file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Topology/ContinuousFunction/Basic.lean
|
ContinuousMap.liftCover_restrict'
|
[] |
[
450,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Analysis/Convex/Join.lean
|
convexJoin_singleton_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.27529\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns✝ t s₁ s₂ t₁ t₂ u : Set E\nx y✝ : E\ns : Set E\ny : E\n⊢ convexJoin 𝕜 s {y} = ⋃ (x : E) (_ : x ∈ s), segment 𝕜 x y",
"tactic": "simp [convexJoin]"
}
] |
[
76,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
cauchy_map_iff'
|
[] |
[
74,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.kroneckerMap_map
|
[] |
[
89,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/FieldTheory/Normal.lean
|
AlgEquiv.restrictNormalHom_surjective
|
[] |
[
458,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
456,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean
|
PadicInt.coe_neg
|
[] |
[
133,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Topology/Basic.lean
|
eventually_nhds_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np✝ p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\np : α → Prop\n⊢ (∃ t, t ⊆ {x | (fun x => p x) x} ∧ IsOpen t ∧ a ∈ t) ↔ ∃ t, (∀ (x : α), x ∈ t → p x) ∧ IsOpen t ∧ a ∈ t",
"tactic": "simp only [subset_def, exists_prop, mem_setOf_eq]"
}
] |
[
894,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
892,
1
] |
Mathlib/Algebra/Associated.lean
|
Irreducible.not_square
|
[
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.95275\nγ : Type ?u.95278\nδ : Type ?u.95281\ninst✝ : CommMonoid α\nb : α\nha : Irreducible (b * b)\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.95275\nγ : Type ?u.95278\nδ : Type ?u.95281\ninst✝ : CommMonoid α\na : α\nha : Irreducible a\n⊢ ¬IsSquare a",
"tactic": "rintro ⟨b, rfl⟩"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.95275\nγ : Type ?u.95278\nδ : Type ?u.95281\ninst✝ : CommMonoid α\nb : α\nha : Irreducible b ∧ IsUnit b\n⊢ False",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.95275\nγ : Type ?u.95278\nδ : Type ?u.95281\ninst✝ : CommMonoid α\nb : α\nha : Irreducible (b * b)\n⊢ False",
"tactic": "simp only [irreducible_mul_iff, or_self_iff] at ha"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.95275\nγ : Type ?u.95278\nδ : Type ?u.95281\ninst✝ : CommMonoid α\nb : α\nha : Irreducible b ∧ IsUnit b\n⊢ False",
"tactic": "exact ha.1.not_unit ha.2"
}
] |
[
302,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/Topology/UniformSpace/Equicontinuity.lean
|
equicontinuousAt_iff_pair
|
[
{
"state_after": "case mp\nι : Type u_1\nκ : Type ?u.26882\nX : Type u_2\nY : Type ?u.26888\nZ : Type ?u.26891\nα : Type u_3\nβ : Type ?u.26897\nγ : Type ?u.26900\n𝓕 : Type ?u.26903\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nF : ι → X → α\nx₀ : X\nH : EquicontinuousAt F x₀\nU : Set (α × α)\nhU : U ∈ 𝓤 α\n⊢ ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : X), x ∈ V → ∀ (y : X), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U\n\ncase mpr\nι : Type u_1\nκ : Type ?u.26882\nX : Type u_2\nY : Type ?u.26888\nZ : Type ?u.26891\nα : Type u_3\nβ : Type ?u.26897\nγ : Type ?u.26900\n𝓕 : Type ?u.26903\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nF : ι → X → α\nx₀ : X\nH : ∀ (U : Set (α × α)), U ∈ 𝓤 α → ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : X), x ∈ V → ∀ (y : X), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U\nU : Set (α × α)\nhU : U ∈ 𝓤 α\n⊢ ∀ᶠ (x : X) in 𝓝 x₀, ∀ (i : ι), (F i x₀, F i x) ∈ U",
"state_before": "ι : Type u_1\nκ : Type ?u.26882\nX : Type u_2\nY : Type ?u.26888\nZ : Type ?u.26891\nα : Type u_3\nβ : Type ?u.26897\nγ : Type ?u.26900\n𝓕 : Type ?u.26903\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nF : ι → X → α\nx₀ : X\n⊢ EquicontinuousAt F x₀ ↔\n ∀ (U : Set (α × α)), U ∈ 𝓤 α → ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : X), x ∈ V → ∀ (y : X), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U",
"tactic": "constructor <;> intro H U hU"
},
{
"state_after": "case mp.intro.intro.intro\nι : Type u_1\nκ : Type ?u.26882\nX : Type u_2\nY : Type ?u.26888\nZ : Type ?u.26891\nα : Type u_3\nβ : Type ?u.26897\nγ : Type ?u.26900\n𝓕 : Type ?u.26903\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nF : ι → X → α\nx₀ : X\nH : EquicontinuousAt F x₀\nU : Set (α × α)\nhU : U ∈ 𝓤 α\nV : Set (α × α)\nhV : V ∈ 𝓤 α\nhVsymm : SymmetricRel V\nhVU : V ○ V ⊆ U\n⊢ ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : X), x ∈ V → ∀ (y : X), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U",
"state_before": "case mp\nι : Type u_1\nκ : Type ?u.26882\nX : Type u_2\nY : Type ?u.26888\nZ : Type ?u.26891\nα : Type u_3\nβ : Type ?u.26897\nγ : Type ?u.26900\n𝓕 : Type ?u.26903\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nF : ι → X → α\nx₀ : X\nH : EquicontinuousAt F x₀\nU : Set (α × α)\nhU : U ∈ 𝓤 α\n⊢ ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : X), x ∈ V → ∀ (y : X), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U",
"tactic": "rcases comp_symm_mem_uniformity_sets hU with ⟨V, hV, hVsymm, hVU⟩"
},
{
"state_after": "case mp.intro.intro.intro\nι : Type u_1\nκ : Type ?u.26882\nX : Type u_2\nY : Type ?u.26888\nZ : Type ?u.26891\nα : Type u_3\nβ : Type ?u.26897\nγ : Type ?u.26900\n𝓕 : Type ?u.26903\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nF : ι → X → α\nx₀ : X\nH : EquicontinuousAt F x₀\nU : Set (α × α)\nhU : U ∈ 𝓤 α\nV : Set (α × α)\nhV : V ∈ 𝓤 α\nhVsymm : SymmetricRel V\nhVU : V ○ V ⊆ U\nx : X\nhx : x ∈ {x | (fun x => ∀ (i : ι), (F i x₀, F i x) ∈ V) x}\ny : X\nhy : y ∈ {x | (fun x => ∀ (i : ι), (F i x₀, F i x) ∈ V) x}\ni : ι\n⊢ (F i x, F i x₀) ∈ V",
"state_before": "case mp.intro.intro.intro\nι : Type u_1\nκ : Type ?u.26882\nX : Type u_2\nY : Type ?u.26888\nZ : Type ?u.26891\nα : Type u_3\nβ : Type ?u.26897\nγ : Type ?u.26900\n𝓕 : Type ?u.26903\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nF : ι → X → α\nx₀ : X\nH : EquicontinuousAt F x₀\nU : Set (α × α)\nhU : U ∈ 𝓤 α\nV : Set (α × α)\nhV : V ∈ 𝓤 α\nhVsymm : SymmetricRel V\nhVU : V ○ V ⊆ U\n⊢ ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : X), x ∈ V → ∀ (y : X), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U",
"tactic": "refine' ⟨_, H V hV, fun x hx y hy i => hVU (prod_mk_mem_compRel _ (hy i))⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro\nι : Type u_1\nκ : Type ?u.26882\nX : Type u_2\nY : Type ?u.26888\nZ : Type ?u.26891\nα : Type u_3\nβ : Type ?u.26897\nγ : Type ?u.26900\n𝓕 : Type ?u.26903\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nF : ι → X → α\nx₀ : X\nH : EquicontinuousAt F x₀\nU : Set (α × α)\nhU : U ∈ 𝓤 α\nV : Set (α × α)\nhV : V ∈ 𝓤 α\nhVsymm : SymmetricRel V\nhVU : V ○ V ⊆ U\nx : X\nhx : x ∈ {x | (fun x => ∀ (i : ι), (F i x₀, F i x) ∈ V) x}\ny : X\nhy : y ∈ {x | (fun x => ∀ (i : ι), (F i x₀, F i x) ∈ V) x}\ni : ι\n⊢ (F i x, F i x₀) ∈ V",
"tactic": "exact hVsymm.mk_mem_comm.mp (hx i)"
},
{
"state_after": "case mpr.intro.intro\nι : Type u_1\nκ : Type ?u.26882\nX : Type u_2\nY : Type ?u.26888\nZ : Type ?u.26891\nα : Type u_3\nβ : Type ?u.26897\nγ : Type ?u.26900\n𝓕 : Type ?u.26903\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nF : ι → X → α\nx₀ : X\nH : ∀ (U : Set (α × α)), U ∈ 𝓤 α → ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : X), x ∈ V → ∀ (y : X), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U\nU : Set (α × α)\nhU : U ∈ 𝓤 α\nV : Set X\nhV : V ∈ 𝓝 x₀\nhVU : ∀ (x : X), x ∈ V → ∀ (y : X), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U\n⊢ ∀ᶠ (x : X) in 𝓝 x₀, ∀ (i : ι), (F i x₀, F i x) ∈ U",
"state_before": "case mpr\nι : Type u_1\nκ : Type ?u.26882\nX : Type u_2\nY : Type ?u.26888\nZ : Type ?u.26891\nα : Type u_3\nβ : Type ?u.26897\nγ : Type ?u.26900\n𝓕 : Type ?u.26903\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nF : ι → X → α\nx₀ : X\nH : ∀ (U : Set (α × α)), U ∈ 𝓤 α → ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : X), x ∈ V → ∀ (y : X), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U\nU : Set (α × α)\nhU : U ∈ 𝓤 α\n⊢ ∀ᶠ (x : X) in 𝓝 x₀, ∀ (i : ι), (F i x₀, F i x) ∈ U",
"tactic": "rcases H U hU with ⟨V, hV, hVU⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nι : Type u_1\nκ : Type ?u.26882\nX : Type u_2\nY : Type ?u.26888\nZ : Type ?u.26891\nα : Type u_3\nβ : Type ?u.26897\nγ : Type ?u.26900\n𝓕 : Type ?u.26903\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nF : ι → X → α\nx₀ : X\nH : ∀ (U : Set (α × α)), U ∈ 𝓤 α → ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : X), x ∈ V → ∀ (y : X), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U\nU : Set (α × α)\nhU : U ∈ 𝓤 α\nV : Set X\nhV : V ∈ 𝓝 x₀\nhVU : ∀ (x : X), x ∈ V → ∀ (y : X), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U\n⊢ ∀ᶠ (x : X) in 𝓝 x₀, ∀ (i : ι), (F i x₀, F i x) ∈ U",
"tactic": "filter_upwards [hV]using fun x hx i => hVU x₀ (mem_of_mem_nhds hV) x hx i"
}
] |
[
142,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.liftNC_mul
|
[
{
"state_after": "k : Type u₁\nG : Type u₂\nR : Type u_2\ninst✝³ : Semiring k\ninst✝² : Mul G\ninst✝¹ : Semiring R\ng_hom : Type u_1\ninst✝ : MulHomClass g_hom G R\nf : k →+* R\ng : g_hom\na b : MonoidAlgebra k G\nh_comm : ∀ {x y : G}, y ∈ a.support → Commute (↑f (↑b x)) (↑g y)\n⊢ ↑(liftNC ↑f ↑g) (a * b) = ↑(liftNC ↑f ↑g) (sum a single) * ↑(liftNC ↑f ↑g) (sum b single)",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type u_2\ninst✝³ : Semiring k\ninst✝² : Mul G\ninst✝¹ : Semiring R\ng_hom : Type u_1\ninst✝ : MulHomClass g_hom G R\nf : k →+* R\ng : g_hom\na b : MonoidAlgebra k G\nh_comm : ∀ {x y : G}, y ∈ a.support → Commute (↑f (↑b x)) (↑g y)\n⊢ ↑(liftNC ↑f ↑g) (a * b) = ↑(liftNC ↑f ↑g) a * ↑(liftNC ↑f ↑g) b",
