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/Order/Closure.lean
|
ClosureOperator.closure_iSup₂_closure
|
[] |
[
286,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/MeasureTheory/Constructions/Pi.lean
|
MeasureTheory.Measure.tprod_cons
|
[] |
[
245,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.root_right_of_root_gcd
|
[
{
"state_after": "case intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : Field R\np✝ q : R[X]\ninst✝ : CommSemiring k\nϕ : R →+* k\nf g : R[X]\nα : k\nhα : eval₂ ϕ α (EuclideanDomain.gcd f g) = 0\np : R[X]\nhp : g = EuclideanDomain.gcd f g * p\n⊢ eval₂ ϕ α g = 0",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : Field R\np q : R[X]\ninst✝ : CommSemiring k\nϕ : R →+* k\nf g : R[X]\nα : k\nhα : eval₂ ϕ α (EuclideanDomain.gcd f g) = 0\n⊢ eval₂ ϕ α g = 0",
"tactic": "cases' EuclideanDomain.gcd_dvd_right f g with p hp"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : Field R\np✝ q : R[X]\ninst✝ : CommSemiring k\nϕ : R →+* k\nf g : R[X]\nα : k\nhα : eval₂ ϕ α (EuclideanDomain.gcd f g) = 0\np : R[X]\nhp : g = EuclideanDomain.gcd f g * p\n⊢ eval₂ ϕ α g = 0",
"tactic": "rw [hp, Polynomial.eval₂_mul, hα, MulZeroClass.zero_mul]"
}
] |
[
346,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.lintegral_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.865618\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nf : β → ℝ≥0∞\nμ : MeasureTheory.Measure β\na : α\n⊢ (∫⁻ (x : β), f x ∂↑(const α μ) a) = ∫⁻ (x : β), f x ∂μ",
"tactic": "rw [kernel.const_apply]"
}
] |
[
450,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Order/WellFounded.lean
|
WellFounded.has_min
|
[] |
[
53,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.edist_le_mul_of_le
|
[] |
[
139,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.succAbove_aux
|
[] |
[
2020,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2016,
1
] |
Mathlib/GroupTheory/GroupAction/ConjAct.lean
|
MulAut.conjNormal_apply
|
[] |
[
339,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean
|
ContDiffAt.dist
|
[
{
"state_after": "𝕜 : Type ?u.276081\nE : Type u_2\nF : Type ?u.276087\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : NormedSpace ℝ E\nG : Type u_1\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf g : G → E\nf' g' : G →L[ℝ] E\ns : Set G\nx : G\nn : ℕ∞\nhf : ContDiffAt ℝ n f x\nhg : ContDiffAt ℝ n g x\nhne : f x ≠ g x\n⊢ ContDiffAt ℝ n (fun y => ‖f y - g y‖) x",
"state_before": "𝕜 : Type ?u.276081\nE : Type u_2\nF : Type ?u.276087\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : NormedSpace ℝ E\nG : Type u_1\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf g : G → E\nf' g' : G →L[ℝ] E\ns : Set G\nx : G\nn : ℕ∞\nhf : ContDiffAt ℝ n f x\nhg : ContDiffAt ℝ n g x\nhne : f x ≠ g x\n⊢ ContDiffAt ℝ n (fun y => Dist.dist (f y) (g y)) x",
"tactic": "simp only [dist_eq_norm]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.276081\nE : Type u_2\nF : Type ?u.276087\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : NormedSpace ℝ E\nG : Type u_1\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf g : G → E\nf' g' : G →L[ℝ] E\ns : Set G\nx : G\nn : ℕ∞\nhf : ContDiffAt ℝ n f x\nhg : ContDiffAt ℝ n g x\nhne : f x ≠ g x\n⊢ ContDiffAt ℝ n (fun y => ‖f y - g y‖) x",
"tactic": "exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne)"
}
] |
[
188,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Algebra/Order/Hom/Monoid.lean
|
OrderMonoidWithZeroHom.mul_comp
|
[] |
[
748,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
747,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
EMetric.infEdist_closure
|
[
{
"state_after": "ι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\n⊢ infEdist x s ≤ infEdist x (closure s)",
"state_before": "ι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\n⊢ infEdist x (closure s) = infEdist x s",
"tactic": "refine' le_antisymm (infEdist_anti subset_closure) _"
},
{
"state_after": "ι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nε : ℝ≥0\nεpos : 0 < ε\nh : infEdist x (closure s) < ⊤\n⊢ infEdist x s ≤ infEdist x (closure s) + ↑ε",
"state_before": "ι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\n⊢ infEdist x s ≤ infEdist x (closure s)",
"tactic": "refine' ENNReal.le_of_forall_pos_le_add fun ε εpos h => _"
},
{
"state_after": "ι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nε : ℝ≥0\nεpos : 0 < ε\nh : infEdist x (closure s) < ⊤\nε0 : 0 < ↑ε / 2\n⊢ infEdist x s ≤ infEdist x (closure s) + ↑ε",
"state_before": "ι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nε : ℝ≥0\nεpos : 0 < ε\nh : infEdist x (closure s) < ⊤\n⊢ infEdist x s ≤ infEdist x (closure s) + ↑ε",
"tactic": "have ε0 : 0 < (ε / 2 : ℝ≥0∞) := by simpa [pos_iff_ne_zero] using εpos"
},
{
"state_after": "ι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nε : ℝ≥0\nεpos : 0 < ε\nh : infEdist x (closure s) < ⊤\nε0 : 0 < ↑ε / 2\nthis : infEdist x (closure s) < infEdist x (closure s) + ↑ε / 2\n⊢ infEdist x s ≤ infEdist x (closure s) + ↑ε",
"state_before": "ι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nε : ℝ≥0\nεpos : 0 < ε\nh : infEdist x (closure s) < ⊤\nε0 : 0 < ↑ε / 2\n⊢ infEdist x s ≤ infEdist x (closure s) + ↑ε",
"tactic": "have : infEdist x (closure s) < infEdist x (closure s) + ε / 2 :=\n ENNReal.lt_add_right h.ne ε0.ne'"
},
{
"state_after": "case intro.intro\nι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y✝ : α\ns t : Set α\nΦ : α → β\nε : ℝ≥0\nεpos : 0 < ε\nh : infEdist x (closure s) < ⊤\nε0 : 0 < ↑ε / 2\nthis : infEdist x (closure s) < infEdist x (closure s) + ↑ε / 2\ny : α\nycs : y ∈ closure s\nhy : edist x y < infEdist x (closure s) + ↑ε / 2\n⊢ infEdist x s ≤ infEdist x (closure s) + ↑ε",
"state_before": "ι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nε : ℝ≥0\nεpos : 0 < ε\nh : infEdist x (closure s) < ⊤\nε0 : 0 < ↑ε / 2\nthis : infEdist x (closure s) < infEdist x (closure s) + ↑ε / 2\n⊢ infEdist x s ≤ infEdist x (closure s) + ↑ε",
"tactic": "rcases infEdist_lt_iff.mp this with ⟨y, ycs, hy⟩"
},
{
"state_after": "case intro.intro.intro.intro\nι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y✝ : α\ns t : Set α\nΦ : α → β\nε : ℝ≥0\nεpos : 0 < ε\nh : infEdist x (closure s) < ⊤\nε0 : 0 < ↑ε / 2\nthis : infEdist x (closure s) < infEdist x (closure s) + ↑ε / 2\ny : α\nycs : y ∈ closure s\nhy : edist x y < infEdist x (closure s) + ↑ε / 2\nz : α\nzs : z ∈ s\ndyz : edist y z < ↑ε / 2\n⊢ infEdist x s ≤ infEdist x (closure s) + ↑ε",
"state_before": "case intro.intro\nι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y✝ : α\ns t : Set α\nΦ : α → β\nε : ℝ≥0\nεpos : 0 < ε\nh : infEdist x (closure s) < ⊤\nε0 : 0 < ↑ε / 2\nthis : infEdist x (closure s) < infEdist x (closure s) + ↑ε / 2\ny : α\nycs : y ∈ closure s\nhy : edist x y < infEdist x (closure s) + ↑ε / 2\n⊢ infEdist x s ≤ infEdist x (closure s) + ↑ε",
"tactic": "rcases EMetric.mem_closure_iff.1 ycs (ε / 2) ε0 with ⟨z, zs, dyz⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y✝ : α\ns t : Set α\nΦ : α → β\nε : ℝ≥0\nεpos : 0 < ε\nh : infEdist x (closure s) < ⊤\nε0 : 0 < ↑ε / 2\nthis : infEdist x (closure s) < infEdist x (closure s) + ↑ε / 2\ny : α\nycs : y ∈ closure s\nhy : edist x y < infEdist x (closure s) + ↑ε / 2\nz : α\nzs : z ∈ s\ndyz : edist y z < ↑ε / 2\n⊢ infEdist x s ≤ infEdist x (closure s) + ↑ε",
"tactic": "calc\n infEdist x s ≤ edist x z := infEdist_le_edist_of_mem zs\n _ ≤ edist x y + edist y z := (edist_triangle _ _ _)\n _ ≤ infEdist x (closure s) + ε / 2 + ε / 2 := (add_le_add (le_of_lt hy) (le_of_lt dyz))\n _ = infEdist x (closure s) + ↑ε := by rw [add_assoc, ENNReal.add_halves]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nε : ℝ≥0\nεpos : 0 < ε\nh : infEdist x (closure s) < ⊤\n⊢ 0 < ↑ε / 2",
"tactic": "simpa [pos_iff_ne_zero] using εpos"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.11409\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y✝ : α\ns t : Set α\nΦ : α → β\nε : ℝ≥0\nεpos : 0 < ε\nh : infEdist x (closure s) < ⊤\nε0 : 0 < ↑ε / 2\nthis : infEdist x (closure s) < infEdist x (closure s) + ↑ε / 2\ny : α\nycs : y ∈ closure s\nhy : edist x y < infEdist x (closure s) + ↑ε / 2\nz : α\nzs : z ∈ s\ndyz : edist y z < ↑ε / 2\n⊢ infEdist x (closure s) + ↑ε / 2 + ↑ε / 2 = infEdist x (closure s) + ↑ε",
"tactic": "rw [add_assoc, ENNReal.add_halves]"
}
] |
[
149,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
Subgroup.smul_sup
|
[] |
[
305,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/Analysis/Normed/Group/AddTorsor.lean
|
edist_vadd_vadd_le
|
[
{
"state_after": "α : Type ?u.26982\nV : Type u_2\nP : Type u_1\nW : Type ?u.26991\nQ : Type ?u.26994\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : NormedAddCommGroup W\ninst✝¹ : MetricSpace Q\ninst✝ : NormedAddTorsor W Q\nv v' : V\np p' : P\n⊢ ↑(nndist (v +ᵥ p) (v' +ᵥ p')) ≤ ↑(nndist v v') + ↑(nndist p p')",
