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/MeasureTheory/MeasurableSpaceDef.lean
|
MeasurableSet.of_compl
|
[] |
[
95,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
11
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.IsNormal.apply_lt_nfp
|
[
{
"state_after": "f✝ : Ordinal → Ordinal\nf : Ordinal → Ordinal\nH : IsNormal f\na b : Ordinal\n⊢ f b < nfpFamily (fun x => f) a ↔ b < nfpFamily (fun x => f) a",
"state_before": "f✝ : Ordinal → Ordinal\nf : Ordinal → Ordinal\nH : IsNormal f\na b : Ordinal\n⊢ f b < nfp f a ↔ b < nfp f a",
"tactic": "unfold nfp"
},
{
"state_after": "f✝ : Ordinal → Ordinal\nf : Ordinal → Ordinal\nH : IsNormal f\na b : Ordinal\n⊢ f b < nfpFamily (fun x => f) a ↔ Unit → f b < nfpFamily (fun x => f) a",
"state_before": "f✝ : Ordinal → Ordinal\nf : Ordinal → Ordinal\nH : IsNormal f\na b : Ordinal\n⊢ f b < nfpFamily (fun x => f) a ↔ b < nfpFamily (fun x => f) a",
"tactic": "rw [← @apply_lt_nfpFamily_iff Unit (fun _ => f) _ (fun _ => H) a b]"
},
{
"state_after": "no goals",
"state_before": "f✝ : Ordinal → Ordinal\nf : Ordinal → Ordinal\nH : IsNormal f\na b : Ordinal\n⊢ f b < nfpFamily (fun x => f) a ↔ Unit → f b < nfpFamily (fun x => f) a",
"tactic": "exact ⟨fun h _ => h, fun h => h Unit.unit⟩"
}
] |
[
475,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.preimage_inr_range_inl
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.85765\nι : Sort ?u.85768\nι' : Sort ?u.85771\nf : ι → α\ns t : Set α\n⊢ Sum.inr ⁻¹' range Sum.inl = ∅",
"tactic": "rw [← image_univ, preimage_inr_image_inl]"
}
] |
[
936,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
935,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.nonempty_ball
|
[] |
[
434,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.biInter_and'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.92502\nγ : Type ?u.92505\nι : Sort u_2\nι' : Sort u_3\nι₂ : Sort ?u.92514\nκ : ι → Sort ?u.92519\nκ₁ : ι → Sort ?u.92524\nκ₂ : ι → Sort ?u.92529\nκ' : ι' → Sort ?u.92534\np : ι' → Prop\nq : ι → ι' → Prop\ns : (x : ι) → (y : ι') → p y ∧ q x y → Set α\n⊢ (⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h) = ⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y (_ : p y ∧ q x y)",
"tactic": "simp only [iInter_and, @iInter_comm _ ι]"
}
] |
[
831,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
828,
1
] |
Mathlib/Algebra/Order/AbsoluteValue.lean
|
AbsoluteValue.map_one_of_isLeftRegular
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nS : Type u_1\ninst✝¹ : Semiring R\ninst✝ : OrderedSemiring S\nabv : AbsoluteValue R S\nh : IsLeftRegular (↑abv 1)\n⊢ (fun x => ↑abv 1 * x) (↑abv 1) = (fun x => ↑abv 1 * x) 1",
"tactic": "simp [← map_mul]"
}
] |
[
140,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Computability/Language.lean
|
Language.map_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nl✝ m : Language α\na b x : List α\ng : β → γ\nf : α → β\nl : Language α\n⊢ ↑(map g) (↑(map f) l) = ↑(map (g ∘ f)) l",
"tactic": "simp [map, image_image]"
}
] |
[
176,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Analysis/Complex/Arg.lean
|
Complex.sameRay_iff_arg_div_eq_zero
|
[
{
"state_after": "x y : ℂ\n⊢ x = 0 ∨ y = 0 ∨ arg x = arg y ↔ ↑(arg (x / y)) = 0",
"state_before": "x y : ℂ\n⊢ SameRay ℝ x y ↔ arg (x / y) = 0",
"tactic": "rw [← Real.Angle.toReal_zero, ← arg_coe_angle_eq_iff_eq_toReal, sameRay_iff]"
},
{
"state_after": "case pos\nx y : ℂ\nhx : x = 0\n⊢ x = 0 ∨ y = 0 ∨ arg x = arg y ↔ ↑(arg (x / y)) = 0\n\ncase neg\nx y : ℂ\nhx : ¬x = 0\n⊢ x = 0 ∨ y = 0 ∨ arg x = arg y ↔ ↑(arg (x / y)) = 0",
"state_before": "x y : ℂ\n⊢ x = 0 ∨ y = 0 ∨ arg x = arg y ↔ ↑(arg (x / y)) = 0",
"tactic": "by_cases hx : x = 0"
},
{
"state_after": "case pos\nx y : ℂ\nhx : ¬x = 0\nhy : y = 0\n⊢ x = 0 ∨ y = 0 ∨ arg x = arg y ↔ ↑(arg (x / y)) = 0\n\ncase neg\nx y : ℂ\nhx : ¬x = 0\nhy : ¬y = 0\n⊢ x = 0 ∨ y = 0 ∨ arg x = arg y ↔ ↑(arg (x / y)) = 0",
"state_before": "case neg\nx y : ℂ\nhx : ¬x = 0\n⊢ x = 0 ∨ y = 0 ∨ arg x = arg y ↔ ↑(arg (x / y)) = 0",
"tactic": "by_cases hy : y = 0"
},
{
"state_after": "no goals",
"state_before": "case neg\nx y : ℂ\nhx : ¬x = 0\nhy : ¬y = 0\n⊢ x = 0 ∨ y = 0 ∨ arg x = arg y ↔ ↑(arg (x / y)) = 0",
"tactic": "simp [hx, hy, arg_div_coe_angle, sub_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case pos\nx y : ℂ\nhx : x = 0\n⊢ x = 0 ∨ y = 0 ∨ arg x = arg y ↔ ↑(arg (x / y)) = 0",
"tactic": "simp [hx]"
},
{
"state_after": "no goals",
"state_before": "case pos\nx y : ℂ\nhx : ¬x = 0\nhy : y = 0\n⊢ x = 0 ∨ y = 0 ∨ arg x = arg y ↔ ↑(arg (x / y)) = 0",
"tactic": "simp [hy]"
}
] |
[
48,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
rank_fun
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV✝ V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη✝ : Type u₁'\nφ : η✝ → Type ?u.526682\ninst✝¹⁸ : Ring K\ninst✝¹⁷ : StrongRankCondition K\ninst✝¹⁶ : AddCommGroup V✝\ninst✝¹⁵ : Module K V✝\ninst✝¹⁴ : Module.Free K V✝\ninst✝¹³ : AddCommGroup V'\ninst✝¹² : Module K V'\ninst✝¹¹ : Module.Free K V'\ninst✝¹⁰ : AddCommGroup V₁\ninst✝⁹ : Module K V₁\ninst✝⁸ : Module.Free K V₁\ninst✝⁷ : (i : η✝) → AddCommGroup (φ i)\ninst✝⁶ : (i : η✝) → Module K (φ i)\ninst✝⁵ : ∀ (i : η✝), Module.Free K (φ i)\ninst✝⁴ : Fintype η✝\nV η : Type u\ninst✝³ : Fintype η\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Free K V\n⊢ Module.rank K (η → V) = ↑(Fintype.card η) * Module.rank K V",
"tactic": "rw [rank_pi, Cardinal.sum_const', Cardinal.mk_fintype]"
}
] |
[
1019,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1017,
1
] |
Mathlib/Data/Rat/Lemmas.lean
|
Rat.mul_den_eq_num
|
[
{
"state_after": "q : ℚ\n⊢ q.num /. ↑q.den * (↑q.den /. 1) = q.num /. 1",
"state_before": "q : ℚ\n⊢ q * ↑q.den = ↑q.num",
"tactic": "suffices (q.num /. ↑q.den) * (↑q.den /. 1) = q.num /. 1 by\n conv => pattern (occs := 1)q ; (rw [← @num_den q])\n simp only [coe_int_eq_divInt, coe_nat_eq_divInt, num_den] at this ⊢; assumption"
},
{
"state_after": "q : ℚ\nthis : ↑q.den ≠ 0\n⊢ q.num /. ↑q.den * (↑q.den /. 1) = q.num /. 1",
"state_before": "q : ℚ\n⊢ q.num /. ↑q.den * (↑q.den /. 1) = q.num /. 1",
"tactic": "have : (q.den : ℤ) ≠ 0 := ne_of_gt (by exact_mod_cast q.pos)"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nthis : ↑q.den ≠ 0\n⊢ q.num /. ↑q.den * (↑q.den /. 1) = q.num /. 1",
"tactic": "rw [Rat.mul_def' this one_ne_zero, mul_comm (q.den : ℤ) 1, divInt_mul_right this]"
},
{
"state_after": "q : ℚ\nthis : q.num /. ↑q.den * (↑q.den /. 1) = q.num /. 1\n⊢ q.num /. ↑q.den * ↑q.den = ↑q.num",
"state_before": "q : ℚ\nthis : q.num /. ↑q.den * (↑q.den /. 1) = q.num /. 1\n⊢ q * ↑q.den = ↑q.num",
"tactic": "conv => pattern (occs := 1)q ; (rw [← @num_den q])"
},
{
"state_after": "q : ℚ\nthis : q * (↑q.den /. 1) = q.num /. 1\n⊢ q * (↑q.den /. 1) = q.num /. 1",
"state_before": "q : ℚ\nthis : q.num /. ↑q.den * (↑q.den /. 1) = q.num /. 1\n⊢ q.num /. ↑q.den * ↑q.den = ↑q.num",
"tactic": "simp only [coe_int_eq_divInt, coe_nat_eq_divInt, num_den] at this ⊢"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nthis : q * (↑q.den /. 1) = q.num /. 1\n⊢ q * (↑q.den /. 1) = q.num /. 1",
"tactic": "assumption"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\n⊢ 0 < ↑q.den",
"tactic": "exact_mod_cast q.pos"
}
] |
[
194,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
totallyBounded_iff_ultrafilter
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nH : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → Cauchy ↑f\n⊢ ∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c, c ≤ f ∧ Cauchy c",
"state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\n⊢ TotallyBounded s ↔ ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → Cauchy ↑f",
"tactic": "refine' ⟨fun hs f => f.cauchy_of_totallyBounded hs, fun H => totallyBounded_iff_filter.2 _⟩"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nH : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → Cauchy ↑f\nf : Filter α\nhf : NeBot f\nhfs : f ≤ 𝓟 s\n⊢ ∃ c, c ≤ f ∧ Cauchy c",
"state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nH : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → Cauchy ↑f\n⊢ ∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c, c ≤ f ∧ Cauchy c",
"tactic": "intro f hf hfs"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nH : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → Cauchy ↑f\nf : Filter α\nhf : NeBot f\nhfs : f ≤ 𝓟 s\n⊢ ∃ c, c ≤ f ∧ Cauchy c",
"tactic": "exact ⟨Ultrafilter.of f, Ultrafilter.of_le f, H _ ((Ultrafilter.of_le f).trans hfs)⟩"
}
] |
[
572,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
568,
1
] |
Mathlib/Data/PFunctor/Univariate/M.lean
|
PFunctor.M.children_mk
|
[
{
"state_after": "no goals",
"state_before": "F : PFunctor\nX : Type ?u.26665\nf : X → Obj F X\na : F.A\nx : B F a → M F\ni : B F (head (M.mk { fst := a, snd := x }))\n⊢ B F (head (M.mk { fst := a, snd := x })) = B F a",
"tactic": "rw [head_mk]"
},
{
"state_after": "case H\nF : PFunctor\nX : Type ?u.26665\nf : X → Obj F X\na : F.A\nx : B F a → M F\ni : B F (head (M.mk { fst := a, snd := x }))\n⊢ ∀ (i_1 : ℕ),\n MIntl.approx (children (M.mk { fst := a, snd := x }) i) i_1 =\n MIntl.approx (x (cast (_ : B F (head (M.mk { fst := a, snd := x })) = B F a) i)) i_1",
"state_before": "F : PFunctor\nX : Type ?u.26665\nf : X → Obj F X\na : F.A\nx : B F a → M F\ni : B F (head (M.mk { fst := a, snd := x }))\n⊢ children (M.mk { fst := a, snd := x }) i = x (cast (_ : B F (head (M.mk { fst := a, snd := x })) = B F a) i)",
"tactic": "apply ext'"
},
{
"state_after": "case H\nF : PFunctor\nX : Type ?u.26665\nf : X → Obj F X\na : F.A\nx : B F a → M F\ni : B F (head (M.mk { fst := a, snd := x }))\nn : ℕ\n⊢ MIntl.approx (children (M.mk { fst := a, snd := x }) i) n =\n MIntl.approx (x (cast (_ : B F (head (M.mk { fst := a, snd := x })) = B F a) i)) n",
"state_before": "case H\nF : PFunctor\nX : Type ?u.26665\nf : X → Obj F X\na : F.A\nx : B F a → M F\ni : B F (head (M.mk { fst := a, snd := x }))\n⊢ ∀ (i_1 : ℕ),\n MIntl.approx (children (M.mk { fst := a, snd := x }) i) i_1 =\n MIntl.approx (x (cast (_ : B F (head (M.mk { fst := a, snd := x })) = B F a) i)) i_1",
"tactic": "intro n"
},
{
"state_after": "no goals",
"state_before": "case H\nF : PFunctor\nX : Type ?u.26665\nf : X → Obj F X\na : F.A\nx : B F a → M F\ni : B F (head (M.mk { fst := a, snd := x }))\nn : ℕ\n⊢ MIntl.approx (children (M.mk { fst := a, snd := x }) i) n =\n MIntl.approx (x (cast (_ : B F (head (M.mk { fst := a, snd := x })) = B F a) i)) n",
"tactic": "rfl"
}
] |
[
547,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
546,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.symm_trans_apply
|
[] |
[
393,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
391,
1
] |
Mathlib/Data/List/Basic.lean
|
