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/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_const_lt_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.76494\nγ : Type ?u.76497\nδ : Type ?u.76500\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : IsFiniteMeasure μ\nc : ℝ≥0∞\nhc : c ≠ ⊤\n⊢ (∫⁻ (x : α), c ∂μ) < ⊤",
"tactic": "simpa only [Measure.restrict_univ] using set_lintegral_const_lt_top (univ : Set α) hc"
}
] |
[
196,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Data/Finsupp/WellFounded.lean
|
Finsupp.Lex.acc
|
[
{
"state_after": "α : Type u_1\nN : Type u_2\ninst✝ : Zero N\nr : α → α → Prop\ns : N → N → Prop\nhbot : ∀ ⦃n : N⦄, ¬s n 0\nhs : WellFounded s\nx : α →₀ N\nh : ∀ (a : α), a ∈ x.support → Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) a\n⊢ Acc (InvImage (Dfinsupp.Lex r fun x => s) toDfinsupp) x",
"state_before": "α : Type u_1\nN : Type u_2\ninst✝ : Zero N\nr : α → α → Prop\ns : N → N → Prop\nhbot : ∀ ⦃n : N⦄, ¬s n 0\nhs : WellFounded s\nx : α →₀ N\nh : ∀ (a : α), a ∈ x.support → Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) a\n⊢ Acc (Finsupp.Lex r s) x",
"tactic": "rw [lex_eq_invImage_dfinsupp_lex]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nN : Type u_2\ninst✝ : Zero N\nr : α → α → Prop\ns : N → N → Prop\nhbot : ∀ ⦃n : N⦄, ¬s n 0\nhs : WellFounded s\nx : α →₀ N\nh : ∀ (a : α), a ∈ x.support → Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) a\n⊢ Acc (InvImage (Dfinsupp.Lex r fun x => s) toDfinsupp) x",
"tactic": "classical\n refine' InvImage.accessible toDfinsupp (Dfinsupp.Lex.acc (fun _ => hbot) (fun _ => hs) _ _)\n simpa only [toDfinsupp_support] using h"
},
{
"state_after": "α : Type u_1\nN : Type u_2\ninst✝ : Zero N\nr : α → α → Prop\ns : N → N → Prop\nhbot : ∀ ⦃n : N⦄, ¬s n 0\nhs : WellFounded s\nx : α →₀ N\nh : ∀ (a : α), a ∈ x.support → Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) a\n⊢ ∀ (i : α), i ∈ Dfinsupp.support (toDfinsupp x) → Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) i",
"state_before": "α : Type u_1\nN : Type u_2\ninst✝ : Zero N\nr : α → α → Prop\ns : N → N → Prop\nhbot : ∀ ⦃n : N⦄, ¬s n 0\nhs : WellFounded s\nx : α →₀ N\nh : ∀ (a : α), a ∈ x.support → Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) a\n⊢ Acc (InvImage (Dfinsupp.Lex r fun x => s) toDfinsupp) x",
"tactic": "refine' InvImage.accessible toDfinsupp (Dfinsupp.Lex.acc (fun _ => hbot) (fun _ => hs) _ _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nN : Type u_2\ninst✝ : Zero N\nr : α → α → Prop\ns : N → N → Prop\nhbot : ∀ ⦃n : N⦄, ¬s n 0\nhs : WellFounded s\nx : α →₀ N\nh : ∀ (a : α), a ∈ x.support → Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) a\n⊢ ∀ (i : α), i ∈ Dfinsupp.support (toDfinsupp x) → Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) i",
"tactic": "simpa only [toDfinsupp_support] using h"
}
] |
[
45,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.ofReal_one
|
[] |
[
178,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean
|
Multiset.nodup_Ico
|
[] |
[
36,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.eq_mul_of_div_eq_right
|
[
{
"state_after": "no goals",
"state_before": "m n k a b c : ℕ\nH1 : b ∣ a\nH2 : a / b = c\n⊢ a = b * c",
"tactic": "rw [← H2, Nat.mul_div_cancel' H1]"
}
] |
[
654,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
653,
11
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
lt_mul_iff_one_lt_right'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.24947\ninst✝³ : MulOneClass α\ninst✝² : LT α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : α\n⊢ a < a * b ↔ a * 1 < a * b",
"tactic": "rw [mul_one]"
}
] |
[
508,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
505,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.rpow_add
|
[
{
"state_after": "case none\ny z : ℝ\nhx : none ≠ 0\nh'x : none ≠ ⊤\n⊢ none ^ (y + z) = none ^ y * none ^ z\n\ncase some\ny z : ℝ\nx : ℝ≥0\nhx : Option.some x ≠ 0\nh'x : Option.some x ≠ ⊤\n⊢ Option.some x ^ (y + z) = Option.some x ^ y * Option.some x ^ z",
"state_before": "x : ℝ≥0∞\ny z : ℝ\nhx : x ≠ 0\nh'x : x ≠ ⊤\n⊢ x ^ (y + z) = x ^ y * x ^ z",
"tactic": "cases' x with x"
},
{
"state_after": "case some\ny z : ℝ\nx : ℝ≥0\nhx : Option.some x ≠ 0\nh'x : Option.some x ≠ ⊤\nthis : x ≠ 0\n⊢ Option.some x ^ (y + z) = Option.some x ^ y * Option.some x ^ z",
"state_before": "case some\ny z : ℝ\nx : ℝ≥0\nhx : Option.some x ≠ 0\nh'x : Option.some x ≠ ⊤\n⊢ Option.some x ^ (y + z) = Option.some x ^ y * Option.some x ^ z",
"tactic": "have : x ≠ 0 := fun h => by simp [h] at hx"
},
{
"state_after": "no goals",
"state_before": "case some\ny z : ℝ\nx : ℝ≥0\nhx : Option.some x ≠ 0\nh'x : Option.some x ≠ ⊤\nthis : x ≠ 0\n⊢ Option.some x ^ (y + z) = Option.some x ^ y * Option.some x ^ z",
"tactic": "simp [coe_rpow_of_ne_zero this, NNReal.rpow_add this]"
},
{
"state_after": "no goals",
"state_before": "case none\ny z : ℝ\nhx : none ≠ 0\nh'x : none ≠ ⊤\n⊢ none ^ (y + z) = none ^ y * none ^ z",
"tactic": "exact (h'x rfl).elim"
},
{
"state_after": "no goals",
"state_before": "y z : ℝ\nx : ℝ≥0\nhx : Option.some x ≠ 0\nh'x : Option.some x ≠ ⊤\nh : x = 0\n⊢ False",
"tactic": "simp [h] at hx"
}
] |
[
434,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderHom.prod_mono
|
[] |
[
413,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
412,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.IsPath.length_lt
|
[
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : Fintype V\nu v : V\np : Walk G u v\nhp : IsPath p\n⊢ List.length (support p) ≤ Fintype.card V",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : Fintype V\nu v : V\np : Walk G u v\nhp : IsPath p\n⊢ length p < Fintype.card V",
"tactic": "rw [Nat.lt_iff_add_one_le, ← length_support]"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : Fintype V\nu v : V\np : Walk G u v\nhp : IsPath p\n⊢ List.length (support p) ≤ Fintype.card V",
"tactic": "exact hp.support_nodup.length_le_card"
}
] |
[
1027,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1024,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.Measure.toENNRealVectorMeasure_add
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.149284\nm : MeasurableSpace α\nμ ν : Measure α\ni : Set α\nhi : MeasurableSet i\n⊢ ↑(toENNRealVectorMeasure (μ + ν)) i = ↑(toENNRealVectorMeasure μ + toENNRealVectorMeasure ν) i",
"state_before": "α : Type u_1\nβ : Type ?u.149284\nm : MeasurableSpace α\nμ ν : Measure α\n⊢ toENNRealVectorMeasure (μ + ν) = toENNRealVectorMeasure μ + toENNRealVectorMeasure ν",
"tactic": "refine' MeasureTheory.VectorMeasure.ext fun i hi => _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.149284\nm : MeasurableSpace α\nμ ν : Measure α\ni : Set α\nhi : MeasurableSet i\n⊢ ↑(toENNRealVectorMeasure (μ + ν)) i = ↑(toENNRealVectorMeasure μ + toENNRealVectorMeasure ν) i",
"tactic": "rw [toENNRealVectorMeasure_apply_measurable hi, add_apply, VectorMeasure.add_apply,\n toENNRealVectorMeasure_apply_measurable hi, toENNRealVectorMeasure_apply_measurable hi]"
}
] |
[
516,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
512,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.eq
|
[] |
[
344,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
342,
11
] |
Mathlib/FieldTheory/Separable.lean
|
Polynomial.separable_iff_derivative_ne_zero
|
[
{
"state_after": "F : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\nf : F[X]\nhf : Irreducible f\nh : ↑derivative f ≠ 0\ng : F[X]\nhg1 : g ∈ nonunits F[X]\n_hg2 : g ≠ 0\nx✝ : g ∣ f\nhg4 : g ∣ ↑derivative f\np : F[X]\nhg3 : f = g * p\nu : F[X]ˣ\nhu : ↑u = p\n⊢ g * ↑u ∣ ↑derivative f",
"state_before": "F : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\nf : F[X]\nhf : Irreducible f\nh : ↑derivative f ≠ 0\ng : F[X]\nhg1 : g ∈ nonunits F[X]\n_hg2 : g ≠ 0\nx✝ : g ∣ f\nhg4 : g ∣ ↑derivative f\np : F[X]\nhg3 : f = g * p\nu : F[X]ˣ\nhu : ↑u = p\n⊢ f ∣ ↑derivative f",
"tactic": "conv_lhs => rw [hg3, ← hu]"
},
{
"state_after": "no goals",
"state_before": "F : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\nf : F[X]\nhf : Irreducible f\nh : ↑derivative f ≠ 0\ng : F[X]\nhg1 : g ∈ nonunits F[X]\n_hg2 : g ≠ 0\nx✝ : g ∣ f\nhg4 : g ∣ ↑derivative f\np : F[X]\nhg3 : f = g * p\nu : F[X]ˣ\nhu : ↑u = p\n⊢ g * ↑u ∣ ↑derivative f",
"tactic": "rwa [Units.mul_right_dvd]"
}
] |
[
286,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Topology/Algebra/ConstMulAction.lean