"state_before": "α : Type ?u.26982\nV : Type u_2\nP : Type u_1\nW : Type ?u.26991\nQ : Type ?u.26994\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : NormedAddCommGroup W\ninst✝¹ : MetricSpace Q\ninst✝ : NormedAddTorsor W Q\nv v' : V\np p' : P\n⊢ edist (v +ᵥ p) (v' +ᵥ p') ≤ edist v v' + edist p p'",
"tactic": "simp only [edist_nndist]"
},
{
"state_after": "α : Type ?u.26982\nV : Type u_2\nP : Type u_1\nW : Type ?u.26991\nQ : Type ?u.26994\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : NormedAddCommGroup W\ninst✝¹ : MetricSpace Q\ninst✝ : NormedAddTorsor W Q\nv v' : V\np p' : P\n⊢ nndist (v +ᵥ p) (v' +ᵥ p') ≤ nndist v v' + nndist p p'",
"state_before": "α : Type ?u.26982\nV : Type u_2\nP : Type u_1\nW : Type ?u.26991\nQ : Type ?u.26994\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : NormedAddCommGroup W\ninst✝¹ : MetricSpace Q\ninst✝ : NormedAddTorsor W Q\nv v' : V\np p' : P\n⊢ ↑(nndist (v +ᵥ p) (v' +ᵥ p')) ≤ ↑(nndist v v') + ↑(nndist p p')",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.26982\nV : Type u_2\nP : Type u_1\nW : Type ?u.26991\nQ : Type ?u.26994\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : NormedAddCommGroup W\ninst✝¹ : MetricSpace Q\ninst✝ : NormedAddTorsor W Q\nv v' : V\np p' : P\n⊢ nndist (v +ᵥ p) (v' +ᵥ p') ≤ nndist v v' + nndist p p'",
"tactic": "apply dist_vadd_vadd_le"
}
] |
[
199,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.mapOfSurjective_eq_mapGen
|
[
{
"state_after": "M : Type u_1\nN : Type u_2\nP : Type ?u.35290\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nc✝ c : Con M\nf : M → N\nH : ∀ (x y : M), f (x * y) = f x * f y\nh : mulKer f H ≤ c\nhf : Surjective f\n⊢ mapGen c f = conGen ↑(mapOfSurjective c f H h hf)",
"state_before": "M : Type u_1\nN : Type u_2\nP : Type ?u.35290\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nc✝ c : Con M\nf : M → N\nH : ∀ (x y : M), f (x * y) = f x * f y\nh : mulKer f H ≤ c\nhf : Surjective f\n⊢ mapGen c f = mapOfSurjective c f H h hf",
"tactic": "rw [← conGen_of_con (c.mapOfSurjective f H h hf)]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type u_2\nP : Type ?u.35290\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nc✝ c : Con M\nf : M → N\nH : ∀ (x y : M), f (x * y) = f x * f y\nh : mulKer f H ≤ c\nhf : Surjective f\n⊢ mapGen c f = conGen ↑(mapOfSurjective c f H h hf)",
"tactic": "rfl"
}
] |
[
660,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/LinearAlgebra/Finrank.lean
|
FiniteDimensional.finrank_eq_card_finset_basis
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝⁵ : Ring K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : StrongRankCondition K\nι : Type w\nb : Finset ι\nh : Basis { x // x ∈ b } K V\n⊢ finrank K V = Finset.card b",
"tactic": "rw [finrank_eq_card_basis h, Fintype.card_coe]"
}
] |
[
135,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.of_pow
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.500881\nE : Type ?u.500884\nF : Type ?u.500887\nG : Type ?u.500890\nE' : Type ?u.500893\nF' : Type ?u.500896\nG' : Type ?u.500899\nE'' : Type ?u.500902\nF'' : Type ?u.500905\nG'' : Type ?u.500908\nR : Type u_3\nR' : Type ?u.500914\n𝕜 : Type u_2\n𝕜' : Type ?u.500920\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nn : ℕ\nf : α → 𝕜\ng : α → R\nh : IsBigOWith c l (f ^ n) (g ^ n)\nhn : n ≠ 0\nhc : c ≤ c' ^ n\nhc' : 0 ≤ c'\nx : α\nhx : ‖(f ^ n) x‖ ≤ c' ^ n * ‖(g ^ n) x‖\n⊢ ‖g x ^ n‖ ≤ ‖g x‖ ^ n",
"state_before": "α : Type u_1\nβ : Type ?u.500881\nE : Type ?u.500884\nF : Type ?u.500887\nG : Type ?u.500890\nE' : Type ?u.500893\nF' : Type ?u.500896\nG' : Type ?u.500899\nE'' : Type ?u.500902\nF'' : Type ?u.500905\nG'' : Type ?u.500908\nR : Type u_3\nR' : Type ?u.500914\n𝕜 : Type u_2\n𝕜' : Type ?u.500920\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nn : ℕ\nf : α → 𝕜\ng : α → R\nh : IsBigOWith c l (f ^ n) (g ^ n)\nhn : n ≠ 0\nhc : c ≤ c' ^ n\nhc' : 0 ≤ c'\nx : α\nhx : ‖(f ^ n) x‖ ≤ c' ^ n * ‖(g ^ n) x‖\n⊢ c' ^ n * ‖g x ^ n‖ ≤ c' ^ n * ‖g x‖ ^ n",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.500881\nE : Type ?u.500884\nF : Type ?u.500887\nG : Type ?u.500890\nE' : Type ?u.500893\nF' : Type ?u.500896\nG' : Type ?u.500899\nE'' : Type ?u.500902\nF'' : Type ?u.500905\nG'' : Type ?u.500908\nR : Type u_3\nR' : Type ?u.500914\n𝕜 : Type u_2\n𝕜' : Type ?u.500920\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nn : ℕ\nf : α → 𝕜\ng : α → R\nh : IsBigOWith c l (f ^ n) (g ^ n)\nhn : n ≠ 0\nhc : c ≤ c' ^ n\nhc' : 0 ≤ c'\nx : α\nhx : ‖(f ^ n) x‖ ≤ c' ^ n * ‖(g ^ n) x‖\n⊢ ‖g x ^ n‖ ≤ ‖g x‖ ^ n",
"tactic": "exact norm_pow_le' _ hn.bot_lt"
}
] |
[
1635,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1626,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.Memℒp.induction
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nthis✝ : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f\nthis : ∀ (f : { x // x ∈ Lp.simpleFunc E p μ }), P ↑↑↑f\n⊢ ∀ ⦃f : α → E⦄, Memℒp f p → P f",
"state_before": "α : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nthis : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f\n⊢ ∀ ⦃f : α → E⦄, Memℒp f p → P f",
"tactic": "have : ∀ f : Lp.simpleFunc E p μ, P f := by\n intro f\n exact\n h_ae (Lp.simpleFunc.toSimpleFunc_eq_toFun f) (Lp.simpleFunc.memℒp f)\n (this (Lp.simpleFunc.toSimpleFunc f) (Lp.simpleFunc.memℒp f))"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nthis✝¹ : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f\nthis✝ : ∀ (f : { x // x ∈ Lp.simpleFunc E p μ }), P ↑↑↑f\nthis : ∀ (f : { x // x ∈ Lp E p }), P ↑↑f\n⊢ ∀ ⦃f : α → E⦄, Memℒp f p → P f",
"state_before": "α : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nthis✝ : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f\nthis : ∀ (f : { x // x ∈ Lp.simpleFunc E p μ }), P ↑↑↑f\n⊢ ∀ ⦃f : α → E⦄, Memℒp f p → P f",
"tactic": "have : ∀ f : Lp E p μ, P f := fun f =>\n (Lp.simpleFunc.denseRange hp_ne_top).induction_on f h_closed this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nthis✝¹ : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f\nthis✝ : ∀ (f : { x // x ∈ Lp.simpleFunc E p μ }), P ↑↑↑f\nthis : ∀ (f : { x // x ∈ Lp E p }), P ↑↑f\n⊢ ∀ ⦃f : α → E⦄, Memℒp f p → P f",
"tactic": "exact fun f hf => h_ae hf.coeFn_toLp (Lp.memℒp _) (this (hf.toLp f))"
},
{
"state_after": "case h_ind\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\n⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s),\n Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p →\n P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))\n\ncase h_add\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\n⊢ ∀ ⦃f g : α →ₛ E⦄,\n Disjoint (support ↑f) (support ↑g) → (Memℒp (↑f) p → P ↑f) → (Memℒp (↑g) p → P ↑g) → Memℒp (↑(f + g)) p → P ↑(f + g)",
"state_before": "α : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\n⊢ ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f",
"tactic": "apply SimpleFunc.induction"
},
{
"state_after": "case h_ind\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nc : E\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\n⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))",
"state_before": "case h_ind\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\n⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s),\n Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p →\n P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))",
"tactic": "intro c s hs h"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nc : E\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\nhc : c = 0\n⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))\n\ncase neg\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nc : E\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\nhc : ¬c = 0\n⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))",
"state_before": "case h_ind\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nc : E\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\n⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))",
"tactic": "by_cases hc : c = 0"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nc : E\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\nhc : ¬c = 0\nhp_pos : p ≠ 0\n⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nc : E\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\nhc : ¬c = 0\n⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))",
"tactic": "have hp_pos : p ≠ 0 := (lt_of_lt_of_le zero_lt_one _i.elim).ne'"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nc : E\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\nhc : ¬c = 0\nhp_pos : p ≠ 0\n⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))",
"tactic": "exact h_ind c hs (SimpleFunc.measure_lt_top_of_memℒp_indicator hp_pos hp_ne_top hc hs h)"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p\n⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nc : E\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p\nhc : c = 0\n⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))",