List.zipLeft_nil_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.476599\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nas : List α\nb : β\nbs : List β\n⊢ zipLeft as [] = map (fun a => (a, none)) as",
"tactic": "cases as <;> rfl"
}
] |
[
4176,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4175,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.vsub_le_vsub_left
|
[] |
[
1148,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1147,
1
] |
Mathlib/Data/Set/Intervals/ProjIcc.lean
|
Set.projIcc_eq_left
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.3590\ninst✝ : LinearOrder α\na b : α\nh✝ : a ≤ b\nx : α\nh : a < b\nh' : projIcc a b (_ : a ≤ b) x = { val := a, property := (_ : a ∈ Icc a b) }\n⊢ x ≤ a",
"state_before": "α : Type u_1\nβ : Type ?u.3590\ninst✝ : LinearOrder α\na b : α\nh✝ : a ≤ b\nx : α\nh : a < b\n⊢ projIcc a b (_ : a ≤ b) x = { val := a, property := (_ : a ∈ Icc a b) } ↔ x ≤ a",
"tactic": "refine' ⟨fun h' => _, projIcc_of_le_left _⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.3590\ninst✝ : LinearOrder α\na b : α\nh✝ : a ≤ b\nx : α\nh : a < b\nh' : x ≤ a\n⊢ x ≤ a",
"state_before": "α : Type u_1\nβ : Type ?u.3590\ninst✝ : LinearOrder α\na b : α\nh✝ : a ≤ b\nx : α\nh : a < b\nh' : projIcc a b (_ : a ≤ b) x = { val := a, property := (_ : a ∈ Icc a b) }\n⊢ x ≤ a",
"tactic": "simp_rw [Subtype.ext_iff_val, projIcc, max_eq_left_iff, min_le_iff, h.not_le, false_or_iff] at h'"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3590\ninst✝ : LinearOrder α\na b : α\nh✝ : a ≤ b\nx : α\nh : a < b\nh' : x ≤ a\n⊢ x ≤ a",
"tactic": "exact h'"
}
] |
[
62,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
UpperSemicontinuous.upperSemicontinuousWithinAt
|
[] |
[
705,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
703,
1
] |
Mathlib/Order/Filter/Pi.lean
|
Filter.map_pi_map_coprodᵢ_le
|
[
{
"state_after": "ι : Type u_1\nα : ι → Type u_3\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\nβ : ι → Type u_2\nm : (i : ι) → α i → β i\n⊢ ∀ (x : Set ((i : ι) → β i)),\n (∀ (i : ι), ∃ t₁, m i ⁻¹' t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ x) →\n ∀ (i : ι), ∃ t₁, t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ (fun k i => m i (k i)) ⁻¹' x",
"state_before": "ι : Type u_1\nα : ι → Type u_3\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\nβ : ι → Type u_2\nm : (i : ι) → α i → β i\n⊢ map (fun k i => m i (k i)) (Filter.coprodᵢ f) ≤ Filter.coprodᵢ fun i => map (m i) (f i)",
"tactic": "simp only [le_def, mem_map, mem_coprodᵢ_iff]"
},
{
"state_after": "ι : Type u_1\nα : ι → Type u_3\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\nβ : ι → Type u_2\nm : (i : ι) → α i → β i\ns : Set ((i : ι) → β i)\nh : ∀ (i : ι), ∃ t₁, m i ⁻¹' t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ s\ni : ι\n⊢ ∃ t₁, t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ (fun k i => m i (k i)) ⁻¹' s",
"state_before": "ι : Type u_1\nα : ι → Type u_3\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\nβ : ι → Type u_2\nm : (i : ι) → α i → β i\n⊢ ∀ (x : Set ((i : ι) → β i)),\n (∀ (i : ι), ∃ t₁, m i ⁻¹' t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ x) →\n ∀ (i : ι), ∃ t₁, t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ (fun k i => m i (k i)) ⁻¹' x",
"tactic": "intro s h i"
},
{
"state_after": "case intro.intro\nι : Type u_1\nα : ι → Type u_3\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\nβ : ι → Type u_2\nm : (i : ι) → α i → β i\ns : Set ((i : ι) → β i)\nh : ∀ (i : ι), ∃ t₁, m i ⁻¹' t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ s\ni : ι\nt : Set (β i)\nH : m i ⁻¹' t ∈ f i\nhH : eval i ⁻¹' t ⊆ s\n⊢ ∃ t₁, t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ (fun k i => m i (k i)) ⁻¹' s",
"state_before": "ι : Type u_1\nα : ι → Type u_3\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\nβ : ι → Type u_2\nm : (i : ι) → α i → β i\ns : Set ((i : ι) → β i)\nh : ∀ (i : ι), ∃ t₁, m i ⁻¹' t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ s\ni : ι\n⊢ ∃ t₁, t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ (fun k i => m i (k i)) ⁻¹' s",
"tactic": "obtain ⟨t, H, hH⟩ := h i"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nι : Type u_1\nα : ι → Type u_3\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\nβ : ι → Type u_2\nm : (i : ι) → α i → β i\ns : Set ((i : ι) → β i)\nh : ∀ (i : ι), ∃ t₁, m i ⁻¹' t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ s\ni : ι\nt : Set (β i)\nH : m i ⁻¹' t ∈ f i\nhH : eval i ⁻¹' t ⊆ s\n⊢ ∃ t₁, t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ (fun k i => m i (k i)) ⁻¹' s",
"tactic": "exact ⟨{ x : α i | m i x ∈ t }, H, fun x hx => hH hx⟩"
}
] |
[
265,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
RingHom.ker_eq
|
[] |
[
2005,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2004,
1
] |
Mathlib/Topology/Order.lean
|
continuous_id_iff_le
|
[] |
[
819,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
818,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.map_copy
|
[
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu' v' : V\np : Walk G u' v'\n⊢ Walk.map f (Walk.copy p (_ : u' = u') (_ : v' = v')) =\n Walk.copy (Walk.map f p) (_ : ↑f u' = ↑f u') (_ : ↑f v' = ↑f v')",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v u' v' : V\np : Walk G u v\nhu : u = u'\nhv : v = v'\n⊢ Walk.map f (Walk.copy p hu hv) = Walk.copy (Walk.map f p) (_ : ↑f u = ↑f u') (_ : ↑f v = ↑f v')",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu' v' : V\np : Walk G u' v'\n⊢ Walk.map f (Walk.copy p (_ : u' = u') (_ : v' = v')) =\n Walk.copy (Walk.map f p) (_ : ↑f u' = ↑f u') (_ : ↑f v' = ↑f v')",
"tactic": "rfl"
}
] |
[
1469,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1466,
1
] |
Mathlib/Control/Traversable/Lemmas.lean
|
Traversable.traverse_map'
|
[
{
"state_after": "case h\nt : Type u → Type u\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nα β γ : Type u\ng✝ : α → F β\nh✝ : β → G γ\nf : β → γ\nx : t β\ng : α → β\nh : β → G γ\nx✝ : t α\n⊢ traverse (h ∘ g) x✝ = (traverse h ∘ map g) x✝",
"state_before": "t : Type u → Type u\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nα β γ : Type u\ng✝ : α → F β\nh✝ : β → G γ\nf : β → γ\nx : t β\ng : α → β\nh : β → G γ\n⊢ traverse (h ∘ g) = traverse h ∘ map g",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nt : Type u → Type u\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nα β γ : Type u\ng✝ : α → F β\nh✝ : β → G γ\nf : β → γ\nx : t β\ng : α → β\nh : β → G γ\nx✝ : t α\n⊢ traverse (h ∘ g) x✝ = (traverse h ∘ map g) x✝",
"tactic": "rw [comp_apply, traverse_map]"
}
] |
[
137,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.mk'_spec
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.742376\ninst✝ : CommMonoid P\nf : LocalizationMap S N\nx : M\ny : { x // x ∈ S }\n⊢ ↑(toMap f) x *\n ↑(↑(IsUnit.liftRight (MonoidHom.restrict (toMap f) S) (_ : ∀ (y : { x // x ∈ S }), IsUnit (↑(toMap f) ↑y)))\n y)⁻¹ *\n ↑(toMap f) ↑y =\n ↑(toMap f) x",
"tactic": "rw [mul_assoc, mul_comm _ (f.toMap y), ← mul_assoc, mul_inv_left, mul_comm]"
}
] |
[
743,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
742,
1
] |
Mathlib/RingTheory/FreeCommRing.lean
|
FreeCommRing.lift_of
|
[] |
[
146,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Data/Finset/Sum.lean
|
Finset.inr_mem_disjSum
|
[] |
[
82,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
ContDiff.rpow
|
[] |
[
517,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/Order/WellFounded.lean
|
Function.not_lt_argmin
|
[] |
[
194,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Topology/MetricSpace/Gluing.lean
|
Metric.eq_of_glueDist_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nε0 : 0 < ε\nx y : X\nh : glueDist Φ Ψ ε (Sum.inl x) (Sum.inl y) = 0\n⊢ Sum.inl x = Sum.inl y",
"tactic": "rw [eq_of_dist_eq_zero h]"
},
{
"state_after": "case h\nX : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nε0 : 0 < ε\nx : X\ny : Y\nh : glueDist Φ Ψ ε (Sum.inl x) (Sum.inr y) = 0\n⊢ False",
"state_before": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nε0 : 0 < ε\nx : X\ny : Y\nh : glueDist Φ Ψ ε (Sum.inl x) (Sum.inr y) = 0\n⊢ Sum.inl x = Sum.inr y",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case h\nX : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nε0 : 0 < ε\nx : X\ny : Y\nh : glueDist Φ Ψ ε (Sum.inl x) (Sum.inr y) = 0\n⊢ False",
"tactic": "linarith [le_glueDist_inl_inr Φ Ψ ε x y]"
},
{
"state_after": "case h\nX : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nε0 : 0 < ε\nx : Y\ny : X\nh : glueDist Φ Ψ ε (Sum.inr x) (Sum.inl y) = 0\n⊢ False",
"state_before": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nε0 : 0 < ε\nx : Y\ny : X\nh : glueDist Φ Ψ ε (Sum.inr x) (Sum.inl y) = 0\n⊢ Sum.inr x = Sum.inl y",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case h\nX : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nε0 : 0 < ε\nx : Y\ny : X\nh : glueDist Φ Ψ ε (Sum.inr x) (Sum.inl y) = 0\n⊢ False",
"tactic": "linarith [le_glueDist_inr_inl Φ Ψ ε x y]"
},
{
"state_after": "no goals",
"state_before": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nε0 : 0 < ε\nx y : Y\nh : glueDist Φ Ψ ε (Sum.inr x) (Sum.inr y) = 0\n⊢ Sum.inr x = Sum.inr y",
"tactic": "rw [eq_of_dist_eq_zero h]"
}
] |
[
162,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
9
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.preimage_Ico
|
[
{
"state_after": "case h\nF : Type ?u.238645\nα : Type u_1\nβ : Type ?u.238651\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\nx✝ : ℤ\n⊢ x✝ ∈ Int.cast ⁻¹' Ico a b ↔ x✝ ∈ Ico ⌈a⌉ ⌈b⌉",
"state_before": "F : Type ?u.238645\nα : Type u_1\nβ : Type ?u.238651\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ Int.cast ⁻¹' Ico a b = Ico ⌈a⌉ ⌈b⌉",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type ?u.238645\nα : Type u_1\nβ : Type ?u.238651\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\nx✝ : ℤ\n⊢ x✝ ∈ Int.cast ⁻¹' Ico a b ↔ x✝ ∈ Ico ⌈a⌉ ⌈b⌉",
"tactic": "simp [ceil_le, lt_ceil]"
}
] |
[
1275,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1273,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.coe_ennreal_zero
|
[] |
[
467,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/Data/Multiset/Dedup.lean
|
Multiset.dedup_zero
|
[] |
[
39,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean
|
hasFDerivWithinAt_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\n⊢ (∀ (i : ι),\n HasFDerivWithinAt (fun x => (↑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i)\n (comp (proj i) (comp (↑(EuclideanSpace.equiv ι 𝕜)) f')) t y) ↔\n ∀ (i : ι), HasFDerivWithinAt (fun x => f x i) (comp (EuclideanSpace.proj i) f') t 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\n⊢ HasFDerivWithinAt f f' t y ↔ ∀ (i : ι), HasFDerivWithinAt (fun x => f x i) (comp (EuclideanSpace.proj i) f') t y",
