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/RingTheory/Ideal/Operations.lean
|
Ideal.radical_mono
|
[] |
[
879,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
879,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.ae_eq_set
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.107971\nγ : Type ?u.107974\nδ : Type ?u.107977\nι : Type ?u.107980\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ s t : Set α\n⊢ s =ᵐ[μ] t ↔ ↑↑μ (s \\ t) = 0 ∧ ↑↑μ (t \\ s) = 0",
"tactic": "simp [eventuallyLE_antisymm_iff, ae_le_set]"
}
] |
[
484,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
FractionalIdeal.mul_right_le_iff
|
[
{
"state_after": "R : Type ?u.697823\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nJ : FractionalIdeal A⁰ K\nhJ : J ≠ 0\nI I' : FractionalIdeal A⁰ K\n⊢ I * J ≤ I' * J ↔ I ≤ I'",
"state_before": "R : Type ?u.697823\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nJ : FractionalIdeal A⁰ K\nhJ : J ≠ 0\n⊢ ∀ {I I' : FractionalIdeal A⁰ K}, I * J ≤ I' * J ↔ I ≤ I'",
"tactic": "intro I I'"
},
{
"state_after": "case mp\nR : Type ?u.697823\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nJ : FractionalIdeal A⁰ K\nhJ : J ≠ 0\nI I' : FractionalIdeal A⁰ K\n⊢ I * J ≤ I' * J → I ≤ I'\n\ncase mpr\nR : Type ?u.697823\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nJ : FractionalIdeal A⁰ K\nhJ : J ≠ 0\nI I' : FractionalIdeal A⁰ K\n⊢ I ≤ I' → I * J ≤ I' * J",
"state_before": "R : Type ?u.697823\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nJ : FractionalIdeal A⁰ K\nhJ : J ≠ 0\nI I' : FractionalIdeal A⁰ K\n⊢ I * J ≤ I' * J ↔ I ≤ I'",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type ?u.697823\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nJ : FractionalIdeal A⁰ K\nhJ : J ≠ 0\nI I' : FractionalIdeal A⁰ K\nh : I * J ≤ I' * J\n⊢ I ≤ I'",
"state_before": "case mp\nR : Type ?u.697823\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nJ : FractionalIdeal A⁰ K\nhJ : J ≠ 0\nI I' : FractionalIdeal A⁰ K\n⊢ I * J ≤ I' * J → I ≤ I'",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mp\nR : Type ?u.697823\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nJ : FractionalIdeal A⁰ K\nhJ : J ≠ 0\nI I' : FractionalIdeal A⁰ K\nh : I * J ≤ I' * J\n⊢ I ≤ I'",
"tactic": "convert mul_right_mono J⁻¹ h <;> dsimp only <;>\nrw [mul_assoc, FractionalIdeal.mul_inv_cancel hJ, mul_one]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type ?u.697823\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nJ : FractionalIdeal A⁰ K\nhJ : J ≠ 0\nI I' : FractionalIdeal A⁰ K\n⊢ I ≤ I' → I * J ≤ I' * J",
"tactic": "exact fun h => mul_right_mono J h"
}
] |
[
558,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
551,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
concaveOn_of_convex_hypograph
|
[] |
[
225,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
Subgroup.set_mul_normal_comm
|
[
{
"state_after": "case h\nα : Type ?u.21855\nG : Type u_1\nA : Type ?u.21861\nS : Type ?u.21864\ninst✝¹ : Group G\ninst✝ : AddGroup A\ns✝ s : Set G\nN : Subgroup G\nhN : Normal N\nx : G\n⊢ x ∈ s * ↑N ↔ x ∈ ↑N * s",
"state_before": "α : Type ?u.21855\nG : Type u_1\nA : Type ?u.21861\nS : Type ?u.21864\ninst✝¹ : Group G\ninst✝ : AddGroup A\ns✝ s : Set G\nN : Subgroup G\nhN : Normal N\n⊢ s * ↑N = ↑N * s",
"tactic": "ext x"
},
{
"state_after": "case h\nα : Type ?u.21855\nG : Type u_1\nA : Type ?u.21861\nS : Type ?u.21864\ninst✝¹ : Group G\ninst✝ : AddGroup A\ns✝ s : Set G\nN : Subgroup G\nhN : Normal N\nx y : G\n⊢ (∃ b, y ∈ s ∧ b ∈ ↑N ∧ (fun x x_1 => x * x_1) y b = x) ↔ ∃ y_1, y_1 ∈ ↑N ∧ y ∈ s ∧ (fun x x_1 => x * x_1) y_1 y = x",
"state_before": "case h\nα : Type ?u.21855\nG : Type u_1\nA : Type ?u.21861\nS : Type ?u.21864\ninst✝¹ : Group G\ninst✝ : AddGroup A\ns✝ s : Set G\nN : Subgroup G\nhN : Normal N\nx : G\n⊢ x ∈ s * ↑N ↔ x ∈ ↑N * s",
"tactic": "refine (exists_congr fun y => ?_).trans exists_swap"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type ?u.21855\nG : Type u_1\nA : Type ?u.21861\nS : Type ?u.21864\ninst✝¹ : Group G\ninst✝ : AddGroup A\ns✝ s : Set G\nN : Subgroup G\nhN : Normal N\nx y : G\n⊢ (∃ b, y ∈ s ∧ b ∈ ↑N ∧ (fun x x_1 => x * x_1) y b = x) ↔ ∃ y_1, y_1 ∈ ↑N ∧ y ∈ s ∧ (fun x x_1 => x * x_1) y_1 y = x",
"tactic": "simp only [exists_and_left, @and_left_comm _ (y ∈ s), ← eq_inv_mul_iff_mul_eq (b := y),\n ← eq_mul_inv_iff_mul_eq (c := y), exists_eq_right, SetLike.mem_coe, hN.mem_comm_iff]"
}
] |
[
165,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Reflexive.lean
|
CategoryTheory.left_comp_retraction
|
[] |
[
99,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
RingHom.coe_addMonoidHom_id
|
[] |
[
681,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
680,
1
] |
Mathlib/Data/Set/Pairwise/Lattice.lean
|
Set.biUnion_diff_biUnion_eq
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.3238\nγ : Type ?u.3241\nι : Type u_1\nι' : Type ?u.3247\nr p q : α → α → Prop\ns t : Set ι\nf : ι → Set α\nh : PairwiseDisjoint (s ∪ t) f\ni : ι\nhi : i ∈ s \\ t\na : α\nha : a ∈ f i\n⊢ ¬a ∈ ⋃ (x : ι) (_ : x ∈ t), f x",
"state_before": "α : Type u_2\nβ : Type ?u.3238\nγ : Type ?u.3241\nι : Type u_1\nι' : Type ?u.3247\nr p q : α → α → Prop\ns t : Set ι\nf : ι → Set α\nh : PairwiseDisjoint (s ∪ t) f\n⊢ ((⋃ (i : ι) (_ : i ∈ s), f i) \\ ⋃ (i : ι) (_ : i ∈ t), f i) = ⋃ (i : ι) (_ : i ∈ s \\ t), f i",
"tactic": "refine'\n (biUnion_diff_biUnion_subset f s t).antisymm\n (iUnion₂_subset fun i hi a ha => (mem_diff _).2 ⟨mem_biUnion hi.1 ha, _⟩)"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.3238\nγ : Type ?u.3241\nι : Type u_1\nι' : Type ?u.3247\nr p q : α → α → Prop\ns t : Set ι\nf : ι → Set α\nh : PairwiseDisjoint (s ∪ t) f\ni : ι\nhi : i ∈ s \\ t\na : α\nha : a ∈ f i\n⊢ ¬∃ i j, a ∈ f i",
"state_before": "α : Type u_2\nβ : Type ?u.3238\nγ : Type ?u.3241\nι : Type u_1\nι' : Type ?u.3247\nr p q : α → α → Prop\ns t : Set ι\nf : ι → Set α\nh : PairwiseDisjoint (s ∪ t) f\ni : ι\nhi : i ∈ s \\ t\na : α\nha : a ∈ f i\n⊢ ¬a ∈ ⋃ (x : ι) (_ : x ∈ t), f x",
"tactic": "rw [mem_iUnion₂]"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type ?u.3238\nγ : Type ?u.3241\nι : Type u_1\nι' : Type ?u.3247\nr p q : α → α → Prop\ns t : Set ι\nf : ι → Set α\nh : PairwiseDisjoint (s ∪ t) f\ni : ι\nhi : i ∈ s \\ t\na : α\nha : a ∈ f i\nj : ι\nhj : j ∈ t\nhaj : a ∈ f j\n⊢ False",
"state_before": "α : Type u_2\nβ : Type ?u.3238\nγ : Type ?u.3241\nι : Type u_1\nι' : Type ?u.3247\nr p q : α → α → Prop\ns t : Set ι\nf : ι → Set α\nh : PairwiseDisjoint (s ∪ t) f\ni : ι\nhi : i ∈ s \\ t\na : α\nha : a ∈ f i\n⊢ ¬∃ i j, a ∈ f i",
"tactic": "rintro ⟨j, hj, haj⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type ?u.3238\nγ : Type ?u.3241\nι : Type u_1\nι' : Type ?u.3247\nr p q : α → α → Prop\ns t : Set ι\nf : ι → Set α\nh : PairwiseDisjoint (s ∪ t) f\ni : ι\nhi : i ∈ s \\ t\na : α\nha : a ∈ f i\nj : ι\nhj : j ∈ t\nhaj : a ∈ f j\n⊢ False",
"tactic": "exact (h (Or.inl hi.1) (Or.inr hj) (ne_of_mem_of_not_mem hj hi.2).symm).le_bot ⟨ha, haj⟩"
}
] |
[
100,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Data/Nat/Choose/Cast.lean
|
Nat.cast_choose_eq_pochhammer_div
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : CharZero K\na b : ℕ\n⊢ ↑(choose a b) = Polynomial.eval (↑(a - (b - 1))) (pochhammer K b) / ↑b !",
"tactic": "rw [eq_div_iff_mul_eq (cast_ne_zero.2 b.factorial_ne_zero : (b ! : K) ≠ 0), ← cast_mul,\n mul_comm, ← descFactorial_eq_factorial_mul_choose, ← cast_descFactorial]"
}
] |
[
41,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/Algebra/Group/Prod.lean
|
Prod.snd_div
|
[] |
[
176,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Topology/Instances/Matrix.lean
|
Continuous.matrix_adjugate
|
[] |
[
230,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Data/Real/Sign.lean
|
Real.sign_mul_nonneg
|
[
{
"state_after": "case inl\nr : ℝ\nhn : r < 0\n⊢ 0 ≤ sign r * r\n\ncase inr.inl\n\n⊢ 0 ≤ sign 0 * 0\n\ncase inr.inr\nr : ℝ\nhp : 0 < r\n⊢ 0 ≤ sign r * r",
"state_before": "r : ℝ\n⊢ 0 ≤ sign r * r",
"tactic": "obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)"
},
{
"state_after": "case inl\nr : ℝ\nhn : r < 0\n⊢ 0 ≤ -1 * r",
"state_before": "case inl\nr : ℝ\nhn : r < 0\n⊢ 0 ≤ sign r * r",
"tactic": "rw [sign_of_neg hn]"
},
{
"state_after": "no goals",
"state_before": "case inl\nr : ℝ\nhn : r < 0\n⊢ 0 ≤ -1 * r",
"tactic": "exact mul_nonneg_of_nonpos_of_nonpos (by norm_num) hn.le"
},
{
"state_after": "no goals",
"state_before": "r : ℝ\nhn : r < 0\n⊢ -1 ≤ 0",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\n\n⊢ 0 ≤ sign 0 * 0",
"tactic": "rw [mul_zero]"
},
{
"state_after": "case inr.inr\nr : ℝ\nhp : 0 < r\n⊢ 0 ≤ r",
"state_before": "case inr.inr\nr : ℝ\nhp : 0 < r\n⊢ 0 ≤ sign r * r",
"tactic": "rw [sign_of_pos hp, one_mul]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nr : ℝ\nhp : 0 < r\n⊢ 0 ≤ r",
"tactic": "exact hp.le"
}
] |
[
98,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
DirectSum.IsInternal.collectedBasis_orthonormal
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3801506\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_3\ndec_ι : DecidableEq ι\nG : ι → Type ?u.3801569\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV✝ : (i : ι) → G i →ₗᵢ[𝕜] E\nhV✝ : OrthogonalFamily 𝕜 G V✝\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nV : ι → Submodule 𝕜 E\nhV : OrthogonalFamily 𝕜 (fun i => { x // x ∈ V i }) fun i => Submodule.subtypeₗᵢ (V i)\nhV_sum : IsInternal fun i => V i\nα : ι → Type u_4\nv_family : (i : ι) → Basis (α i) 𝕜 { x // x ∈ V i }\nhv_family : ∀ (i : ι), Orthonormal 𝕜 ↑(v_family i)\n⊢ Orthonormal 𝕜 ↑(collectedBasis hV_sum v_family)",