"tactic": "subst hc"
},
{
"state_after": "case h.e'_1\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p\n⊢ ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) = Set.indicator ∅ fun x => 0",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p\n⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))",
"tactic": "convert h_ind 0 MeasurableSet.empty (by simp) using 1"
},
{
"state_after": "case h.e'_1.h\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p\nx✝ : α\n⊢ ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) x✝ = Set.indicator ∅ (fun x => 0) x✝",
"state_before": "case h.e'_1\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p\n⊢ ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) = Set.indicator ∅ fun x => 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.e'_1.h\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p\nx✝ : α\n⊢ ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) x✝ = Set.indicator ∅ (fun x => 0) x✝",
"tactic": "simp [const]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\ns : Set α\nhs : MeasurableSet s\nh : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p\n⊢ ↑↑μ ∅ < ⊤",
"tactic": "simp"
},
{
"state_after": "case h_add\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nf g : α →ₛ E\nhfg : Disjoint (support ↑f) (support ↑g)\nhf : Memℒp (↑f) p → P ↑f\nhg : Memℒp (↑g) p → P ↑g\nint_fg : Memℒp (↑(f + g)) p\n⊢ P ↑(f + g)",
"state_before": "case h_add\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\n⊢ ∀ ⦃f g : α →ₛ E⦄,\n Disjoint (support ↑f) (support ↑g) → (Memℒp (↑f) p → P ↑f) → (Memℒp (↑g) p → P ↑g) → Memℒp (↑(f + g)) p → P ↑(f + g)",
"tactic": "intro f g hfg hf hg int_fg"
},
{
"state_after": "case h_add\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nf g : α →ₛ E\nhfg : Disjoint (support ↑f) (support ↑g)\nhf : Memℒp (↑f) p → P ↑f\nhg : Memℒp (↑g) p → P ↑g\nint_fg : Memℒp (↑f) p ∧ Memℒp (↑g) p\n⊢ P ↑(f + g)",
"state_before": "case h_add\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nf g : α →ₛ E\nhfg : Disjoint (support ↑f) (support ↑g)\nhf : Memℒp (↑f) p → P ↑f\nhg : Memℒp (↑g) p → P ↑g\nint_fg : Memℒp (↑(f + g)) p\n⊢ P ↑(f + g)",
"tactic": "rw [SimpleFunc.coe_add,\n memℒp_add_of_disjoint hfg f.stronglyMeasurable g.stronglyMeasurable] at int_fg"
},
{
"state_after": "no goals",
"state_before": "case h_add\nα : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nf g : α →ₛ E\nhfg : Disjoint (support ↑f) (support ↑g)\nhf : Memℒp (↑f) p → P ↑f\nhg : Memℒp (↑g) p → P ↑g\nint_fg : Memℒp (↑f) p ∧ Memℒp (↑g) p\n⊢ P ↑(f + g)",
"tactic": "refine' h_add hfg int_fg.1 int_fg.2 (hf int_fg.1) (hg int_fg.2)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nthis : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f\nf : { x // x ∈ Lp.simpleFunc E p μ }\n⊢ P ↑↑↑f",
"state_before": "α : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nthis : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f\n⊢ ∀ (f : { x // x ∈ Lp.simpleFunc E p μ }), P ↑↑↑f",
"tactic": "intro f"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3297285\nι : Type ?u.3297288\nE : Type u_2\nF : Type ?u.3297294\n𝕜 : Type ?u.3297297\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf✝ : α → E\np : ℝ≥0∞\nμ : Measure α\n_i : Fact (1 ≤ p)\nhp_ne_top : p ≠ ⊤\nP : (α → E) → Prop\nh_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c)\nh_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g)\nh_closed : IsClosed {f | P ↑↑f}\nh_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g\nthis : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f\nf : { x // x ∈ Lp.simpleFunc E p μ }\n⊢ P ↑↑↑f",
"tactic": "exact\n h_ae (Lp.simpleFunc.toSimpleFunc_eq_toFun f) (Lp.simpleFunc.memℒp f)\n (this (Lp.simpleFunc.toSimpleFunc f) (Lp.simpleFunc.memℒp f))"
}
] |
[
1020,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
994,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.coeFn_injective
|
[] |
[
154,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Std/Logic.lean
|
and_congr_right_eq
|
[] |
[
172,
51
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
171,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.get_multiplicity_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na : α\nha : Finite a a\n⊢ a ^ 1 ∣ a",
"tactic": "simp"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na : α\nha : Finite a a\nx✝ : a ^ (1 + 1) ∣ a\nb : α\nhb : 1 = 1 * (a * 1 * b)\n⊢ False",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na : α\nha : Finite a a\nx✝ : a ^ (1 + 1) ∣ a\nb : α\nhb : a = a ^ (1 + 1) * b\n⊢ False",
"tactic": "rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,\n mul_right_inj' (ne_zero_of_finite ha)] at hb"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na : α\nha : Finite a a\nx✝ : a ^ (1 + 1) ∣ a\nb : α\nhb : 1 = 1 * (a * 1 * b)\n⊢ False",
"tactic": "exact\n mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na : α\nha : Finite a a\nx✝ : a ^ (1 + 1) ∣ a\nb : α\nhb : 1 = 1 * (a * 1 * b)\n⊢ 1 = a * b",
"tactic": "simp_all"
}
] |
[
543,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
536,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.add_lt_add_of_le_of_lt
|
[] |
[
790,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
789,
11
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.toAlgHom_eq_coe
|
[] |
[
256,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/CategoryTheory/Limits/ConcreteCategory.lean
|
CategoryTheory.Limits.Concrete.widePushout_exists_rep
|
[
{
"state_after": "case intro.none.intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type v\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesColimit F (forget C)\nB : C\nα : Type v\nX : α → C\nf : (j : α) → B ⟶ X j\ninst✝¹ : HasWidePushout B X f\ninst✝ : PreservesColimit (wideSpan B X f) (forget C)\ny : (forget C).obj ((wideSpan B X f).obj none)\n⊢ (∃ y_1, (forget C).map (head f) y_1 = (forget C).map (colimit.ι (wideSpan B X f) none) y) ∨\n ∃ i y_1, (forget C).map (ι f i) y_1 = (forget C).map (colimit.ι (wideSpan B X f) none) y\n\ncase intro.some.intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type v\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesColimit F (forget C)\nB : C\nα : Type v\nX : α → C\nf : (j : α) → B ⟶ X j\ninst✝¹ : HasWidePushout B X f\ninst✝ : PreservesColimit (wideSpan B X f) (forget C)\nj : α\ny : (forget C).obj ((wideSpan B X f).obj (some j))\n⊢ (∃ y_1, (forget C).map (head f) y_1 = (forget C).map (colimit.ι (wideSpan B X f) (some j)) y) ∨\n ∃ i y_1, (forget C).map (ι f i) y_1 = (forget C).map (colimit.ι (wideSpan B X f) (some j)) y",
"state_before": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type v\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesColimit F (forget C)\nB : C\nα : Type v\nX : α → C\nf : (j : α) → B ⟶ X j\ninst✝¹ : HasWidePushout B X f\ninst✝ : PreservesColimit (wideSpan B X f) (forget C)\nx : (forget C).obj (widePushout B X f)\n⊢ (∃ y, (forget C).map (head f) y = x) ∨ ∃ i y, (forget C).map (ι f i) y = x",
"tactic": "obtain ⟨_ | j, y, rfl⟩ := Concrete.colimit_exists_rep _ x"
},
{
"state_after": "case intro.none.intro.h\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type v\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesColimit F (forget C)\nB : C\nα : Type v\nX : α → C\nf : (j : α) → B ⟶ X j\ninst✝¹ : HasWidePushout B X f\ninst✝ : PreservesColimit (wideSpan B X f) (forget C)\ny : (forget C).obj ((wideSpan B X f).obj none)\n⊢ ∃ y_1, (forget C).map (head f) y_1 = (forget C).map (colimit.ι (wideSpan B X f) none) y",
"state_before": "case intro.none.intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type v\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesColimit F (forget C)\nB : C\nα : Type v\nX : α → C\nf : (j : α) → B ⟶ X j\ninst✝¹ : HasWidePushout B X f\ninst✝ : PreservesColimit (wideSpan B X f) (forget C)\ny : (forget C).obj ((wideSpan B X f).obj none)\n⊢ (∃ y_1, (forget C).map (head f) y_1 = (forget C).map (colimit.ι (wideSpan B X f) none) y) ∨\n ∃ i y_1, (forget C).map (ι f i) y_1 = (forget C).map (colimit.ι (wideSpan B X f) none) y",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case intro.none.intro.h\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type v\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesColimit F (forget C)\nB : C\nα : Type v\nX : α → C\nf : (j : α) → B ⟶ X j\ninst✝¹ : HasWidePushout B X f\ninst✝ : PreservesColimit (wideSpan B X f) (forget C)\ny : (forget C).obj ((wideSpan B X f).obj none)\n⊢ ∃ y_1, (forget C).map (head f) y_1 = (forget C).map (colimit.ι (wideSpan B X f) none) y",
"tactic": "use y"
},
{
"state_after": "case intro.some.intro.h\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type v\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesColimit F (forget C)\nB : C\nα : Type v\nX : α → C\nf : (j : α) → B ⟶ X j\ninst✝¹ : HasWidePushout B X f\ninst✝ : PreservesColimit (wideSpan B X f) (forget C)\nj : α\ny : (forget C).obj ((wideSpan B X f).obj (some j))\n⊢ ∃ i y_1, (forget C).map (ι f i) y_1 = (forget C).map (colimit.ι (wideSpan B X f) (some j)) y",