"tactic": "rw [← (EuclideanSpace.equiv ι 𝕜).comp_hasFDerivWithinAt_iff, hasFDerivWithinAt_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\n⊢ (∀ (i : ι),\n HasFDerivWithinAt (fun x => (↑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i)\n (comp (proj i) (comp (↑(EuclideanSpace.equiv ι 𝕜)) f')) t y) ↔\n ∀ (i : ι), HasFDerivWithinAt (fun x => f x i) (comp (EuclideanSpace.proj i) f') t y",
"tactic": "rfl"
}
] |
[
333,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
1
] |
Mathlib/Data/Finset/Sym.lean
|
Finset.sym_inter
|
[
{
"state_after": "case a\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\na b : α\nn✝ : ℕ\nm✝ : Sym α n✝\ns t : Finset α\nn : ℕ\nm : Sym α n\n⊢ m ∈ Finset.sym (s ∩ t) n ↔ m ∈ Finset.sym s n ∩ Finset.sym t n",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\na b : α\nn✝ : ℕ\nm : Sym α n✝\ns t : Finset α\nn : ℕ\n⊢ Finset.sym (s ∩ t) n = Finset.sym s n ∩ Finset.sym t n",
"tactic": "ext m"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\na b : α\nn✝ : ℕ\nm✝ : Sym α n✝\ns t : Finset α\nn : ℕ\nm : Sym α n\n⊢ m ∈ Finset.sym (s ∩ t) n ↔ m ∈ Finset.sym s n ∩ Finset.sym t n",
"tactic": "simp only [mem_inter, mem_sym_iff, imp_and, forall_and]"
}
] |
[
220,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.comp_mk
|
[] |
[
220,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/NumberTheory/LSeries.lean
|
Nat.ArithmeticFunction.LSeries_eq_zero_of_not_LSeriesSummable
|
[] |
[
56,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/RingTheory/Adjoin/FG.lean
|
Subalgebra.fg_def
|
[] |
[
105,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
UniqueFactorizationMonoid.factors_prod
|
[
{
"state_after": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\na : α\nane0 : a ≠ 0\n⊢ Multiset.prod (Classical.choose (_ : ∃ f, (∀ (b : α), b ∈ f → Prime b) ∧ Multiset.prod f ~ᵤ a)) ~ᵤ a",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\na : α\nane0 : a ≠ 0\n⊢ Multiset.prod (factors a) ~ᵤ a",
"tactic": "rw [factors, dif_neg ane0]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\na : α\nane0 : a ≠ 0\n⊢ Multiset.prod (Classical.choose (_ : ∃ f, (∀ (b : α), b ∈ f → Prime b) ∧ Multiset.prod f ~ᵤ a)) ~ᵤ a",
"tactic": "exact (Classical.choose_spec (exists_prime_factors a ane0)).2"
}
] |
[
450,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/Order/MinMax.lean
|
max_min_distrib_left
|
[] |
[
116,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.mem_map_iff_exists_image
|
[] |
[
1852,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1850,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.map_top
|
[] |
[
1385,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1384,
1
] |
Mathlib/Topology/Sheaves/Stalks.lean
|
TopCat.Presheaf.stalkSpecializes_comp
|
[
{
"state_after": "no goals",
"state_before": "C✝ : Type u\ninst✝³ : Category C✝\ninst✝² : HasColimits C✝\nX✝ Y Z : TopCat\nC : Type u_1\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX : TopCat\nF : Presheaf C X\nx y z : ↑X\nh : x ⤳ y\nh' : y ⤳ z\nx✝¹ : Opens ↑X\nx✝ : z ∈ x✝¹\n⊢ germ F { val := z, property := x✝ } ≫ stalkSpecializes F h' ≫ stalkSpecializes F h =\n germ F { val := z, property := x✝ } ≫ stalkSpecializes F (_ : x ⤳ z)",
"tactic": "simp"
}
] |
[
353,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.to_hasBasis
|
[] |
[
358,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean
|
GroupSeminorm.comp_assoc
|
[] |
[
367,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/Analysis/Quaternion.lean
|
Quaternion.coe_ofComplex
|
[] |
[
173,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Iio_union_Ioo
|
[
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.106002\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : c ≤ d\nh : c < b\n⊢ Iio b ∪ Ioo c d = Iio (max b d)\n\ncase inr\nα : Type u_1\nβ : Type ?u.106002\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : d ≤ c\nh : d < b\n⊢ Iio b ∪ Ioo c d = Iio (max b d)",
"state_before": "α : Type u_1\nβ : Type ?u.106002\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh : min c d < b\n⊢ Iio b ∪ Ioo c d = Iio (max b d)",
"tactic": "cases' le_total c d with hcd hcd <;> simp [hcd] at h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.106002\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : c ≤ d\nh : c < b\n⊢ Iio b ∪ Ioo c d = Iio (max b d)",
"tactic": "exact Iio_union_Ioo' h"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type ?u.106002\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : d ≤ c\nh : d < b\n⊢ Iio b ∪ Ioo c d = Iio (max d b)",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.106002\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : d ≤ c\nh : d < b\n⊢ Iio b ∪ Ioo c d = Iio (max b d)",
"tactic": "rw [max_comm]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.106002\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : d ≤ c\nh : d < b\n⊢ Iio b ∪ Ioo c d = Iio (max d b)",
"tactic": "simp [*, max_eq_right_of_lt h]"
}
] |
[
1447,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1443,
1
] |
Mathlib/Data/Set/Function.lean
|
Function.Semiconj.injOn_image
|
[
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.111902\nι : Sort ?u.111905\nπ : α → Type ?u.111910\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : InjOn fa s\nhf : InjOn f (fa '' s)\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nH : fb (f x) = fb (f y)\n⊢ f x = f y",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.111902\nι : Sort ?u.111905\nπ : α → Type ?u.111910\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : InjOn fa s\nhf : InjOn f (fa '' s)\n⊢ InjOn fb (f '' s)",
"tactic": "rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩ H"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.111902\nι : Sort ?u.111905\nπ : α → Type ?u.111910\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : InjOn fa s\nhf : InjOn f (fa '' s)\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nH : f (fa x) = f (fa y)\n⊢ f x = f y",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.111902\nι : Sort ?u.111905\nπ : α → Type ?u.111910\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : InjOn fa s\nhf : InjOn f (fa '' s)\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nH : fb (f x) = fb (f y)\n⊢ f x = f y",
"tactic": "simp only [← h.eq] at H"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.111902\nι : Sort ?u.111905\nπ : α → Type ?u.111910\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : InjOn fa s\nhf : InjOn f (fa '' s)\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nH : f (fa x) = f (fa y)\n⊢ f x = f y",
"tactic": "exact congr_arg f (ha hx hy <| hf (mem_image_of_mem fa hx) (mem_image_of_mem fa hy) H)"
}
] |
[
1643,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1639,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean
|
MeasureTheory.measure_preimage_mul
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.369584\nG : Type u_1\nH : Type ?u.369590\ninst✝⁴ : MeasurableSpace G\ninst✝³ : MeasurableSpace H\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\nμ : Measure G\ninst✝ : IsMulLeftInvariant μ\ng : G\nA : Set G\n⊢ ↑↑(Measure.map (fun h => g * h) μ) A = ↑↑μ A",
"tactic": "rw [map_mul_left_eq_self μ g]"
}
] |
[
284,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean
|
TensorAlgebra.ι_leftInverse
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝² : CommSemiring R\nM : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\n⊢ ↑ιInv (↑(ι R) x) = x",
"tactic": "simp [ιInv, (AlgHom.toLinearMap_apply), toTrivSqZeroExt_ι _]"
}
] |
[
248,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Deprecated/Subfield.lean
|
Field.closure_subset
|
[
{
"state_after": "case intro.intro.intro.intro.refl\nF : Type u_1\ninst✝ : Field F\nS T : Set F\nhT : IsSubfield T\nH : S ⊆ T\np : F\nhp : p ∈ Ring.closure S\nq : F\nhq : q ∈ Ring.closure S\n⊢ p / q ∈ T",
"state_before": "F : Type u_1\ninst✝ : Field F\nS T : Set F\nhT : IsSubfield T\nH : S ⊆ T\n⊢ closure S ⊆ T",
"tactic": "rintro _ ⟨p, hp, q, hq, hq0, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refl\nF : Type u_1\ninst✝ : Field F\nS T : Set F\nhT : IsSubfield T\nH : S ⊆ T\np : F\nhp : p ∈ Ring.closure S\nq : F\nhq : q ∈ Ring.closure S\n⊢ p / q ∈ T",
"tactic": "exact hT.div_mem (Ring.closure_subset hT.toIsSubring H hp)\n (Ring.closure_subset hT.toIsSubring H hq)"
}
] |
[
139,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Antiperiodic.funext'
|
[] |
[
374,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
11
] |
Mathlib/Data/Set/Basic.lean
|
Set.disjoint_iff_inter_eq_empty
|
[] |
[
1522,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1521,
1
] |
Mathlib/ModelTheory/LanguageMap.lean
|
FirstOrder.Language.lhomWithConstants_injective
|
[] |
[
473,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
IsMin.Iio_eq
|
[] |
[
711,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
710,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.monotone_iff_le_succ
|
[] |
[
1891,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1889,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.iterate_succ
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₁ : Type ?u.170956\nR₂ : Type ?u.170959\nR₃ : Type ?u.170962\nR₄ : Type ?u.170965\nS : Type ?u.170968\nK : Type ?u.170971\nK₂ : Type ?u.170974\nM : Type u_1\nM' : Type ?u.170980\nM₁ : Type ?u.170983\nM₂ : Type ?u.170986\nM₃ : Type ?u.170989\nM₄ : Type ?u.170992\nN : Type ?u.170995\nN₂ : Type ?u.170998\nι : Type ?u.171001\nV : Type ?u.171004\nV₂ : Type ?u.171007\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\nf' : M →ₗ[R] M\nn : ℕ\n⊢ f' ^ (n + 1) = comp (f' ^ n) f'",
"tactic": "rw [pow_succ', mul_eq_comp]"
}
] |
[
371,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/RingTheory/Localization/Module.lean
|
LinearIndependent.localization
|
[
{
"state_after": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\nhli : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0\n⊢ ∀ (s : Finset ι) (g : ι → Rₛ), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\nhli : LinearIndependent R b\n⊢ LinearIndependent Rₛ b",
"tactic": "rw [linearIndependent_iff'] at hli⊢"
},
{
"state_after": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\nhli : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni : ι\nhi : i ∈ s\n⊢ g i = 0",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\nhli : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0\n⊢ ∀ (s : Finset ι) (g : ι → Rₛ), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0",
"tactic": "intro s g hg i hi"
},
{