"tactic": "simpa only [hV_sum.collectedBasis_coe] using hV.orthonormal_sigma_orthonormal hv_family"
}
] |
[
2180,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2175,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
MonoidHom.iterate_map_frobenius
|
[] |
[
387,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/LinearAlgebra/Matrix/IsDiag.lean
|
Matrix.IsDiag.transpose
|
[] |
[
120,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
inner_add_add_self
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1804599\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ inner x x + inner y x + (inner x y + inner y y) = inner x x + inner x y + inner y x + inner y y",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1804599\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ inner (x + y) (x + y) = inner x x + inner x y + inner y x + inner y y",
"tactic": "simp only [inner_add_left, inner_add_right]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1804599\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ inner x x + inner y x + (inner x y + inner y y) = inner x x + inner x y + inner y x + inner y y",
"tactic": "ring"
}
] |
[
666,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
665,
1
] |
Mathlib/Data/Int/SuccPred.lean
|
Int.pos_iff_one_le
|
[] |
[
50,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Lp.stronglyMeasurable
|
[] |
[
211,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
11
] |
Std/Logic.lean
|
Decidable.peirce
|
[] |
[
572,
53
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
571,
1
] |
Mathlib/Algebra/Module/LinearMap.lean
|
AddMonoidHom.coe_toIntLinearMap
|
[] |
[
774,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
772,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.Reaches₀.tail'
|
[] |
[
819,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
817,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
sq_le_sq
|
[
{
"state_after": "no goals",
"state_before": "β : Type ?u.260389\nA : Type ?u.260392\nG : Type ?u.260395\nM : Type ?u.260398\nR : Type u_1\ninst✝ : LinearOrderedRing R\nx y : R\n⊢ x ^ 2 ≤ y ^ 2 ↔ abs x ≤ abs y",
"tactic": "simpa only [sq_abs] using\n (@strictMonoOn_pow R _ _ two_pos).le_iff_le (abs_nonneg x) (abs_nonneg y)"
}
] |
[
702,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
700,
1
] |
Mathlib/Algebra/Order/Module.lean
|
smul_neg_iff_of_neg
|
[
{
"state_after": "k : Type u_1\nM : Type u_2\nN : Type ?u.32115\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n⊢ 0 < -c • a ↔ 0 < a",
"state_before": "k : Type u_1\nM : Type u_2\nN : Type ?u.32115\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n⊢ c • a < 0 ↔ 0 < a",
"tactic": "rw [← neg_neg c, neg_smul, neg_neg_iff_pos]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nM : Type u_2\nN : Type ?u.32115\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n⊢ 0 < -c • a ↔ 0 < a",
"tactic": "exact smul_pos_iff_of_pos (neg_pos_of_neg hc)"
}
] |
[
83,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
|
blimsup_cthickening_mul_ae_eq
|
[
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"tactic": "let r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / ((i : ℝ) + 1)"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"tactic": "have h₀ : ∀ i, p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i) := by\n rintro i ⟨-, hi⟩; congr! 1; change r i = ite (0 < r i) (r i) _; simp [hi]"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"tactic": "have h₁ : ∀ i, p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) := by\n rintro i ⟨-, hi⟩; simp only [hi, mul_ite, if_true]"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"tactic": "have h₂ : ∀ i, p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i) := by\n rintro i ⟨-, hi⟩\n have hi' : M * r i ≤ 0 := mul_nonpos_of_nonneg_of_nonpos hM.le hi\n rw [cthickening_of_nonpos hi, cthickening_of_nonpos hi']"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"tactic": "have hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0 := by\n ext i; simp [← and_or_left, lt_or_le 0 (r i)]"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0\n⊢ ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ 0 < r i) ⊔\n blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ r i ≤ 0) =ᵐ[μ]\n (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ 0 < r i) ⊔\n blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"tactic": "rw [hp, blimsup_or_eq_sup, blimsup_or_eq_sup]"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0\n⊢ ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ 0 < r i) ∪\n blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ r i ≤ 0) =ᵐ[μ]\n (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ 0 < r i) ∪\n blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0\n⊢ ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ 0 < r i) ⊔\n blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ r i ≤ 0) =ᵐ[μ]\n (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ 0 < r i) ⊔\n blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0",
"tactic": "simp only [sup_eq_union]"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0\n⊢ ((blimsup (fun x => cthickening (M * r' x) (s x)) atTop fun x => p x ∧ 0 < r x) ∪\n blimsup (fun x => cthickening (r x) (s x)) atTop fun x => p x ∧ r x ≤ 0) =ᵐ[μ]\n (blimsup (fun x => cthickening (r' x) (s x)) atTop fun x => p x ∧ 0 < r x) ∪\n blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0\n⊢ ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ 0 < r i) ∪\n blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ r i ≤ 0) =ᵐ[μ]\n (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ 0 < r i) ∪\n blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0",
"tactic": "rw [blimsup_congr (eventually_of_forall h₀), blimsup_congr (eventually_of_forall h₁),\n blimsup_congr (eventually_of_forall h₂)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)\nhp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0\n⊢ ((blimsup (fun x => cthickening (M * r' x) (s x)) atTop fun x => p x ∧ 0 < r x) ∪\n blimsup (fun x => cthickening (r x) (s x)) atTop fun x => p x ∧ r x ≤ 0) =ᵐ[μ]\n (blimsup (fun x => cthickening (r' x) (s x)) atTop fun x => p x ∧ 0 < r x) ∪\n blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0",
"tactic": "exact ae_eq_set_union (this (fun i => p i ∧ 0 < r i) hr') (ae_eq_refl _)"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\n⊢ ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\n⊢ ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"tactic": "clear p hr r"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\np : ℕ → Prop\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝[Ioi 0] 0)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\n⊢ ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"tactic": "intro p r hr"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\np : ℕ → Prop\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\np : ℕ → Prop\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝[Ioi 0] 0)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"tactic": "have hr' : Tendsto (fun i => M * r i) atTop (𝓝[>] 0) := by\n convert TendstoNhdsWithinIoi.const_mul hM hr <;> simp only [MulZeroClass.mul_zero]"
},
{
"state_after": "case refine'_1\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\np : ℕ → Prop\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\n\ncase refine'_2\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\np : ℕ → Prop\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)\n⊢ blimsup (fun i => cthickening (r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (M * r i) (s i)) atTop p",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\np : ℕ → Prop\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"tactic": "refine' eventuallyLE_antisymm_iff.mpr ⟨_, _⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\np : ℕ → Prop\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝[Ioi 0] 0)\n⊢ Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)",
"tactic": "convert TendstoNhdsWithinIoi.const_mul hM hr <;> simp only [MulZeroClass.mul_zero]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\np : ℕ → Prop\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)\n⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p",
"tactic": "exact blimsup_cthickening_ae_le_of_eventually_mul_le μ p (inv_pos.mpr hM) hr'\n (eventually_of_forall fun i => by rw [inv_mul_cancel_left₀ hM.ne' (r i)])"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\np : ℕ → Prop\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)\ni : ℕ\n⊢ M⁻¹ * (M * r i) ≤ r i",
"tactic": "rw [inv_mul_cancel_left₀ hM.ne' (r i)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\np : ℕ → Prop\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝[Ioi 0] 0)\nhr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)\n⊢ blimsup (fun i => cthickening (r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (M * r i) (s i)) atTop p",
"tactic": "exact blimsup_cthickening_ae_le_of_eventually_mul_le μ p hM hr\n (eventually_of_forall fun i => le_refl _)"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\ni : ℕ\n⊢ r' i ∈ Ioi 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\n⊢ Tendsto r' atTop (𝓝[Ioi 0] 0)",
