢ (f.inv ⊗ g) ≫ (f.hom ⊗ h) = (𝟙 W ⊗ g) ≫ (𝟙 W ⊗ h)",
"tactic": "rw [← tensor_comp, f.inv_hom_id, id_tensor_comp]"
}
] |
[
362,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
360,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.subset_countable_closure_of_compact
|
[
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.280595\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε✝ ε₁ ε₂ : ℝ≥0∞\ns✝ t s : Set α\nhs : IsCompact s\nε : ℝ≥0∞\nhε : ε > 0\n⊢ ∃ t, Set.Countable t ∧ s ⊆ ⋃ (x : α) (_ : x ∈ t), closedBall x ε",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.280595\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε ε₁ ε₂ : ℝ≥0∞\ns✝ t s : Set α\nhs : IsCompact s\n⊢ ∃ t, t ⊆ s ∧ Set.Countable t ∧ s ⊆ closure t",
"tactic": "refine' subset_countable_closure_of_almost_dense_set s fun ε hε => _"
},
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.280595\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε✝ ε₁ ε₂ : ℝ≥0∞\ns✝ t✝ s : Set α\nhs : IsCompact s\nε : ℝ≥0∞\nhε : ε > 0\nt : Set α\nhtf : Set.Finite t\nhst : s ⊆ ⋃ (y : α) (_ : y ∈ t), ball y ε\n⊢ ∃ t, Set.Countable t ∧ s ⊆ ⋃ (x : α) (_ : x ∈ t), closedBall x ε",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.280595\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε✝ ε₁ ε₂ : ℝ≥0∞\ns✝ t s : Set α\nhs : IsCompact s\nε : ℝ≥0∞\nhε : ε > 0\n⊢ ∃ t, Set.Countable t ∧ s ⊆ ⋃ (x : α) (_ : x ∈ t), closedBall x ε",
"tactic": "rcases totallyBounded_iff'.1 hs.totallyBounded ε hε with ⟨t, -, htf, hst⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.280595\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε✝ ε₁ ε₂ : ℝ≥0∞\ns✝ t✝ s : Set α\nhs : IsCompact s\nε : ℝ≥0∞\nhε : ε > 0\nt : Set α\nhtf : Set.Finite t\nhst : s ⊆ ⋃ (y : α) (_ : y ∈ t), ball y ε\n⊢ ∃ t, Set.Countable t ∧ s ⊆ ⋃ (x : α) (_ : x ∈ t), closedBall x ε",
"tactic": "exact ⟨t, htf.countable, hst.trans <| iUnion₂_mono fun _ _ => ball_subset_closedBall⟩"
}
] |
[
833,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
829,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.ext_iff
|
[] |
[
112,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPushout.flip
|
[] |
[
667,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
666,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.multiplicity_pow_self
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np : α\nh0 : p ≠ 0\nhu : ¬IsUnit p\nn : ℕ\n⊢ p ^ n ∣ p ^ n ∧ ¬p ^ (n + 1) ∣ p ^ n",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np : α\nh0 : p ≠ 0\nhu : ¬IsUnit p\nn : ℕ\n⊢ multiplicity p (p ^ n) = ↑n",
"tactic": "rw [eq_coe_iff]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np : α\nh0 : p ≠ 0\nhu : ¬IsUnit p\nn : ℕ\n⊢ ¬p ^ (n + 1) ∣ p ^ n",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np : α\nh0 : p ≠ 0\nhu : ¬IsUnit p\nn : ℕ\n⊢ p ^ n ∣ p ^ n ∧ ¬p ^ (n + 1) ∣ p ^ n",
"tactic": "use dvd_rfl"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np : α\nh0 : p ≠ 0\nhu : ¬IsUnit p\nn : ℕ\n⊢ ¬n + 1 ≤ n",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np : α\nh0 : p ≠ 0\nhu : ¬IsUnit p\nn : ℕ\n⊢ ¬p ^ (n + 1) ∣ p ^ n",
"tactic": "rw [pow_dvd_pow_iff h0 hu]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np : α\nh0 : p ≠ 0\nhu : ¬IsUnit p\nn : ℕ\n⊢ ¬n + 1 ≤ n",
"tactic": "apply Nat.not_succ_le_self"
}
] |
[
619,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
614,
1
] |
Mathlib/Data/Nat/Bits.lean
|
Nat.div2_bit0
|
[] |
[
57,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.ae_le_toMeasurable
|
[] |
[
605,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
604,
1
] |
Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean
|
TopCat.Presheaf.SheafConditionEqualizerProducts.fork_pt
|
[] |
[
119,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.max_eq_left
|
[
{
"state_after": "a b : Int\nh : b ≤ a\n⊢ max b a = a",
"state_before": "a b : Int\nh : b ≤ a\n⊢ max a b = a",
