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/Data/Finmap.lean
|
Finmap.union_comm_of_disjoint
|
[
{
"state_after": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\ns₁✝ s₂✝ : Finmap β\ns₁ s₂ : AList β\nh : Disjoint ⟦s₁⟧ ⟦s₂⟧\n⊢ ⟦s₁⟧ ∪ ⟦s₂⟧ = ⟦s₂⟧ ∪ ⟦s₁⟧",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\ns₁✝ s₂✝ : Finmap β\ns₁ s₂ : AList β\n⊢ Disjoint ⟦s₁⟧ ⟦s₂⟧ → ⟦s₁⟧ ∪ ⟦s₂⟧ = ⟦s₂⟧ ∪ ⟦s₁⟧",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\ns₁✝ s₂✝ : Finmap β\ns₁ s₂ : AList β\nh : Disjoint ⟦s₁⟧ ⟦s₂⟧\n⊢ ⟦s₁⟧ ∪ ⟦s₂⟧ = ⟦s₂⟧ ∪ ⟦s₁⟧",
"tactic": "simp only [AList.toFinmap_eq, union_toFinmap, AList.union_comm_of_disjoint h]"
}
] |
[
685,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
682,
1
] |
Mathlib/Data/Sym/Sym2.lean
|
Sym2.relBool_spec
|
[
{
"state_after": "case mk\nα : Type u_1\nβ : Type ?u.59102\nγ : Type ?u.59105\ninst✝ : DecidableEq α\ny : α × α\nx₁ x₂ : α\n⊢ relBool (x₁, x₂) y = true ↔ Rel α (x₁, x₂) y",
"state_before": "α : Type u_1\nβ : Type ?u.59102\nγ : Type ?u.59105\ninst✝ : DecidableEq α\nx y : α × α\n⊢ relBool x y = true ↔ Rel α x y",
"tactic": "cases' x with x₁ x₂"
},
{
"state_after": "case mk.mk\nα : Type u_1\nβ : Type ?u.59102\nγ : Type ?u.59105\ninst✝ : DecidableEq α\nx₁ x₂ y₁ y₂ : α\n⊢ relBool (x₁, x₂) (y₁, y₂) = true ↔ Rel α (x₁, x₂) (y₁, y₂)",
"state_before": "case mk\nα : Type u_1\nβ : Type ?u.59102\nγ : Type ?u.59105\ninst✝ : DecidableEq α\ny : α × α\nx₁ x₂ : α\n⊢ relBool (x₁, x₂) y = true ↔ Rel α (x₁, x₂) y",
"tactic": "cases' y with y₁ y₂"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nα : Type u_1\nβ : Type ?u.59102\nγ : Type ?u.59105\ninst✝ : DecidableEq α\nx₁ x₂ y₁ y₂ : α\n⊢ relBool (x₁, x₂) (y₁, y₂) = true ↔ Rel α (x₁, x₂) (y₁, y₂)",
"tactic": "aesop (rule_sets [Sym2]) (add norm unfold [relBool])"
}
] |
[
647,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
645,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
extChartAt_preimage_mem_nhdsWithin
|
[] |
[
1215,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1213,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.HasFiniteIntegral.mono
|
[
{
"state_after": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.704421\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\ng : α → γ\nh : ∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ ‖g a‖\nhg : (∫⁻ (a : α), ENNReal.ofReal ‖g a‖ ∂μ) < ⊤\n⊢ (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ) < ⊤",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.704421\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\ng : α → γ\nhg : HasFiniteIntegral g\nh : ∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ ‖g a‖\n⊢ HasFiniteIntegral f",
"tactic": "simp only [hasFiniteIntegral_iff_norm] at *"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.704421\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\ng : α → γ\nh : ∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ ‖g a‖\nhg : (∫⁻ (a : α), ENNReal.ofReal ‖g a‖ ∂μ) < ⊤\n⊢ (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ) < ⊤",
"tactic": "calc\n (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) ≤ ∫⁻ a : α, ENNReal.ofReal ‖g a‖ ∂μ :=\n lintegral_mono_ae (h.mono fun a h => ofReal_le_ofReal h)\n _ < ∞ := hg"
}
] |
[
146,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
strictConcaveOn_of_deriv2_neg'
|
[] |
[
1247,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1245,
1
] |
Mathlib/Tactic/Abel.lean
|
Mathlib.Tactic.Abel.unfold_smul
|
[] |
[
255,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Logic/Equiv/Fin.lean
|
finSuccEquiv'_at
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\ni : Fin (n + 1)\n⊢ ↑(finSuccEquiv' i) i = none",
"tactic": "simp [finSuccEquiv']"
}
] |
[
142,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Topology/Perfect.lean
|
exists_countable_union_perfect_of_isClosed
|
[