"state_before": "case intro.some.intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type v\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesColimit F (forget C)\nB : C\nα : Type v\nX : α → C\nf : (j : α) → B ⟶ X j\ninst✝¹ : HasWidePushout B X f\ninst✝ : PreservesColimit (wideSpan B X f) (forget C)\nj : α\ny : (forget C).obj ((wideSpan B X f).obj (some j))\n⊢ (∃ y_1, (forget C).map (head f) y_1 = (forget C).map (colimit.ι (wideSpan B X f) (some j)) y) ∨\n ∃ i y_1, (forget C).map (ι f i) y_1 = (forget C).map (colimit.ι (wideSpan B X f) (some j)) y",
"tactic": "right"
},
{
"state_after": "no goals",
"state_before": "case intro.some.intro.h\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type v\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesColimit F (forget C)\nB : C\nα : Type v\nX : α → C\nf : (j : α) → B ⟶ X j\ninst✝¹ : HasWidePushout B X f\ninst✝ : PreservesColimit (wideSpan B X f) (forget C)\nj : α\ny : (forget C).obj ((wideSpan B X f).obj (some j))\n⊢ ∃ i y_1, (forget C).map (ι f i) y_1 = (forget C).map (colimit.ι (wideSpan B X f) (some j)) y",
"tactic": "use j, y"
}
] |
[
320,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/CategoryTheory/Equivalence.lean
|
CategoryTheory.Equivalence.inverse_map_inj_iff
|
[] |
[
751,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
749,
1
] |
Std/Data/String/Lemmas.lean
|
Substring.Valid.dropWhile
|
[] |
[
1076,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1075,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
OrderIso.map_csInf'
|
[] |
[
1351,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1349,
1
] |
Mathlib/RingTheory/Noetherian.lean
|
IsNoetherian.injective_of_surjective_endomorphism
|
[
{
"state_after": "case intro.intro\nR : Type u_1\nM : Type u_2\nP : Type ?u.268814\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ninst✝ : IsNoetherian R M\nf : M →ₗ[R] M\ns : Surjective ↑f\nn : ℕ\nne : n ≠ 0\nw : LinearMap.ker (f ^ n) ⊓ LinearMap.range (f ^ n) = ⊥\n⊢ Injective ↑f",
"state_before": "R : Type u_1\nM : Type u_2\nP : Type ?u.268814\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ninst✝ : IsNoetherian R M\nf : M →ₗ[R] M\ns : Surjective ↑f\n⊢ Injective ↑f",
"tactic": "obtain ⟨n, ne, w⟩ := IsNoetherian.exists_endomorphism_iterate_ker_inf_range_eq_bot f"
},
{
"state_after": "case intro.intro\nR : Type u_1\nM : Type u_2\nP : Type ?u.268814\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ninst✝ : IsNoetherian R M\nf : M →ₗ[R] M\ns : Surjective ↑f\nn : ℕ\nne : n ≠ 0\nw : Injective ↑(f ^ n)\n⊢ Injective ↑f",
"state_before": "case intro.intro\nR : Type u_1\nM : Type u_2\nP : Type ?u.268814\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ninst✝ : IsNoetherian R M\nf : M →ₗ[R] M\ns : Surjective ↑f\nn : ℕ\nne : n ≠ 0\nw : LinearMap.ker (f ^ n) ⊓ LinearMap.range (f ^ n) = ⊥\n⊢ Injective ↑f",
"tactic": "rw [LinearMap.range_eq_top.mpr (LinearMap.iterate_surjective s n), inf_top_eq,\n LinearMap.ker_eq_bot] at w"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\nM : Type u_2\nP : Type ?u.268814\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ninst✝ : IsNoetherian R M\nf : M →ₗ[R] M\ns : Surjective ↑f\nn : ℕ\nne : n ≠ 0\nw : Injective ↑(f ^ n)\n⊢ Injective ↑f",
"tactic": "exact LinearMap.injective_of_iterate_injective ne w"
}
] |
[
437,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.of_b_eq_zero'
|
[] |
[
153,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Data/Set/Intervals/Disjoint.lean
|
Set.Ici_disjoint_Iic
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort u\nα : Type v\nβ : Type w\ninst✝ : Preorder α\na b c : α\n⊢ Disjoint (Ici a) (Iic b) ↔ ¬a ≤ b",
"tactic": "rw [Set.disjoint_iff_inter_eq_empty, Ici_inter_Iic, Icc_eq_empty_iff]"
}
] |
[
58,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
Subgroup.pointwise_smul_toSubmonoid
|
[] |
[
287,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.cast_sub_one
|
[
{
"state_after": "case inl\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\nk : ZMod n\nhk : k = 0\n⊢ ↑(k - 1) = ↑n - 1\n\ncase inr\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\nk : ZMod n\nhk : ¬k = 0\n⊢ ↑(k - 1) = ↑k - 1",
"state_before": "R : Type u_1\ninst✝ : Ring R\nn : ℕ\nk : ZMod n\n⊢ ↑(k - 1) = (if k = 0 then ↑n else ↑k) - 1",
"tactic": "split_ifs with hk"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\nk : ZMod n\nhk : k = 0\n⊢ ↑(k - 1) = ↑n - 1",
"tactic": "rw [hk, zero_sub, ZMod.cast_neg_one]"
},
{
"state_after": "case inr.zero\nR : Type u_1\ninst✝ : Ring R\nk : ZMod Nat.zero\nhk : ¬k = 0\n⊢ ↑(k - 1) = ↑k - 1\n\ncase inr.succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nk : ZMod (Nat.succ n✝)\nhk : ¬k = 0\n⊢ ↑(k - 1) = ↑k - 1",
"state_before": "case inr\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\nk : ZMod n\nhk : ¬k = 0\n⊢ ↑(k - 1) = ↑k - 1",
"tactic": "cases n"
},
{
"state_after": "case inr.zero\nR : Type u_1\ninst✝ : Ring R\nk : ZMod Nat.zero\nhk : ¬k = 0\n⊢ ↑(k - 1) = ↑k - 1",
"state_before": "case inr.zero\nR : Type u_1\ninst✝ : Ring R\nk : ZMod Nat.zero\nhk : ¬k = 0\n⊢ ↑(k - 1) = ↑k - 1",
"tactic": "dsimp [ZMod, ZMod.cast]"
},
{
"state_after": "no goals",
"state_before": "case inr.zero\nR : Type u_1\ninst✝ : Ring R\nk : ZMod Nat.zero\nhk : ¬k = 0\n⊢ ↑(k - 1) = ↑k - 1",
"tactic": "rw [Int.cast_sub, Int.cast_one]"
},
{
"state_after": "case inr.succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nk : ZMod (Nat.succ n✝)\nhk : ¬k = 0\n⊢ ↑↑(k - 1) = ↑↑k - 1",
"state_before": "case inr.succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nk : ZMod (Nat.succ n✝)\nhk : ¬k = 0\n⊢ ↑(k - 1) = ↑k - 1",
"tactic": "dsimp [ZMod, ZMod.cast, ZMod.val]"
},
{
"state_after": "case inr.succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nk : ZMod (Nat.succ n✝)\nhk : ¬k = 0\n⊢ ↑(↑k - 1) = ↑↑k - 1\n\ncase inr.succ.hnc\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nk : ZMod (Nat.succ n✝)\nhk : ¬k = 0\n⊢ ¬k = 0",
"state_before": "case inr.succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nk : ZMod (Nat.succ n✝)\nhk : ¬k = 0\n⊢ ↑↑(k - 1) = ↑↑k - 1",
"tactic": "rw [Fin.coe_sub_one, if_neg]"
},
{
"state_after": "case inr.succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nk : ZMod (Nat.succ n✝)\nhk : ¬k = 0\n⊢ 1 ≤ ↑k",
"state_before": "case inr.succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nk : ZMod (Nat.succ n✝)\nhk : ¬k = 0\n⊢ ↑(↑k - 1) = ↑↑k - 1",
"tactic": "rw [Nat.cast_sub, Nat.cast_one]"
},
{
"state_after": "no goals",
"state_before": "case inr.succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nk : ZMod (Nat.succ n✝)\nhk : ¬k = 0\n⊢ 1 ≤ ↑k",
"tactic": "rwa [Fin.ext_iff, Fin.val_zero, ← Ne, ← Nat.one_le_iff_ne_zero] at hk"
},
{
"state_after": "no goals",
"state_before": "case inr.succ.hnc\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nk : ZMod (Nat.succ n✝)\nhk : ¬k = 0\n⊢ ¬k = 0",
"tactic": "exact hk"
}
] |
[
526,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
AddLECancellable.tsub_lt_tsub_right_of_le
|
[
{
"state_after": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhc : AddLECancellable c\nh : c ≤ a\nh2 : a < b\n⊢ c + (a - c) < b",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhc : AddLECancellable c\nh : c ≤ a\nh2 : a < b\n⊢ a - c < b - c",
"tactic": "apply hc.lt_tsub_of_add_lt_left"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhc : AddLECancellable c\nh : c ≤ a\nh2 : a < b\n⊢ c + (a - c) < b",
"tactic": "rwa [add_tsub_cancel_of_le h]"
}
] |
[
175,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
11
] |
Mathlib/Data/List/Dedup.lean
|
List.replicate_dedup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : DecidableEq α\nx : α\nn : ℕ\nx✝ : n + 2 ≠ 0\n⊢ dedup (replicate (n + 2) x) = [x]",
"tactic": "rw [replicate_succ, dedup_cons_of_mem (mem_replicate.2 ⟨n.succ_ne_zero, rfl⟩),\n replicate_dedup n.succ_ne_zero]"
}
] |
[
106,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Topology/LocallyFinite.lean
|
locallyFinite_iff_smallSets
|
[] |
[
66,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Data/Bool/Basic.lean
|
Bool.not_and
|
[
{
"state_after": "no goals",
"state_before": "⊢ ∀ (a b : Bool), (!decide ((a && b) = (!a || !b))) = true",
"tactic": "decide"
}
] |
[
307,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
Matrix.toLin_apply
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\ninst✝⁷ : CommSemiring R\nl : Type ?u.1631986\nm : Type u_1\nn : Type u_2\ninst✝⁶ : Fintype n\ninst✝⁵ : Fintype m\ninst✝⁴ : DecidableEq n\nM₁ : Type u_5\nM₂ : Type u_4\ninst✝³ : AddCommMonoid M₁\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM : Matrix m n R\nv : M₁\n⊢ ↑(LinearEquiv.symm (Basis.equivFun v₂)) (↑(↑toLin' M) ↑(↑v₁.repr v)) = ∑ j : m, mulVec M (↑(↑v₁.repr v)) j • ↑v₂ j",