"tactic": "rw [← Int.max_comm b a]"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nh : b ≤ a\n⊢ max b a = a",
"tactic": "exact Int.max_eq_right h"
}
] |
[
726,
52
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
725,
11
] |
Mathlib/Topology/UniformSpace/UniformEmbedding.lean
|
uniformEmbedding_subtype_val
|
[] |
[
161,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieSubmodule.coe_smul
|
[] |
[
213,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Init/Data/Nat/Bitwise.lean
|
Nat.bitwise'_swap
|
[
{
"state_after": "case h.h\nf : Bool → Bool → Bool\nh : f false false = false\nm n : ℕ\n⊢ bitwise' (Function.swap f) m n = Function.swap (bitwise' f) m n",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\n⊢ bitwise' (Function.swap f) = Function.swap (bitwise' f)",
"tactic": "funext m n"
},
{
"state_after": "case h.h\nf : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ ∀ (n : ℕ), bitwise' (Function.swap f) m n = Function.swap (bitwise' f) m n",
"state_before": "case h.h\nf : Bool → Bool → Bool\nh : f false false = false\nm n : ℕ\n⊢ bitwise' (Function.swap f) m n = Function.swap (bitwise' f) m n",
"tactic": "revert n"
},
{
"state_after": "case h.h\nf : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ ∀ (n : ℕ), bitwise' (fun y x => f x y) m n = bitwise' f n m",
"state_before": "case h.h\nf : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ ∀ (n : ℕ), bitwise' (Function.swap f) m n = Function.swap (bitwise' f) m n",
"tactic": "dsimp [Function.swap]"
},
{
"state_after": "f : Bool → Bool → Bool\nh : f false false = false\nm n : ℕ\n⊢ bitwise' (fun y x => f x y) 0 n = bitwise' f n 0\n\nf : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nn : Bool\n⊢ ∀ (n_1 : ℕ),\n (∀ (n : ℕ), bitwise' (fun y x => f x y) n_1 n = bitwise' f n n_1) →\n ∀ (n_2 : ℕ), bitwise' (fun y x => f x y) (bit n n_1) n_2 = bitwise' f n_2 (bit n n_1)",
"state_before": "case h.h\nf : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ ∀ (n : ℕ), bitwise' (fun y x => f x y) m n = bitwise' f n m",
"tactic": "apply binaryRec _ _ m <;> intro n"
},
{
"state_after": "case h\nf : Bool → Bool → Bool\nh : f false false = false\nm n : ℕ\n⊢ f false false = false",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm n : ℕ\n⊢ bitwise' (fun y x => f x y) 0 n = bitwise' f n 0",
"tactic": "rw [bitwise'_zero_left, bitwise'_zero_right]"
},
{
"state_after": "no goals",
"state_before": "case h\nf : Bool → Bool → Bool\nh : f false false = false\nm n : ℕ\n⊢ f false false = false",
"tactic": "exact h"
},
{
"state_after": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nn : Bool\na : ℕ\nih : ∀ (n : ℕ), bitwise' (fun y x => f x y) a n = bitwise' f n a\nm' : ℕ\n⊢ bitwise' (fun y x => f x y) (bit n a) m' = bitwise' f m' (bit n a)",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nn : Bool\n⊢ ∀ (n_1 : ℕ),\n (∀ (n : ℕ), bitwise' (fun y x => f x y) n_1 n = bitwise' f n n_1) →\n ∀ (n_2 : ℕ), bitwise' (fun y x => f x y) (bit n n_1) n_2 = bitwise' f n_2 (bit n n_1)",
"tactic": "intros a ih m'"
},
{
"state_after": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nn : Bool\na : ℕ\nih : ∀ (n : ℕ), bitwise' (fun y x => f x y) a n = bitwise' f n a\nm' : ℕ\n⊢ ∀ (b : Bool) (n_1 : ℕ), bitwise' (fun y x => f x y) (bit n a) (bit b n_1) = bitwise' f (bit b n_1) (bit n a)",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nn : Bool\na : ℕ\nih : ∀ (n : ℕ), bitwise' (fun y x => f x y) a n = bitwise' f n a\nm' : ℕ\n⊢ bitwise' (fun y x => f x y) (bit n a) m' = bitwise' f m' (bit n a)",
"tactic": "apply bitCasesOn m'"
},
{
"state_after": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nn : Bool\na : ℕ\nih : ∀ (n : ℕ), bitwise' (fun y x => f x y) a n = bitwise' f n a\nm' : ℕ\nb : Bool\nn' : ℕ\n⊢ bitwise' (fun y x => f x y) (bit n a) (bit b n') = bitwise' f (bit b n') (bit n a)",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nn : Bool\na : ℕ\nih : ∀ (n : ℕ), bitwise' (fun y x => f x y) a n = bitwise' f n a\nm' : ℕ\n⊢ ∀ (b : Bool) (n_1 : ℕ), bitwise' (fun y x => f x y) (bit n a) (bit b n_1) = bitwise' f (bit b n_1) (bit n a)",