"tactic": "conv_rhs => rw [← sum_single a, ← sum_single b]"
},
{
"state_after": "k : Type u₁\nG : Type u₂\nR : Type u_2\ninst✝³ : Semiring k\ninst✝² : Mul G\ninst✝¹ : Semiring R\ng_hom : Type u_1\ninst✝ : MulHomClass g_hom G R\nf : k →+* R\ng : g_hom\na b : MonoidAlgebra k G\nh_comm : ∀ {x y : G}, y ∈ a.support → Commute (↑f (↑b x)) (↑g y)\n⊢ (sum a fun a b_1 => sum b fun a_1 b => ↑↑f (b_1 * b) * ↑g (a * a_1)) =\n sum a fun a c => sum b fun a_1 c_1 => ↑↑f c * ↑g a * (↑↑f c_1 * ↑g a_1)",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type u_2\ninst✝³ : Semiring k\ninst✝² : Mul G\ninst✝¹ : Semiring R\ng_hom : Type u_1\ninst✝ : MulHomClass g_hom G R\nf : k →+* R\ng : g_hom\na b : MonoidAlgebra k G\nh_comm : ∀ {x y : G}, y ∈ a.support → Commute (↑f (↑b x)) (↑g y)\n⊢ ↑(liftNC ↑f ↑g) (a * b) = ↑(liftNC ↑f ↑g) (sum a single) * ↑(liftNC ↑f ↑g) (sum b single)",
"tactic": "simp_rw [mul_def, map_finsupp_sum, liftNC_single, Finsupp.sum_mul, Finsupp.mul_sum]"
},
{
"state_after": "k : Type u₁\nG : Type u₂\nR : Type u_2\ninst✝³ : Semiring k\ninst✝² : Mul G\ninst✝¹ : Semiring R\ng_hom : Type u_1\ninst✝ : MulHomClass g_hom G R\nf : k →+* R\ng : g_hom\na b : MonoidAlgebra k G\nh_comm : ∀ {x y : G}, y ∈ a.support → Commute (↑f (↑b x)) (↑g y)\ny : G\nhy : y ∈ a.support\nx : G\n_hx : x ∈ b.support\n⊢ (fun a_1 b => ↑↑f (↑a y * b) * ↑g (y * a_1)) x (↑b x) = (fun a_1 c => ↑↑f (↑a y) * ↑g y * (↑↑f c * ↑g a_1)) x (↑b x)",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type u_2\ninst✝³ : Semiring k\ninst✝² : Mul G\ninst✝¹ : Semiring R\ng_hom : Type u_1\ninst✝ : MulHomClass g_hom G R\nf : k →+* R\ng : g_hom\na b : MonoidAlgebra k G\nh_comm : ∀ {x y : G}, y ∈ a.support → Commute (↑f (↑b x)) (↑g y)\n⊢ (sum a fun a b_1 => sum b fun a_1 b => ↑↑f (b_1 * b) * ↑g (a * a_1)) =\n sum a fun a c => sum b fun a_1 c_1 => ↑↑f c * ↑g a * (↑↑f c_1 * ↑g a_1)",
"tactic": "refine Finset.sum_congr rfl fun y hy => Finset.sum_congr rfl fun x _hx => ?_"
},
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type u_2\ninst✝³ : Semiring k\ninst✝² : Mul G\ninst✝¹ : Semiring R\ng_hom : Type u_1\ninst✝ : MulHomClass g_hom G R\nf : k →+* R\ng : g_hom\na b : MonoidAlgebra k G\nh_comm : ∀ {x y : G}, y ∈ a.support → Commute (↑f (↑b x)) (↑g y)\ny : G\nhy : y ∈ a.support\nx : G\n_hx : x ∈ b.support\n⊢ (fun a_1 b => ↑↑f (↑a y * b) * ↑g (y * a_1)) x (↑b x) = (fun a_1 c => ↑↑f (↑a y) * ↑g y * (↑↑f c * ↑g a_1)) x (↑b x)",
"tactic": "simp [mul_assoc, (h_comm hy).left_comm]"
}
] |
[
204,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
WeakBilin.coeFn_continuous
|
[] |
[
121,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
Subtype.irreducibleSpace
|
[] |
[
1891,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1888,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
tendsto_self_mul_const_pow_of_abs_lt_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.129259\nβ : Type ?u.129262\nι : Type ?u.129265\nr : ℝ\nhr : abs r < 1\n⊢ Tendsto (fun n => ↑n * r ^ n) atTop (𝓝 0)",
"tactic": "simpa only [pow_one] using tendsto_pow_const_mul_const_pow_of_abs_lt_one 1 hr"
}
] |
[
257,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.mem_sup_left
|
[] |
[
636,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
635,
1
] |
Mathlib/Algebra/Tropical/Basic.lean
|
Tropical.trop_untrop
|
[] |
[
97,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/MeasureTheory/MeasurableSpaceDef.lean
|
MeasurableSet.biInter
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.7316\nδ : Type ?u.7319\nδ' : Type ?u.7322\nι : Sort ?u.7325\ns✝ t u : Set α\nm : MeasurableSpace α\nf : β → Set α\ns : Set β\nhs : Set.Countable s\nh : ∀ (b : β), b ∈ s → MeasurableSet (f b)\n⊢ MeasurableSet (⋃ (i : β) (_ : i ∈ s), f iᶜ)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.7316\nδ : Type ?u.7319\nδ' : Type ?u.7322\nι : Sort ?u.7325\ns✝ t u : Set α\nm : MeasurableSpace α\nf : β → Set α\ns : Set β\nhs : Set.Countable s\nh : ∀ (b : β), b ∈ s → MeasurableSet (f b)\n⊢ MeasurableSet ((⋂ (b : β) (_ : b ∈ s), f b)ᶜ)",
"tactic": "rw [compl_iInter₂]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.7316\nδ : Type ?u.7319\nδ' : Type ?u.7322\nι : Sort ?u.7325\ns✝ t u : Set α\nm : MeasurableSpace α\nf : β → Set α\ns : Set β\nhs : Set.Countable s\nh : ∀ (b : β), b ∈ s → MeasurableSet (f b)\n⊢ MeasurableSet (⋃ (i : β) (_ : i ∈ s), f iᶜ)",
"tactic": "exact .biUnion hs fun b hb => (h b hb).compl"
}
] |
[
169,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean
|
Set.infsep_nonneg
|
[] |
[
346,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.restrict_compl_add_restrict
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.298946\nγ : Type ?u.298949\nδ : Type ?u.298952\nι : Type ?u.298955\nR : Type ?u.298958\nR' : Type ?u.298961\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nhs : MeasurableSet s\n⊢ restrict μ (sᶜ) + restrict μ s = μ",
"tactic": "rw [add_comm, restrict_add_restrict_compl hs]"
}
] |
[
1746,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1745,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.C_0
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ ↑C 0 = 0",
"tactic": "simp"
}
] |
[
499,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
499,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.coe_embedding
|
[] |
[
319,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/Data/Real/Irrational.lean
|
irrational_mul_rat_iff
|
[
{
"state_after": "no goals",
"state_before": "q : ℚ\nm : ℤ\nn : ℕ\nx : ℝ\n⊢ Irrational (x * ↑q) ↔ q ≠ 0 ∧ Irrational x",
"tactic": "rw [mul_comm, irrational_rat_mul_iff]"
}
] |
[
609,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
608,
1
] |
Mathlib/Analysis/Convex/Hull.lean
|
convexHull_neg
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.48966\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ s : Set E\n⊢ ↑(convexHull 𝕜).toOrderHom (Neg.neg '' s) = Neg.neg '' ↑(convexHull 𝕜).toOrderHom s",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.48966\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ s : Set E\n⊢ ↑(convexHull 𝕜).toOrderHom (-s) = -↑(convexHull 𝕜).toOrderHom s",
"tactic": "simp_rw [← image_neg]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.48966\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ s : Set E\n⊢ ↑(convexHull 𝕜).toOrderHom (Neg.neg '' s) = Neg.neg '' ↑(convexHull 𝕜).toOrderHom s",
"tactic": "exact (AffineMap.image_convexHull _ <| -1).symm"
}
] |
[
230,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sSup_image
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nβ₂ : Type ?u.111484\nγ : Type ?u.111487\nι : Sort ?u.111490\nι' : Sort ?u.111493\nκ : ι → Sort ?u.111498\nκ' : ι' → Sort ?u.111503\ninst✝ : CompleteLattice α\nf✝ g s✝ t : ι → α\na b : α\ns : Set β\nf : β → α\n⊢ sSup (f '' s) = ⨆ (a : β) (_ : a ∈ s), f a",
"tactic": "rw [← iSup_subtype'', sSup_image']"
}
] |
[
1392,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1391,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.rel_cons_left
|
[
{
"state_after": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nas : Multiset α\nbs : Multiset β\n⊢ Rel r (a ::ₘ as) bs → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'\n\ncase mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nas : Multiset α\nbs : Multiset β\n⊢ (∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs') → Rel r (a ::ₘ as) bs",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nas : Multiset α\nbs : Multiset β\n⊢ Rel r (a ::ₘ as) bs ↔ ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nas : Multiset α\nbs : Multiset β\nm : Multiset α\nhm : a ::ₘ as = m\n⊢ Rel r m bs → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'",
"state_before": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nas : Multiset α\nbs : Multiset β\n⊢ Rel r (a ::ₘ as) bs → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'",
"tactic": "generalize hm : a ::ₘ as = m"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nas : Multiset α\nbs : Multiset β\nm : Multiset α\nhm : a ::ₘ as = m\nh : Rel r m bs\n⊢ ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'",
"state_before": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nas : Multiset α\nbs : Multiset β\nm : Multiset α\nhm : a ::ₘ as = m\n⊢ Rel r m bs → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'",
"tactic": "intro h"
},
{
"state_after": "case mp.zero\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs : Multiset β\nm as : Multiset α\nhm : a ::ₘ as = 0\n⊢ ∃ b bs', r a b ∧ Rel r as bs' ∧ 0 = b ::ₘ bs'\n\ncase mp.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs : Multiset β\nm : Multiset α\na✝² : α\nb✝ : β\nas✝ : Multiset α\nbs✝ : Multiset β\na✝¹ : r a✝² b✝\na✝ : Rel r as✝ bs✝\na_ih✝ : ∀ {as : Multiset α}, a ::ₘ as = as✝ → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs✝ = b ::ₘ bs'\nas : Multiset α\nhm : a ::ₘ as = a✝² ::ₘ as✝\n⊢ ∃ b bs', r a b ∧ Rel r as bs' ∧ b✝ ::ₘ bs✝ = b ::ₘ bs'",
"state_before": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nas : Multiset α\nbs : Multiset β\nm : Multiset α\nhm : a ::ₘ as = m\nh : Rel r m bs\n⊢ ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'",
"tactic": "induction h generalizing as"
},
{
