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/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigO.sup
|
[] |
[
645,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
642,
1
] |
Mathlib/Algebra/Order/Nonneg/Ring.lean
|
Nonneg.coe_nsmul
|
[] |
[
157,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
11
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.fin_param
|
[] |
[
1088,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1085,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.span_singleton_eq_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nI : Ideal α\nx : α\n⊢ span {x} = ⊤ ↔ IsUnit x",
"tactic": "rw [isUnit_iff_dvd_one, ← span_singleton_le_span_singleton, span_singleton_one, eq_top_iff]"
}
] |
[
522,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
isClopen_empty
|
[] |
[
1561,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1561,
9
] |
Mathlib/MeasureTheory/Measure/GiryMonad.lean
|
MeasureTheory.Measure.bind_zero_right'
|
[] |
[
173,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Algebra/Group/Semiconj.lean
|
SemiconjBy.reflexive
|
[] |
[
109,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
11
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.union_subset_union
|
[] |
[
1381,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1380,
1
] |
Mathlib/CategoryTheory/Subobject/Basic.lean
|
CategoryTheory.Subobject.ofMkLE_comp_ofLE
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category D\nB A₁ : C\nf : A₁ ⟶ B\ninst✝ : Mono f\nX Y : Subobject B\nh₁ : mk f ≤ X\nh₂ : X ≤ Y\n⊢ (underlyingIso f).inv ≫ underlying.map (LE.le.hom h₁ ≫ LE.le.hom h₂) =\n (underlyingIso f).inv ≫ underlying.map (LE.le.hom (_ : mk f ≤ Y))",
"state_before": "C : Type u₁\ninst✝² : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category D\nB A₁ : C\nf : A₁ ⟶ B\ninst✝ : Mono f\nX Y : Subobject B\nh₁ : mk f ≤ X\nh₂ : X ≤ Y\n⊢ ofMkLE f X h₁ ≫ ofLE X Y h₂ = ofMkLE f Y (_ : mk f ≤ Y)",
"tactic": "simp only [ofMkLE, ofLEMk, ofLE, ofMkLEMk, ← Functor.map_comp underlying,\n assoc]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category D\nB A₁ : C\nf : A₁ ⟶ B\ninst✝ : Mono f\nX Y : Subobject B\nh₁ : mk f ≤ X\nh₂ : X ≤ Y\n⊢ (underlyingIso f).inv ≫ underlying.map (LE.le.hom h₁ ≫ LE.le.hom h₂) =\n (underlyingIso f).inv ≫ underlying.map (LE.le.hom (_ : mk f ≤ Y))",
"tactic": "congr 1"
}
] |
[
423,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
419,
1
] |
Mathlib/LinearAlgebra/Vandermonde.lean
|
Matrix.det_vandermonde_eq_zero_iff
|
[
{
"state_after": "case mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nv : Fin n → R\n⊢ det (vandermonde v) = 0 → ∃ i j, v i = v j ∧ i ≠ j\n\ncase mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nv : Fin n → R\n⊢ (∃ i j, v i = v j ∧ i ≠ j) → det (vandermonde v) = 0",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nv : Fin n → R\n⊢ det (vandermonde v) = 0 ↔ ∃ i j, v i = v j ∧ i ≠ j",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nv : Fin n → R\n⊢ ∀ (x : Fin n), (x ∈ univ ∧ ∃ a, a ∈ Ioi x ∧ v a = v x) → ∃ i j, v i = v j ∧ i ≠ j",
"state_before": "case mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nv : Fin n → R\n⊢ det (vandermonde v) = 0 → ∃ i j, v i = v j ∧ i ≠ j",
"tactic": "simp only [det_vandermonde v, Finset.prod_eq_zero_iff, sub_eq_zero, forall_exists_index]"
},
{
"state_after": "case mp.intro.intro.intro\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nv : Fin n → R\ni : Fin n\nleft✝ : i ∈ univ\nj : Fin n\nh₁ : j ∈ Ioi i\nh₂ : v j = v i\n⊢ ∃ i j, v i = v j ∧ i ≠ j",
"state_before": "case mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nv : Fin n → R\n⊢ ∀ (x : Fin n), (x ∈ univ ∧ ∃ a, a ∈ Ioi x ∧ v a = v x) → ∃ i j, v i = v j ∧ i ≠ j",
"tactic": "rintro i ⟨_, j, h₁, h₂⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nv : Fin n → R\ni : Fin n\nleft✝ : i ∈ univ\nj : Fin n\nh₁ : j ∈ Ioi i\nh₂ : v j = v i\n⊢ ∃ i j, v i = v j ∧ i ≠ j",
"tactic": "exact ⟨j, i, h₂, (mem_Ioi.mp h₁).ne'⟩"
},
{
"state_after": "case mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nv : Fin n → R\n⊢ ∀ (x x_1 : Fin n), v x = v x_1 → ¬x = x_1 → det (vandermonde v) = 0",
"state_before": "case mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nv : Fin n → R\n⊢ (∃ i j, v i = v j ∧ i ≠ j) → det (vandermonde v) = 0",
"tactic": "simp only [Ne.def, forall_exists_index, and_imp]"
},
{
"state_after": "case mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nv : Fin n → R\ni j : Fin n\nh₁ : v i = v j\nh₂ : ¬i = j\nk : Fin n\n⊢ vandermonde v i k = vandermonde v j k",
"state_before": "case mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nv : Fin n → R\n⊢ ∀ (x x_1 : Fin n), v x = v x_1 → ¬x = x_1 → det (vandermonde v) = 0",
"tactic": "refine' fun i j h₁ h₂ => Matrix.det_zero_of_row_eq h₂ (funext fun k => _)"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nv : Fin n → R\ni j : Fin n\nh₁ : v i = v j\nh₂ : ¬i = j\nk : Fin n\n⊢ vandermonde v i k = vandermonde v j k",
"tactic": "rw [vandermonde_apply, vandermonde_apply, h₁]"
}
] |
[
151,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Algebra/CovariantAndContravariant.lean
|
covariant_flip_mul_iff
|
[
{
"state_after": "no goals",
"state_before": "M : Type ?u.6553\nN : Type u_1\nμ : M → N → N\nr : N → N → Prop\ninst✝ : CommSemigroup N\n⊢ Covariant N N (flip fun x x_1 => x * x_1) r ↔ Covariant N N (fun x x_1 => x * x_1) r",
"tactic": "rw [flip_mul]"
}
] |
[
320,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.diam_union
|
[
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.535752\nι : Type ?u.535755\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nt : Set α\nxs : x ∈ s\nyt : y ∈ t\n⊢ ENNReal.toReal (EMetric.diam (s ∪ t)) ≤\n ENNReal.toReal (EMetric.diam s) + ENNReal.toReal (edist x y) + ENNReal.toReal (EMetric.diam t)",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.535752\nι : Type ?u.535755\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nt : Set α\nxs : x ∈ s\nyt : y ∈ t\n⊢ diam (s ∪ t) ≤ diam s + dist x y + diam t",
"tactic": "simp only [diam, dist_edist]"
},
{
"state_after": "case refine_1\nα : Type u\nβ : Type v\nX : Type ?u.535752\nι : Type ?u.535755\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nt : Set α\nxs : x ∈ s\nyt : y ∈ t\n⊢ EMetric.diam s + edist x y = ⊤ → EMetric.diam (s ∪ t) = ⊤\n\ncase refine_2\nα : Type u\nβ : Type v\nX : Type ?u.535752\nι : Type ?u.535755\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nt : Set α\nxs : x ∈ s\nyt : y ∈ t\n⊢ EMetric.diam t = ⊤ → EMetric.diam (s ∪ t) = ⊤",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.535752\nι : Type ?u.535755\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nt : Set α\nxs : x ∈ s\nyt : y ∈ t\n⊢ ENNReal.toReal (EMetric.diam (s ∪ t)) ≤\n ENNReal.toReal (EMetric.diam s) + ENNReal.toReal (edist x y) + ENNReal.toReal (EMetric.diam t)",
"tactic": "refine (ENNReal.toReal_le_add' (EMetric.diam_union xs yt) ?_ ?_).trans\n (add_le_add_right ENNReal.toReal_add_le _)"
},
{
"state_after": "case refine_1\nα : Type u\nβ : Type v\nX : Type ?u.535752\nι : Type ?u.535755\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nt : Set α\nxs : x ∈ s\nyt : y ∈ t\n⊢ EMetric.diam s = ⊤ → EMetric.diam (s ∪ t) = ⊤",
"state_before": "case refine_1\nα : Type u\nβ : Type v\nX : Type ?u.535752\nι : Type ?u.535755\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nt : Set α\nxs : x ∈ s\nyt : y ∈ t\n⊢ EMetric.diam s + edist x y = ⊤ → EMetric.diam (s ∪ t) = ⊤",
"tactic": "simp only [ENNReal.add_eq_top, edist_ne_top, or_false]"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nα : Type u\nβ : Type v\nX : Type ?u.535752\nι : Type ?u.535755\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nt : Set α\nxs : x ∈ s\nyt : y ∈ t\n⊢ EMetric.diam s = ⊤ → EMetric.diam (s ∪ t) = ⊤",
"tactic": "exact fun h ↦ top_unique <| h ▸ EMetric.diam_mono (subset_union_left _ _)"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nα : Type u\nβ : Type v\nX : Type ?u.535752\nι : Type ?u.535755\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nt : Set α\nxs : x ∈ s\nyt : y ∈ t\n⊢ EMetric.diam t = ⊤ → EMetric.diam (s ∪ t) = ⊤",
"tactic": "exact fun h ↦ top_unique <| h ▸ EMetric.diam_mono (subset_union_right _ _)"
}
] |
[
2705,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2698,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderIso.symm_apply_le
|
[] |
[
994,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
993,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.uncurry_curryLeft
|
[] |
[
1382,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1380,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.ext'
|
[
{
"state_after": "case h\nι : Type u_5\nι' : Type ?u.184951\nR : Type u_1\nR₂ : Type ?u.184957\nK : Type ?u.184960\nM : Type u_3\nM' : Type ?u.184966\nM'' : Type ?u.184969\nV : Type u\nV' : Type ?u.184974\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nR₁ : Type u_2\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type u_4\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf₁ f₂ : M ≃ₛₗ[σ] M₁\nh : ∀ (i : ι), ↑f₁ (↑b i) = ↑f₂ (↑b i)\nx : M\n⊢ ↑f₁ x = ↑f₂ x",
"state_before": "ι : Type u_5\nι' : Type ?u.184951\nR : Type u_1\nR₂ : Type ?u.184957\nK : Type ?u.184960\nM : Type u_3\nM' : Type ?u.184966\nM'' : Type ?u.184969\nV : Type u\nV' : Type ?u.184974\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx : M\nR₁ : Type u_2\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type u_4\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf₁ f₂ : M ≃ₛₗ[σ] M₁\nh : ∀ (i : ι), ↑f₁ (↑b i) = ↑f₂ (↑b i)\n⊢ f₁ = f₂",
"tactic": "ext x"
},
{
"state_after": "case h\nι : Type u_5\nι' : Type ?u.184951\nR : Type u_1\nR₂ : Type ?u.184957\nK : Type ?u.184960\nM : Type u_3\nM' : Type ?u.184966\nM'' : Type ?u.184969\nV : Type u\nV' : Type ?u.184974\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nR₁ : Type u_2\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type u_4\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf₁ f₂ : M ≃ₛₗ[σ] M₁\nh : ∀ (i : ι), ↑f₁ (↑b i) = ↑f₂ (↑b i)\nx : M\n⊢ ↑f₁ (∑ a in (↑b.repr x).support, ↑(↑b.repr x) a • ↑b a) = ↑f₂ (∑ a in (↑b.repr x).support, ↑(↑b.repr x) a • ↑b a)",
"state_before": "case h\nι : Type u_5\nι' : Type ?u.184951\nR : Type u_1\nR₂ : Type ?u.184957\nK : Type ?u.184960\nM : Type u_3\nM' : Type ?u.184966\nM'' : Type ?u.184969\nV : Type u\nV' : Type ?u.184974\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nR₁ : Type u_2\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type u_4\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf₁ f₂ : M ≃ₛₗ[σ] M₁\nh : ∀ (i : ι), ↑f₁ (↑b i) = ↑f₂ (↑b i)\nx : M\n⊢ ↑f₁ x = ↑f₂ x",
"tactic": "rw [← b.total_repr x, Finsupp.total_apply, Finsupp.sum]"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_5\nι' : Type ?u.184951\nR : Type u_1\nR₂ : Type ?u.184957\nK : Type ?u.184960\nM : Type u_3\nM' : Type ?u.184966\nM'' : Type ?u.184969\nV : Type u\nV' : Type ?u.184974\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nR₁ : Type u_2\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type u_4\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf₁ f₂ : M ≃ₛₗ[σ] M₁\nh : ∀ (i : ι), ↑f₁ (↑b i) = ↑f₂ (↑b i)\nx : M\n⊢ ↑f₁ (∑ a in (↑b.repr x).support, ↑(↑b.repr x) a • ↑b a) = ↑f₂ (∑ a in (↑b.repr x).support, ↑(↑b.repr x) a • ↑b a)",
"tactic": "simp only [LinearEquiv.map_sum, LinearEquiv.map_smulₛₗ, h]"
}
] |
[
288,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiffWithinAt_prod'
|
[] |
[
1410,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1407,
1
] |
Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
|
LiouvilleWith.nat_add
|
[] |
[
245,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/Algebra/Module/Basic.lean
|
smul_eq_zero
|
[] |
[
618,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
616,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.diam_eq_zero_of_unbounded
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.535436\nι : Type ?u.535439\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nh : ¬Bounded s\n⊢ diam s = 0",
"tactic": "rw [diam, ediam_of_unbounded h, ENNReal.top_toReal]"
}
] |
[