"tactic": "intro b n'"
},
{
"state_after": "no goals",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nn : Bool\na : ℕ\nih : ∀ (n : ℕ), bitwise' (fun y x => f x y) a n = bitwise' f n a\nm' : ℕ\nb : Bool\nn' : ℕ\n⊢ bitwise' (fun y x => f x y) (bit n a) (bit b n') = bitwise' f (bit b n') (bit n a)",
"tactic": "rw [bitwise'_bit, bitwise'_bit, ih] <;> exact h"
}
] |
[
455,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean
|
sSupHom.id_comp
|
[] |
[
340,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.right_inv_eq_left_inv
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.261753\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B C : Matrix n n α\nh : A ⬝ B = 1\ng : C ⬝ A = 1\n⊢ B = C",
"tactic": "rw [← inv_eq_right_inv h, ← inv_eq_left_inv g]"
}
] |
[
473,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/Data/Set/Pairwise/Basic.lean
|
pairwise_disjoint_on
|
[] |
[
60,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.sups_subset_iff
|
[] |
[
114,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.lineMap_vsub_left
|
[] |
[
636,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
635,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.one_le_iff_pos
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.179673\nβ : Type ?u.179676\nγ : Type ?u.179679\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\n⊢ 1 ≤ o ↔ 0 < o",
"tactic": "rw [← succ_zero, succ_le_iff]"
}
] |
[
1077,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1077,
1
] |
Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean
|
SimpleGraph.FarFromTriangleFree.not_cliqueFree
|
[] |
[
82,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Data/Nat/Cast/Basic.lean
|
Nat.cast_pos
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.13001\ninst✝¹ : OrderedSemiring α\ninst✝ : Nontrivial α\nn : ℕ\n⊢ 0 < ↑n ↔ 0 < n",
"tactic": "cases n <;> simp [cast_add_one_pos]"
}
] |
[
113,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Algebra/CharP/MixedCharZero.lean
|
EqualCharZero.to_not_mixedCharZero
|
[
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nh : ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)\np : ℕ\np_pos : p > 0\n⊢ ¬MixedCharZero R p",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nh : ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)\n⊢ ∀ (p : ℕ), p > 0 → ¬MixedCharZero R p",
"tactic": "intro p p_pos"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nh : ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)\np : ℕ\np_pos : p > 0\nhp_mixedChar : MixedCharZero R p\n⊢ False",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nh : ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)\np : ℕ\np_pos : p > 0\n⊢ ¬MixedCharZero R p",
"tactic": "by_contra hp_mixedChar"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝ : CommRing R\nh : ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)\np : ℕ\np_pos : p > 0\nhp_mixedChar : MixedCharZero R p\nI : Ideal R\nhI_ne_top : I ≠ ⊤\nhI_p : CharP (R ⧸ I) p\n⊢ False",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nh : ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)\np : ℕ\np_pos : p > 0\nhp_mixedChar : MixedCharZero R p\n⊢ False",
"tactic": "rcases hp_mixedChar.charP_quotient with ⟨I, hI_ne_top, hI_p⟩"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝ : CommRing R\nh : ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)\np : ℕ\np_pos : p > 0\nhp_mixedChar : MixedCharZero R p\nI : Ideal R\nhI_ne_top : I ≠ ⊤\nhI_p : CharP (R ⧸ I) p\nhI_zero : CharP (R ⧸ I) 0\n⊢ False",
"state_before": "case intro.intro\nR : Type u_1\ninst✝ : CommRing R\nh : ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)\np : ℕ\np_pos : p > 0\nhp_mixedChar : MixedCharZero R p\nI : Ideal R\nhI_ne_top : I ≠ ⊤\nhI_p : CharP (R ⧸ I) p\n⊢ False",