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\n⊢ ∃ V D, Set.Countable V ∧ Perfect D ∧ C = V ∪ D",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\n⊢ ∃ V D, Set.Countable V ∧ Perfect D ∧ C = V ∪ D",
"tactic": "obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\n⊢ ∃ V D, Set.Countable V ∧ Perfect D ∧ C = V ∪ D",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\n⊢ ∃ V D, Set.Countable V ∧ Perfect D ∧ C = V ∪ D",
"tactic": "let v := { U ∈ b | (U ∩ C).Countable }"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\n⊢ ∃ V D, Set.Countable V ∧ Perfect D ∧ C = V ∪ D",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\n⊢ ∃ V D, Set.Countable V ∧ Perfect D ∧ C = V ∪ D",
"tactic": "let V := ⋃ U ∈ v, U"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\n⊢ ∃ V D, Set.Countable V ∧ Perfect D ∧ C = V ∪ D",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\n⊢ ∃ V D, Set.Countable V ∧ Perfect D ∧ C = V ∪ D",
"tactic": "let D := C \\ V"
},
{
"state_after": "case intro.intro.intro.refine'_1\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\n⊢ IsClosed D\n\ncase intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\n⊢ Preperfect D\n\ncase intro.intro.intro.refine'_3\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\n⊢ C = V ∩ C ∪ D",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\n⊢ ∃ V D, Set.Countable V ∧ Perfect D ∧ C = V ∪ D",
"tactic": "refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩"
},
{
"state_after": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\n⊢ Set.Countable (⋃ (i : Set α) (_ : i ∈ b ∧ Set.Countable (i ∩ C)), i ∩ C)",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\n⊢ Set.Countable (V ∩ C)",
"tactic": "simp only [iUnion_inter, mem_sep_iff]"
},
{
"state_after": "case hs\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\n⊢ Set.Countable fun a => a ∈ b ∧ Set.Countable (a ∩ C)\n\ncase a\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\n⊢ ∀ (a : Set α), (a ∈ fun a => a ∈ b ∧ Set.Countable (a ∩ C)) → Set.Countable (a ∩ C)",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\n⊢ Set.Countable (⋃ (i : Set α) (_ : i ∈ b ∧ Set.Countable (i ∩ C)), i ∩ C)",
"tactic": "apply Countable.biUnion"
},
{
"state_after": "no goals",
"state_before": "case hs\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\n⊢ Set.Countable fun a => a ∈ b ∧ Set.Countable (a ∩ C)",
"tactic": "exact Countable.mono (inter_subset_left _ _) bct"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\n⊢ ∀ (a : Set α), (a ∈ fun a => a ∈ b ∧ Set.Countable (a ∩ C)) → Set.Countable (a ∩ C)",
"tactic": "exact inter_subset_right _ _"
},
{
"state_after": "case intro.intro.intro.refine'_1\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx✝ : Set α\n⊢ x✝ ∈ v → IsOpen x✝",
"state_before": "case intro.intro.intro.refine'_1\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\n⊢ IsClosed D",
"tactic": "refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.refine'_1\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx✝ : Set α\n⊢ x✝ ∈ v → IsOpen x✝",
"tactic": "exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub"
},
{
"state_after": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\n⊢ ∀ (x : α), x ∈ D → ∀ (U : Set α), U ∈ 𝓝 x → ∃ y, y ∈ U ∩ D ∧ y ≠ x",
"state_before": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\n⊢ Preperfect D",
"tactic": "rw [preperfect_iff_nhds]"
},
{
"state_after": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\n⊢ ∃ y, y ∈ E ∩ D ∧ y ≠ x",
"state_before": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\n⊢ ∀ (x : α), x ∈ D → ∀ (U : Set α), U ∈ 𝓝 x → ∃ y, y ∈ U ∩ D ∧ y ≠ x",
"tactic": "intro x xD E xE"
},
{
"state_after": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nthis : ¬Set.Countable (E ∩ D)\nh : ¬∃ y, y ∈ E ∩ D ∧ y ≠ x\n⊢ False",
"state_before": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nthis : ¬Set.Countable (E ∩ D)\n⊢ ∃ y, y ∈ E ∩ D ∧ y ≠ x",
"tactic": "by_contra h"
},
{