2687,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2686,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
IsOpen.ite'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.385864\nγ : Type ?u.385867\nδ : Type ?u.385870\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns s' t : Set α\nhs : IsOpen s\nhs' : IsOpen s'\nht : ∀ (x : α), x ∈ frontier t → (x ∈ s ↔ x ∈ s')\n⊢ IsOpen (Set.ite t s s')",
"tactic": "classical\n simp only [isOpen_iff_continuous_mem, Set.ite] at *\n convert continuous_piecewise (fun x hx => propext (ht x hx)) hs.continuousOn hs'.continuousOn\n rename_i x\n by_cases hx : x ∈ t <;> simp [hx]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.385864\nγ : Type ?u.385867\nδ : Type ?u.385870\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns s' t : Set α\nht : ∀ (x : α), x ∈ frontier t → (x ∈ s ↔ x ∈ s')\nhs : Continuous fun x => x ∈ s\nhs' : Continuous fun x => x ∈ s'\n⊢ Continuous fun x => x ∈ s ∩ t ∪ s' \\ t",
"state_before": "α : Type u_1\nβ : Type ?u.385864\nγ : Type ?u.385867\nδ : Type ?u.385870\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns s' t : Set α\nhs : IsOpen s\nhs' : IsOpen s'\nht : ∀ (x : α), x ∈ frontier t → (x ∈ s ↔ x ∈ s')\n⊢ IsOpen (Set.ite t s s')",
"tactic": "simp only [isOpen_iff_continuous_mem, Set.ite] at *"
},
{
"state_after": "case h.e'_5.h.a\nα : Type u_1\nβ : Type ?u.385864\nγ : Type ?u.385867\nδ : Type ?u.385870\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns s' t : Set α\nht : ∀ (x : α), x ∈ frontier t → (x ∈ s ↔ x ∈ s')\nhs : Continuous fun x => x ∈ s\nhs' : Continuous fun x => x ∈ s'\nx✝ : α\n⊢ x✝ ∈ s ∩ t ∪ s' \\ t ↔ piecewise t (fun x => x ∈ s) (fun x => x ∈ s') x✝",
"state_before": "α : Type u_1\nβ : Type ?u.385864\nγ : Type ?u.385867\nδ : Type ?u.385870\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns s' t : Set α\nht : ∀ (x : α), x ∈ frontier t → (x ∈ s ↔ x ∈ s')\nhs : Continuous fun x => x ∈ s\nhs' : Continuous fun x => x ∈ s'\n⊢ Continuous fun x => x ∈ s ∩ t ∪ s' \\ t",
"tactic": "convert continuous_piecewise (fun x hx => propext (ht x hx)) hs.continuousOn hs'.continuousOn"
},
{
"state_after": "case h.e'_5.h.a\nα : Type u_1\nβ : Type ?u.385864\nγ : Type ?u.385867\nδ : Type ?u.385870\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns s' t : Set α\nht : ∀ (x : α), x ∈ frontier t → (x ∈ s ↔ x ∈ s')\nhs : Continuous fun x => x ∈ s\nhs' : Continuous fun x => x ∈ s'\nx : α\n⊢ x ∈ s ∩ t ∪ s' \\ t ↔ piecewise t (fun x => x ∈ s) (fun x => x ∈ s') x",
"state_before": "case h.e'_5.h.a\nα : Type u_1\nβ : Type ?u.385864\nγ : Type ?u.385867\nδ : Type ?u.385870\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns s' t : Set α\nht : ∀ (x : α), x ∈ frontier t → (x ∈ s ↔ x ∈ s')\nhs : Continuous fun x => x ∈ s\nhs' : Continuous fun x => x ∈ s'\nx✝ : α\n⊢ x✝ ∈ s ∩ t ∪ s' \\ t ↔ piecewise t (fun x => x ∈ s) (fun x => x ∈ s') x✝",
"tactic": "rename_i x"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.a\nα : Type u_1\nβ : Type ?u.385864\nγ : Type ?u.385867\nδ : Type ?u.385870\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns s' t : Set α\nht : ∀ (x : α), x ∈ frontier t → (x ∈ s ↔ x ∈ s')\nhs : Continuous fun x => x ∈ s\nhs' : Continuous fun x => x ∈ s'\nx : α\n⊢ x ∈ s ∩ t ∪ s' \\ t ↔ piecewise t (fun x => x ∈ s) (fun x => x ∈ s') x",
"tactic": "by_cases hx : x ∈ t <;> simp [hx]"
}
] |
[
1220,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1214,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.map_iInf_comap_of_surjective
|
[] |
[
1592,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1591,
1
] |
Mathlib/Data/Matrix/Hadamard.lean
|
Matrix.hadamard_comm
|
[] |
[
68,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Analysis/Convex/Strict.lean
|
StrictConvex.smul_mem_of_zero_mem
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕝 : Type ?u.190924\nE : Type u_2\nF : Type ?u.190930\nβ : Type ?u.190933\ninst✝⁶ : OrderedRing 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t✝ : Set E\nx y : E\nhs : StrictConvex 𝕜 s\nzero_mem : 0 ∈ s\nhx : x ∈ s\nhx₀ : x ≠ 0\nt : 𝕜\nht₀ : 0 < t\nht₁ : t < 1\n⊢ t • x ∈ interior s",
"tactic": "simpa using hs.add_smul_mem zero_mem (by simpa using hx) hx₀ ht₀ ht₁"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕝 : Type ?u.190924\nE : Type u_2\nF : Type ?u.190930\nβ : Type ?u.190933\ninst✝⁶ : OrderedRing 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t✝ : Set E\nx y : E\nhs : StrictConvex 𝕜 s\nzero_mem : 0 ∈ s\nhx : x ∈ s\nhx₀ : x ≠ 0\nt : 𝕜\nht₀ : 0 < t\nht₁ : t < 1\n⊢ 0 + x ∈ s",
"tactic": "simpa using hx"
}
] |
[
356,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/LinearAlgebra/Trace.lean
|
LinearMap.trace_eq_matrix_trace
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type ?u.237802\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb : Basis ι R M\nc : Basis κ R M\nf : M →ₗ[R] M\n⊢ ↑(trace R M) f = Matrix.trace (↑(toMatrix b b) f)",
"tactic": "rw [trace_eq_matrix_trace_of_finset R b.reindexFinsetRange, ← traceAux_def, ← traceAux_def,\n traceAux_eq R b b.reindexFinsetRange]"
}
] |
[
108,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Std/Data/List/Basic.lean
|
List.tail_cons
|
[] |
[
311,
56
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
311,
9
] |
Mathlib/CategoryTheory/FintypeCat.lean
|
FintypeCat.Skeleton.ext
|
[] |
[
144,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Order/Disjoint.lean
|
disjoint_iff
|
[] |
[
129,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Algebra/Group/Prod.lean
|
MonoidHom.prod_apply
|
[] |
[
565,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
564,
1
] |
Mathlib/RingTheory/Ideal/Prod.lean
|
Ideal.prod.ext_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I'✝ : Ideal R\nJ✝ J'✝ : Ideal S\nI I' : Ideal R\nJ J' : Ideal S\n⊢ prod I J = prod I' J' ↔ I = I' ∧ J = J'",
"tactic": "simp only [← idealProdEquiv_symm_apply, idealProdEquiv.symm.injective.eq_iff, Prod.mk.inj_iff]"
}
] |
[
109,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.congr_apply
|
[] |
[
676,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
674,
1
] |
Mathlib/Data/Set/Intervals/Monotone.lean
|
iUnion_Ioo_of_mono_of_isGLB_of_isLUB
|
[] |
[
195,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/Logic/Basic.lean
|
not_xor
|
[
{
"state_after": "no goals",
"state_before": "P Q : Prop\n⊢ ¬Xor' P Q ↔ (P ↔ Q)",
"tactic": "simp only [not_and, Xor', not_or, not_not, ← iff_iff_implies_and_implies]"
}
] |
[
476,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
475,
9
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.lineMap_apply_one
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV1 : Type u_3\nP1 : Type u_1\nV2 : Type ?u.456425\nP2 : Type ?u.456428\nV3 : Type ?u.456431\nP3 : Type ?u.456434\nV4 : Type ?u.456437\nP4 : Type ?u.456440\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np₀ p₁ : P1\n⊢ ↑(lineMap p₀ p₁) 1 = p₁",
"tactic": "simp [(lineMap_apply), (vsub_vadd)]"
}
] |
[
568,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
566,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_pos_iff_support_of_nonneg_ae
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.1114206\nF : Type ?u.1114209\n𝕜 : Type ?u.1114212\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1116903\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfi : Integrable f\n⊢ (0 < ∫ (x : α), f x ∂μ) ↔ 0 < ↑↑μ (Function.support f)",
"tactic": "simp_rw [(integral_nonneg_of_ae hf).lt_iff_ne, pos_iff_ne_zero, Ne.def, @eq_comm ℝ 0,\n integral_eq_zero_iff_of_nonneg_ae hf hfi, Filter.EventuallyEq, ae_iff, Pi.zero_apply,\n Function.support]"
}
] |
[
1243,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1239,
1
] |
Mathlib/Data/Real/Irrational.lean
|
Irrational.div_rat
|
[
{
"state_after": "q✝ : ℚ\nx y : ℝ\nh : Irrational x\nq : ℚ\nhq : q ≠ 0\n⊢ Irrational (x * ↑q⁻¹)",
"state_before": "q✝ : ℚ\nx y : ℝ\nh : Irrational x\nq : ℚ\nhq : q ≠ 0\n⊢ Irrational (x / ↑q)",
"tactic": "rw [div_eq_mul_inv, ← cast_inv]"
},
{
"state_after": "no goals",
"state_before": "q✝ : ℚ\nx y : ℝ\nh : Irrational x\nq : ℚ\nhq : q ≠ 0\n⊢ Irrational (x * ↑q⁻¹)",
"tactic": "exact h.mul_rat (inv_ne_zero hq)"
}
] |
[
424,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
422,
1
] |
Mathlib/RingTheory/TensorProduct.lean
|
Algebra.TensorProduct.basisAux_tmul
|
[
{
"state_after": "case h\nk : Type u_3\ninst✝⁴ : CommRing k\nR : Type u_1\ninst✝³ : Ring R\ninst✝² : Algebra k R\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module k M\nι : Type u_2\nb : Basis ι k M\nr : R\nm : M\na✝ : ι\n⊢ ↑(↑(basisAux R b) (r ⊗ₜ[k] m)) a✝ =\n ↑(r • Finsupp.mapRange ↑(algebraMap k R) (_ : ↑(algebraMap k R) 0 = 0) (↑b.repr m)) a✝",
"state_before": "k : Type u_3\ninst✝⁴ : CommRing k\nR : Type u_1\ninst✝³ : Ring R\ninst✝² : Algebra k R\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module k M\nι : Type u_2\nb : Basis ι k M\nr : R\nm : M\n⊢ ↑(basisAux R b) (r ⊗ₜ[k] m) = r • Finsupp.mapRange ↑(algebraMap k R) (_ : ↑(algebraMap k R) 0 = 0) (↑b.repr m)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nk : Type u_3\ninst✝⁴ : CommRing k\nR : Type u_1\ninst✝³ : Ring R\ninst✝² : Algebra k R\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module k M\nι : Type u_2\nb : Basis ι k M\nr : R\nm : M\na✝ : ι\n⊢ ↑(↑(basisAux R b) (r ⊗ₜ[k] m)) a✝ =\n ↑(r • Finsupp.mapRange ↑(algebraMap k R) (_ : ↑(algebraMap k R) 0 = 0) (↑b.repr m)) a✝",
"tactic": "simp [basisAux, ← Algebra.commutes, Algebra.smul_def]"
}
] |
[
1037,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1034,
1
] |
Mathlib/Topology/UniformSpace/Separation.lean
|
eq_of_forall_symmetric
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : UniformSpace α✝\ninst✝³ : UniformSpace β\ninst✝² : UniformSpace γ\nα : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : SeparatedSpace α\nx y : α\nh : ∀ {V : Set (α × α)}, V ∈ 𝓤 α → SymmetricRel V → (x, y) ∈ V\n⊢ ∀ {i : Set (α × α)}, i ∈ 𝓤 α ∧ SymmetricRel i → (x, y) ∈ id i",
"tactic": "simpa [and_imp]"
}
] |
[
160,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Data/Set/Intervals/Instances.lean
|
Set.Icc.coe_eq_zero
|
[
{
"state_after": "α : Type u_1\ninst✝ : OrderedSemiring α\nx : ↑(Icc 0 1)\n⊢ x = 0 ↔ ↑x = 0",
"state_before": "α : Type u_1\ninst✝ : OrderedSemiring α\nx : ↑(Icc 0 1)\n⊢ ↑x = 0 ↔ x = 0",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedSemiring α\nx : ↑(Icc 0 1)\n⊢ x = 0 ↔ ↑x = 0",
"tactic": "exact Subtype.ext_iff"
}
] |
[
82,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
Convex.inter
|
[] |
[
98,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
hasFDerivAtFilter_id
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type ?u.658148\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.658243\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.658338\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\nx : E\nL : Filter E\n⊢ ∀ (x_1 : E), 0 = _root_.id x_1 - _root_.id x - ↑(ContinuousLinearMap.id 𝕜 E) (x_1 - x)",
"tactic": "simp"
}
] |
[
995,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
994,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
lt_csSup_of_lt
|
[] |
[
630,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
629,
1
] |
Std/Logic.lean
|
and_not_self_iff
|
[] |
[
239,
85
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
239,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.span_eq_iSup_of_singleton_spans
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₂ : Type ?u.64934\nK : Type ?u.64937\nM : Type u_1\nM₂ : Type ?u.64943\nV : Type ?u.64946\nS : Type ?u.64949\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns✝ t s : Set M\n⊢ span R s = ⨆ (x : M) (_ : x ∈ s), span R {x}",
"tactic": "simp only [← span_iUnion, Set.biUnion_of_singleton s]"
}
] |
[
267,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Order/Height.lean
|
Set.chainHeight_eq_top_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.7005\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nh : ∀ (n : ℕ), ∃ l, l ∈ subchain s ∧ length l = n\n⊢ chainHeight s = ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.7005\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\n⊢ chainHeight s = ⊤ ↔ ∀ (n : ℕ), ∃ l, l ∈ subchain s ∧ length l = n",
