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/Logic/Basic.lean
|
exists_unique_eq'
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.16719\nα : Sort u_1\nκ : ι → Sort ?u.16721\np q : α → Prop\na' : α\n⊢ ∃! a, a' = a",
"tactic": "simp only [ExistsUnique, and_self, forall_eq', exists_eq']"
}
] |
[
770,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
769,
9
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPushout.of_is_coproduct
|
[
{
"state_after": "case mk.right\nC : Type u₁\ninst✝ : Category C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nc : BinaryCofan X Y\nh : IsColimit c\nt : IsInitial Z\n⊢ BinaryCofan.inr c ≫ 𝟙 c.pt = BinaryCofan.inr c",
"state_before": "case mk.right\nC : Type u₁\ninst✝ : Category C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nc : BinaryCofan X Y\nh : IsColimit c\nt : IsInitial Z\n⊢ c.ι.app { as := WalkingPair.right } ≫ (Iso.refl c.pt).hom =\n (BinaryCofan.mk (c.ι.1 { as := WalkingPair.left }) (c.ι.1 { as := WalkingPair.right })).ι.app\n { as := WalkingPair.right }",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case mk.right\nC : Type u₁\ninst✝ : Category C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nc : BinaryCofan X Y\nh : IsColimit c\nt : IsInitial Z\n⊢ BinaryCofan.inr c ≫ 𝟙 c.pt = BinaryCofan.inr c",
"tactic": "simp"
}
] |
[
388,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Order/Interval.lean
|
NonemptyInterval.coe_pure
|
[] |
[
272,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/ModelTheory/LanguageMap.lean
|
FirstOrder.Language.LHom.mk₂_funext
|
[] |
[
143,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Analysis/Calculus/LHopital.lean
|
deriv.lhopital_zero_atTop
|
[
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in atTop, deriv g x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdg : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ g x\n⊢ Tendsto (fun x => f x / g x) atTop l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in atTop, deriv g x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "have hdg : ∀ᶠ x in atTop, DifferentiableAt ℝ g x := hg'.mp\n (eventually_of_forall fun _ hg' =>\n by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))"
},
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in atTop, deriv g x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdg : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ g x\nhdf' : ∀ᶠ (x : ℝ) in atTop, HasDerivAt f (deriv f x) x\n⊢ Tendsto (fun x => f x / g x) atTop l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in atTop, deriv g x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdg : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ g x\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "have hdf' : ∀ᶠ x in atTop, HasDerivAt f (deriv f x) x :=\n hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)"
},
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in atTop, deriv g x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdg : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ g x\nhdf' : ∀ᶠ (x : ℝ) in atTop, HasDerivAt f (deriv f x) x\nhdg' : ∀ᶠ (x : ℝ) in atTop, HasDerivAt g (deriv g x) x\n⊢ Tendsto (fun x => f x / g x) atTop l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in atTop, deriv g x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdg : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ g x\nhdf' : ∀ᶠ (x : ℝ) in atTop, HasDerivAt f (deriv f x) x\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "have hdg' : ∀ᶠ x in atTop, HasDerivAt g (deriv g x) x :=\n hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in atTop, deriv g x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdg : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ g x\nhdf' : ∀ᶠ (x : ℝ) in atTop, HasDerivAt f (deriv f x) x\nhdg' : ∀ᶠ (x : ℝ) in atTop, HasDerivAt g (deriv g x) x\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "exact HasDerivAt.lhopital_zero_atTop hdf' hdg' hg' hftop hgtop hdiv"
}
] |
[
448,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Algebra/Ring/Commute.lean
|
Commute.neg_right_iff
|
[] |
[
100,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.sameCycle_inv
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.33030\nα : Type u_1\nβ : Type ?u.33036\nf g : Perm α\np : α → Prop\nx y z : α\n⊢ (∃ b, ↑(f⁻¹ ^ ↑(Equiv.neg ℤ).symm b) x = y) ↔ SameCycle f x y",
"tactic": "simp [SameCycle]"
}
] |
[
112,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
DifferentiableAt.derivWithin
|
[
{
"state_after": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nh : DifferentiableAt 𝕜 f x\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ ↑(fderivWithin 𝕜 f s x) 1 = ↑(fderiv 𝕜 f x) 1",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nh : DifferentiableAt 𝕜 f x\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ _root_.derivWithin f s x = deriv f x",
"tactic": "unfold derivWithin deriv"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nh : DifferentiableAt 𝕜 f x\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ ↑(fderivWithin 𝕜 f s x) 1 = ↑(fderiv 𝕜 f x) 1",
"tactic": "rw [h.fderivWithin hxs]"
}
] |
[
483,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
480,
1
] |
Mathlib/GroupTheory/Perm/Option.lean
|
Finset.univ_perm_option
|
[] |
[
89,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
LinearMap.span_singleton_eq_range
|
[
{
"state_after": "R : Type u_2\nR₂ : Type ?u.329641\nK : Type ?u.329644\nM : Type u_1\nM₂ : Type ?u.329650\nV : Type ?u.329653\nS : Type ?u.329656\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\n⊢ y ∈ span R {x} ↔ ∃ y_1, ↑(toSpanSingleton R M x) y_1 = y",
"state_before": "R : Type u_2\nR₂ : Type ?u.329641\nK : Type ?u.329644\nM : Type u_1\nM₂ : Type ?u.329650\nV : Type ?u.329653\nS : Type ?u.329656\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\n⊢ y ∈ span R {x} ↔ y ∈ range (toSpanSingleton R M x)",
"tactic": "refine' Iff.trans _ LinearMap.mem_range.symm"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₂ : Type ?u.329641\nK : Type ?u.329644\nM : Type u_1\nM₂ : Type ?u.329650\nV : Type ?u.329653\nS : Type ?u.329656\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\n⊢ y ∈ span R {x} ↔ ∃ y_1, ↑(toSpanSingleton R M x) y_1 = y",
"tactic": "exact mem_span_singleton"
}
] |
[
922,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
919,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snorm'_const_smul_le
|
[] |
[
1470,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1467,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.BoundedFormula.realize_foldr_sup
|
[
{
"state_after": "case nil\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.66568\nP : Type ?u.66571\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\nv : α → M\nxs : Fin n → M\n⊢ Realize (List.foldr (fun x x_1 => x ⊔ x_1) ⊥ []) v xs ↔ ∃ φ, φ ∈ [] ∧ Realize φ v xs\n\ncase cons\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.66568\nP : Type ?u.66571\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn l✝ : ℕ\nφ✝ ψ : BoundedFormula L α l✝\nθ : BoundedFormula L α (Nat.succ l✝)\nv✝ : α → M\nxs✝ : Fin l✝ → M\nv : α → M\nxs : Fin n → M\nφ : BoundedFormula L α n\nl : List (BoundedFormula L α n)\nih : Realize (List.foldr (fun x x_1 => x ⊔ x_1) ⊥ l) v xs ↔ ∃ φ, φ ∈ l ∧ Realize φ v xs\n⊢ Realize (List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (φ :: l)) v xs ↔ ∃ φ_1, φ_1 ∈ φ :: l ∧ Realize φ_1 v xs",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.66568\nP : Type ?u.66571\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn l✝ : ℕ\nφ ψ : BoundedFormula L α l✝\nθ : BoundedFormula L α (Nat.succ l✝)\nv✝ : α → M\nxs✝ : Fin l✝ → M\nl : List (BoundedFormula L α n)\nv : α → M\nxs : Fin n → M\n⊢ Realize (List.foldr (fun x x_1 => x ⊔ x_1) ⊥ l) v xs ↔ ∃ φ, φ ∈ l ∧ Realize φ v xs",
"tactic": "induction' l with φ l ih"
},
{
"state_after": "no goals",
"state_before": "case nil\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.66568\nP : Type ?u.66571\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\nv : α → M\nxs : Fin n → M\n⊢ Realize (List.foldr (fun x x_1 => x ⊔ x_1) ⊥ []) v xs ↔ ∃ φ, φ ∈ [] ∧ Realize φ v xs",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.66568\nP : Type ?u.66571\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn l✝ : ℕ\nφ✝ ψ : BoundedFormula L α l✝\nθ : BoundedFormula L α (Nat.succ l✝)\nv✝ : α → M\nxs✝ : Fin l✝ → M\nv : α → M\nxs : Fin n → M\nφ : BoundedFormula L α n\nl : List (BoundedFormula L α n)\nih : Realize (List.foldr (fun x x_1 => x ⊔ x_1) ⊥ l) v xs ↔ ∃ φ, φ ∈ l ∧ Realize φ v xs\n⊢ Realize (List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (φ :: l)) v xs ↔ ∃ φ_1, φ_1 ∈ φ :: l ∧ Realize φ_1 v xs",
"tactic": "simp_rw [List.foldr_cons, realize_sup, ih, exists_prop, List.mem_cons, or_and_right, exists_or,\n exists_eq_left]"
}
] |
[
340,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Data/Polynomial/Module.lean
|
PolynomialModule.add_apply
|
[] |
[
81,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Algebra/BigOperators/Ring.lean
|
Finset.mul_sum
|
[] |
[
59,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Algebra/BigOperators/Fin.lean
|
List.prod_ofFn
|
[
{
"state_after": "case h.e'_2.h.e'_4\nα : Type u_1\nβ : Type ?u.226607\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\n⊢ ofFn f = take n (ofFn f)\n\ncase h.e'_3.h\nα : Type u_1\nβ : Type ?u.226607\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\n⊢ univ = Finset.filter (fun j => ↑j < n) univ",
"state_before": "α : Type u_1\nβ : Type ?u.226607\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\n⊢ prod (ofFn f) = ∏ i : Fin n, f i",
"tactic": "convert prod_take_ofFn f n"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_4\nα : Type u_1\nβ : Type ?u.226607\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\n⊢ ofFn f = take n (ofFn f)",
"tactic": "rw [take_all_of_le (le_of_eq (length_ofFn f))]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h\nα : Type u_1\nβ : Type ?u.226607\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\n⊢ univ = Finset.filter (fun j => ↑j < n) univ",
"tactic": "simp"
}
] |
[
485,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Mathlib/Algebra/Module/Submodule/Pointwise.lean
|
Submodule.neg_toAddSubmonoid
|
[] |
[
73,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.subsingleton_of_preimage
|
[
{
"state_after": "case intro.intro\nα✝ : Type ?u.101676\nβ✝ : Type ?u.101679\nγ : Type ?u.101682\nι : Sort ?u.101685\nι' : Sort ?u.101688\ns✝ : Set α✝\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Surjective f\ns : Set β\nhs : Set.Subsingleton (f ⁻¹' s)\nx : α\nhx : f x ∈ s\ny : α\nhy : f y ∈ s\n⊢ f x = f y",
"state_before": "α✝ : Type ?u.101676\nβ✝ : Type ?u.101679\nγ : Type ?u.101682\nι : Sort ?u.101685\nι' : Sort ?u.101688\ns✝ : Set α✝\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Surjective f\ns : Set β\nhs : Set.Subsingleton (f ⁻¹' s)\nfx : β\nhx : fx ∈ s\nfy : β\nhy : fy ∈ s\n⊢ fx = fy",
"tactic": "rcases hf fx, hf fy with ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα✝ : Type ?u.101676\nβ✝ : Type ?u.101679\nγ : Type ?u.101682\nι : Sort ?u.101685\nι' : Sort ?u.101688\ns✝ : Set α✝\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Surjective f\ns : Set β\nhs : Set.Subsingleton (f ⁻¹' s)\nx : α\nhx : f x ∈ s\ny : α\nhy : f y ∈ s\n⊢ f x = f y",
"tactic": "exact congr_arg f (hs hx hy)"
}
] |
[
1238,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1235,
1
] |
Mathlib/Order/Filter/Extr.lean
|
IsMinOn.min
|
[] |
[
600,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
598,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Succ.rec
|
[
{
"state_after": "case intro\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\nP : α → Prop\nm : α\nh0 : P m\nh1 : ∀ (n : α), m ≤ n → P n → P (succ n)\nn : ℕ\nhmn : m ≤ (succ^[n]) m\n⊢ P ((succ^[n]) m)",
"state_before": "α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\nP : α → Prop\nm : α\nh0 : P m\nh1 : ∀ (n : α), m ≤ n → P n → P (succ n)\nn : α\nhmn : m ≤ n\n⊢ P n",