"state_after": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nthis : ¬Set.Countable (E ∩ D)\nh : ∀ (x_1 : α), ¬(x_1 ∈ E ∩ D ∧ x_1 ≠ x)\n⊢ False",
"state_before": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nthis : ¬Set.Countable (E ∩ D)\nh : ¬∃ y, y ∈ E ∩ D ∧ y ≠ x\n⊢ False",
"tactic": "rw [not_exists] at h"
},
{
"state_after": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nthis : ¬Set.Countable (E ∩ D)\nh : ∀ (x_1 : α), x_1 ∈ E ∩ (C \\ V) → x_1 = x\n⊢ False",
"state_before": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nthis : ¬Set.Countable (E ∩ D)\nh : ∀ (x_1 : α), ¬(x_1 ∈ E ∩ D ∧ x_1 ≠ x)\n⊢ False",
"tactic": "push_neg at h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nthis : ¬Set.Countable (E ∩ D)\nh : ∀ (x_1 : α), x_1 ∈ E ∩ (C \\ V) → x_1 = x\n⊢ False",
"tactic": "exact absurd (Countable.mono h (Set.countable_singleton _)) this"
},
{
"state_after": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\n⊢ False",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\n⊢ ¬Set.Countable (E ∩ D)",
"tactic": "intro h"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\n⊢ False",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\n⊢ False",
"tactic": "obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=\n (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\nhU_cnt : Set.Countable (U ∩ C)\nthis : U ∈ v\n⊢ False",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\nhU_cnt : Set.Countable (U ∩ C)\n⊢ False",
"tactic": "have : U ∈ v := ⟨hUb, hU_cnt⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\nhU_cnt : Set.Countable (U ∩ C)\nthis : U ∈ v\n⊢ x ∈ V",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\nhU_cnt : Set.Countable (U ∩ C)\nthis : U ∈ v\n⊢ False",
"tactic": "apply xD.2"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\nhU_cnt : Set.Countable (U ∩ C)\nthis : U ∈ v\n⊢ x ∈ V",
"tactic": "exact mem_biUnion this xU"
},
{
"state_after": "case h\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\n⊢ U ∩ C ⊆ E ∩ D ∪ V ∩ C\n\ncase a\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\n⊢ Set.Countable (E ∩ D ∪ V ∩ C)",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\n⊢ Set.Countable (U ∩ C)",
"tactic": "apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\n⊢ Set.Countable (E ∩ D ∪ V ∩ C)",
"tactic": "exact Countable.union h Vct"
},
{
"state_after": "case h.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\ny : α\nyU : y ∈ U\nyC : y ∈ C\n⊢ y ∈ E ∩ D ∪ V ∩ C",
"state_before": "case h\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\n⊢ U ∩ C ⊆ E ∩ D ∪ V ∩ C",
"tactic": "rintro y ⟨yU, yC⟩"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh✝ : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\ny : α\nyU : y ∈ U\nyC : y ∈ C\nh : y ∈ V\n⊢ y ∈ E ∩ D ∪ V ∩ C\n\ncase neg\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh✝ : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\ny : α\nyU : y ∈ U\nyC : y ∈ C\nh : ¬y ∈ V\n⊢ y ∈ E ∩ D ∪ V ∩ C",
"state_before": "case h.intro\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\ny : α\nyU : y ∈ U\nyC : y ∈ C\n⊢ y ∈ E ∩ D ∪ V ∩ C",
"tactic": "by_cases h : y ∈ V"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh✝ : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\ny : α\nyU : y ∈ U\nyC : y ∈ C\nh : y ∈ V\n⊢ y ∈ E ∩ D ∪ V ∩ C",
"tactic": "exact mem_union_right _ (mem_inter h yC)"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\nx : α\nxD : x ∈ D\nE : Set α\nxE : E ∈ 𝓝 x\nh✝ : Set.Countable (E ∩ D)\nU : Set α\nhUb : U ∈ b\nxU : x ∈ U\nhU : U ⊆ E\ny : α\nyU : y ∈ U\nyC : y ∈ C\nh : ¬y ∈ V\n⊢ y ∈ E ∩ D ∪ V ∩ C",
"tactic": "exact mem_union_left _ (mem_inter (hU yU) ⟨yC, h⟩)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.refine'_3\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed C\nb : Set (Set α)\nbct : Set.Countable b\nleft✝ : ¬∅ ∈ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}\nV : Set α := ⋃ (U : Set α) (_ : U ∈ v), U\nD : Set α := C \\ V\nVct : Set.Countable (V ∩ C)\n⊢ C = V ∩ C ∪ D",