"state_after": "case mp.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs : Multiset β\nm : Multiset α\na✝² : α\nb✝ : β\nas✝ : Multiset α\nbs✝ : Multiset β\na✝¹ : r a✝² b✝\na✝ : Rel r as✝ bs✝\na_ih✝ : ∀ {as : Multiset α}, a ::ₘ as = as✝ → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs✝ = b ::ₘ bs'\nas : Multiset α\nhm : a ::ₘ as = a✝² ::ₘ as✝\n⊢ ∃ b bs', r a b ∧ Rel r as bs' ∧ b✝ ::ₘ bs✝ = b ::ₘ bs'",
"state_before": "case mp.zero\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs : Multiset β\nm as : Multiset α\nhm : a ::ₘ as = 0\n⊢ ∃ b bs', r a b ∧ Rel r as bs' ∧ 0 = b ::ₘ bs'\n\ncase mp.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs : Multiset β\nm : Multiset α\na✝² : α\nb✝ : β\nas✝ : Multiset α\nbs✝ : Multiset β\na✝¹ : r a✝² b✝\na✝ : Rel r as✝ bs✝\na_ih✝ : ∀ {as : Multiset α}, a ::ₘ as = as✝ → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs✝ = b ::ₘ bs'\nas : Multiset α\nhm : a ::ₘ as = a✝² ::ₘ as✝\n⊢ ∃ b bs', r a b ∧ Rel r as bs' ∧ b✝ ::ₘ bs✝ = b ::ₘ bs'",
"tactic": "case zero => simp at hm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs : Multiset β\nm as : Multiset α\nhm : a ::ₘ as = 0\n⊢ ∃ b bs', r a b ∧ Rel r as bs' ∧ 0 = b ::ₘ bs'",
"tactic": "simp at hm"
},
{
"state_after": "case inl.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs✝ : Multiset β\nm : Multiset α\na' : α\nb : β\nas' : Multiset α\nbs : Multiset β\nha'b : r a' b\nh : Rel r as' bs\nih : ∀ {as : Multiset α}, a ::ₘ as = as' → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'\nas : Multiset α\nhm : a ::ₘ as = a' ::ₘ as'\neq₁ : a = a'\neq₂ : as = as'\n⊢ ∃ b_1 bs', r a b_1 ∧ Rel r as bs' ∧ b ::ₘ bs = b_1 ::ₘ bs'\n\ncase inr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs✝ : Multiset β\nm : Multiset α\na' : α\nb : β\nas' : Multiset α\nbs : Multiset β\nha'b : r a' b\nh : Rel r as' bs\nih : ∀ {as : Multiset α}, a ::ₘ as = as' → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'\nas : Multiset α\nhm : a ::ₘ as = a' ::ₘ as'\n_h : a ≠ a'\ncs : Multiset α\neq₁ : as = a' ::ₘ cs\neq₂ : as' = a ::ₘ cs\n⊢ ∃ b_1 bs', r a b_1 ∧ Rel r as bs' ∧ b ::ₘ bs = b_1 ::ₘ bs'",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs✝ : Multiset β\nm : Multiset α\na' : α\nb : β\nas' : Multiset α\nbs : Multiset β\nha'b : r a' b\nh : Rel r as' bs\nih : ∀ {as : Multiset α}, a ::ₘ as = as' → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'\nas : Multiset α\nhm : a ::ₘ as = a' ::ₘ as'\n⊢ ∃ b_1 bs', r a b_1 ∧ Rel r as bs' ∧ b ::ₘ bs = b_1 ::ₘ bs'",
"tactic": "rcases cons_eq_cons.1 hm with (⟨eq₁, eq₂⟩ | ⟨_h, cs, eq₁, eq₂⟩)"
},
{
"state_after": "case inl.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs✝ : Multiset β\nm : Multiset α\nb : β\nas' : Multiset α\nbs : Multiset β\nh : Rel r as' bs\nih : ∀ {as : Multiset α}, a ::ₘ as = as' → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'\nas : Multiset α\neq₂ : as = as'\nha'b : r a b\nhm : a ::ₘ as = a ::ₘ as'\n⊢ ∃ b_1 bs', r a b_1 ∧ Rel r as bs' ∧ b ::ₘ bs = b_1 ::ₘ bs'",
"state_before": "case inl.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs✝ : Multiset β\nm : Multiset α\na' : α\nb : β\nas' : Multiset α\nbs : Multiset β\nha'b : r a' b\nh : Rel r as' bs\nih : ∀ {as : Multiset α}, a ::ₘ as = as' → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'\nas : Multiset α\nhm : a ::ₘ as = a' ::ₘ as'\neq₁ : a = a'\neq₂ : as = as'\n⊢ ∃ b_1 bs', r a b_1 ∧ Rel r as bs' ∧ b ::ₘ bs = b_1 ::ₘ bs'",
"tactic": "subst eq₁"
},
{
"state_after": "case inl.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs✝ : Multiset β\nm : Multiset α\nb : β\nbs : Multiset β\nas : Multiset α\nha'b : r a b\nh : Rel r as bs\nih : ∀ {as_1 : Multiset α}, a ::ₘ as_1 = as → ∃ b bs', r a b ∧ Rel r as_1 bs' ∧ bs = b ::ₘ bs'\nhm : a ::ₘ as = a ::ₘ as\n⊢ ∃ b_1 bs', r a b_1 ∧ Rel r as bs' ∧ b ::ₘ bs = b_1 ::ₘ bs'",
"state_before": "case inl.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs✝ : Multiset β\nm : Multiset α\nb : β\nas' : Multiset α\nbs : Multiset β\nh : Rel r as' bs\nih : ∀ {as : Multiset α}, a ::ₘ as = as' → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'\nas : Multiset α\neq₂ : as = as'\nha'b : r a b\nhm : a ::ₘ as = a ::ₘ as'\n⊢ ∃ b_1 bs', r a b_1 ∧ Rel r as bs' ∧ b ::ₘ bs = b_1 ::ₘ bs'",
"tactic": "subst eq₂"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs✝ : Multiset β\nm : Multiset α\nb : β\nbs : Multiset β\nas : Multiset α\nha'b : r a b\nh : Rel r as bs\nih : ∀ {as_1 : Multiset α}, a ::ₘ as_1 = as → ∃ b bs', r a b ∧ Rel r as_1 bs' ∧ bs = b ::ₘ bs'\nhm : a ::ₘ as = a ::ₘ as\n⊢ ∃ b_1 bs', r a b_1 ∧ Rel r as bs' ∧ b ::ₘ bs = b_1 ::ₘ bs'",
"tactic": "exact ⟨b, bs, ha'b, h, rfl⟩"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs✝ : Multiset β\nm : Multiset α\na' : α\nb : β\nas' : Multiset α\nbs : Multiset β\nha'b : r a' b\nh : Rel r as' bs\nih : ∀ {as : Multiset α}, a ::ₘ as = as' → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'\nas : Multiset α\nhm : a ::ₘ as = a' ::ₘ as'\n_h : a ≠ a'\ncs : Multiset α\neq₁ : as = a' ::ₘ cs\neq₂ : as' = a ::ₘ cs\nb' : β\nbs' : Multiset β\nh₁ : r a b'\nh₂ : Rel r cs bs'\neq : bs = b' ::ₘ bs'\n⊢ ∃ b_1 bs', r a b_1 ∧ Rel r as bs' ∧ b ::ₘ bs = b_1 ::ₘ bs'",
"state_before": "case inr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs✝ : Multiset β\nm : Multiset α\na' : α\nb : β\nas' : Multiset α\nbs : Multiset β\nha'b : r a' b\nh : Rel r as' bs\nih : ∀ {as : Multiset α}, a ::ₘ as = as' → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'\nas : Multiset α\nhm : a ::ₘ as = a' ::ₘ as'\n_h : a ≠ a'\ncs : Multiset α\neq₁ : as = a' ::ₘ cs\neq₂ : as' = a ::ₘ cs\n⊢ ∃ b_1 bs', r a b_1 ∧ Rel r as bs' ∧ b ::ₘ bs = b_1 ::ₘ bs'",
"tactic": "rcases ih eq₂.symm with ⟨b', bs', h₁, h₂, eq⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nbs✝ : Multiset β\nm : Multiset α\na' : α\nb : β\nas' : Multiset α\nbs : Multiset β\nha'b : r a' b\nh : Rel r as' bs\nih : ∀ {as : Multiset α}, a ::ₘ as = as' → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'\nas : Multiset α\nhm : a ::ₘ as = a' ::ₘ as'\n_h : a ≠ a'\ncs : Multiset α\neq₁ : as = a' ::ₘ cs\neq₂ : as' = a ::ₘ cs\nb' : β\nbs' : Multiset β\nh₁ : r a b'\nh₂ : Rel r cs bs'\neq : bs = b' ::ₘ bs'\n⊢ ∃ b_1 bs', r a b_1 ∧ Rel r as bs' ∧ b ::ₘ bs = b_1 ::ₘ bs'",
"tactic": "exact ⟨b', b ::ₘ bs', h₁, eq₁.symm ▸ Rel.cons ha'b h₂, eq.symm ▸ cons_swap _ _ _⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.444789\nδ : Type ?u.444792\nr : α → β → Prop\np : γ → δ → Prop\na : α\nas : Multiset α\nbs : Multiset β\n⊢ (∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs') → Rel r (a ::ₘ as) bs",
"tactic": "exact fun ⟨b, bs', hab, h, Eq⟩ => Eq.symm ▸ Rel.cons hab h"
}
] |
[
2735,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2720,
1
] |
Mathlib/Data/PNat/Xgcd.lean
|
PNat.XgcdType.flip_w
|
[] |
[
192,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean
|
CategoryTheory.HomOrthogonal.eq_zero
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\nι : Type u_1\ns : ι → C\ninst✝ : HasZeroMorphisms C\no : HomOrthogonal s\ni j : ι\nw : i ≠ j\nf : s i ⟶ s j\nthis : Subsingleton (s i ⟶ s j)\n⊢ f = 0",
"state_before": "C : Type u\ninst✝¹ : Category C\nι : Type u_1\ns : ι → C\ninst✝ : HasZeroMorphisms C\no : HomOrthogonal s\ni j : ι\nw : i ≠ j\nf : s i ⟶ s j\n⊢ f = 0",
"tactic": "haveI := o i j w"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nι : Type u_1\ns : ι → C\ninst✝ : HasZeroMorphisms C\no : HomOrthogonal s\ni j : ι\nw : i ≠ j\nf : s i ⟶ s j\nthis : Subsingleton (s i ⟶ s j)\n⊢ f = 0",
"tactic": "apply Subsingleton.elim"
}
] |
[
66,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
Rat.toNNRat_mul
|
[
{
"state_after": "case inl\np q : ℚ\nhp : 0 ≤ p\nhq : 0 ≤ q\n⊢ toNNRat (p * q) = toNNRat p * toNNRat q\n\ncase inr\np q : ℚ\nhp : 0 ≤ p\nhq : q ≤ 0\n⊢ toNNRat (p * q) = toNNRat p * toNNRat q",
"state_before": "p q : ℚ\nhp : 0 ≤ p\n⊢ toNNRat (p * q) = toNNRat p * toNNRat q",
"tactic": "cases' le_total 0 q with hq hq"
},
{
"state_after": "no goals",
"state_before": "case inl\np q : ℚ\nhp : 0 ≤ p\nhq : 0 ≤ q\n⊢ toNNRat (p * q) = toNNRat p * toNNRat q",
"tactic": "ext <;> simp [toNNRat, hp, hq, max_eq_left, mul_nonneg]"
},
{
"state_after": "case inr\np q : ℚ\nhp : 0 ≤ p\nhq : q ≤ 0\nhpq : p * q ≤ 0\n⊢ toNNRat (p * q) = toNNRat p * toNNRat q",
"state_before": "case inr\np q : ℚ\nhp : 0 ≤ p\nhq : q ≤ 0\n⊢ toNNRat (p * q) = toNNRat p * toNNRat q",
"tactic": "have hpq := mul_nonpos_of_nonneg_of_nonpos hp hq"
},
{
"state_after": "no goals",
"state_before": "case inr\np q : ℚ\nhp : 0 ≤ p\nhq : q ≤ 0\nhpq : p * q ≤ 0\n⊢ toNNRat (p * q) = toNNRat p * toNNRat q",
"tactic": "rw [toNNRat_eq_zero.2 hq, toNNRat_eq_zero.2 hpq, mul_zero]"
}
] |
[
415,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
411,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
Real.le_sqrt
|
[] |
[
305,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/GroupTheory/Submonoid/Basic.lean
|
Submonoid.coe_sInf
|
[] |
[
318,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/Algebra/Hom/NonUnitalAlg.lean
|
AlgHom.toNonUnitalAlgHom_eq_coe
|
[] |
[
436,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
435,
1
] |
Mathlib/CategoryTheory/IsConnected.lean
|
CategoryTheory.IsConnected.of_induct
|
[
{
"state_after": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nj₀ : J\nh : ∀ (p : Set J), j₀ ∈ p → (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) → ∀ (j : J), j ∈ p\nα : Type u₁\nF : J → α\na : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂\nw : ∀ (j : J), j ∈ {j | F j = F j₀}\n⊢ ∀ (j j' : J), F j = F j'",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nj₀ : J\nh : ∀ (p : Set J), j₀ ∈ p → (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) → ∀ (j : J), j ∈ p\nα : Type u₁\nF : J → α\na : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂\n⊢ ∀ (j j' : J), F j = F j'",