"tactic": "refine' tendsto_nhdsWithin_iff.mpr\n ⟨Tendsto.if' hr tendsto_one_div_add_atTop_nhds_0_nat, eventually_of_forall fun i => _⟩"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\ni : ℕ\nhi : 0 < r i\n⊢ r' i ∈ Ioi 0\n\ncase neg\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\ni : ℕ\nhi : ¬0 < r i\n⊢ r' i ∈ Ioi 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\ni : ℕ\n⊢ r' i ∈ Ioi 0",
"tactic": "by_cases hi : 0 < r i"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\ni : ℕ\nhi : 0 < r i\n⊢ r' i ∈ Ioi 0",
"tactic": "simp [hi]"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\ni : ℕ\nhi : ¬0 < r i\n⊢ 0 < ↑i + 1",
"state_before": "case neg\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\ni : ℕ\nhi : ¬0 < r i\n⊢ r' i ∈ Ioi 0",
"tactic": "simp only [hi, one_div, mem_Ioi, if_false, inv_pos]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\ni : ℕ\nhi : ¬0 < r i\n⊢ 0 < ↑i + 1",
"tactic": "positivity"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\ni : ℕ\nhi : 0 < r i\n⊢ cthickening (r i) (s i) = cthickening (r' i) (s i)",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\n⊢ ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)",
"tactic": "rintro i ⟨-, hi⟩"
},
{
"state_after": "case intro.h.e'_3\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\ni : ℕ\nhi : 0 < r i\n⊢ r i = r' i",
"state_before": "case intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\ni : ℕ\nhi : 0 < r i\n⊢ cthickening (r i) (s i) = cthickening (r' i) (s i)",
"tactic": "congr! 1"
},
{
"state_after": "case intro.h.e'_3\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\ni : ℕ\nhi : 0 < r i\n⊢ r i = if 0 < r i then r i else 1 / (↑i + 1)",
"state_before": "case intro.h.e'_3\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\ni : ℕ\nhi : 0 < r i\n⊢ r i = r' i",
"tactic": "change r i = ite (0 < r i) (r i) _"
},
{
"state_after": "no goals",
"state_before": "case intro.h.e'_3\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\ni : ℕ\nhi : 0 < r i\n⊢ r i = if 0 < r i then r i else 1 / (↑i + 1)",
"tactic": "simp [hi]"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\ni : ℕ\nhi : 0 < r i\n⊢ cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\n⊢ ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)",
"tactic": "rintro i ⟨-, hi⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\ni : ℕ\nhi : 0 < r i\n⊢ cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)",
"tactic": "simp only [hi, mul_ite, if_true]"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\ni : ℕ\nhi : r i ≤ 0\n⊢ cthickening (M * r i) (s i) = cthickening (r i) (s i)",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\n⊢ ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)",
"tactic": "rintro i ⟨-, hi⟩"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\ni : ℕ\nhi : r i ≤ 0\nhi' : M * r i ≤ 0\n⊢ cthickening (M * r i) (s i) = cthickening (r i) (s i)",
"state_before": "case intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\ni : ℕ\nhi : r i ≤ 0\n⊢ cthickening (M * r i) (s i) = cthickening (r i) (s i)",
"tactic": "have hi' : M * r i ≤ 0 := mul_nonpos_of_nonneg_of_nonpos hM.le hi"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\ni : ℕ\nhi : r i ≤ 0\nhi' : M * r i ≤ 0\n⊢ cthickening (M * r i) (s i) = cthickening (r i) (s i)",
"tactic": "rw [cthickening_of_nonpos hi, cthickening_of_nonpos hi']"
},
{
"state_after": "case h.a\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)\ni : ℕ\n⊢ p i ↔ p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)\n⊢ p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case h.a\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : MeasureTheory.Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nthis :\n ∀ (p : ℕ → Prop) {r : ℕ → ℝ},\n Tendsto r atTop (𝓝[Ioi 0] 0) →\n blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p\nr' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)\nhr' : Tendsto r' atTop (𝓝[Ioi 0] 0)\nh₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)\nh₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)\nh₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)\ni : ℕ\n⊢ p i ↔ p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0",
"tactic": "simp [← and_or_left, lt_or_le 0 (r i)]"
}
] |
[
230,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Topology/Algebra/Polynomial.lean
|
Polynomial.tendsto_abv_aeval_atTop
|
[] |
[
131,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Data/List/Intervals.lean
|
List.Ico.filter_lt
|
[
{
"state_after": "case inl\nn m l : ℕ\nhml : m ≤ l\n⊢ filter (fun x => decide (x < l)) (Ico n m) = Ico n (min m l)\n\ncase inr\nn m l : ℕ\nhlm : l ≤ m\n⊢ filter (fun x => decide (x < l)) (Ico n m) = Ico n (min m l)",
"state_before": "n m l : ℕ\n⊢ filter (fun x => decide (x < l)) (Ico n m) = Ico n (min m l)",
"tactic": "cases' le_total m l with hml hlm"
},
{
"state_after": "no goals",
"state_before": "case inl\nn m l : ℕ\nhml : m ≤ l\n⊢ filter (fun x => decide (x < l)) (Ico n m) = Ico n (min m l)",
"tactic": "rw [min_eq_left hml, filter_lt_of_top_le hml]"
},
{
"state_after": "no goals",
"state_before": "case inr\nn m l : ℕ\nhlm : l ≤ m\n⊢ filter (fun x => decide (x < l)) (Ico n m) = Ico n (min m l)",
"tactic": "rw [min_eq_right hlm, filter_lt_of_ge hlm]"
}
] |
[
184,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.toSubring_injective
|
[
{
"state_after": "no goals",
"state_before": "R' : Type u'\nR✝ : Type u\nA✝ : Type v\nB : Type w\nC : Type w'\ninst✝⁹ : CommSemiring R✝\ninst✝⁸ : Semiring A✝\ninst✝⁷ : Algebra R✝ A✝\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R✝ B\ninst✝⁴ : Semiring C\ninst✝³ : Algebra R✝ C\nS✝ : Subalgebra R✝ A✝\nR : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nh : toSubring S = toSubring T\nx : A\n⊢ x ∈ S ↔ x ∈ T",
"tactic": "rw [← mem_toSubring, ← mem_toSubring, h]"
}
] |
[
242,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
|
LiouvilleWith.add_nat_iff
|
[
{
"state_after": "no goals",
"state_before": "p q x y : ℝ\nr : ℚ\nm : ℤ\nn : ℕ\n⊢ LiouvilleWith p (x + ↑n) ↔ LiouvilleWith p x",
"tactic": "rw [← Rat.cast_coe_nat n, add_rat_iff]"
}
] |
[
225,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.Disjoint.mul_right
|
[
{
"state_after": "α : Type u_1\nf g h : Perm α\nH1 : Disjoint f g\nH2 : Disjoint f h\n⊢ Disjoint (g * h) f",
"state_before": "α : Type u_1\nf g h : Perm α\nH1 : Disjoint f g\nH2 : Disjoint f h\n⊢ Disjoint f (g * h)",
"tactic": "rw [disjoint_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf g h : Perm α\nH1 : Disjoint f g\nH2 : Disjoint f h\n⊢ Disjoint (g * h) f",
"tactic": "exact H1.symm.mul_left H2.symm"
}
] |
[
117,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.embDomain_one
|
[] |
[
1049,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1047,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.biUnion_empty
|
[] |
[
964,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
963,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
tendsto_nat_ceil_mul_div_atTop
|
[
{
"state_after": "α : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 (a + 0))\n⊢ Tendsto (fun x => ↑⌈a * x⌉₊ / x) atTop (𝓝 a)",
"state_before": "α : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\n⊢ Tendsto (fun x => ↑⌈a * x⌉₊ / x) atTop (𝓝 a)",
"tactic": "have A : Tendsto (fun x : R => a + x⁻¹) atTop (𝓝 (a + 0)) :=\n tendsto_const_nhds.add tendsto_inv_atTop_zero"
},
{
"state_after": "α : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)\n⊢ Tendsto (fun x => ↑⌈a * x⌉₊ / x) atTop (𝓝 a)",
"state_before": "α : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 (a + 0))\n⊢ Tendsto (fun x => ↑⌈a * x⌉₊ / x) atTop (𝓝 a)",
"tactic": "rw [add_zero] at A"
},
{
"state_after": "case hgf\nα : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)\n⊢ ∀ᶠ (b : R) in atTop, a ≤ ↑⌈a * b⌉₊ / b\n\ncase hfh\nα : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)\n⊢ ∀ᶠ (b : R) in atTop, ↑⌈a * b⌉₊ / b ≤ a + b⁻¹",
"state_before": "α : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)\n⊢ Tendsto (fun x => ↑⌈a * x⌉₊ / x) atTop (𝓝 a)",
"tactic": "apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds A"
},
{
"state_after": "case hgf\nα : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ a ≤ ↑⌈a * x⌉₊ / x",
"state_before": "case hgf\nα : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)\n⊢ ∀ᶠ (b : R) in atTop, a ≤ ↑⌈a * b⌉₊ / b",
"tactic": "refine' eventually_atTop.2 ⟨1, fun x hx => _⟩"
},
{
"state_after": "case hgf\nα : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ a * x ≤ ↑⌈a * x⌉₊",
"state_before": "case hgf\nα : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ a ≤ ↑⌈a * x⌉₊ / x",
"tactic": "rw [le_div_iff (zero_lt_one.trans_le hx)]"
},
{
"state_after": "no goals",
"state_before": "case hgf\nα : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ a * x ≤ ↑⌈a * x⌉₊",
"tactic": "exact Nat.le_ceil _"
},
{
"state_after": "case hfh\nα : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ ↑⌈a * x⌉₊ / x ≤ a + x⁻¹",
"state_before": "case hfh\nα : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)\n⊢ ∀ᶠ (b : R) in atTop, ↑⌈a * b⌉₊ / b ≤ a + b⁻¹",
"tactic": "refine' eventually_atTop.2 ⟨1, fun x hx => _⟩"
},
{
"state_after": "no goals",
"state_before": "case hfh\nα : Type ?u.516691\nβ : Type ?u.516694\nι : Type ?u.516697\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ ↑⌈a * x⌉₊ / x ≤ a + x⁻¹",
"tactic": "simp [div_le_iff (zero_lt_one.trans_le hx), inv_mul_cancel (zero_lt_one.trans_le hx).ne',\n (Nat.ceil_lt_add_one (mul_nonneg ha (zero_le_one.trans hx))).le, add_mul]"
}
] |
[
610,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