"tactic": "rw [Matrix.toLin'_apply, v₂.equivFun_symm_apply]"
}
] |
[
605,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
602,
1
] |
Mathlib/Topology/PathConnected.lean
|
Path.continuous_uncurry_iff
|
[] |
[
232,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Order/Zorn.lean
|
zorn_partialOrder₀
|
[] |
[
177,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Order/SuccPred/LinearLocallyFinite.lean
|
LinearLocallyFiniteOrder.le_succFn
|
[
{
"state_after": "ι : Type u_1\ninst✝ : LinearOrder ι\ni : ι\n⊢ ∀ (x : ι), x ∈ Set.Ioi i → i ≤ x",
"state_before": "ι : Type u_1\ninst✝ : LinearOrder ι\ni : ι\n⊢ i ≤ succFn i",
"tactic": "rw [le_isGLB_iff (succFn_spec i), mem_lowerBounds]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝ : LinearOrder ι\ni : ι\n⊢ ∀ (x : ι), x ∈ Set.Ioi i → i ≤ x",
"tactic": "exact fun x hx ↦ le_of_lt hx"
}
] |
[
78,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/FieldTheory/IsAlgClosed/Basic.lean
|
IsAlgClosed.exists_aeval_eq_zero
|
[] |
[
131,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.diam_le_of_subset_closedBall
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.538230\nι : Type ?u.538233\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nr : ℝ\nhr : 0 ≤ r\nh : s ⊆ closedBall x r\na : α\nha : a ∈ s\nb : α\nhb : b ∈ s\n⊢ r + r = 2 * r",
"tactic": "simp [mul_two, mul_comm]"
}
] |
[
2720,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2714,
1
] |
Mathlib/CategoryTheory/Over.lean
|
CategoryTheory.Over.map_obj_left
|
[] |
[
176,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_eq_one_of_forall_eq_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.210357\nι : Type ?u.210360\nG : Type ?u.210363\nM : Type u_1\nN : Type ?u.210369\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns t : Set α\nf : α → M\nh : ∀ (x : α), f x = 1\n⊢ (∏ᶠ (i : α), f i) = 1",
"tactic": "simp (config := { contextual := true }) [h]"
}
] |
[
588,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
587,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.withDensity_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1772355\nγ : Type ?u.1772358\nδ : Type ?u.1772361\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\nh : withDensity μ f = 0\n⊢ f =ᶠ[ae μ] 0",
"tactic": "rw [← lintegral_eq_zero_iff' hf, ← set_lintegral_univ, ← withDensity_apply _ MeasurableSet.univ,\n h, Measure.coe_zero, Pi.zero_apply]"
}
] |
[
1672,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1669,
1
] |
Std/Data/List/Lemmas.lean
|
List.eraseP_nil
|
[] |
[
926,
53
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
926,
9
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.coe_comap
|
[] |
[
1363,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1362,
1
] |
Mathlib/CategoryTheory/Abelian/Pseudoelements.lean
|
CategoryTheory.Abelian.Pseudoelement.pseudoZero_aux
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : Over P\nx✝ : f ≈ Over.mk 0\nR : C\np : R ⟶ f.left\nq : R ⟶ (Over.mk 0).left\nep : Epi p\nw✝ : Epi q\ncomm : p ≫ f.hom = q ≫ (Over.mk 0).hom\n⊢ p ≫ f.hom = 0",
"tactic": "simp [comm]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : Over P\nhf : f.hom = 0\n⊢ biprod.fst ≫ f.hom = biprod.snd ≫ (Over.mk 0).hom",
"tactic": "rw [hf, Over.coe_hom, HasZeroMorphisms.comp_zero, HasZeroMorphisms.comp_zero]"
}
] |
[
224,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
ContinuousOrderHom.comp_id
|
[] |
[
182,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
Sbtw.angle₃₁₂_eq_zero
|
[] |
[
356,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
MeasureTheory.uniformIntegrable_iff
|
[] |
[
919,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
913,
1
] |
Mathlib/MeasureTheory/Measure/GiryMonad.lean
|
MeasureTheory.Measure.measurable_map
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ Measurable fun b => ↑↑(map f b) s",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : Measurable f\n⊢ Measurable fun μ => map f μ",
"tactic": "refine' measurable_of_measurable_coe _ fun s hs => _"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ Measurable fun b => ↑↑b (f ⁻¹' s)",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ Measurable fun b => ↑↑(map f b) s",
"tactic": "simp_rw [map_apply hf hs]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ Measurable fun b => ↑↑b (f ⁻¹' s)",
"tactic": "exact measurable_coe (hf hs)"
}
] |
[
83,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
HasFDerivAt.cos
|
[] |
[
930,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
928,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.SimpleFunc.setToSimpleFunc_add_left
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.222342\nF : Type u_2\nF' : Type u_3\nG : Type ?u.222351\n𝕜 : Type ?u.222354\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\nT T' : Set α → F →L[ℝ] F'\nf : α →ₛ F\n⊢ ∑ x in SimpleFunc.range f, ↑(T (↑f ⁻¹' {x}) + T' (↑f ⁻¹' {x})) x =\n ∑ x in SimpleFunc.range f, ↑(T (↑f ⁻¹' {x})) x + ∑ x in SimpleFunc.range f, ↑(T' (↑f ⁻¹' {x})) x",
"state_before": "α : Type u_1\nE : Type ?u.222342\nF : Type u_2\nF' : Type u_3\nG : Type ?u.222351\n𝕜 : Type ?u.222354\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\nT T' : Set α → F →L[ℝ] F'\nf : α →ₛ F\n⊢ setToSimpleFunc (T + T') f = setToSimpleFunc T f + setToSimpleFunc T' f",
"tactic": "simp_rw [setToSimpleFunc, Pi.add_apply]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.222342\nF : Type u_2\nF' : Type u_3\nG : Type ?u.222351\n𝕜 : Type ?u.222354\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\nT T' : Set α → F →L[ℝ] F'\nf : α →ₛ F\n⊢ ∑ x in SimpleFunc.range f, (↑(T (↑f ⁻¹' {x})) + ↑(T' (↑f ⁻¹' {x}))) x =\n ∑ x in SimpleFunc.range f, ↑(T (↑f ⁻¹' {x})) x + ∑ x in SimpleFunc.range f, ↑(T' (↑f ⁻¹' {x})) x",
"state_before": "α : Type u_1\nE : Type ?u.222342\nF : Type u_2\nF' : Type u_3\nG : Type ?u.222351\n𝕜 : Type ?u.222354\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\nT T' : Set α → F →L[ℝ] F'\nf : α →ₛ F\n⊢ ∑ x in SimpleFunc.range f, ↑(T (↑f ⁻¹' {x}) + T' (↑f ⁻¹' {x})) x =\n ∑ x in SimpleFunc.range f, ↑(T (↑f ⁻¹' {x})) x + ∑ x in SimpleFunc.range f, ↑(T' (↑f ⁻¹' {x})) x",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.222342\nF : Type u_2\nF' : Type u_3\nG : Type ?u.222351\n𝕜 : Type ?u.222354\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\nT T' : Set α → F →L[ℝ] F'\nf : α →ₛ F\n⊢ ∑ x in SimpleFunc.range f, (↑(T (↑f ⁻¹' {x})) + ↑(T' (↑f ⁻¹' {x}))) x =\n ∑ x in SimpleFunc.range f, ↑(T (↑f ⁻¹' {x})) x + ∑ x in SimpleFunc.range f, ↑(T' (↑f ⁻¹' {x})) x",
"tactic": "simp_rw [Pi.add_apply, sum_add_distrib]"
}
] |
[
410,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
406,
1
] |
Mathlib/Algebra/Lie/Abelian.lean
|
LieModule.maxTrivEquiv_of_refl_eq_refl
|
[
{
"state_after": "case h.a\nR : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nm✝ : { x // x ∈ ↑(maxTrivSubmodule R L M) }\n⊢ ↑(↑(maxTrivEquiv LieModuleEquiv.refl) m✝) = ↑(↑LieModuleEquiv.refl m✝)",
"state_before": "R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\n⊢ maxTrivEquiv LieModuleEquiv.refl = LieModuleEquiv.refl",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.a\nR : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nm✝ : { x // x ∈ ↑(maxTrivSubmodule R L M) }\n⊢ ↑(↑(maxTrivEquiv LieModuleEquiv.refl) m✝) = ↑(↑LieModuleEquiv.refl m✝)",
"tactic": "simp only [coe_maxTrivEquiv_apply, LieModuleEquiv.refl_apply]"
}
] |
[
205,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
|
FreeAbelianGroup.mem_support_iff
|
[
{
"state_after": "X : Type u_1\nx : X\na : FreeAbelianGroup X\n⊢ ↑(↑toFinsupp a) x ≠ 0 ↔ ↑(coeff x) a ≠ 0",
"state_before": "X : Type u_1\nx : X\na : FreeAbelianGroup X\n⊢ x ∈ support a ↔ ↑(coeff x) a ≠ 0",
"tactic": "rw [support, Finsupp.mem_support_iff]"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nx : X\na : FreeAbelianGroup X\n⊢ ↑(↑toFinsupp a) x ≠ 0 ↔ ↑(coeff x) a ≠ 0",
"tactic": "exact Iff.rfl"
}
] |
[
158,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/RingTheory/Subsemiring/Pointwise.lean
|
Subsemiring.mem_inv_pointwise_smul_iff₀
|
[] |
[
158,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean
|
contDiffAt_euclidean
|
[
{
"state_after": "𝕜 : Type u_1\nι : Type u_3\nH : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup H\ninst✝¹ : NormedSpace 𝕜 H\ninst✝ : Fintype ι\nf : H → EuclideanSpace 𝕜 ι\nf' : H →L[𝕜] EuclideanSpace 𝕜 ι\nt : Set H\ny : H\nn : ℕ∞\n⊢ (∀ (i : ι), ContDiffAt 𝕜 n (fun x => (↑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) y) ↔\n ∀ (i : ι), ContDiffAt 𝕜 n (fun x => f x i) y",
"state_before": "𝕜 : Type u_1\nι : Type u_3\nH : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup H\ninst✝¹ : NormedSpace 𝕜 H\ninst✝ : Fintype ι\nf : H → EuclideanSpace 𝕜 ι\nf' : H →L[𝕜] EuclideanSpace 𝕜 ι\nt : Set H\ny : H\nn : ℕ∞\n⊢ ContDiffAt 𝕜 n f y ↔ ∀ (i : ι), ContDiffAt 𝕜 n (fun x => f x i) y",