|
tendsto_const_smul_iff₀
|
[] |
[
299,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/MeasureTheory/Group/FundamentalDomain.lean
|
MeasureTheory.IsFundamentalDomain.fundamentalInterior
|
[
{
"state_after": "G : Type u_1\nH : Type ?u.546154\nα : Type u_2\nβ : Type ?u.546160\nE : Type ?u.546163\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\n⊢ ↑↑μ ((⋃ (i : G), i⁻¹ • fundamentalInterior G s)ᶜ) = 0",
"state_before": "G : Type u_1\nH : Type ?u.546154\nα : Type u_2\nβ : Type ?u.546160\nE : Type ?u.546163\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\n⊢ ∀ᵐ (x : α) ∂μ, ∃ g, g • x ∈ fundamentalInterior G s",
"tactic": "simp_rw [ae_iff, not_exists, ← mem_inv_smul_set_iff, setOf_forall, ← compl_setOf,\n setOf_mem_eq, ← compl_iUnion]"
},
{
"state_after": "G : Type u_1\nH : Type ?u.546154\nα : Type u_2\nβ : Type ?u.546160\nE : Type ?u.546163\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\nthis : ((⋃ (g : G), g⁻¹ • s) \\ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s) ⊆ ⋃ (g : G), g⁻¹ • fundamentalInterior G s\n⊢ ↑↑μ ((⋃ (i : G), i⁻¹ • fundamentalInterior G s)ᶜ) = 0",
"state_before": "G : Type u_1\nH : Type ?u.546154\nα : Type u_2\nβ : Type ?u.546160\nE : Type ?u.546163\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\n⊢ ↑↑μ ((⋃ (i : G), i⁻¹ • fundamentalInterior G s)ᶜ) = 0",
"tactic": "have :\n ((⋃ g : G, g⁻¹ • s) \\ ⋃ g : G, g⁻¹ • fundamentalFrontier G s) ⊆\n ⋃ g : G, g⁻¹ • fundamentalInterior G s := by\n simp_rw [diff_subset_iff, ← iUnion_union_distrib, ← smul_set_union (α := G) (β := α),\n fundamentalFrontier_union_fundamentalInterior]; rfl"
},
{
"state_after": "G : Type u_1\nH : Type ?u.546154\nα : Type u_2\nβ : Type ?u.546160\nE : Type ?u.546163\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\nthis : ((⋃ (g : G), g⁻¹ • s) \\ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s) ⊆ ⋃ (g : G), g⁻¹ • fundamentalInterior G s\n⊢ ↑↑μ (((⋃ (g : G), g⁻¹ • s) \\ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s)ᶜ) = ⊥",
"state_before": "G : Type u_1\nH : Type ?u.546154\nα : Type u_2\nβ : Type ?u.546160\nE : Type ?u.546163\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\nthis : ((⋃ (g : G), g⁻¹ • s) \\ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s) ⊆ ⋃ (g : G), g⁻¹ • fundamentalInterior G s\n⊢ ↑↑μ ((⋃ (i : G), i⁻¹ • fundamentalInterior G s)ᶜ) = 0",
"tactic": "refine' eq_bot_mono (μ.mono <| compl_subset_compl.2 this) _"
},
{
"state_after": "G : Type u_1\nH : Type ?u.546154\nα : Type u_2\nβ : Type ?u.546160\nE : Type ?u.546163\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\nthis : ((⋃ (g : G), g⁻¹ • s) \\ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s) ⊆ ⋃ (g : G), g⁻¹ • fundamentalInterior G s\n⊢ ↑↑μ ((⋃ (g : G), g • fundamentalFrontier G s) ∪ {a | ∃ g, g • a ∈ s}ᶜ) = 0",
"state_before": "G : Type u_1\nH : Type ?u.546154\nα : Type u_2\nβ : Type ?u.546160\nE : Type ?u.546163\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\nthis : ((⋃ (g : G), g⁻¹ • s) \\ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s) ⊆ ⋃ (g : G), g⁻¹ • fundamentalInterior G s\n⊢ ↑↑μ (((⋃ (g : G), g⁻¹ • s) \\ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s)ᶜ) = ⊥",
"tactic": "simp only [iUnion_inv_smul, compl_sdiff, ENNReal.bot_eq_zero, himp_eq, sup_eq_union,\n @iUnion_smul_eq_setOf_exists _ _ _ _ s]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nH : Type ?u.546154\nα : Type u_2\nβ : Type ?u.546160\nE : Type ?u.546163\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\nthis : ((⋃ (g : G), g⁻¹ • s) \\ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s) ⊆ ⋃ (g : G), g⁻¹ • fundamentalInterior G s\n⊢ ↑↑μ ((⋃ (g : G), g • fundamentalFrontier G s) ∪ {a | ∃ g, g • a ∈ s}ᶜ) = 0",
"tactic": "exact measure_union_null\n (measure_iUnion_null fun _ => measure_smul_null hs.measure_fundamentalFrontier _) hs.ae_covers"
},
{
"state_after": "G : Type u_1\nH : Type ?u.546154\nα : Type u_2\nβ : Type ?u.546160\nE : Type ?u.546163\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\n⊢ (⋃ (g : G), g⁻¹ • s) ⊆ ⋃ (g : G), g⁻¹ • s",
"state_before": "G : Type u_1\nH : Type ?u.546154\nα : Type u_2\nβ : Type ?u.546160\nE : Type ?u.546163\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\n⊢ ((⋃ (g : G), g⁻¹ • s) \\ ⋃ (g : G), g⁻¹ • fundamentalFrontier G s) ⊆ ⋃ (g : G), g⁻¹ • fundamentalInterior G s",
"tactic": "simp_rw [diff_subset_iff, ← iUnion_union_distrib, ← smul_set_union (α := G) (β := α),\n fundamentalFrontier_union_fundamentalInterior]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nH : Type ?u.546154\nα : Type u_2\nβ : Type ?u.546160\nE : Type ?u.546163\ninst✝⁶ : Countable G\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\ninst✝ : SMulInvariantMeasure G α μ\n⊢ (⋃ (g : G), g⁻¹ • s) ⊆ ⋃ (g : G), g⁻¹ • s",
"tactic": "rfl"
}
] |
[
702,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
687,
11
] |
Mathlib/Topology/Algebra/ConstMulAction.lean
|
HasCompactMulSupport.comp_smul
|
[] |
[
385,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
LocallyFinite.finite_nonempty_of_compact
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι✝ : Type ?u.89027\nπ : ι✝ → Type ?u.89032\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns t : Set α\nι : Type u_1\ninst✝ : CompactSpace α\nf : ι → Set α\nhf : LocallyFinite f\n⊢ Set.Finite {i | Set.Nonempty (f i)}",
"tactic": "simpa only [inter_univ] using hf.finite_nonempty_inter_compact isCompact_univ"
}
] |
[
819,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
817,
1
] |
Mathlib/Topology/Separation.lean
|
ClosedEmbedding.normalSpace
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : NormalSpace β\nf : α → β\nhf : ClosedEmbedding f\ns t : Set α\nhs : IsClosed s\nht : IsClosed t\nhst : Disjoint s t\nH : SeparatedNhds (f '' s) (f '' t)\n⊢ SeparatedNhds s t",
"state_before": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : NormalSpace β\nf : α → β\nhf : ClosedEmbedding f\ns t : Set α\nhs : IsClosed s\nht : IsClosed t\nhst : Disjoint s t\n⊢ SeparatedNhds s t",
"tactic": "have H : SeparatedNhds (f '' s) (f '' t) :=\n NormalSpace.normal (f '' s) (f '' t) (hf.isClosedMap s hs) (hf.isClosedMap t ht)\n (disjoint_image_of_injective hf.inj hst)"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : NormalSpace β\nf : α → β\nhf : ClosedEmbedding f\ns t : Set α\nhs : IsClosed s\nht : IsClosed t\nhst : Disjoint s t\nH : SeparatedNhds (f '' s) (f '' t)\n⊢ SeparatedNhds s t",
"tactic": "exact (H.preimage hf.continuous).mono (subset_preimage_image _ _) (subset_preimage_image _ _)"
}
] |
[
1743,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1736,
11
] |
Mathlib/Algebra/Star/StarAlgHom.lean
|
StarAlgEquiv.ofBijective_apply
|
[] |
[
969,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
967,
1
] |
Mathlib/Algebra/Group/Units.lean
|
isUnit_one
|
[] |
[
624,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
623,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
|
Real.le_tan
|
[
{
"state_after": "case inl\nh1 : 0 ≤ 0\nh2 : 0 < π / 2\n⊢ 0 ≤ tan 0\n\ncase inr\nx : ℝ\nh1 : 0 ≤ x\nh2 : x < π / 2\nh1' : 0 < x\n⊢ x ≤ tan x",
"state_before": "x : ℝ\nh1 : 0 ≤ x\nh2 : x < π / 2\n⊢ x ≤ tan x",
"tactic": "rcases eq_or_lt_of_le h1 with (rfl | h1')"
},
{
"state_after": "no goals",
"state_before": "case inl\nh1 : 0 ≤ 0\nh2 : 0 < π / 2\n⊢ 0 ≤ tan 0",
"tactic": "rw [tan_zero]"
},
{
"state_after": "no goals",
"state_before": "case inr\nx : ℝ\nh1 : 0 ≤ x\nh2 : x < π / 2\nh1' : 0 < x\n⊢ x ≤ tan x",
"tactic": "exact le_of_lt (lt_tan h1' h2)"
}
] |
[
126,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConcaveOn.lt_right_of_left_lt
|
[] |
[
822,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
820,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.pointed_iff_not_blunt
|
[] |
[
356,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/Data/PFun.lean
|
PFun.fixInduction'_stop
|
[