599,
1
] |
Mathlib/Algebra/BigOperators/Pi.lean
|
Finset.univ_prod_mulSingle
|
[
{
"state_after": "case h\nI : Type u_1\ninst✝² : DecidableEq I\nZ : I → Type u_2\ninst✝¹ : (i : I) → CommMonoid (Z i)\ninst✝ : Fintype I\nf : (i : I) → Z i\na : I\n⊢ Finset.prod univ (fun i => Pi.mulSingle i (f i)) a = f a",
"state_before": "I : Type u_1\ninst✝² : DecidableEq I\nZ : I → Type u_2\ninst✝¹ : (i : I) → CommMonoid (Z i)\ninst✝ : Fintype I\nf : (i : I) → Z i\n⊢ ∏ i : I, Pi.mulSingle i (f i) = f",
"tactic": "ext a"
},
{
"state_after": "no goals",
"state_before": "case h\nI : Type u_1\ninst✝² : DecidableEq I\nZ : I → Type u_2\ninst✝¹ : (i : I) → CommMonoid (Z i)\ninst✝ : Fintype I\nf : (i : I) → Z i\na : I\n⊢ Finset.prod univ (fun i => Pi.mulSingle i (f i)) a = f a",
"tactic": "simp"
}
] |
[
84,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.Tendsto.not_tendsto
|
[] |
[
3083,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3081,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.add_lt_of_lt_sub_left
|
[
{
"state_after": "a b c : Int\nh✝ : b < c - a\nh : a + b < a + (c - a)\n⊢ a + b < c",
"state_before": "a b c : Int\nh : b < c - a\n⊢ a + b < c",
"tactic": "have h := Int.add_lt_add_left h a"
},
{
"state_after": "no goals",
"state_before": "a b c : Int\nh✝ : b < c - a\nh : a + b < a + (c - a)\n⊢ a + b < c",
"tactic": "rwa [← Int.add_sub_assoc, Int.add_comm a c, Int.add_sub_cancel] at h"
}
] |
[
1049,
71
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1047,
11
] |
Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean
|
CategoryTheory.Limits.isIso_π_of_isTerminal
|
[
{
"state_after": "case w\nC : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsTerminal j\nF : J ⥤ C\ninst✝¹ : HasLimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\nj✝ : J\n⊢ (limit.π F j ≫ limit.lift F (coneOfDiagramTerminal I F)) ≫ limit.π F j✝ = 𝟙 (limit F) ≫ limit.π F j✝",
"state_before": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsTerminal j\nF : J ⥤ C\ninst✝¹ : HasLimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ limit.π F j ≫ limit.lift F (coneOfDiagramTerminal I F) = 𝟙 (limit F)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsTerminal j\nF : J ⥤ C\ninst✝¹ : HasLimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\nj✝ : J\n⊢ (limit.π F j ≫ limit.lift F (coneOfDiagramTerminal I F)) ≫ limit.π F j✝ = 𝟙 (limit F) ≫ limit.π F j✝",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsTerminal j\nF : J ⥤ C\ninst✝¹ : HasLimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ limit.lift F (coneOfDiagramTerminal I F) ≫ limit.π F j = 𝟙 (F.obj j)",
"tactic": "simp"
}
] |
[
721,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
719,
1
] |
Mathlib/Init/Data/Bool/Lemmas.lean
|
Bool.or_eq_true_eq_eq_true_or_eq_true
|
[
{
"state_after": "no goals",
"state_before": "a b : Bool\n⊢ ((a || b) = true) = (a = true ∨ b = true)",
"tactic": "simp"
}
] |
[
91,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean
|
geometric_hahn_banach_compact_closed
|
[
{
"state_after": "case inl\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\nt : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nht₁ : Convex ℝ t\nht₂ : IsClosed t\nhs₁ : Convex ℝ ∅\nhs₂ : IsCompact ∅\ndisj : Disjoint ∅ t\n⊢ ∃ f u v, (∀ (a : E), a ∈ ∅ → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b\n\ncase inr\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"state_before": "𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"tactic": "obtain rfl | hs := s.eq_empty_or_nonempty"
},
{
"state_after": "case inr.inl\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nhs : Set.Nonempty s\nht₁ : Convex ℝ ∅\nht₂ : IsClosed ∅\ndisj : Disjoint s ∅\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ ∅ → v < ↑f b\n\ncase inr.inr\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"state_before": "case inr\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"tactic": "obtain rfl | _ht := t.eq_empty_or_nonempty"
},
{
"state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU₁ : Convex ℝ U\nhV₁ : Convex ℝ V\nsU : s ⊆ U\ntV : t ⊆ V\ndisj' : Disjoint U V\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"state_before": "case inr.inr\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"tactic": "obtain ⟨U, V, hU, hV, hU₁, hV₁, sU, tV, disj'⟩ := disj.exists_open_convexes hs₁ hs₂ ht₁ ht₂"
},
{
"state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU₁ : Convex ℝ U\nhV₁ : Convex ℝ V\nsU : s ⊆ U\ntV : t ⊆ V\ndisj' : Disjoint U V\nf : E →L[ℝ] ℝ\nu : ℝ\nhf₁ : ∀ (a : E), a ∈ U → ↑f a < u\nhf₂ : ∀ (b : E), b ∈ V → u < ↑f b\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU₁ : Convex ℝ U\nhV₁ : Convex ℝ V\nsU : s ⊆ U\ntV : t ⊆ V\ndisj' : Disjoint U V\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"tactic": "obtain ⟨f, u, hf₁, hf₂⟩ := geometric_hahn_banach_open_open hU₁ hU hV₁ hV disj'"
},
{
"state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx✝ y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU₁ : Convex ℝ U\nhV₁ : Convex ℝ V\nsU : s ⊆ U\ntV : t ⊆ V\ndisj' : Disjoint U V\nf : E →L[ℝ] ℝ\nu : ℝ\nhf₁ : ∀ (a : E), a ∈ U → ↑f a < u\nhf₂ : ∀ (b : E), b ∈ V → u < ↑f b\nx : E\nhx₁ : x ∈ s\nhx₂ : ∀ (y : E), y ∈ s → ↑f y ≤ ↑f x\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU₁ : Convex ℝ U\nhV₁ : Convex ℝ V\nsU : s ⊆ U\ntV : t ⊆ V\ndisj' : Disjoint U V\nf : E →L[ℝ] ℝ\nu : ℝ\nhf₁ : ∀ (a : E), a ∈ U → ↑f a < u\nhf₂ : ∀ (b : E), b ∈ V → u < ↑f b\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"tactic": "obtain ⟨x, hx₁, hx₂⟩ := hs₂.exists_forall_ge hs f.continuous.continuousOn"
},
{
"state_after": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx✝ y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU₁ : Convex ℝ U\nhV₁ : Convex ℝ V\nsU : s ⊆ U\ntV : t ⊆ V\ndisj' : Disjoint U V\nf : E →L[ℝ] ℝ\nu : ℝ\nhf₁ : ∀ (a : E), a ∈ U → ↑f a < u\nhf₂ : ∀ (b : E), b ∈ V → u < ↑f b\nx : E\nhx₁ : x ∈ s\nhx₂ : ∀ (y : E), y ∈ s → ↑f y ≤ ↑f x\nthis : ↑f x < u\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx✝ y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU₁ : Convex ℝ U\nhV₁ : Convex ℝ V\nsU : s ⊆ U\ntV : t ⊆ V\ndisj' : Disjoint U V\nf : E →L[ℝ] ℝ\nu : ℝ\nhf₁ : ∀ (a : E), a ∈ U → ↑f a < u\nhf₂ : ∀ (b : E), b ∈ V → u < ↑f b\nx : E\nhx₁ : x ∈ s\nhx₂ : ∀ (y : E), y ∈ s → ↑f y ≤ ↑f x\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"tactic": "have : f x < u := hf₁ x (sU hx₁)"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx✝ y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU₁ : Convex ℝ U\nhV₁ : Convex ℝ V\nsU : s ⊆ U\ntV : t ⊆ V\ndisj' : Disjoint U V\nf : E →L[ℝ] ℝ\nu : ℝ\nhf₁ : ∀ (a : E), a ∈ U → ↑f a < u\nhf₂ : ∀ (b : E), b ∈ V → u < ↑f b\nx : E\nhx₁ : x ∈ s\nhx₂ : ∀ (y : E), y ∈ s → ↑f y ≤ ↑f x\nthis : ↑f x < u\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"tactic": "exact\n ⟨f, (f x + u) / 2, u, fun a ha => by linarith [hx₂ a ha], by linarith, fun b hb =>\n hf₂ b (tV hb)⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\nt : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nht₁ : Convex ℝ t\nht₂ : IsClosed t\nhs₁ : Convex ℝ ∅\nhs₂ : IsCompact ∅\ndisj : Disjoint ∅ t\n⊢ ∃ f u v, (∀ (a : E), a ∈ ∅ → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ t → v < ↑f b",
"tactic": "exact ⟨0, -2, -1, by simp, by norm_num, fun b _hb => by norm_num⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\nt : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nht₁ : Convex ℝ t\nht₂ : IsClosed t\nhs₁ : Convex ℝ ∅\nhs₂ : IsCompact ∅\ndisj : Disjoint ∅ t\n⊢ ∀ (a : E), a ∈ ∅ → ↑0 a < -2",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\nt : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nht₁ : Convex ℝ t\nht₂ : IsClosed t\nhs₁ : Convex ℝ ∅\nhs₂ : IsCompact ∅\ndisj : Disjoint ∅ t\n⊢ -2 < -1",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\nt : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nht₁ : Convex ℝ t\nht₂ : IsClosed t\nhs₁ : Convex ℝ ∅\nhs₂ : IsCompact ∅\ndisj : Disjoint ∅ t\nb : E\n_hb : b ∈ t\n⊢ -1 < ↑0 b",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\n𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nhs : Set.Nonempty s\nht₁ : Convex ℝ ∅\nht₂ : IsClosed ∅\ndisj : Disjoint s ∅\n⊢ ∃ f u v, (∀ (a : E), a ∈ s → ↑f a < u) ∧ u < v ∧ ∀ (b : E), b ∈ ∅ → v < ↑f b",
"tactic": "exact ⟨0, 1, 2, fun a _ha => by norm_num, by norm_num, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nhs : Set.Nonempty s\nht₁ : Convex ℝ ∅\nht₂ : IsClosed ∅\ndisj : Disjoint s ∅\na : E\n_ha : a ∈ s\n⊢ ↑0 a < 1",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nhs : Set.Nonempty s\nht₁ : Convex ℝ ∅\nht₂ : IsClosed ∅\ndisj : Disjoint s ∅\n⊢ 1 < 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nhs : Set.Nonempty s\nht₁ : Convex ℝ ∅\nht₂ : IsClosed ∅\ndisj : Disjoint s ∅\n⊢ ∀ (b : E), b ∈ ∅ → 2 < ↑0 b",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx✝ y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU₁ : Convex ℝ U\nhV₁ : Convex ℝ V\nsU : s ⊆ U\ntV : t ⊆ V\ndisj' : Disjoint U V\nf : E →L[ℝ] ℝ\nu : ℝ\nhf₁ : ∀ (a : E), a ∈ U → ↑f a < u\nhf₂ : ∀ (b : E), b ∈ V → u < ↑f b\nx : E\nhx₁ : x ∈ s\nhx₂ : ∀ (y : E), y ∈ s → ↑f y ≤ ↑f x\nthis : ↑f x < u\na : E\nha : a ∈ s\n⊢ ↑f a < (↑f x + u) / 2",
"tactic": "linarith [hx₂ a ha]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.82757\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx✝ y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU₁ : Convex ℝ U\nhV₁ : Convex ℝ V\nsU : s ⊆ U\ntV : t ⊆ V\ndisj' : Disjoint U V\nf : E →L[ℝ] ℝ\nu : ℝ\nhf₁ : ∀ (a : E), a ∈ U → ↑f a < u\nhf₂ : ∀ (b : E), b ∈ V → u < ↑f b\nx : E\nhx₁ : x ∈ s\nhx₂ : ∀ (y : E), y ∈ s → ↑f y ≤ ↑f x\nthis : ↑f x < u\n⊢ (↑f x + u) / 2 < u",