"tactic": "obtain ⟨n, rfl⟩ := hmn.exists_succ_iterate"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\nP : α → Prop\nm : α\nh0 : P m\nh1 : ∀ (n : α), m ≤ n → P n → P (succ n)\nn : ℕ\n⊢ P ((succ^[n]) m)",
"state_before": "case intro\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\nP : α → Prop\nm : α\nh0 : P m\nh1 : ∀ (n : α), m ≤ n → P n → P (succ n)\nn : ℕ\nhmn : m ≤ (succ^[n]) m\n⊢ P ((succ^[n]) m)",
"tactic": "clear hmn"
},
{
"state_after": "case intro.zero\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\nP : α → Prop\nm : α\nh0 : P m\nh1 : ∀ (n : α), m ≤ n → P n → P (succ n)\n⊢ P ((succ^[Nat.zero]) m)\n\ncase intro.succ\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\nP : α → Prop\nm : α\nh0 : P m\nh1 : ∀ (n : α), m ≤ n → P n → P (succ n)\nn : ℕ\nih : P ((succ^[n]) m)\n⊢ P ((succ^[Nat.succ n]) m)",
"state_before": "case intro\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\nP : α → Prop\nm : α\nh0 : P m\nh1 : ∀ (n : α), m ≤ n → P n → P (succ n)\nn : ℕ\n⊢ P ((succ^[n]) m)",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case intro.zero\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\nP : α → Prop\nm : α\nh0 : P m\nh1 : ∀ (n : α), m ≤ n → P n → P (succ n)\n⊢ P ((succ^[Nat.zero]) m)",
"tactic": "exact h0"
},
{
"state_after": "case intro.succ\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\nP : α → Prop\nm : α\nh0 : P m\nh1 : ∀ (n : α), m ≤ n → P n → P (succ n)\nn : ℕ\nih : P ((succ^[n]) m)\n⊢ P (succ ((succ^[n]) m))",
"state_before": "case intro.succ\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\nP : α → Prop\nm : α\nh0 : P m\nh1 : ∀ (n : α), m ≤ n → P n → P (succ n)\nn : ℕ\nih : P ((succ^[n]) m)\n⊢ P ((succ^[Nat.succ n]) m)",
"tactic": "rw [Function.iterate_succ_apply']"
},
{
"state_after": "no goals",
"state_before": "case intro.succ\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\na b : α\nP : α → Prop\nm : α\nh0 : P m\nh1 : ∀ (n : α), m ≤ n → P n → P (succ n)\nn : ℕ\nih : P ((succ^[n]) m)\n⊢ P (succ ((succ^[n]) m))",
"tactic": "exact h1 _ (id_le_iterate_of_id_le le_succ n m) ih"
}
] |
[
1393,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1387,
1
] |
Mathlib/Algebra/Lie/SkewAdjoint.lean
|
skewAdjointMatricesLieSubalgebraEquiv_apply
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nn : Type w\ninst✝² : CommRing R\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nJ P : Matrix n n R\nh : Invertible P\nA : { x // x ∈ skewAdjointMatricesLieSubalgebra J }\n⊢ ↑(↑(skewAdjointMatricesLieSubalgebraEquiv J P h) A) = P⁻¹ ⬝ ↑A ⬝ P",
"tactic": "simp [skewAdjointMatricesLieSubalgebraEquiv]"
}
] |
[
148,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
fderivWithin_const_apply
|
[
{
"state_after": "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.1390092\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.1390187\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\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderiv 𝕜 (fun x => c) x = 0",
"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.1390092\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.1390187\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\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderivWithin 𝕜 (fun x => c) s x = 0",
"tactic": "rw [DifferentiableAt.fderivWithin (differentiableAt_const _) hxs]"
},
{
"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.1390092\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.1390187\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\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderiv 𝕜 (fun x => c) x = 0",
"tactic": "exact fderiv_const_apply _"
}
] |
[
1100,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1097,
1
] |
Mathlib/Order/Filter/Cofinite.lean
|
Filter.eventually_cofinite
|
[] |
[
49,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/Topology/Algebra/Order/ExtrClosure.lean
|
IsMinOn.closure
|
[] |
[
37,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
11
] |
Mathlib/Probability/ProbabilityMassFunction/Basic.lean
|
Pmf.toMeasure_apply_inter_support
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.84659\nγ : Type ?u.84662\ninst✝ : MeasurableSpace α\np : Pmf α\ns t : Set α\nhs : MeasurableSet s\nhp : MeasurableSet (support p)\n⊢ ↑↑(toMeasure p) (s ∩ support p) = ↑↑(toMeasure p) s",
"tactic": "simp [p.toMeasure_apply_eq_toOuterMeasure_apply s hs,\n p.toMeasure_apply_eq_toOuterMeasure_apply _ (hs.inter hp)]"
}
] |
[
281,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.coeff_mul_X'
|
[
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns✝ : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\ninst✝ : DecidableEq σ\nm : σ →₀ ℕ\ns : σ\np : MvPolynomial σ R\n⊢ (if Finsupp.single s 1 ≤ m then coeff (m - Finsupp.single s 1) p * 1 else 0) =\n if s ∈ m.support then coeff (m - Finsupp.single s 1) p else 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns✝ : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\ninst✝ : DecidableEq σ\nm : σ →₀ ℕ\ns : σ\np : MvPolynomial σ R\n⊢ coeff m (p * X s) = if s ∈ m.support then coeff (m - Finsupp.single s 1) p else 0",
"tactic": "refine' (coeff_mul_monomial' _ _ _ _).trans _"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns✝ : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\ninst✝ : DecidableEq σ\nm : σ →₀ ℕ\ns : σ\np : MvPolynomial σ R\n⊢ (if Finsupp.single s 1 ≤ m then coeff (m - Finsupp.single s 1) p * 1 else 0) =\n if s ∈ m.support then coeff (m - Finsupp.single s 1) p else 0",
"tactic": "simp_rw [Finsupp.single_le_iff, Finsupp.mem_support_iff, Nat.succ_le_iff, pos_iff_ne_zero,\n mul_one]"
}
] |
[
797,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
793,
1
] |
Mathlib/Data/Setoid/Basic.lean
|
Setoid.ker_iff_mem_preimage
|
[] |
[
290,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean
|
Polynomial.Chebyshev.T_complex_cos
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\n⊢ eval (cos θ) (T ℂ 0) = cos (↑0 * θ)",
"tactic": "simp only [T_zero, eval_one, Nat.cast_zero, MulZeroClass.zero_mul, cos_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\n⊢ eval (cos θ) (T ℂ 1) = cos (↑1 * θ)",
"tactic": "simp only [eval_X, one_mul, T_one, Nat.cast_one]"
},
{
"state_after": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\nn : ℕ\nthis : 2 = ↑2\n⊢ eval (cos θ) (T ℂ (n + 2)) = cos (↑(n + 2) * θ)",
"state_before": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\nn : ℕ\n⊢ eval (cos θ) (T ℂ (n + 2)) = cos (↑(n + 2) * θ)",
"tactic": "have : (2 : ℂ[X]) = (2 : ℕ) := by norm_num"
},
{
"state_after": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\nn : ℕ\nthis : 2 = ↑2\n⊢ ↑2 * cos θ * eval (cos θ) (T ℂ (n + 1)) - eval (cos θ) (T ℂ n) = cos (↑(n + 2) * θ)",
"state_before": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\nn : ℕ\nthis : 2 = ↑2\n⊢ eval (cos θ) (T ℂ (n + 2)) = cos (↑(n + 2) * θ)",
"tactic": "simp only [this, eval_X, eval_one, T_add_two, eval_sub, eval_mul, eval_nat_cast]"
},
{
"state_after": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\nn : ℕ\nthis : 2 = ↑2\n⊢ 2 * cos θ * eval (cos θ) (T ℂ (n + 1)) - eval (cos θ) (T ℂ n) = cos ((↑n + 2) * θ)",
"state_before": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\nn : ℕ\nthis : 2 = ↑2\n⊢ ↑2 * cos θ * eval (cos θ) (T ℂ (n + 1)) - eval (cos θ) (T ℂ n) = cos (↑(n + 2) * θ)",
"tactic": "simp only [Nat.cast_ofNat, Nat.cast_add]"
},
{
"state_after": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\nn : ℕ\nthis : 2 = ↑2\n⊢ 2 * cos θ * cos (↑(n + 1) * θ) - cos (↑n * θ) = cos ((↑n + 2) * θ)",
"state_before": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\nn : ℕ\nthis : 2 = ↑2\n⊢ 2 * cos θ * eval (cos θ) (T ℂ (n + 1)) - eval (cos θ) (T ℂ n) = cos ((↑n + 2) * θ)",
"tactic": "rw [T_complex_cos (n + 1), T_complex_cos n]"
},
{
"state_after": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\nn : ℕ\nthis : 2 = ↑2\n⊢ 2 * (cos θ * (cos (↑n * θ) * cos θ - sin (↑n * θ) * sin θ)) - cos (↑n * θ) =\n cos (↑n * θ) * (2 * cos θ ^ 2 - 1) - sin (↑n * θ) * (2 * (sin θ * cos θ))",
"state_before": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\nn : ℕ\nthis : 2 = ↑2\n⊢ 2 * cos θ * cos (↑(n + 1) * θ) - cos (↑n * θ) = cos ((↑n + 2) * θ)",
"tactic": "simp only [Nat.cast_add, Nat.cast_one, add_mul, cos_add, one_mul, mul_assoc, sin_two_mul,\n cos_two_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\nn : ℕ\nthis : 2 = ↑2\n⊢ 2 * (cos θ * (cos (↑n * θ) * cos θ - sin (↑n * θ) * sin θ)) - cos (↑n * θ) =\n cos (↑n * θ) * (2 * cos θ ^ 2 - 1) - sin (↑n * θ) * (2 * (sin θ * cos θ))",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.105956\nA : Type ?u.105959\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nθ : ℂ\nn : ℕ\n⊢ 2 = ↑2",
"tactic": "norm_num"
}
] |
[
88,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.basicOpen_le_basicOpen_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf g : R\n⊢ basicOpen f ≤ basicOpen g ↔ f ∈ Ideal.radical (Ideal.span {g})",
"tactic": "rw [← SetLike.coe_subset_coe, basicOpen_eq_zeroLocus_compl, basicOpen_eq_zeroLocus_compl,\n Set.compl_subset_compl, zeroLocus_subset_zeroLocus_singleton_iff]"
}
] |
[
799,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
796,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.EqOnSource.source_inter_preimage_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.57966\nδ : Type ?u.57969\ne✝ : LocalEquiv α β\ne'✝ : LocalEquiv β γ\ne e' : LocalEquiv α β\nhe : e ≈ e'\ns : Set β\n⊢ e.source ∩ ↑e ⁻¹' s = e'.source ∩ ↑e' ⁻¹' s",
"tactic": "rw [he.eqOn.inter_preimage_eq, source_eq he]"
}
] |
[
886,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
885,
1
] |
Mathlib/Analysis/Calculus/IteratedDeriv.lean
|
iteratedDeriv_succ'
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.162489\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\n⊢ (deriv^[n + 1]) f = (deriv^[n]) (deriv f)",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.162489\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\n⊢ iteratedDeriv (n + 1) f = iteratedDeriv n (deriv f)",
"tactic": "rw [iteratedDeriv_eq_iterate, iteratedDeriv_eq_iterate]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.162489\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\n⊢ (deriv^[n + 1]) f = (deriv^[n]) (deriv f)",
"tactic": "rfl"
}
] |
[
308,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
collinear_iff_of_mem
|
[
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\n⊢ (∃ v₀, ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v) ↔ ∃ v, ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\n⊢ Collinear k s ↔ ∃ v, ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀",
"tactic": "simp_rw [collinear_iff_rank_le_one, rank_submodule_le_one_iff', Submodule.le_span_singleton_iff]"
},
{
"state_after": "case mp\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\n⊢ (∃ v₀, ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v) → ∃ v, ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\n\ncase mpr\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\n⊢ (∃ v, ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀) → ∃ v₀, ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\n⊢ (∃ v₀, ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v) ↔ ∃ v, ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv₀ : V\nhv : ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v\n⊢ ∃ v, ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀",
"state_before": "case mp\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\n⊢ (∃ v₀, ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v) → ∃ v, ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀",
"tactic": "rintro ⟨v₀, hv⟩"
},
{
"state_after": "case mp.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv₀ : V\nhv : ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v\n⊢ ∀ (p : P), p ∈ s → ∃ r, p = r • v₀ +ᵥ p₀",
"state_before": "case mp.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv₀ : V\nhv : ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v\n⊢ ∃ v, ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀",
"tactic": "use v₀"
},
{