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.33763\nδ : Type ?u.33766\nε : Type ?u.33769\nι : Type ?u.33772\nC : α → Sort u_1\nf : α →. β ⊕ α\nb : β\na : α\nh : b ∈ fix f a\nfa : Sum.inl b ∈ f a\nhbase : (a_final : α) → Sum.inl b ∈ f a_final → C a_final\nhind : (a₀ a₁ : α) → b ∈ fix f a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀\n⊢ (fixInduction h fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x → C a') (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : ∃ h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b ∈ f a'))\n (fun a'' e =>\n let_fun e := (_ : ∃ h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b ∈ fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom))) =\n hbase a fa",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.33763\nδ : Type ?u.33766\nε : Type ?u.33769\nι : Type ?u.33772\nC : α → Sort u_1\nf : α →. β ⊕ α\nb : β\na : α\nh : b ∈ fix f a\nfa : Sum.inl b ∈ f a\nhbase : (a_final : α) → Sum.inl b ∈ f a_final → C a_final\nhind : (a₀ a₁ : α) → b ∈ fix f a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀\n⊢ fixInduction' h hbase hind = hbase a fa",
"tactic": "unfold fixInduction'"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.33763\nδ : Type ?u.33766\nε : Type ?u.33769\nι : Type ?u.33772\nC : α → Sort u_1\nf : α →. β ⊕ α\nb : β\na : α\nh : b ∈ fix f a\nfa : Sum.inl b ∈ f a\nhbase : (a_final : α) → Sum.inl b ∈ f a_final → C a_final\nhind : (a₀ a₁ : α) → b ∈ fix f a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀\n⊢ Sum.casesOn (motive := fun x => Part.get (f a) (_ : (f a).Dom) = x → C a) (Part.get (f a) (_ : (f a).Dom))\n (fun b' e =>\n let_fun e := (_ : ∃ h, Part.get (f a) h = Sum.inl b');\n hbase a (_ : Sum.inl b ∈ f a))\n (fun a'' e =>\n let_fun e := (_ : ∃ h, Part.get (f a) h = Sum.inr a'');\n hind a a'' (_ : b ∈ fix f a'') e\n ((fun a' h' =>\n fixInduction (_ : b ∈ fix f a') fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x → C a')\n (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : ∃ h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b ∈ f a'))\n (fun a'' e =>\n let_fun e := (_ : ∃ h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b ∈ fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom)))\n a'' e))\n (_ : Part.get (f a) (_ : (f a).Dom) = Part.get (f a) (_ : (f a).Dom)) =\n hbase a fa",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.33763\nδ : Type ?u.33766\nε : Type ?u.33769\nι : Type ?u.33772\nC : α → Sort u_1\nf : α →. β ⊕ α\nb : β\na : α\nh : b ∈ fix f a\nfa : Sum.inl b ∈ f a\nhbase : (a_final : α) → Sum.inl b ∈ f a_final → C a_final\nhind : (a₀ a₁ : α) → b ∈ fix f a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀\n⊢ (fixInduction h fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x → C a') (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : ∃ h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b ∈ f a'))\n (fun a'' e =>\n let_fun e := (_ : ∃ h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b ∈ fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom))) =\n hbase a fa",
"tactic": "rw [fixInduction_spec]"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.33763\nδ : Type ?u.33766\nε : Type ?u.33769\nι : Type ?u.33772\nC : α → Sort u_1\nf : α →. β ⊕ α\nb : β\na : α\nh : b ∈ fix f a\nfa : Sum.inl b ∈ f a\nhbase : (a_final : α) → Sum.inl b ∈ f a_final → C a_final\nhind : (a₀ a₁ : α) → b ∈ fix f a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀\n⊢ (fun x e =>\n Sum.casesOn (motive := fun y => Part.get (f a) (_ : (f a).Dom) = y → C a) x\n (fun val =>\n (fun b' e =>\n let_fun e := (_ : ∃ h, Part.get (f a) h = Sum.inl b');\n hbase a (_ : Sum.inl b ∈ f a))\n val)\n (fun val =>\n (fun a'' e =>\n let_fun e := (_ : ∃ h, Part.get (f a) h = Sum.inr a'');\n hind a a'' (_ : b ∈ fix f a'') e\n ((fun a' h' =>\n fixInduction (_ : b ∈ fix f a') fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x → C a')\n (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : ∃ h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b ∈ f a'))\n (fun a'' e =>\n let_fun e := (_ : ∃ h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b ∈ fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom)))\n a'' e))\n val)\n (_ : Part.get (f a) (_ : (f a).Dom) = x) =\n hbase a fa)\n (Sum.inl b) (_ : Sum.inl b = Sum.inl b)",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.33763\nδ : Type ?u.33766\nε : Type ?u.33769\nι : Type ?u.33772\nC : α → Sort u_1\nf : α →. β ⊕ α\nb : β\na : α\nh : b ∈ fix f a\nfa : Sum.inl b ∈ f a\nhbase : (a_final : α) → Sum.inl b ∈ f a_final → C a_final\nhind : (a₀ a₁ : α) → b ∈ fix f a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀\n⊢ Sum.casesOn (motive := fun x => Part.get (f a) (_ : (f a).Dom) = x → C a) (Part.get (f a) (_ : (f a).Dom))\n (fun b' e =>\n let_fun e := (_ : ∃ h, Part.get (f a) h = Sum.inl b');\n hbase a (_ : Sum.inl b ∈ f a))\n (fun a'' e =>\n let_fun e := (_ : ∃ h, Part.get (f a) h = Sum.inr a'');\n hind a a'' (_ : b ∈ fix f a'') e\n ((fun a' h' =>\n fixInduction (_ : b ∈ fix f a') fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x → C a')\n (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : ∃ h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b ∈ f a'))\n (fun a'' e =>\n let_fun e := (_ : ∃ h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b ∈ fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom)))\n a'' e))\n (_ : Part.get (f a) (_ : (f a).Dom) = Part.get (f a) (_ : (f a).Dom)) =\n hbase a fa",
"tactic": "refine' Eq.rec (motive := fun x e =>\n Sum.casesOn (motive := fun y => (f a).get (dom_of_mem_fix h) = y → C a) x _ _\n (Eq.trans (Part.get_eq_of_mem fa (dom_of_mem_fix h)) e) = hbase a fa) _\n (Part.get_eq_of_mem fa (dom_of_mem_fix h)).symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.33763\nδ : Type ?u.33766\nε : Type ?u.33769\nι : Type ?u.33772\nC : α → Sort u_1\nf : α →. β ⊕ α\nb : β\na : α\nh : b ∈ fix f a\nfa : Sum.inl b ∈ f a\nhbase : (a_final : α) → Sum.inl b ∈ f a_final → C a_final\nhind : (a₀ a₁ : α) → b ∈ fix f a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀\n⊢ (fun x e =>\n Sum.casesOn (motive := fun y => Part.get (f a) (_ : (f a).Dom) = y → C a) x\n (fun val =>\n (fun b' e =>\n let_fun e := (_ : ∃ h, Part.get (f a) h = Sum.inl b');\n hbase a (_ : Sum.inl b ∈ f a))\n val)\n (fun val =>\n (fun a'' e =>\n let_fun e := (_ : ∃ h, Part.get (f a) h = Sum.inr a'');\n hind a a'' (_ : b ∈ fix f a'') e\n ((fun a' h' =>\n fixInduction (_ : b ∈ fix f a') fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x → C a')\n (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : ∃ h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b ∈ f a'))\n (fun a'' e =>\n let_fun e := (_ : ∃ h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b ∈ fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom)))\n a'' e))\n val)\n (_ : Part.get (f a) (_ : (f a).Dom) = x) =\n hbase a fa)\n (Sum.inl b) (_ : Sum.inl b = Sum.inl b)",
"tactic": "simp"
}
] |
[
379,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
368,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
deriv_add_const'
|
[] |
[
117,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
NNRat.coe_inv
|
[] |
[
140,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.comap_abs_atTop
|
[
{
"state_after": "ι : Type ?u.201690\nι' : Type ?u.201693\nα : Type u_1\nβ : Type ?u.201699\nγ : Type ?u.201702\ninst✝ : LinearOrderedAddCommGroup α\n⊢ ∀ (i' : α × α), True ∧ True → ∃ i, True ∧ abs ⁻¹' Ici i ⊆ Iic i'.fst ∪ Ici i'.snd",
"state_before": "ι : Type ?u.201690\nι' : Type ?u.201693\nα : Type u_1\nβ : Type ?u.201699\nγ : Type ?u.201702\ninst✝ : LinearOrderedAddCommGroup α\n⊢ comap abs atTop = atBot ⊔ atTop",
"tactic": "refine'\n le_antisymm (((atTop_basis.comap _).le_basis_iff (atBot_basis.sup atTop_basis)).2 _)\n (sup_le tendsto_abs_atBot_atTop.le_comap tendsto_abs_atTop_atTop.le_comap)"
},
{
"state_after": "case mk\nι : Type ?u.201690\nι' : Type ?u.201693\nα : Type u_1\nβ : Type ?u.201699\nγ : Type ?u.201702\ninst✝ : LinearOrderedAddCommGroup α\na b : α\n⊢ ∃ i, True ∧ abs ⁻¹' Ici i ⊆ Iic (a, b).fst ∪ Ici (a, b).snd",
"state_before": "ι : Type ?u.201690\nι' : Type ?u.201693\nα : Type u_1\nβ : Type ?u.201699\nγ : Type ?u.201702\ninst✝ : LinearOrderedAddCommGroup α\n⊢ ∀ (i' : α × α), True ∧ True → ∃ i, True ∧ abs ⁻¹' Ici i ⊆ Iic i'.fst ∪ Ici i'.snd",
"tactic": "rintro ⟨a, b⟩ -"
},
{
"state_after": "case mk\nι : Type ?u.201690\nι' : Type ?u.201693\nα : Type u_1\nβ : Type ?u.201699\nγ : Type ?u.201702\ninst✝ : LinearOrderedAddCommGroup α\na b x : α\nhx : x ∈ abs ⁻¹' Ici (max (-a) b)\n⊢ x ∈ Iic (a, b).fst ∪ Ici (a, b).snd",
"state_before": "case mk\nι : Type ?u.201690\nι' : Type ?u.201693\nα : Type u_1\nβ : Type ?u.201699\nγ : Type ?u.201702\ninst✝ : LinearOrderedAddCommGroup α\na b : α\n⊢ ∃ i, True ∧ abs ⁻¹' Ici i ⊆ Iic (a, b).fst ∪ Ici (a, b).snd",