"tactic": "linarith"
}
] |
[
169,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.add_eq_one_iff
|
[
{
"state_after": "no goals",
"state_before": "m n k l : ℕ\n⊢ m + n = 1 ↔ m = 0 ∧ n = 1 ∨ m = 1 ∧ n = 0",
"tactic": "cases n <;> simp [succ_eq_add_one, ← add_assoc, succ_inj']"
}
] |
[
196,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.int_cast_re
|
[
{
"state_after": "no goals",
"state_before": "n : ℤ\n⊢ (↑n).re = ↑n",
"tactic": "rw [← ofReal_int_cast, ofReal_re]"
}
] |
[
852,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
852,
1
] |
Mathlib/FieldTheory/PerfectClosure.lean
|
pthRoot_pow_p'
|
[] |
[
82,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
reflection_mem_subspace_eq_self
|
[] |
[
704,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
703,
1
] |
Mathlib/Tactic/Ring/Basic.lean
|
Mathlib.Tactic.Ring.neg_one_mul
|
[
{
"state_after": "u : Lean.Level\nR✝ : Type ?u.203351\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝¹ : CommSemiring R✝\nR : Type u_1\ninst✝ : Ring R\na : R\n⊢ -a = Int.rawCast (Int.negOfNat 1) * a",
"state_before": "u : Lean.Level\nR✝ : Type ?u.203351\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝¹ : CommSemiring R✝\nR : Type u_1\ninst✝ : Ring R\na b : R\nx✝ : Int.rawCast (Int.negOfNat 1) * a = b\n⊢ -a = b",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "u : Lean.Level\nR✝ : Type ?u.203351\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝¹ : CommSemiring R✝\nR : Type u_1\ninst✝ : Ring R\na : R\n⊢ -a = Int.rawCast (Int.negOfNat 1) * a",
"tactic": "simp [Int.negOfNat]"
}
] |
[
527,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
526,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
Subgroup.mem_smul_pointwise_iff_exists
|
[] |
[
296,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.concat_append
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w x : V\np : Walk G u v\nh : Adj G v w\nq : Walk G w x\n⊢ append (concat p h) q = append p (cons h q)",
"tactic": "rw [concat_eq_append, ← append_assoc, cons_nil_append]"
}
] |
[
298,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Data/Setoid/Basic.lean
|
Setoid.inf_iff_and
|
[] |
[
146,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.le_eq
|
[] |
[
88,
52
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
88,
9
] |
Std/Data/List/Lemmas.lean
|
List.filter_sublist
|
[
{
"state_after": "α : Type u_1\np : α → Bool\na : α\nl : List α\n⊢ (match p a with\n | true => a :: filter p l\n | false => filter p l) <+\n a :: l",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl : List α\n⊢ filter p (a :: l) <+ a :: l",
"tactic": "rw [filter]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl : List α\n⊢ (match p a with\n | true => a :: filter p l\n | false => filter p l) <+\n a :: l",
"tactic": "split <;> simp [Sublist.cons, Sublist.cons₂, filter_sublist l]"
}
] |
[
1121,
93
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1119,
9
] |
Mathlib/Analysis/BoxIntegral/Box/Basic.lean
|
BoxIntegral.Box.isCompact_Icc
|
[] |
[
220,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
11
] |
Mathlib/Data/Int/ModEq.lean
|
Int.modEq_add_fac_self
|
[] |
[
293,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.integral_eq_zero_iff_of_le_of_nonneg_ae
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.19335522\n𝕜 : Type ?u.19335525\nE : Type ?u.19335528\nF : Type ?u.19335531\nA : Type ?u.19335534\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf g : ℝ → ℝ\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhab : a ≤ b\nhf : 0 ≤ᵐ[Measure.restrict μ (Ioc a b)] f\nhfi : IntervalIntegrable f μ a b\n⊢ (∫ (x : ℝ) in a..b, f x ∂μ) = 0 ↔ f =ᵐ[Measure.restrict μ (Ioc a b)] 0",
"tactic": "rw [integral_of_le hab, integral_eq_zero_iff_of_nonneg_ae hf hfi.1]"
}
] |
[
1264,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1262,
1
] |
Mathlib/Order/Atoms.lean
|
isCoatomic_iff_forall_isCoatomic_Ici
|
[] |
[
310,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.iInf_empty
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort u_1\nf✝ : ι → ℝ≥0\ninst✝ : IsEmpty ι\nf : ι → ℝ≥0\n⊢ (⨅ (i : ι), f i) = 0",
"tactic": "rw [iInf_of_empty', sInf_empty]"
}
] |
[
944,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
943,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.Prime.factorization
|
[
{
"state_after": "case h\np : ℕ\nhp : Prime p\nq : ℕ\n⊢ ↑(Nat.factorization p) q = ↑(single p 1) q",
"state_before": "p : ℕ\nhp : Prime p\n⊢ Nat.factorization p = single p 1",
"tactic": "ext q"
},
{
"state_after": "no goals",
"state_before": "case h\np : ℕ\nhp : Prime p\nq : ℕ\n⊢ ↑(Nat.factorization p) q = ↑(single p 1) q",
"tactic": "rw [← factors_count_eq, factors_prime hp, single_apply, count_singleton', if_congr eq_comm] <;>\n rfl"
}
] |
[
276,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/Data/Real/Irrational.lean
|
Irrational.of_add_int
|
[] |
[
234,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.one_le_coe_iff
|
[] |
[
663,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
663,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.coe_zpow
|
[] |
[
305,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean
|
Asymptotics.SuperpolynomialDecay.mul_param_zpow
|
[] |
[
269,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.ret_orElse
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\na : α\nc₂ : Computation α\n⊢ destruct ({ hOrElse := fun a b => OrElse.orElse a b }.1 (pure a) fun x => c₂) = Sum.inl a",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nc₂ : Computation α\n⊢ destruct (HOrElse.hOrElse (pure a) fun x => c₂) = Sum.inl a",
"tactic": "unfold HOrElse.hOrElse instHOrElse"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\na : α\nc₂ : Computation α\n⊢ destruct\n ({\n hOrElse := fun a b =>\n {\n orElse :=\n (let src := monad;\n Alternative.mk empty @orElse).3 }.1\n a b }.1\n (pure a) fun x => c₂) =\n Sum.inl a",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nc₂ : Computation α\n⊢ destruct ({ hOrElse := fun a b => OrElse.orElse a b }.1 (pure a) fun x => c₂) = Sum.inl a",
"tactic": "unfold OrElse.orElse instOrElse Alternative.orElse instAlternativeComputation"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nc₂ : Computation α\n⊢ destruct\n ({\n hOrElse := fun a b =>\n {\n orElse :=\n (let src := monad;\n Alternative.mk empty @orElse).3 }.1\n a b }.1\n (pure a) fun x => c₂) =\n Sum.inl a",
"tactic": "simp [orElse]"
}
] |
[
940,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
936,
1
] |
Mathlib/Analysis/Convex/Segment.lean
|
image_segment
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.97868\nι : Type ?u.97871\nπ : ι → Type ?u.97876\ninst✝⁵ : OrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : AddCommGroup G\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nf : E →ᵃ[𝕜] F\na b : E\nx : F\n⊢ x ∈ ↑f '' [a-[𝕜]b] ↔ x ∈ [↑f a-[𝕜]↑f b]",
"tactic": "simp_rw [segment_eq_image_lineMap, mem_image, exists_exists_and_eq_and, AffineMap.apply_lineMap]"
}
] |
[
239,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/CategoryTheory/Subobject/MonoOver.lean
|
CategoryTheory.MonoOver.mk'_coe'
|
[] |
[
87,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/GroupTheory/Perm/Basic.lean
|
Equiv.Perm.trans_one
|
[] |
[
137,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Algebra/Module/Submodule/Bilinear.lean
|
Submodule.map₂_sup_right
|
[] |
[
122,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.eventually_or_distrib_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.143807\nι : Sort x\nf : Filter α\np : Prop\nq : α → Prop\nh : p\n⊢ (∀ᶠ (x : α) in f, p ∨ q x) ↔ p ∨ ∀ᶠ (x : α) in f, q x",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.143807\nι : Sort x\nf : Filter α\np : Prop\nq : α → Prop\nh : ¬p\n⊢ (∀ᶠ (x : α) in f, p ∨ q x) ↔ p ∨ ∀ᶠ (x : α) in f, q x",
"tactic": "simp [h]"
}
] |
[
1187,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1185,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.toLinearMap_injective
|
[] |
[
376,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/Analysis/Calculus/Deriv/ZPow.lean
|
deriv_zpow'
|
[] |
[
103,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
HasFDerivAtFilter.add
|
[
{
"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.127337\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.127432\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\nhf : HasFDerivAtFilter f f' x L\nhg : HasFDerivAtFilter g g' x L\nx✝ : E\n⊢ f x✝ - f x - (↑f' x✝ - ↑f' x) + (g x✝ - g x - (↑g' x✝ - ↑g' x)) =\n f x✝ + g x✝ - (f x + g x) - (↑f' x✝ + ↑g' x✝ - (↑f' x + ↑g' x))",
"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.127337\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.127432\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\nhf : HasFDerivAtFilter f f' x L\nhg : HasFDerivAtFilter g g' x L\nx✝ : E\n⊢ f x✝ - f x - ↑f' (x✝ - x) + (g x✝ - g x - ↑g' (x✝ - x)) =\n (fun y => f y + g y) x✝ - (fun y => f y + g y) x - ↑(f' + g') (x✝ - x)",
"tactic": "simp only [LinearMap.sub_apply, LinearMap.add_apply, map_sub, map_add, add_apply]"
},
{
"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.127337\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.127432\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\nhf : HasFDerivAtFilter f f' x L\nhg : HasFDerivAtFilter g g' x L\nx✝ : E\n⊢ f x✝ - f x - (↑f' x✝ - ↑f' x) + (g x✝ - g x - (↑g' x✝ - ↑g' x)) =\n f x✝ + g x✝ - (f x + g x) - (↑f' x✝ + ↑g' x✝ - (↑f' x + ↑g' x))",