"state_after": "case mp.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv₀ : V\nhv : ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v\np : P\nhp : p ∈ s\n⊢ ∃ r, p = r • v₀ +ᵥ p₀",
"state_before": "case mp.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv₀ : V\nhv : ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v\n⊢ ∀ (p : P), p ∈ s → ∃ r, p = r • v₀ +ᵥ p₀",
"tactic": "intro p hp"
},
{
"state_after": "case mp.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv₀ : V\nhv : ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v\np : P\nhp : p ∈ s\nr : k\nhr : r • v₀ = p -ᵥ p₀\n⊢ ∃ r, p = r • v₀ +ᵥ p₀",
"state_before": "case mp.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv₀ : V\nhv : ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v\np : P\nhp : p ∈ s\n⊢ ∃ r, p = r • v₀ +ᵥ p₀",
"tactic": "obtain ⟨r, hr⟩ := hv (p -ᵥ p₀) (vsub_mem_vectorSpan k hp h)"
},
{
"state_after": "case mp.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv₀ : V\nhv : ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v\np : P\nhp : p ∈ s\nr : k\nhr : r • v₀ = p -ᵥ p₀\n⊢ p = r • v₀ +ᵥ p₀",
"state_before": "case mp.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv₀ : V\nhv : ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v\np : P\nhp : p ∈ s\nr : k\nhr : r • v₀ = p -ᵥ p₀\n⊢ ∃ r, p = r • v₀ +ᵥ p₀",
"tactic": "use r"
},
{
"state_after": "case mp.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv₀ : V\nhv : ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v\np : P\nhp : p ∈ s\nr : k\nhr : r • v₀ = p -ᵥ p₀\n⊢ p -ᵥ p₀ = r • v₀",
"state_before": "case mp.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv₀ : V\nhv : ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v\np : P\nhp : p ∈ s\nr : k\nhr : r • v₀ = p -ᵥ p₀\n⊢ p = r • v₀ +ᵥ p₀",
"tactic": "rw [eq_vadd_iff_vsub_eq]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv₀ : V\nhv : ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v\np : P\nhp : p ∈ s\nr : k\nhr : r • v₀ = p -ᵥ p₀\n⊢ p -ᵥ p₀ = r • v₀",
"tactic": "exact hr.symm"
},
{
"state_after": "case mpr.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\n⊢ ∃ v₀, ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v",
"state_before": "case mpr\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\n⊢ (∃ v, ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀) → ∃ v₀, ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v",
"tactic": "rintro ⟨v, hp₀v⟩"
},
{
"state_after": "case mpr.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\n⊢ ∀ (v_1 : V), v_1 ∈ vectorSpan k s → ∃ r, r • v = v_1",
"state_before": "case mpr.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\n⊢ ∃ v₀, ∀ (v : V), v ∈ vectorSpan k s → ∃ r, r • v₀ = v",
"tactic": "use v"
},
{
"state_after": "case mpr.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\n⊢ ∃ r, r • v = w",
"state_before": "case mpr.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\n⊢ ∀ (v_1 : V), v_1 ∈ vectorSpan k s → ∃ r, r • v = v_1",
"tactic": "intro w hw"
},
{
"state_after": "case mpr.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nhs : vectorSpan k s ≤ Submodule.span k {v}\n⊢ ∃ r, r • v = w",
"state_before": "case mpr.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\n⊢ ∃ r, r • v = w",
"tactic": "have hs : vectorSpan k s ≤ k ∙ v := by\n rw [vectorSpan_eq_span_vsub_set_right k h, Submodule.span_le, Set.subset_def]\n intro x hx\n rw [SetLike.mem_coe, Submodule.mem_span_singleton]\n rw [Set.mem_image] at hx\n rcases hx with ⟨p, hp, rfl⟩\n rcases hp₀v p hp with ⟨r, rfl⟩\n use r\n simp"
},
{
"state_after": "case mpr.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nhs : vectorSpan k s ≤ Submodule.span k {v}\nhw' : w ∈ Submodule.span k {v}\n⊢ ∃ r, r • v = w",
"state_before": "case mpr.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nhs : vectorSpan k s ≤ Submodule.span k {v}\n⊢ ∃ r, r • v = w",
"tactic": "have hw' := SetLike.le_def.1 hs hw"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nhs : vectorSpan k s ≤ Submodule.span k {v}\nhw' : w ∈ Submodule.span k {v}\n⊢ ∃ r, r • v = w",
"tactic": "rwa [Submodule.mem_span_singleton] at hw'"
},
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\n⊢ ∀ (x : V), x ∈ (fun x => x -ᵥ p₀) '' s → x ∈ ↑(Submodule.span k {v})",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\n⊢ vectorSpan k s ≤ Submodule.span k {v}",
"tactic": "rw [vectorSpan_eq_span_vsub_set_right k h, Submodule.span_le, Set.subset_def]"
},
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nx : V\nhx : x ∈ (fun x => x -ᵥ p₀) '' s\n⊢ x ∈ ↑(Submodule.span k {v})",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\n⊢ ∀ (x : V), x ∈ (fun x => x -ᵥ p₀) '' s → x ∈ ↑(Submodule.span k {v})",
"tactic": "intro x hx"
},
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nx : V\nhx : x ∈ (fun x => x -ᵥ p₀) '' s\n⊢ ∃ a, a • v = x",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nx : V\nhx : x ∈ (fun x => x -ᵥ p₀) '' s\n⊢ x ∈ ↑(Submodule.span k {v})",
"tactic": "rw [SetLike.mem_coe, Submodule.mem_span_singleton]"
},
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nx : V\nhx : ∃ x_1, x_1 ∈ s ∧ x_1 -ᵥ p₀ = x\n⊢ ∃ a, a • v = x",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nx : V\nhx : x ∈ (fun x => x -ᵥ p₀) '' s\n⊢ ∃ a, a • v = x",
"tactic": "rw [Set.mem_image] at hx"
},
{
"state_after": "case intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\np : P\nhp : p ∈ s\n⊢ ∃ a, a • v = p -ᵥ p₀",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nx : V\nhx : ∃ x_1, x_1 ∈ s ∧ x_1 -ᵥ p₀ = x\n⊢ ∃ a, a • v = x",
"tactic": "rcases hx with ⟨p, hp, rfl⟩"
},
{
"state_after": "case intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nr : k\nhp : r • v +ᵥ p₀ ∈ s\n⊢ ∃ a, a • v = r • v +ᵥ p₀ -ᵥ p₀",
"state_before": "case intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\np : P\nhp : p ∈ s\n⊢ ∃ a, a • v = p -ᵥ p₀",
"tactic": "rcases hp₀v p hp with ⟨r, rfl⟩"
},
{
"state_after": "case intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nr : k\nhp : r • v +ᵥ p₀ ∈ s\n⊢ r • v = r • v +ᵥ p₀ -ᵥ p₀",
"state_before": "case intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nr : k\nhp : r • v +ᵥ p₀ ∈ s\n⊢ ∃ a, a • v = r • v +ᵥ p₀ -ᵥ p₀",
"tactic": "use r"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.250748\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\nr : k\nhp : r • v +ᵥ p₀ ∈ s\n⊢ r • v = r • v +ᵥ p₀ -ᵥ p₀",
"tactic": "simp"
}
] |
[
405,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
ContinuousWithinAt.lowerSemicontinuousWithinAt
|
[] |
[
286,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/FieldTheory/Laurent.lean
|
RatFunc.taylor_mem_nonZeroDivisors
|
[
{
"state_after": "R : Type u\ninst✝ : CommRing R\nhdomain : IsDomain R\nr s : R\np q : R[X]\nf : RatFunc R\nhp : p ∈ R[X]⁰\n⊢ ∀ (x : (fun x => R[X]) p), x * ↑(taylor r) p = 0 → x = 0",
"state_before": "R : Type u\ninst✝ : CommRing R\nhdomain : IsDomain R\nr s : R\np q : R[X]\nf : RatFunc R\nhp : p ∈ R[X]⁰\n⊢ ↑(taylor r) p ∈ R[X]⁰",
"tactic": "rw [mem_nonZeroDivisors_iff]"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\nhdomain : IsDomain R\nr s : R\np q : R[X]\nf : RatFunc R\nhp : p ∈ R[X]⁰\nx : R[X]\nhx : x * ↑(taylor r) p = 0\n⊢ x = 0",
"state_before": "R : Type u\ninst✝ : CommRing R\nhdomain : IsDomain R\nr s : R\np q : R[X]\nf : RatFunc R\nhp : p ∈ R[X]⁰\n⊢ ∀ (x : (fun x => R[X]) p), x * ↑(taylor r) p = 0 → x = 0",
"tactic": "intro x hx"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\nhdomain : IsDomain R\nr s : R\np q : R[X]\nf : RatFunc R\nhp : p ∈ R[X]⁰\nx : R[X]\nhx : x * ↑(taylor r) p = 0\nthis : x = ↑(taylor (r - r)) x\n⊢ x = 0",
"state_before": "R : Type u\ninst✝ : CommRing R\nhdomain : IsDomain R\nr s : R\np q : R[X]\nf : RatFunc R\nhp : p ∈ R[X]⁰\nx : R[X]\nhx : x * ↑(taylor r) p = 0\n⊢ x = 0",
"tactic": "have : x = taylor (r - r) x := by simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nhdomain : IsDomain R\nr s : R\np q : R[X]\nf : RatFunc R\nhp : p ∈ R[X]⁰\nx : R[X]\nhx : x * ↑(taylor r) p = 0\nthis : x = ↑(taylor (r - r)) x\n⊢ x = 0",
"tactic": "rwa [this, sub_eq_add_neg, ← taylor_taylor, ← taylor_mul,\n LinearMap.map_eq_zero_iff _ (taylor_injective _), mul_right_mem_nonZeroDivisors_eq_zero_iff hp,\n LinearMap.map_eq_zero_iff _ (taylor_injective _)] at hx"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nhdomain : IsDomain R\nr s : R\np q : R[X]\nf : RatFunc R\nhp : p ∈ R[X]⁰\nx : R[X]\nhx : x * ↑(taylor r) p = 0\n⊢ x = ↑(taylor (r - r)) x",
"tactic": "simp"
}
] |
[
48,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/ModelTheory/Types.lean
|
FirstOrder.Language.Theory.CompleteType.setOf_subset_eq_empty_iff
|
[
{
"state_after": "L : Language\nT : Theory L\nα : Type w\nS : Theory (L[[α]])\n⊢ IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ S) ↔ ∃ x, x ∈ {p | S ⊆ ↑p}",
"state_before": "L : Language\nT : Theory L\nα : Type w\nS : Theory (L[[α]])\n⊢ {p | S ⊆ ↑p} = ∅ ↔ ¬IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ S)",
"tactic": "rw [iff_not_comm, ← not_nonempty_iff_eq_empty, Classical.not_not, Set.Nonempty]"
},
{
"state_after": "L : Language\nT : Theory L\nα : Type w\nS : Theory (L[[α]])\n⊢ (∃ x, x ∈ {p | S ⊆ ↑p}) → IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ S)",
"state_before": "L : Language\nT : Theory L\nα : Type w\nS : Theory (L[[α]])\n⊢ IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ S) ↔ ∃ x, x ∈ {p | S ⊆ ↑p}",
"tactic": "refine'\n ⟨fun h =>\n ⟨⟨L[[α]].completeTheory h.some, (subset_union_left _ S).trans completeTheory.subset,\n completeTheory.isMaximal (L[[α]]) h.some⟩,\n (subset_union_right ((L.lhomWithConstants α).onTheory T) _).trans completeTheory.subset⟩,\n _⟩"
},
{
"state_after": "case intro\nL : Language\nT : Theory L\nα : Type w\nS : Theory (L[[α]])\np : CompleteType T α\nhp : p ∈ {p | S ⊆ ↑p}\n⊢ IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ S)",
"state_before": "L : Language\nT : Theory L\nα : Type w\nS : Theory (L[[α]])\n⊢ (∃ x, x ∈ {p | S ⊆ ↑p}) → IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ S)",
"tactic": "rintro ⟨p, hp⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nL : Language\nT : Theory L\nα : Type w\nS : Theory (L[[α]])\np : CompleteType T α\nhp : p ∈ {p | S ⊆ ↑p}\n⊢ IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ S)",
"tactic": "exact p.isMaximal.1.mono (union_subset p.subset hp)"
}
] |
[
128,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
mul_finsum
|
[] |
[
1193,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1191,
1
] |
Mathlib/Data/ENat/Basic.lean
|
ENat.some_eq_coe
|
[] |
[
63,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
9
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.coequalizer.π_of_eq
|
[] |
[
1056,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1055,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.top_rpow_of_pos
|
[
{
"state_after": "no goals",
"state_before": "y : ℝ\nh : 0 < y\n⊢ ⊤ ^ y = ⊤",
"tactic": "simp [top_rpow_def, h]"
}
] |
[
315,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/Data/Sum/Order.lean
|
Sum.Lex.inr_strictMono
|
[] |
[
418,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
417,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.map_pow
|
[] |
[
274,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
11
] |
Std/Data/List/Lemmas.lean
|
List.subset_append_left
|
[] |
[
249,
98
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
249,
9
] |
Mathlib/Data/List/AList.lean
|
AList.toAList_cons
|
[] |
[
345,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/Data/Set/Function.lean
|
Function.Semiconj.injOn_range
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.112312\nι : Sort ?u.112315\nπ : α → Type ?u.112320\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : Injective fa\nhf : InjOn f (fa '' univ)\n⊢ InjOn fb (f '' univ)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.112312\nι : Sort ?u.112315\nπ : α → Type ?u.112320\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : Injective fa\nhf : InjOn f (range fa)\n⊢ InjOn fb (range f)",