"tactic": "refine' ⟨max (-a) b, trivial, fun x hx => _⟩"
},
{
"state_after": "case mk\nι : Type ?u.201690\nι' : Type ?u.201693\nα : Type u_1\nβ : Type ?u.201699\nγ : Type ?u.201702\ninst✝ : LinearOrderedAddCommGroup α\na b x : α\nhx : x ≤ a ∧ x ≤ -b ∨ -a ≤ x ∧ b ≤ x\n⊢ x ∈ Iic (a, b).fst ∪ Ici (a, b).snd",
"state_before": "case mk\nι : Type ?u.201690\nι' : Type ?u.201693\nα : Type u_1\nβ : Type ?u.201699\nγ : Type ?u.201702\ninst✝ : LinearOrderedAddCommGroup α\na b x : α\nhx : x ∈ abs ⁻¹' Ici (max (-a) b)\n⊢ x ∈ Iic (a, b).fst ∪ Ici (a, b).snd",
"tactic": "rw [mem_preimage, mem_Ici, le_abs', max_le_iff, ← min_neg_neg, le_min_iff, neg_neg] at hx"
},
{
"state_after": "no goals",
"state_before": "case mk\nι : Type ?u.201690\nι' : Type ?u.201693\nα : Type u_1\nβ : Type ?u.201699\nγ : Type ?u.201702\ninst✝ : LinearOrderedAddCommGroup α\na b x : α\nhx : x ≤ a ∧ x ≤ -b ∨ -a ≤ x ∧ b ≤ x\n⊢ x ∈ Iic (a, b).fst ∪ Ici (a, b).snd",
"tactic": "exact hx.imp And.left And.right"
}
] |
[
950,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
943,
1
] |
Mathlib/Data/List/Sublists.lean
|
List.nodup_sublists'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ Nodup (sublists' l) ↔ Nodup l",
"tactic": "rw [sublists'_eq_sublists, nodup_map_iff reverse_injective, nodup_sublists, nodup_reverse]"
}
] |
[
380,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Topology/Order.lean
|
singletons_open_iff_discrete
|
[] |
[
325,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
RingHom.liftOfRightInverse_comp
|
[] |
[
2286,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2283,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.mk'_toSubmonoid
|
[] |
[
291,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.mul_pos_of_neg_of_neg
|
[
{
"state_after": "a b : Int\nha : a < 0\nhb : b < 0\nthis : 0 * b < a * b\n⊢ 0 < a * b",
"state_before": "a b : Int\nha : a < 0\nhb : b < 0\n⊢ 0 < a * b",
"tactic": "have : 0 * b < a * b := Int.mul_lt_mul_of_neg_right ha hb"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nha : a < 0\nhb : b < 0\nthis : 0 * b < a * b\n⊢ 0 < a * b",
"tactic": "rwa [Int.zero_mul] at this"
}
] |
[
1231,
29
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1229,
11
] |
src/lean/Init/Data/List/Basic.lean
|
List.reverse_nil
|
[] |
[
172,
6
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
171,
9
] |
Mathlib/FieldTheory/Normal.lean
|
AlgEquiv.transfer_normal
|
[] |
[
154,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Analysis/Calculus/FDerivMeasurable.lean
|
measurableSet_of_differentiableWithinAt_Ioi
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\ninst✝ : CompleteSpace F\n⊢ MeasurableSet {x | DifferentiableWithinAt ℝ f (Ioi x) x}",
"tactic": "simpa [differentiableWithinAt_Ioi_iff_Ici] using measurableSet_of_differentiableWithinAt_Ici f"
}
] |
[
808,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
806,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieSubmodule.lowerCentralSeries_map_eq_lcs
|
[
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\n⊢ lcs k N ≤ N",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\n⊢ map (incl N) (lowerCentralSeries R L { x // x ∈ ↑N } k) = lcs k N",
"tactic": "rw [lowerCentralSeries_eq_lcs_comap, LieSubmodule.map_comap_incl, inf_eq_right]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\n⊢ lcs k N ≤ N",
"tactic": "apply lcs_le_self"
}
] |
[
121,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Order/Filter/Extr.lean
|
isMaxOn_dual_iff
|
[] |
[
238,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Geometry/Manifold/ChartedSpace.lean
|
StructureGroupoid.symm
|
[] |
[
188,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Algebra/Invertible.lean
|
IsUnit.nonempty_invertible
|
[] |
[
213,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Topology/Constructions.lean
|
ContinuousAt.fst''
|
[] |
[
359,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Std/Data/AssocList.lean
|
Std.AssocList.any_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\np : α → β → Bool\nl : AssocList α β\n⊢ any p l =\n List.any (toList l) fun x =>\n match x with\n | (a, b) => p a b",
"tactic": "induction l <;> simp [any, *]"
}
] |
[
134,
83
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
133,
9
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.image_symm_eq_preimage
|
[] |
[
571,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
570,
11
] |
Mathlib/CategoryTheory/Linear/Yoneda.lean
|
CategoryTheory.whiskering_linearYoneda₂
|
[] |
[
82,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Data/Sum/Order.lean
|
OrderIso.sumLexAssoc_symm_apply_inr_inr
|
[] |
[
687,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
686,
1
] |
Std/Data/String/Lemmas.lean
|
Substring.ValidFor.contains
|
[
{
"state_after": "no goals",
"state_before": "l m r : List Char\nc : Char\nx✝ : Substring\nh : ValidFor l m r x✝\n⊢ Substring.contains x✝ c = true ↔ c ∈ m",
"tactic": "simp [Substring.contains, h.any, String.contains]"
}
] |
[
935,
65
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
934,
1
] |
Mathlib/Topology/Instances/Matrix.lean
|
HasSum.matrix_conjTranspose
|
[] |
[
320,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Prod.uIcc_eq
|
[] |
[
1003,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1000,
1
] |
Mathlib/Data/Fintype/Perm.lean
|
mem_perms_of_finset_iff
|
[
{
"state_after": "case mk.mk\nα : Type u_1\nβ : Type ?u.50561\nγ : Type ?u.50564\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nval✝ : Multiset α\nl : List α\nhs : Multiset.Nodup (Quot.mk Setoid.r l)\nf : Equiv.Perm α\n⊢ f ∈ permsOfFinset { val := Quot.mk Setoid.r l, nodup := hs } ↔\n ∀ {x : α}, ↑f x ≠ x → x ∈ { val := Quot.mk Setoid.r l, nodup := hs }",
"state_before": "α : Type u_1\nβ : Type ?u.50561\nγ : Type ?u.50564\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\n⊢ ∀ {s : Finset α} {f : Equiv.Perm α}, f ∈ permsOfFinset s ↔ ∀ {x : α}, ↑f x ≠ x → x ∈ s",
"tactic": "rintro ⟨⟨l⟩, hs⟩ f"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nα : Type u_1\nβ : Type ?u.50561\nγ : Type ?u.50564\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nval✝ : Multiset α\nl : List α\nhs : Multiset.Nodup (Quot.mk Setoid.r l)\nf : Equiv.Perm α\n⊢ f ∈ permsOfFinset { val := Quot.mk Setoid.r l, nodup := hs } ↔\n ∀ {x : α}, ↑f x ≠ x → x ∈ { val := Quot.mk Setoid.r l, nodup := hs }",
"tactic": "exact mem_permsOfList_iff"
}
] |
[
145,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
|
ContinuousLinearEquiv.comp_right_fderiv
|
[
{
"state_after": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_2\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.247750\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : F → G\nx : E\n⊢ UniqueDiffWithinAt 𝕜 (↑iso ⁻¹' univ) x",
"state_before": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_2\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.247750\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : F → G\nx : E\n⊢ fderiv 𝕜 (f ∘ ↑iso) x = comp (fderiv 𝕜 f (↑iso x)) ↑iso",
"tactic": "rw [← fderivWithin_univ, ← fderivWithin_univ, ← iso.comp_right_fderivWithin, preimage_univ]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_2\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.247750\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : F → G\nx : E\n⊢ UniqueDiffWithinAt 𝕜 (↑iso ⁻¹' univ) x",
"tactic": "exact uniqueDiffWithinAt_univ"
}
] |
[
253,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
TendstoUniformlyOn.continuousOn
|
[] |
[
895,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
893,
11
] |
Std/Control/ForInStep/Lemmas.lean
|
ForInStep.bind_bindList_assoc
|
[
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nβ : Type u_1\nα : Type u_3\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : β → m (ForInStep β)\ng : α → β → m (ForInStep β)\ns : ForInStep β\nl : List α\n⊢ (do\n let x ← ForInStep.bind s f\n bindList g l x) =\n ForInStep.bind s fun b => do\n let x ← f b\n bindList g l x",
"tactic": "cases s <;> simp"
}
] |
[
38,
19
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
35,
9
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.powerlt_min
|
[] |
[
2307,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2306,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.mapDomain_zero
|
[] |
[
490,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/Topology/ContinuousFunction/Basic.lean