"tactic": "replace hI_zero : CharP (R ⧸ I) 0 := @CharP.ofCharZero _ _ (h I hI_ne_top)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\ninst✝ : CommRing R\nh : ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)\np : ℕ\np_pos : p > 0\nhp_mixedChar : MixedCharZero R p\nI : Ideal R\nhI_ne_top : I ≠ ⊤\nhI_p : CharP (R ⧸ I) p\nhI_zero : CharP (R ⧸ I) 0\n⊢ False",
"tactic": "exact absurd (CharP.eq (R ⧸ I) hI_p hI_zero) (ne_of_gt p_pos)"
}
] |
[
275,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiff.neg
|
[] |
[
1280,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1279,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
AEMeasurable.pow
|
[] |
[
227,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.cos_conj
|
[
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ cos (↑(starRingEnd ℂ) x) = ↑(starRingEnd ℂ) (cos x)",
"tactic": "rw [← cosh_mul_I, ← conj_neg_I, ← RingHom.map_mul, ← cosh_mul_I, cosh_conj, mul_neg, cosh_neg]"
}
] |
[
951,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
950,
1
] |
Mathlib/Algebra/Symmetrized.lean
|
SymAlg.sym_comp_unsym
|
[] |
[
76,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.beth_zero
|
[] |
[
418,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
417,
1
] |
Mathlib/Data/Nat/GCD/Basic.lean
|
Nat.coprime_mul_left_add_left
|
[
{
"state_after": "no goals",
"state_before": "m n k : ℕ\n⊢ coprime (n * k + m) n ↔ coprime m n",
"tactic": "rw [coprime, coprime, gcd_mul_left_add_left]"
}
] |
[
189,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Polynomial.support_restriction
|
[
{
"state_after": "case a\nR : Type u\nS : Type ?u.69179\ninst✝ : Ring R\np : R[X]\ni : ℕ\n⊢ i ∈ support (restriction p) ↔ i ∈ support p",
"state_before": "R : Type u\nS : Type ?u.69179\ninst✝ : Ring R\np : R[X]\n⊢ support (restriction p) = support p",
"tactic": "ext i"
},
{
"state_after": "case a\nR : Type u\nS : Type ?u.69179\ninst✝ : Ring R\np : R[X]\ni : ℕ\n⊢ coeff (restriction p) i = 0 ↔ coeff p i = 0",
"state_before": "case a\nR : Type u\nS : Type ?u.69179\ninst✝ : Ring R\np : R[X]\ni : ℕ\n⊢ i ∈ support (restriction p) ↔ i ∈ support p",
"tactic": "simp only [mem_support_iff, not_iff_not, Ne.def]"
},
{
"state_after": "case a\nR : Type u\nS : Type ?u.69179\ninst✝ : Ring R\np : R[X]\ni : ℕ\n⊢ coeff (restriction p) i = 0 ↔ ↑(coeff (restriction p) i) = 0",
"state_before": "case a\nR : Type u\nS : Type ?u.69179\ninst✝ : Ring R\np : R[X]\ni : ℕ\n⊢ coeff (restriction p) i = 0 ↔ coeff p i = 0",
"tactic": "conv_rhs => rw [← coeff_restriction]"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\nS : Type ?u.69179\ninst✝ : Ring R\np : R[X]\ni : ℕ\n⊢ coeff (restriction p) i = 0 ↔ ↑(coeff (restriction p) i) = 0",
"tactic": "exact ⟨fun H => by rw [H, ZeroMemClass.coe_zero], fun H => Subtype.coe_injective H⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.69179\ninst✝ : Ring R\np : R[X]\ni : ℕ\nH : coeff (restriction p) i = 0\n⊢ ↑(coeff (restriction p) i) = 0",
"tactic": "rw [H, ZeroMemClass.coe_zero]"
}
] |
[
296,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.logb_lt_logb_of_base_lt_one
|
[
{
"state_after": "b x y : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\nhxy : x < y\n⊢ log x < log y",
"state_before": "b x y : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\nhxy : x < y\n⊢ logb b y < logb b x",
"tactic": "rw [logb, logb, div_lt_div_right_of_neg (log_neg b_pos b_lt_one)]"
},
{
"state_after": "no goals",
"state_before": "b x y : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\nhxy : x < y\n⊢ log x < log y",
"tactic": "exact log_lt_log hx hxy"
}
] |
[
262,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/ModelTheory/ElementaryMaps.lean
|
FirstOrder.Language.ElementaryEmbedding.coe_toHom
|
[] |
[
188,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
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