"tactic": "have w := h { j | F j = F j₀ } rfl (fun {j₁} {j₂} f => by\n change F j₁ = F j₀ ↔ F j₂ = F j₀\n simp [a f];)"
},
{
"state_after": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nj₀ : J\nh : ∀ (p : Set J), j₀ ∈ p → (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) → ∀ (j : J), j ∈ p\nα : Type u₁\nF : J → α\na : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂\nw : ∀ (j : J), j ∈ {j | F j = F j₀}\n⊢ ∀ (j j' : J), F j = F j'",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nj₀ : J\nh : ∀ (p : Set J), j₀ ∈ p → (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) → ∀ (j : J), j ∈ p\nα : Type u₁\nF : J → α\na : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂\nw : ∀ (j : J), j ∈ {j | F j = F j₀}\n⊢ ∀ (j j' : J), F j = F j'",
"tactic": "dsimp at w"
},
{
"state_after": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nj₀ : J\nh : ∀ (p : Set J), j₀ ∈ p → (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) → ∀ (j : J), j ∈ p\nα : Type u₁\nF : J → α\na : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂\nw : ∀ (j : J), j ∈ {j | F j = F j₀}\nj j' : J\n⊢ F j = F j'",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nj₀ : J\nh : ∀ (p : Set J), j₀ ∈ p → (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) → ∀ (j : J), j ∈ p\nα : Type u₁\nF : J → α\na : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂\nw : ∀ (j : J), j ∈ {j | F j = F j₀}\n⊢ ∀ (j j' : J), F j = F j'",
"tactic": "intro j j'"
},
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nj₀ : J\nh : ∀ (p : Set J), j₀ ∈ p → (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) → ∀ (j : J), j ∈ p\nα : Type u₁\nF : J → α\na : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂\nw : ∀ (j : J), j ∈ {j | F j = F j₀}\nj j' : J\n⊢ F j = F j'",
"tactic": "rw [w j, w j']"
},
{
"state_after": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nj₀ : J\nh : ∀ (p : Set J), j₀ ∈ p → (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) → ∀ (j : J), j ∈ p\nα : Type u₁\nF : J → α\na : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂\nj₁ j₂ : J\nf : j₁ ⟶ j₂\n⊢ F j₁ = F j₀ ↔ F j₂ = F j₀",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nj₀ : J\nh : ∀ (p : Set J), j₀ ∈ p → (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) → ∀ (j : J), j ∈ p\nα : Type u₁\nF : J → α\na : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂\nj₁ j₂ : J\nf : j₁ ⟶ j₂\n⊢ j₁ ∈ {j | F j = F j₀} ↔ j₂ ∈ {j | F j = F j₀}",
"tactic": "change F j₁ = F j₀ ↔ F j₂ = F j₀"
},
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nj₀ : J\nh : ∀ (p : Set J), j₀ ∈ p → (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) → ∀ (j : J), j ∈ p\nα : Type u₁\nF : J → α\na : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂\nj₁ j₂ : J\nf : j₁ ⟶ j₂\n⊢ F j₁ = F j₀ ↔ F j₂ = F j₀",
"tactic": "simp [a f]"
}
] |
[
174,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.embedding
|
[] |
[
246,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
11
] |
Mathlib/Analysis/Normed/Group/AddTorsor.lean
|
nndist_vadd_cancel_right
|
[] |
[
113,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.single_sub
|
[] |
[
1152,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1151,
1
] |
Mathlib/Combinatorics/Quiver/Push.lean
|
Quiver.Push.of_obj
|
[] |
[
55,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
|
DifferentiableAt.log
|
[] |
[
162,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.one_le_exp_iff
|
[] |
[
1557,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1556,
1
] |
Mathlib/Data/Finset/Prod.lean
|
Finset.Nonempty.product
|
[] |
[
192,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.mem_Icc_of_Ioc
|
[] |
[
671,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
670,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean
|
mul_inv_eq_one₀
|
[] |
[
52,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Combinatorics/Additive/SalemSpencer.lean
|
rothNumberNat_add_le
|
[
{
"state_after": "F : Type ?u.152613\nα : Type ?u.152616\nβ : Type ?u.152619\n𝕜 : Type ?u.152622\nE : Type ?u.152625\ns : Finset ℕ\nk n M N : ℕ\n⊢ ↑addRothNumber (range (M + N)) ≤ ↑addRothNumber (range M) + ↑addRothNumber (range N)",
"state_before": "F : Type ?u.152613\nα : Type ?u.152616\nβ : Type ?u.152619\n𝕜 : Type ?u.152622\nE : Type ?u.152625\ns : Finset ℕ\nk n M N : ℕ\n⊢ ↑rothNumberNat (M + N) ≤ ↑rothNumberNat M + ↑rothNumberNat N",
"tactic": "simp_rw [rothNumberNat_def]"
},
{
"state_after": "F : Type ?u.152613\nα : Type ?u.152616\nβ : Type ?u.152619\n𝕜 : Type ?u.152622\nE : Type ?u.152625\ns : Finset ℕ\nk n M N : ℕ\n⊢ ↑addRothNumber (range M ∪ map (addLeftEmbedding M) (range N)) ≤\n ↑addRothNumber (range M) + ↑addRothNumber (map (addLeftEmbedding M) (range N))",
"state_before": "F : Type ?u.152613\nα : Type ?u.152616\nβ : Type ?u.152619\n𝕜 : Type ?u.152622\nE : Type ?u.152625\ns : Finset ℕ\nk n M N : ℕ\n⊢ ↑addRothNumber (range (M + N)) ≤ ↑addRothNumber (range M) + ↑addRothNumber (range N)",
"tactic": "rw [range_add_eq_union, ← addRothNumber_map_add_left (range N) M]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.152613\nα : Type ?u.152616\nβ : Type ?u.152619\n𝕜 : Type ?u.152622\nE : Type ?u.152625\ns : Finset ℕ\nk n M N : ℕ\n⊢ ↑addRothNumber (range M ∪ map (addLeftEmbedding M) (range N)) ≤\n ↑addRothNumber (range M) + ↑addRothNumber (map (addLeftEmbedding M) (range N))",
"tactic": "exact addRothNumber_union_le _ _"
}
] |
[
496,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
492,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.periodicOrbit_apply_eq
|
[] |
[
558,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
1
] |
Mathlib/Data/Fintype/BigOperators.lean
|
Fintype.eq_of_subsingleton_of_prod_eq
|
[] |
[
101,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.eq_zero_of_mem_range_zero
|
[] |
[
560,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
559,
1
] |
Mathlib/Order/Basic.lean
|
lt_of_strongLT
|
[
{
"state_after": "ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type u_2\nr : α → α → Prop\ninst✝¹ : (i : ι) → Preorder (π i)\na b c : (i : ι) → π i\ninst✝ : Nonempty ι\nh : a ≺ b\ninhabited_h : Inhabited ι\n⊢ a < b",
"state_before": "ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type u_2\nr : α → α → Prop\ninst✝¹ : (i : ι) → Preorder (π i)\na b c : (i : ι) → π i\ninst✝ : Nonempty ι\nh : a ≺ b\n⊢ a < b",
"tactic": "inhabit ι"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type u_2\nr : α → α → Prop\ninst✝¹ : (i : ι) → Preorder (π i)\na b c : (i : ι) → π i\ninst✝ : Nonempty ι\nh : a ≺ b\ninhabited_h : Inhabited ι\n⊢ a < b",
"tactic": "exact Pi.lt_def.2 ⟨le_of_strongLT h, default, h _⟩"
}
] |
[
834,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
832,
1
] |
Mathlib/Data/Finmap.lean
|
Finmap.disjoint_union_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nx y z : Finmap β\n⊢ Disjoint (x ∪ y) z ↔ Disjoint x z ∧ Disjoint y z",
"tactic": "simp [Disjoint, Finmap.mem_union, or_imp, forall_and]"
}
] |
[
674,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
673,
1
] |
Mathlib/Data/Set/Intervals/Monotone.lean
|
Monotone.Ici
|
[] |
[
41,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
11
] |
Mathlib/Analysis/Convex/Hull.lean
|
convexHull_eq_iInter
|
[] |
[
66,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.mul_max
|
[] |
[
996,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
996,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.hasPushout_assoc
|
[] |
[
2553,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2552,
1
] |
Mathlib/Analysis/Convex/Body.lean
|
ConvexBody.isCompact
|
[] |
[
72,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
11
] |
Mathlib/Algebra/Order/Ring/Canonical.lean
|
tsub_mul
|
[] |
[
156,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/GroupTheory/FreeProduct.lean
|
FreeProduct.Word.cons_eq_smul
|
[
{
"state_after": "ι : Type u_2\nM : ι → Type u_1\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.440478\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni : ι\nm : M i\nls : List ((i : ι) × M i)\nh1 : ∀ (l : (i : ι) × M i), l ∈ { fst := i, snd := m } :: ls → l.snd ≠ 1\nh2 : List.Chain' (fun l l' => l.fst ≠ l'.fst) ({ fst := i, snd := m } :: ls)\n⊢ rcons\n { head := m, tail := FreeProduct.Word.mkAux ls h1 h2,\n fstIdx_ne := (_ : fstIdx (FreeProduct.Word.mkAux ls h1 h2) ≠ some i) } =\n rcons\n (let src := { head := 1, tail := FreeProduct.Word.mkAux ls h1 h2, fstIdx_ne := ?m.453010 };\n { head := m * { head := 1, tail := FreeProduct.Word.mkAux ls h1 h2, fstIdx_ne := ?m.453010 }.head,\n tail := src.tail, fstIdx_ne := (_ : fstIdx src.tail ≠ some i) })\n\nι : Type u_2\nM : ι → Type u_1\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.440478\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni : ι\nm : M i\nls : List ((i : ι) × M i)\nh1 : ∀ (l : (i : ι) × M i), l ∈ { fst := i, snd := m } :: ls → l.snd ≠ 1\nh2 : List.Chain' (fun l l' => l.fst ≠ l'.fst) ({ fst := i, snd := m } :: ls)\n⊢ fstIdx (FreeProduct.Word.mkAux ls h1 h2) ≠ some i",
"state_before": "ι : Type u_2\nM : ι → Type u_1\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.440478\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni : ι\nm : M i\nls : List ((i : ι) × M i)\nh1 : ∀ (l : (i : ι) × M i), l ∈ { fst := i, snd := m } :: ls → l.snd ≠ 1\nh2 : List.Chain' (fun l l' => l.fst ≠ l'.fst) ({ fst := i, snd := m } :: ls)\n⊢ { toList := { fst := i, snd := m } :: ls, ne_one := h1, chain_ne := h2 } = ↑of m • FreeProduct.Word.mkAux ls h1 h2",