"tactic": "abel"
}
] |
[
133,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
8
] |
Mathlib/GroupTheory/Perm/Basic.lean
|
Equiv.Perm.sumCongr_one
|
[] |
[
202,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
pi_norm_lt_iff'
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.1270061\n𝕜 : Type ?u.1270064\nα : Type ?u.1270067\nι : Type u_1\nκ : Type ?u.1270073\nE : Type ?u.1270076\nF : Type ?u.1270079\nG : Type ?u.1270082\nπ : ι → Type u_2\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → SeminormedGroup (π i)\ninst✝ : SeminormedGroup E\nf x : (i : ι) → π i\nr : ℝ\nhr : 0 < r\n⊢ ‖x‖ < r ↔ ∀ (i : ι), ‖x i‖ < r",
"tactic": "simp only [← dist_one_right, dist_pi_lt_iff hr, Pi.one_apply]"
}
] |
[
2499,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2498,
1
] |
Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean
|
CategoryTheory.GrothendieckTopology.whiskerRight_toSheafify_sheafifyCompIso_hom
|
[
{
"state_after": "C : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ whiskerRight (toSheafify J P) F ≫ (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom =\n toSheafify J (P ⋙ F)",
"state_before": "C : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ whiskerRight (toSheafify J P) F ≫ (sheafifyCompIso J F P).hom = toSheafify J (P ⋙ F)",
"tactic": "dsimp [sheafifyCompIso]"
},
{
"state_after": "C : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ whiskerRight (toPlus J P) F ≫\n whiskerRight (plusMap J (toPlus J P)) F ≫\n (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom =\n toSheafify J (P ⋙ F)",
"state_before": "C : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ whiskerRight (toSheafify J P) F ≫ (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom =\n toSheafify J (P ⋙ F)",
"tactic": "erw [whiskerRight_comp, Category.assoc]"
},
{
"state_after": "C : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ whiskerRight (toPlus J P) F ≫\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerRight (toPlus J P) F)) ≫ plusMap J (plusCompIso J F P).hom =\n toSheafify J (P ⋙ F)",
"state_before": "C : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ whiskerRight (toPlus J P) F ≫\n whiskerRight (plusMap J (toPlus J P)) F ≫\n (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom =\n toSheafify J (P ⋙ F)",
"tactic": "slice_lhs 2 3 => rw [plusCompIso_whiskerRight]"
},
{
"state_after": "C : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ toPlus J (P ⋙ F) ≫ plusMap J (toPlus J (P ⋙ F)) = toSheafify J (P ⋙ F)",
"state_before": "C : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ whiskerRight (toPlus J P) F ≫\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerRight (toPlus J P) F)) ≫ plusMap J (plusCompIso J F P).hom =\n toSheafify J (P ⋙ F)",
"tactic": "rw [Category.assoc, ← J.plusMap_comp, whiskerRight_toPlus_comp_plusCompIso_hom, ←\n Category.assoc, whiskerRight_toPlus_comp_plusCompIso_hom]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁸ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁷ : Category D\nE : Type w₂\ninst✝⁶ : Category E\nF : D ⥤ E\ninst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝³ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\n⊢ toPlus J (P ⋙ F) ≫ plusMap J (toPlus J (P ⋙ F)) = toSheafify J (P ⋙ F)",
"tactic": "rfl"
}
] |
[
137,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
rank_le_one_iff
|
[
{
"state_after": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\n⊢ Module.rank K V ≤ 1 ↔ ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\n⊢ Module.rank K V ≤ 1 ↔ ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v",
"tactic": "let b := Basis.ofVectorSpace K V"
},
{
"state_after": "case mp\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\n⊢ Module.rank K V ≤ 1 → ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v\n\ncase mpr\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\n⊢ (∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v) → Module.rank K V ≤ 1",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\n⊢ Module.rank K V ≤ 1 ↔ ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v",
"tactic": "constructor"
},
{
"state_after": "case mp\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Module.rank K V ≤ 1\n⊢ ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v",
"state_before": "case mp\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\n⊢ Module.rank K V ≤ 1 → ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v",
"tactic": "intro hd"
},
{
"state_after": "case mp\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\n⊢ ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v",
"state_before": "case mp\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Module.rank K V ≤ 1\n⊢ ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v",
"tactic": "rw [← b.mk_eq_rank'', Cardinal.le_one_iff_subsingleton, subsingleton_coe] at hd"
},
{
"state_after": "case mp.inl\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nhb : ofVectorSpaceIndex K V = ∅\n⊢ ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v\n\ncase mp.inr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nv₀ : V\nhv₀ : v₀ ∈ ofVectorSpaceIndex K V\n⊢ ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v",
"state_before": "case mp\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\n⊢ ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v",
"tactic": "rcases eq_empty_or_nonempty (ofVectorSpaceIndex K V) with (hb | ⟨⟨v₀, hv₀⟩⟩)"
},
{
"state_after": "case mp.inl\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nhb : ofVectorSpaceIndex K V = ∅\n⊢ ∀ (v : V), ∃ r, r • 0 = v",
"state_before": "case mp.inl\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nhb : ofVectorSpaceIndex K V = ∅\n⊢ ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v",
"tactic": "use 0"
},
{
"state_after": "case mp.inl\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nhb : ofVectorSpaceIndex K V = ∅\nh' : ∀ (v : V), v = 0\n⊢ ∀ (v : V), ∃ r, r • 0 = v",
"state_before": "case mp.inl\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nhb : ofVectorSpaceIndex K V = ∅\n⊢ ∀ (v : V), ∃ r, r • 0 = v",
"tactic": "have h' : ∀ v : V, v = 0 := by simpa [hb, Submodule.eq_bot_iff] using b.span_eq.symm"
},
{
"state_after": "case mp.inl\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nhb : ofVectorSpaceIndex K V = ∅\nh' : ∀ (v : V), v = 0\nv : V\n⊢ ∃ r, r • 0 = v",
"state_before": "case mp.inl\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nhb : ofVectorSpaceIndex K V = ∅\nh' : ∀ (v : V), v = 0\n⊢ ∀ (v : V), ∃ r, r • 0 = v",
"tactic": "intro v"
},
{
"state_after": "no goals",
"state_before": "case mp.inl\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nhb : ofVectorSpaceIndex K V = ∅\nh' : ∀ (v : V), v = 0\nv : V\n⊢ ∃ r, r • 0 = v",
"tactic": "simp [h' v]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nhb : ofVectorSpaceIndex K V = ∅\n⊢ ∀ (v : V), v = 0",
"tactic": "simpa [hb, Submodule.eq_bot_iff] using b.span_eq.symm"
},
{
"state_after": "case mp.inr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nv₀ : V\nhv₀ : v₀ ∈ ofVectorSpaceIndex K V\n⊢ ∀ (v : V), ∃ r, r • v₀ = v",
"state_before": "case mp.inr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nv₀ : V\nhv₀ : v₀ ∈ ofVectorSpaceIndex K V\n⊢ ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v",
"tactic": "use v₀"
},
{
"state_after": "case mp.inr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nv₀ : V\nhv₀ : v₀ ∈ ofVectorSpaceIndex K V\nh' : span K {v₀} = ⊤\n⊢ ∀ (v : V), ∃ r, r • v₀ = v",
"state_before": "case mp.inr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nv₀ : V\nhv₀ : v₀ ∈ ofVectorSpaceIndex K V\n⊢ ∀ (v : V), ∃ r, r • v₀ = v",
"tactic": "have h' : (K ∙ v₀) = ⊤ := by simpa [hd.eq_singleton_of_mem hv₀] using b.span_eq"
},
{
"state_after": "case mp.inr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nv₀ : V\nhv₀ : v₀ ∈ ofVectorSpaceIndex K V\nh' : span K {v₀} = ⊤\nv : V\n⊢ ∃ r, r • v₀ = v",
"state_before": "case mp.inr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nv₀ : V\nhv₀ : v₀ ∈ ofVectorSpaceIndex K V\nh' : span K {v₀} = ⊤\n⊢ ∀ (v : V), ∃ r, r • v₀ = v",
"tactic": "intro v"
},
{
"state_after": "case mp.inr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nv₀ : V\nhv₀ : v₀ ∈ ofVectorSpaceIndex K V\nh' : span K {v₀} = ⊤\nv : V\nhv : v ∈ ⊤\n⊢ ∃ r, r • v₀ = v",
"state_before": "case mp.inr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nv₀ : V\nhv₀ : v₀ ∈ ofVectorSpaceIndex K V\nh' : span K {v₀} = ⊤\nv : V\n⊢ ∃ r, r • v₀ = v",
"tactic": "have hv : v ∈ (⊤ : Submodule K V) := mem_top"
},
{
"state_after": "no goals",
"state_before": "case mp.inr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nv₀ : V\nhv₀ : v₀ ∈ ofVectorSpaceIndex K V\nh' : span K {v₀} = ⊤\nv : V\nhv : v ∈ ⊤\n⊢ ∃ r, r • v₀ = v",
"tactic": "rwa [← h', mem_span_singleton] at hv"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nhd : Set.Subsingleton (ofVectorSpaceIndex K V)\nv₀ : V\nhv₀ : v₀ ∈ ofVectorSpaceIndex K V\n⊢ span K {v₀} = ⊤",
"tactic": "simpa [hd.eq_singleton_of_mem hv₀] using b.span_eq"
},
{
"state_after": "case mpr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nv₀ : V\nhv₀ : ∀ (v : V), ∃ r, r • v₀ = v\n⊢ Module.rank K V ≤ 1",
"state_before": "case mpr\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\n⊢ (∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v) → Module.rank K V ≤ 1",