"tactic": "rw [← image_univ] at *"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.112312\nι : Sort ?u.112315\nπ : α → Type ?u.112320\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : Injective fa\nhf : InjOn f (fa '' univ)\n⊢ InjOn fb (f '' univ)",
"tactic": "exact h.injOn_image (ha.injOn univ) hf"
}
] |
[
1649,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1646,
1
] |
Mathlib/CategoryTheory/Subobject/Limits.lean
|
CategoryTheory.Limits.kernelSubobjectMap_id
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nX Y Z : C\ninst✝² : HasZeroMorphisms C\nf : X ⟶ Y\ninst✝¹ : HasKernel f\nX' Y' : C\nf' : X' ⟶ Y'\ninst✝ : HasKernel f'\n⊢ kernelSubobjectMap (𝟙 (Arrow.mk f)) = 𝟙 (underlying.obj (kernelSubobject f))",
"tactic": "aesop_cat"
}
] |
[
167,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Order/RelClasses.lean
|
Set.not_bounded_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nr✝ : α → α → Prop\ns✝ : β → β → Prop\nr : α → α → Prop\ns : Set α\n⊢ ¬Bounded r s ↔ Unbounded r s",
"tactic": "simp only [Bounded, Unbounded, not_forall, not_exists, exists_prop, not_and, not_not]"
}
] |
[
538,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
537,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.sin_add
|
[
{
"state_after": "case h\nθ₂ : Angle\nx✝ : ℝ\n⊢ sin (↑x✝ + θ₂) = sin ↑x✝ * cos θ₂ + cos ↑x✝ * sin θ₂",
"state_before": "θ₁ θ₂ : Angle\n⊢ sin (θ₁ + θ₂) = sin θ₁ * cos θ₂ + cos θ₁ * sin θ₂",
"tactic": "induction θ₁ using Real.Angle.induction_on"
},
{
"state_after": "case h.h\nx✝¹ x✝ : ℝ\n⊢ sin (↑x✝¹ + ↑x✝) = sin ↑x✝¹ * cos ↑x✝ + cos ↑x✝¹ * sin ↑x✝",
"state_before": "case h\nθ₂ : Angle\nx✝ : ℝ\n⊢ sin (↑x✝ + θ₂) = sin ↑x✝ * cos θ₂ + cos ↑x✝ * sin θ₂",
"tactic": "induction θ₂ using Real.Angle.induction_on"
},
{
"state_after": "no goals",
"state_before": "case h.h\nx✝¹ x✝ : ℝ\n⊢ sin (↑x✝¹ + ↑x✝) = sin ↑x✝¹ * cos ↑x✝ + cos ↑x✝¹ * sin ↑x✝",
"tactic": "exact Real.sin_add _ _"
}
] |
[
434,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
431,
1
] |
Mathlib/Data/Finset/Preimage.lean
|
Finset.image_subset_iff_subset_preimage
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝ : DecidableEq β\nf : α → β\ns : Finset α\nt : Finset β\nhf : InjOn f (f ⁻¹' ↑t)\n⊢ (∀ (x : α), x ∈ s → f x ∈ t) ↔ s ⊆ preimage t f hf",
"tactic": "simp only [subset_iff, mem_preimage]"
}
] |
[
89,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.inv_zero
|
[
{
"state_after": "n : ℕ\n⊢ ↑(Nat.gcdA 0 (n + 1)) = 0",
"state_before": "n : ℕ\n⊢ ↑(Nat.gcdA (val 0) (n + 1)) = 0",
"tactic": "rw [val_zero]"
},
{
"state_after": "n : ℕ\n⊢ ↑(n + 1, 0, 1).snd.fst = 0",
"state_before": "n : ℕ\n⊢ ↑(Nat.gcdA 0 (n + 1)) = 0",
"tactic": "unfold Nat.gcdA Nat.xgcd Nat.xgcdAux"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ↑(n + 1, 0, 1).snd.fst = 0",
"tactic": "rfl"
}
] |
[
642,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
636,
1
] |
src/lean/Init/SimpLemmas.lean
|
and_false
|
[] |
[
85,
77
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
85,
9
] |
Mathlib/Algebra/Order/CompleteField.lean
|
LinearOrderedField.cutMap_bddAbove
|
[
{
"state_after": "case intro\nF : Type ?u.11263\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.11272\ninst✝² : LinearOrderedField α\ninst✝¹ : LinearOrderedField β\na✝ a₁ a₂ : α\nb : β\nq✝ : ℚ\ninst✝ : Archimedean α\na : α\nq : ℚ\nhq : a < ↑q\n⊢ BddAbove (cutMap β a)",
"state_before": "F : Type ?u.11263\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.11272\ninst✝² : LinearOrderedField α\ninst✝¹ : LinearOrderedField β\na✝ a₁ a₂ : α\nb : β\nq : ℚ\ninst✝ : Archimedean α\na : α\n⊢ BddAbove (cutMap β a)",
"tactic": "obtain ⟨q, hq⟩ := exists_rat_gt a"
},
{
"state_after": "no goals",
"state_before": "case intro\nF : Type ?u.11263\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.11272\ninst✝² : LinearOrderedField α\ninst✝¹ : LinearOrderedField β\na✝ a₁ a₂ : α\nb : β\nq✝ : ℚ\ninst✝ : Archimedean α\na : α\nq : ℚ\nhq : a < ↑q\n⊢ BddAbove (cutMap β a)",
"tactic": "exact ⟨q, ball_image_iff.2 fun r hr => by exact_mod_cast (hq.trans' hr).le⟩"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.11263\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.11272\ninst✝² : LinearOrderedField α\ninst✝¹ : LinearOrderedField β\na✝ a₁ a₂ : α\nb : β\nq✝ : ℚ\ninst✝ : Archimedean α\na : α\nq : ℚ\nhq : a < ↑q\nr : ℚ\nhr : r ∈ {t | ↑t < a}\n⊢ ↑r ≤ ↑q",
"tactic": "exact_mod_cast (hq.trans' hr).le"
}
] |
[
149,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.infinitePos_add_not_infinite
|
[] |
[
574,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
572,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
edist_dist
|
[] |
[
196,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.comp_assoc
|
[
{
"state_after": "case refine_3\nR✝ : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R✝\nm n : ℕ\ninst✝¹ : Semiring R✝\np q r : R✝[X]\nR : Type u_1\ninst✝ : CommSemiring R\nφ ψ χ : R[X]\nn✝ : ℕ\na✝¹ : R\na✝ : comp (comp (↑C a✝¹ * X ^ n✝) ψ) χ = comp (↑C a✝¹ * X ^ n✝) (comp ψ χ)\n⊢ comp (comp (↑C a✝¹ * X ^ (n✝ + 1)) ψ) χ = comp (↑C a✝¹ * X ^ (n✝ + 1)) (comp ψ χ)",
"state_before": "case refine_3\nR✝ : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R✝\nm n : ℕ\ninst✝¹ : Semiring R✝\np q r : R✝[X]\nR : Type u_1\ninst✝ : CommSemiring R\nφ ψ χ : R[X]\n⊢ ∀ (n : ℕ) (a : R),\n comp (comp (↑C a * X ^ n) ψ) χ = comp (↑C a * X ^ n) (comp ψ χ) →\n comp (comp (↑C a * X ^ (n + 1)) ψ) χ = comp (↑C a * X ^ (n + 1)) (comp ψ χ)",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "case refine_3\nR✝ : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R✝\nm n : ℕ\ninst✝¹ : Semiring R✝\np q r : R✝[X]\nR : Type u_1\ninst✝ : CommSemiring R\nφ ψ χ : R[X]\nn✝ : ℕ\na✝¹ : R\na✝ : comp (comp (↑C a✝¹ * X ^ n✝) ψ) χ = comp (↑C a✝¹ * X ^ n✝) (comp ψ χ)\n⊢ comp (comp (↑C a✝¹ * X ^ (n✝ + 1)) ψ) χ = comp (↑C a✝¹ * X ^ (n✝ + 1)) (comp ψ χ)",
"tactic": "simp_all only [add_comp, mul_comp, C_comp, X_comp, pow_succ', ← mul_assoc]"
}
] |
[
662,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/Data/Multiset/Nodup.lean
|
Multiset.nodup_range
|
[] |
[
207,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.Injective.surjective_comp_right
|
[] |
[
791,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
789,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
Function.Bijective.cauchySeq_comp_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\nf : ℕ → ℕ\nhf : Bijective f\nu : ℕ → α\nH : CauchySeq (u ∘ f)\n⊢ CauchySeq u",
"state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\nf : ℕ → ℕ\nhf : Bijective f\nu : ℕ → α\n⊢ CauchySeq (u ∘ f) ↔ CauchySeq u",
"tactic": "refine' ⟨fun H => _, fun H => H.comp_injective hf.injective⟩"
},
{
"state_after": "case intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\nu : ℕ → α\nf : ℕ ≃ ℕ\nH : CauchySeq (u ∘ ↑f)\n⊢ CauchySeq u",
"state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\nf : ℕ → ℕ\nhf : Bijective f\nu : ℕ → α\nH : CauchySeq (u ∘ f)\n⊢ CauchySeq u",
"tactic": "lift f to ℕ ≃ ℕ using hf"
}
] |
[
216,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
src/lean/Init/Control/Lawful.lean
|
ExceptT.ext
|
[
{
"state_after": "m : Type u_1 → Type u_2\nε α : Type u_1\ninst✝ : Monad m\nx y : ExceptT ε m α\nh : x = y\n⊢ x = y",
"state_before": "m : Type u_1 → Type u_2\nε α : Type u_1\ninst✝ : Monad m\nx y : ExceptT ε m α\nh : run x = run y\n⊢ x = y",
"tactic": "simp [run] at h"
},
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nε α : Type u_1\ninst✝ : Monad m\nx y : ExceptT ε m α\nh : x = y\n⊢ x = y",
"tactic": "assumption"
}
] |
[
106,
13
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
104,
1
] |
Mathlib/Analysis/NormedSpace/QuaternionExponential.lean
|
Quaternion.norm_exp
|
[
{
"state_after": "no goals",
"state_before": "q : ℍ\n⊢ ‖exp ℝ q‖ = ‖exp ℝ q.re‖",
"tactic": "rw [norm_eq_sqrt_real_inner (exp ℝ q), inner_self, normSq_exp, Real.sqrt_sq_eq_abs,\n Real.norm_eq_abs]"
}
] |
[
142,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
Set.Subsingleton.isCompact
|
[] |
[
402,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.refl_trans
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.47735\nδ : Type ?u.47738\ne : LocalEquiv α β\ne' : LocalEquiv β γ\n⊢ (LocalEquiv.trans (LocalEquiv.refl α) e).source = e.source",
"tactic": "simp [trans_source, preimage_id]"
}
] |
[
754,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
753,
1
] |
Mathlib/Algebra/CovariantAndContravariant.lean
|
act_rel_of_act_rel_of_rel_act_rel
|
[] |
[
235,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Data/Nat/Squarefree.lean
|
Nat.squarefree_iff_prime_squarefree
|
[] |
[
38,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
le_ciInf_set_iff
|
[] |
[
529,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/Algebra/Hom/Freiman.lean
|
MonoidHom.toFreimanHom_injective
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.80415\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.80424\nδ : Type ?u.80427\nG : Type ?u.80430\ninst✝⁵ : FunLike F α fun x => β\ninst✝⁴ : CommMonoid α\ninst✝³ : CommMonoid β\ninst✝² : CommMonoid γ\ninst✝¹ : CommMonoid δ\ninst✝ : CommGroup G\nA : Set α\nB : Set β\nC : Set γ\nn : ℕ\na b c d : α\nf g : α →* β\nh : toFreimanHom A n f = toFreimanHom A n g\n⊢ f = g",
"tactic": "rwa [toFreimanHom, toFreimanHom, FreimanHom.mk.injEq, FunLike.coe_fn_eq] at h"
}
] |
[
483,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.pow_succ
|
[] |
[
474,
6
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
473,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
Localization.monoidOf_eq_algebraMap
|
[] |
[
1021,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1020,
1
] |
Mathlib/RingTheory/PowerSeries/WellKnown.lean
|
PowerSeries.exp_pow_sum
|
[
{
"state_after": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ (Finset.sum (Finset.range n) fun x =>\n ↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x_1 y : PowerSeries A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n (x_1 + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n x_1 +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n y) }\n (exp A)) =\n mk fun p => Finset.sum (Finset.range n) fun x => ↑(x ^ p) * ↑(algebraMap ℚ A) (↑p !)⁻¹",
"state_before": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ (Finset.sum (Finset.range n) fun k => exp A ^ k) =\n mk fun p => Finset.sum (Finset.range n) fun k => ↑(k ^ p) * ↑(algebraMap ℚ A) (↑p !)⁻¹",
"tactic": "simp only [exp_pow_eq_rescale_exp, rescale]"
},
{
"state_after": "case h\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn n✝ : ℕ\n⊢ ↑(coeff A n✝)\n (Finset.sum (Finset.range n) fun x =>\n ↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x_1 y : PowerSeries A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n (x_1 + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n x_1 +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n y) }\n (exp A)) =\n ↑(coeff A n✝) (mk fun p => Finset.sum (Finset.range n) fun x => ↑(x ^ p) * ↑(algebraMap ℚ A) (↑p !)⁻¹)",
"state_before": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ (Finset.sum (Finset.range n) fun x =>\n ↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x_1 y : PowerSeries A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n (x_1 + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n x_1 +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n y) }\n (exp A)) =\n mk fun p => Finset.sum (Finset.range n) fun x => ↑(x ^ p) * ↑(algebraMap ℚ A) (↑p !)⁻¹",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn n✝ : ℕ\n⊢ ↑(coeff A n✝)\n (Finset.sum (Finset.range n) fun x =>\n ↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x_1 y : PowerSeries A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n (x_1 + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n x_1 +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => ↑x ^ n * ↑(coeff A n) f) 1 = 1) }\n g) })\n y) }\n (exp A)) =\n ↑(coeff A n✝) (mk fun p => Finset.sum (Finset.range n) fun x => ↑(x ^ p) * ↑(algebraMap ℚ A) (↑p !)⁻¹)",
"tactic": "simp only [one_div, coeff_mk, cast_pow, coe_mk, MonoidHom.coe_mk, OneHom.coe_mk,\n coeff_exp, factorial, LinearMap.map_sum]"
}
] |
[
209,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