"tactic": "rw [inter_comm, inter_union_diff]"
}
] |
[
202,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Algebra/Group/OrderSynonym.lean
|
ofLex_pow
|
[] |
[
290,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Order/BoundedOrder.lean
|
OrderDual.toDual_bot
|
[] |
[
305,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/NumberTheory/Multiplicity.lean
|
multiplicity.pow_prime_sub_pow_prime
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nn : ℕ\ninst✝² : CommRing R\na b x y : R\np : ℕ\ninst✝¹ : IsDomain R\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nhp : Prime ↑p\nhp1 : Odd p\nhxy : ↑p ∣ x - y\nhx : ¬↑p ∣ x\n⊢ multiplicity (↑p) (x ^ p - y ^ p) = multiplicity (↑p) (x - y) + 1",
"tactic": "rw [← geom_sum₂_mul, multiplicity.mul hp, geom_sum₂_eq_one hp hp1 hxy hx, add_comm]"
}
] |
[
184,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
borel_eq_generateFrom_Ioi
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.118128\nγ : Type ?u.118131\nγ₂ : Type ?u.118134\nδ : Type ?u.118137\nι : Sort y\ns t u : Set α\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\n⊢ SecondCountableTopology α",
"tactic": "infer_instance"
}
] |
[
157,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.Memℒp.of_nnnorm_le_mul
|
[] |
[
1320,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1316,
1
] |
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
|
MeasureTheory.LocallyIntegrable.locallyIntegrableOn
|
[] |
[
133,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.mk_eq_coe
|
[] |
[
473,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
Convex.monotoneOn_of_deriv_nonneg
|
[
{
"state_after": "no goals",
"state_before": "E : Type ?u.415583\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.415679\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : DifferentiableOn ℝ f (interior D)\nhf'_nonneg : ∀ (x : ℝ), x ∈ interior D → 0 ≤ deriv f x\nx : ℝ\nhx : x ∈ D\ny : ℝ\nhy : y ∈ D\nhxy : x ≤ y\n⊢ f x ≤ f y",
"tactic": "simpa only [zero_mul, sub_nonneg] using\n hD.mul_sub_le_image_sub_of_le_deriv hf hf' hf'_nonneg x hx y hy hxy"
}
] |
[
918,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
914,
1
] |
Mathlib/Topology/Separation.lean
|
minimal_nonempty_open_subsingleton
|
[
{
"state_after": "α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\n⊢ Set.Subsingleton s",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\n⊢ Set.Subsingleton s",
"tactic": "clear β"
},
{
"state_after": "α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : ¬x = y\n⊢ False",
"state_before": "α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\n⊢ Set.Subsingleton s",
"tactic": "refine' fun x hx y hy => of_not_not fun hxy => _"
},
{
"state_after": "case intro.intro\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : ¬x = y\nU : Set α\nhUo : IsOpen U\nhU : Xor' (x ∈ U) (y ∈ U)\n⊢ False",
"state_before": "α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : ¬x = y\n⊢ False",
"tactic": "rcases exists_isOpen_xor'_mem hxy with ⟨U, hUo, hU⟩"
},
{
"state_after": "case intro.intro.inr\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : ¬x = y\nU : Set α\nhUo : IsOpen U\nhU : Xor' (x ∈ U) (y ∈ U)\nthis :\n ∀ {α : Type u} [inst : TopologicalSpace α] [inst_1 : T0Space α] {s : Set α},\n IsOpen s →\n (∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s) →\n ∀ (x : α),\n x ∈ s → ∀ (y : α), y ∈ s → ¬x = y → ∀ (U : Set α), IsOpen U → Xor' (x ∈ U) (y ∈ U) → x ∈ U ∧ ¬y ∈ U → False\nh : ¬(x ∈ U ∧ ¬y ∈ U)\n⊢ False\n\nα✝ : Type u\ninst✝² : TopologicalSpace α✝\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : ¬x = y\nU : Set α\nhUo : IsOpen U\nhU : Xor' (x ∈ U) (y ∈ U)\nh : x ∈ U ∧ ¬y ∈ U\n⊢ False",
"state_before": "case intro.intro\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : ¬x = y\nU : Set α\nhUo : IsOpen U\nhU : Xor' (x ∈ U) (y ∈ U)\n⊢ False",