"state_after": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\nhli : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni : ι\nhi : i ∈ s\na : { x // x ∈ S }\ng' : ι → R\nhg' : ∀ (i : ι), i ∈ s → ↑(algebraMap R Rₛ) (g' i) = ↑a • g i\n⊢ g i = 0",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\nhli : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni : ι\nhi : i ∈ s\n⊢ g i = 0",
"tactic": "choose! a g' hg' using IsLocalization.exist_integer_multiples S s g"
},
{
"state_after": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\nhli : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni : ι\nhi : i ∈ s\na : { x // x ∈ S }\ng' : ι → R\nhg' : ∀ (i : ι), i ∈ s → ↑(algebraMap R Rₛ) (g' i) = ↑a • g i\n⊢ ∑ i in s, g' i • b i = 0\n\nR : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni : ι\nhi : i ∈ s\na : { x // x ∈ S }\ng' : ι → R\nhg' : ∀ (i : ι), i ∈ s → ↑(algebraMap R Rₛ) (g' i) = ↑a • g i\nhli : g' i = 0\n⊢ g i = 0",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\nhli : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni : ι\nhi : i ∈ s\na : { x // x ∈ S }\ng' : ι → R\nhg' : ∀ (i : ι), i ∈ s → ↑(algebraMap R Rₛ) (g' i) = ↑a • g i\n⊢ g i = 0",
"tactic": "specialize hli s g' _ i hi"
},
{
"state_after": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni : ι\nhi : i ∈ s\na : { x // x ∈ S }\ng' : ι → R\nhg' : ∀ (i : ι), i ∈ s → ↑(algebraMap R Rₛ) (g' i) = ↑a • g i\nhli : g' i = 0\n⊢ ↑(algebraMap R Rₛ) ↑a * g i = 0",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni : ι\nhi : i ∈ s\na : { x // x ∈ S }\ng' : ι → R\nhg' : ∀ (i : ι), i ∈ s → ↑(algebraMap R Rₛ) (g' i) = ↑a • g i\nhli : g' i = 0\n⊢ g i = 0",
"tactic": "refine' (IsLocalization.map_units Rₛ a).mul_right_eq_zero.mp _"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni : ι\nhi : i ∈ s\na : { x // x ∈ S }\ng' : ι → R\nhg' : ∀ (i : ι), i ∈ s → ↑(algebraMap R Rₛ) (g' i) = ↑a • g i\nhli : g' i = 0\n⊢ ↑(algebraMap R Rₛ) ↑a * g i = 0",
"tactic": "rw [← Algebra.smul_def, ← map_zero (algebraMap R Rₛ), ← hli, hg' i hi]"
},
{
"state_after": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\nhli : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni : ι\nhi : i ∈ s\na : { x // x ∈ S }\ng' : ι → R\nhg' : ∀ (i : ι), i ∈ s → ↑(algebraMap R Rₛ) (g' i) = ↑a • g i\n⊢ ∑ i in s, g' i • b i = ∑ x in s, ↑a • g x • b x",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\nhli : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni : ι\nhi : i ∈ s\na : { x // x ∈ S }\ng' : ι → R\nhg' : ∀ (i : ι), i ∈ s → ↑(algebraMap R Rₛ) (g' i) = ↑a • g i\n⊢ ∑ i in s, g' i • b i = 0",
"tactic": "rw [← @smul_zero _ M _ _ (a : R), ← hg, Finset.smul_sum]"
},
{
"state_after": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\nhli : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni✝ : ι\nhi✝ : i✝ ∈ s\na : { x // x ∈ S }\ng' : ι → R\nhg' : ∀ (i : ι), i ∈ s → ↑(algebraMap R Rₛ) (g' i) = ↑a • g i\ni : ι\nhi : i ∈ s\n⊢ g' i • b i = ↑a • g i • b i",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\nhli : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni : ι\nhi : i ∈ s\na : { x // x ∈ S }\ng' : ι → R\nhg' : ∀ (i : ι), i ∈ s → ↑(algebraMap R Rₛ) (g' i) = ↑a • g i\n⊢ ∑ i in s, g' i • b i = ∑ x in s, ↑a • g x • b x",
"tactic": "refine' Finset.sum_congr rfl fun i hi => _"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nRₛ : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing Rₛ\ninst✝⁴ : Algebra R Rₛ\nS : Submonoid R\nhT : IsLocalization S Rₛ\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rₛ M\ninst✝ : IsScalarTower R Rₛ M\nι : Type u_1\nb : ι → M\nhli : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • b i = 0 → ∀ (i : ι), i ∈ s → g i = 0\ns : Finset ι\ng : ι → Rₛ\nhg : ∑ i in s, g i • b i = 0\ni✝ : ι\nhi✝ : i✝ ∈ s\na : { x // x ∈ S }\ng' : ι → R\nhg' : ∀ (i : ι), i ∈ s → ↑(algebraMap R Rₛ) (g' i) = ↑a • g i\ni : ι\nhi : i ∈ s\n⊢ g' i • b i = ↑a • g i • b i",
"tactic": "rw [← IsScalarTower.algebraMap_smul Rₛ, hg' i hi, smul_assoc]"
}
] |
[
57,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.fuzzy_congr_imp
|
[] |
[
963,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
962,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.coe_mks
|
[] |
[
163,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/ModelTheory/ElementaryMaps.lean
|
FirstOrder.Language.Equiv.coe_toElementaryEmbedding
|
[] |
[
346,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
iteratedFDerivWithin_succ_eq_comp_right
|
[
{
"state_after": "case H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\nm : Fin (n + 1) → E\n⊢ ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑((↑(continuousMultilinearCurryRightEquiv' 𝕜 n E F) ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) x)\n m",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\n⊢ iteratedFDerivWithin 𝕜 (n + 1) f s x =\n (↑(continuousMultilinearCurryRightEquiv' 𝕜 n E F) ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) x",
"tactic": "ext m"
},
{
"state_after": "case H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\nm : Fin (n + 1) → E\n⊢ ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n)) =\n ↑((↑(continuousMultilinearCurryRightEquiv' 𝕜 n E F) ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) x)\n m",
"state_before": "case H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\nm : Fin (n + 1) → E\n⊢ ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑((↑(continuousMultilinearCurryRightEquiv' 𝕜 n E F) ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) x)\n m",
"tactic": "rw [iteratedFDerivWithin_succ_apply_right hs hx]"
},
{
"state_after": "no goals",
"state_before": "case H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\nm : Fin (n + 1) → E\n⊢ ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n)) =\n ↑((↑(continuousMultilinearCurryRightEquiv' 𝕜 n E F) ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) x)\n m",
"tactic": "rfl"
}
] |
[
860,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
855,
1
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.mapQ_comp
|
[
{
"state_after": "case h.h\nR : Type u_5\nM : Type u_6\nr : R\nx y : M\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\np p' : Submodule R M\nR₂ : Type u_3\nM₂ : Type u_4\ninst✝⁶ : Ring R₂\ninst✝⁵ : AddCommGroup M₂\ninst✝⁴ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nR₃ : Type u_1\nM₃ : Type u_2\ninst✝³ : Ring R₃\ninst✝² : AddCommGroup M₃\ninst✝¹ : Module R₃ M₃\np₂ : Submodule R₂ M₂\np₃ : Submodule R₃ M₃\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nf : M →ₛₗ[τ₁₂] M₂\ng : M₂ →ₛₗ[τ₂₃] M₃\nhf : p ≤ comap f p₂\nhg : p₂ ≤ comap g p₃\nh : optParam (p ≤ comap f (comap g p₃)) (_ : p ≤ comap f (comap g p₃))\nx✝ : M\n⊢ ↑(comp (mapQ p p₃ (comp g f) h) (mkQ p)) x✝ = ↑(comp (comp (mapQ p₂ p₃ g hg) (mapQ p p₂ f hf)) (mkQ p)) x✝",
"state_before": "R : Type u_5\nM : Type u_6\nr : R\nx y : M\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\np p' : Submodule R M\nR₂ : Type u_3\nM₂ : Type u_4\ninst✝⁶ : Ring R₂\ninst✝⁵ : AddCommGroup M₂\ninst✝⁴ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nR₃ : Type u_1\nM₃ : Type u_2\ninst✝³ : Ring R₃\ninst✝² : AddCommGroup M₃\ninst✝¹ : Module R₃ M₃\np₂ : Submodule R₂ M₂\np₃ : Submodule R₃ M₃\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nf : M →ₛₗ[τ₁₂] M₂\ng : M₂ →ₛₗ[τ₂₃] M₃\nhf : p ≤ comap f p₂\nhg : p₂ ≤ comap g p₃\nh : optParam (p ≤ comap f (comap g p₃)) (_ : p ≤ comap f (comap g p₃))\n⊢ mapQ p p₃ (comp g f) h = comp (mapQ p₂ p₃ g hg) (mapQ p p₂ f hf)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h\nR : Type u_5\nM : Type u_6\nr : R\nx y : M\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\np p' : Submodule R M\nR₂ : Type u_3\nM₂ : Type u_4\ninst✝⁶ : Ring R₂\ninst✝⁵ : AddCommGroup M₂\ninst✝⁴ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nR₃ : Type u_1\nM₃ : Type u_2\ninst✝³ : Ring R₃\ninst✝² : AddCommGroup M₃\ninst✝¹ : Module R₃ M₃\np₂ : Submodule R₂ M₂\np₃ : Submodule R₃ M₃\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nf : M →ₛₗ[τ₁₂] M₂\ng : M₂ →ₛₗ[τ₂₃] M₃\nhf : p ≤ comap f p₂\nhg : p₂ ≤ comap g p₃\nh : optParam (p ≤ comap f (comap g p₃)) (_ : p ≤ comap f (comap g p₃))\nx✝ : M\n⊢ ↑(comp (mapQ p p₃ (comp g f) h) (mkQ p)) x✝ = ↑(comp (comp (mapQ p₂ p₃ g hg) (mapQ p p₂ f hf)) (mkQ p)) x✝",
"tactic": "simp"
}
] |
[
438,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean
|
PadicInt.norm_le_pow_iff_norm_lt_pow_add_one
|
[
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nn : ℤ\n⊢ ‖↑x‖ ≤ ↑p ^ n ↔ ‖↑x‖ < ↑p ^ (n + 1)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nn : ℤ\n⊢ ‖x‖ ≤ ↑p ^ n ↔ ‖x‖ < ↑p ^ (n + 1)",
"tactic": "rw [norm_def]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nn : ℤ\n⊢ ‖↑x‖ ≤ ↑p ^ n ↔ ‖↑x‖ < ↑p ^ (n + 1)",
"tactic": "exact Padic.norm_le_pow_iff_norm_lt_pow_add_one _ _"
}
] |
[
563,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
561,
1
] |
Mathlib/Topology/MetricSpace/Antilipschitz.lean
|
AntilipschitzWith.edist_lt_top
|
[] |
[
42,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/CategoryTheory/CofilteredSystem.lean
|
nonempty_sections_of_finite_cofiltered_system
|
[
{
"state_after": "J : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\n⊢ Set.Nonempty (Functor.sections F)",
"state_before": "J : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\n⊢ Set.Nonempty (Functor.sections F)",
"tactic": "let J' : Type max w v u := AsSmall.{max w v} J"
},
{
"state_after": "J : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\n⊢ Set.Nonempty (Functor.sections F)",
"state_before": "J : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\n⊢ Set.Nonempty (Functor.sections F)",
"tactic": "let down : J' ⥤ J := AsSmall.down"
},
{
"state_after": "J : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\n⊢ Set.Nonempty (Functor.sections F)",
"state_before": "J : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\n⊢ Set.Nonempty (Functor.sections F)",
"tactic": "let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}"
},
{
"state_after": "J : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis : ∀ (i : J'), _root_.Nonempty (F'.obj i)\n⊢ Set.Nonempty (Functor.sections F)",
"state_before": "J : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\n⊢ Set.Nonempty (Functor.sections F)",
"tactic": "haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩"
},
{
"state_after": "J : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis : ∀ (i : J'), Finite (F'.obj i)\n⊢ Set.Nonempty (Functor.sections F)",
"state_before": "J : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis : ∀ (i : J'), _root_.Nonempty (F'.obj i)\n⊢ Set.Nonempty (Functor.sections F)",
"tactic": "haveI : ∀ i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm"
},
{