"tactic": "refine' ⟨fun h n ↦ le_chainHeight_iff.1 (le_top.trans_eq h.symm), fun h ↦ _⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.7005\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nh : chainHeight s ≠ ⊤\n⊢ ∃ n, ∀ (l : List α), l ∈ subchain s → length l ≠ n",
"state_before": "α : Type u_1\nβ : Type ?u.7005\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nh : ∀ (n : ℕ), ∃ l, l ∈ subchain s ∧ length l = n\n⊢ chainHeight s = ⊤",
"tactic": "contrapose! h"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.7005\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nh : chainHeight s ≠ ⊤\nn : ℕ\nhn : ↑n = chainHeight s\n⊢ ∃ n, ∀ (l : List α), l ∈ subchain s → length l ≠ n",
"state_before": "α : Type u_1\nβ : Type ?u.7005\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nh : chainHeight s ≠ ⊤\n⊢ ∃ n, ∀ (l : List α), l ∈ subchain s → length l ≠ n",
"tactic": "obtain ⟨n, hn⟩ := WithTop.ne_top_iff_exists.1 h"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.7005\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nh : chainHeight s ≠ ⊤\nn : ℕ\nhn : ↑n = chainHeight s\n⊢ ∃ n, ∀ (l : List α), l ∈ subchain s → length l ≠ n",
"tactic": "exact ⟨n + 1, fun l hs ↦ (Nat.lt_succ_iff.2 <| Nat.cast_le.1 <|\n (length_le_chainHeight_of_mem_subchain hs).trans_eq hn.symm).ne⟩"
}
] |
[
134,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/GroupTheory/Commutator.lean
|
Subgroup.commutator_commutator_eq_bot_of_rotate
|
[
{
"state_after": "G : Type u_1\nG' : Type ?u.56694\nF : Type ?u.56697\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : MonoidHomClass F G G'\nf : F\ng₁ g₂ g₃ g : G\nH₁ H₂ H₃ K₁ K₂ : Subgroup G\nh1 : ∀ (g₁ : G), g₁ ∈ H₂ → ∀ (g₂ : G), g₂ ∈ H₃ → ∀ (h : G), h ∈ H₁ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nh2 : ∀ (g₁ : G), g₁ ∈ H₃ → ∀ (g₂ : G), g₂ ∈ H₁ → ∀ (h : G), h ∈ H₂ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\n⊢ ∀ (g₁ : G), g₁ ∈ H₁ → ∀ (g₂ : G), g₂ ∈ H₂ → ∀ (h : G), h ∈ H₃ → ⁅h, ⁅g₁, g₂⁆⁆ = 1",
"state_before": "G : Type u_1\nG' : Type ?u.56694\nF : Type ?u.56697\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : MonoidHomClass F G G'\nf : F\ng₁ g₂ g₃ g : G\nH₁ H₂ H₃ K₁ K₂ : Subgroup G\nh1 : ⁅⁅H₂, H₃⁆, H₁⁆ = ⊥\nh2 : ⁅⁅H₃, H₁⁆, H₂⁆ = ⊥\n⊢ ⁅⁅H₁, H₂⁆, H₃⁆ = ⊥",
"tactic": "simp_rw [commutator_eq_bot_iff_le_centralizer, commutator_le,\n mem_centralizer_iff_commutator_eq_one, ← commutatorElement_def] at h1 h2⊢"
},
{
"state_after": "G : Type u_1\nG' : Type ?u.56694\nF : Type ?u.56697\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : MonoidHomClass F G G'\nf : F\ng₁ g₂ g₃ g : G\nH₁ H₂ H₃ K₁ K₂ : Subgroup G\nh1 : ∀ (g₁ : G), g₁ ∈ H₂ → ∀ (g₂ : G), g₂ ∈ H₃ → ∀ (h : G), h ∈ H₁ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nh2 : ∀ (g₁ : G), g₁ ∈ H₃ → ∀ (g₂ : G), g₂ ∈ H₁ → ∀ (h : G), h ∈ H₂ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nx : G\nhx : x ∈ H₁\ny : G\nhy : y ∈ H₂\nz : G\nhz : z ∈ H₃\n⊢ ⁅z, ⁅x, y⁆⁆ = 1",
"state_before": "G : Type u_1\nG' : Type ?u.56694\nF : Type ?u.56697\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : MonoidHomClass F G G'\nf : F\ng₁ g₂ g₃ g : G\nH₁ H₂ H₃ K₁ K₂ : Subgroup G\nh1 : ∀ (g₁ : G), g₁ ∈ H₂ → ∀ (g₂ : G), g₂ ∈ H₃ → ∀ (h : G), h ∈ H₁ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nh2 : ∀ (g₁ : G), g₁ ∈ H₃ → ∀ (g₂ : G), g₂ ∈ H₁ → ∀ (h : G), h ∈ H₂ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\n⊢ ∀ (g₁ : G), g₁ ∈ H₁ → ∀ (g₂ : G), g₂ ∈ H₂ → ∀ (h : G), h ∈ H₃ → ⁅h, ⁅g₁, g₂⁆⁆ = 1",
"tactic": "intro x hx y hy z hz"
},
{
"state_after": "G : Type u_1\nG' : Type ?u.56694\nF : Type ?u.56697\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : MonoidHomClass F G G'\nf : F\ng₁ g₂ g₃ g : G\nH₁ H₂ H₃ K₁ K₂ : Subgroup G\nh1 : ∀ (g₁ : G), g₁ ∈ H₂ → ∀ (g₂ : G), g₂ ∈ H₃ → ∀ (h : G), h ∈ H₁ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nh2 : ∀ (g₁ : G), g₁ ∈ H₃ → ∀ (g₂ : G), g₂ ∈ H₁ → ∀ (h : G), h ∈ H₂ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nx : G\nhx : x ∈ H₁\ny : G\nhy : y ∈ H₂\nz : G\nhz : z ∈ H₃\n⊢ ⁅z, ⁅x, y⁆⁆ = x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹\n\nG : Type u_1\nG' : Type ?u.56694\nF : Type ?u.56697\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : MonoidHomClass F G G'\nf : F\ng₁ g₂ g₃ g : G\nH₁ H₂ H₃ K₁ K₂ : Subgroup G\nh1 : ∀ (g₁ : G), g₁ ∈ H₂ → ∀ (g₂ : G), g₂ ∈ H₃ → ∀ (h : G), h ∈ H₁ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nh2 : ∀ (g₁ : G), g₁ ∈ H₃ → ∀ (g₂ : G), g₂ ∈ H₁ → ∀ (h : G), h ∈ H₂ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nx : G\nhx : x ∈ H₁\ny : G\nhy : y ∈ H₂\nz : G\nhz : z ∈ H₃\n⊢ x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹ = 1",
"state_before": "G : Type u_1\nG' : Type ?u.56694\nF : Type ?u.56697\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : MonoidHomClass F G G'\nf : F\ng₁ g₂ g₃ g : G\nH₁ H₂ H₃ K₁ K₂ : Subgroup G\nh1 : ∀ (g₁ : G), g₁ ∈ H₂ → ∀ (g₂ : G), g₂ ∈ H₃ → ∀ (h : G), h ∈ H₁ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nh2 : ∀ (g₁ : G), g₁ ∈ H₃ → ∀ (g₂ : G), g₂ ∈ H₁ → ∀ (h : G), h ∈ H₂ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nx : G\nhx : x ∈ H₁\ny : G\nhy : y ∈ H₂\nz : G\nhz : z ∈ H₃\n⊢ ⁅z, ⁅x, y⁆⁆ = 1",
"tactic": "trans x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.56694\nF : Type ?u.56697\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : MonoidHomClass F G G'\nf : F\ng₁ g₂ g₃ g : G\nH₁ H₂ H₃ K₁ K₂ : Subgroup G\nh1 : ∀ (g₁ : G), g₁ ∈ H₂ → ∀ (g₂ : G), g₂ ∈ H₃ → ∀ (h : G), h ∈ H₁ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nh2 : ∀ (g₁ : G), g₁ ∈ H₃ → ∀ (g₂ : G), g₂ ∈ H₁ → ∀ (h : G), h ∈ H₂ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nx : G\nhx : x ∈ H₁\ny : G\nhy : y ∈ H₂\nz : G\nhz : z ∈ H₃\n⊢ ⁅z, ⁅x, y⁆⁆ = x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹",
"tactic": "group"
},
{
"state_after": "G : Type u_1\nG' : Type ?u.56694\nF : Type ?u.56697\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : MonoidHomClass F G G'\nf : F\ng₁ g₂ g₃ g : G\nH₁ H₂ H₃ K₁ K₂ : Subgroup G\nh1 : ∀ (g₁ : G), g₁ ∈ H₂ → ∀ (g₂ : G), g₂ ∈ H₃ → ∀ (h : G), h ∈ H₁ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nh2 : ∀ (g₁ : G), g₁ ∈ H₃ → ∀ (g₂ : G), g₂ ∈ H₁ → ∀ (h : G), h ∈ H₂ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nx : G\nhx : x ∈ H₁\ny : G\nhy : y ∈ H₂\nz : G\nhz : z ∈ H₃\n⊢ x * z * 1⁻¹ * z⁻¹ * y * 1⁻¹ * y⁻¹ * x⁻¹ = 1",
"state_before": "G : Type u_1\nG' : Type ?u.56694\nF : Type ?u.56697\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : MonoidHomClass F G G'\nf : F\ng₁ g₂ g₃ g : G\nH₁ H₂ H₃ K₁ K₂ : Subgroup G\nh1 : ∀ (g₁ : G), g₁ ∈ H₂ → ∀ (g₂ : G), g₂ ∈ H₃ → ∀ (h : G), h ∈ H₁ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nh2 : ∀ (g₁ : G), g₁ ∈ H₃ → ∀ (g₂ : G), g₂ ∈ H₁ → ∀ (h : G), h ∈ H₂ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nx : G\nhx : x ∈ H₁\ny : G\nhy : y ∈ H₂\nz : G\nhz : z ∈ H₃\n⊢ x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹ = 1",
"tactic": "rw [h1 _ (H₂.inv_mem hy) _ hz _ (H₁.inv_mem hx), h2 _ (H₃.inv_mem hz) _ (H₁.inv_mem hx) _ hy]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.56694\nF : Type ?u.56697\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : MonoidHomClass F G G'\nf : F\ng₁ g₂ g₃ g : G\nH₁ H₂ H₃ K₁ K₂ : Subgroup G\nh1 : ∀ (g₁ : G), g₁ ∈ H₂ → ∀ (g₂ : G), g₂ ∈ H₃ → ∀ (h : G), h ∈ H₁ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nh2 : ∀ (g₁ : G), g₁ ∈ H₃ → ∀ (g₂ : G), g₂ ∈ H₁ → ∀ (h : G), h ∈ H₂ → ⁅h, ⁅g₁, g₂⁆⁆ = 1\nx : G\nhx : x ∈ H₁\ny : G\nhy : y ∈ H₂\nz : G\nhz : z ∈ H₃\n⊢ x * z * 1⁻¹ * z⁻¹ * y * 1⁻¹ * y⁻¹ * x⁻¹ = 1",
"tactic": "group"
}
] |
[
117,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.ofSet_toLocalEquiv
|
[] |
[
766,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
765,
1
] |
Mathlib/Data/Sym/Sym2.lean
|
Sym2.IsDiag.mem_range_diag
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.40966\nγ : Type ?u.40969\nx y : α\n⊢ IsDiag (Quotient.mk (Rel.setoid α) (x, y)) → Quotient.mk (Rel.setoid α) (x, y) ∈ Set.range diag",
"state_before": "α : Type u_1\nβ : Type ?u.40966\nγ : Type ?u.40969\nz : Sym2 α\n⊢ IsDiag z → z ∈ Set.range diag",
"tactic": "induction' z using Sym2.ind with x y"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.40966\nγ : Type ?u.40969\nx : α\n⊢ Quotient.mk (Rel.setoid α) (x, x) ∈ Set.range diag",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.40966\nγ : Type ?u.40969\nx y : α\n⊢ IsDiag (Quotient.mk (Rel.setoid α) (x, y)) → Quotient.mk (Rel.setoid α) (x, y) ∈ Set.range diag",
"tactic": "rintro (rfl : x = y)"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.40966\nγ : Type ?u.40969\nx : α\n⊢ Quotient.mk (Rel.setoid α) (x, x) ∈ Set.range diag",
"tactic": "exact ⟨_, rfl⟩"
}
] |
[
457,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
ModelWithCorners.continuous_symm
|
[] |
[
236,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean
|
midpoint_vsub_right
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\nV : Type u_1\nV' : Type ?u.44515\nP : Type u_2\nP' : Type ?u.44521\ninst✝⁷ : Ring R\ninst✝⁶ : Invertible 2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z p₁ p₂ : P\n⊢ midpoint R p₁ p₂ -ᵥ p₂ = ⅟2 • (p₁ -ᵥ p₂)",
"tactic": "rw [midpoint_comm, midpoint_vsub_left]"
}
] |
[
114,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Data/Rel.lean
|
Rel.core_comp
|
[
{
"state_after": "case h\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nr : Rel α β\ns : Rel β γ\nt : Set γ\nx : α\n⊢ x ∈ core (r • s) t ↔ x ∈ core r (core s t)",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nr : Rel α β\ns : Rel β γ\nt : Set γ\n⊢ core (r • s) t = core r (core s t)",
"tactic": "ext x"
},
{
"state_after": "case h\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nr : Rel α β\ns : Rel β γ\nt : Set γ\nx : α\n⊢ (∀ (y : γ) (x_1 : β), r x x_1 → s x_1 y → y ∈ t) ↔ ∀ (y : β), r x y → ∀ (y_1 : γ), s y y_1 → y_1 ∈ t",
"state_before": "case h\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nr : Rel α β\ns : Rel β γ\nt : Set γ\nx : α\n⊢ x ∈ core (r • s) t ↔ x ∈ core r (core s t)",
"tactic": "simp [core, comp]"
},
{
"state_after": "case h.mp\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nr : Rel α β\ns : Rel β γ\nt : Set γ\nx : α\n⊢ (∀ (y : γ) (x_1 : β), r x x_1 → s x_1 y → y ∈ t) → ∀ (y : β), r x y → ∀ (y_1 : γ), s y y_1 → y_1 ∈ t\n\ncase h.mpr\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nr : Rel α β\ns : Rel β γ\nt : Set γ\nx : α\n⊢ (∀ (y : β), r x y → ∀ (y_1 : γ), s y y_1 → y_1 ∈ t) → ∀ (y : γ) (x_1 : β), r x x_1 → s x_1 y → y ∈ t",
"state_before": "case h\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nr : Rel α β\ns : Rel β γ\nt : Set γ\nx : α\n⊢ (∀ (y : γ) (x_1 : β), r x x_1 → s x_1 y → y ∈ t) ↔ ∀ (y : β), r x y → ∀ (y_1 : γ), s y y_1 → y_1 ∈ t",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case h.mp\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nr : Rel α β\ns : Rel β γ\nt : Set γ\nx : α\n⊢ (∀ (y : γ) (x_1 : β), r x x_1 → s x_1 y → y ∈ t) → ∀ (y : β), r x y → ∀ (y_1 : γ), s y y_1 → y_1 ∈ t",
"tactic": "exact fun h y rxy z => h z y rxy"
},
{
"state_after": "no goals",
"state_before": "case h.mpr\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nr : Rel α β\ns : Rel β γ\nt : Set γ\nx : α\n⊢ (∀ (y : β), r x y → ∀ (y_1 : γ), s y y_1 → y_1 ∈ t) → ∀ (y : γ) (x_1 : β), r x x_1 → s x_1 y → y ∈ t",
"tactic": "exact fun h z y rzy => h y rzy z"
}
] |
[
245,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Data/Set/Intervals/Pi.lean
|
Set.image_mulSingle_Ico_right
|
[] |
[
263,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieSubmodule.sInf_glb
|
[
{
"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\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nS : Set (LieSubmodule R L M)\nh : ∀ {N N' : LieSubmodule R L M}, ↑N ≤ ↑N' ↔ N ≤ N'\n⊢ IsGLB S (sInf S)",
"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\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nS : Set (LieSubmodule R L M)\n⊢ IsGLB S (sInf S)",
"tactic": "have h : ∀ {N N' : LieSubmodule R L M}, (N : Set M) ≤ N' ↔ N ≤ N' := fun {_ _} ↦ Iff.rfl"
},
{
"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\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nS : Set (LieSubmodule R L M)\nh : ∀ {N N' : LieSubmodule R L M}, ↑N ≤ ↑N' ↔ N ≤ N'\n⊢ IsGLB ((fun {x} => ↑x) '' S) ↑(sInf S)",