"tactic": "rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffAt_iff, contDiffAt_pi]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nι : Type u_3\nH : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup H\ninst✝¹ : NormedSpace 𝕜 H\ninst✝ : Fintype ι\nf : H → EuclideanSpace 𝕜 ι\nf' : H →L[𝕜] EuclideanSpace 𝕜 ι\nt : Set H\ny : H\nn : ℕ∞\n⊢ (∀ (i : ι), ContDiffAt 𝕜 n (fun x => (↑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) y) ↔\n ∀ (i : ι), ContDiffAt 𝕜 n (fun x => f x i) y",
"tactic": "rfl"
}
] |
[
345,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
342,
1
] |
Mathlib/RingTheory/FiniteType.lean
|
MonoidAlgebra.finiteType_iff_group_fg
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nM : Type ?u.721203\ninst✝³ : CommMonoid M\nG : Type u_1\ninst✝² : CommGroup G\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\n⊢ FiniteType R (MonoidAlgebra R G) ↔ Group.FG G",
"tactic": "simpa [Group.fg_iff_monoid_fg] using finiteType_iff_fg"
}
] |
[
623,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.logb_abs
|
[
{
"state_after": "no goals",
"state_before": "b x✝ y x : ℝ\n⊢ logb b (abs x) = logb b x",
"tactic": "rw [logb, logb, log_abs]"
}
] |
[
60,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.isProbabilityMeasure_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.655475\nδ : Type ?u.655478\nι : Type ?u.655481\nR : Type ?u.655484\nR' : Type ?u.655487\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : IsProbabilityMeasure μ\nf : α → β\nhf : AEMeasurable f\n⊢ ↑↑(Measure.map f μ) univ = 1",
"tactic": "simp [map_apply_of_aemeasurable, hf]"
}
] |
[
3238,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3236,
1
] |
Mathlib/Algebra/GeomSum.lean
|
geom_sum₂_with_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Semiring α\nx : α\nn i : ℕ\nx✝ : i ∈ range n\n⊢ x ^ i * 1 ^ (n - 1 - i) = x ^ i",
"tactic": "rw [one_pow, mul_one]"
}
] |
[
103,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.reindexFinsetRange_repr
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nι' : Type ?u.435910\nR : Type u_3\nR₂ : Type ?u.435916\nK : Type ?u.435919\nM : Type u_1\nM' : Type ?u.435925\nM'' : Type ?u.435928\nV : Type u\nV' : Type ?u.435933\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq M\nx : M\ni : ι\nh : optParam (↑b i ∈ Finset.image (↑b) Finset.univ) (_ : ↑b i ∈ Finset.image (↑b) Finset.univ)\n⊢ ↑(↑(reindexFinsetRange b).repr x) { val := ↑b i, property := h } = ↑(↑b.repr x) i",
"tactic": "simp [reindexFinsetRange]"
}
] |
[
551,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
549,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.single_embedding
|
[
{
"state_after": "case h\nn✝ : ℕ\nc : Composition n✝\nn : ℕ\nh : 0 < n\ni : Fin n\n⊢ ↑(↑(embedding (single n h) 0) i) = ↑i",
"state_before": "n✝ : ℕ\nc : Composition n✝\nn : ℕ\nh : 0 < n\ni : Fin n\n⊢ ↑(embedding (single n h) 0) i = i",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nn✝ : ℕ\nc : Composition n✝\nn : ℕ\nh : 0 < n\ni : Fin n\n⊢ ↑(↑(embedding (single n h) 0) i) = ↑i",
"tactic": "simp"
}
] |
[
595,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
592,
1
] |
Mathlib/Topology/Algebra/Module/Multilinear.lean
|
ContinuousMultilinearMap.ext_iff
|
[
{
"state_after": "R : Type u\nι : Type v\nn : ℕ\nM : Fin (Nat.succ n) → Type w\nM₁ : ι → Type w₁\nM₁' : ι → Type w₁'\nM₂ : Type w₂\nM₃ : Type w₃\nM₄ : Type w₄\ninst✝¹⁸ : Semiring R\ninst✝¹⁷ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝¹⁶ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝¹⁵ : (i : ι) → AddCommMonoid (M₁' i)\ninst✝¹⁴ : AddCommMonoid M₂\ninst✝¹³ : AddCommMonoid M₃\ninst✝¹² : AddCommMonoid M₄\ninst✝¹¹ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝¹⁰ : (i : ι) → Module R (M₁ i)\ninst✝⁹ : (i : ι) → Module R (M₁' i)\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M₃\ninst✝⁶ : Module R M₄\ninst✝⁵ : (i : Fin (Nat.succ n)) → TopologicalSpace (M i)\ninst✝⁴ : (i : ι) → TopologicalSpace (M₁ i)\ninst✝³ : (i : ι) → TopologicalSpace (M₁' i)\ninst✝² : TopologicalSpace M₂\ninst✝¹ : TopologicalSpace M₃\ninst✝ : TopologicalSpace M₄\nf✝ f'✝ f f' : ContinuousMultilinearMap R M₁ M₂\n⊢ (∀ (x : (i : ι) → M₁ i), ↑f.toMultilinearMap x = ↑f'.toMultilinearMap x) ↔ ∀ (x : (i : ι) → M₁ i), ↑f x = ↑f' x",
"state_before": "R : Type u\nι : Type v\nn : ℕ\nM : Fin (Nat.succ n) → Type w\nM₁ : ι → Type w₁\nM₁' : ι → Type w₁'\nM₂ : Type w₂\nM₃ : Type w₃\nM₄ : Type w₄\ninst✝¹⁸ : Semiring R\ninst✝¹⁷ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝¹⁶ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝¹⁵ : (i : ι) → AddCommMonoid (M₁' i)\ninst✝¹⁴ : AddCommMonoid M₂\ninst✝¹³ : AddCommMonoid M₃\ninst✝¹² : AddCommMonoid M₄\ninst✝¹¹ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝¹⁰ : (i : ι) → Module R (M₁ i)\ninst✝⁹ : (i : ι) → Module R (M₁' i)\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M₃\ninst✝⁶ : Module R M₄\ninst✝⁵ : (i : Fin (Nat.succ n)) → TopologicalSpace (M i)\ninst✝⁴ : (i : ι) → TopologicalSpace (M₁ i)\ninst✝³ : (i : ι) → TopologicalSpace (M₁' i)\ninst✝² : TopologicalSpace M₂\ninst✝¹ : TopologicalSpace M₃\ninst✝ : TopologicalSpace M₄\nf✝ f'✝ f f' : ContinuousMultilinearMap R M₁ M₂\n⊢ f = f' ↔ ∀ (x : (i : ι) → M₁ i), ↑f x = ↑f' x",
"tactic": "rw [← toMultilinearMap_injective.eq_iff, MultilinearMap.ext_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type v\nn : ℕ\nM : Fin (Nat.succ n) → Type w\nM₁ : ι → Type w₁\nM₁' : ι → Type w₁'\nM₂ : Type w₂\nM₃ : Type w₃\nM₄ : Type w₄\ninst✝¹⁸ : Semiring R\ninst✝¹⁷ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝¹⁶ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝¹⁵ : (i : ι) → AddCommMonoid (M₁' i)\ninst✝¹⁴ : AddCommMonoid M₂\ninst✝¹³ : AddCommMonoid M₃\ninst✝¹² : AddCommMonoid M₄\ninst✝¹¹ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝¹⁰ : (i : ι) → Module R (M₁ i)\ninst✝⁹ : (i : ι) → Module R (M₁' i)\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M₃\ninst✝⁶ : Module R M₄\ninst✝⁵ : (i : Fin (Nat.succ n)) → TopologicalSpace (M i)\ninst✝⁴ : (i : ι) → TopologicalSpace (M₁ i)\ninst✝³ : (i : ι) → TopologicalSpace (M₁' i)\ninst✝² : TopologicalSpace M₂\ninst✝¹ : TopologicalSpace M₃\ninst✝ : TopologicalSpace M₄\nf✝ f'✝ f f' : ContinuousMultilinearMap R M₁ M₂\n⊢ (∀ (x : (i : ι) → M₁ i), ↑f.toMultilinearMap x = ↑f'.toMultilinearMap x) ↔ ∀ (x : (i : ι) → M₁ i), ↑f x = ↑f' x",
"tactic": "rfl"
}
] |
[
117,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Algebra/Squarefree.lean
|
Squarefree.of_mul_left
|
[] |
[
80,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Topology/Compactification/OnePoint.lean
|
OnePoint.isOpen_image_coe
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\ninst✝ : TopologicalSpace X\ns✝ : Set (OnePoint X)\nt s : Set X\n⊢ IsOpen (some '' s) ↔ IsOpen s",
"tactic": "rw [isOpen_iff_of_not_mem infty_not_mem_image_coe, preimage_image_eq _ coe_injective]"
}
] |
[
240,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/Logic/Relation.lean
|
Relation.TransGen.head'
|
[] |
[
376,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
|
Complex.inv_cpow
|
[
{
"state_after": "no goals",
"state_before": "x n : ℂ\nhx : arg x ≠ π\n⊢ x⁻¹ ^ n = (x ^ n)⁻¹",
"tactic": "rw [inv_cpow_eq_ite, if_neg hx]"
}
] |
[
190,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.sub_one
|
[] |
[
141,
60
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
141,
11
] |
Mathlib/Data/List/Sublists.lean
|
List.sublists'_reverse
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ sublists' (reverse l) = map reverse (sublists l)",
"tactic": "simp only [sublists_eq_sublists', map_map, map_id' reverse_reverse, Function.comp]"
}
] |
[
194,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Localization.ind
|
[] |
[
372,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.BlankRel.refl
|
[] |
[
126,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.coe_coeHom
|
[] |
[
71,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
Finset.card_le_card_biUnion_add_card_fiber
|
[
{
"state_after": "ι : Type u_1\nα : Type u_2\nβ : Type ?u.112786\nM : Type ?u.112789\nN : Type ?u.112792\nG : Type ?u.112795\nk : Type ?u.112798\nR : Type ?u.112801\ninst✝ : DecidableEq α\ns✝ : Finset α\nB : Finset (Finset α)\nn : ℕ\ns : Finset ι\nf : ι → Finset α\nhs : Set.PairwiseDisjoint (↑s) f\n⊢ card (filter (fun a => ¬f a = ∅) s) + card (filter (fun i => f i = ∅) s) ≤\n card (Finset.biUnion s f) + card (filter (fun i => f i = ∅) s)",
"state_before": "ι : Type u_1\nα : Type u_2\nβ : Type ?u.112786\nM : Type ?u.112789\nN : Type ?u.112792\nG : Type ?u.112795\nk : Type ?u.112798\nR : Type ?u.112801\ninst✝ : DecidableEq α\ns✝ : Finset α\nB : Finset (Finset α)\nn : ℕ\ns : Finset ι\nf : ι → Finset α\nhs : Set.PairwiseDisjoint (↑s) f\n⊢ card s ≤ card (Finset.biUnion s f) + card (filter (fun i => f i = ∅) s)",