"state_after": "case inl\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis : ∀ (i : J'), Finite (F'.obj i)\nh✝ : IsEmpty J\n⊢ Set.Nonempty (Functor.sections F)\n\ncase inr\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\n⊢ Set.Nonempty (Functor.sections F)",
"state_before": "J : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis : ∀ (i : J'), Finite (F'.obj i)\n⊢ Set.Nonempty (Functor.sections F)",
"tactic": "cases isEmpty_or_nonempty J"
},
{
"state_after": "case inr\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝¹ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis✝ : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\nthis : IsCofiltered J\n⊢ Set.Nonempty (Functor.sections F)",
"state_before": "case inr\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\n⊢ Set.Nonempty (Functor.sections F)",
"tactic": "haveI : IsCofiltered J := ⟨⟩"
},
{
"state_after": "case inr.intro\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝¹ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis✝ : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') → F'.obj j\nhu : u ∈ Functor.sections F'\n⊢ Set.Nonempty (Functor.sections F)",
"state_before": "case inr\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝¹ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis✝ : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\nthis : IsCofiltered J\n⊢ Set.Nonempty (Functor.sections F)",
"tactic": "obtain ⟨u, hu⟩ := nonempty_sections_of_finite_cofiltered_system.init F'"
},
{
"state_after": "case inr.intro\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝¹ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis✝ : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') → F'.obj j\nhu : u ∈ Functor.sections F'\n⊢ (fun j => (u { down := j }).down) ∈ Functor.sections F",
"state_before": "case inr.intro\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝¹ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis✝ : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') → F'.obj j\nhu : u ∈ Functor.sections F'\n⊢ Set.Nonempty (Functor.sections F)",
"tactic": "use fun j => (u ⟨j⟩).down"
},
{
"state_after": "case inr.intro\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝¹ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis✝ : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') → F'.obj j\nhu : u ∈ Functor.sections F'\nj j' : J\nf : j ⟶ j'\n⊢ F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'",
"state_before": "case inr.intro\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝¹ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis✝ : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') → F'.obj j\nhu : u ∈ Functor.sections F'\n⊢ (fun j => (u { down := j }).down) ∈ Functor.sections F",
"tactic": "intro j j' f"
},
{
"state_after": "case inr.intro\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝¹ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis✝ : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') → F'.obj j\nhu : u ∈ Functor.sections F'\nj j' : J\nf : j ⟶ j'\nh : F'.map { down := f } (u { down := j }) = u { down := j' }\n⊢ F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'",
"state_before": "case inr.intro\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝¹ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis✝ : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') → F'.obj j\nhu : u ∈ Functor.sections F'\nj j' : J\nf : j ⟶ j'\n⊢ F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'",
"tactic": "have h := @hu (⟨j⟩ : J') (⟨j'⟩ : J') (ULift.up f)"
},
{
"state_after": "case inr.intro\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝¹ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis✝ : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') → F'.obj j\nhu : u ∈ Functor.sections F'\nj j' : J\nf : j ⟶ j'\nh : { down := F.map f (u { down := j }).down } = u { down := j' }\n⊢ F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'",
"state_before": "case inr.intro\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝¹ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis✝ : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') → F'.obj j\nhu : u ∈ Functor.sections F'\nj j' : J\nf : j ⟶ j'\nh : F'.map { down := f } (u { down := j }) = u { down := j' }\n⊢ F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'",
"tactic": "simp only [AsSmall.down, Functor.comp_map, uliftFunctor_map, Functor.op_map] at h"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝¹ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis✝ : ∀ (i : J'), Finite (F'.obj i)\nh✝ : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') → F'.obj j\nhu : u ∈ Functor.sections F'\nj j' : J\nf : j ⟶ j'\nh : { down := F.map f (u { down := j }).down } = u { down := j' }\n⊢ F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'",
"tactic": "simp_rw [← h]"
},
{
"state_after": "no goals",
"state_before": "case inl\nJ : Type u\ninst✝³ : Category J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor\nthis✝ : ∀ (i : J'), _root_.Nonempty (F'.obj i)\nthis : ∀ (i : J'), Finite (F'.obj i)\nh✝ : IsEmpty J\n⊢ Set.Nonempty (Functor.sections F)",
"tactic": "fconstructor <;> apply isEmptyElim"
}
] |
[
102,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_cons
|
[] |
[
100,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.weightedSMul_congr
|
[
{
"state_after": "case h\nα : Type u_1\nE : Type ?u.42851\nF : Type u_2\n𝕜 : Type ?u.42857\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhst : ↑↑μ s = ↑↑μ t\nx : F\n⊢ ↑(weightedSMul μ s) x = ↑(weightedSMul μ t) x",
"state_before": "α : Type u_1\nE : Type ?u.42851\nF : Type u_2\n𝕜 : Type ?u.42857\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhst : ↑↑μ s = ↑↑μ t\n⊢ weightedSMul μ s = weightedSMul μ t",
"tactic": "ext1 x"
},
{
"state_after": "case h\nα : Type u_1\nE : Type ?u.42851\nF : Type u_2\n𝕜 : Type ?u.42857\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhst : ↑↑μ s = ↑↑μ t\nx : F\n⊢ ENNReal.toReal (↑↑μ s) • x = ENNReal.toReal (↑↑μ t) • x",
"state_before": "case h\nα : Type u_1\nE : Type ?u.42851\nF : Type u_2\n𝕜 : Type ?u.42857\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhst : ↑↑μ s = ↑↑μ t\nx : F\n⊢ ↑(weightedSMul μ s) x = ↑(weightedSMul μ t) x",
"tactic": "simp_rw [weightedSMul_apply]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nE : Type ?u.42851\nF : Type u_2\n𝕜 : Type ?u.42857\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhst : ↑↑μ s = ↑↑μ t\nx : F\n⊢ ENNReal.toReal (↑↑μ s) • x = ENNReal.toReal (↑↑μ t) • x",
"tactic": "congr 2"
}
] |
[
208,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean
|
MeasureTheory.map_div_right_eq_self
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.311574\nG : Type u_1\nH : Type ?u.311580\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : DivInvMonoid G\nμ : Measure G\ninst✝ : IsMulRightInvariant μ\ng : G\n⊢ Measure.map (fun x => x / g) μ = μ",
"tactic": "simp_rw [div_eq_mul_inv, map_mul_right_eq_self μ g⁻¹]"
}
] |
[
258,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Function.Surjective.iUnion_comp
|
[] |
[
2051,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2050,
1
] |
Mathlib/GroupTheory/NoncommPiCoprod.lean
|
Subgroup.eq_one_of_noncommProd_eq_one_of_independent
|
[
{
"state_after": "G : Type u_2\ninst✝ : Group G\nι : Type u_1\ns : Finset ι\nf : ι → G\ncomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem : ∀ (x : ι), x ∈ s → f x ∈ K x\n⊢ Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1",
"state_before": "G : Type u_2\ninst✝ : Group G\nι : Type u_1\ns : Finset ι\nf : ι → G\ncomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem : ∀ (x : ι), x ∈ s → f x ∈ K x\nheq1 : Finset.noncommProd s f comm = 1\n⊢ ∀ (i : ι), i ∈ s → f i = 1",
"tactic": "revert heq1"
},
{
"state_after": "case empty\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s → f x ∈ K x\ncomm : Set.Pairwise ↑∅ fun a b => Commute (f a) (f b)\nhmem : ∀ (x : ι), x ∈ ∅ → f x ∈ K x\n⊢ Finset.noncommProd ∅ f comm = 1 → ∀ (i : ι), i ∈ ∅ → f i = 1\n\ncase insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhmem : ∀ (x : ι), x ∈ insert i s → f x ∈ K x\n⊢ Finset.noncommProd (insert i s) f comm = 1 → ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"state_before": "G : Type u_2\ninst✝ : Group G\nι : Type u_1\ns : Finset ι\nf : ι → G\ncomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem : ∀ (x : ι), x ∈ s → f x ∈ K x\n⊢ Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1",
"tactic": "induction' s using Finset.induction_on with i s hnmem ih"
},
{
"state_after": "no goals",
"state_before": "case empty\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s → f x ∈ K x\ncomm : Set.Pairwise ↑∅ fun a b => Commute (f a) (f b)\nhmem : ∀ (x : ι), x ∈ ∅ → f x ∈ K x\n⊢ Finset.noncommProd ∅ f comm = 1 → ∀ (i : ι), i ∈ ∅ → f i = 1",
"tactic": "simp"
},
{
"state_after": "case insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhmem : ∀ (x : ι), x ∈ insert i s → f x ∈ K x\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\n⊢ Finset.noncommProd (insert i s) f comm = 1 → ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"state_before": "case insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhmem : ∀ (x : ι), x ∈ insert i s → f x ∈ K x\n⊢ Finset.noncommProd (insert i s) f comm = 1 → ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"tactic": "have hcomm := comm.mono (Finset.coe_subset.2 <| Finset.subset_insert _ _)"
},
{
"state_after": "case insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\n⊢ Finset.noncommProd (insert i s) f comm = 1 → ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"state_before": "case insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhmem : ∀ (x : ι), x ∈ insert i s → f x ∈ K x\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\n⊢ Finset.noncommProd (insert i s) f comm = 1 → ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"tactic": "simp only [Finset.forall_mem_insert] at hmem"
},
{
"state_after": "case insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\n⊢ Finset.noncommProd (insert i s) f comm = 1 → ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"state_before": "case insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\n⊢ Finset.noncommProd (insert i s) f comm = 1 → ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"tactic": "have hmem_bsupr : s.noncommProd f hcomm ∈ ⨆ i ∈ (s : Set ι), K i := by\n refine' Subgroup.noncommProd_mem _ _ _\n intro x hx\n have : K x ≤ ⨆ i ∈ (s : Set ι), K i := le_iSup₂ (f := fun i _ => K i) x hx\n exact this (hmem.2 x hx)"
},
{
"state_after": "case insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : Finset.noncommProd (insert i s) f comm = 1\n⊢ ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"state_before": "case insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\n⊢ Finset.noncommProd (insert i s) f comm = 1 → ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"tactic": "intro heq1"