"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\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nS : Set (LieSubmodule R L M)\nh : ∀ {N N' : LieSubmodule R L M}, ↑N ≤ ↑N' ↔ N ≤ N'\n⊢ IsGLB S (sInf S)",
"tactic": "apply IsGLB.of_image h"
},
{
"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\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nS : Set (LieSubmodule R L M)\nh : ∀ {N N' : LieSubmodule R L M}, ↑N ≤ ↑N' ↔ N ≤ N'\n⊢ IsGLB ((fun {x} => ↑x) '' S) (⋂ (s : LieSubmodule R L M) (_ : s ∈ S), ↑s)",
"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\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nS : Set (LieSubmodule R L M)\nh : ∀ {N N' : LieSubmodule R L M}, ↑N ≤ ↑N' ↔ N ≤ N'\n⊢ IsGLB ((fun {x} => ↑x) '' S) ↑(sInf S)",
"tactic": "simp only [sInf_coe]"
},
{
"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\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nS : Set (LieSubmodule R L M)\nh : ∀ {N N' : LieSubmodule R L M}, ↑N ≤ ↑N' ↔ N ≤ N'\n⊢ IsGLB ((fun {x} => ↑x) '' S) (⋂ (s : LieSubmodule R L M) (_ : s ∈ S), ↑s)",
"tactic": "exact isGLB_biInf"
}
] |
[
437,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.iUnion_smul_left_image
|
[] |
[
232,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Icc_subset_Icc
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.5978\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\nha : a₂ ≤ a₁\nhb : b₁ ≤ b₂\n⊢ Icc a₁ b₁ ⊆ Icc a₂ b₂",
"tactic": "simpa [← coe_subset] using Set.Icc_subset_Icc ha hb"
}
] |
[
163,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Topology/Constructions.lean
|
ULift.closedEmbedding_down
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.564674\nδ : Type ?u.564677\nε : Type ?u.564680\nζ : Type ?u.564683\ninst✝ : TopologicalSpace α\n⊢ IsClosed (range down)",
"tactic": "simp only [ULift.down_surjective.range_eq, isClosed_univ]"
}
] |
[
1595,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1593,
1
] |
Mathlib/Algebra/Associated.lean
|
associated_iff_eq
|
[
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.248864\nγ : Type ?u.248867\nδ : Type ?u.248870\ninst✝¹ : Monoid α\ninst✝ : Unique αˣ\nx y : α\n⊢ x ~ᵤ y → x = y\n\ncase mpr\nα : Type u_1\nβ : Type ?u.248864\nγ : Type ?u.248867\nδ : Type ?u.248870\ninst✝¹ : Monoid α\ninst✝ : Unique αˣ\nx y : α\n⊢ x = y → x ~ᵤ y",
"state_before": "α : Type u_1\nβ : Type ?u.248864\nγ : Type ?u.248867\nδ : Type ?u.248870\ninst✝¹ : Monoid α\ninst✝ : Unique αˣ\nx y : α\n⊢ x ~ᵤ y ↔ x = y",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nα : Type u_1\nβ : Type ?u.248864\nγ : Type ?u.248867\nδ : Type ?u.248870\ninst✝¹ : Monoid α\ninst✝ : Unique αˣ\nx : α\nc : αˣ\n⊢ x = x * ↑c",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.248864\nγ : Type ?u.248867\nδ : Type ?u.248870\ninst✝¹ : Monoid α\ninst✝ : Unique αˣ\nx y : α\n⊢ x ~ᵤ y → x = y",
"tactic": "rintro ⟨c, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nα : Type u_1\nβ : Type ?u.248864\nγ : Type ?u.248867\nδ : Type ?u.248870\ninst✝¹ : Monoid α\ninst✝ : Unique αˣ\nx : α\nc : αˣ\n⊢ x = x * ↑c",
"tactic": "rw [units_eq_one c, Units.val_one, mul_one]"
},
{
"state_after": "case mpr\nα : Type u_1\nβ : Type ?u.248864\nγ : Type ?u.248867\nδ : Type ?u.248870\ninst✝¹ : Monoid α\ninst✝ : Unique αˣ\nx : α\n⊢ x ~ᵤ x",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.248864\nγ : Type ?u.248867\nδ : Type ?u.248870\ninst✝¹ : Monoid α\ninst✝ : Unique αˣ\nx y : α\n⊢ x = y → x ~ᵤ y",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.248864\nγ : Type ?u.248867\nδ : Type ?u.248870\ninst✝¹ : Monoid α\ninst✝ : Unique αˣ\nx : α\n⊢ x ~ᵤ x",
"tactic": "rfl"
}
] |
[
697,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
692,
1
] |
Std/Data/String/Lemmas.lean
|
String.Pos.valid_zero
|
[] |
[
151,
62
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
151,
9
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.ball_finset_sup
|
[
{
"state_after": "R : Type ?u.1093501\nR' : Type ?u.1093504\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1093510\n𝕜₃ : Type ?u.1093513\n𝕝 : Type ?u.1093516\nE : Type u_2\nE₂ : Type ?u.1093522\nE₃ : Type ?u.1093525\nF : Type ?u.1093528\nG : Type ?u.1093531\nι : Type u_3\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ : Seminorm 𝕜 E\np : ι → Seminorm 𝕜 E\ns : Finset ι\nx : E\nr : ℝ\nhr : 0 < r\n⊢ ball (Finset.sup s p) x r = ⨅ (a : ι) (_ : a ∈ s), ball (p a) x r",
"state_before": "R : Type ?u.1093501\nR' : Type ?u.1093504\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1093510\n𝕜₃ : Type ?u.1093513\n𝕝 : Type ?u.1093516\nE : Type u_2\nE₂ : Type ?u.1093522\nE₃ : Type ?u.1093525\nF : Type ?u.1093528\nG : Type ?u.1093531\nι : Type u_3\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ : Seminorm 𝕜 E\np : ι → Seminorm 𝕜 E\ns : Finset ι\nx : E\nr : ℝ\nhr : 0 < r\n⊢ ball (Finset.sup s p) x r = Finset.inf s fun i => ball (p i) x r",
"tactic": "rw [Finset.inf_eq_iInf]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1093501\nR' : Type ?u.1093504\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1093510\n𝕜₃ : Type ?u.1093513\n𝕝 : Type ?u.1093516\nE : Type u_2\nE₂ : Type ?u.1093522\nE₃ : Type ?u.1093525\nF : Type ?u.1093528\nG : Type ?u.1093531\nι : Type u_3\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ : Seminorm 𝕜 E\np : ι → Seminorm 𝕜 E\ns : Finset ι\nx : E\nr : ℝ\nhr : 0 < r\n⊢ ball (Finset.sup s p) x r = ⨅ (a : ι) (_ : a ∈ s), ball (p a) x r",
"tactic": "exact ball_finset_sup_eq_iInter _ _ _ hr"
}
] |
[
873,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
870,
1
] |
Mathlib/Data/Matrix/Rank.lean
|
Matrix.rank_conjTranspose_mul_self
|
[
{
"state_after": "l : Type ?u.244698\nm : Type u_1\nn : Type u_2\no : Type ?u.244707\nR : Type u_3\nm_fin : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\nA : Matrix m n R\n⊢ finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ ⬝ A)) } = finrank R { x // x ∈ LinearMap.range (mulVecLin A) }",
"state_before": "l : Type ?u.244698\nm : Type u_1\nn : Type u_2\no : Type ?u.244707\nR : Type u_3\nm_fin : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\nA : Matrix m n R\n⊢ rank (Aᴴ ⬝ A) = rank A",
"tactic": "dsimp only [rank]"
},
{
"state_after": "l : Type ?u.244698\nm : Type u_1\nn : Type u_2\no : Type ?u.244707\nR : Type u_3\nm_fin : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\nA : Matrix m n R\n⊢ (fun x => x + finrank R { x // x ∈ LinearMap.ker (mulVecLin A) })\n (finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ ⬝ A)) }) =\n (fun x => x + finrank R { x // x ∈ LinearMap.ker (mulVecLin A) })\n (finrank R { x // x ∈ LinearMap.range (mulVecLin A) })",
"state_before": "l : Type ?u.244698\nm : Type u_1\nn : Type u_2\no : Type ?u.244707\nR : Type u_3\nm_fin : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\nA : Matrix m n R\n⊢ finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ ⬝ A)) } = finrank R { x // x ∈ LinearMap.range (mulVecLin A) }",
"tactic": "refine' add_left_injective (finrank R (LinearMap.ker (mulVecLin A))) _"
},
{
"state_after": "l : Type ?u.244698\nm : Type u_1\nn : Type u_2\no : Type ?u.244707\nR : Type u_3\nm_fin : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\nA : Matrix m n R\n⊢ finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ ⬝ A)) } + finrank R { x // x ∈ LinearMap.ker (mulVecLin A) } =\n finrank R { x // x ∈ LinearMap.range (mulVecLin A) } + finrank R { x // x ∈ LinearMap.ker (mulVecLin A) }",
"state_before": "l : Type ?u.244698\nm : Type u_1\nn : Type u_2\no : Type ?u.244707\nR : Type u_3\nm_fin : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\nA : Matrix m n R\n⊢ (fun x => x + finrank R { x // x ∈ LinearMap.ker (mulVecLin A) })\n (finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ ⬝ A)) }) =\n (fun x => x + finrank R { x // x ∈ LinearMap.ker (mulVecLin A) })\n (finrank R { x // x ∈ LinearMap.range (mulVecLin A) })",
"tactic": "dsimp only"
},
{
"state_after": "l : Type ?u.244698\nm : Type u_1\nn : Type u_2\no : Type ?u.244707\nR : Type u_3\nm_fin : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\nA : Matrix m n R\n⊢ finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ ⬝ A)) } + finrank R { x // x ∈ LinearMap.ker (mulVecLin A) } =\n finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ ⬝ A)) } +\n finrank R { x // x ∈ LinearMap.ker (mulVecLin (Aᴴ ⬝ A)) }\n\nl : Type ?u.244698\nm : Type u_1\nn : Type u_2\no : Type ?u.244707\nR : Type u_3\nm_fin : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\nA : Matrix m n R\n⊢ finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ ⬝ A)) } +\n finrank R { x // x ∈ LinearMap.ker (mulVecLin (Aᴴ ⬝ A)) } =\n finrank R { x // x ∈ LinearMap.range (mulVecLin A) } + finrank R { x // x ∈ LinearMap.ker (mulVecLin A) }",
"state_before": "l : Type ?u.244698\nm : Type u_1\nn : Type u_2\no : Type ?u.244707\nR : Type u_3\nm_fin : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\nA : Matrix m n R\n⊢ finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ ⬝ A)) } + finrank R { x // x ∈ LinearMap.ker (mulVecLin A) } =\n finrank R { x // x ∈ LinearMap.range (mulVecLin A) } + finrank R { x // x ∈ LinearMap.ker (mulVecLin A) }",
"tactic": "trans finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ ⬝ A)) } +\n finrank R { x // x ∈ LinearMap.ker (mulVecLin (Aᴴ ⬝ A)) }"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.244698\nm : Type u_1\nn : Type u_2\no : Type ?u.244707\nR : Type u_3\nm_fin : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\nA : Matrix m n R\n⊢ finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ ⬝ A)) } + finrank R { x // x ∈ LinearMap.ker (mulVecLin A) } =\n finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ ⬝ A)) } +\n finrank R { x // x ∈ LinearMap.ker (mulVecLin (Aᴴ ⬝ A)) }",
"tactic": "rw [ker_mulVecLin_conjTranspose_mul_self]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.244698\nm : Type u_1\nn : Type u_2\no : Type ?u.244707\nR : Type u_3\nm_fin : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\nA : Matrix m n R\n⊢ finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ ⬝ A)) } +\n finrank R { x // x ∈ LinearMap.ker (mulVecLin (Aᴴ ⬝ A)) } =\n finrank R { x // x ∈ LinearMap.range (mulVecLin A) } + finrank R { x // x ∈ LinearMap.ker (mulVecLin A) }",
"tactic": "simp only [LinearMap.finrank_range_add_finrank_ker]"
}
] |
[
210,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Topology/UniformSpace/Equiv.lean
|
UniformEquiv.symm_apply_apply
|
[] |
[
179,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/FieldTheory/Finite/Basic.lean
|
FiniteField.unit_isSquare_iff
|
[
{
"state_after": "case intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\n⊢ IsSquare a ↔ a ^ (Fintype.card F / 2) = 1",
"state_before": "K : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : Fˣ\n⊢ IsSquare a ↔ a ^ (Fintype.card F / 2) = 1",
"tactic": "obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := Fˣ)"
},
{
"state_after": "case intro.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : (fun x x_1 => x ^ x_1) g n = a\n⊢ IsSquare a ↔ a ^ (Fintype.card F / 2) = 1",
"state_before": "case intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\n⊢ IsSquare a ↔ a ^ (Fintype.card F / 2) = 1",
"tactic": "obtain ⟨n, hn⟩ : a ∈ Submonoid.powers g := by rw [mem_powers_iff_mem_zpowers]; apply hg"
},
{
"state_after": "case intro.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : (fun x x_1 => x ^ x_1) g n = a\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\n⊢ IsSquare a ↔ a ^ (Fintype.card F / 2) = 1",
"state_before": "case intro.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : (fun x x_1 => x ^ x_1) g n = a\n⊢ IsSquare a ↔ a ^ (Fintype.card F / 2) = 1",
"tactic": "have hodd := Nat.two_mul_odd_div_two (FiniteField.odd_card_of_char_ne_two hF)"
},
{
"state_after": "case intro.intro.mp\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : (fun x x_1 => x ^ x_1) g n = a\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\n⊢ IsSquare a → a ^ (Fintype.card F / 2) = 1\n\ncase intro.intro.mpr\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : (fun x x_1 => x ^ x_1) g n = a\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\n⊢ a ^ (Fintype.card F / 2) = 1 → IsSquare a",