"tactic": "rw [← Finset.filter_card_add_filter_neg_card_eq_card fun i ↦ f i = ∅, add_comm]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\nβ : Type ?u.112786\nM : Type ?u.112789\nN : Type ?u.112792\nG : Type ?u.112795\nk : Type ?u.112798\nR : Type ?u.112801\ninst✝ : DecidableEq α\ns✝ : Finset α\nB : Finset (Finset α)\nn : ℕ\ns : Finset ι\nf : ι → Finset α\nhs : Set.PairwiseDisjoint (↑s) f\n⊢ card (filter (fun a => ¬f a = ∅) s) + card (filter (fun i => f i = ∅) s) ≤\n card (Finset.biUnion s f) + card (filter (fun i => f i = ∅) s)",
"tactic": "exact\n add_le_add_right\n ((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi ↦\n nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|\n card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)\n _"
}
] |
[
384,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Algebra/Lie/Matrix.lean
|
Matrix.reindexLieEquiv_apply
|
[] |
[
101,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/FieldTheory/Minpoly/Field.lean
|
minpoly.sub_algebraMap
|
[
{
"state_after": "no goals",
"state_before": "A : Type u_2\nB✝ : Type ?u.356948\ninst✝⁴ : Field A\ninst✝³ : Ring B✝\ninst✝² : Algebra A B✝\nx✝ : B✝\nB : Type u_1\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\na : A\n⊢ minpoly A (x - ↑(algebraMap A B) a) = Polynomial.comp (minpoly A x) (X + ↑C a)",
"tactic": "simpa [sub_eq_add_neg] using add_algebraMap hx (-a)"
}
] |
[
163,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Order/Filter/Cofinite.lean
|
Filter.eventually_cofinite_ne
|
[] |
[
94,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/RingTheory/MvPolynomial/Ideal.lean
|
MvPolynomial.mem_ideal_span_X_image
|
[
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set σ\nthis :\n x ∈ Ideal.span ((fun s => ↑(monomial s) 1) '' ((fun i => Finsupp.single i 1) '' s)) ↔\n ∀ (xi : σ →₀ ℕ), xi ∈ support x → ∃ si, si ∈ (fun i => Finsupp.single i 1) '' s ∧ si ≤ xi\n⊢ x ∈ Ideal.span (X '' s) ↔ ∀ (m : σ →₀ ℕ), m ∈ support x → ∃ i, i ∈ s ∧ ↑m i ≠ 0",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set σ\n⊢ x ∈ Ideal.span (X '' s) ↔ ∀ (m : σ →₀ ℕ), m ∈ support x → ∃ i, i ∈ s ∧ ↑m i ≠ 0",
"tactic": "have := @mem_ideal_span_monomial_image σ R _ x ((fun i => Finsupp.single i 1) '' s)"
},
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set σ\nthis :\n x ∈ Ideal.span ((fun x => ↑(monomial (Finsupp.single x 1)) 1) '' s) ↔\n ∀ (xi : σ →₀ ℕ), xi ∈ support x → ∃ si, si ∈ (fun i => Finsupp.single i 1) '' s ∧ si ≤ xi\n⊢ x ∈ Ideal.span (X '' s) ↔ ∀ (m : σ →₀ ℕ), m ∈ support x → ∃ i, i ∈ s ∧ ↑m i ≠ 0",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set σ\nthis :\n x ∈ Ideal.span ((fun s => ↑(monomial s) 1) '' ((fun i => Finsupp.single i 1) '' s)) ↔\n ∀ (xi : σ →₀ ℕ), xi ∈ support x → ∃ si, si ∈ (fun i => Finsupp.single i 1) '' s ∧ si ≤ xi\n⊢ x ∈ Ideal.span (X '' s) ↔ ∀ (m : σ →₀ ℕ), m ∈ support x → ∃ i, i ∈ s ∧ ↑m i ≠ 0",
"tactic": "rw [Set.image_image] at this"
},
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set σ\nthis :\n x ∈ Ideal.span ((fun x => ↑(monomial (Finsupp.single x 1)) 1) '' s) ↔\n ∀ (xi : σ →₀ ℕ), xi ∈ support x → ∃ si, si ∈ (fun i => Finsupp.single i 1) '' s ∧ si ≤ xi\n⊢ (∀ (xi : σ →₀ ℕ), xi ∈ support x → ∃ si, si ∈ (fun i => Finsupp.single i 1) '' s ∧ si ≤ xi) ↔\n ∀ (m : σ →₀ ℕ), m ∈ support x → ∃ i, i ∈ s ∧ ↑m i ≠ 0",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set σ\nthis :\n x ∈ Ideal.span ((fun x => ↑(monomial (Finsupp.single x 1)) 1) '' s) ↔\n ∀ (xi : σ →₀ ℕ), xi ∈ support x → ∃ si, si ∈ (fun i => Finsupp.single i 1) '' s ∧ si ≤ xi\n⊢ x ∈ Ideal.span (X '' s) ↔ ∀ (m : σ →₀ ℕ), m ∈ support x → ∃ i, i ∈ s ∧ ↑m i ≠ 0",
"tactic": "refine' this.trans _"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set σ\nthis :\n x ∈ Ideal.span ((fun x => ↑(monomial (Finsupp.single x 1)) 1) '' s) ↔\n ∀ (xi : σ →₀ ℕ), xi ∈ support x → ∃ si, si ∈ (fun i => Finsupp.single i 1) '' s ∧ si ≤ xi\n⊢ (∀ (xi : σ →₀ ℕ), xi ∈ support x → ∃ si, si ∈ (fun i => Finsupp.single i 1) '' s ∧ si ≤ xi) ↔\n ∀ (m : σ →₀ ℕ), m ∈ support x → ∃ i, i ∈ s ∧ ↑m i ≠ 0",
"tactic": "simp [Nat.one_le_iff_ne_zero]"
}
] |
[
57,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
LinearMap.splittingOfFunOnFintypeSurjective_splits
|
[
{
"state_after": "R : Type u_2\nM : Type u_3\nN : Type ?u.781740\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nα : Type u_1\ninst✝ : Fintype α\nf : M →ₗ[R] α → R\ns : Surjective ↑f\nx y : α\n⊢ ↑(comp (comp f (splittingOfFunOnFintypeSurjective f s)) (single x)) 1 y = ↑(comp id (single x)) 1 y",
"state_before": "R : Type u_2\nM : Type u_3\nN : Type ?u.781740\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nα : Type u_1\ninst✝ : Fintype α\nf : M →ₗ[R] α → R\ns : Surjective ↑f\n⊢ comp f (splittingOfFunOnFintypeSurjective f s) = id",
"tactic": "refine pi_ext' fun x => ext_ring <| funext fun y => ?_"
},
{
"state_after": "R : Type u_2\nM : Type u_3\nN : Type ?u.781740\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nα : Type u_1\ninst✝ : Fintype α\nf : M →ₗ[R] α → R\ns : Surjective ↑f\nx y : α\n⊢ ↑f\n (sum (↑(LinearEquiv.symm (linearEquivFunOnFinite R R α)) (Pi.single x 1)) fun x r =>\n r • Exists.choose (_ : ∃ a, ↑f a = ↑(Finsupp.single x 1)))\n y =\n Pi.single x 1 y",
"state_before": "R : Type u_2\nM : Type u_3\nN : Type ?u.781740\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nα : Type u_1\ninst✝ : Fintype α\nf : M →ₗ[R] α → R\ns : Surjective ↑f\nx y : α\n⊢ ↑(comp (comp f (splittingOfFunOnFintypeSurjective f s)) (single x)) 1 y = ↑(comp id (single x)) 1 y",
"tactic": "dsimp [splittingOfFunOnFintypeSurjective]"
},
{
"state_after": "R : Type u_2\nM : Type u_3\nN : Type ?u.781740\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nα : Type u_1\ninst✝ : Fintype α\nf : M →ₗ[R] α → R\ns : Surjective ↑f\nx y : α\n⊢ 0 • Exists.choose (_ : ∃ a, ↑f a = ↑(Finsupp.single x 1)) = 0",
"state_before": "R : Type u_2\nM : Type u_3\nN : Type ?u.781740\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nα : Type u_1\ninst✝ : Fintype α\nf : M →ₗ[R] α → R\ns : Surjective ↑f\nx y : α\n⊢ ↑f\n (sum (↑(LinearEquiv.symm (linearEquivFunOnFinite R R α)) (Pi.single x 1)) fun x r =>\n r • Exists.choose (_ : ∃ a, ↑f a = ↑(Finsupp.single x 1)))\n y =\n Pi.single x 1 y",
"tactic": "rw [linearEquivFunOnFinite_symm_single, Finsupp.sum_single_index, one_smul,\n (s (Finsupp.single x 1)).choose_spec, Finsupp.single_eq_pi_single]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nM : Type u_3\nN : Type ?u.781740\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nα : Type u_1\ninst✝ : Fintype α\nf : M →ₗ[R] α → R\ns : Surjective ↑f\nx y : α\n⊢ 0 • Exists.choose (_ : ∃ a, ↑f a = ↑(Finsupp.single x 1)) = 0",
"tactic": "rw [zero_smul]"
}
] |
[
1257,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1250,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.coe_sInf
|
[] |
[
728,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
727,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.prod_coe
|
[] |
[
772,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
771,
1
] |
Mathlib/Order/PFilter.lean
|
Order.PFilter.mem_principal
|
[] |
[
124,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
9
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsLittleO.mono
|
[] |
[
456,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/Algebra/Support.lean
|
Function.support_nat_cast
|
[] |
[
325,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Topology/Order.lean
|
continuous_top
|
[] |
[
815,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
814,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
InfTopHom.comp_id
|
[] |
[
968,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
968,
9
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
StrictMono.div_const
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.114417\nα : Type u_2\nβ✝ : Type ?u.114423\ninst✝¹ : LinearOrderedSemifield α\na b c✝ d e : α\nm n : ℤ\nβ : Type u_1\ninst✝ : Preorder β\nf : β → α\nhf : StrictMono f\nc : α\nhc : 0 < c\n⊢ StrictMono fun x => f x / c",
"tactic": "simpa only [div_eq_mul_inv] using hf.mul_const (inv_pos.2 hc)"
}
] |
[
587,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
585,
1
] |
Mathlib/Data/Bool/Count.lean
|
List.Chain.count_not
|
[
{
"state_after": "x : Bool\nl : List Bool\nh : Chain (fun x x_1 => x ≠ x_1) (!x) (x :: l)\n⊢ count (!!x) (x :: l) = count (!x) (x :: l) + length (x :: l) % 2",
"state_before": "b x : Bool\nl : List Bool\nh : Chain (fun x x_1 => x ≠ x_1) b (x :: l)\n⊢ count (!b) (x :: l) = count b (x :: l) + length (x :: l) % 2",
"tactic": "obtain rfl : b = !x := Bool.eq_not_iff.2 (rel_of_chain_cons h)"
},
{
"state_after": "no goals",
"state_before": "x : Bool\nl : List Bool\nh : Chain (fun x x_1 => x ≠ x_1) (!x) (x :: l)\n⊢ count (!!x) (x :: l) = count (!x) (x :: l) + length (x :: l) % 2",
"tactic": "rw [Bool.not_not, count_cons_self, count_cons_of_ne x.not_ne_self,\n Chain.count_not (chain_of_chain_cons h), length, add_assoc, Nat.mod_two_add_succ_mod_two]"