LinearMap.isAdjointPair_id
|
[] |
[
450,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
450,
1
] |
Mathlib/Data/PNat/Defs.lean
|
PNat.ne_zero
|
[] |
[
160,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.translationNumber_eq_of_dist_bounded
|
[] |
[
708,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
705,
1
] |
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
rightLim_eq_of_tendsto
|
[] |
[
67,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean
|
AddLECancellable.lt_add_of_tsub_lt_left
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.38298\ninst✝³ : PartialOrder α\ninst✝² : AddCommSemigroup α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nh : a - b < c\n⊢ a - b ≤ c ∧ a ≠ b + c",
"state_before": "α : Type u_1\nβ : Type ?u.38298\ninst✝³ : PartialOrder α\ninst✝² : AddCommSemigroup α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nh : a - b < c\n⊢ a < b + c",
"tactic": "rw [lt_iff_le_and_ne, ← tsub_le_iff_left]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.38298\ninst✝³ : PartialOrder α\ninst✝² : AddCommSemigroup α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nh : a - b < c\n⊢ a ≠ b + c",
"state_before": "α : Type u_1\nβ : Type ?u.38298\ninst✝³ : PartialOrder α\ninst✝² : AddCommSemigroup α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nh : a - b < c\n⊢ a - b ≤ c ∧ a ≠ b + c",
"tactic": "refine' ⟨h.le, _⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.38298\ninst✝³ : PartialOrder α\ninst✝² : AddCommSemigroup α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\nb c d : α\nhb : AddLECancellable b\nh : b + c - b < c\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.38298\ninst✝³ : PartialOrder α\ninst✝² : AddCommSemigroup α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nh : a - b < c\n⊢ a ≠ b + c",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.38298\ninst✝³ : PartialOrder α\ninst✝² : AddCommSemigroup α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\nb c d : α\nhb : AddLECancellable b\nh : b + c - b < c\n⊢ False",
"tactic": "simp [hb] at h"
}
] |
[
315,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
11
] |
Mathlib/LinearAlgebra/Prod.lean
|
Submodule.prod_le_iff
|
[
{
"state_after": "case mp\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\n⊢ prod p₁ p₂ ≤ q → map (inl R M M₂) p₁ ≤ q ∧ map (inr R M M₂) p₂ ≤ q\n\ncase mpr\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\n⊢ map (inl R M M₂) p₁ ≤ q ∧ map (inr R M M₂) p₂ ≤ q → prod p₁ p₂ ≤ q",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\n⊢ prod p₁ p₂ ≤ q ↔ map (inl R M M₂) p₁ ≤ q ∧ map (inr R M M₂) p₂ ≤ q",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\n⊢ map (inl R M M₂) p₁ ≤ q ∧ map (inr R M M₂) p₂ ≤ q",
"state_before": "case mp\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\n⊢ prod p₁ p₂ ≤ q → map (inl R M M₂) p₁ ≤ q ∧ map (inr R M M₂) p₂ ≤ q",
"tactic": "intro h"
},
{
"state_after": "case mp.left\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\n⊢ map (inl R M M₂) p₁ ≤ q\n\ncase mp.right\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\n⊢ map (inr R M M₂) p₂ ≤ q",
"state_before": "case mp\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\n⊢ map (inl R M M₂) p₁ ≤ q ∧ map (inr R M M₂) p₂ ≤ q",
"tactic": "constructor"
},
{
"state_after": "case mp.left.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\nx : M\nhx : x ∈ ↑p₁\n⊢ ↑(inl R M M₂) x ∈ q",
"state_before": "case mp.left\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\n⊢ map (inl R M M₂) p₁ ≤ q",
"tactic": "rintro _ ⟨x, hx, rfl⟩"
},
{
"state_after": "case mp.left.intro.intro.a\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\nx : M\nhx : x ∈ ↑p₁\n⊢ ↑(inl R M M₂) x ∈ prod p₁ p₂",
"state_before": "case mp.left.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\nx : M\nhx : x ∈ ↑p₁\n⊢ ↑(inl R M M₂) x ∈ q",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case mp.left.intro.intro.a\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\nx : M\nhx : x ∈ ↑p₁\n⊢ ↑(inl R M M₂) x ∈ prod p₁ p₂",
"tactic": "exact ⟨hx, zero_mem p₂⟩"
},
{
"state_after": "case mp.right.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\nx : M₂\nhx : x ∈ ↑p₂\n⊢ ↑(inr R M M₂) x ∈ q",
"state_before": "case mp.right\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\n⊢ map (inr R M M₂) p₂ ≤ q",
"tactic": "rintro _ ⟨x, hx, rfl⟩"
},
{
"state_after": "case mp.right.intro.intro.a\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\nx : M₂\nhx : x ∈ ↑p₂\n⊢ ↑(inr R M M₂) x ∈ prod p₁ p₂",
"state_before": "case mp.right.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\nx : M₂\nhx : x ∈ ↑p₂\n⊢ ↑(inr R M M₂) x ∈ q",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case mp.right.intro.intro.a\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : prod p₁ p₂ ≤ q\nx : M₂\nhx : x ∈ ↑p₂\n⊢ ↑(inr R M M₂) x ∈ prod p₁ p₂",
"tactic": "exact ⟨zero_mem p₁, hx⟩"
},
{
"state_after": "case mpr.intro.mk.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nhH : map (inl R M M₂) p₁ ≤ q\nhK : map (inr R M M₂) p₂ ≤ q\nx1 : M\nx2 : M₂\nh1 : (x1, x2).fst ∈ ↑p₁\nh2 : (x1, x2).snd ∈ ↑p₂\n⊢ (x1, x2) ∈ q",
"state_before": "case mpr\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\n⊢ map (inl R M M₂) p₁ ≤ q ∧ map (inr R M M₂) p₂ ≤ q → prod p₁ p₂ ≤ q",
"tactic": "rintro ⟨hH, hK⟩ ⟨x1, x2⟩ ⟨h1, h2⟩"
},
{
"state_after": "case mpr.intro.mk.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nhH : map (inl R M M₂) p₁ ≤ q\nhK : map (inr R M M₂) p₂ ≤ q\nx1 : M\nx2 : M₂\nh1 : (x1, x2).fst ∈ ↑p₁\nh2 : (x1, x2).snd ∈ ↑p₂\nh1' : ↑(inl R M M₂) x1 ∈ q\n⊢ (x1, x2) ∈ q",
"state_before": "case mpr.intro.mk.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nhH : map (inl R M M₂) p₁ ≤ q\nhK : map (inr R M M₂) p₂ ≤ q\nx1 : M\nx2 : M₂\nh1 : (x1, x2).fst ∈ ↑p₁\nh2 : (x1, x2).snd ∈ ↑p₂\n⊢ (x1, x2) ∈ q",
"tactic": "have h1' : (LinearMap.inl R _ _) x1 ∈ q := by\n apply hH\n simpa using h1"
},
{
"state_after": "case mpr.intro.mk.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nhH : map (inl R M M₂) p₁ ≤ q\nhK : map (inr R M M₂) p₂ ≤ q\nx1 : M\nx2 : M₂\nh1 : (x1, x2).fst ∈ ↑p₁\nh2 : (x1, x2).snd ∈ ↑p₂\nh1' : ↑(inl R M M₂) x1 ∈ q\nh2' : ↑(inr R M M₂) x2 ∈ q\n⊢ (x1, x2) ∈ q",
"state_before": "case mpr.intro.mk.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nhH : map (inl R M M₂) p₁ ≤ q\nhK : map (inr R M M₂) p₂ ≤ q\nx1 : M\nx2 : M₂\nh1 : (x1, x2).fst ∈ ↑p₁\nh2 : (x1, x2).snd ∈ ↑p₂\nh1' : ↑(inl R M M₂) x1 ∈ q\n⊢ (x1, x2) ∈ q",
"tactic": "have h2' : (LinearMap.inr R _ _) x2 ∈ q := by\n apply hK\n simpa using h2"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.mk.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nhH : map (inl R M M₂) p₁ ≤ q\nhK : map (inr R M M₂) p₂ ≤ q\nx1 : M\nx2 : M₂\nh1 : (x1, x2).fst ∈ ↑p₁\nh2 : (x1, x2).snd ∈ ↑p₂\nh1' : ↑(inl R M M₂) x1 ∈ q\nh2' : ↑(inr R M M₂) x2 ∈ q\n⊢ (x1, x2) ∈ q",
"tactic": "simpa using add_mem h1' h2'"
},
{
"state_after": "case a\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nhH : map (inl R M M₂) p₁ ≤ q\nhK : map (inr R M M₂) p₂ ≤ q\nx1 : M\nx2 : M₂\nh1 : (x1, x2).fst ∈ ↑p₁\nh2 : (x1, x2).snd ∈ ↑p₂\n⊢ ↑(inl R M M₂) x1 ∈ map (inl R M M₂) p₁",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nhH : map (inl R M M₂) p₁ ≤ q\nhK : map (inr R M M₂) p₂ ≤ q\nx1 : M\nx2 : M₂\nh1 : (x1, x2).fst ∈ ↑p₁\nh2 : (x1, x2).snd ∈ ↑p₂\n⊢ ↑(inl R M M₂) x1 ∈ q",
"tactic": "apply hH"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nhH : map (inl R M M₂) p₁ ≤ q\nhK : map (inr R M M₂) p₂ ≤ q\nx1 : M\nx2 : M₂\nh1 : (x1, x2).fst ∈ ↑p₁\nh2 : (x1, x2).snd ∈ ↑p₂\n⊢ ↑(inl R M M₂) x1 ∈ map (inl R M M₂) p₁",
"tactic": "simpa using h1"
},
{
"state_after": "case a\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nhH : map (inl R M M₂) p₁ ≤ q\nhK : map (inr R M M₂) p₂ ≤ q\nx1 : M\nx2 : M₂\nh1 : (x1, x2).fst ∈ ↑p₁\nh2 : (x1, x2).snd ∈ ↑p₂\nh1' : ↑(inl R M M₂) x1 ∈ q\n⊢ ↑(inr R M M₂) x2 ∈ map (inr R M M₂) p₂",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nhH : map (inl R M M₂) p₁ ≤ q\nhK : map (inr R M M₂) p₂ ≤ q\nx1 : M\nx2 : M₂\nh1 : (x1, x2).fst ∈ ↑p₁\nh2 : (x1, x2).snd ∈ ↑p₂\nh1' : ↑(inl R M M₂) x1 ∈ q\n⊢ ↑(inr R M M₂) x2 ∈ q",
"tactic": "apply hK"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.377212\nM₆ : Type ?u.377215\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq✝ : Submodule R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nhH : map (inl R M M₂) p₁ ≤ q\nhK : map (inr R M M₂) p₂ ≤ q\nx1 : M\nx2 : M₂\nh1 : (x1, x2).fst ∈ ↑p₁\nh2 : (x1, x2).snd ∈ ↑p₂\nh1' : ↑(inl R M M₂) x1 ∈ q\n⊢ ↑(inr R M M₂) x2 ∈ map (inr R M M₂) p₂",
"tactic": "simpa using h2"
}
] |
[
723,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
705,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
isPreirreducible_iff_closed_union_closed
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.198320\nπ : ι → Type ?u.198325\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s z₁ z₂ : Set α\n⊢ IsOpen (z₁ᶜ) → IsOpen (z₂ᶜ) → Set.Nonempty (s ∩ z₁ᶜ) → Set.Nonempty (s ∩ z₂ᶜ) → Set.Nonempty (s ∩ (z₁ᶜ ∩ z₂ᶜ)) ↔\n IsClosed z₁ → IsClosed z₂ → s ⊆ z₁ ∪ z₂ → s ⊆ z₁ ∨ s ⊆ z₂",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.198320\nπ : ι → Type ?u.198325\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s : Set α\n⊢ IsPreirreducible s ↔ ∀ (z₁ z₂ : Set α), IsClosed z₁ → IsClosed z₂ → s ⊆ z₁ ∪ z₂ → s ⊆ z₁ ∨ s ⊆ z₂",
"tactic": "refine compl_surjective.forall.trans <| forall_congr' fun z₁ => compl_surjective.forall.trans <|\n forall_congr' fun z₂ => ?_"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.198320\nπ : ι → Type ?u.198325\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s z₁ z₂ : Set α\n⊢ IsClosed z₁ → IsClosed z₂ → ¬s ⊆ z₁ → ¬s ⊆ z₂ → ¬s ⊆ z₁ ∪ z₂ ↔\n IsClosed z₁ → IsClosed z₂ → s ⊆ z₁ ∪ z₂ → s ⊆ z₁ ∨ s ⊆ z₂",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.198320\nπ : ι → Type ?u.198325\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s z₁ z₂ : Set α\n⊢ IsOpen (z₁ᶜ) → IsOpen (z₂ᶜ) → Set.Nonempty (s ∩ z₁ᶜ) → Set.Nonempty (s ∩ z₂ᶜ) → Set.Nonempty (s ∩ (z₁ᶜ ∩ z₂ᶜ)) ↔\n IsClosed z₁ → IsClosed z₂ → s ⊆ z₁ ∪ z₂ → s ⊆ z₁ ∨ s ⊆ z₂",
"tactic": "simp only [isOpen_compl_iff, ← compl_union, inter_compl_nonempty_iff]"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.198320\nπ : ι → Type ?u.198325\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s z₁ z₂ : Set α\nx✝¹ : IsClosed z₁\nx✝ : IsClosed z₂\n⊢ ¬s ⊆ z₁ → ¬s ⊆ z₂ → ¬s ⊆ z₁ ∪ z₂ ↔ s ⊆ z₁ ∪ z₂ → s ⊆ z₁ ∨ s ⊆ z₂",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.198320\nπ : ι → Type ?u.198325\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s z₁ z₂ : Set α\n⊢ IsClosed z₁ → IsClosed z₂ → ¬s ⊆ z₁ → ¬s ⊆ z₂ → ¬s ⊆ z₁ ∪ z₂ ↔\n IsClosed z₁ → IsClosed z₂ → s ⊆ z₁ ∪ z₂ → s ⊆ z₁ ∨ s ⊆ z₂",
"tactic": "refine forall₂_congr fun _ _ => ?_"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.198320\nπ : ι → Type ?u.198325\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s z₁ z₂ : Set α\nx✝¹ : IsClosed z₁\nx✝ : IsClosed z₂\n⊢ ¬s ⊆ z₁ → ¬s ⊆ z₂ → ¬s ⊆ z₁ ∪ z₂ ↔ s ⊆ z₁ ∪ z₂ → s ⊆ z₁ ∨ s ⊆ z₂",
"tactic": "rw [← and_imp, ← not_or, not_imp_not]"
}
] |
[
1932,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1925,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
IsCompl.eq_compl
|
[] |
[
864,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
863,
1
] |
Mathlib/Data/Sym/Basic.lean
|
Sym.exists_mem
|
[] |
[
299,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.C_pow
|
[] |
[
235,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.insert_eq_self
|
[] |
[
1148,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1147,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
NNReal.summable_of_le
|
[] |
[
1120,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1117,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.coe_add
|
[] |
[
481,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
480,
1
] |
Mathlib/Algebra/Lie/SkewAdjoint.lean
|
Matrix.lie_transpose
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nn : Type w\ninst✝² : CommRing R\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nJ A B : Matrix n n R\n⊢ (A * B - B * A)ᵀ = Bᵀ * Aᵀ - Aᵀ * Bᵀ",