"state_before": "case intro.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : (fun x x_1 => x ^ x_1) g n = a\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\n⊢ IsSquare a ↔ a ^ (Fintype.card F / 2) = 1",
"tactic": "constructor"
},
{
"state_after": "K : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\n⊢ a ∈ Subgroup.zpowers g",
"state_before": "K : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\n⊢ a ∈ Submonoid.powers g",
"tactic": "rw [mem_powers_iff_mem_zpowers]"
},
{
"state_after": "no goals",
"state_before": "K : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\n⊢ a ∈ Subgroup.zpowers g",
"tactic": "apply hg"
},
{
"state_after": "case intro.intro.mp.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\ny : Fˣ\nhn : (fun x x_1 => x ^ x_1) g n = y * y\n⊢ (y * y) ^ (Fintype.card F / 2) = 1",
"state_before": "case intro.intro.mp\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : (fun x x_1 => x ^ x_1) g n = a\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\n⊢ IsSquare a → a ^ (Fintype.card F / 2) = 1",
"tactic": "rintro ⟨y, rfl⟩"
},
{
"state_after": "case intro.intro.mp.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\ny : Fˣ\nhn : (fun x x_1 => x ^ x_1) g n = y * y\n⊢ y ^ (Fintype.card F - 1) = 1",
"state_before": "case intro.intro.mp.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\ny : Fˣ\nhn : (fun x x_1 => x ^ x_1) g n = y * y\n⊢ (y * y) ^ (Fintype.card F / 2) = 1",
"tactic": "rw [← pow_two, ← pow_mul, hodd]"
},
{
"state_after": "case intro.intro.mp.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\ny : Fˣ\nhn : (fun x x_1 => x ^ x_1) g n = y * y\n⊢ ↑(y ^ (Fintype.card F - 1)) = ↑1",
"state_before": "case intro.intro.mp.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\ny : Fˣ\nhn : (fun x x_1 => x ^ x_1) g n = y * y\n⊢ y ^ (Fintype.card F - 1) = 1",
"tactic": "apply_fun Units.val using Units.ext (α := F)"
},
{
"state_after": "case intro.intro.mp.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\ny : Fˣ\nhn : (fun x x_1 => x ^ x_1) g n = y * y\n⊢ ↑y ^ (Fintype.card F - 1) = 1",
"state_before": "case intro.intro.mp.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\ny : Fˣ\nhn : (fun x x_1 => x ^ x_1) g n = y * y\n⊢ ↑(y ^ (Fintype.card F - 1)) = ↑1",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.mp.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\ny : Fˣ\nhn : (fun x x_1 => x ^ x_1) g n = y * y\n⊢ ↑y ^ (Fintype.card F - 1) = 1",
"tactic": "exact FiniteField.pow_card_sub_one_eq_one (y : F) (Units.ne_zero y)"
},
{
"state_after": "case intro.intro.mpr\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\n⊢ (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1 → IsSquare ((fun x x_1 => x ^ x_1) g n)",
"state_before": "case intro.intro.mpr\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : (fun x x_1 => x ^ x_1) g n = a\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\n⊢ a ^ (Fintype.card F / 2) = 1 → IsSquare a",
"tactic": "subst a"
},
{
"state_after": "case intro.intro.mpr\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1\n⊢ IsSquare ((fun x x_1 => x ^ x_1) g n)",
"state_before": "case intro.intro.mpr\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\n⊢ (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1 → IsSquare ((fun x x_1 => x ^ x_1) g n)",
"tactic": "intro h"
},
{
"state_after": "case intro.intro.mpr\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) ∣ n * (Fintype.card F / 2)\n⊢ IsSquare ((fun x x_1 => x ^ x_1) g n)",
"state_before": "case intro.intro.mpr\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1\n⊢ IsSquare ((fun x x_1 => x ^ x_1) g n)",
"tactic": "have key : 2 * (Fintype.card F / 2) ∣ n * (Fintype.card F / 2) := by\n rw [← pow_mul] at h\n rw [hodd, ← Fintype.card_units, ← orderOf_eq_card_of_forall_mem_zpowers hg]\n apply orderOf_dvd_of_pow_eq_one h"
},
{
"state_after": "case intro.intro.mpr\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) ∣ n * (Fintype.card F / 2)\nthis : 0 < Fintype.card F / 2\n⊢ IsSquare ((fun x x_1 => x ^ x_1) g n)",
"state_before": "case intro.intro.mpr\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) ∣ n * (Fintype.card F / 2)\n⊢ IsSquare ((fun x x_1 => x ^ x_1) g n)",
"tactic": "have : 0 < Fintype.card F / 2 := Nat.div_pos Fintype.one_lt_card (by norm_num)"
},
{
"state_after": "case intro.intro.mpr.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nthis : 0 < Fintype.card F / 2\nm : ℕ\nh : (fun x x_1 => x ^ x_1) g (2 * m) ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) ∣ 2 * m * (Fintype.card F / 2)\n⊢ IsSquare ((fun x x_1 => x ^ x_1) g (2 * m))",
"state_before": "case intro.intro.mpr\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) ∣ n * (Fintype.card F / 2)\nthis : 0 < Fintype.card F / 2\n⊢ IsSquare ((fun x x_1 => x ^ x_1) g n)",
"tactic": "obtain ⟨m, rfl⟩ := Nat.dvd_of_mul_dvd_mul_right this key"
},
{
"state_after": "case intro.intro.mpr.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nthis : 0 < Fintype.card F / 2\nm : ℕ\nh : (fun x x_1 => x ^ x_1) g (2 * m) ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) ∣ 2 * m * (Fintype.card F / 2)\n⊢ (fun x x_1 => x ^ x_1) g (2 * m) = g ^ m * g ^ m",
"state_before": "case intro.intro.mpr.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nthis : 0 < Fintype.card F / 2\nm : ℕ\nh : (fun x x_1 => x ^ x_1) g (2 * m) ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) ∣ 2 * m * (Fintype.card F / 2)\n⊢ IsSquare ((fun x x_1 => x ^ x_1) g (2 * m))",
"tactic": "refine' ⟨g ^ m, _⟩"
},
{
"state_after": "case intro.intro.mpr.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nthis : 0 < Fintype.card F / 2\nm : ℕ\nh : (fun x x_1 => x ^ x_1) g (2 * m) ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) ∣ 2 * m * (Fintype.card F / 2)\n⊢ g ^ (2 * m) = g ^ m * g ^ m",
"state_before": "case intro.intro.mpr.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nthis : 0 < Fintype.card F / 2\nm : ℕ\nh : (fun x x_1 => x ^ x_1) g (2 * m) ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) ∣ 2 * m * (Fintype.card F / 2)\n⊢ (fun x x_1 => x ^ x_1) g (2 * m) = g ^ m * g ^ m",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.mpr.intro\nK : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nthis : 0 < Fintype.card F / 2\nm : ℕ\nh : (fun x x_1 => x ^ x_1) g (2 * m) ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) ∣ 2 * m * (Fintype.card F / 2)\n⊢ g ^ (2 * m) = g ^ m * g ^ m",
"tactic": "rw [mul_comm, pow_mul, pow_two]"
},
{
"state_after": "K : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : g ^ (n * (Fintype.card F / 2)) = 1\n⊢ 2 * (Fintype.card F / 2) ∣ n * (Fintype.card F / 2)",
"state_before": "K : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1\n⊢ 2 * (Fintype.card F / 2) ∣ n * (Fintype.card F / 2)",
"tactic": "rw [← pow_mul] at h"
},
{
"state_after": "K : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : g ^ (n * (Fintype.card F / 2)) = 1\n⊢ orderOf g ∣ n * (Fintype.card F / 2)",
"state_before": "K : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : g ^ (n * (Fintype.card F / 2)) = 1\n⊢ 2 * (Fintype.card F / 2) ∣ n * (Fintype.card F / 2)",
"tactic": "rw [hodd, ← Fintype.card_units, ← orderOf_eq_card_of_forall_mem_zpowers hg]"
},
{
"state_after": "no goals",
"state_before": "K : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : g ^ (n * (Fintype.card F / 2)) = 1\n⊢ orderOf g ∣ n * (Fintype.card F / 2)",
"tactic": "apply orderOf_dvd_of_pow_eq_one h"
},
{
"state_after": "no goals",
"state_before": "K : Type ?u.1100843\nR : Type ?u.1100846\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ng : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) ∣ n * (Fintype.card F / 2)\n⊢ 0 < 2",
"tactic": "norm_num"
}
] |
[
556,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
pow_le_of_le_one
|
[] |
[
677,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
676,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
RingHom.is_integral_neg
|
[] |
[
504,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
1
] |
Mathlib/Data/Vector/Basic.lean
|
Vector.traverse_def
|
[
{
"state_after": "case mk\nF G : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : Applicative G\nα β : Type u\nf : α → F β\nx : α\nxs : List α\n⊢ Vector.traverse f (x ::ᵥ { val := xs, property := (_ : List.length xs = List.length xs) }) =\n Seq.seq (cons <$> f x) fun x => Vector.traverse f { val := xs, property := (_ : List.length xs = List.length xs) }",
"state_before": "n : ℕ\nF G : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : Applicative G\nα β : Type u\nf : α → F β\nx : α\n⊢ ∀ (xs : Vector α n), Vector.traverse f (x ::ᵥ xs) = Seq.seq (cons <$> f x) fun x => Vector.traverse f xs",
"tactic": "rintro ⟨xs, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mk\nF G : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : Applicative G\nα β : Type u\nf : α → F β\nx : α\nxs : List α\n⊢ Vector.traverse f (x ::ᵥ { val := xs, property := (_ : List.length xs = List.length xs) }) =\n Seq.seq (cons <$> f x) fun x => Vector.traverse f { val := xs, property := (_ : List.length xs = List.length xs) }",
"tactic": "rfl"
}
] |
[
660,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
11
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_inter_distrib_left
|
[] |
[
1082,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1081,
1
] |
Mathlib/Data/ENat/Basic.lean
|
ENat.le_of_lt_add_one
|
[] |
[
205,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Std/Data/List/Lemmas.lean
|
List.prefix_or_prefix_of_prefix
|
[] |
[
1669,
39
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1667,
1
] |
Mathlib/MeasureTheory/Measure/Portmanteau.lean
|
MeasureTheory.limsup_measure_closed_le_iff_liminf_measure_open_ge
|
[
{
"state_after": "case mp\nΩ : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\n⊢ (∀ (F : Set Ω), IsClosed F → limsup (fun i => ↑↑(μs i) F) L ≤ ↑↑μ F) →\n ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) L\n\ncase mpr\nΩ : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\n⊢ (∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) L) →\n ∀ (F : Set Ω), IsClosed F → limsup (fun i => ↑↑(μs i) F) L ≤ ↑↑μ F",
"state_before": "Ω : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\n⊢ (∀ (F : Set Ω), IsClosed F → limsup (fun i => ↑↑(μs i) F) L ≤ ↑↑μ F) ↔\n ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) L",
"tactic": "constructor"
},
{
"state_after": "case mp\nΩ : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (F : Set Ω), IsClosed F → limsup (fun i => ↑↑(μs i) F) L ≤ ↑↑μ F\nG : Set Ω\nG_open : IsOpen G\n⊢ ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) L",
"state_before": "case mp\nΩ : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\n⊢ (∀ (F : Set Ω), IsClosed F → limsup (fun i => ↑↑(μs i) F) L ≤ ↑↑μ F) →\n ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) L",
"tactic": "intro h G G_open"
},
{
"state_after": "no goals",
"state_before": "case mp\nΩ : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (F : Set Ω), IsClosed F → limsup (fun i => ↑↑(μs i) F) L ≤ ↑↑μ F\nG : Set Ω\nG_open : IsOpen G\n⊢ ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) L",
"tactic": "exact le_measure_liminf_of_limsup_measure_compl_le\n G_open.measurableSet (h (Gᶜ) (isClosed_compl_iff.mpr G_open))"
},
{
"state_after": "case mpr\nΩ : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) L\nF : Set Ω\nF_closed : IsClosed F\n⊢ limsup (fun i => ↑↑(μs i) F) L ≤ ↑↑μ F",
"state_before": "case mpr\nΩ : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\n⊢ (∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) L) →\n ∀ (F : Set Ω), IsClosed F → limsup (fun i => ↑↑(μs i) F) L ≤ ↑↑μ F",
"tactic": "intro h F F_closed"
},
{
"state_after": "no goals",
"state_before": "case mpr\nΩ : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) L\nF : Set Ω\nF_closed : IsClosed F\n⊢ limsup (fun i => ↑↑(μs i) F) L ≤ ↑↑μ F",
"tactic": "exact limsup_measure_le_of_le_liminf_measure_compl\n F_closed.measurableSet (h (Fᶜ) (isOpen_compl_iff.mpr F_closed))"