"tactic": "rintro ⟨v₀, hv₀⟩"
},
{
"state_after": "case mpr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nv₀ : V\nhv₀ : ∀ (v : V), ∃ r, r • v₀ = v\nh : span K {v₀} = ⊤\n⊢ Module.rank K V ≤ 1",
"state_before": "case mpr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nv₀ : V\nhv₀ : ∀ (v : V), ∃ r, r • v₀ = v\n⊢ Module.rank K V ≤ 1",
"tactic": "have h : (K ∙ v₀) = ⊤ := by\n ext\n simp [mem_span_singleton, hv₀]"
},
{
"state_after": "case mpr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nv₀ : V\nhv₀ : ∀ (v : V), ∃ r, r • v₀ = v\nh : span K {v₀} = ⊤\n⊢ Module.rank K { x // x ∈ span K {v₀} } ≤ 1",
"state_before": "case mpr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nv₀ : V\nhv₀ : ∀ (v : V), ∃ r, r • v₀ = v\nh : span K {v₀} = ⊤\n⊢ Module.rank K V ≤ 1",
"tactic": "rw [← rank_top, ← h]"
},
{
"state_after": "case mpr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nv₀ : V\nhv₀ : ∀ (v : V), ∃ r, r • v₀ = v\nh : span K {v₀} = ⊤\n⊢ (#↑{v₀}) = 1",
"state_before": "case mpr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nv₀ : V\nhv₀ : ∀ (v : V), ∃ r, r • v₀ = v\nh : span K {v₀} = ⊤\n⊢ Module.rank K { x // x ∈ span K {v₀} } ≤ 1",
"tactic": "refine' (rank_span_le _).trans_eq _"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nv₀ : V\nhv₀ : ∀ (v : V), ∃ r, r • v₀ = v\nh : span K {v₀} = ⊤\n⊢ (#↑{v₀}) = 1",
"tactic": "simp"
},
{
"state_after": "case h\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nv₀ : V\nhv₀ : ∀ (v : V), ∃ r, r • v₀ = v\nx✝ : V\n⊢ x✝ ∈ span K {v₀} ↔ x✝ ∈ ⊤",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nv₀ : V\nhv₀ : ∀ (v : V), ∃ r, r • v₀ = v\n⊢ span K {v₀} = ⊤",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1044072\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nb : Basis (↑(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\nv₀ : V\nhv₀ : ∀ (v : V), ∃ r, r • v₀ = v\nx✝ : V\n⊢ x✝ ∈ span K {v₀} ↔ x✝ ∈ ⊤",
"tactic": "simp [mem_span_singleton, hv₀]"
}
] |
[
1231,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1210,
1
] |
Mathlib/CategoryTheory/Conj.lean
|
CategoryTheory.Iso.homCongr_symm
|
[] |
[
74,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/RingTheory/PolynomialAlgebra.lean
|
matPolyEquiv_smul_one
|
[
{
"state_after": "case a.a.h\nR : Type u_1\nA : Type ?u.620220\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\np : R[X]\nm : ℕ\ni j : n\n⊢ coeff (↑matPolyEquiv (p • 1)) m i j = coeff (Polynomial.map (algebraMap R (Matrix n n R)) p) m i j",
"state_before": "R : Type u_1\nA : Type ?u.620220\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\np : R[X]\n⊢ ↑matPolyEquiv (p • 1) = Polynomial.map (algebraMap R (Matrix n n R)) p",
"tactic": "ext (m i j)"
},
{
"state_after": "case a.a.h\nR : Type u_1\nA : Type ?u.620220\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\np : R[X]\nm : ℕ\ni j : n\n⊢ coeff ((p • 1) i j) m = if i = j then ↑(algebraMap R R) (coeff p m) else 0",
"state_before": "case a.a.h\nR : Type u_1\nA : Type ?u.620220\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\np : R[X]\nm : ℕ\ni j : n\n⊢ coeff (↑matPolyEquiv (p • 1)) m i j = coeff (Polynomial.map (algebraMap R (Matrix n n R)) p) m i j",
"tactic": "simp only [coeff_map, one_apply, algebraMap_matrix_apply, mul_boole, Pi.smul_apply,\n matPolyEquiv_coeff_apply]"
},
{
"state_after": "no goals",
"state_before": "case a.a.h\nR : Type u_1\nA : Type ?u.620220\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\np : R[X]\nm : ℕ\ni j : n\n⊢ coeff ((p • 1) i j) m = if i = j then ↑(algebraMap R R) (coeff p m) else 0",
"tactic": "split_ifs <;> simp <;> rename_i h <;> simp [h]"
}
] |
[
289,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Std/Data/RBMap/Lemmas.lean
|
Std.RBNode.balance2_toList
|
[
{
"state_after": "α : Type u_1\nl : RBNode α\nv : α\nr : RBNode α\n⊢ toList\n (match l, v, r with\n | a, x, node red (node red b y c) z d => node red (node black a x b) y (node black c z d)\n | a, x, node red b y (node red c z d) => node red (node black a x b) y (node black c z d)\n | a, x, b => node black a x b) =\n toList l ++ v :: toList r",
"state_before": "α : Type u_1\nl : RBNode α\nv : α\nr : RBNode α\n⊢ toList (balance2 l v r) = toList l ++ v :: toList r",
"tactic": "unfold balance2"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : RBNode α\nv : α\nr : RBNode α\n⊢ toList\n (match l, v, r with\n | a, x, node red (node red b y c) z d => node red (node black a x b) y (node black c z d)\n | a, x, node red b y (node red c z d) => node red (node black a x b) y (node black c z d)\n | a, x, b => node black a x b) =\n toList l ++ v :: toList r",
"tactic": "split <;> simp"
}
] |
[
425,
34
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
423,
9
] |
Mathlib/Order/Height.lean
|
Set.one_le_chainHeight_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.7989\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\n⊢ (∃ l, l ∈ subchain s ∧ length l = 1) ↔ Set.Nonempty s",
"state_before": "α : Type u_1\nβ : Type ?u.7989\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\n⊢ 1 ≤ chainHeight s ↔ Set.Nonempty s",
"tactic": "rw [← Nat.cast_one, Set.le_chainHeight_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7989\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\n⊢ (∃ l, l ∈ subchain s ∧ length l = 1) ↔ Set.Nonempty s",
"tactic": "simp only [length_eq_one, @and_comm (_ ∈ _), @eq_comm _ _ [_], exists_exists_eq_and,\n singleton_mem_subchain_iff, Set.Nonempty]"
}
] |
[
141,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.findGreatest_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "m n k l : ℕ\nP Q : ℕ → Prop\ninst✝ : DecidablePred P\n⊢ Nat.findGreatest P k = 0 ↔ ∀ ⦃n : ℕ⦄, 0 < n → n ≤ k → ¬P n",
"tactic": "simp [findGreatest_eq_iff]"
}
] |
[
628,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
627,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.le_add_of_nonneg_right
|
[
{
"state_after": "a b : Int\nh : 0 ≤ b\nthis : a + b ≥ a + 0\n⊢ a ≤ a + b",
"state_before": "a b : Int\nh : 0 ≤ b\n⊢ a ≤ a + b",
"tactic": "have : a + b ≥ a + 0 := Int.add_le_add_left h a"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nh : 0 ≤ b\nthis : a + b ≥ a + 0\n⊢ a ≤ a + b",
"tactic": "rwa [Int.add_zero] at this"
}
] |
[
796,
29
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
794,
11
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
comp_open_symm_mem_uniformity_sets
|
[
{
"state_after": "case intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.94197\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nt : Set (α × α)\nht₁ : t ∈ 𝓤 α\nht₂ : t ○ t ⊆ s\n⊢ ∃ t, t ∈ 𝓤 α ∧ IsOpen t ∧ SymmetricRel t ∧ t ○ t ⊆ s",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.94197\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\n⊢ ∃ t, t ∈ 𝓤 α ∧ IsOpen t ∧ SymmetricRel t ∧ t ○ t ⊆ s",
"tactic": "obtain ⟨t, ht₁, ht₂⟩ := comp_mem_uniformity_sets hs"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.94197\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nt : Set (α × α)\nht₁ : t ∈ 𝓤 α\nht₂ : t ○ t ⊆ s\nu : Set (α × α)\nhu₄ : u ⊆ t\nhu₁ : u ∈ 𝓤 α\nhu₂ : IsOpen u\nhu₃ : SymmetricRel u\n⊢ ∃ t, t ∈ 𝓤 α ∧ IsOpen t ∧ SymmetricRel t ∧ t ○ t ⊆ s",
"state_before": "case intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.94197\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nt : Set (α × α)\nht₁ : t ∈ 𝓤 α\nht₂ : t ○ t ⊆ s\n⊢ ∃ t, t ∈ 𝓤 α ∧ IsOpen t ∧ SymmetricRel t ∧ t ○ t ⊆ s",
"tactic": "obtain ⟨u, ⟨hu₁, hu₂, hu₃⟩, hu₄ : u ⊆ t⟩ := uniformity_hasBasis_open_symmetric.mem_iff.mp ht₁"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.94197\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nt : Set (α × α)\nht₁ : t ∈ 𝓤 α\nht₂ : t ○ t ⊆ s\nu : Set (α × α)\nhu₄ : u ⊆ t\nhu₁ : u ∈ 𝓤 α\nhu₂ : IsOpen u\nhu₃ : SymmetricRel u\n⊢ ∃ t, t ∈ 𝓤 α ∧ IsOpen t ∧ SymmetricRel t ∧ t ○ t ⊆ s",
"tactic": "exact ⟨u, hu₁, hu₂, hu₃, (compRel_mono hu₄ hu₄).trans ht₂⟩"
}
] |
[
1060,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1056,
1
] |
Mathlib/Algebra/Hom/GroupAction.lean
|
MulActionHom.map_smul
|
[] |
[
121,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
11
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.kroneckerTMul_zero
|
[] |
[
474,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
473,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConcaveOn.add_strictConcaveOn
|
[] |
[
532,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
530,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.IsMultiplicative.ppow
|
[
{
"state_after": "case zero\nR : Type u_1\ninst✝ : CommSemiring R\nf : ArithmeticFunction R\nhf : IsMultiplicative f\n⊢ IsMultiplicative (ArithmeticFunction.ppow f Nat.zero)\n\ncase succ\nR : Type u_1\ninst✝ : CommSemiring R\nf : ArithmeticFunction R\nhf : IsMultiplicative f\nk : ℕ\nhi : IsMultiplicative (ArithmeticFunction.ppow f k)\n⊢ IsMultiplicative (ArithmeticFunction.ppow f (succ k))",
"state_before": "R : Type u_1\ninst✝ : CommSemiring R\nf : ArithmeticFunction R\nhf : IsMultiplicative f\nk : ℕ\n⊢ IsMultiplicative (ArithmeticFunction.ppow f k)",
"tactic": "induction' k with k hi"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u_1\ninst✝ : CommSemiring R\nf : ArithmeticFunction R\nhf : IsMultiplicative f\n⊢ IsMultiplicative (ArithmeticFunction.ppow f Nat.zero)",
"tactic": "exact isMultiplicative_zeta.nat_cast"
},
{