"tactic": "simp"
}
] |
[
102,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.AbsTheory.abs_conj
|
[
{
"state_after": "no goals",
"state_before": "z : ℂ\n⊢ Real.sqrt (↑normSq (↑(starRingEnd ℂ) z)) = Real.sqrt (↑normSq z)",
"tactic": "simp"
}
] |
[
913,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
913,
1
] |
Mathlib/Combinatorics/Additive/SalemSpencer.lean
|
MulSalemSpencer.mul_left
|
[
{
"state_after": "case intro.intro.intro.intro.intro.intro\nF : Type ?u.77288\nα : Type u_1\nβ : Type ?u.77294\n𝕜 : Type ?u.77297\nE : Type ?u.77300\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : (fun x x_1 => x * x_1) a b * (fun x x_1 => x * x_1) a c = (fun x x_1 => x * x_1) a d * (fun x x_1 => x * x_1) a d\n⊢ (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c",
"state_before": "F : Type ?u.77288\nα : Type u_1\nβ : Type ?u.77294\n𝕜 : Type ?u.77297\nE : Type ?u.77300\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\n⊢ MulSalemSpencer ((fun x x_1 => x * x_1) a '' s)",
"tactic": "rintro _ _ _ ⟨b, hb, rfl⟩ ⟨c, hc, rfl⟩ ⟨d, hd, rfl⟩ h"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nF : Type ?u.77288\nα : Type u_1\nβ : Type ?u.77294\n𝕜 : Type ?u.77297\nE : Type ?u.77300\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : a * a * (b * c) = a * a * (d * d)\n⊢ (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c",
"state_before": "case intro.intro.intro.intro.intro.intro\nF : Type ?u.77288\nα : Type u_1\nβ : Type ?u.77294\n𝕜 : Type ?u.77297\nE : Type ?u.77300\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : (fun x x_1 => x * x_1) a b * (fun x x_1 => x * x_1) a c = (fun x x_1 => x * x_1) a d * (fun x x_1 => x * x_1) a d\n⊢ (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c",
"tactic": "rw [mul_mul_mul_comm, mul_mul_mul_comm a d] at h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nF : Type ?u.77288\nα : Type u_1\nβ : Type ?u.77294\n𝕜 : Type ?u.77297\nE : Type ?u.77300\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : a * a * (b * c) = a * a * (d * d)\n⊢ (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c",
"tactic": "rw [hs hb hc hd (mul_left_cancel h)]"
}
] |
[
183,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
src/lean/Init/SimpLemmas.lean
|
bne_self_eq_false
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\na : α\n⊢ (a != a) = false",
"tactic": "simp [bne]"
}
] |
[
151,
100
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
151,
9
] |
Mathlib/Topology/SubsetProperties.lean
|
isCompact_iUnion
|
[] |
[
433,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
431,
1
] |
Mathlib/SetTheory/Ordinal/Principal.lean
|
Ordinal.mul_lt_omega_opow
|
[
{
"state_after": "case inl\na b : Ordinal\nhb : b < ω\nc0 : 0 < 0\nha : a < ω ^ 0\n⊢ a * b < ω ^ 0\n\ncase inr.inl.intro\na b : Ordinal\nhb : b < ω\nc : Ordinal\nc0 : 0 < succ c\nha : a < ω ^ succ c\n⊢ a * b < ω ^ succ c\n\ncase inr.inr\na b c : Ordinal\nc0 : 0 < c\nha : a < ω ^ c\nhb : b < ω\nl : IsLimit c\n⊢ a * b < ω ^ c",
"state_before": "a b c : Ordinal\nc0 : 0 < c\nha : a < ω ^ c\nhb : b < ω\n⊢ a * b < ω ^ c",
"tactic": "rcases zero_or_succ_or_limit c with (rfl | ⟨c, rfl⟩ | l)"
},
{
"state_after": "no goals",
"state_before": "case inl\na b : Ordinal\nhb : b < ω\nc0 : 0 < 0\nha : a < ω ^ 0\n⊢ a * b < ω ^ 0",
"tactic": "exact (lt_irrefl _).elim c0"
},
{
"state_after": "case inr.inl.intro\na b : Ordinal\nhb : b < ω\nc : Ordinal\nc0 : 0 < succ c\nha : a < ω ^ c * ω\n⊢ a * b < ω ^ succ c",
"state_before": "case inr.inl.intro\na b : Ordinal\nhb : b < ω\nc : Ordinal\nc0 : 0 < succ c\nha : a < ω ^ succ c\n⊢ a * b < ω ^ succ c",
"tactic": "rw [opow_succ] at ha"
},
{
"state_after": "case inr.inl.intro.intro.intro\na b : Ordinal\nhb : b < ω\nc : Ordinal\nc0 : 0 < succ c\nha : a < ω ^ c * ω\nn : Ordinal\nhn : n < ω\nan : a < (fun x x_1 => x * x_1) (ω ^ c) n\n⊢ a * b < ω ^ succ c",
"state_before": "case inr.inl.intro\na b : Ordinal\nhb : b < ω\nc : Ordinal\nc0 : 0 < succ c\nha : a < ω ^ c * ω\n⊢ a * b < ω ^ succ c",
"tactic": "rcases((mul_isNormal <| opow_pos _ omega_pos).limit_lt omega_isLimit).1 ha with ⟨n, hn, an⟩"
},
{
"state_after": "case inr.inl.intro.intro.intro\na b : Ordinal\nhb : b < ω\nc : Ordinal\nc0 : 0 < succ c\nha : a < ω ^ c * ω\nn : Ordinal\nhn : n < ω\nan : a < (fun x x_1 => x * x_1) (ω ^ c) n\n⊢ (fun x x_1 => x * x_1) (ω ^ c) n * b < ω ^ succ c",
"state_before": "case inr.inl.intro.intro.intro\na b : Ordinal\nhb : b < ω\nc : Ordinal\nc0 : 0 < succ c\nha : a < ω ^ c * ω\nn : Ordinal\nhn : n < ω\nan : a < (fun x x_1 => x * x_1) (ω ^ c) n\n⊢ a * b < ω ^ succ c",
"tactic": "apply (mul_le_mul_right' (le_of_lt an) _).trans_lt"
},
{
"state_after": "case inr.inl.intro.intro.intro\na b : Ordinal\nhb : b < ω\nc : Ordinal\nc0 : 0 < succ c\nha : a < ω ^ c * ω\nn : Ordinal\nhn : n < ω\nan : a < (fun x x_1 => x * x_1) (ω ^ c) n\n⊢ n * b < ω",
"state_before": "case inr.inl.intro.intro.intro\na b : Ordinal\nhb : b < ω\nc : Ordinal\nc0 : 0 < succ c\nha : a < ω ^ c * ω\nn : Ordinal\nhn : n < ω\nan : a < (fun x x_1 => x * x_1) (ω ^ c) n\n⊢ (fun x x_1 => x * x_1) (ω ^ c) n * b < ω ^ succ c",
"tactic": "rw [opow_succ, mul_assoc, mul_lt_mul_iff_left (opow_pos _ omega_pos)]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.intro.intro.intro\na b : Ordinal\nhb : b < ω\nc : Ordinal\nc0 : 0 < succ c\nha : a < ω ^ c * ω\nn : Ordinal\nhn : n < ω\nan : a < (fun x x_1 => x * x_1) (ω ^ c) n\n⊢ n * b < ω",
"tactic": "exact principal_mul_omega hn hb"
},
{
"state_after": "case inr.inr.intro.intro\na b c : Ordinal\nc0 : 0 < c\nha : a < ω ^ c\nhb : b < ω\nl : IsLimit c\nx : Ordinal\nhx : x < c\nax : a < (fun x x_1 => x ^ x_1) ω x\n⊢ a * b < ω ^ c",
"state_before": "case inr.inr\na b c : Ordinal\nc0 : 0 < c\nha : a < ω ^ c\nhb : b < ω\nl : IsLimit c\n⊢ a * b < ω ^ c",
"tactic": "rcases((opow_isNormal one_lt_omega).limit_lt l).1 ha with ⟨x, hx, ax⟩"
},
{
"state_after": "case inr.inr.intro.intro\na b c : Ordinal\nc0 : 0 < c\nha : a < ω ^ c\nhb : b < ω\nl : IsLimit c\nx : Ordinal\nhx : x < c\nax : a < (fun x x_1 => x ^ x_1) ω x\n⊢ (fun x x_1 => x ^ x_1) ω x * ω < ω ^ c",
"state_before": "case inr.inr.intro.intro\na b c : Ordinal\nc0 : 0 < c\nha : a < ω ^ c\nhb : b < ω\nl : IsLimit c\nx : Ordinal\nhx : x < c\nax : a < (fun x x_1 => x ^ x_1) ω x\n⊢ a * b < ω ^ c",
"tactic": "refine' (mul_le_mul' (le_of_lt ax) (le_of_lt hb)).trans_lt _"
},
{
"state_after": "case inr.inr.intro.intro\na b c : Ordinal\nc0 : 0 < c\nha : a < ω ^ c\nhb : b < ω\nl : IsLimit c\nx : Ordinal\nhx : x < c\nax : a < (fun x x_1 => x ^ x_1) ω x\n⊢ succ x < c",
"state_before": "case inr.inr.intro.intro\na b c : Ordinal\nc0 : 0 < c\nha : a < ω ^ c\nhb : b < ω\nl : IsLimit c\nx : Ordinal\nhx : x < c\nax : a < (fun x x_1 => x ^ x_1) ω x\n⊢ (fun x x_1 => x ^ x_1) ω x * ω < ω ^ c",
"tactic": "rw [← opow_succ, opow_lt_opow_iff_right one_lt_omega]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.intro.intro\na b c : Ordinal\nc0 : 0 < c\nha : a < ω ^ c\nhb : b < ω\nl : IsLimit c\nx : Ordinal\nhx : x < c\nax : a < (fun x x_1 => x ^ x_1) ω x\n⊢ succ x < c",
"tactic": "exact l.2 _ hx"
}
] |
[
363,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
351,
1
] |
Mathlib/RingTheory/Localization/NumDen.lean
|
IsFractionRing.exists_reduced_fraction
|
[
{
"state_after": "case intro.mk.intro\nR : Type ?u.2417\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.2623\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.2877\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nb : A\nb_nonzero : b ∈ nonZeroDivisors A\na : A\nhab : ↑(algebraMap A K) a = ↑{ val := b, property := b_nonzero } • x\n⊢ ∃ a b, (∀ {d : A}, d ∣ a → d ∣ ↑b → IsUnit d) ∧ mk' K a b = x",
"state_before": "R : Type ?u.2417\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.2623\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.2877\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\n⊢ ∃ a b, (∀ {d : A}, d ∣ a → d ∣ ↑b → IsUnit d) ∧ mk' K a b = x",
"tactic": "obtain ⟨⟨b, b_nonzero⟩, a, hab⟩ := exists_integer_multiple (nonZeroDivisors A) x"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.intro.intro\nR : Type ?u.2417\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.2623\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.2877\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : ∀ {d : A}, d ∣ a' → d ∣ b' → IsUnit d\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : ↑(algebraMap A K) (c' * a') = ↑{ val := c' * b', property := b_nonzero } • x\n⊢ ∃ a b, (∀ {d : A}, d ∣ a → d ∣ ↑b → IsUnit d) ∧ mk' K a b = x",
"state_before": "case intro.mk.intro\nR : Type ?u.2417\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.2623\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.2877\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nb : A\nb_nonzero : b ∈ nonZeroDivisors A\na : A\nhab : ↑(algebraMap A K) a = ↑{ val := b, property := b_nonzero } • x\n⊢ ∃ a b, (∀ {d : A}, d ∣ a → d ∣ ↑b → IsUnit d) ∧ mk' K a b = x",
"tactic": "obtain ⟨a', b', c', no_factor, rfl, rfl⟩ :=\n UniqueFactorizationMonoid.exists_reduced_factors' a b\n (mem_nonZeroDivisors_iff_ne_zero.mp b_nonzero)"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type ?u.2417\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.2623\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.2877\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : ∀ {d : A}, d ∣ a' → d ∣ b' → IsUnit d\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : ↑(algebraMap A K) (c' * a') = ↑{ val := c' * b', property := b_nonzero } • x\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\n⊢ ∃ a b, (∀ {d : A}, d ∣ a → d ∣ ↑b → IsUnit d) ∧ mk' K a b = x",
"state_before": "case intro.mk.intro.intro.intro.intro.intro.intro\nR : Type ?u.2417\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.2623\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.2877\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : ∀ {d : A}, d ∣ a' → d ∣ b' → IsUnit d\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : ↑(algebraMap A K) (c' * a') = ↑{ val := c' * b', property := b_nonzero } • x\n⊢ ∃ a b, (∀ {d : A}, d ∣ a → d ∣ ↑b → IsUnit d) ∧ mk' K a b = x",
"tactic": "obtain ⟨_, b'_nonzero⟩ := mul_mem_nonZeroDivisors.mp b_nonzero"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type ?u.2417\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.2623\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.2877\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : ∀ {d : A}, d ∣ a' → d ∣ b' → IsUnit d\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : ↑(algebraMap A K) (c' * a') = ↑{ val := c' * b', property := b_nonzero } • x\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\n⊢ mk' K a' { val := b', property := b'_nonzero } = x",
"state_before": "case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type ?u.2417\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.2623\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.2877\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : ∀ {d : A}, d ∣ a' → d ∣ b' → IsUnit d\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : ↑(algebraMap A K) (c' * a') = ↑{ val := c' * b', property := b_nonzero } • x\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\n⊢ ∃ a b, (∀ {d : A}, d ∣ a → d ∣ ↑b → IsUnit d) ∧ mk' K a b = x",
"tactic": "refine' ⟨a', ⟨b', b'_nonzero⟩, no_factor, _⟩"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type ?u.2417\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.2623\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.2877\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : ∀ {d : A}, d ∣ a' → d ∣ b' → IsUnit d\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : ↑(algebraMap A K) (c' * a') = ↑{ val := c' * b', property := b_nonzero } • x\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\n⊢ ↑(algebraMap A K) (c' * b') * mk' K a' { val := b', property := b'_nonzero } = ↑(algebraMap A K) (c' * b') * x",