}
] |
[
190,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Algebra/FreeMonoid/Basic.lean
|
FreeMonoid.toList_of
|
[] |
[
135,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Dynamics/Ergodic/Ergodic.lean
|
PreErgodic.measure_self_or_compl_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : MeasureTheory.Measure α\nhf : PreErgodic f\nhs : MeasurableSet s\nhs' : f ⁻¹' s = s\n⊢ ↑↑μ s = 0 ∨ ↑↑μ (sᶜ) = 0",
"tactic": "simpa using hf.ae_empty_or_univ hs hs'"
}
] |
[
69,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Analysis/Calculus/LocalExtr.lean
|
IsLocalMin.hasFDerivAt_eq_zero
|
[
{
"state_after": "case h\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\nh : IsLocalMin f a\nhf : HasFDerivAt f f' a\ny : E\n⊢ ↑f' y = ↑0 y",
"state_before": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\nh : IsLocalMin f a\nhf : HasFDerivAt f f' a\n⊢ f' = 0",
"tactic": "ext y"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\nh : IsLocalMin f a\nhf : HasFDerivAt f f' a\ny : E\n⊢ ↑f' y = ↑0 y",
"tactic": "apply (h.on univ).hasFDerivWithinAt_eq_zero hf.hasFDerivWithinAt <;>\n rw [posTangentConeAt_univ] <;>\n apply mem_univ"
}
] |
[
193,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.rotate'_length_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nl : List α\n⊢ rotate' l (length l * 0) = l",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ rotate' l (length l * (n + 1)) = rotate' (rotate' l (length l * n)) (length (rotate' l (length l * n)))",
"tactic": "simp [-rotate'_length, Nat.mul_succ, rotate'_rotate']"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ rotate' (rotate' l (length l * n)) (length (rotate' l (length l * n))) = l",
"tactic": "rw [rotate'_length, rotate'_length_mul l n]"
}
] |
[
100,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Topology/Connected.lean
|
IsConnected.biUnion_of_chain
|
[] |
[
281,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Analysis/Calculus/Deriv/Inv.lean
|
fderivWithin_inv
|
[
{
"state_after": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nx_ne_zero : x ≠ 0\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderiv 𝕜 (fun x => x⁻¹) x = smulRight 1 (-(x ^ 2)⁻¹)",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nx_ne_zero : x ≠ 0\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderivWithin 𝕜 (fun x => x⁻¹) s x = smulRight 1 (-(x ^ 2)⁻¹)",
"tactic": "rw [DifferentiableAt.fderivWithin (differentiableAt_inv.2 x_ne_zero) hxs]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nx_ne_zero : x ≠ 0\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderiv 𝕜 (fun x => x⁻¹) x = smulRight 1 (-(x ^ 2)⁻¹)",
"tactic": "exact fderiv_inv"
}
] |
[
130,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Computability/Reduce.lean
|
toNat_manyOneEquiv
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝⁵ : Primcodable α\ninst✝⁴ : Inhabited α\nβ : Type v\ninst✝³ : Primcodable β\ninst✝² : Inhabited β\nγ : Type w\ninst✝¹ : Primcodable γ\ninst✝ : Inhabited γ\np : Set α\n⊢ ManyOneEquiv (toNat p) p",
"tactic": "simp [ManyOneEquiv]"
}
] |
[
357,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.mul_self_sub_mul_self_eq
|
[
{
"state_after": "no goals",
"state_before": "a b : Nat\n⊢ a * a - b * b = (a + b) * (a - b)",
"tactic": "rw [Nat.mul_sub_left_distrib, Nat.right_distrib, Nat.right_distrib,\n Nat.mul_comm b a, Nat.add_comm (a*a) (a*b), Nat.add_sub_add_left]"
}
] |
[
419,
72
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
417,
11
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.coe_ofInvolutive
|
[] |
[
603,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
601,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
isLittleO_pow_pow_of_lt_left
|
[] |
[
100,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.inter_div_subset
|
[] |
[
720,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
719,
1
] |
Mathlib/LinearAlgebra/Matrix/IsDiag.lean
|
Matrix.mul_transpose_self_isDiag_iff_hasOrthogonalRows
|
[] |
[
196,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Topology/NhdsSet.lean
|
Continuous.tendsto_nhdsSet
|
[] |
[
148,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.IsIso.of_isIso_fac_left
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝ : C\nf✝ g✝ : X✝ ⟶ Y✝\nh✝ : Y✝ ⟶ Z✝\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : X ⟶ Z\ninst✝ : IsIso f\nhh : IsIso (f ≫ g)\nw : f ≫ g = h\n⊢ IsIso g",
"state_before": "C : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝ : C\nf✝ g✝ : X✝ ⟶ Y✝\nh✝ : Y✝ ⟶ Z✝\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : X ⟶ Z\ninst✝ : IsIso f\nhh : IsIso h\nw : f ≫ g = h\n⊢ IsIso g",
"tactic": "rw [← w] at hh"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝ : C\nf✝ g✝ : X✝ ⟶ Y✝\nh✝ : Y✝ ⟶ Z✝\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : X ⟶ Z\ninst✝ : IsIso f\nhh : IsIso (f ≫ g)\nw : f ≫ g = h\nthis : IsIso (f ≫ g)\n⊢ IsIso g",
"state_before": "C : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝ : C\nf✝ g✝ : X✝ ⟶ Y✝\nh✝ : Y✝ ⟶ Z✝\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : X ⟶ Z\ninst✝ : IsIso f\nhh : IsIso (f ≫ g)\nw : f ≫ g = h\n⊢ IsIso g",
"tactic": "haveI := hh"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝ : C\nf✝ g✝ : X✝ ⟶ Y✝\nh✝ : Y✝ ⟶ Z✝\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : X ⟶ Z\ninst✝ : IsIso f\nhh : IsIso (f ≫ g)\nw : f ≫ g = h\nthis : IsIso (f ≫ g)\n⊢ IsIso g",
"tactic": "exact of_isIso_comp_left f g"
}
] |
[
458,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/CategoryTheory/Sites/CoverLifting.lean
|
CategoryTheory.idCoverLifting
|
[
{
"state_after": "no goals",
"state_before": "C : Type u_1\ninst✝² : Category C\nD : Type ?u.741\ninst✝¹ : Category D\nE : Type ?u.748\ninst✝ : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nU✝ : C\nS✝ : Sieve ((𝟭 C).obj U✝)\nh : S✝ ∈ GrothendieckTopology.sieves J ((𝟭 C).obj U✝)\n⊢ Sieve.functorPullback (𝟭 C) S✝ ∈ GrothendieckTopology.sieves J U✝",
"tactic": "simpa using h"
}
] |
[
84,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.bit0_of_bit0
|
[] |
[
229,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.set_integral_piecewise
|
[
{
"state_after": "α : Type u_3\nβ : Type u_2\nι : Type ?u.1474570\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : { x // x ∈ kernel α β }\ns : Set α\nhs : MeasurableSet s\ninst✝³ : DecidablePred fun x => x ∈ s\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na : α\ng : β → E\nt : Set β\n⊢ (∫ (b : β) in t, g b ∂if a ∈ s then ↑κ a else ↑η a) =\n if a ∈ s then ∫ (b : β) in t, g b ∂↑κ a else ∫ (b : β) in t, g b ∂↑η a",
"state_before": "α : Type u_3\nβ : Type u_2\nι : Type ?u.1474570\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : { x // x ∈ kernel α β }\ns : Set α\nhs : MeasurableSet s\ninst✝³ : DecidablePred fun x => x ∈ s\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na : α\ng : β → E\nt : Set β\n⊢ (∫ (b : β) in t, g b ∂↑(piecewise hs κ η) a) = if a ∈ s then ∫ (b : β) in t, g b ∂↑κ a else ∫ (b : β) in t, g b ∂↑η a",
"tactic": "simp_rw [piecewise_apply]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_2\nι : Type ?u.1474570\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : { x // x ∈ kernel α β }\ns : Set α\nhs : MeasurableSet s\ninst✝³ : DecidablePred fun x => x ∈ s\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na : α\ng : β → E\nt : Set β\n⊢ (∫ (b : β) in t, g b ∂if a ∈ s then ↑κ a else ↑η a) =\n if a ∈ s then ∫ (b : β) in t, g b ∂↑κ a else ∫ (b : β) in t, g b ∂↑η a",
"tactic": "split_ifs <;> rfl"
}
] |
[
669,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
665,
1
] |
Mathlib/Order/Concept.lean
|
Concept.ext'
|
[
{
"state_after": "case mk.mk\nι : Sort ?u.6130\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.6139\nκ : ι → Sort ?u.6144\nr : α → β → Prop\ns s₁✝ s₂ : Set α\nt t₁✝ t₂ : Set β\nd : Concept α β r\ns₁ : Set α\nt₁ : Set β\nclosure_fst✝ : intentClosure r (s₁, t₁).fst = (s₁, t₁).snd\nh₁ : extentClosure r (s₁, t₁).snd = (s₁, t₁).fst\nh : { toProd := (s₁, t₁), closure_fst := closure_fst✝, closure_snd := h₁ }.toProd.snd = d.snd\n⊢ { toProd := (s₁, t₁), closure_fst := closure_fst✝, closure_snd := h₁ } = d",
"state_before": "ι : Sort ?u.6130\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.6139\nκ : ι → Sort ?u.6144\nr : α → β → Prop\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nc d : Concept α β r\nh : c.snd = d.snd\n⊢ c = d",
"tactic": "obtain ⟨⟨s₁, t₁⟩, _, h₁⟩ := c"
},
{
"state_after": "case mk.mk.mk.mk\nι : Sort ?u.6130\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.6139\nκ : ι → Sort ?u.6144\nr : α → β → Prop\ns s₁✝ s₂✝ : Set α\nt t₁✝ t₂✝ : Set β\ns₁ : Set α\nt₁ : Set β\nclosure_fst✝¹ : intentClosure r (s₁, t₁).fst = (s₁, t₁).snd\nh₁ : extentClosure r (s₁, t₁).snd = (s₁, t₁).fst\ns₂ : Set α\nt₂ : Set β\nclosure_fst✝ : intentClosure r (s₂, t₂).fst = (s₂, t₂).snd\nh₂ : extentClosure r (s₂, t₂).snd = (s₂, t₂).fst\nh :\n { toProd := (s₁, t₁), closure_fst := closure_fst✝¹, closure_snd := h₁ }.toProd.snd =\n { toProd := (s₂, t₂), closure_fst := closure_fst✝, closure_snd := h₂ }.toProd.snd\n⊢ { toProd := (s₁, t₁), closure_fst := closure_fst✝¹, closure_snd := h₁ } =\n { toProd := (s₂, t₂), closure_fst := closure_fst✝, closure_snd := h₂ }",
"state_before": "case mk.mk\nι : Sort ?u.6130\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.6139\nκ : ι → Sort ?u.6144\nr : α → β → Prop\ns s₁✝ s₂ : Set α\nt t₁✝ t₂ : Set β\nd : Concept α β r\ns₁ : Set α\nt₁ : Set β\nclosure_fst✝ : intentClosure r (s₁, t₁).fst = (s₁, t₁).snd\nh₁ : extentClosure r (s₁, t₁).snd = (s₁, t₁).fst\nh : { toProd := (s₁, t₁), closure_fst := closure_fst✝, closure_snd := h₁ }.toProd.snd = d.snd\n⊢ { toProd := (s₁, t₁), closure_fst := closure_fst✝, closure_snd := h₁ } = d",
"tactic": "obtain ⟨⟨s₂, t₂⟩, _, h₂⟩ := d"
},
{
"state_after": "case mk.mk.mk.mk\nι : Sort ?u.6130\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.6139\nκ : ι → Sort ?u.6144\nr : α → β → Prop\ns s₁✝ s₂✝ : Set α\nt t₁✝ t₂✝ : Set β\ns₁ : Set α\nt₁ : Set β\nclosure_fst✝¹ : intentClosure r (s₁, t₁).fst = (s₁, t₁).snd\nh₁ : extentClosure r t₁ = s₁\ns₂ : Set α\nt₂ : Set β\nclosure_fst✝ : intentClosure r (s₂, t₂).fst = (s₂, t₂).snd\nh₂ : extentClosure r t₂ = s₂\nh : t₁ = t₂\n⊢ { toProd := (s₁, t₁), closure_fst := closure_fst✝¹, closure_snd := h₁ } =\n { toProd := (s₂, t₂), closure_fst := closure_fst✝, closure_snd := h₂ }",
"state_before": "case mk.mk.mk.mk\nι : Sort ?u.6130\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.6139\nκ : ι → Sort ?u.6144\nr : α → β → Prop\ns s₁✝ s₂✝ : Set α\nt t₁✝ t₂✝ : Set β\ns₁ : Set α\nt₁ : Set β\nclosure_fst✝¹ : intentClosure r (s₁, t₁).fst = (s₁, t₁).snd\nh₁ : extentClosure r (s₁, t₁).snd = (s₁, t₁).fst\ns₂ : Set α\nt₂ : Set β\nclosure_fst✝ : intentClosure r (s₂, t₂).fst = (s₂, t₂).snd\nh₂ : extentClosure r (s₂, t₂).snd = (s₂, t₂).fst\nh :\n { toProd := (s₁, t₁), closure_fst := closure_fst✝¹, closure_snd := h₁ }.toProd.snd =\n { toProd := (s₂, t₂), closure_fst := closure_fst✝, closure_snd := h₂ }.toProd.snd\n⊢ { toProd := (s₁, t₁), closure_fst := closure_fst✝¹, closure_snd := h₁ } =\n { toProd := (s₂, t₂), closure_fst := closure_fst✝, closure_snd := h₂ }",
"tactic": "dsimp at h₁ h₂ h"
},
{
"state_after": "case mk.mk.mk.mk\nι : Sort ?u.6130\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.6139\nκ : ι → Sort ?u.6144\nr : α → β → Prop\ns s₁ s₂ : Set α\nt t₁✝ t₂ t₁ : Set β\nclosure_fst✝¹ closure_fst✝ : intentClosure r (extentClosure r t₁, t₁).fst = (extentClosure r t₁, t₁).snd\n⊢ { toProd := (extentClosure r t₁, t₁), closure_fst := closure_fst✝¹,\n closure_snd := (_ : extentClosure r t₁ = extentClosure r t₁) } =\n { toProd := (extentClosure r t₁, t₁), closure_fst := closure_fst✝,\n closure_snd := (_ : extentClosure r t₁ = extentClosure r t₁) }",
"state_before": "case mk.mk.mk.mk\nι : Sort ?u.6130\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.6139\nκ : ι → Sort ?u.6144\nr : α → β → Prop\ns s₁✝ s₂✝ : Set α\nt t₁✝ t₂✝ : Set β\ns₁ : Set α\nt₁ : Set β\nclosure_fst✝¹ : intentClosure r (s₁, t₁).fst = (s₁, t₁).snd\nh₁ : extentClosure r t₁ = s₁\ns₂ : Set α\nt₂ : Set β\nclosure_fst✝ : intentClosure r (s₂, t₂).fst = (s₂, t₂).snd\nh₂ : extentClosure r t₂ = s₂\nh : t₁ = t₂\n⊢ { toProd := (s₁, t₁), closure_fst := closure_fst✝¹, closure_snd := h₁ } =\n { toProd := (s₂, t₂), closure_fst := closure_fst✝, closure_snd := h₂ }",