}
] |
[
56,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
convex_Ioi
|
[] |
[
291,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
iteratedFDerivWithin_succ_apply_left
|
[] |
[
799,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
796,
1
] |
Mathlib/Data/Polynomial/Mirror.lean
|
Polynomial.mirror_natDegree
|
[
{
"state_after": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nhp : p = 0\n⊢ natDegree (mirror p) = natDegree p\n\ncase neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nhp : ¬p = 0\n⊢ natDegree (mirror p) = natDegree p",
"state_before": "R : Type u_1\ninst✝ : Semiring R\np q : R[X]\n⊢ natDegree (mirror p) = natDegree p",
"tactic": "by_cases hp : p = 0"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\np q : R[X]\nhp : ¬p = 0\n✝ : Nontrivial R\n⊢ natDegree (mirror p) = natDegree p",
"state_before": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nhp : ¬p = 0\n⊢ natDegree (mirror p) = natDegree p",
"tactic": "nontriviality R"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\np q : R[X]\nhp : ¬p = 0\n✝ : Nontrivial R\n⊢ leadingCoeff (reverse p) * leadingCoeff (X ^ natTrailingDegree p) ≠ 0",
"state_before": "R : Type u_1\ninst✝ : Semiring R\np q : R[X]\nhp : ¬p = 0\n✝ : Nontrivial R\n⊢ natDegree (mirror p) = natDegree p",
"tactic": "rw [mirror, natDegree_mul', reverse_natDegree, natDegree_X_pow,\n tsub_add_cancel_of_le p.natTrailingDegree_le_natDegree]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\np q : R[X]\nhp : ¬p = 0\n✝ : Nontrivial R\n⊢ leadingCoeff (reverse p) * leadingCoeff (X ^ natTrailingDegree p) ≠ 0",
"tactic": "rwa [leadingCoeff_X_pow, mul_one, reverse_leadingCoeff, Ne, trailingCoeff_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nhp : p = 0\n⊢ natDegree (mirror p) = natDegree p",
"tactic": "rw [hp, mirror_zero]"
}
] |
[
75,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.sup_singleton''
|
[
{
"state_after": "case a\nF : Type ?u.416774\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.416783\nι : Type ?u.416786\nκ : Type ?u.416789\ninst✝ : DecidableEq α\ns : Finset β\nf : β → α\na : α\n⊢ (a ∈ sup s fun b => {f b}) ↔ a ∈ image f s",
"state_before": "F : Type ?u.416774\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.416783\nι : Type ?u.416786\nκ : Type ?u.416789\ninst✝ : DecidableEq α\ns : Finset β\nf : β → α\n⊢ (sup s fun b => {f b}) = image f s",
"tactic": "ext a"
},
{
"state_after": "case a\nF : Type ?u.416774\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.416783\nι : Type ?u.416786\nκ : Type ?u.416789\ninst✝ : DecidableEq α\ns : Finset β\nf : β → α\na : α\n⊢ (∃ v, v ∈ s ∧ a ∈ {f v}) ↔ ∃ a_1, a_1 ∈ s ∧ f a_1 = a",
"state_before": "case a\nF : Type ?u.416774\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.416783\nι : Type ?u.416786\nκ : Type ?u.416789\ninst✝ : DecidableEq α\ns : Finset β\nf : β → α\na : α\n⊢ (a ∈ sup s fun b => {f b}) ↔ a ∈ image f s",
"tactic": "rw [mem_sup, mem_image]"
},
{
"state_after": "no goals",
"state_before": "case a\nF : Type ?u.416774\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.416783\nι : Type ?u.416786\nκ : Type ?u.416789\ninst✝ : DecidableEq α\ns : Finset β\nf : β → α\na : α\n⊢ (∃ v, v ∈ s ∧ a ∈ {f v}) ↔ ∃ a_1, a_1 ∈ s ∧ f a_1 = a",
"tactic": "simp only [mem_singleton, eq_comm]"
}
] |
[
1798,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1794,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.not_nonempty_iff_eq_empty
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns✝ s₁ s₂ t t₁ t₂ u s : Set α\n⊢ ¬Set.Nonempty s ↔ s = ∅",
"tactic": "simp only [Set.Nonempty, not_exists, eq_empty_iff_forall_not_mem]"
}
] |
[
604,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
603,
1
] |
Mathlib/Data/Real/Pi/Wallis.lean
|
Real.tendsto_prod_pi_div_two
|
[] |
[
130,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/RingTheory/Noetherian.lean
|
isNoetherian_of_injective
|
[] |
[
147,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean
|
MeasureTheory.map_mul_right_ae
|
[] |
[
308,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Topology/Order/Basic.lean
|
mem_nhdsWithin_Ici_iff_exists_Ico_subset'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na u' : α\ns : Set α\nhu' : a < u'\n⊢ List.get?\n [s ∈ 𝓝[Ici a] a, s ∈ 𝓝[Icc a u'] a, s ∈ 𝓝[Ico a u'] a, ∃ u, u ∈ Ioc a u' ∧ Ico a u ⊆ s,\n ∃ u, u ∈ Ioi a ∧ Ico a u ⊆ s]\n 0 =\n some (s ∈ 𝓝[Ici a] a)",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na u' : α\ns : Set α\nhu' : a < u'\n⊢ List.get?\n [s ∈ 𝓝[Ici a] a, s ∈ 𝓝[Icc a u'] a, s ∈ 𝓝[Ico a u'] a, ∃ u, u ∈ Ioc a u' ∧ Ico a u ⊆ s,\n ∃ u, u ∈ Ioi a ∧ Ico a u ⊆ s]\n 4 =\n some (∃ u, u ∈ Ioi a ∧ Ico a u ⊆ s)",
"tactic": "norm_num"
}
] |
[
1765,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1763,
1
] |
Mathlib/Order/SymmDiff.lean
|
bot_symmDiff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.18813\nα : Type u_1\nβ : Type ?u.18819\nπ : ι → Type ?u.18824\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\n⊢ ⊥ ∆ a = a",
"tactic": "rw [symmDiff_comm, symmDiff_bot]"
}
] |
[
133,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Std/Data/List/Lemmas.lean
|
List.dropLast_nil
|
[] |
[
485,
58
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
485,
9
] |
Mathlib/Topology/Sequences.lean
|
tendsto_nhds_iff_seq_tendsto
|
[
{
"state_after": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : FrechetUrysohnSpace X\nf : X → Y\na : X\nb : Y\nh : ∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b)\n⊢ ∀ (ib : Set Y), ¬b ∈ ib ∧ IsClosed ib → ∃ ia, (¬a ∈ ia ∧ IsClosed ia) ∧ ∀ (x : X), x ∈ iaᶜ → f x ∈ ibᶜ",
"state_before": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : FrechetUrysohnSpace X\nf : X → Y\na : X\nb : Y\n⊢ Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b)",
"tactic": "refine'\n ⟨fun hf u hu => hf.comp hu, fun h =>\n ((nhds_basis_closeds _).tendsto_iff (nhds_basis_closeds _)).2 _⟩"
},
{
"state_after": "case intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : FrechetUrysohnSpace X\nf : X → Y\na : X\nb : Y\nh : ∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b)\ns : Set Y\nhbs : ¬b ∈ s\nhsc : IsClosed s\n⊢ ∃ ia, (¬a ∈ ia ∧ IsClosed ia) ∧ ∀ (x : X), x ∈ iaᶜ → f x ∈ sᶜ",
"state_before": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : FrechetUrysohnSpace X\nf : X → Y\na : X\nb : Y\nh : ∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b)\n⊢ ∀ (ib : Set Y), ¬b ∈ ib ∧ IsClosed ib → ∃ ia, (¬a ∈ ia ∧ IsClosed ia) ∧ ∀ (x : X), x ∈ iaᶜ → f x ∈ ibᶜ",
"tactic": "rintro s ⟨hbs, hsc⟩"
},
{
"state_after": "case intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : FrechetUrysohnSpace X\nf : X → Y\na : X\nb : Y\nh : ∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b)\ns : Set Y\nhbs : ¬b ∈ s\nhsc : IsClosed s\n⊢ a ∈ closure (f ⁻¹' s) → b ∈ s",
"state_before": "case intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : FrechetUrysohnSpace X\nf : X → Y\na : X\nb : Y\nh : ∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b)\ns : Set Y\nhbs : ¬b ∈ s\nhsc : IsClosed s\n⊢ ∃ ia, (¬a ∈ ia ∧ IsClosed ia) ∧ ∀ (x : X), x ∈ iaᶜ → f x ∈ sᶜ",
"tactic": "refine' ⟨closure (f ⁻¹' s), ⟨mt _ hbs, isClosed_closure⟩, fun x => mt fun hx => subset_closure hx⟩"
},
{
"state_after": "case intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : FrechetUrysohnSpace X\nf : X → Y\na : X\nb : Y\nh : ∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b)\ns : Set Y\nhbs : ¬b ∈ s\nhsc : IsClosed s\n⊢ a ∈ seqClosure (f ⁻¹' s) → b ∈ s",
"state_before": "case intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : FrechetUrysohnSpace X\nf : X → Y\na : X\nb : Y\nh : ∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b)\ns : Set Y\nhbs : ¬b ∈ s\nhsc : IsClosed s\n⊢ a ∈ closure (f ⁻¹' s) → b ∈ s",
"tactic": "rw [← seqClosure_eq_closure]"
},
{
"state_after": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : FrechetUrysohnSpace X\nf : X → Y\na : X\nb : Y\nh : ∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b)\ns : Set Y\nhbs : ¬b ∈ s\nhsc : IsClosed s\nu : ℕ → X\nhus : ∀ (n : ℕ), u n ∈ f ⁻¹' s\nhu : Tendsto u atTop (𝓝 a)\n⊢ b ∈ s",
"state_before": "case intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : FrechetUrysohnSpace X\nf : X → Y\na : X\nb : Y\nh : ∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b)\ns : Set Y\nhbs : ¬b ∈ s\nhsc : IsClosed s\n⊢ a ∈ seqClosure (f ⁻¹' s) → b ∈ s",
"tactic": "rintro ⟨u, hus, hu⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : FrechetUrysohnSpace X\nf : X → Y\na : X\nb : Y\nh : ∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b)\ns : Set Y\nhbs : ¬b ∈ s\nhsc : IsClosed s\nu : ℕ → X\nhus : ∀ (n : ℕ), u n ∈ f ⁻¹' s\nhu : Tendsto u atTop (𝓝 a)\n⊢ b ∈ s",
"tactic": "exact hsc.mem_of_tendsto (h u hu) (eventually_of_forall hus)"
}
] |
[
155,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.