"state_before": "case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type ?u.2417\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.2623\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.2877\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : ∀ {d : A}, d ∣ a' → d ∣ b' → IsUnit d\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : ↑(algebraMap A K) (c' * a') = ↑{ val := c' * b', property := b_nonzero } • x\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\n⊢ mk' K a' { val := b', property := b'_nonzero } = x",
"tactic": "refine' mul_left_cancel₀ (IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors b_nonzero) _"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type ?u.2417\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.2623\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.2877\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : ∀ {d : A}, d ∣ a' → d ∣ b' → IsUnit d\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\nhab : ↑(algebraMap A K) c' * ↑(algebraMap A K) a' = ↑(algebraMap A K) c' * ↑(algebraMap A K) b' * x\n⊢ ↑(algebraMap A K) c' * ↑(algebraMap A K) b' * mk' K a' { val := b', property := b'_nonzero } =\n ↑(algebraMap A K) c' * ↑(algebraMap A K) b' * x",
"state_before": "case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type ?u.2417\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.2623\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.2877\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : ∀ {d : A}, d ∣ a' → d ∣ b' → IsUnit d\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nhab : ↑(algebraMap A K) (c' * a') = ↑{ val := c' * b', property := b_nonzero } • x\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\n⊢ ↑(algebraMap A K) (c' * b') * mk' K a' { val := b', property := b'_nonzero } = ↑(algebraMap A K) (c' * b') * x",
"tactic": "simp only [Subtype.coe_mk, RingHom.map_mul, Algebra.smul_def] at *"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type ?u.2417\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.2623\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.2877\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : ∀ {d : A}, d ∣ a' → d ∣ b' → IsUnit d\nb_nonzero : c' * b' ∈ nonZeroDivisors A\nleft✝ : c' ∈ nonZeroDivisors A\nb'_nonzero : b' ∈ nonZeroDivisors A\nhab : ↑(algebraMap A K) c' * ↑(algebraMap A K) a' = ↑(algebraMap A K) c' * ↑(algebraMap A K) b' * x\n⊢ ↑(algebraMap A K) c' * ↑(algebraMap A K) b' * mk' K a' { val := b', property := b'_nonzero } =\n ↑(algebraMap A K) c' * ↑(algebraMap A K) b' * x",
"tactic": "erw [← hab, mul_assoc, mk'_spec' _ a' ⟨b', b'_nonzero⟩]"
}
] |
[
52,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/SetTheory/Ordinal/NaturalOps.lean
|
Ordinal.nadd_lt_nadd_right
|
[] |
[
231,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/Topology/Separation.lean
|
nhds_basis_clopen
|
[
{
"state_after": "case mp\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\n⊢ U ∈ 𝓝 x → ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U\n\ncase mpr\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\n⊢ (∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U) → U ∈ 𝓝 x",
"state_before": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\n⊢ U ∈ 𝓝 x ↔ ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : connectedComponent x = {x}\n⊢ U ∈ 𝓝 x → ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U",
"state_before": "case mp\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\n⊢ U ∈ 𝓝 x → ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U",
"tactic": "have hx : connectedComponent x = {x} :=\n totallyDisconnectedSpace_iff_connectedComponent_singleton.mp ‹_› x"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\n⊢ U ∈ 𝓝 x → ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U",
"state_before": "case mp\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : connectedComponent x = {x}\n⊢ U ∈ 𝓝 x → ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U",
"tactic": "rw [connectedComponent_eq_iInter_clopen] at hx"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\n⊢ ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U",
"state_before": "case mp\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\n⊢ U ∈ 𝓝 x → ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U",
"tactic": "intro hU"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\n⊢ ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U",
"state_before": "case mp\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\n⊢ ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U",
"tactic": "let N := { Z // IsClopen Z ∧ x ∈ Z }"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : ∃ Z, ↑Z ⊆ U\n⊢ ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U\n\ncase this\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\n⊢ ∃ Z, ↑Z ⊆ U",
"state_before": "case mp\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\n⊢ ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U",
"tactic": "suffices : ∃ Z : N, Z.val ⊆ U"
},
{
"state_after": "case this\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\n⊢ ∃ Z, ↑Z ⊆ U",
"state_before": "case this\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\n⊢ ∃ Z, ↑Z ⊆ U",
"tactic": "haveI : Nonempty N := ⟨⟨univ, isClopen_univ, mem_univ x⟩⟩"
},
{
"state_after": "case this\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\nhNcl : ∀ (Z : N), IsClosed ↑Z\n⊢ ∃ Z, ↑Z ⊆ U",
"state_before": "case this\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\n⊢ ∃ Z, ↑Z ⊆ U",
"tactic": "have hNcl : ∀ Z : N, IsClosed Z.val := fun Z => Z.property.1.2"
},
{
"state_after": "case this\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\nhNcl : ∀ (Z : N), IsClosed ↑Z\nhdir : Directed Superset fun Z => ↑Z\n⊢ ∃ Z, ↑Z ⊆ U",
"state_before": "case this\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\nhNcl : ∀ (Z : N), IsClosed ↑Z\n⊢ ∃ Z, ↑Z ⊆ U",
"tactic": "have hdir : Directed Superset fun Z : N => Z.val := by\n rintro ⟨s, hs, hxs⟩ ⟨t, ht, hxt⟩\n exact ⟨⟨s ∩ t, hs.inter ht, ⟨hxs, hxt⟩⟩, inter_subset_left s t, inter_subset_right s t⟩"
},
{
"state_after": "case this\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\nhNcl : ∀ (Z : N), IsClosed ↑Z\nhdir : Directed Superset fun Z => ↑Z\nh_nhd : ∀ (y : α), (y ∈ ⋂ (Z : N), ↑Z) → U ∈ 𝓝 y\n⊢ ∃ Z, ↑Z ⊆ U",
"state_before": "case this\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\nhNcl : ∀ (Z : N), IsClosed ↑Z\nhdir : Directed Superset fun Z => ↑Z\n⊢ ∃ Z, ↑Z ⊆ U",
"tactic": "have h_nhd : ∀ y ∈ ⋂ Z : N, Z.val, U ∈ 𝓝 y := fun y y_in => by\n erw [hx, mem_singleton_iff] at y_in\n rwa [y_in]"
},
{
"state_after": "no goals",
"state_before": "case this\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\nhNcl : ∀ (Z : N), IsClosed ↑Z\nhdir : Directed Superset fun Z => ↑Z\nh_nhd : ∀ (y : α), (y ∈ ⋂ (Z : N), ↑Z) → U ∈ 𝓝 y\n⊢ ∃ Z, ↑Z ⊆ U",
"tactic": "exact exists_subset_nhds_of_compactSpace hdir hNcl h_nhd"
},
{
"state_after": "case mp.intro.mk.intro\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\ns : Set α\nhs : IsClopen s\nhs' : x ∈ s\nhs'' : ↑{ val := s, property := (_ : IsClopen s ∧ x ∈ s) } ⊆ U\n⊢ ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U",
"state_before": "case mp\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : ∃ Z, ↑Z ⊆ U\n⊢ ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U",
"tactic": "rcases this with ⟨⟨s, hs, hs'⟩, hs''⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.mk.intro\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\ns : Set α\nhs : IsClopen s\nhs' : x ∈ s\nhs'' : ↑{ val := s, property := (_ : IsClopen s ∧ x ∈ s) } ⊆ U\n⊢ ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U",
"tactic": "exact ⟨s, ⟨hs', hs⟩, hs''⟩"
},
{
"state_after": "case mk.intro.mk.intro\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\nhNcl : ∀ (Z : N), IsClosed ↑Z\ns : Set α\nhs : IsClopen s\nhxs : x ∈ s\nt : Set α\nht : IsClopen t\nhxt : x ∈ t\n⊢ ∃ z,\n (fun Z => ↑Z) { val := s, property := (_ : IsClopen s ∧ x ∈ s) } ⊇ (fun Z => ↑Z) z ∧\n (fun Z => ↑Z) { val := t, property := (_ : IsClopen t ∧ x ∈ t) } ⊇ (fun Z => ↑Z) z",
"state_before": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\nhNcl : ∀ (Z : N), IsClosed ↑Z\n⊢ Directed Superset fun Z => ↑Z",
"tactic": "rintro ⟨s, hs, hxs⟩ ⟨t, ht, hxt⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.mk.intro\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\nhNcl : ∀ (Z : N), IsClosed ↑Z\ns : Set α\nhs : IsClopen s\nhxs : x ∈ s\nt : Set α\nht : IsClopen t\nhxt : x ∈ t\n⊢ ∃ z,\n (fun Z => ↑Z) { val := s, property := (_ : IsClopen s ∧ x ∈ s) } ⊇ (fun Z => ↑Z) z ∧\n (fun Z => ↑Z) { val := t, property := (_ : IsClopen t ∧ x ∈ t) } ⊇ (fun Z => ↑Z) z",
"tactic": "exact ⟨⟨s ∩ t, hs.inter ht, ⟨hxs, hxt⟩⟩, inter_subset_left s t, inter_subset_right s t⟩"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\nhNcl : ∀ (Z : N), IsClosed ↑Z\nhdir : Directed Superset fun Z => ↑Z\ny : α\ny_in : y = x\n⊢ U ∈ 𝓝 y",
"state_before": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\nhNcl : ∀ (Z : N), IsClosed ↑Z\nhdir : Directed Superset fun Z => ↑Z\ny : α\ny_in : y ∈ ⋂ (Z : N), ↑Z\n⊢ U ∈ 𝓝 y",
"tactic": "erw [hx, mem_singleton_iff] at y_in"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\nhx : (⋂ (Z : { Z // IsClopen Z ∧ x ∈ Z }), ↑Z) = {x}\nhU : U ∈ 𝓝 x\nN : Type u := { Z // IsClopen Z ∧ x ∈ Z }\nthis : Nonempty N\nhNcl : ∀ (Z : N), IsClosed ↑Z\nhdir : Directed Superset fun Z => ↑Z\ny : α\ny_in : y = x\n⊢ U ∈ 𝓝 y",
"tactic": "rwa [y_in]"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU V : Set α\nhUV : V ⊆ U\nhxV : x ∈ V\nV_op : IsOpen V\n⊢ U ∈ 𝓝 x",
"state_before": "case mpr\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU : Set α\n⊢ (∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U) → U ∈ 𝓝 x",
"tactic": "rintro ⟨V, ⟨hxV, V_op, -⟩, hUV : V ⊆ U⟩"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU V : Set α\nhUV : V ⊆ U\nhxV : x ∈ V\nV_op : IsOpen V\n⊢ ∃ t, t ⊆ U ∧ IsOpen t ∧ x ∈ t",
"state_before": "case mpr.intro.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU V : Set α\nhUV : V ⊆ U\nhxV : x ∈ V\nV_op : IsOpen V\n⊢ U ∈ 𝓝 x",
"tactic": "rw [mem_nhds_iff]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : CompactSpace α\ninst✝ : TotallyDisconnectedSpace α\nx : α\nU V : Set α\nhUV : V ⊆ U\nhxV : x ∈ V\nV_op : IsOpen V\n⊢ ∃ t, t ⊆ U ∧ IsOpen t ∧ x ∈ t",
"tactic": "exact ⟨V, hUV, V_op, hxV⟩"
}
] |
[
1975,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1954,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
Subgroup.smul_mem_pointwise_smul
|
[] |
[
291,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
MvPolynomial.isNoetherianRing_fin_0
|
[
{
"state_after": "case f\nR : Type u\nS : Type ?u.780222\nσ : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\n⊢ R ≃+* MvPolynomial (Fin 0) R",
"state_before": "R : Type u\nS : Type ?u.780222\nσ : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\n⊢ IsNoetherianRing (MvPolynomial (Fin 0) R)",
"tactic": "apply isNoetherianRing_of_ringEquiv R"
},
{
"state_after": "case f\nR : Type u\nS : Type ?u.780222\nσ : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\n⊢ MvPolynomial (Fin 0) R ≃+* R",
"state_before": "case f\nR : Type u\nS : Type ?u.780222\nσ : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\n⊢ R ≃+* MvPolynomial (Fin 0) R",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "case f\nR : Type u\nS : Type ?u.780222\nσ : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\n⊢ MvPolynomial (Fin 0) R ≃+* R",
"tactic": "apply MvPolynomial.isEmptyRingEquiv R (Fin 0)"
}
] |
[
1064,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1061,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
WithTop.sum_lt_top
|
[] |
[
733,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
731,
1
] |
Mathlib/Topology/UniformSpace/CompleteSeparated.lean
|
IsComplete.isClosed
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : SeparatedSpace α\ns : Set α\nh : IsComplete s\na : α\nha : ClusterPt a (𝓟 s)\nf : Filter α := 𝓝[s] a\n⊢ a ∈ s",
"state_before": "α : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : SeparatedSpace α\ns : Set α\nh : IsComplete s\na : α\nha : ClusterPt a (𝓟 s)\n⊢ a ∈ s",