"tactic": "substs h h₁ h₂"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.mk.mk\nι : Sort ?u.6130\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.6139\nκ : ι → Sort ?u.6144\nr : α → β → Prop\ns s₁ s₂ : Set α\nt t₁✝ t₂ t₁ : Set β\nclosure_fst✝¹ closure_fst✝ : intentClosure r (extentClosure r t₁, t₁).fst = (extentClosure r t₁, t₁).snd\n⊢ { toProd := (extentClosure r t₁, t₁), closure_fst := closure_fst✝¹,\n closure_snd := (_ : extentClosure r t₁ = extentClosure r t₁) } =\n { toProd := (extentClosure r t₁, t₁), closure_fst := closure_fst✝,\n closure_snd := (_ : extentClosure r t₁ = extentClosure r t₁) }",
"tactic": "rfl"
}
] |
[
204,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.re_add_im
|
[] |
[
298,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/Algebra/Homology/Homotopy.lean
|
dNext_eq
|
[
{
"state_after": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nf : (i j : ι) → X C i ⟶ X D j\ni : ι\nw : ComplexShape.Rel c i (ComplexShape.next c i)\n⊢ ↑(dNext i) f = d C i (ComplexShape.next c i) ≫ f (ComplexShape.next c i) i",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nf : (i j : ι) → X C i ⟶ X D j\ni i' : ι\nw : ComplexShape.Rel c i i'\n⊢ ↑(dNext i) f = d C i i' ≫ f i' i",
"tactic": "obtain rfl := c.next_eq' w"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nf : (i j : ι) → X C i ⟶ X D j\ni : ι\nw : ComplexShape.Rel c i (ComplexShape.next c i)\n⊢ ↑(dNext i) f = d C i (ComplexShape.next c i) ≫ f (ComplexShape.next c i) i",
"tactic": "rfl"
}
] |
[
59,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Order/Lattice.lean
|
left_eq_inf
|
[] |
[
444,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.biUnion'
|
[
{
"state_after": "case intro\nα : Type u\nβ : Type v\nι✝ : Sort w\nγ : Type x\nι : Type u_1\ns : Set ι\nt : (i : ι) → i ∈ s → Set α\nht : ∀ (i : ι) (hi : i ∈ s), Set.Finite (t i hi)\na✝ : Fintype ↑s\n⊢ Set.Finite (⋃ (i : ι) (h : i ∈ s), t i h)",
"state_before": "α : Type u\nβ : Type v\nι✝ : Sort w\nγ : Type x\nι : Type u_1\ns : Set ι\nhs : Set.Finite s\nt : (i : ι) → i ∈ s → Set α\nht : ∀ (i : ι) (hi : i ∈ s), Set.Finite (t i hi)\n⊢ Set.Finite (⋃ (i : ι) (h : i ∈ s), t i h)",
"tactic": "cases hs"
},
{
"state_after": "case intro\nα : Type u\nβ : Type v\nι✝ : Sort w\nγ : Type x\nι : Type u_1\ns : Set ι\nt : (i : ι) → i ∈ s → Set α\nht : ∀ (i : ι) (hi : i ∈ s), Set.Finite (t i hi)\na✝ : Fintype ↑s\n⊢ Set.Finite (⋃ (x : ↑s), t ↑x (_ : ↑x ∈ s))",
"state_before": "case intro\nα : Type u\nβ : Type v\nι✝ : Sort w\nγ : Type x\nι : Type u_1\ns : Set ι\nt : (i : ι) → i ∈ s → Set α\nht : ∀ (i : ι) (hi : i ∈ s), Set.Finite (t i hi)\na✝ : Fintype ↑s\n⊢ Set.Finite (⋃ (i : ι) (h : i ∈ s), t i h)",
"tactic": "rw [biUnion_eq_iUnion]"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nβ : Type v\nι✝ : Sort w\nγ : Type x\nι : Type u_1\ns : Set ι\nt : (i : ι) → i ∈ s → Set α\nht : ∀ (i : ι) (hi : i ∈ s), Set.Finite (t i hi)\na✝ : Fintype ↑s\n⊢ Set.Finite (⋃ (x : ↑s), t ↑x (_ : ↑x ∈ s))",
"tactic": "apply finite_iUnion fun i : s => ht i.1 i.2"
}
] |
[
800,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
796,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.surj_on_of_inj_on_of_card_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : card t ≤ card s\n⊢ ∀ (b : β), b ∈ t → ∃ a ha, b = f a ha",
"tactic": "classical\n intro b hb\n have h : (s.attach.image fun a : { a // a ∈ s } => f a a.prop).card = s.card :=\n @card_attach _ s ▸\n card_image_of_injective _ fun ⟨a₁, ha₁⟩ ⟨a₂, ha₂⟩ h => Subtype.eq <| hinj _ _ _ _ h\n have h' : image (fun a : { a // a ∈ s } => f a a.prop) s.attach = t :=\n eq_of_subset_of_card_le\n (fun b h =>\n let ⟨a, _, ha₂⟩ := mem_image.1 h\n ha₂ ▸ hf _ _)\n (by simp [hst, h])\n rw [← h'] at hb\n obtain ⟨a, _, ha₂⟩ := mem_image.1 hb\n exact ⟨a, a.2, ha₂.symm⟩"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : card t ≤ card s\nb : β\nhb : b ∈ t\n⊢ ∃ a ha, b = f a ha",
"state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : card t ≤ card s\n⊢ ∀ (b : β), b ∈ t → ∃ a ha, b = f a ha",
"tactic": "intro b hb"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : card t ≤ card s\nb : β\nhb : b ∈ t\nh : card (image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s)) = card s\n⊢ ∃ a ha, b = f a ha",
"state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : card t ≤ card s\nb : β\nhb : b ∈ t\n⊢ ∃ a ha, b = f a ha",
"tactic": "have h : (s.attach.image fun a : { a // a ∈ s } => f a a.prop).card = s.card :=\n @card_attach _ s ▸\n card_image_of_injective _ fun ⟨a₁, ha₁⟩ ⟨a₂, ha₂⟩ h => Subtype.eq <| hinj _ _ _ _ h"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : card t ≤ card s\nb : β\nhb : b ∈ t\nh : card (image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s)) = card s\nh' : image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s) = t\n⊢ ∃ a ha, b = f a ha",
"state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : card t ≤ card s\nb : β\nhb : b ∈ t\nh : card (image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s)) = card s\n⊢ ∃ a ha, b = f a ha",
"tactic": "have h' : image (fun a : { a // a ∈ s } => f a a.prop) s.attach = t :=\n eq_of_subset_of_card_le\n (fun b h =>\n let ⟨a, _, ha₂⟩ := mem_image.1 h\n ha₂ ▸ hf _ _)\n (by simp [hst, h])"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : card t ≤ card s\nb : β\nhb : b ∈ image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s)\nh : card (image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s)) = card s\nh' : image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s) = t\n⊢ ∃ a ha, b = f a ha",
"state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : card t ≤ card s\nb : β\nhb : b ∈ t\nh : card (image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s)) = card s\nh' : image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s) = t\n⊢ ∃ a ha, b = f a ha",
"tactic": "rw [← h'] at hb"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : card t ≤ card s\nb : β\nhb : b ∈ image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s)\nh : card (image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s)) = card s\nh' : image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s) = t\na : { a // a ∈ s }\nleft✝ : a ∈ attach s\nha₂ : f ↑a (_ : ↑a ∈ s) = b\n⊢ ∃ a ha, b = f a ha",
"state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : card t ≤ card s\nb : β\nhb : b ∈ image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s)\nh : card (image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s)) = card s\nh' : image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s) = t\n⊢ ∃ a ha, b = f a ha",
"tactic": "obtain ⟨a, _, ha₂⟩ := mem_image.1 hb"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : card t ≤ card s\nb : β\nhb : b ∈ image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s)\nh : card (image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s)) = card s\nh' : image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s) = t\na : { a // a ∈ s }\nleft✝ : a ∈ attach s\nha₂ : f ↑a (_ : ↑a ∈ s) = b\n⊢ ∃ a ha, b = f a ha",
"tactic": "exact ⟨a, a.2, ha₂.symm⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : card t ≤ card s\nb : β\nhb : b ∈ t\nh : card (image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s)) = card s\n⊢ card t ≤ card (image (fun a => f ↑a (_ : ↑a ∈ s)) (attach s))",
"tactic": "simp [hst, h]"
}
] |
[
372,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
AddSubmonoid.smul_mem_pointwise_smul
|
[] |
[
377,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
376,
1
] |
Mathlib/Data/Part.lean
|
Part.map_id'
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.49680\nγ : Type ?u.49683\nf : α → α\nH : ∀ (x : α), f x = x\no : Part α\n⊢ map id o = o",
"state_before": "α : Type u_1\nβ : Type ?u.49680\nγ : Type ?u.49683\nf : α → α\nH : ∀ (x : α), f x = x\no : Part α\n⊢ map f o = o",
"tactic": "rw [show f = id from funext H]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.49680\nγ : Type ?u.49683\nf : α → α\nH : ∀ (x : α), f x = x\no : Part α\n⊢ map id o = o",
"tactic": "exact id_map o"
}
] |
[
593,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
592,
1
] |
Mathlib/Topology/Basic.lean
|
not_isOpen_singleton
|
[] |
[
1325,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1324,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.isDetecting_op_iff
|
[
{
"state_after": "case refine'_1\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsDetecting (Set.op 𝒢)\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : X ⟶ G), ∃! h', f ≫ h' = h\n⊢ IsIso f\n\ncase refine'_2\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsCodetecting 𝒢\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : ∀ (G : Cᵒᵖ), G ∈ Set.op 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\n⊢ IsIso f",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\n⊢ IsDetecting (Set.op 𝒢) ↔ IsCodetecting 𝒢",
"tactic": "refine' ⟨fun h𝒢 X Y f hf => _, fun h𝒢 X Y f hf => _⟩"
},
{
"state_after": "case refine'_1\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsDetecting (Set.op 𝒢)\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : X ⟶ G), ∃! h', f ≫ h' = h\nG : Cᵒᵖ\nhG : G ∈ Set.op 𝒢\nh : G ⟶ X.op\n⊢ ∃! h', h' ≫ f.op = h",
"state_before": "case refine'_1\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsDetecting (Set.op 𝒢)\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : X ⟶ G), ∃! h', f ≫ h' = h\n⊢ IsIso f",
"tactic": "refine' (isIso_op_iff _).1 (h𝒢 _ fun G hG h => _)"
},
{
"state_after": "case refine'_1.intro.intro\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsDetecting (Set.op 𝒢)\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : X ⟶ G), ∃! h', f ≫ h' = h\nG : Cᵒᵖ\nhG : G ∈ Set.op 𝒢\nh : G ⟶ X.op\nt : Y ⟶ G.unop\nht : f ≫ t = h.unop\nht' : ∀ (y : Y ⟶ G.unop), (fun h' => f ≫ h' = h.unop) y → y = t\n⊢ ∃! h', h' ≫ f.op = h",
"state_before": "case refine'_1\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsDetecting (Set.op 𝒢)\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : X ⟶ G), ∃! h', f ≫ h' = h\nG : Cᵒᵖ\nhG : G ∈ Set.op 𝒢\nh : G ⟶ X.op\n⊢ ∃! h', h' ≫ f.op = h",
"tactic": "obtain ⟨t, ht, ht'⟩ := hf (unop G) (Set.mem_op.1 hG) h.unop"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsDetecting (Set.op 𝒢)\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : X ⟶ G), ∃! h', f ≫ h' = h\nG : Cᵒᵖ\nhG : G ∈ Set.op 𝒢\nh : G ⟶ X.op\nt : Y ⟶ G.unop\nht : f ≫ t = h.unop\nht' : ∀ (y : Y ⟶ G.unop), (fun h' => f ≫ h' = h.unop) y → y = t\n⊢ ∃! h', h' ≫ f.op = h",
"tactic": "exact\n ⟨t.op, Quiver.Hom.unop_inj ht, fun y hy => Quiver.Hom.unop_inj (ht' _ (Quiver.Hom.op_inj hy))⟩"
},
{
"state_after": "case refine'_2\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsCodetecting 𝒢\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : ∀ (G : Cᵒᵖ), G ∈ Set.op 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nG : C\nhG : G ∈ 𝒢\nh : Y.unop ⟶ G\n⊢ ∃! h', f.unop ≫ h' = h",
"state_before": "case refine'_2\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsCodetecting 𝒢\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : ∀ (G : Cᵒᵖ), G ∈ Set.op 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\n⊢ IsIso f",
"tactic": "refine' (isIso_unop_iff _).1 (h𝒢 _ fun G hG h => _)"
},
{
"state_after": "case refine'_2.intro.intro\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsCodetecting 𝒢\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : ∀ (G : Cᵒᵖ), G ∈ Set.op 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nG : C\nhG : G ∈ 𝒢\nh : Y.unop ⟶ G\nt : G.op ⟶ X\nht : t ≫ f = h.op\nht' : ∀ (y : G.op ⟶ X), (fun h' => h' ≫ f = h.op) y → y = t\n⊢ ∃! h', f.unop ≫ h' = h",
"state_before": "case refine'_2\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsCodetecting 𝒢\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : ∀ (G : Cᵒᵖ), G ∈ Set.op 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nG : C\nhG : G ∈ 𝒢\nh : Y.unop ⟶ G\n⊢ ∃! h', f.unop ≫ h' = h",
"tactic": "obtain ⟨t, ht, ht'⟩ := hf (op G) (Set.op_mem_op.2 hG) h.op"