},
{
"state_after": "case insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : f i * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\n⊢ ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"state_before": "case insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : Finset.noncommProd (insert i s) f comm = 1\n⊢ ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"tactic": "rw [Finset.noncommProd_insert_of_not_mem _ _ _ _ hnmem] at heq1"
},
{
"state_after": "case insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : f i * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\nhnmem' : ¬i ∈ ↑s\n⊢ ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"state_before": "case insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : f i * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\n⊢ ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"tactic": "have hnmem' : i ∉ (s : Set ι) := by simpa"
},
{
"state_after": "case insert.intro\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : f i * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\nhnmem' : ¬i ∈ ↑s\nheq1i : f i = 1\nheq1S : Finset.noncommProd s f hcomm = 1\n⊢ ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"state_before": "case insert\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : f i * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\nhnmem' : ¬i ∈ ↑s\n⊢ ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"tactic": "obtain ⟨heq1i : f i = 1, heq1S : s.noncommProd f _ = 1⟩ :=\n Subgroup.disjoint_iff_mul_eq_one.mp (hind.disjoint_biSup hnmem') hmem.1 hmem_bsupr heq1"
},
{
"state_after": "case insert.intro\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni✝ : ι\ns : Finset ι\nhnmem : ¬i✝ ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i✝ s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i✝ ∈ K i✝ ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : f i✝ * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\nhnmem' : ¬i✝ ∈ ↑s\nheq1i : f i✝ = 1\nheq1S : Finset.noncommProd s f hcomm = 1\ni : ι\nh : i ∈ insert i✝ s\n⊢ f i = 1",
"state_before": "case insert.intro\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : f i * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\nhnmem' : ¬i ∈ ↑s\nheq1i : f i = 1\nheq1S : Finset.noncommProd s f hcomm = 1\n⊢ ∀ (i_1 : ι), i_1 ∈ insert i s → f i_1 = 1",
"tactic": "intro i h"
},
{
"state_after": "case insert.intro\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni✝ : ι\ns : Finset ι\nhnmem : ¬i✝ ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i✝ s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i✝ ∈ K i✝ ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : f i✝ * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\nhnmem' : ¬i✝ ∈ ↑s\nheq1i : f i✝ = 1\nheq1S : Finset.noncommProd s f hcomm = 1\ni : ι\nh : i = i✝ ∨ i ∈ s\n⊢ f i = 1",
"state_before": "case insert.intro\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni✝ : ι\ns : Finset ι\nhnmem : ¬i✝ ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i✝ s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i✝ ∈ K i✝ ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : f i✝ * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\nhnmem' : ¬i✝ ∈ ↑s\nheq1i : f i✝ = 1\nheq1S : Finset.noncommProd s f hcomm = 1\ni : ι\nh : i ∈ insert i✝ s\n⊢ f i = 1",
"tactic": "simp only [Finset.mem_insert] at h"
},
{
"state_after": "case insert.intro.inl\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ns : Finset ι\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1S : Finset.noncommProd s f hcomm = 1\ni : ι\nhnmem : ¬i ∈ s\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nheq1 : f i * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\nhnmem' : ¬i ∈ ↑s\nheq1i : f i = 1\n⊢ f i = 1\n\ncase insert.intro.inr\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni✝ : ι\ns : Finset ι\nhnmem : ¬i✝ ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i✝ s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i✝ ∈ K i✝ ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : f i✝ * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\nhnmem' : ¬i✝ ∈ ↑s\nheq1i : f i✝ = 1\nheq1S : Finset.noncommProd s f hcomm = 1\ni : ι\nh : i ∈ s\n⊢ f i = 1",
"state_before": "case insert.intro\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni✝ : ι\ns : Finset ι\nhnmem : ¬i✝ ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i✝ s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i✝ ∈ K i✝ ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : f i✝ * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\nhnmem' : ¬i✝ ∈ ↑s\nheq1i : f i✝ = 1\nheq1S : Finset.noncommProd s f hcomm = 1\ni : ι\nh : i = i✝ ∨ i ∈ s\n⊢ f i = 1",
"tactic": "rcases h with (rfl | h)"
},
{
"state_after": "G : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\n⊢ ∀ (c : ι), c ∈ s → f c ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i",
"state_before": "G : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\n⊢ Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i",
"tactic": "refine' Subgroup.noncommProd_mem _ _ _"
},
{
"state_after": "G : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nx : ι\nhx : x ∈ s\n⊢ f x ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i",
"state_before": "G : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\n⊢ ∀ (c : ι), c ∈ s → f c ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i",
"tactic": "intro x hx"
},
{
"state_after": "G : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nx : ι\nhx : x ∈ s\nthis : K x ≤ ⨆ (i : ι) (_ : i ∈ ↑s), K i\n⊢ f x ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i",
"state_before": "G : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nx : ι\nhx : x ∈ s\n⊢ f x ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i",
"tactic": "have : K x ≤ ⨆ i ∈ (s : Set ι), K i := le_iSup₂ (f := fun i _ => K i) x hx"
},
{
"state_after": "no goals",
"state_before": "G : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nx : ι\nhx : x ∈ s\nthis : K x ≤ ⨆ (i : ι) (_ : i ∈ ↑s), K i\n⊢ f x ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i",
"tactic": "exact this (hmem.2 x hx)"
},
{
"state_after": "no goals",
"state_before": "G : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni : ι\ns : Finset ι\nhnmem : ¬i ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : f i * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\n⊢ ¬i ∈ ↑s",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "case insert.intro.inl\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ns : Finset ι\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1S : Finset.noncommProd s f hcomm = 1\ni : ι\nhnmem : ¬i ∈ s\ncomm : Set.Pairwise ↑(insert i s) fun a b => Commute (f a) (f b)\nhmem : f i ∈ K i ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nheq1 : f i * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\nhnmem' : ¬i ∈ ↑s\nheq1i : f i = 1\n⊢ f i = 1",
"tactic": "exact heq1i"
},
{
"state_after": "no goals",
"state_before": "case insert.intro.inr\nG : Type u_2\ninst✝ : Group G\nι : Type u_1\ns✝ : Finset ι\nf : ι → G\ncomm✝ : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)\nK : ι → Subgroup G\nhind : CompleteLattice.Independent K\nhmem✝ : ∀ (x : ι), x ∈ s✝ → f x ∈ K x\ni✝ : ι\ns : Finset ι\nhnmem : ¬i✝ ∈ s\nih :\n ∀ (comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)),\n (∀ (x : ι), x ∈ s → f x ∈ K x) → Finset.noncommProd s f comm = 1 → ∀ (i : ι), i ∈ s → f i = 1\ncomm : Set.Pairwise ↑(insert i✝ s) fun a b => Commute (f a) (f b)\nhcomm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)\nhmem : f i✝ ∈ K i✝ ∧ ∀ (x : ι), x ∈ s → f x ∈ K x\nhmem_bsupr : Finset.noncommProd s f hcomm ∈ ⨆ (i : ι) (_ : i ∈ ↑s), K i\nheq1 : f i✝ * Finset.noncommProd s f (_ : Set.Pairwise ↑s fun a b => Commute (f a) (f b)) = 1\nhnmem' : ¬i✝ ∈ ↑s\nheq1i : f i✝ = 1\nheq1S : Finset.noncommProd s f hcomm = 1\ni : ι\nh : i ∈ s\n⊢ f i = 1",
"tactic": "refine' ih hcomm hmem.2 heq1S _ h"
}
] |
[
83,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Order/UpperLower/Hom.lean
|
UpperSet.coe_icisSupHom
|
[] |
[
59,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Data/Rel.lean
|
Rel.core_mono
|
[] |
[
223,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Data/Rat/Defs.lean
|
Rat.sub_def''
|
[] |
[
174,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
|
CategoryTheory.Limits.WalkingParallelFamily.hom_id
|
[] |
[
117,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Analysis/BoxIntegral/Basic.lean
|
BoxIntegral.Integrable.to_subbox
|
[] |
[
539,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
538,
1
] |
Std/Classes/Order.lean
|
Std.TransCmp.cmp_congr_right
|
[
{
"state_after": "no goals",
"state_before": "cmp✝ : ?m.3864 → ?m.3864 → Ordering\ninst✝² : TransCmp cmp✝\nx✝ : Sort u_1\ncmp : x✝ → x✝ → Ordering\ninst✝¹ : TransCmp cmp\ny z x : x✝\ninst✝ : TransCmp cmp\nyz : cmp y z = Ordering.eq\n⊢ cmp x y = cmp x z",
"tactic": "rw [← Ordering.swap_inj, symm, symm, cmp_congr_left yz]"
}
] |
[
86,
58
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
85,
1
] |
Mathlib/RingTheory/Int/Basic.lean
|
induction_on_primes
|
[
{
"state_after": "case h₁\nP : ℕ → Prop\nh₀ : P 0\nh₁ : P 1\nh : ∀ (p a : ℕ), Nat.Prime p → P a → P (p * a)\nn : ℕ\n⊢ P 0\n\ncase h₂\nP : ℕ → Prop\nh₀ : P 0\nh₁ : P 1\nh : ∀ (p a : ℕ), Nat.Prime p → P a → P (p * a)\nn : ℕ\n⊢ ∀ (x : ℕ), IsUnit x → P x\n\ncase h₃\nP : ℕ → Prop\nh₀ : P 0\nh₁ : P 1\nh : ∀ (p a : ℕ), Nat.Prime p → P a → P (p * a)\nn : ℕ\n⊢ ∀ (a p : ℕ), a ≠ 0 → Prime p → P a → P (p * a)",
"state_before": "P : ℕ → Prop\nh₀ : P 0\nh₁ : P 1\nh : ∀ (p a : ℕ), Nat.Prime p → P a → P (p * a)\nn : ℕ\n⊢ P n",
"tactic": "apply UniqueFactorizationMonoid.induction_on_prime"
},
{
"state_after": "no goals",
"state_before": "case h₁\nP : ℕ → Prop\nh₀ : P 0\nh₁ : P 1\nh : ∀ (p a : ℕ), Nat.Prime p → P a → P (p * a)\nn : ℕ\n⊢ P 0",
"tactic": "exact h₀"
},
{
"state_after": "case h₂\nP : ℕ → Prop\nh₀ : P 0\nh₁ : P 1\nh✝ : ∀ (p a : ℕ), Nat.Prime p → P a → P (p * a)\nn✝ n : ℕ\nh : IsUnit n\n⊢ P n",
"state_before": "case h₂\nP : ℕ → Prop\nh₀ : P 0\nh₁ : P 1\nh : ∀ (p a : ℕ), Nat.Prime p → P a → P (p * a)\nn : ℕ\n⊢ ∀ (x : ℕ), IsUnit x → P x",
"tactic": "intro n h"
},
{
"state_after": "case h₂\nP : ℕ → Prop\nh₀ : P 0\nh₁ : P 1\nh✝ : ∀ (p a : ℕ), Nat.Prime p → P a → P (p * a)\nn✝ n : ℕ\nh : IsUnit n\n⊢ P 1",
"state_before": "case h₂\nP : ℕ → Prop\nh₀ : P 0\nh₁ : P 1\nh✝ : ∀ (p a : ℕ), Nat.Prime p → P a → P (p * a)\nn✝ n : ℕ\nh : IsUnit n\n⊢ P n",
"tactic": "rw [Nat.isUnit_iff.1 h]"
},
{
"state_after": "no goals",