"tactic": "let f := 𝓝[s] a"
},
{
"state_after": "α : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : SeparatedSpace α\ns : Set α\nh : IsComplete s\na : α\nha : ClusterPt a (𝓟 s)\nf : Filter α := 𝓝[s] a\nthis : Cauchy f\n⊢ a ∈ s",
"state_before": "α : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : SeparatedSpace α\ns : Set α\nh : IsComplete s\na : α\nha : ClusterPt a (𝓟 s)\nf : Filter α := 𝓝[s] a\n⊢ a ∈ s",
"tactic": "have : Cauchy f := cauchy_nhds.mono' ha inf_le_left"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : SeparatedSpace α\ns : Set α\nh : IsComplete s\na : α\nha : ClusterPt a (𝓟 s)\nf : Filter α := 𝓝[s] a\nthis : Cauchy f\ny : α\nys : y ∈ s\nfy : f ≤ 𝓝 y\n⊢ a ∈ s",
"state_before": "α : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : SeparatedSpace α\ns : Set α\nh : IsComplete s\na : α\nha : ClusterPt a (𝓟 s)\nf : Filter α := 𝓝[s] a\nthis : Cauchy f\n⊢ a ∈ s",
"tactic": "rcases h f this inf_le_right with ⟨y, ys, fy⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : SeparatedSpace α\ns : Set α\nh : IsComplete s\na : α\nha : ClusterPt a (𝓟 s)\nf : Filter α := 𝓝[s] a\nthis : Cauchy f\ny : α\nys : y ∈ s\nfy : f ≤ 𝓝 y\n⊢ a ∈ s",
"tactic": "rwa [(tendsto_nhds_unique' ha inf_le_left fy : a = y)]"
}
] |
[
35,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
29,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.update_piecewise_of_not_mem
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.308822\nγ : Type ?u.308825\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ninst✝ : DecidableEq α\ni : α\nhi : ¬i ∈ s\nv : δ i\n⊢ piecewise s (update f i v) (update g i v) = piecewise s f (update g i v)",
"state_before": "α : Type u_1\nβ : Type ?u.308822\nγ : Type ?u.308825\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ninst✝ : DecidableEq α\ni : α\nhi : ¬i ∈ s\nv : δ i\n⊢ update (piecewise s f g) i v = piecewise s f (update g i v)",
"tactic": "rw [update_piecewise]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.308822\nγ : Type ?u.308825\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ninst✝ : DecidableEq α\ni : α\nhi : ¬i ∈ s\nv : δ i\nj : α\nhj : j ∈ s\n⊢ j ≠ i",
"state_before": "α : Type u_1\nβ : Type ?u.308822\nγ : Type ?u.308825\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ninst✝ : DecidableEq α\ni : α\nhi : ¬i ∈ s\nv : δ i\n⊢ piecewise s (update f i v) (update g i v) = piecewise s f (update g i v)",
"tactic": "refine' s.piecewise_congr (fun j hj => update_noteq _ _ _) fun _ _ => rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.308822\nγ : Type ?u.308825\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ninst✝ : DecidableEq α\ni : α\nhi : ¬i ∈ s\nv : δ i\nj : α\nhj : j ∈ s\n⊢ j ≠ i",
"tactic": "exact fun h => hi (h ▸ hj)"
}
] |
[
2554,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2550,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.weaken
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.29021\nE : Type u_2\nF : Type ?u.29027\nG : Type ?u.29030\nE' : Type ?u.29033\nF' : Type u_3\nG' : Type ?u.29039\nE'' : Type ?u.29042\nF'' : Type ?u.29045\nG'' : Type ?u.29048\nR : Type ?u.29051\nR' : Type ?u.29054\n𝕜 : Type ?u.29057\n𝕜' : Type ?u.29060\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\nh : IsBigOWith c l f g'\nhc : c ≤ c'\nx : α\nhx : x ∈ {x | (fun x => ‖f x‖ ≤ c * ‖g' x‖) x}\n⊢ c * ‖g' x‖ ≤ c' * ‖g' x‖",
"tactic": "gcongr"
}
] |
[
199,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Data/Nat/Hyperoperation.lean
|
hyperoperation_two_two_eq_four
|
[
{
"state_after": "case zero\n\n⊢ hyperoperation (Nat.zero + 1) 2 2 = 4\n\ncase succ\nnn : ℕ\nnih : hyperoperation (nn + 1) 2 2 = 4\n⊢ hyperoperation (Nat.succ nn + 1) 2 2 = 4",
"state_before": "n : ℕ\n⊢ hyperoperation (n + 1) 2 2 = 4",
"tactic": "induction' n with nn nih"
},
{
"state_after": "no goals",
"state_before": "case zero\n\n⊢ hyperoperation (Nat.zero + 1) 2 2 = 4",
"tactic": "rw [hyperoperation_one]"
},
{
"state_after": "no goals",
"state_before": "case succ\nnn : ℕ\nnih : hyperoperation (nn + 1) 2 2 = 4\n⊢ hyperoperation (Nat.succ nn + 1) 2 2 = 4",
"tactic": "rw [hyperoperation_recursion, hyperoperation_ge_two_eq_self, nih]"
}
] |
[
106,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/MeasureTheory/Constructions/Pi.lean
|
MeasureTheory.Measure.pi_Ico_ae_eq_pi_Icc
|
[] |
[
536,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
534,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.L1.SimpleFunc.setToL1S_mono
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.748247\nF : Type ?u.748250\nF' : Type ?u.748253\nG : Type ?u.748256\n𝕜 : Type ?u.748259\np : ℝ≥0∞\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : NormedAddCommGroup F'\ninst✝⁷ : NormedSpace ℝ F'\ninst✝⁶ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\nG'' : Type u_2\nG' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G'\ninst✝² : NormedSpace ℝ G'\ninst✝¹ : NormedLatticeAddCommGroup G''\ninst✝ : NormedSpace ℝ G''\nT : Set α → G'' →L[ℝ] G'\nh_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G''), 0 ≤ x → 0 ≤ ↑(T s) x\nf g : { x // x ∈ simpleFunc G'' 1 μ }\nhfg✝ : f ≤ g\nhfg : 0 ≤ g - f\n⊢ 0 ≤ setToL1S T g - setToL1S T f",
"state_before": "α : Type u_1\nE : Type ?u.748247\nF : Type ?u.748250\nF' : Type ?u.748253\nG : Type ?u.748256\n𝕜 : Type ?u.748259\np : ℝ≥0∞\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : NormedAddCommGroup F'\ninst✝⁷ : NormedSpace ℝ F'\ninst✝⁶ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\nG'' : Type u_2\nG' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G'\ninst✝² : NormedSpace ℝ G'\ninst✝¹ : NormedLatticeAddCommGroup G''\ninst✝ : NormedSpace ℝ G''\nT : Set α → G'' →L[ℝ] G'\nh_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G''), 0 ≤ x → 0 ≤ ↑(T s) x\nf g : { x // x ∈ simpleFunc G'' 1 μ }\nhfg : f ≤ g\n⊢ setToL1S T f ≤ setToL1S T g",
"tactic": "rw [← sub_nonneg] at hfg ⊢"
},
{
"state_after": "α : Type u_1\nE : Type ?u.748247\nF : Type ?u.748250\nF' : Type ?u.748253\nG : Type ?u.748256\n𝕜 : Type ?u.748259\np : ℝ≥0∞\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : NormedAddCommGroup F'\ninst✝⁷ : NormedSpace ℝ F'\ninst✝⁶ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\nG'' : Type u_2\nG' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G'\ninst✝² : NormedSpace ℝ G'\ninst✝¹ : NormedLatticeAddCommGroup G''\ninst✝ : NormedSpace ℝ G''\nT : Set α → G'' →L[ℝ] G'\nh_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G''), 0 ≤ x → 0 ≤ ↑(T s) x\nf g : { x // x ∈ simpleFunc G'' 1 μ }\nhfg✝ : f ≤ g\nhfg : 0 ≤ g - f\n⊢ 0 ≤ setToL1S (fun s => T s) (g - f)",
"state_before": "α : Type u_1\nE : Type ?u.748247\nF : Type ?u.748250\nF' : Type ?u.748253\nG : Type ?u.748256\n𝕜 : Type ?u.748259\np : ℝ≥0∞\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : NormedAddCommGroup F'\ninst✝⁷ : NormedSpace ℝ F'\ninst✝⁶ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\nG'' : Type u_2\nG' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G'\ninst✝² : NormedSpace ℝ G'\ninst✝¹ : NormedLatticeAddCommGroup G''\ninst✝ : NormedSpace ℝ G''\nT : Set α → G'' →L[ℝ] G'\nh_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G''), 0 ≤ x → 0 ≤ ↑(T s) x\nf g : { x // x ∈ simpleFunc G'' 1 μ }\nhfg✝ : f ≤ g\nhfg : 0 ≤ g - f\n⊢ 0 ≤ setToL1S T g - setToL1S T f",
"tactic": "rw [← setToL1S_sub h_zero h_add]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.748247\nF : Type ?u.748250\nF' : Type ?u.748253\nG : Type ?u.748256\n𝕜 : Type ?u.748259\np : ℝ≥0∞\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : NormedAddCommGroup F'\ninst✝⁷ : NormedSpace ℝ F'\ninst✝⁶ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\nG'' : Type u_2\nG' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G'\ninst✝² : NormedSpace ℝ G'\ninst✝¹ : NormedLatticeAddCommGroup G''\ninst✝ : NormedSpace ℝ G''\nT : Set α → G'' →L[ℝ] G'\nh_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G''), 0 ≤ x → 0 ≤ ↑(T s) x\nf g : { x // x ∈ simpleFunc G'' 1 μ }\nhfg✝ : f ≤ g\nhfg : 0 ≤ g - f\n⊢ 0 ≤ setToL1S (fun s => T s) (g - f)",
"tactic": "exact setToL1S_nonneg h_zero h_add hT_nonneg hfg"
}
] |
[
864,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
858,
1
] |
Mathlib/Data/Finsupp/AList.lean
|
AList.lookupFinsupp_support
|
[
{
"state_after": "α : Type u_1\nM : Type u_2\ninst✝² : Zero M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq M\nl : AList fun _x => M\n⊢ toFinset (List.keys (filter (fun x => decide ¬x.snd = 0) l.entries)) =\n toFinset (List.keys (filter (fun x => decide ¬x.snd = 0) l.entries))",
"state_before": "α : Type u_1\nM : Type u_2\ninst✝² : Zero M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq M\nl : AList fun _x => M\n⊢ (lookupFinsupp l).support = toFinset (List.keys (filter (fun x => decide (x.snd ≠ 0)) l.entries))",
"tactic": "simp only [lookupFinsupp, ne_eq, Finsupp.coe_mk]"
},
{
"state_after": "case h.e_2.h\nα : Type u_1\nM : Type u_2\ninst✝² : Zero M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq M\nl : AList fun _x => M\n⊢ (fun a b => Classical.decEq α a b) = fun a b => inst✝¹ a b\n\ncase h.e_3.h.e_a.e_p\nα : Type u_1\nM : Type u_2\ninst✝² : Zero M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq M\nl : AList fun _x => M\n⊢ (fun x => decide ¬x.snd = 0) = fun x => decide ¬x.snd = 0",
"state_before": "α : Type u_1\nM : Type u_2\ninst✝² : Zero M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq M\nl : AList fun _x => M\n⊢ toFinset (List.keys (filter (fun x => decide ¬x.snd = 0) l.entries)) =\n toFinset (List.keys (filter (fun x => decide ¬x.snd = 0) l.entries))",
"tactic": "congr"
},
{
"state_after": "case h.e_3.h.e_a.e_p\nα : Type u_1\nM : Type u_2\ninst✝² : Zero M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq M\nl : AList fun _x => M\n⊢ (fun x => decide ¬x.snd = 0) = fun x => decide ¬x.snd = 0",
"state_before": "case h.e_2.h\nα : Type u_1\nM : Type u_2\ninst✝² : Zero M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq M\nl : AList fun _x => M\n⊢ (fun a b => Classical.decEq α a b) = fun a b => inst✝¹ a b\n\ncase h.e_3.h.e_a.e_p\nα : Type u_1\nM : Type u_2\ninst✝² : Zero M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq M\nl : AList fun _x => M\n⊢ (fun x => decide ¬x.snd = 0) = fun x => decide ¬x.snd = 0",
"tactic": ". apply Subsingleton.elim"
},
{
"state_after": "no goals",
"state_before": "case h.e_3.h.e_a.e_p\nα : Type u_1\nM : Type u_2\ninst✝² : Zero M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq M\nl : AList fun _x => M\n⊢ (fun x => decide ¬x.snd = 0) = fun x => decide ¬x.snd = 0",
"tactic": ". funext ; congr"
},
{
"state_after": "no goals",
"state_before": "case h.e_2.h\nα : Type u_1\nM : Type u_2\ninst✝² : Zero M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq M\nl : AList fun _x => M\n⊢ (fun a b => Classical.decEq α a b) = fun a b => inst✝¹ a b",
"tactic": "apply Subsingleton.elim"
},
{
"state_after": "case h.e_3.h.e_a.e_p.h\nα : Type u_1\nM : Type u_2\ninst✝² : Zero M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq M\nl : AList fun _x => M\nx✝ : (_ : α) × M\n⊢ (decide ¬x✝.snd = 0) = decide ¬x✝.snd = 0",
"state_before": "case h.e_3.h.e_a.e_p\nα : Type u_1\nM : Type u_2\ninst✝² : Zero M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq M\nl : AList fun _x => M\n⊢ (fun x => decide ¬x.snd = 0) = fun x => decide ¬x.snd = 0",
"tactic": "funext"
},
{
"state_after": "no goals",
"state_before": "case h.e_3.h.e_a.e_p.h\nα : Type u_1\nM : Type u_2\ninst✝² : Zero M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq M\nl : AList fun _x => M\nx✝ : (_ : α) × M\n⊢ (decide ¬x✝.snd = 0) = decide ¬x✝.snd = 0",
"tactic": "congr"
}
] |
[
92,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Analysis/SpecialFunctions/Stirling.lean
|
Stirling.stirlingSeq_one
|
[
{
"state_after": "no goals",
"state_before": "⊢ stirlingSeq 1 = exp 1 / Real.sqrt 2",
"tactic": "rw [stirlingSeq, pow_one, factorial_one, cast_one, mul_one, mul_one_div, one_div_div]"
}
] |
[
67,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
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