"tactic": "rw [cons_eq_rcons, of_smul_def, equivPair_eq_of_fstIdx_ne _]"
},
{
"state_after": "ι : Type u_2\nM : ι → Type u_1\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.440478\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni : ι\nm : M i\nls : List ((i : ι) × M i)\nh1 : ∀ (l : (i : ι) × M i), l ∈ { fst := i, snd := m } :: ls → l.snd ≠ 1\nh2 : List.Chain' (fun l l' => l.fst ≠ l'.fst) ({ fst := i, snd := m } :: ls)\n⊢ fstIdx (FreeProduct.Word.mkAux ls h1 h2) ≠ some i",
"state_before": "ι : Type u_2\nM : ι → Type u_1\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.440478\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni : ι\nm : M i\nls : List ((i : ι) × M i)\nh1 : ∀ (l : (i : ι) × M i), l ∈ { fst := i, snd := m } :: ls → l.snd ≠ 1\nh2 : List.Chain' (fun l l' => l.fst ≠ l'.fst) ({ fst := i, snd := m } :: ls)\n⊢ rcons\n { head := m, tail := FreeProduct.Word.mkAux ls h1 h2,\n fstIdx_ne := (_ : fstIdx (FreeProduct.Word.mkAux ls h1 h2) ≠ some i) } =\n rcons\n (let src := { head := 1, tail := FreeProduct.Word.mkAux ls h1 h2, fstIdx_ne := ?m.453010 };\n { head := m * { head := 1, tail := FreeProduct.Word.mkAux ls h1 h2, fstIdx_ne := ?m.453010 }.head,\n tail := src.tail, fstIdx_ne := (_ : fstIdx src.tail ≠ some i) })\n\nι : Type u_2\nM : ι → Type u_1\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.440478\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni : ι\nm : M i\nls : List ((i : ι) × M i)\nh1 : ∀ (l : (i : ι) × M i), l ∈ { fst := i, snd := m } :: ls → l.snd ≠ 1\nh2 : List.Chain' (fun l l' => l.fst ≠ l'.fst) ({ fst := i, snd := m } :: ls)\n⊢ fstIdx (FreeProduct.Word.mkAux ls h1 h2) ≠ some i",
"tactic": ". simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nM : ι → Type u_1\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.440478\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni : ι\nm : M i\nls : List ((i : ι) × M i)\nh1 : ∀ (l : (i : ι) × M i), l ∈ { fst := i, snd := m } :: ls → l.snd ≠ 1\nh2 : List.Chain' (fun l l' => l.fst ≠ l'.fst) ({ fst := i, snd := m } :: ls)\n⊢ fstIdx (FreeProduct.Word.mkAux ls h1 h2) ≠ some i",
"tactic": ". rw [fstIdx_ne_iff]\n exact (List.chain'_cons'.1 h2).1"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nM : ι → Type u_1\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.440478\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni : ι\nm : M i\nls : List ((i : ι) × M i)\nh1 : ∀ (l : (i : ι) × M i), l ∈ { fst := i, snd := m } :: ls → l.snd ≠ 1\nh2 : List.Chain' (fun l l' => l.fst ≠ l'.fst) ({ fst := i, snd := m } :: ls)\n⊢ rcons\n { head := m, tail := FreeProduct.Word.mkAux ls h1 h2,\n fstIdx_ne := (_ : fstIdx (FreeProduct.Word.mkAux ls h1 h2) ≠ some i) } =\n rcons\n (let src := { head := 1, tail := FreeProduct.Word.mkAux ls h1 h2, fstIdx_ne := ?m.453010 };\n { head := m * { head := 1, tail := FreeProduct.Word.mkAux ls h1 h2, fstIdx_ne := ?m.453010 }.head,\n tail := src.tail, fstIdx_ne := (_ : fstIdx src.tail ≠ some i) })",
"tactic": "simp"
},
{
"state_after": "ι : Type u_2\nM : ι → Type u_1\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.440478\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni : ι\nm : M i\nls : List ((i : ι) × M i)\nh1 : ∀ (l : (i : ι) × M i), l ∈ { fst := i, snd := m } :: ls → l.snd ≠ 1\nh2 : List.Chain' (fun l l' => l.fst ≠ l'.fst) ({ fst := i, snd := m } :: ls)\n⊢ ∀ (l : (i : ι) × M i), l ∈ List.head? (FreeProduct.Word.mkAux ls h1 h2).toList → i ≠ l.fst",
"state_before": "ι : Type u_2\nM : ι → Type u_1\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.440478\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni : ι\nm : M i\nls : List ((i : ι) × M i)\nh1 : ∀ (l : (i : ι) × M i), l ∈ { fst := i, snd := m } :: ls → l.snd ≠ 1\nh2 : List.Chain' (fun l l' => l.fst ≠ l'.fst) ({ fst := i, snd := m } :: ls)\n⊢ fstIdx (FreeProduct.Word.mkAux ls h1 h2) ≠ some i",
"tactic": "rw [fstIdx_ne_iff]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nM : ι → Type u_1\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.440478\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni : ι\nm : M i\nls : List ((i : ι) × M i)\nh1 : ∀ (l : (i : ι) × M i), l ∈ { fst := i, snd := m } :: ls → l.snd ≠ 1\nh2 : List.Chain' (fun l l' => l.fst ≠ l'.fst) ({ fst := i, snd := m } :: ls)\n⊢ ∀ (l : (i : ι) × M i), l ∈ List.head? (FreeProduct.Word.mkAux ls h1 h2).toList → i ≠ l.fst",
"tactic": "exact (List.chain'_cons'.1 h2).1"
}
] |
[
433,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
LinearMap.mul_toMatrix₂'_mul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.928393\nR₂ : Type ?u.928396\nM✝ : Type ?u.928399\nM₁ : Type ?u.928402\nM₂ : Type ?u.928405\nM₁' : Type ?u.928408\nM₂' : Type ?u.928411\nn : Type u_2\nm : Type u_3\nn' : Type u_4\nm' : Type u_5\nι : Type ?u.928426\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing R₁\ninst✝⁸ : CommRing R₂\ninst✝⁷ : Fintype n\ninst✝⁶ : Fintype m\ninst✝⁵ : DecidableEq n\ninst✝⁴ : DecidableEq m\nσ₁ : R₁ →+* R\nσ₂ : R₂ →+* R\ninst✝³ : Fintype n'\ninst✝² : Fintype m'\ninst✝¹ : DecidableEq n'\ninst✝ : DecidableEq m'\nB : (n → R) →ₗ[R] (m → R) →ₗ[R] R\nM : Matrix n' n R\nN : Matrix m m' R\n⊢ M ⬝ ↑toMatrix₂' B ⬝ N = ↑toMatrix₂' (compl₁₂ B (↑toLin' Mᵀ) (↑toLin' N))",
"tactic": "simp"
}
] |
[
322,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/RingTheory/Polynomial/Quotient.lean
|
Ideal.isDomain_map_C_quotient
|
[] |
[
160,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Data/Real/ConjugateExponents.lean
|
Real.IsConjugateExponent.mul_eq_add
|
[
{
"state_after": "no goals",
"state_before": "p q : ℝ\nh : IsConjugateExponent p q\n⊢ p * q = p + q",
"tactic": "simpa only [sub_mul, sub_eq_iff_eq_add, one_mul] using h.sub_one_mul_conj"
}
] |
[
88,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/LinearAlgebra/FreeModule/Basic.lean
|
Module.free_def
|
[] |
[
61,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConcaveOn.right_le_of_le_left
|
[] |
[
761,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
759,
1
] |
src/lean/Init/Prelude.lean
|
usize_size_eq
|
[
{
"state_after": "no goals",
"state_before": "⊢ Eq (hPow 2 32) 4294967296",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "⊢ Eq (hPow 2 64) 18446744073709551616",
"tactic": "decide"
}
] |
[
1979,
40
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
1975,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.preimage_Ioo
|
[
{
"state_after": "case h\nF : Type ?u.236506\nα : Type u_1\nβ : Type ?u.236512\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\nx✝ : ℤ\n⊢ x✝ ∈ Int.cast ⁻¹' Ioo a b ↔ x✝ ∈ Ioo ⌊a⌋ ⌈b⌉",
"state_before": "F : Type ?u.236506\nα : Type u_1\nβ : Type ?u.236512\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ Int.cast ⁻¹' Ioo a b = Ioo ⌊a⌋ ⌈b⌉",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type ?u.236506\nα : Type u_1\nβ : Type ?u.236512\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\nx✝ : ℤ\n⊢ x✝ ∈ Int.cast ⁻¹' Ioo a b ↔ x✝ ∈ Ioo ⌊a⌋ ⌈b⌉",
"tactic": "simp [floor_lt, lt_ceil]"
}
] |
[
1268,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1266,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.mem_direction_iff_eq_vsub_right
|
[
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\nv : V\n⊢ v ∈ (fun x => x -ᵥ p) '' ↑s ↔ ∃ p2, p2 ∈ s ∧ v = p2 -ᵥ p",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\nv : V\n⊢ v ∈ direction s ↔ ∃ p2, p2 ∈ s ∧ v = p2 -ᵥ p",
"tactic": "rw [← SetLike.mem_coe, coe_direction_eq_vsub_set_right hp]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\nv : V\n⊢ v ∈ (fun x => x -ᵥ p) '' ↑s ↔ ∃ p2, p2 ∈ s ∧ v = p2 -ᵥ p",
"tactic": "exact ⟨fun ⟨p2, hp2, hv⟩ => ⟨p2, hp2, hv.symm⟩, fun ⟨p2, hp2, hv⟩ => ⟨p2, hp2, hv.symm⟩⟩"
}
] |
[
323,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/MeasureTheory/PiSystem.lean
|
MeasurableSpace.DynkinSystem.has_univ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nd : DynkinSystem α\n⊢ Has d univ",
"tactic": "simpa using d.has_compl d.has_empty"
}
] |
[
572,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
572,
1
] |
Mathlib/Data/Vector/Basic.lean
|
Vector.last_def
|
[] |
[
289,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.translate_apply
|
[] |
[
299,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.insert_eq_of_mem
|
[] |
[
1099,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1098,
1
] |
Mathlib/Analysis/Normed/Group/Quotient.lean
|
Ideal.Quotient.norm_mk_lt
|
[] |
[
493,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
491,
8
] |
Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean
|
TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst
|
[] |
[
493,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.dist_map
|
[] |
[
981,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
980,
1
] |
Mathlib/Data/MvPolynomial/Comap.lean
|
MvPolynomial.comap_apply
|
[] |
[
47,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Algebra/Order/CompleteField.lean
|
LinearOrderedField.inducedMap_mono
|
[] |
[
190,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/CategoryTheory/StructuredArrow.lean
|
CategoryTheory.CostructuredArrow.ext_iff
|
[] |
[
372,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.SimpleFunc.integrable_of_isFiniteMeasure
|
[] |
[
397,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