},
{
"state_after": "case refine'_2.intro.intro\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsCodetecting 𝒢\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : ∀ (G : Cᵒᵖ), G ∈ Set.op 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nG : C\nhG : G ∈ 𝒢\nh : Y.unop ⟶ G\nt : G.op ⟶ X\nht : t ≫ f = h.op\nht' : ∀ (y : G.op ⟶ X), (fun h' => h' ≫ f = h.op) y → y = t\ny : X.unop ⟶ G\nhy : (fun h' => f.unop ≫ h' = h) y\n⊢ (fun h' => h' ≫ f = h.op) y.op",
"state_before": "case refine'_2.intro.intro\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsCodetecting 𝒢\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : ∀ (G : Cᵒᵖ), G ∈ Set.op 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nG : C\nhG : G ∈ 𝒢\nh : Y.unop ⟶ G\nt : G.op ⟶ X\nht : t ≫ f = h.op\nht' : ∀ (y : G.op ⟶ X), (fun h' => h' ≫ f = h.op) y → y = t\n⊢ ∃! h', f.unop ≫ h' = h",
"tactic": "refine' ⟨t.unop, Quiver.Hom.op_inj ht, fun y hy => Quiver.Hom.op_inj (ht' _ _)⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsCodetecting 𝒢\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : ∀ (G : Cᵒᵖ), G ∈ Set.op 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nG : C\nhG : G ∈ 𝒢\nh : Y.unop ⟶ G\nt : G.op ⟶ X\nht : t ≫ f = h.op\nht' : ∀ (y : G.op ⟶ X), (fun h' => h' ≫ f = h.op) y → y = t\ny : X.unop ⟶ G\nhy : (fun h' => f.unop ≫ h' = h) y\n⊢ (fun h' => h' ≫ f = h.op) y.op",
"tactic": "exact Quiver.Hom.unop_inj (by simpa only using hy)"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\n𝒢 : Set C\nh𝒢 : IsCodetecting 𝒢\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : ∀ (G : Cᵒᵖ), G ∈ Set.op 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nG : C\nhG : G ∈ 𝒢\nh : Y.unop ⟶ G\nt : G.op ⟶ X\nht : t ≫ f = h.op\nht' : ∀ (y : G.op ⟶ X), (fun h' => h' ≫ f = h.op) y → y = t\ny : X.unop ⟶ G\nhy : (fun h' => f.unop ≫ h' = h) y\n⊢ (y.op ≫ f).unop = h.op.unop",
"tactic": "simpa only using hy"
}
] |
[
128,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.ne_zero_iff
|
[
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q p : MvPolynomial σ R\n⊢ (¬∀ (d : σ →₀ ℕ), coeff d p = 0) ↔ ∃ d, coeff d p ≠ 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q p : MvPolynomial σ R\n⊢ p ≠ 0 ↔ ∃ d, coeff d p ≠ 0",
"tactic": "rw [Ne.def, eq_zero_iff]"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q p : MvPolynomial σ R\n⊢ (∃ d, coeff d p ≠ 0) ↔ ∃ d, coeff d p ≠ 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q p : MvPolynomial σ R\n⊢ (¬∀ (d : σ →₀ ℕ), coeff d p = 0) ↔ ∃ d, coeff d p ≠ 0",
"tactic": "push_neg"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q p : MvPolynomial σ R\n⊢ (∃ d, coeff d p ≠ 0) ↔ ∃ d, coeff d p ≠ 0",
"tactic": "rfl"
}
] |
[
815,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
812,
1
] |
Mathlib/Algebra/BigOperators/Fin.lean
|
Fin.prod_univ_succ
|
[] |
[
85,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Combinatorics/Catalan.lean
|
catalan_eq_centralBinom_div
|
[
{
"state_after": "n : ℕ\n⊢ ↑(catalan n) = ↑(Nat.centralBinom n) / (↑n + 1)",
"state_before": "n : ℕ\n⊢ catalan n = Nat.centralBinom n / (n + 1)",
"tactic": "suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by\n have h := Nat.succ_dvd_centralBinom n\n exact_mod_cast this"
},
{
"state_after": "case hz\n\n⊢ ↑(catalan 0) = ↑(Nat.centralBinom 0) / (↑0 + 1)\n\ncase hi\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ↑(catalan (Nat.succ d)) = ↑(Nat.centralBinom (Nat.succ d)) / (↑(Nat.succ d) + 1)",
"state_before": "n : ℕ\n⊢ ↑(catalan n) = ↑(Nat.centralBinom n) / (↑n + 1)",
"tactic": "induction' n using Nat.case_strong_induction_on with d hd"
},
{
"state_after": "n : ℕ\nthis : ↑(catalan n) = ↑(Nat.centralBinom n) / (↑n + 1)\nh : n + 1 ∣ Nat.centralBinom n\n⊢ catalan n = Nat.centralBinom n / (n + 1)",
"state_before": "n : ℕ\nthis : ↑(catalan n) = ↑(Nat.centralBinom n) / (↑n + 1)\n⊢ catalan n = Nat.centralBinom n / (n + 1)",
"tactic": "have h := Nat.succ_dvd_centralBinom n"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nthis : ↑(catalan n) = ↑(Nat.centralBinom n) / (↑n + 1)\nh : n + 1 ∣ Nat.centralBinom n\n⊢ catalan n = Nat.centralBinom n / (n + 1)",
"tactic": "exact_mod_cast this"
},
{
"state_after": "no goals",
"state_before": "case hz\n\n⊢ ↑(catalan 0) = ↑(Nat.centralBinom 0) / (↑0 + 1)",
"tactic": "simp"
},
{
"state_after": "case hi\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ∑ x : Fin (Nat.succ d), ↑(catalan ↑x) * ↑(catalan (d - ↑x)) = ↑(Nat.centralBinom (Nat.succ d)) / (↑(Nat.succ d) + 1)",
"state_before": "case hi\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ↑(catalan (Nat.succ d)) = ↑(Nat.centralBinom (Nat.succ d)) / (↑(Nat.succ d) + 1)",
"tactic": "simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]"
},
{
"state_after": "d : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ∑ x : Fin (Nat.succ d), ↑(catalan ↑x) * ↑(catalan (d - ↑x)) =\n ∑ i : Fin (Nat.succ d), ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1))\n\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ∑ i : Fin (Nat.succ d), ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1)) =\n ↑(Nat.centralBinom (Nat.succ d)) / (↑(Nat.succ d) + 1)",
"state_before": "case hi\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ∑ x : Fin (Nat.succ d), ↑(catalan ↑x) * ↑(catalan (d - ↑x)) = ↑(Nat.centralBinom (Nat.succ d)) / (↑(Nat.succ d) + 1)",
"tactic": "trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *\n (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)"
},
{
"state_after": "case e_f\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ (fun x => ↑(catalan ↑x) * ↑(catalan (d - ↑x))) = fun i =>\n ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1))",
"state_before": "d : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ∑ x : Fin (Nat.succ d), ↑(catalan ↑x) * ↑(catalan (d - ↑x)) =\n ∑ i : Fin (Nat.succ d), ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1))",
"tactic": "congr"
},
{
"state_after": "case e_f.h\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\nx : Fin (Nat.succ d)\n⊢ ↑(catalan ↑x) * ↑(catalan (d - ↑x)) =\n ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑d - ↑↑x + 1))",
"state_before": "case e_f\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ (fun x => ↑(catalan ↑x) * ↑(catalan (d - ↑x))) = fun i =>\n ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1))",
"tactic": "ext1 x"
},
{
"state_after": "case e_f.h\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\nx : Fin (Nat.succ d)\nm_le_d : ↑x ≤ d\n⊢ ↑(catalan ↑x) * ↑(catalan (d - ↑x)) =\n ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑d - ↑↑x + 1))",
"state_before": "case e_f.h\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\nx : Fin (Nat.succ d)\n⊢ ↑(catalan ↑x) * ↑(catalan (d - ↑x)) =\n ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑d - ↑↑x + 1))",
"tactic": "have m_le_d : x.val ≤ d := by apply Nat.le_of_lt_succ; apply x.2"
},
{
"state_after": "case e_f.h\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\nx : Fin (Nat.succ d)\nm_le_d : ↑x ≤ d\nd_minus_x_le_d : d - ↑x ≤ d\n⊢ ↑(catalan ↑x) * ↑(catalan (d - ↑x)) =\n ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑d - ↑↑x + 1))",
"state_before": "case e_f.h\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\nx : Fin (Nat.succ d)\nm_le_d : ↑x ≤ d\n⊢ ↑(catalan ↑x) * ↑(catalan (d - ↑x)) =\n ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑d - ↑↑x + 1))",
"tactic": "have d_minus_x_le_d : (d - x.val) ≤ d := tsub_le_self"
},
{
"state_after": "case e_f.h\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\nx : Fin (Nat.succ d)\nm_le_d : ↑x ≤ d\nd_minus_x_le_d : d - ↑x ≤ d\n⊢ ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑(d - ↑x) + 1)) =\n ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑d - ↑↑x + 1))",
"state_before": "case e_f.h\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\nx : Fin (Nat.succ d)\nm_le_d : ↑x ≤ d\nd_minus_x_le_d : d - ↑x ≤ d\n⊢ ↑(catalan ↑x) * ↑(catalan (d - ↑x)) =\n ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑d - ↑↑x + 1))",
"tactic": "rw [hd _ m_le_d, hd _ d_minus_x_le_d]"
},
{
"state_after": "no goals",
"state_before": "case e_f.h\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\nx : Fin (Nat.succ d)\nm_le_d : ↑x ≤ d\nd_minus_x_le_d : d - ↑x ≤ d\n⊢ ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑(d - ↑x) + 1)) =\n ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑d - ↑↑x + 1))",
"tactic": "norm_cast"
},
{
"state_after": "case a\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\nx : Fin (Nat.succ d)\n⊢ ↑x < Nat.succ d",
"state_before": "d : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\nx : Fin (Nat.succ d)\n⊢ ↑x ≤ d",
"tactic": "apply Nat.le_of_lt_succ"
},
{
"state_after": "no goals",
"state_before": "case a\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\nx : Fin (Nat.succ d)\n⊢ ↑x < Nat.succ d",
"tactic": "apply x.2"
},
{
"state_after": "d : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ∑ i : Fin (Nat.succ d), ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1)) =\n ∑ i : Fin (Nat.succ d), (gosperCatalan (d + 1) (↑i + 1) - gosperCatalan (d + 1) ↑i)\n\nd : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ∑ i : Fin (Nat.succ d), (gosperCatalan (d + 1) (↑i + 1) - gosperCatalan (d + 1) ↑i) =\n ↑(Nat.centralBinom (Nat.succ d)) / (↑(Nat.succ d) + 1)",
"state_before": "d : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ∑ i : Fin (Nat.succ d), ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1)) =\n ↑(Nat.centralBinom (Nat.succ d)) / (↑(Nat.succ d) + 1)",
"tactic": "trans (∑ i : Fin d.succ, (gosperCatalan (d + 1) (i + 1) - gosperCatalan (d + 1) i))"
},
{
"state_after": "d : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\ni : Fin (Nat.succ d)\nx✝ : i ∈ univ\n⊢ ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1)) =\n gosperCatalan (d + 1) (↑i + 1) - gosperCatalan (d + 1) ↑i",
"state_before": "d : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ∑ i : Fin (Nat.succ d), ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1)) =\n ∑ i : Fin (Nat.succ d), (gosperCatalan (d + 1) (↑i + 1) - gosperCatalan (d + 1) ↑i)",
"tactic": "refine' sum_congr rfl fun i _ => _"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\ni : Fin (Nat.succ d)\nx✝ : i ∈ univ\n⊢ ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1)) =\n gosperCatalan (d + 1) (↑i + 1) - gosperCatalan (d + 1) ↑i",
"tactic": "rw [gosper_trick i.is_le, mul_div]"
},
{
"state_after": "d : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ gosperCatalan (d + 1) (d + 1) - gosperCatalan (d + 1) 0 = ↑(Nat.centralBinom (d + 1)) / (↑(d + 1) + 1)",
"state_before": "d : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ∑ i : Fin (Nat.succ d), (gosperCatalan (d + 1) (↑i + 1) - gosperCatalan (d + 1) ↑i) =\n ↑(Nat.centralBinom (Nat.succ d)) / (↑(Nat.succ d) + 1)",
"tactic": "rw [← sum_range fun i => gosperCatalan (d + 1) (i + 1) - gosperCatalan (d + 1) i,\n sum_range_sub, Nat.succ_eq_add_one]"
},
{
"state_after": "d : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ↑(Nat.centralBinom (d + 1)) / (↑d + 2) = ↑(Nat.centralBinom (d + 1)) / (↑(d + 1) + 1)",
"state_before": "d : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ gosperCatalan (d + 1) (d + 1) - gosperCatalan (d + 1) 0 = ↑(Nat.centralBinom (d + 1)) / (↑(d + 1) + 1)",
"tactic": "rw [gosper_catalan_sub_eq_central_binom_div d]"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\nhd : ∀ (m : ℕ), m ≤ d → ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)\n⊢ ↑(Nat.centralBinom (d + 1)) / (↑d + 2) = ↑(Nat.centralBinom (d + 1)) / (↑(d + 1) + 1)",
"tactic": "norm_cast"
}
] |
[
141,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
round_one
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.250745\nα : Type u_1\nβ : Type ?u.250751\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\n⊢ round 1 = 1",
"tactic": "simp [round]"
}
] |
[
1343,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1343,
1
] |
Mathlib/Algebra/Group/Units.lean
|
Units.inv_mk
|
[] |
[
244,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
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