"state_before": "case h₂\nP : ℕ → Prop\nh₀ : P 0\nh₁ : P 1\nh✝ : ∀ (p a : ℕ), Nat.Prime p → P a → P (p * a)\nn✝ n : ℕ\nh : IsUnit n\n⊢ P 1",
"tactic": "exact h₁"
},
{
"state_after": "case h₃\nP : ℕ → Prop\nh₀ : P 0\nh₁ : P 1\nh : ∀ (p a : ℕ), Nat.Prime p → P a → P (p * a)\nn a p : ℕ\na✝ : a ≠ 0\nhp : Prime p\nha : P a\n⊢ P (p * a)",
"state_before": "case h₃\nP : ℕ → Prop\nh₀ : P 0\nh₁ : P 1\nh : ∀ (p a : ℕ), Nat.Prime p → P a → P (p * a)\nn : ℕ\n⊢ ∀ (a p : ℕ), a ≠ 0 → Prime p → P a → P (p * a)",
"tactic": "intro a p _ hp ha"
},
{
"state_after": "no goals",
"state_before": "case h₃\nP : ℕ → Prop\nh₀ : P 0\nh₁ : P 1\nh : ∀ (p a : ℕ), Nat.Prime p → P a → P (p * a)\nn a p : ℕ\na✝ : a ≠ 0\nhp : Prime p\nha : P a\n⊢ P (p * a)",
"tactic": "exact h p a hp.nat_prime ha"
}
] |
[
362,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/MeasureTheory/Function/ContinuousMapDense.lean
|
MeasureTheory.Integrable.exists_hasCompactSupport_lintegral_sub_le
|
[
{
"state_after": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\nf : α → E\nε : ℝ≥0∞\nhε : ε ≠ 0\nhf : Memℒp f 1\n⊢ ∃ g, HasCompactSupport g ∧ snorm (fun x => f x - g x) 1 μ ≤ ε ∧ Continuous g ∧ Memℒp g 1",
"state_before": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\nf : α → E\nhf : Integrable f\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∃ g, HasCompactSupport g ∧ (∫⁻ (x : α), ↑‖f x - g x‖₊ ∂μ) ≤ ε ∧ Continuous g ∧ Integrable g",
"tactic": "simp only [← memℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf ⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\nf : α → E\nε : ℝ≥0∞\nhε : ε ≠ 0\nhf : Memℒp f 1\n⊢ ∃ g, HasCompactSupport g ∧ snorm (fun x => f x - g x) 1 μ ≤ ε ∧ Continuous g ∧ Memℒp g 1",
"tactic": "exact hf.exists_hasCompactSupport_snorm_sub_le ENNReal.one_ne_top hε"
}
] |
[
224,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Order/InitialSeg.lean
|
RelEmbedding.collapseF.not_lt
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.71243\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\ninst✝ : IsWellOrder β s\nf : r ↪r s\na : α\nb : β\nh : ∀ (a' : α), r a' a → s (↑(collapseF f a')) b\n⊢ ¬s b\n ↑(WellFounded.fix (_ : WellFounded r)\n (fun a IH =>\n let S := {b | ∀ (a_1 : α) (h : r a_1 a), s (↑(IH a_1 h)) b};\n let_fun this := (_ : ∀ (a' : α) (h : r a' a), s (↑(IH a' h)) (↑f a));\n { val := WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S),\n property := (_ : ¬s (↑f a) (WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S))) })\n a)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.71243\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\ninst✝ : IsWellOrder β s\nf : r ↪r s\na : α\nb : β\nh : ∀ (a' : α), r a' a → s (↑(collapseF f a')) b\n⊢ ¬s b ↑(collapseF f a)",
"tactic": "unfold collapseF"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.71243\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\ninst✝ : IsWellOrder β s\nf : r ↪r s\na : α\nb : β\nh : ∀ (a' : α), r a' a → s (↑(collapseF f a')) b\n⊢ ¬s b\n ↑(let S :=\n {b |\n ∀ (a_1 : α) (h : r a_1 a),\n s\n (↑((fun y x =>\n WellFounded.fix (_ : WellFounded r)\n (fun a IH =>\n let S := {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b};\n let_fun this := (_ : ∀ (a' : α) (h : r a' a), s (↑(IH a' h)) (↑f a));\n { val := WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S),\n property := (_ : ¬s (↑f a) (WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S))) })\n y)\n a_1 h))\n b};\n let_fun this :=\n (_ :\n ∀ (a' : α),\n r a' a →\n s\n (↑(WellFounded.fix (_ : WellFounded r)\n (fun a IH =>\n let S := {b | ∀ (a_1 : α) (h : r a_1 a), s (↑(IH a_1 h)) b};\n let_fun this := (_ : ∀ (a' : α) (h : r a' a), s (↑(IH a' h)) (↑f a));\n { val := WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S),\n property := (_ : ¬s (↑f a) (WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S))) })\n a'))\n (↑f a));\n { val := WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S),\n property := (_ : ¬s (↑f a) (WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S))) })",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.71243\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\ninst✝ : IsWellOrder β s\nf : r ↪r s\na : α\nb : β\nh : ∀ (a' : α), r a' a → s (↑(collapseF f a')) b\n⊢ ¬s b\n ↑(WellFounded.fix (_ : WellFounded r)\n (fun a IH =>\n let S := {b | ∀ (a_1 : α) (h : r a_1 a), s (↑(IH a_1 h)) b};\n let_fun this := (_ : ∀ (a' : α) (h : r a' a), s (↑(IH a' h)) (↑f a));\n { val := WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S),\n property := (_ : ¬s (↑f a) (WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S))) })\n a)",
"tactic": "rw [WellFounded.fix_eq]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.71243\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\ninst✝ : IsWellOrder β s\nf : r ↪r s\na : α\nb : β\nh : ∀ (a' : α), r a' a → s (↑(collapseF f a')) b\n⊢ ¬s b\n (WellFounded.min (_ : WellFounded s)\n {b |\n ∀ (a_1 : α),\n r a_1 a →\n s\n (↑(WellFounded.fix (_ : WellFounded r)\n (fun a IH =>\n {\n val :=\n WellFounded.min (_ : WellFounded s) {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}\n (_ : ∃ x, x ∈ {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}),\n property :=\n (_ :\n ¬s (↑f a)\n (WellFounded.min (_ : WellFounded s) {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}\n (_ : ∃ x, x ∈ {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}))) })\n a_1))\n b}\n (_ :\n ∃ x,\n x ∈\n {b |\n ∀ (a_1 : α),\n r a_1 a →\n s\n (↑(WellFounded.fix (_ : WellFounded r)\n (fun a IH =>\n {\n val :=\n WellFounded.min (_ : WellFounded s) {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}\n (_ : ∃ x, x ∈ {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}),\n property :=\n (_ :\n ¬s (↑f a)\n (WellFounded.min (_ : WellFounded s)\n {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}\n (_ : ∃ x, x ∈ {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}))) })\n a_1))\n b}))",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.71243\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\ninst✝ : IsWellOrder β s\nf : r ↪r s\na : α\nb : β\nh : ∀ (a' : α), r a' a → s (↑(collapseF f a')) b\n⊢ ¬s b\n ↑(let S :=\n {b |\n ∀ (a_1 : α) (h : r a_1 a),\n s\n (↑((fun y x =>\n WellFounded.fix (_ : WellFounded r)\n (fun a IH =>\n let S := {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b};\n let_fun this := (_ : ∀ (a' : α) (h : r a' a), s (↑(IH a' h)) (↑f a));\n { val := WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S),\n property := (_ : ¬s (↑f a) (WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S))) })\n y)\n a_1 h))\n b};\n let_fun this :=\n (_ :\n ∀ (a' : α),\n r a' a →\n s\n (↑(WellFounded.fix (_ : WellFounded r)\n (fun a IH =>\n let S := {b | ∀ (a_1 : α) (h : r a_1 a), s (↑(IH a_1 h)) b};\n let_fun this := (_ : ∀ (a' : α) (h : r a' a), s (↑(IH a' h)) (↑f a));\n { val := WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S),\n property := (_ : ¬s (↑f a) (WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S))) })\n a'))\n (↑f a));\n { val := WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S),\n property := (_ : ¬s (↑f a) (WellFounded.min (_ : WellFounded s) S (_ : ∃ x, x ∈ S))) })",
"tactic": "dsimp only"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.71243\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\ninst✝ : IsWellOrder β s\nf : r ↪r s\na : α\nb : β\nh : ∀ (a' : α), r a' a → s (↑(collapseF f a')) b\n⊢ ¬s b\n (WellFounded.min (_ : WellFounded s)\n {b |\n ∀ (a_1 : α),\n r a_1 a →\n s\n (↑(WellFounded.fix (_ : WellFounded r)\n (fun a IH =>\n {\n val :=\n WellFounded.min (_ : WellFounded s) {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}\n (_ : ∃ x, x ∈ {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}),\n property :=\n (_ :\n ¬s (↑f a)\n (WellFounded.min (_ : WellFounded s) {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}\n (_ : ∃ x, x ∈ {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}))) })\n a_1))\n b}\n (_ :\n ∃ x,\n x ∈\n {b |\n ∀ (a_1 : α),\n r a_1 a →\n s\n (↑(WellFounded.fix (_ : WellFounded r)\n (fun a IH =>\n {\n val :=\n WellFounded.min (_ : WellFounded s) {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}\n (_ : ∃ x, x ∈ {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}),\n property :=\n (_ :\n ¬s (↑f a)\n (WellFounded.min (_ : WellFounded s)\n {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}\n (_ : ∃ x, x ∈ {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}))) })\n a_1))\n b}))",
"tactic": "exact WellFounded.not_lt_min _ _ _ h"
}
] |
[
527,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.drop_const
|
[] |
[
265,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.VectorMeasure.le_restrict_empty
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.545966\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv w : VectorMeasure α M\nj : Set α\na✝ : MeasurableSet j\n⊢ ↑(restrict v ∅) j ≤ ↑(restrict w ∅) j",
"state_before": "α : Type u_1\nβ : Type ?u.545966\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv w : VectorMeasure α M\n⊢ restrict v ∅ ≤ restrict w ∅",
"tactic": "intro j _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.545966\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv w : VectorMeasure α M\nj : Set α\na✝ : MeasurableSet j\n⊢ ↑(restrict v ∅) j ≤ ↑(restrict w ∅) j",
"tactic": "rw [restrict_empty, restrict_empty]"
}
] |
[
902,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
900,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.cast_mul
|
[] |
[
230,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/Analysis/Convex/Join.lean
|
Convex.convexHull_union
|
[] |
[
198,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
11
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.natDegree_sub_eq_right_of_natDegree_lt
|
[] |
[
1344,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1342,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.coeToSubmodule_injective
|
[] |
[
205,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.isIrreducible_iff_vanishingIdeal_isPrime
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ns : Set (PrimeSpectrum R)\n⊢ IsIrreducible s ↔ Ideal.IsPrime (vanishingIdeal s)",
"tactic": "rw [← isIrreducible_iff_closure, ← zeroLocus_vanishingIdeal_eq_closure,\n isIrreducible_zeroLocus_iff_of_radical _ (isRadical_vanishingIdeal s)]"
}
] |
[
561,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
558,
1
] |
Mathlib/Data/Set/Pointwise/Finite.lean
|
Set.Finite.smul
|
[] |
[
96,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Data/Dfinsupp/Interval.lean
|
Dfinsupp.card_Ioo