irreducible_pow_sup
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.914699\nA : Type ?u.914702\nK : Type ?u.914705\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : Irreducible J\nn : ℕ\n⊢ J ^ n ⊔ I = J ^ min (count J (normalizedFactors I)) n",
"tactic": "rw [sup_eq_prod_inf_factors (pow_ne_zero n hJ.ne_zero) hI, min_comm,\n normalizedFactors_of_irreducible_pow hJ, normalize_eq J, replicate_inter, prod_replicate]"
}
] |
[
943,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
940,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.add_one_le_of_lt
|
[
{
"state_after": "case a\nx y : PartENat\nh✝ : x < y\nh : x < ⊤\n⊢ x + 1 ≤ ⊤\n\ncase a\nx y : PartENat\nh✝ : x < y\nn : ℕ\nh : x < ↑n\n⊢ x + 1 ≤ ↑n",
"state_before": "x y : PartENat\nh : x < y\n⊢ x + 1 ≤ y",
"tactic": "induction' y using PartENat.casesOn with n"
},
{
"state_after": "case a.intro\ny : PartENat\nn m : ℕ\nh✝ : ↑m < y\nh : ↑m < ↑n\n⊢ ↑m + 1 ≤ ↑n",
"state_before": "case a\nx y : PartENat\nh✝ : x < y\nn : ℕ\nh : x < ↑n\n⊢ x + 1 ≤ ↑n",
"tactic": "rcases ne_top_iff.mp (ne_top_of_lt h) with ⟨m, rfl⟩"
},
{
"state_after": "case a.intro\ny : PartENat\nn m : ℕ\nh✝ : ↑m < y\nh : ↑m < ↑n\n⊢ m + 1 ≤ n",
"state_before": "case a.intro\ny : PartENat\nn m : ℕ\nh✝ : ↑m < y\nh : ↑m < ↑n\n⊢ ↑m + 1 ≤ ↑n",
"tactic": "norm_cast"
},
{
"state_after": "case a.intro.h\ny : PartENat\nn m : ℕ\nh✝ : ↑m < y\nh : ↑m < ↑n\n⊢ m < n",
"state_before": "case a.intro\ny : PartENat\nn m : ℕ\nh✝ : ↑m < y\nh : ↑m < ↑n\n⊢ m + 1 ≤ n",
"tactic": "apply Nat.succ_le_of_lt"
},
{
"state_after": "no goals",
"state_before": "case a.intro.h\ny : PartENat\nn m : ℕ\nh✝ : ↑m < y\nh : ↑m < ↑n\n⊢ m < n",
"tactic": "norm_cast at h"
},
{
"state_after": "no goals",
"state_before": "case a\nx y : PartENat\nh✝ : x < y\nh : x < ⊤\n⊢ x + 1 ≤ ⊤",
"tactic": "apply le_top"
}
] |
[
503,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
498,
1
] |
Mathlib/Topology/Maps.lean
|
inducing_induced
|
[] |
[
72,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Order/Basic.lean
|
le_update_self_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type u_2\nr : α → α → Prop\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → Preorder (π i)\nx y : (i : ι) → π i\ni : ι\na b : π i\n⊢ x ≤ update x i a ↔ x i ≤ a",
"tactic": "simp [le_update_iff]"
}
] |
[
886,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
886,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.forall_range_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.12286\nδ : Type ?u.12289\ninst✝ : MeasurableSpace α\nf : α →ₛ β\np : β → Prop\n⊢ (∀ (y : β), y ∈ SimpleFunc.range f → p y) ↔ ∀ (x : α), p (↑f x)",
"tactic": "simp only [mem_range, Set.forall_range_iff]"
}
] |
[
126,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Data/List/Infix.lean
|
List.eq_of_infix_of_length_eq
|
[] |
[
119,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Analysis/LocallyConvex/AbsConvex.lean
|
gaugeSeminormFamily_ball
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.63454\nG : Type ?u.63457\nι : Type ?u.63460\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ Seminorm.ball (gaugeSeminorm (_ : Balanced 𝕜 ↑s) (_ : Convex ℝ ↑s) (_ : Absorbent ℝ ↑s)) 0 1 = ↑s",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.63454\nG : Type ?u.63457\nι : Type ?u.63460\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ Seminorm.ball (gaugeSeminormFamily 𝕜 E s) 0 1 = ↑s",
"tactic": "dsimp only [gaugeSeminormFamily]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.63454\nG : Type ?u.63457\nι : Type ?u.63460\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ {y | ↑(gaugeSeminorm (_ : Balanced 𝕜 ↑s) (_ : Convex ℝ ↑s) (_ : Absorbent ℝ ↑s)) y < 1} = ↑s",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.63454\nG : Type ?u.63457\nι : Type ?u.63460\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ Seminorm.ball (gaugeSeminorm (_ : Balanced 𝕜 ↑s) (_ : Convex ℝ ↑s) (_ : Absorbent ℝ ↑s)) 0 1 = ↑s",
"tactic": "rw [Seminorm.ball_zero_eq]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.63454\nG : Type ?u.63457\nι : Type ?u.63460\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ {y | gauge (↑s) y < 1} = ↑s",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.63454\nG : Type ?u.63457\nι : Type ?u.63460\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ {y | ↑(gaugeSeminorm (_ : Balanced 𝕜 ↑s) (_ : Convex ℝ ↑s) (_ : Absorbent ℝ ↑s)) y < 1} = ↑s",
"tactic": "simp_rw [gaugeSeminorm_toFun]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.63454\nG : Type ?u.63457\nι : Type ?u.63460\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ {y | gauge (↑s) y < 1} = ↑s",
"tactic": "exact gauge_lt_one_eq_self_of_open s.coe_convex s.coe_zero_mem s.coe_isOpen"
}
] |
[
158,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.mul_le_of_le_div'
|
[] |
[
1603,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1602,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
StrictMono.inv
|
[] |
[
1318,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1317,
1
] |
Mathlib/Computability/PartrecCode.lean
|
Nat.Partrec.Code.fixed_point
|
[
{
"state_after": "no goals",
"state_before": "f : Code → Code\nhf : Computable f\ng : ℕ → ℕ → Part ℕ :=\n fun x y => do\n let b ← eval (ofNat Code x) x\n eval (ofNat Code b) y\nthis : Partrec₂ g\ncg : Code\neg : eval cg = fun n => Part.bind ↑(decode n) fun a => Part.map encode ((fun p => g p.fst p.snd) a)\n⊢ ∀ (a n : ℕ), eval cg (Nat.pair a n) = Part.map encode (g a n)",
"tactic": "simp [eg]"
},
{
"state_after": "no goals",
"state_before": "f : Code → Code\nhf : Computable f\ng : ℕ → ℕ → Part ℕ :=\n fun x y => do\n let b ← eval (ofNat Code x) x\n eval (ofNat Code b) y\nthis✝ : Partrec₂ g\ncg : Code\neg : eval cg = fun n => Part.bind ↑(decode n) fun a => Part.map encode ((fun p => g p.fst p.snd) a)\neg' : ∀ (a n : ℕ), eval cg (Nat.pair a n) = Part.map encode (g a n)\nF : ℕ → Code := fun x => f (curry cg x)\nthis : Computable F\ncF : Code\neF : eval cF = fun n => Part.bind ↑(decode n) fun a => Part.map encode (↑F a)\n⊢ eval cF (encode cF) = Part.some (encode (F (encode cF)))",
"tactic": "simp [eF]"
},
{
"state_after": "no goals",
"state_before": "f : Code → Code\nhf : Computable f\ng : ℕ → ℕ → Part ℕ :=\n fun x y => do\n let b ← eval (ofNat Code x) x\n eval (ofNat Code b) y\nthis✝ : Partrec₂ g\ncg : Code\neg : eval cg = fun n => Part.bind ↑(decode n) fun a => Part.map encode ((fun p => g p.fst p.snd) a)\neg' : ∀ (a n : ℕ), eval cg (Nat.pair a n) = Part.map encode (g a n)\nF : ℕ → Code := fun x => f (curry cg x)\nthis : Computable F\ncF : Code\neF : eval cF = fun n => Part.bind ↑(decode n) fun a => Part.map encode (↑F a)\neF' : eval cF (encode cF) = Part.some (encode (F (encode cF)))\nn : ℕ\n⊢ eval (f (curry cg (encode cF))) n = eval (curry cg (encode cF)) n",
"tactic": "simp [eg', eF', Part.map_id']"
}
] |
[
1186,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1171,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isLittleO_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nE : Type u_3\nF : Type u_4\nG : Type ?u.71439\nE' : Type ?u.71442\nF' : Type ?u.71445\nG' : Type ?u.71448\nE'' : Type ?u.71451\nF'' : Type ?u.71454\nG'' : Type ?u.71457\nR : Type ?u.71460\nR' : Type ?u.71463\n𝕜 : Type ?u.71466\n𝕜' : Type ?u.71469\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk✝ : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl✝ l' : Filter α\nk : β → α\nl : Filter β\n⊢ f =o[map k l] g ↔ (f ∘ k) =o[l] (g ∘ k)",
"tactic": "simp only [IsLittleO_def, isBigOWith_map]"
}
] |
[
444,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.mem_skewAdjointSubmodule
|
[
{
"state_after": "R : Type ?u.1256417\nM : Type ?u.1256420\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1256456\nM₁ : Type ?u.1256459\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type ?u.1257068\nM₂ : Type ?u.1257071\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type u_1\nM₃ : Type u_2\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1257849\nK : Type ?u.1257852\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1259066\nM₂'' : Type ?u.1259069\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1259380\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1259817\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1262296\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\nB₃ : BilinForm R₃ M₃\nf : Module.End R₃ M₃\n⊢ f ∈ skewAdjointSubmodule B₃ ↔ IsAdjointPair (-B₃) B₃ f f",
"state_before": "R : Type ?u.1256417\nM : Type ?u.1256420\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1256456\nM₁ : Type ?u.1256459\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type ?u.1257068\nM₂ : Type ?u.1257071\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type u_1\nM₃ : Type u_2\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1257849\nK : Type ?u.1257852\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1259066\nM₂'' : Type ?u.1259069\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1259380\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1259817\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1262296\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\nB₃ : BilinForm R₃ M₃\nf : Module.End R₃ M₃\n⊢ f ∈ skewAdjointSubmodule B₃ ↔ IsSkewAdjoint B₃ f",