"state_after": "case succ\nR : Type u_1\ninst✝ : CommSemiring R\nf : ArithmeticFunction R\nhf : IsMultiplicative f\nk : ℕ\nhi : IsMultiplicative (ArithmeticFunction.ppow f k)\n⊢ IsMultiplicative (ArithmeticFunction.pmul f (ArithmeticFunction.ppow f k))",
"state_before": "case succ\nR : Type u_1\ninst✝ : CommSemiring R\nf : ArithmeticFunction R\nhf : IsMultiplicative f\nk : ℕ\nhi : IsMultiplicative (ArithmeticFunction.ppow f k)\n⊢ IsMultiplicative (ArithmeticFunction.ppow f (succ k))",
"tactic": "rw [ppow_succ]"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u_1\ninst✝ : CommSemiring R\nf : ArithmeticFunction R\nhf : IsMultiplicative f\nk : ℕ\nhi : IsMultiplicative (ArithmeticFunction.ppow f k)\n⊢ IsMultiplicative (ArithmeticFunction.pmul f (ArithmeticFunction.ppow f k))",
"tactic": "apply hf.pmul hi"
}
] |
[
832,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
827,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoMod_sub_zsmul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ toIcoMod hp a (b - m • p) = toIcoMod hp a b",
"tactic": "rw [sub_eq_add_neg, ← neg_smul, toIcoMod_add_zsmul]"
}
] |
[
452,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
451,
1
] |
Mathlib/CategoryTheory/Sums/Basic.lean
|
CategoryTheory.Sum.swap_map_inl
|
[] |
[
132,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
dist_toDual
|
[] |
[
3243,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3243,
9
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.Measure.ext_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.10223\nγ : Type ?u.10226\nδ : Type ?u.10229\nι : Type ?u.10232\ninst✝ : MeasurableSpace α\nμ μ₁ : Measure α\ns✝ s₁ s₂ t s : Set α\n_hs : MeasurableSet s\n⊢ ↑↑μ₁ s = ↑↑μ₁ s",
"state_before": "α : Type u_1\nβ : Type ?u.10223\nγ : Type ?u.10226\nδ : Type ?u.10229\nι : Type ?u.10232\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\n⊢ μ₁ = μ₂ → ∀ (s : Set α), MeasurableSet s → ↑↑μ₁ s = ↑↑μ₂ s",
"tactic": "rintro rfl s _hs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.10223\nγ : Type ?u.10226\nδ : Type ?u.10229\nι : Type ?u.10232\ninst✝ : MeasurableSpace α\nμ μ₁ : Measure α\ns✝ s₁ s₂ t s : Set α\n_hs : MeasurableSet s\n⊢ ↑↑μ₁ s = ↑↑μ₁ s",
"tactic": "rfl"
}
] |
[
146,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Algebra/CharP/ExpChar.lean
|
char_eq_expChar_iff
|
[
{
"state_after": "case zero\nR : Type u\ninst✝ : Semiring R\np : ℕ\nhp : CharP R p\nq : CharZero R\n⊢ p = 1 ↔ Nat.Prime p\n\ncase prime\nR : Type u\ninst✝ : Semiring R\np q : ℕ\nhp : CharP R p\nhq_prime : Nat.Prime q\nhq_hchar : CharP R q\n⊢ p = q ↔ Nat.Prime p",
"state_before": "R : Type u\ninst✝ : Semiring R\np q : ℕ\nhp : CharP R p\nhq : ExpChar R q\n⊢ p = q ↔ Nat.Prime p",
"tactic": "cases' hq with q hq_one hq_prime hq_hchar"
},
{
"state_after": "case zero\nR : Type u\ninst✝ : Semiring R\np : ℕ\nhp : CharP R p\nq : CharZero R\n⊢ 0 = 1 ↔ Nat.Prime 0",
"state_before": "case zero\nR : Type u\ninst✝ : Semiring R\np : ℕ\nhp : CharP R p\nq : CharZero R\n⊢ p = 1 ↔ Nat.Prime p",
"tactic": "rw [(CharP.eq R hp inferInstance : p = 0)]"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\ninst✝ : Semiring R\np : ℕ\nhp : CharP R p\nq : CharZero R\n⊢ 0 = 1 ↔ Nat.Prime 0",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "case prime\nR : Type u\ninst✝ : Semiring R\np q : ℕ\nhp : CharP R p\nhq_prime : Nat.Prime q\nhq_hchar : CharP R q\n⊢ p = q ↔ Nat.Prime p",
"tactic": "exact ⟨fun hpq => hpq.symm ▸ hq_prime, fun _ => CharP.eq R hp hq_hchar⟩"
}
] |
[
63,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingHom.toFun_eq_coe
|
[] |
[
196,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/LinearAlgebra/Projection.lean
|
LinearMap.ker_id_sub_eq_of_proj
|
[
{
"state_after": "case h\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.9010\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.9526\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.10489\ninst✝² : Semiring S\nM : Type ?u.10495\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\n⊢ x ∈ ker (id - comp (Submodule.subtype p) f) ↔ x ∈ p",
"state_before": "R : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.9010\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.9526\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.10489\ninst✝² : Semiring S\nM : Type ?u.10495\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\n⊢ ker (id - comp (Submodule.subtype p) f) = p",
"tactic": "ext x"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.9010\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.9526\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.10489\ninst✝² : Semiring S\nM : Type ?u.10495\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\n⊢ x = ↑(↑f x) ↔ x ∈ p",
"state_before": "case h\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.9010\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.9526\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.10489\ninst✝² : Semiring S\nM : Type ?u.10495\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\n⊢ x ∈ ker (id - comp (Submodule.subtype p) f) ↔ x ∈ p",
"tactic": "simp only [comp_apply, mem_ker, subtype_apply, sub_apply, id_apply, sub_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.9010\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.9526\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.10489\ninst✝² : Semiring S\nM : Type ?u.10495\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\n⊢ x = ↑(↑f x) ↔ x ∈ p",
"tactic": "exact ⟨fun h => h.symm ▸ Submodule.coe_mem _, fun hx => by erw [hf ⟨x, hx⟩, Subtype.coe_mk]⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.9010\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.9526\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.10489\ninst✝² : Semiring S\nM : Type ?u.10495\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\nhx : x ∈ p\n⊢ x = ↑(↑f x)",
"tactic": "erw [hf ⟨x, hx⟩, Subtype.coe_mk]"
}
] |
[
48,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/RingTheory/Ideal/Quotient.lean
|
Ideal.Quotient.ringHom_ext
|
[] |
[
121,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.add_le_add_iff_right
|
[] |
[
778,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
777,
11
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.liftMonoid_coe
|
[] |
[
541,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
1
] |
Mathlib/Algebra/Order/Nonneg/Field.lean
|
Nonneg.coe_div
|
[] |
[
60,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
11
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
isOpenMap_mul_right
|
[] |
[
128,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean
|
CompleteLatticeHom.symm_dual_comp
|
[] |
[
902,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
899,
1
] |
Mathlib/Topology/Order/Basic.lean
|
isOpen_Ioi
|
[] |
[
302,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Algebra/Star/Basic.lean
|
star_zpow₀
|
[] |
[
440,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
439,
1
] |
Mathlib/Data/Set/Countable.lean
|
Set.Countable.preimage
|
[] |
[
153,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
11
] |
Mathlib/Analysis/BoxIntegral/Partition/Split.lean
|
BoxIntegral.Prepartition.eventually_not_disjoint_imp_le_of_mem_splitMany
|
[
{
"state_after": "case intro\nι : Type u_1\nM : Type ?u.49727\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\ninst✝ : Finite ι\ns : Finset (Box ι)\nval✝ : Fintype ι\n⊢ ∀ᶠ (t : Finset (ι × ℝ)) in atTop,\n ∀ (I J : Box ι), J ∈ s → ∀ (J' : Box ι), J' ∈ splitMany I t → ¬Disjoint ↑J ↑J' → J' ≤ J",
"state_before": "ι : Type u_1\nM : Type ?u.49727\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\ninst✝ : Finite ι\ns : Finset (Box ι)\n⊢ ∀ᶠ (t : Finset (ι × ℝ)) in atTop,\n ∀ (I J : Box ι), J ∈ s → ∀ (J' : Box ι), J' ∈ splitMany I t → ¬Disjoint ↑J ↑J' → J' ≤ J",
"tactic": "cases nonempty_fintype ι"
},
{
"state_after": "case intro\nι : Type u_1\nM : Type ?u.49727\nn : ℕ\nI✝ J✝ : Box ι\ni✝ : ι\nx : ℝ\ninst✝ : Finite ι\ns : Finset (Box ι)\nval✝ : Fintype ι\nt : Finset (ι × ℝ)\nht : t ≥ Finset.biUnion s fun J => Finset.biUnion Finset.univ fun i => {(i, Box.lower J i), (i, Box.upper J i)}\nI J : Box ι\nhJ : J ∈ s\nJ' : Box ι\nhJ' : J' ∈ splitMany I t\ni : ι\n⊢ {(i, Box.lower J i), (i, Box.upper J i)} ⊆ t",
"state_before": "case intro\nι : Type u_1\nM : Type ?u.49727\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\ninst✝ : Finite ι\ns : Finset (Box ι)\nval✝ : Fintype ι\n⊢ ∀ᶠ (t : Finset (ι × ℝ)) in atTop,\n ∀ (I J : Box ι), J ∈ s → ∀ (J' : Box ι), J' ∈ splitMany I t → ¬Disjoint ↑J ↑J' → J' ≤ J",
"tactic": "refine' eventually_atTop.2\n ⟨s.biUnion fun J => Finset.univ.biUnion fun i => {(i, J.lower i), (i, J.upper i)},\n fun t ht I J hJ J' hJ' => not_disjoint_imp_le_of_subset_of_mem_splitMany (fun i => _) hJ'⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type u_1\nM : Type ?u.49727\nn : ℕ\nI✝ J✝ : Box ι\ni✝ : ι\nx : ℝ\ninst✝ : Finite ι\ns : Finset (Box ι)\nval✝ : Fintype ι\nt : Finset (ι × ℝ)\nht : t ≥ Finset.biUnion s fun J => Finset.biUnion Finset.univ fun i => {(i, Box.lower J i), (i, Box.upper J i)}\nI J : Box ι\nhJ : J ∈ s\nJ' : Box ι\nhJ' : J' ∈ splitMany I t\ni : ι\n⊢ {(i, Box.lower J i), (i, Box.upper J i)} ⊆ t",
"tactic": "exact fun p hp =>\n ht (Finset.mem_biUnion.2 ⟨J, hJ, Finset.mem_biUnion.2 ⟨i, Finset.mem_univ _, hp⟩⟩)"
}
] |
[
318,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
310,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.ne_nil_of_length_eq_succ
|
[] |
[
47,
88
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
47,
1
] |
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