|
ContinuousMap.pi_eval
|
[] |
[
340,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/LinearAlgebra/Matrix/Circulant.lean
|
Matrix.Fin.circulant_inj
|
[] |
[
80,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Analysis/Convex/Hull.lean
|
IsLinearMap.convexHull_image
|
[] |
[
176,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Algebra/Order/Kleene.lean
|
Prod.snd_kstar
|
[] |
[
322,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioc_union_Ioc_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.211771\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\n⊢ Ioc a c ∪ Ioc b c = Ioc (min a b) c",
"tactic": "rw [Ioc_union_Ioc, max_self] <;> exact (min_le_right _ _).trans (le_max_right _ _)"
}
] |
[
1847,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1846,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
UniqueFactorizationMonoid.exists_prime_factors
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\n⊢ a ≠ 0 → ∃ f, (∀ (b : α), b ∈ f → Irreducible b) ∧ Multiset.prod f ~ᵤ a",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\n⊢ a ≠ 0 → ∃ f, (∀ (b : α), b ∈ f → Prime b) ∧ Multiset.prod f ~ᵤ a",
"tactic": "simp_rw [← UniqueFactorizationMonoid.irreducible_iff_prime]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\n⊢ a ≠ 0 → ∃ f, (∀ (b : α), b ∈ f → Irreducible b) ∧ Multiset.prod f ~ᵤ a",
"tactic": "apply WfDvdMonoid.exists_factors a"
}
] |
[
206,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/CategoryTheory/Limits/FullSubcategory.lean
|
CategoryTheory.Limits.ClosedUnderColimitsOfShape.colimit
|
[] |
[
56,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Init/CcLemmas.lean
|
and_eq_of_eq_false_left
|
[] |
[
28,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
27,
1
] |
Mathlib/Control/Applicative.lean
|
Functor.Comp.seq_pure
|
[] |
[
101,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Order/Filter/Germ.lean
|
Filter.EventuallyEq.comp_tendsto
|
[] |
[
72,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearEquiv.ofSubmodule'_apply
|
[] |
[
2082,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2080,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
inv_injective
|
[] |
[
253,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.symm_comp
|
[
{
"state_after": "case H\nR : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A₁\ninst✝⁴ : Semiring A₂\ninst✝³ : Semiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne✝ e : A₁ ≃ₐ[R] A₂\nx✝ : A₁\n⊢ ↑(AlgHom.comp ↑(symm e) ↑e) x✝ = ↑(AlgHom.id R A₁) x✝",
"state_before": "R : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A₁\ninst✝⁴ : Semiring A₂\ninst✝³ : Semiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne✝ e : A₁ ≃ₐ[R] A₂\n⊢ AlgHom.comp ↑(symm e) ↑e = AlgHom.id R A₁",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H\nR : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A₁\ninst✝⁴ : Semiring A₂\ninst✝³ : Semiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne✝ e : A₁ ≃ₐ[R] A₂\nx✝ : A₁\n⊢ ↑(AlgHom.comp ↑(symm e) ↑e) x✝ = ↑(AlgHom.id R A₁) x✝",
"tactic": "simp"
}
] |
[
429,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
427,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.surjOn_iUnion_iUnion
|
[] |
[
1592,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1590,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
ContDiff.continuous_fderiv_apply
|
[] |
[
1701,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1696,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.IsImage.symm_iff
|
[] |
[
491,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
490,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoMod_toIocMod
|
[] |
[
736,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
735,
1
] |
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
|
ProjectiveSpectrum.vanishingIdeal_singleton
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nx : ProjectiveSpectrum 𝒜\n⊢ vanishingIdeal {x} = x.asHomogeneousIdeal",
"tactic": "simp [vanishingIdeal]"
}
] |
[
121,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Topology/Order/Basic.lean
|
isClosed_le'
|
[] |
[
135,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean
|
AlgebraicTopology.DoldKan.compatibility_Γ₂N₁_Γ₂N₂_natTrans
|
[
{
"state_after": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : SimplicialObject C\n⊢ Γ₂N₁.natTrans.app X =\n (compatibility_Γ₂N₁_Γ₂N₂.app X).inv ≫\n (N₂ ⋙ Γ₂).map (Karoubi.decompId_i ((toKaroubi (SimplicialObject C)).obj X)) ≫\n (compatibility_Γ₂N₁_Γ₂N₂.hom ≫ Γ₂N₁.natTrans).app ((toKaroubi (SimplicialObject C)).obj X).X ≫\n Karoubi.decompId_p ((toKaroubi (SimplicialObject C)).obj X)",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : SimplicialObject C\n⊢ Γ₂N₁.natTrans.app X = (compatibility_Γ₂N₁_Γ₂N₂.app X).inv ≫ Γ₂N₂.natTrans.app ((toKaroubi (SimplicialObject C)).obj X)",
"tactic": "rw [Γ₂N₂.natTrans_app_f_app]"
},
{
"state_after": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : SimplicialObject C\n⊢ Γ₂N₁.natTrans.app X =\n (compatibility_Γ₂N₁_Γ₂N₂.app X).inv ≫\n Γ₂.map (N₂.map (𝟙 ((toKaroubi (SimplicialObject C)).obj X))) ≫\n (compatibility_Γ₂N₁_Γ₂N₂.hom.app ((toKaroubi (SimplicialObject C)).obj X).X ≫\n Γ₂N₁.natTrans.app ((toKaroubi (SimplicialObject C)).obj X).X) ≫\n 𝟙 (Karoubi.mk ((toKaroubi (SimplicialObject C)).obj X).X (𝟙 ((toKaroubi (SimplicialObject C)).obj X).X))",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : SimplicialObject C\n⊢ Γ₂N₁.natTrans.app X =\n (compatibility_Γ₂N₁_Γ₂N₂.app X).inv ≫\n (N₂ ⋙ Γ₂).map (Karoubi.decompId_i ((toKaroubi (SimplicialObject C)).obj X)) ≫\n (compatibility_Γ₂N₁_Γ₂N₂.hom ≫ Γ₂N₁.natTrans).app ((toKaroubi (SimplicialObject C)).obj X).X ≫\n Karoubi.decompId_p ((toKaroubi (SimplicialObject C)).obj X)",
"tactic": "dsimp only [Karoubi.decompId_i_toKaroubi, Karoubi.decompId_p_toKaroubi, Functor.comp_map,\n NatTrans.comp_app]"
},
{
"state_after": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : SimplicialObject C\n⊢ Γ₂N₁.natTrans.app X =\n compatibility_Γ₂N₁_Γ₂N₂.inv.app X ≫\n 𝟙 (Γ₂.obj (N₂.obj ((toKaroubi (SimplicialObject C)).obj X))) ≫\n (compatibility_Γ₂N₁_Γ₂N₂.hom.app ((toKaroubi (SimplicialObject C)).obj X).X ≫\n Γ₂N₁.natTrans.app ((toKaroubi (SimplicialObject C)).obj X).X) ≫\n 𝟙 (Karoubi.mk ((toKaroubi (SimplicialObject C)).obj X).X (𝟙 ((toKaroubi (SimplicialObject C)).obj X).X))",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : SimplicialObject C\n⊢ Γ₂N₁.natTrans.app X =\n (compatibility_Γ₂N₁_Γ₂N₂.app X).inv ≫\n Γ₂.map (N₂.map (𝟙 ((toKaroubi (SimplicialObject C)).obj X))) ≫\n (compatibility_Γ₂N₁_Γ₂N₂.hom.app ((toKaroubi (SimplicialObject C)).obj X).X ≫\n Γ₂N₁.natTrans.app ((toKaroubi (SimplicialObject C)).obj X).X) ≫\n 𝟙 (Karoubi.mk ((toKaroubi (SimplicialObject C)).obj X).X (𝟙 ((toKaroubi (SimplicialObject C)).obj X).X))",
"tactic": "rw [N₂.map_id, Γ₂.map_id, Iso.app_inv]"
},
{
"state_after": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : SimplicialObject C\n⊢ Γ₂N₁.natTrans.app X =\n compatibility_Γ₂N₁_Γ₂N₂.inv.app X ≫\n 𝟙 (Γ₂.obj (N₂.obj (Karoubi.mk X (𝟙 X)))) ≫\n (compatibility_Γ₂N₁_Γ₂N₂.hom.app X ≫ Γ₂N₁.natTrans.app X) ≫ 𝟙 (Karoubi.mk X (𝟙 X))",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : SimplicialObject C\n⊢ Γ₂N₁.natTrans.app X =\n compatibility_Γ₂N₁_Γ₂N₂.inv.app X ≫\n 𝟙 (Γ₂.obj (N₂.obj ((toKaroubi (SimplicialObject C)).obj X))) ≫\n (compatibility_Γ₂N₁_Γ₂N₂.hom.app ((toKaroubi (SimplicialObject C)).obj X).X ≫\n Γ₂N₁.natTrans.app ((toKaroubi (SimplicialObject C)).obj X).X) ≫\n 𝟙 (Karoubi.mk ((toKaroubi (SimplicialObject C)).obj X).X (𝟙 ((toKaroubi (SimplicialObject C)).obj X).X))",
"tactic": "dsimp only [toKaroubi]"
},
{
"state_after": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : SimplicialObject C\n⊢ Γ₂N₁.natTrans.app X =\n compatibility_Γ₂N₁_Γ₂N₂.inv.app X ≫\n (compatibility_Γ₂N₁_Γ₂N₂.hom.app X ≫ Γ₂N₁.natTrans.app X) ≫ 𝟙 (Karoubi.mk X (𝟙 X))",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : SimplicialObject C\n⊢ Γ₂N₁.natTrans.app X =\n compatibility_Γ₂N₁_Γ₂N₂.inv.app X ≫\n 𝟙 (Γ₂.obj (N₂.obj (Karoubi.mk X (𝟙 X)))) ≫\n (compatibility_Γ₂N₁_Γ₂N₂.hom.app X ≫ Γ₂N₁.natTrans.app X) ≫ 𝟙 (Karoubi.mk X (𝟙 X))",
"tactic": "erw [id_comp]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : SimplicialObject C\n⊢ Γ₂N₁.natTrans.app X =\n compatibility_Γ₂N₁_Γ₂N₂.inv.app X ≫\n (compatibility_Γ₂N₁_Γ₂N₂.hom.app X ≫ Γ₂N₁.natTrans.app X) ≫ 𝟙 (Karoubi.mk X (𝟙 X))",