"tactic": "rw [isSkewAdjoint_iff_neg_self_adjoint]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1256417\nM : Type ?u.1256420\ninst✝²⁴ : Semiring R\ninst✝²³ : AddCommMonoid M\ninst✝²² : Module R M\nR₁ : Type ?u.1256456\nM₁ : Type ?u.1256459\ninst✝²¹ : Ring R₁\ninst✝²⁰ : AddCommGroup M₁\ninst✝¹⁹ : Module R₁ M₁\nR₂ : Type ?u.1257068\nM₂ : Type ?u.1257071\ninst✝¹⁸ : CommSemiring R₂\ninst✝¹⁷ : AddCommMonoid M₂\ninst✝¹⁶ : Module R₂ M₂\nR₃ : Type u_1\nM₃ : Type u_2\ninst✝¹⁵ : CommRing R₃\ninst✝¹⁴ : AddCommGroup M₃\ninst✝¹³ : Module R₃ M₃\nV : Type ?u.1257849\nK : Type ?u.1257852\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1259066\nM₂'' : Type ?u.1259069\ninst✝⁹ : AddCommMonoid M₂'\ninst✝⁸ : AddCommMonoid M₂''\ninst✝⁷ : Module R₂ M₂'\ninst✝⁶ : Module R₂ M₂''\nF : BilinForm R M\nM' : Type ?u.1259380\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nM₁' : Type ?u.1259817\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R₂ M₂'\nf₂ f₂' : M₂ →ₗ[R₂] M₂'\ng₂ g₂' : M₂' →ₗ[R₂] M₂\nM'' : Type ?u.1262296\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nF₂ : BilinForm R₂ M₂\nB₃ : BilinForm R₃ M₃\nf : Module.End R₃ M₃\n⊢ f ∈ skewAdjointSubmodule B₃ ↔ IsAdjointPair (-B₃) B₃ f f",
"tactic": "exact Iff.rfl"
}
] |
[
1164,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1161,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
InnerProductSpace.Core.normSq_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type ?u.540761\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx : F\n⊢ normSq x = 0 ↔ inner x x = 0",
"tactic": "simp only [normSq, ext_iff, map_zero, inner_self_im, eq_self_iff_true, and_true_iff]"
}
] |
[
267,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/Algebra/Hom/Equiv/Basic.lean
|
MulEquiv.invFun_eq_symm
|
[] |
[
272,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.coe_coe
|
[] |
[
936,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
935,
1
] |
Mathlib/MeasureTheory/Function/Jacobian.lean
|
MeasureTheory.integral_image_eq_integral_abs_det_fderiv_smul
|
[
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\n⊢ (∫ (x : ↑s),\n g\n (Set.restrict s f\n x) ∂Measure.comap Subtype.val\n (withDensity μ fun x => ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))))) =\n ∫ (x : E) in s, abs (ContinuousLinearMap.det (f' x)) • g (f x) ∂μ",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\n⊢ (∫ (x : E) in f '' s, g x ∂μ) = ∫ (x : E) in s, abs (ContinuousLinearMap.det (f' x)) • g (f x) ∂μ",
"tactic": "rw [← restrict_map_withDensity_abs_det_fderiv_eq_addHaar μ hs hf' hf,\n (measurableEmbedding_of_fderivWithin hs hf' hf).integral_map]"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\nthis : ∀ (x : ↑s), g (Set.restrict s f x) = (g ∘ f) ↑x\n⊢ (∫ (x : ↑s),\n g\n (Set.restrict s f\n x) ∂Measure.comap Subtype.val\n (withDensity μ fun x => ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))))) =\n ∫ (x : E) in s, abs (ContinuousLinearMap.det (f' x)) • g (f x) ∂μ",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\n⊢ (∫ (x : ↑s),\n g\n (Set.restrict s f\n x) ∂Measure.comap Subtype.val\n (withDensity μ fun x => ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))))) =\n ∫ (x : E) in s, abs (ContinuousLinearMap.det (f' x)) • g (f x) ∂μ",
"tactic": "have : ∀ x : s, g (s.restrict f x) = (g ∘ f) x := fun x => rfl"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\nthis : ∀ (x : ↑s), g (Set.restrict s f x) = (g ∘ f) ↑x\n⊢ (∫ (x : ↑s),\n (g ∘ f)\n ↑x ∂Measure.comap Subtype.val\n (withDensity μ fun x => ↑(Real.toNNReal (abs (ContinuousLinearMap.det (f' x)))))) =\n ∫ (x : E) in s, abs (ContinuousLinearMap.det (f' x)) • g (f x) ∂μ",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\nthis : ∀ (x : ↑s), g (Set.restrict s f x) = (g ∘ f) ↑x\n⊢ (∫ (x : ↑s),\n g\n (Set.restrict s f\n x) ∂Measure.comap Subtype.val\n (withDensity μ fun x => ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))))) =\n ∫ (x : E) in s, abs (ContinuousLinearMap.det (f' x)) • g (f x) ∂μ",
"tactic": "simp only [this, ENNReal.ofReal]"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\nthis : ∀ (x : ↑s), g (Set.restrict s f x) = (g ∘ f) ↑x\n⊢ (∫ (a : E) in s, Real.toNNReal (abs (ContinuousLinearMap.det (f' a))) • (g ∘ f) a ∂μ) =\n ∫ (x : E) in s, abs (ContinuousLinearMap.det (f' x)) • g (f x) ∂μ",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\nthis : ∀ (x : ↑s), g (Set.restrict s f x) = (g ∘ f) ↑x\n⊢ (∫ (x : ↑s),\n (g ∘ f)\n ↑x ∂Measure.comap Subtype.val\n (withDensity μ fun x => ↑(Real.toNNReal (abs (ContinuousLinearMap.det (f' x)))))) =\n ∫ (x : E) in s, abs (ContinuousLinearMap.det (f' x)) • g (f x) ∂μ",
"tactic": "rw [← (MeasurableEmbedding.subtype_coe hs).integral_map, map_comap_subtype_coe hs,\n set_integral_withDensity_eq_set_integral_smul₀\n (aemeasurable_toNNReal_abs_det_fderivWithin μ hs hf') _ hs]"
},
{
"state_after": "case e_f.h\nE : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\nthis : ∀ (x : ↑s), g (Set.restrict s f x) = (g ∘ f) ↑x\nx : E\n⊢ Real.toNNReal (abs (ContinuousLinearMap.det (f' x))) • (g ∘ f) x = abs (ContinuousLinearMap.det (f' x)) • g (f x)",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\nthis : ∀ (x : ↑s), g (Set.restrict s f x) = (g ∘ f) ↑x\n⊢ (∫ (a : E) in s, Real.toNNReal (abs (ContinuousLinearMap.det (f' a))) • (g ∘ f) a ∂μ) =\n ∫ (x : E) in s, abs (ContinuousLinearMap.det (f' x)) • g (f x) ∂μ",
"tactic": "congr with x"
},
{
"state_after": "no goals",
"state_before": "case e_f.h\nE : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\nthis : ∀ (x : ↑s), g (Set.restrict s f x) = (g ∘ f) ↑x\nx : E\n⊢ Real.toNNReal (abs (ContinuousLinearMap.det (f' x))) • (g ∘ f) x = abs (ContinuousLinearMap.det (f' x)) • g (f x)",
"tactic": "conv_rhs => rw [← Real.coe_toNNReal _ (abs_nonneg (f' x).det)]"
}
] |
[
1221,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1210,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
continuousMultilinearCurryFin1_symm_apply
|
[] |
[
1741,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1739,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.atTop_finset_eq_iInf
|
[
{
"state_after": "ι : Type ?u.267659\nι' : Type ?u.267662\nα : Type u_1\nβ : Type ?u.267668\nγ : Type ?u.267671\n⊢ (⨅ (x : α), 𝓟 (Ici {x})) ≤ atTop",
"state_before": "ι : Type ?u.267659\nι' : Type ?u.267662\nα : Type u_1\nβ : Type ?u.267668\nγ : Type ?u.267671\n⊢ atTop = ⨅ (x : α), 𝓟 (Ici {x})",
"tactic": "refine' le_antisymm (le_iInf fun i => le_principal_iff.2 <| mem_atTop {i}) _"
},
{
"state_after": "ι : Type ?u.267659\nι' : Type ?u.267662\nα : Type u_1\nβ : Type ?u.267668\nγ : Type ?u.267671\ns : Finset α\n⊢ (⋂ (i : ↑↑s), Ici {↑i}) ⊆ Ici s",
"state_before": "ι : Type ?u.267659\nι' : Type ?u.267662\nα : Type u_1\nβ : Type ?u.267668\nγ : Type ?u.267671\n⊢ (⨅ (x : α), 𝓟 (Ici {x})) ≤ atTop",
"tactic": "refine'\n le_iInf fun s =>\n le_principal_iff.2 <| mem_iInf_of_iInter s.finite_toSet (fun i => mem_principal_self _) _"
},
{
"state_after": "ι : Type ?u.267659\nι' : Type ?u.267662\nα : Type u_1\nβ : Type ?u.267668\nγ : Type ?u.267671\ns : Finset α\n⊢ ∀ (x : Finset α), (∀ (x_1 : α), x_1 ∈ ↑s → x_1 ∈ x) → ∀ ⦃x_1 : α⦄, x_1 ∈ s → x_1 ∈ x",
"state_before": "ι : Type ?u.267659\nι' : Type ?u.267662\nα : Type u_1\nβ : Type ?u.267668\nγ : Type ?u.267671\ns : Finset α\n⊢ (⋂ (i : ↑↑s), Ici {↑i}) ⊆ Ici s",
"tactic": "simp only [subset_def, mem_iInter, SetCoe.forall, mem_Ici, Finset.le_iff_subset,\n Finset.mem_singleton, Finset.subset_iff, forall_eq]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.267659\nι' : Type ?u.267662\nα : Type u_1\nβ : Type ?u.267668\nγ : Type ?u.267671\ns : Finset α\n⊢ ∀ (x : Finset α), (∀ (x_1 : α), x_1 ∈ ↑s → x_1 ∈ x) → ∀ ⦃x_1 : α⦄, x_1 ∈ s → x_1 ∈ x",
"tactic": "exact fun t => id"
}
] |
[
1369,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1362,
1
] |
Mathlib/Algebra/Order/Rearrangement.lean
|
MonovaryOn.sum_mul_comp_perm_eq_sum_mul_iff
|
[] |
[
364,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConvexOn.openSegment_subset_strict_epigraph
|
[
{
"state_after": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.344708\nα : Type ?u.344711\nβ : Type u_3\nι : Type ?u.344717\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedCancelAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf : E → β\nhf : ConvexOn 𝕜 s f\np q : E × β\nhp : p.fst ∈ s ∧ f p.fst < p.snd\nhq : q.fst ∈ s ∧ f q.fst ≤ q.snd\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • p + b • q ∈ {p | p.fst ∈ s ∧ f p.fst < p.snd}",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.344708\nα : Type ?u.344711\nβ : Type u_3\nι : Type ?u.344717\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedCancelAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf : E → β\nhf : ConvexOn 𝕜 s f\np q : E × β\nhp : p.fst ∈ s ∧ f p.fst < p.snd\nhq : q.fst ∈ s ∧ f q.fst ≤ q.snd\n⊢ openSegment 𝕜 p q ⊆ {p | p.fst ∈ s ∧ f p.fst < p.snd}",
"tactic": "rintro _ ⟨a, b, ha, hb, hab, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.344708\nα : Type ?u.344711\nβ : Type u_3\nι : Type ?u.344717\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedCancelAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf : E → β\nhf : ConvexOn 𝕜 s f\np q : E × β\nhp : p.fst ∈ s ∧ f p.fst < p.snd\nhq : q.fst ∈ s ∧ f q.fst ≤ q.snd\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ f (a • p + b • q).fst < (a • p + b • q).snd",
"state_before": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.344708\nα : Type ?u.344711\nβ : Type u_3\nι : Type ?u.344717\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedCancelAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf : E → β\nhf : ConvexOn 𝕜 s f\np q : E × β\nhp : p.fst ∈ s ∧ f p.fst < p.snd\nhq : q.fst ∈ s ∧ f q.fst ≤ q.snd\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • p + b • q ∈ {p | p.fst ∈ s ∧ f p.fst < p.snd}",