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : ι → Type u_2\ninst✝⁴ : DecidableEq ι\ninst✝³ : (i : ι) → DecidableEq (α i)\ninst✝² : (i : ι) → PartialOrder (α i)\ninst✝¹ : (i : ι) → Zero (α i)\ninst✝ : (i : ι) → LocallyFiniteOrder (α i)\nf g : Π₀ (i : ι), α i\n⊢ card (Ioo f g) = ∏ i in support f ∪ support g, card (Icc (↑f i) (↑g i)) - 2",
"tactic": "rw [card_Ioo_eq_card_Icc_sub_two, card_Icc]"
}
] |
[
194,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.Quotient.restrictScalarsEquiv_symm_mk
|
[] |
[
246,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.sub_self_add
|
[
{
"state_after": "n m : Nat\n⊢ n + 0 - (n + m) = 0",
"state_before": "n m : Nat\n⊢ n - (n + m) = 0",
"tactic": "show (n + 0) - (n + m) = 0"
},
{
"state_after": "no goals",
"state_before": "n m : Nat\n⊢ n + 0 - (n + m) = 0",
"tactic": "rw [Nat.add_sub_add_left, Nat.zero_sub]"
}
] |
[
636,
42
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
634,
11
] |
Mathlib/RingTheory/Polynomial/Content.lean
|
Polynomial.IsPrimitive.dvd_primPart_iff_dvd
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : q ≠ 0\nh : p ∣ q\n⊢ p ∣ primPart q",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : q ≠ 0\n⊢ p ∣ primPart q ↔ p ∣ q",
"tactic": "refine' ⟨fun h => h.trans (Dvd.intro_left _ q.eq_C_content_mul_primPart.symm), fun h => _⟩"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nhp : IsPrimitive p\nr : R[X]\nhq : p * r ≠ 0\n⊢ p ∣ primPart (p * r)",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : IsPrimitive p\nhq : q ≠ 0\nh : p ∣ q\n⊢ p ∣ primPart q",
"tactic": "rcases h with ⟨r, rfl⟩"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nhp : IsPrimitive p\nr : R[X]\nhq : p * r ≠ 0\n⊢ p * ?m.597922 = primPart (p * r)\n\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nhp : IsPrimitive p\nr : R[X]\nhq : p * r ≠ 0\n⊢ R[X]",
"state_before": "case intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nhp : IsPrimitive p\nr : R[X]\nhq : p * r ≠ 0\n⊢ p ∣ primPart (p * r)",
"tactic": "apply Dvd.intro _"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nhp : IsPrimitive p\nr : R[X]\nhq : p * r ≠ 0\n⊢ p * ?m.597922 = primPart (p * r)\n\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nhp : IsPrimitive p\nr : R[X]\nhq : p * r ≠ 0\n⊢ R[X]",
"tactic": "rw [primPart_mul hq, hp.primPart_eq]"
}
] |
[
433,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/Topology/MetricSpace/Closeds.lean
|
EMetric.isClosed_subsets_of_isClosed
|
[
{
"state_after": "α : Type u\ninst✝ : EMetricSpace α\ns : Set α\nhs : IsClosed s\nt : Closeds α\nht : t ∈ closure {t | ↑t ⊆ s}\nx : α\nhx : x ∈ ↑t\n⊢ x ∈ s",
"state_before": "α : Type u\ninst✝ : EMetricSpace α\ns : Set α\nhs : IsClosed s\n⊢ IsClosed {t | ↑t ⊆ s}",
"tactic": "refine' isClosed_of_closure_subset fun t ht x hx => _"
},
{
"state_after": "α : Type u\ninst✝ : EMetricSpace α\ns : Set α\nhs : IsClosed s\nt : Closeds α\nht : t ∈ closure {t | ↑t ⊆ s}\nx : α\nhx : x ∈ ↑t\nthis : x ∈ closure s\n⊢ x ∈ s",
"state_before": "α : Type u\ninst✝ : EMetricSpace α\ns : Set α\nhs : IsClosed s\nt : Closeds α\nht : t ∈ closure {t | ↑t ⊆ s}\nx : α\nhx : x ∈ ↑t\n⊢ x ∈ s",
"tactic": "have : x ∈ closure s := by\n refine' mem_closure_iff.2 fun ε εpos => _\n rcases mem_closure_iff.1 ht ε εpos with ⟨u, hu, Dtu⟩\n rcases exists_edist_lt_of_hausdorffEdist_lt hx Dtu with ⟨y, hy, Dxy⟩\n exact ⟨y, hu hy, Dxy⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : EMetricSpace α\ns : Set α\nhs : IsClosed s\nt : Closeds α\nht : t ∈ closure {t | ↑t ⊆ s}\nx : α\nhx : x ∈ ↑t\nthis : x ∈ closure s\n⊢ x ∈ s",
"tactic": "rwa [hs.closure_eq] at this"
},
{
"state_after": "α : Type u\ninst✝ : EMetricSpace α\ns : Set α\nhs : IsClosed s\nt : Closeds α\nht : t ∈ closure {t | ↑t ⊆ s}\nx : α\nhx : x ∈ ↑t\nε : ℝ≥0∞\nεpos : ε > 0\n⊢ ∃ y, y ∈ s ∧ edist x y < ε",
"state_before": "α : Type u\ninst✝ : EMetricSpace α\ns : Set α\nhs : IsClosed s\nt : Closeds α\nht : t ∈ closure {t | ↑t ⊆ s}\nx : α\nhx : x ∈ ↑t\n⊢ x ∈ closure s",
"tactic": "refine' mem_closure_iff.2 fun ε εpos => _"
},
{
"state_after": "case intro.intro\nα : Type u\ninst✝ : EMetricSpace α\ns : Set α\nhs : IsClosed s\nt : Closeds α\nht : t ∈ closure {t | ↑t ⊆ s}\nx : α\nhx : x ∈ ↑t\nε : ℝ≥0∞\nεpos : ε > 0\nu : Closeds α\nhu : u ∈ {t | ↑t ⊆ s}\nDtu : edist t u < ε\n⊢ ∃ y, y ∈ s ∧ edist x y < ε",
"state_before": "α : Type u\ninst✝ : EMetricSpace α\ns : Set α\nhs : IsClosed s\nt : Closeds α\nht : t ∈ closure {t | ↑t ⊆ s}\nx : α\nhx : x ∈ ↑t\nε : ℝ≥0∞\nεpos : ε > 0\n⊢ ∃ y, y ∈ s ∧ edist x y < ε",
"tactic": "rcases mem_closure_iff.1 ht ε εpos with ⟨u, hu, Dtu⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u\ninst✝ : EMetricSpace α\ns : Set α\nhs : IsClosed s\nt : Closeds α\nht : t ∈ closure {t | ↑t ⊆ s}\nx : α\nhx : x ∈ ↑t\nε : ℝ≥0∞\nεpos : ε > 0\nu : Closeds α\nhu : u ∈ {t | ↑t ⊆ s}\nDtu : edist t u < ε\ny : α\nhy : y ∈ ↑u\nDxy : edist x y < ε\n⊢ ∃ y, y ∈ s ∧ edist x y < ε",
"state_before": "case intro.intro\nα : Type u\ninst✝ : EMetricSpace α\ns : Set α\nhs : IsClosed s\nt : Closeds α\nht : t ∈ closure {t | ↑t ⊆ s}\nx : α\nhx : x ∈ ↑t\nε : ℝ≥0∞\nεpos : ε > 0\nu : Closeds α\nhu : u ∈ {t | ↑t ⊆ s}\nDtu : edist t u < ε\n⊢ ∃ y, y ∈ s ∧ edist x y < ε",
"tactic": "rcases exists_edist_lt_of_hausdorffEdist_lt hx Dtu with ⟨y, hy, Dxy⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u\ninst✝ : EMetricSpace α\ns : Set α\nhs : IsClosed s\nt : Closeds α\nht : t ∈ closure {t | ↑t ⊆ s}\nx : α\nhx : x ∈ ↑t\nε : ℝ≥0∞\nεpos : ε > 0\nu : Closeds α\nhu : u ∈ {t | ↑t ⊆ s}\nDtu : edist t u < ε\ny : α\nhy : y ∈ ↑u\nDxy : edist x y < ε\n⊢ ∃ y, y ∈ s ∧ edist x y < ε",
"tactic": "exact ⟨y, hu hy, Dxy⟩"
}
] |
[
88,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
IsLowerSet.ordConnected
|
[] |
[
242,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/CategoryTheory/Over.lean
|
CategoryTheory.Under.under_left
|
[
{
"state_after": "no goals",
"state_before": "T : Type u₁\ninst✝ : Category T\nX : T\nU : Under X\n⊢ U.left = { as := PUnit.unit }",
"tactic": "simp only"
}
] |
[
336,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Data/PNat/Prime.lean
|
PNat.gcd_mul_lcm
|
[] |
[
103,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
lowerSemicontinuous_iSup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nβ : Type ?u.128024\ninst✝² : Preorder β\nf✝ g : α → β\nx : α\ns t : Set α\ny z : β\nι : Sort u_3\nδ : Type u_2\nδ' : Type ?u.128056\ninst✝¹ : CompleteLinearOrder δ\ninst✝ : ConditionallyCompleteLinearOrder δ'\nf : ι → α → δ\nh : ∀ (i : ι), LowerSemicontinuous (f i)\n⊢ ∀ (x : α), BddAbove (range fun i => f i x)",
"tactic": "simp"
}
] |
[
619,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
617,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingIso.coe_mk
|
[] |
[
400,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
399,
1
] |
Mathlib/GroupTheory/GroupAction/Group.lean
|
MulAction.injective
|
[] |
[
154,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
11
] |
Mathlib/Topology/Bases.lean
|
TopologicalSpace.IsTopologicalBasis.open_eq_sUnion'
|
[] |
[
191,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Data/Int/Bitwise.lean
|
Int.bitwise_and
|
[
{
"state_after": "case h.h\nm n : ℤ\n⊢ bitwise and m n = land m n",
"state_before": "⊢ bitwise and = land",
"tactic": "funext m n"
},
{
"state_after": "case h.h.negSucc.ofNat\nm n : ℕ\n⊢ ↑(Nat.bitwise' (fun x y => !x && y) m n) = land -[m+1] (ofNat n)\n\ncase h.h.negSucc.negSucc\nm n : ℕ\n⊢ -[Nat.bitwise' (fun x y => !(!x && !y)) m n+1] = land -[m+1] -[n+1]",
"state_before": "case h.h\nm n : ℤ\n⊢ bitwise and m n = land m n",
"tactic": "cases' m with m m <;> cases' n with n n <;> try {rfl}\n <;> simp only [bitwise, natBitwise, Bool.not_false, Bool.or_true,\n cond_false, cond_true, lor, Nat.ldiff', Bool.and_true, negSucc.injEq,\n Bool.and_false, Nat.land']"
},
{
"state_after": "case h.h.negSucc.negSucc\nm n : ℕ\n⊢ -[Nat.bitwise' (fun x y => !(!x && !y)) m n+1] = land -[m+1] -[n+1]",
"state_before": "case h.h.negSucc.ofNat\nm n : ℕ\n⊢ ↑(Nat.bitwise' (fun x y => !x && y) m n) = land -[m+1] (ofNat n)\n\ncase h.h.negSucc.negSucc\nm n : ℕ\n⊢ -[Nat.bitwise' (fun x y => !(!x && !y)) m n+1] = land -[m+1] -[n+1]",
"tactic": ". rw [Nat.bitwise'_swap, Function.swap]\n congr\n funext x y\n cases x <;> cases y <;> rfl\n rfl"
},
{
"state_after": "no goals",
"state_before": "case h.h.negSucc.negSucc\nm n : ℕ\n⊢ -[Nat.bitwise' (fun x y => !(!x && !y)) m n+1] = land -[m+1] -[n+1]",
"tactic": ". congr\n funext x y\n cases x <;> cases y <;> rfl"
},
{
"state_after": "case h.h.negSucc.ofNat\nm n : ℕ\n⊢ ↑(Nat.bitwise' (fun y x => !x && y) n m) = land -[m+1] (ofNat n)\n\ncase h.h.negSucc.ofNat\nm n : ℕ\n⊢ (!false && false) = false",
"state_before": "case h.h.negSucc.ofNat\nm n : ℕ\n⊢ ↑(Nat.bitwise' (fun x y => !x && y) m n) = land -[m+1] (ofNat n)",
"tactic": "rw [Nat.bitwise'_swap, Function.swap]"
},
{
"state_after": "case h.h.negSucc.ofNat.e_a.e_f\nm n : ℕ\n⊢ (fun y x => !x && y) = fun a b => a && !b\n\ncase h.h.negSucc.ofNat\nm n : ℕ\n⊢ (!false && false) = false",
"state_before": "case h.h.negSucc.ofNat\nm n : ℕ\n⊢ ↑(Nat.bitwise' (fun y x => !x && y) n m) = land -[m+1] (ofNat n)\n\ncase h.h.negSucc.ofNat\nm n : ℕ\n⊢ (!false && false) = false",
"tactic": "congr"
},
{
"state_after": "case h.h.negSucc.ofNat.e_a.e_f.h.h\nm n : ℕ\nx y : Bool\n⊢ (!y && x) = (x && !y)\n\ncase h.h.negSucc.ofNat\nm n : ℕ\n⊢ (!false && false) = false",
"state_before": "case h.h.negSucc.ofNat.e_a.e_f\nm n : ℕ\n⊢ (fun y x => !x && y) = fun a b => a && !b\n\ncase h.h.negSucc.ofNat\nm n : ℕ\n⊢ (!false && false) = false",
"tactic": "funext x y"
},
{
"state_after": "case h.h.negSucc.ofNat\nm n : ℕ\n⊢ (!false && false) = false",
"state_before": "case h.h.negSucc.ofNat.e_a.e_f.h.h\nm n : ℕ\nx y : Bool\n⊢ (!y && x) = (x && !y)\n\ncase h.h.negSucc.ofNat\nm n : ℕ\n⊢ (!false && false) = false",
"tactic": "cases x <;> cases y <;> rfl"
},
{
"state_after": "no goals",
"state_before": "case h.h.negSucc.ofNat\nm n : ℕ\n⊢ (!false && false) = false",
"tactic": "rfl"
},
{
"state_after": "case h.h.negSucc.negSucc.e_a.e_f\nm n : ℕ\n⊢ (fun x y => !(!x && !y)) = or",
"state_before": "case h.h.negSucc.negSucc\nm n : ℕ\n⊢ -[Nat.bitwise' (fun x y => !(!x && !y)) m n+1] = land -[m+1] -[n+1]",
"tactic": "congr"
},
{
"state_after": "case h.h.negSucc.negSucc.e_a.e_f.h.h\nm n : ℕ\nx y : Bool\n⊢ (!(!x && !y)) = (x || y)",
"state_before": "case h.h.negSucc.negSucc.e_a.e_f\nm n : ℕ\n⊢ (fun x y => !(!x && !y)) = or",
"tactic": "funext x y"
},
{
"state_after": "no goals",
"state_before": "case h.h.negSucc.negSucc.e_a.e_f.h.h\nm n : ℕ\nx y : Bool\n⊢ (!(!x && !y)) = (x || y)",
"tactic": "cases x <;> cases y <;> rfl"
}
] |
[
255,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
BotHom.id_comp
|
[] |
[
473,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
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