"tactic": "rw [comp_id, Iso.inv_hom_id_app_assoc]"
}
] |
[
225,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.map_toDeleteEdges_eq
|
[
{
"state_after": "case hp'\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\ns : Set (Sym2 V)\np : Walk G v w\nhp : ∀ (e : Sym2 V), e ∈ edges p → ¬e ∈ s\n⊢ ∀ (e : Sym2 V),\n e ∈ edges (Walk.transfer p (deleteEdges G s) (_ : ∀ (e : Sym2 V), e ∈ edges p → e ∈ edgeSet (deleteEdges G s))) →\n e ∈ edgeSet G\n\ncase hp'\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\ns : Set (Sym2 V)\np : Walk G v w\nhp : ∀ (e : Sym2 V), e ∈ edges p → ¬e ∈ s\n⊢ ∀ (e : Sym2 V),\n e ∈ edges (Walk.transfer p (deleteEdges G s) (_ : ∀ (e : Sym2 V), e ∈ edges p → e ∈ edgeSet (deleteEdges G s))) →\n e ∈ edgeSet G",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\ns : Set (Sym2 V)\np : Walk G v w\nhp : ∀ (e : Sym2 V), e ∈ edges p → ¬e ∈ s\n⊢ Walk.map (Hom.mapSpanningSubgraphs (_ : deleteEdges G s ≤ G)) (toDeleteEdges s p hp) = p",
"tactic": "rw [← transfer_eq_map_of_le, transfer_transfer, transfer_self]"
},
{
"state_after": "case hp'\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\ns : Set (Sym2 V)\np : Walk G v w\nhp : ∀ (e : Sym2 V), e ∈ edges p → ¬e ∈ s\ne : Sym2 V\n⊢ e ∈ edges (Walk.transfer p (deleteEdges G s) (_ : ∀ (e : Sym2 V), e ∈ edges p → e ∈ edgeSet (deleteEdges G s))) →\n e ∈ edgeSet G",
"state_before": "case hp'\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\ns : Set (Sym2 V)\np : Walk G v w\nhp : ∀ (e : Sym2 V), e ∈ edges p → ¬e ∈ s\n⊢ ∀ (e : Sym2 V),\n e ∈ edges (Walk.transfer p (deleteEdges G s) (_ : ∀ (e : Sym2 V), e ∈ edges p → e ∈ edgeSet (deleteEdges G s))) →\n e ∈ edgeSet G",
"tactic": "intros e"
},
{
"state_after": "case hp'\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\ns : Set (Sym2 V)\np : Walk G v w\nhp : ∀ (e : Sym2 V), e ∈ edges p → ¬e ∈ s\ne : Sym2 V\n⊢ e ∈ edges p → e ∈ edgeSet G",
"state_before": "case hp'\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\ns : Set (Sym2 V)\np : Walk G v w\nhp : ∀ (e : Sym2 V), e ∈ edges p → ¬e ∈ s\ne : Sym2 V\n⊢ e ∈ edges (Walk.transfer p (deleteEdges G s) (_ : ∀ (e : Sym2 V), e ∈ edges p → e ∈ edgeSet (deleteEdges G s))) →\n e ∈ edgeSet G",
"tactic": "rw [edges_transfer]"
},
{
"state_after": "no goals",
"state_before": "case hp'\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\ns : Set (Sym2 V)\np : Walk G v w\nhp : ∀ (e : Sym2 V), e ∈ edges p → ¬e ∈ s\ne : Sym2 V\n⊢ e ∈ edges p → e ∈ edgeSet G",
"tactic": "apply edges_subset_edgeSet p"
}
] |
[
1811,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1806,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Wbtw.collinear
|
[
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n⊢ ∃ p₀ v, ∀ (p : P), p ∈ {x, y, z} → ∃ r, p = r • v +ᵥ p₀",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n⊢ Collinear R {x, y, z}",
"tactic": "rw [collinear_iff_exists_forall_eq_smul_vadd]"
},
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n⊢ ∀ (p : P), p ∈ {x, y, z} → ∃ r, p = r • (z -ᵥ x) +ᵥ x",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n⊢ ∃ p₀ v, ∀ (p : P), p ∈ {x, y, z} → ∃ r, p = r • v +ᵥ p₀",
"tactic": "refine' ⟨x, z -ᵥ x, _⟩"
},
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\np : P\nhp : p ∈ {x, y, z}\n⊢ ∃ r, p = r • (z -ᵥ x) +ᵥ x",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n⊢ ∀ (p : P), p ∈ {x, y, z} → ∃ r, p = r • (z -ᵥ x) +ᵥ x",
"tactic": "intro p hp"
},
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\np : P\nhp : p = x ∨ p = y ∨ p = z\n⊢ ∃ r, p = r • (z -ᵥ x) +ᵥ x",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\np : P\nhp : p ∈ {x, y, z}\n⊢ ∃ r, p = r • (z -ᵥ x) +ᵥ x",
"tactic": "simp_rw [Set.mem_insert_iff, Set.mem_singleton_iff] at hp"
},
{
"state_after": "case inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ny z p : P\nh : Wbtw R p y z\n⊢ ∃ r, p = r • (z -ᵥ p) +ᵥ p\n\ncase inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z p : P\nh : Wbtw R x p z\n⊢ ∃ r, p = r • (z -ᵥ x) +ᵥ x\n\ncase inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y p : P\nh : Wbtw R x y p\n⊢ ∃ r, p = r • (p -ᵥ x) +ᵥ x",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\np : P\nhp : p = x ∨ p = y ∨ p = z\n⊢ ∃ r, p = r • (z -ᵥ x) +ᵥ x",
"tactic": "rcases hp with (rfl | rfl | rfl)"
},
{
"state_after": "case inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ny z p : P\nh : Wbtw R p y z\n⊢ p = 0 • (z -ᵥ p) +ᵥ p",
"state_before": "case inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ny z p : P\nh : Wbtw R p y z\n⊢ ∃ r, p = r • (z -ᵥ p) +ᵥ p",
"tactic": "refine' ⟨0, _⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ny z p : P\nh : Wbtw R p y z\n⊢ p = 0 • (z -ᵥ p) +ᵥ p",
"tactic": "simp"
},
{
"state_after": "case inr.inl.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nt : R\n⊢ ∃ r, ↑(lineMap x z) t = r • (z -ᵥ x) +ᵥ x",
"state_before": "case inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z p : P\nh : Wbtw R x p z\n⊢ ∃ r, p = r • (z -ᵥ x) +ᵥ x",
"tactic": "rcases h with ⟨t, -, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nt : R\n⊢ ∃ r, ↑(lineMap x z) t = r • (z -ᵥ x) +ᵥ x",
"tactic": "exact ⟨t, rfl⟩"
},
{
"state_after": "case inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y p : P\nh : Wbtw R x y p\n⊢ p = 1 • (p -ᵥ x) +ᵥ x",
"state_before": "case inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y p : P\nh : Wbtw R x y p\n⊢ ∃ r, p = r • (p -ᵥ x) +ᵥ x",
"tactic": "refine' ⟨1, _⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.589540\nP : Type u_3\nP' : Type ?u.589546\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y p : P\nh : Wbtw R x y p\n⊢ p = 1 • (p -ᵥ x) +ᵥ x",
"tactic": "simp"
}
] |
[
860,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
849,
1
] |
Mathlib/Data/Int/NatPrime.lean
|
Int.not_prime_of_int_mul
|
[] |
[
24,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
22,
1
] |
Mathlib/CategoryTheory/Functor/FullyFaithful.lean
|
CategoryTheory.Faithful.of_comp
|
[] |
[
288,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/CategoryTheory/Monoidal/Preadditive.lean
|
CategoryTheory.tensor_sum
|
[
{
"state_after": "C : Type u_3\ninst✝³ : Category C\ninst✝² : Preadditive C\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalPreadditive C\nP Q R S : C\nJ : Type u_1\ns : Finset J\nf : P ⟶ Q\ng : J → (R ⟶ S)\n⊢ (f ⊗ 𝟙 R) ≫ (𝟙 Q ⊗ ∑ j in s, g j) = ∑ j in s, f ⊗ g j",
"state_before": "C : Type u_3\ninst✝³ : Category C\ninst✝² : Preadditive C\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalPreadditive C\nP Q R S : C\nJ : Type u_1\ns : Finset J\nf : P ⟶ Q\ng : J → (R ⟶ S)\n⊢ f ⊗ ∑ j in s, g j = ∑ j in s, f ⊗ g j",
"tactic": "rw [← tensor_id_comp_id_tensor]"
},
{
"state_after": "C : Type u_3\ninst✝³ : Category C\ninst✝² : Preadditive C\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalPreadditive C\nP Q R S : C\nJ : Type u_1\ns : Finset J\nf : P ⟶ Q\ng : J → (R ⟶ S)\ntQ : (R ⟶ S) →+ (((tensoringLeft C).obj Q).obj R ⟶ ((tensoringLeft C).obj Q).obj S) :=\n Functor.mapAddHom ((tensoringLeft C).obj Q)\n⊢ (f ⊗ 𝟙 R) ≫ (𝟙 Q ⊗ ∑ j in s, g j) = ∑ j in s, f ⊗ g j",
"state_before": "C : Type u_3\ninst✝³ : Category C\ninst✝² : Preadditive C\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalPreadditive C\nP Q R S : C\nJ : Type u_1\ns : Finset J\nf : P ⟶ Q\ng : J → (R ⟶ S)\n⊢ (f ⊗ 𝟙 R) ≫ (𝟙 Q ⊗ ∑ j in s, g j) = ∑ j in s, f ⊗ g j",
"tactic": "let tQ := (((tensoringLeft C).obj Q).mapAddHom : (R ⟶ S) →+ _)"
},
{
"state_after": "C : Type u_3\ninst✝³ : Category C\ninst✝² : Preadditive C\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalPreadditive C\nP Q R S : C\nJ : Type u_1\ns : Finset J\nf : P ⟶ Q\ng : J → (R ⟶ S)\ntQ : (R ⟶ S) →+ (((tensoringLeft C).obj Q).obj R ⟶ ((tensoringLeft C).obj Q).obj S) :=\n Functor.mapAddHom ((tensoringLeft C).obj Q)\n⊢ (f ⊗ 𝟙 R) ≫ ↑tQ (∑ j in s, g j) = ∑ j in s, f ⊗ g j",
"state_before": "C : Type u_3\ninst✝³ : Category C\ninst✝² : Preadditive C\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalPreadditive C\nP Q R S : C\nJ : Type u_1\ns : Finset J\nf : P ⟶ Q\ng : J → (R ⟶ S)\ntQ : (R ⟶ S) →+ (((tensoringLeft C).obj Q).obj R ⟶ ((tensoringLeft C).obj Q).obj S) :=\n Functor.mapAddHom ((tensoringLeft C).obj Q)\n⊢ (f ⊗ 𝟙 R) ≫ (𝟙 Q ⊗ ∑ j in s, g j) = ∑ j in s, f ⊗ g j",
"tactic": "change _ ≫ tQ _ = _"
},
{