"tactic": "refine' ⟨hf.1 hp.1 hq.1 ha.le hb.le hab, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.344708\nα : Type ?u.344711\nβ : Type u_3\nι : Type ?u.344717\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedCancelAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf : E → β\nhf : ConvexOn 𝕜 s f\np q : E × β\nhp : p.fst ∈ s ∧ f p.fst < p.snd\nhq : q.fst ∈ s ∧ f q.fst ≤ q.snd\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ f (a • p + b • q).fst < (a • p + b • q).snd",
"tactic": "calc\n f (a • p.1 + b • q.1) ≤ a • f p.1 + b • f q.1 := hf.2 hp.1 hq.1 ha.le hb.le hab\n _ < a • p.2 + b • q.2 :=\n add_lt_add_of_lt_of_le (smul_lt_smul_of_pos hp.2 ha) (smul_le_smul_of_nonneg hq.2 hb.le)"
}
] |
[
569,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
561,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
continuousWithinAt_singleton
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.325137\nδ : Type ?u.325140\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\nx : α\n⊢ ContinuousWithinAt f {x} x",
"tactic": "simp only [ContinuousWithinAt, nhdsWithin_singleton, tendsto_pure_nhds]"
}
] |
[
759,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
758,
1
] |
Mathlib/Data/Option/Basic.lean
|
Option.bind_eq_bind
|
[] |
[
104,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Data/Int/Order/Basic.lean
|
Int.toNat_le_toNat
|
[
{
"state_after": "a✝ b✝ : ℤ\nn : ℕ\na b : ℤ\nh : a ≤ b\n⊢ a ≤ ↑(toNat b)",
"state_before": "a✝ b✝ : ℤ\nn : ℕ\na b : ℤ\nh : a ≤ b\n⊢ toNat a ≤ toNat b",
"tactic": "rw [toNat_le]"
},
{
"state_after": "no goals",
"state_before": "a✝ b✝ : ℤ\nn : ℕ\na b : ℤ\nh : a ≤ b\n⊢ a ≤ ↑(toNat b)",
"tactic": "exact le_trans h (self_le_toNat b)"
}
] |
[
517,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/Analysis/Calculus/Inverse.lean
|
ApproximatesLinearOn.continuous
|
[] |
[
173,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
11
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.measure_empty
|
[] |
[
188,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
finite_of_fin_dim_affineIndependent
|
[
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : FiniteDimensional k V\np : ι → P\nhi : AffineIndependent k p\n✝ : Nontrivial ι\n⊢ _root_.Finite ι",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : FiniteDimensional k V\np : ι → P\nhi : AffineIndependent k p\n⊢ _root_.Finite ι",
"tactic": "nontriviality ι"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : FiniteDimensional k V\np : ι → P\nhi : AffineIndependent k p\n✝ : Nontrivial ι\ninhabited_h : Inhabited ι\n⊢ _root_.Finite ι",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : FiniteDimensional k V\np : ι → P\nhi : AffineIndependent k p\n✝ : Nontrivial ι\n⊢ _root_.Finite ι",
"tactic": "inhabit ι"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : FiniteDimensional k V\np : ι → P\n✝ : Nontrivial ι\ninhabited_h : Inhabited ι\nhi : LinearIndependent k fun i => p ↑i -ᵥ p default\n⊢ _root_.Finite ι",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : FiniteDimensional k V\np : ι → P\nhi : AffineIndependent k p\n✝ : Nontrivial ι\ninhabited_h : Inhabited ι\n⊢ _root_.Finite ι",
"tactic": "rw [affineIndependent_iff_linearIndependent_vsub k p default] at hi"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : FiniteDimensional k V\np : ι → P\n✝ : Nontrivial ι\ninhabited_h : Inhabited ι\nhi : LinearIndependent k fun i => p ↑i -ᵥ p default\nthis : IsNoetherian k V := Iff.mpr IsNoetherian.iff_fg inferInstance\n⊢ _root_.Finite ι",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : FiniteDimensional k V\np : ι → P\n✝ : Nontrivial ι\ninhabited_h : Inhabited ι\nhi : LinearIndependent k fun i => p ↑i -ᵥ p default\n⊢ _root_.Finite ι",
"tactic": "letI : IsNoetherian k V := IsNoetherian.iff_fg.2 inferInstance"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : FiniteDimensional k V\np : ι → P\n✝ : Nontrivial ι\ninhabited_h : Inhabited ι\nhi : LinearIndependent k fun i => p ↑i -ᵥ p default\nthis : IsNoetherian k V := Iff.mpr IsNoetherian.iff_fg inferInstance\n⊢ _root_.Finite ι",
"tactic": "exact\n (Set.finite_singleton default).finite_of_compl (Set.finite_coe_iff.1 hi.finite_of_isNoetherian)"
}
] |
[
90,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
differentiableWithinAt_const_add_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.257762\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.257857\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\nc : F\nh : DifferentiableWithinAt 𝕜 (fun y => c + f y) s x\n⊢ DifferentiableWithinAt 𝕜 f s x",
"tactic": "simpa using h.const_add (-c)"
}
] |
[
280,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/GroupTheory/Sylow.lean
|
not_dvd_index_sylow'
|
[
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nh : p ∣ index ↑P\n⊢ False",
"state_before": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\n⊢ ¬p ∣ index ↑P",
"tactic": "intro h"
},
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nh : p ∣ index ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\n⊢ False",
"state_before": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nh : p ∣ index ↑P\n⊢ False",
"tactic": "letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex"
},
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh : p ∣ card (G ⧸ ↑P)\n⊢ False",
"state_before": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nh : p ∣ index ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\n⊢ False",
"tactic": "rw [index_eq_card (P : Subgroup G)] at h"
},
{
"state_after": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\n⊢ False",
"state_before": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh : p ∣ card (G ⧸ ↑P)\n⊢ False",
"tactic": "obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h"
},
{
"state_after": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\n⊢ False",
"state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\n⊢ False",
"tactic": "have h := IsPGroup.of_card ((orderOf_eq_card_zpowers.symm.trans hx).trans (pow_one p).symm)"
},
{
"state_after": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\nQ : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)\n⊢ False",
"state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\n⊢ False",
"tactic": "let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))"
},
{
"state_after": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\nQ : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)\nhQ : IsPGroup p { x // x ∈ Q }\n⊢ False",
"state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\nQ : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)\n⊢ False",
"tactic": "have hQ : IsPGroup p Q := by\n apply h.comap_of_ker_isPGroup\n rw [QuotientGroup.ker_mk']\n exact P.2"
},
{
"state_after": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\nQ : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)\nhQ : IsPGroup p { x // x ∈ Q }\nhp : ¬x = 1\n⊢ False",
"state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\nQ : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)\nhQ : IsPGroup p { x // x ∈ Q }\n⊢ False",
"tactic": "replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)"
},
{
"state_after": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\nQ : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)\nhQ : IsPGroup p { x // x ∈ Q }\nhp : ↑P < comap (QuotientGroup.mk' ↑P) (zpowers x)\n⊢ False",
"state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\nQ : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)\nhQ : IsPGroup p { x // x ∈ Q }\nhp : ¬x = 1\n⊢ False",
"tactic": "rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←\n comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,\n QuotientGroup.ker_mk'] at hp"
},
{
"state_after": "no goals",
"state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\nQ : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)\nhQ : IsPGroup p { x // x ∈ Q }\nhp : ↑P < comap (QuotientGroup.mk' ↑P) (zpowers x)\n⊢ False",
"tactic": "exact hp.ne' (P.3 hQ hp.le)"
},
{
"state_after": "case hϕ\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\nQ : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)\n⊢ IsPGroup p { x // x ∈ MonoidHom.ker (QuotientGroup.mk' ↑P) }",
"state_before": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\nQ : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)\n⊢ IsPGroup p { x // x ∈ Q }",
"tactic": "apply h.comap_of_ker_isPGroup"
},
{
"state_after": "case hϕ\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\nQ : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)\n⊢ IsPGroup p { x // x ∈ ↑P }",
"state_before": "case hϕ\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\nQ : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)\n⊢ IsPGroup p { x // x ∈ MonoidHom.ker (QuotientGroup.mk' ↑P) }",
"tactic": "rw [QuotientGroup.ker_mk']"
},
{
"state_after": "no goals",
"state_before": "case hϕ\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : Normal ↑P\nfP : FiniteIndex ↑P\nthis : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P\nh✝ : p ∣ card (G ⧸ ↑P)\nx : G ⧸ ↑P\nhx : orderOf x = p\nh : IsPGroup p { x_1 // x_1 ∈ zpowers x }\nQ : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)\n⊢ IsPGroup p { x // x ∈ ↑P }",
"tactic": "exact P.2"
}
] |
[
445,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
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