"state_after": "C : Type u_3\ninst✝³ : Category C\ninst✝² : Preadditive C\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalPreadditive C\nP Q R S : C\nJ : Type u_1\ns : Finset J\nf : P ⟶ Q\ng : J → (R ⟶ S)\ntQ : (R ⟶ S) →+ (((tensoringLeft C).obj Q).obj R ⟶ ((tensoringLeft C).obj Q).obj S) :=\n Functor.mapAddHom ((tensoringLeft C).obj Q)\n⊢ ∑ j in s, (f ⊗ 𝟙 R) ≫ ↑tQ (g j) = ∑ j in s, f ⊗ g j",
"state_before": "C : Type u_3\ninst✝³ : Category C\ninst✝² : Preadditive C\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalPreadditive C\nP Q R S : C\nJ : Type u_1\ns : Finset J\nf : P ⟶ Q\ng : J → (R ⟶ S)\ntQ : (R ⟶ S) →+ (((tensoringLeft C).obj Q).obj R ⟶ ((tensoringLeft C).obj Q).obj S) :=\n Functor.mapAddHom ((tensoringLeft C).obj Q)\n⊢ (f ⊗ 𝟙 R) ≫ ↑tQ (∑ j in s, g j) = ∑ j in s, f ⊗ g j",
"tactic": "rw [tQ.map_sum, Preadditive.comp_sum]"
},
{
"state_after": "C : Type u_3\ninst✝³ : Category C\ninst✝² : Preadditive C\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalPreadditive C\nP Q R S : C\nJ : Type u_1\ns : Finset J\nf : P ⟶ Q\ng : J → (R ⟶ S)\ntQ : (R ⟶ S) →+ (((tensoringLeft C).obj Q).obj R ⟶ ((tensoringLeft C).obj Q).obj S) :=\n Functor.mapAddHom ((tensoringLeft C).obj Q)\n⊢ ∑ j in s, (f ⊗ 𝟙 R) ≫ (𝟙 Q ⊗ g j) = ∑ j in s, f ⊗ g j",
"state_before": "C : Type u_3\ninst✝³ : Category C\ninst✝² : Preadditive C\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalPreadditive C\nP Q R S : C\nJ : Type u_1\ns : Finset J\nf : P ⟶ Q\ng : J → (R ⟶ S)\ntQ : (R ⟶ S) →+ (((tensoringLeft C).obj Q).obj R ⟶ ((tensoringLeft C).obj Q).obj S) :=\n Functor.mapAddHom ((tensoringLeft C).obj Q)\n⊢ ∑ j in s, (f ⊗ 𝟙 R) ≫ ↑tQ (g j) = ∑ j in s, f ⊗ g j",
"tactic": "dsimp [Functor.mapAddHom]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_3\ninst✝³ : Category C\ninst✝² : Preadditive C\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalPreadditive C\nP Q R S : C\nJ : Type u_1\ns : Finset J\nf : P ⟶ Q\ng : J → (R ⟶ S)\ntQ : (R ⟶ S) →+ (((tensoringLeft C).obj Q).obj R ⟶ ((tensoringLeft C).obj Q).obj S) :=\n Functor.mapAddHom ((tensoringLeft C).obj Q)\n⊢ ∑ j in s, (f ⊗ 𝟙 R) ≫ (𝟙 Q ⊗ g j) = ∑ j in s, f ⊗ g j",
"tactic": "simp only [tensor_id_comp_id_tensor]"
}
] |
[
105,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.le_iff_derivBFamily
|
[
{
"state_after": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\n⊢ (∀ (i : Ordinal) (hi : i < o), f i hi a ≤ a) ↔ ∃ b, derivFamily (familyOfBFamily o f) b = a",
"state_before": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\n⊢ (∀ (i : Ordinal) (hi : i < o), f i hi a ≤ a) ↔ ∃ b, derivBFamily o f b = a",
"tactic": "unfold derivBFamily"
},
{
"state_after": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\n⊢ (∀ (i : Ordinal) (hi : i < o), f i hi a ≤ a) ↔ ∀ (i : (Quotient.out o).α), familyOfBFamily o f i a ≤ a\n\ncase H\no : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\n⊢ ∀ (i : (Quotient.out o).α), IsNormal (familyOfBFamily o f i)",
"state_before": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\n⊢ (∀ (i : Ordinal) (hi : i < o), f i hi a ≤ a) ↔ ∃ b, derivFamily (familyOfBFamily o f) b = a",
"tactic": "rw [← le_iff_derivFamily]"
},
{
"state_after": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\nh : ∀ (i : (Quotient.out o).α), familyOfBFamily o f i a ≤ a\ni : Ordinal\nhi : i < o\n⊢ f i hi a ≤ a",
"state_before": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\n⊢ (∀ (i : Ordinal) (hi : i < o), f i hi a ≤ a) ↔ ∀ (i : (Quotient.out o).α), familyOfBFamily o f i a ≤ a",
"tactic": "refine' ⟨fun h i => h _ _, fun h i hi => _⟩"
},
{
"state_after": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\nh : ∀ (i : (Quotient.out o).α), familyOfBFamily o f i a ≤ a\ni : Ordinal\nhi : i < o\n⊢ familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)) a ≤ a",
"state_before": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\nh : ∀ (i : (Quotient.out o).α), familyOfBFamily o f i a ≤ a\ni : Ordinal\nhi : i < o\n⊢ f i hi a ≤ a",
"tactic": "rw [← familyOfBFamily_enum o f]"
},
{
"state_after": "no goals",
"state_before": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\nh : ∀ (i : (Quotient.out o).α), familyOfBFamily o f i a ≤ a\ni : Ordinal\nhi : i < o\n⊢ familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)) a ≤ a",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case H\no : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\n⊢ ∀ (i : (Quotient.out o).α), IsNormal (familyOfBFamily o f i)",
"tactic": "exact fun _ => H _ _"
}
] |
[
384,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical
|
[
{
"state_after": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nf : R\n⊢ (∀ (x : PrimeSpectrum R), x ∈ zeroLocus ↑I → f ∈ x.asIdeal) ↔\n ∀ (p : Submodule R R), p ∈ {J | I ≤ J ∧ Ideal.IsPrime J} → f ∈ p",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nf : R\n⊢ f ∈ vanishingIdeal (zeroLocus ↑I) ↔ f ∈ Ideal.radical I",
"tactic": "rw [mem_vanishingIdeal, Ideal.radical_eq_sInf, Submodule.mem_sInf]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\nf : R\n⊢ (∀ (x : PrimeSpectrum R), x ∈ zeroLocus ↑I → f ∈ x.asIdeal) ↔\n ∀ (p : Submodule R R), p ∈ {J | I ≤ J ∧ Ideal.IsPrime J} → f ∈ p",
"tactic": "exact ⟨fun h x hx => h ⟨x, hx.2⟩ hx.1, fun h x hx => h x.1 ⟨hx, x.2⟩⟩"
}
] |
[
226,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.le_map_of_comap_le_of_surjective
|
[] |
[
1610,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1609,
1
] |
src/lean/Init/Data/Nat/Gcd.lean
|
Nat.gcd_zero_right
|
[
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ gcd n 0 = n",
"tactic": "cases n with\n| zero => simp [gcd_succ]\n| succ n =>\n rw [gcd_succ]\n exact gcd_zero_left _"
},
{
"state_after": "no goals",
"state_before": "case zero\n\n⊢ gcd zero 0 = zero",
"tactic": "simp [gcd_succ]"
},
{
"state_after": "case succ\nn : Nat\n⊢ gcd (0 % succ n) (succ n) = succ n",
"state_before": "case succ\nn : Nat\n⊢ gcd (succ n) 0 = succ n",
"tactic": "rw [gcd_succ]"
},
{
"state_after": "no goals",
"state_before": "case succ\nn : Nat\n⊢ gcd (0 % succ n) (succ n) = succ n",
"tactic": "exact gcd_zero_left _"
}
] |
[
36,
26
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
30,
9
] |
Mathlib/CategoryTheory/Monoidal/Functor.lean
|
CategoryTheory.MonoidalFunctor.ε_inv_hom_id
|
[] |
[
262,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Reachable.symm
|
[] |
[
1877,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1876,
11
] |
Mathlib/Data/Finset/Prod.lean
|
Finset.diag_empty
|
[] |
[
355,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
Real.sqrt_div_self'
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ sqrt x / x = 1 / sqrt x",
"tactic": "rw [← div_sqrt, one_div_div, div_sqrt]"
}
] |
[
426,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
1
] |
Mathlib/GroupTheory/Index.lean
|
Subgroup.index_comap_of_surjective
|
[
{
"state_after": "G : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis : Setoid G := QuotientGroup.leftRel H\n⊢ index (comap f H) = index H",
"state_before": "G : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\n⊢ index (comap f H) = index H",
"tactic": "letI := QuotientGroup.leftRel H"
},
{
"state_after": "G : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\n⊢ index (comap f H) = index H",
"state_before": "G : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis : Setoid G := QuotientGroup.leftRel H\n⊢ index (comap f H) = index H",
"tactic": "letI := QuotientGroup.leftRel (H.comap f)"
},
{
"state_after": "case refine'_1\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\n⊢ Function.Injective (Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)))\n\ncase refine'_2\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\n⊢ Function.Surjective (Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)))",
"state_before": "G : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\n⊢ index (comap f H) = index H",
"tactic": "refine' Cardinal.toNat_congr (Equiv.ofBijective (Quotient.map' f fun x y => (key x y).mp) ⟨_, _⟩)"
},
{
"state_after": "G : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\n⊢ ∀ (x y : G'), x⁻¹ * y ∈ comap f H ↔ (↑f x)⁻¹ * ↑f y ∈ H",
"state_before": "G : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\n⊢ ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)",
"tactic": "simp only [QuotientGroup.leftRel_apply]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nx y : G'\n⊢ ↑f (x⁻¹ * y) = (↑f x)⁻¹ * ↑f y",
"tactic": "rw [f.map_mul, f.map_inv]"
},
{
"state_after": "case refine'_1\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey✝ : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\nkey : ∀ (x y : G'), Quotient.mk'' x = Quotient.mk'' y ↔ Quotient.mk'' (↑f x) = Quotient.mk'' (↑f y)\n⊢ Function.Injective (Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)))",
"state_before": "case refine'_1\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\n⊢ Function.Injective (Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)))",
"tactic": "simp_rw [← Quotient.eq''] at key"
},
{
"state_after": "case refine'_1\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey✝ : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\nkey : ∀ (x y : G'), Quotient.mk'' x = Quotient.mk'' y ↔ Quotient.mk'' (↑f x) = Quotient.mk'' (↑f y)\nx : G'\n⊢ ∀ ⦃a₂ : G' ⧸ comap f H⦄,\n Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)) (Quotient.mk'' x) =\n Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)) a₂ →\n Quotient.mk'' x = a₂",
"state_before": "case refine'_1\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey✝ : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\nkey : ∀ (x y : G'), Quotient.mk'' x = Quotient.mk'' y ↔ Quotient.mk'' (↑f x) = Quotient.mk'' (↑f y)\n⊢ Function.Injective (Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)))",
"tactic": "refine' Quotient.ind' fun x => _"
},
{
"state_after": "case refine'_1\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey✝ : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\nkey : ∀ (x y : G'), Quotient.mk'' x = Quotient.mk'' y ↔ Quotient.mk'' (↑f x) = Quotient.mk'' (↑f y)\nx y : G'\n⊢ Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)) (Quotient.mk'' x) =\n Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)) (Quotient.mk'' y) →\n Quotient.mk'' x = Quotient.mk'' y",
"state_before": "case refine'_1\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey✝ : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\nkey : ∀ (x y : G'), Quotient.mk'' x = Quotient.mk'' y ↔ Quotient.mk'' (↑f x) = Quotient.mk'' (↑f y)\nx : G'\n⊢ ∀ ⦃a₂ : G' ⧸ comap f H⦄,\n Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)) (Quotient.mk'' x) =\n Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)) a₂ →\n Quotient.mk'' x = a₂",
"tactic": "refine' Quotient.ind' fun y => _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey✝ : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\nkey : ∀ (x y : G'), Quotient.mk'' x = Quotient.mk'' y ↔ Quotient.mk'' (↑f x) = Quotient.mk'' (↑f y)\nx y : G'\n⊢ Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)) (Quotient.mk'' x) =\n Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)) (Quotient.mk'' y) →\n Quotient.mk'' x = Quotient.mk'' y",
"tactic": "exact (key x y).mpr"
},
{
"state_after": "case refine'_2\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\nx : G\n⊢ ∃ a, Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)) a = Quotient.mk'' x",
"state_before": "case refine'_2\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\n⊢ Function.Surjective (Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)))",
"tactic": "refine' Quotient.ind' fun x => _"
},
{
"state_after": "case refine'_2.intro\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\nx : G\ny : G'\nhy : ↑f y = x\n⊢ ∃ a, Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)) a = Quotient.mk'' x",
"state_before": "case refine'_2\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\nx : G\n⊢ ∃ a, Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)) a = Quotient.mk'' x",
"tactic": "obtain ⟨y, hy⟩ := hf x"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro\nG : Type u_2\ninst✝¹ : Group G\nH K L : Subgroup G\nG' : Type u_1\ninst✝ : Group G'\nf : G' →* G\nhf : Function.Surjective ↑f\nthis✝ : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : ∀ (x y : G'), Setoid.r x y ↔ Setoid.r (↑f x) (↑f y)\nx : G\ny : G'\nhy : ↑f y = x\n⊢ ∃ a, Quotient.map' ↑f (_ : ∀ (x y : G'), Setoid.r x y → Setoid.r (↑f x) (↑f y)) a = Quotient.mk'' x",
"tactic": "exact ⟨y, (Quotient.map'_mk'' f _ y).trans (congr_arg Quotient.mk'' hy)⟩"
}
] |
[
78,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.natDegree_map
|
[] |
[
284,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean
|
lt_add_of_tsub_lt_left
|
[] |
[
368,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/Topology/Sequences.lean
|
CompactSpace.tendsto_subseq
|
[] |
[
313,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
uniformity_setCoe
|
[] |
[
1481,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1479,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithBot.add_lt_add_right
|
[] |
[
691,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
689,
11
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
expSeries_div_summable
|
[] |
[
594,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
593,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Algebra.inf_toSubsemiring
|
[] |
[
851,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
849,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.colimit.ι_post
|
[
{
"state_after": "J : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasColimit F\nG : C ⥤ D\ninst✝ : HasColimit (F ⋙ G)\nj : J\n⊢ (G.mapCocone (cocone F)).ι.app j = G.map (ι F j)",
"state_before": "J : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasColimit F\nG : C ⥤ D\ninst✝ : HasColimit (F ⋙ G)\nj : J\n⊢ ι (F ⋙ G) j ≫ post F G = G.map (ι F j)",
"tactic": "erw [IsColimit.fac]"
},
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasColimit F\nG : C ⥤ D\ninst✝ : HasColimit (F ⋙ G)\nj : J\n⊢ (G.mapCocone (cocone F)).ι.app j = G.map (ι F j)",
"tactic": "rfl"
}
] |
[
1031,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1028,
1
] |
Mathlib/Data/ENat/Basic.lean
|
ENat.top_sub_coe
|
[] |
[
149,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Algebra/Category/GroupCat/Basic.lean
|
AddCommGroupCat.int_hom_ext
|
[] |
[
358,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.centralizer_univ
|
[] |
[
791,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
790,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
exists_pos_left_iff_sameRay_and_ne_zero
|
[
{
"state_after": "case mp\nR : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhx : x ≠ 0\n⊢ (∃ r, 0 < r ∧ r • x = y) → SameRay R x y ∧ y ≠ 0\n\ncase mpr\nR : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhx : x ≠ 0\n⊢ SameRay R x y ∧ y ≠ 0 → ∃ r, 0 < r ∧ r • x = y",
"state_before": "R : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhx : x ≠ 0\n⊢ (∃ r, 0 < r ∧ r • x = y) ↔ SameRay R x y ∧ y ≠ 0",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro\nR : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : M\nhx : x ≠ 0\nr : R\nhr : 0 < r\n⊢ SameRay R x (r • x) ∧ r • x ≠ 0",
"state_before": "case mp\nR : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhx : x ≠ 0\n⊢ (∃ r, 0 < r ∧ r • x = y) → SameRay R x y ∧ y ≠ 0",
"tactic": "rintro ⟨r, hr, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro\nR : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : M\nhx : x ≠ 0\nr : R\nhr : 0 < r\n⊢ SameRay R x (r • x) ∧ r • x ≠ 0",
"tactic": "simp [hx, hr.le, hr.ne']"
},
{
"state_after": "case mpr.intro\nR : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhx : x ≠ 0\nhxy : SameRay R x y\nhy : y ≠ 0\n⊢ ∃ r, 0 < r ∧ r • x = y",
"state_before": "case mpr\nR : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhx : x ≠ 0\n⊢ SameRay R x y ∧ y ≠ 0 → ∃ r, 0 < r ∧ r • x = y",
"tactic": "rintro ⟨hxy, hy⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nR : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhx : x ≠ 0\nhxy : SameRay R x y\nhy : y ≠ 0\n⊢ ∃ r, 0 < r ∧ r • x = y",
"tactic": "exact (exists_pos_left_iff_sameRay hx hy).2 hxy"
}
] |
[
724,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
718,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
DifferentiableWithinAt.congr_mono
|
[] |
[
904,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
902,
1
] |
Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean
|
ZMod.χ₈_nat_mod_eight
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ↑χ₈ ↑n = ↑χ₈ ↑(n % 8)",
"tactic": "rw [← ZMod.nat_cast_mod n 8]"
}
] |
[
154,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
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