"tactic": "wlog h : x ∈ U ∧ y ∉ U"
},
{
"state_after": "case intro\nα✝ : Type u\ninst✝² : TopologicalSpace α✝\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : ¬x = y\nU : Set α\nhUo : IsOpen U\nhU : Xor' (x ∈ U) (y ∈ U)\nhxU : x ∈ U\nhyU : ¬y ∈ U\n⊢ False",
"state_before": "α✝ : Type u\ninst✝² : TopologicalSpace α✝\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : ¬x = y\nU : Set α\nhUo : IsOpen U\nhU : Xor' (x ∈ U) (y ∈ U)\nh : x ∈ U ∧ ¬y ∈ U\n⊢ False",
"tactic": "cases' h with hxU hyU"
},
{
"state_after": "case intro\nα✝ : Type u\ninst✝² : TopologicalSpace α✝\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : ¬x = y\nU : Set α\nhUo : IsOpen U\nhU : Xor' (x ∈ U) (y ∈ U)\nhxU : x ∈ U\nhyU : ¬y ∈ U\nthis : s ∩ U = s\n⊢ False",
"state_before": "case intro\nα✝ : Type u\ninst✝² : TopologicalSpace α✝\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : ¬x = y\nU : Set α\nhUo : IsOpen U\nhU : Xor' (x ∈ U) (y ∈ U)\nhxU : x ∈ U\nhyU : ¬y ∈ U\n⊢ False",
"tactic": "have : s ∩ U = s := hmin (s ∩ U) (inter_subset_left _ _) ⟨x, hx, hxU⟩ (hs.inter hUo)"
},
{
"state_after": "no goals",
"state_before": "case intro\nα✝ : Type u\ninst✝² : TopologicalSpace α✝\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : ¬x = y\nU : Set α\nhUo : IsOpen U\nhU : Xor' (x ∈ U) (y ∈ U)\nhxU : x ∈ U\nhyU : ¬y ∈ U\nthis : s ∩ U = s\n⊢ False",
"tactic": "exact hyU (this.symm.subset hy).2"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inr\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : T0Space α\ns : Set α\nhs : IsOpen s\nhmin : ∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nhxy : ¬x = y\nU : Set α\nhUo : IsOpen U\nhU : Xor' (x ∈ U) (y ∈ U)\nthis :\n ∀ {α : Type u} [inst : TopologicalSpace α] [inst_1 : T0Space α] {s : Set α},\n IsOpen s →\n (∀ (t : Set α), t ⊆ s → Set.Nonempty t → IsOpen t → t = s) →\n ∀ (x : α),\n x ∈ s → ∀ (y : α), y ∈ s → ¬x = y → ∀ (U : Set α), IsOpen U → Xor' (x ∈ U) (y ∈ U) → x ∈ U ∧ ¬y ∈ U → False\nh : ¬(x ∈ U ∧ ¬y ∈ U)\n⊢ False",
"tactic": "exact this hs hmin y hy x hx (Ne.symm hxy) U hUo hU.symm (hU.resolve_left h)"
}
] |
[
292,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.C_inv
|
[] |
[
2192,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2191,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.surjInv_eq
|
[] |
[
491,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
490,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Homeomorph.isLittleO_congr
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nE : Type u_3\ninst✝¹ : Norm E\nF : Type u_4\ninst✝ : Norm F\ne : α ≃ₜ β\nb : β\nf : β → E\ng : β → F\n⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c (𝓝 b) f g) ↔\n ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c (𝓝 (↑(Homeomorph.symm e) b)) (f ∘ ↑e) (g ∘ ↑e)",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nE : Type u_3\ninst✝¹ : Norm E\nF : Type u_4\ninst✝ : Norm F\ne : α ≃ₜ β\nb : β\nf : β → E\ng : β → F\n⊢ f =o[𝓝 b] g ↔ (f ∘ ↑e) =o[𝓝 (↑(Homeomorph.symm e) b)] (g ∘ ↑e)",
"tactic": "simp only [IsLittleO_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nE : Type u_3\ninst✝¹ : Norm E\nF : Type u_4\ninst✝ : Norm F\ne : α ≃ₜ β\nb : β\nf : β → E\ng : β → F\n⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c (𝓝 b) f g) ↔\n ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c (𝓝 (↑(Homeomorph.symm e) b)) (f ∘ ↑e) (g ∘ ↑e)",
"tactic": "exact forall₂_congr fun c _hc => e.isBigOWith_congr"
}
] |
[
2237,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2234,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
monotoneOn_id
|
[] |
[
514,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
514,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.mul_inv_rev
|
[
{
"state_after": "l : Type ?u.349525\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B✝ A B : Matrix n n α\n⊢ Ring.inverse (det (A ⬝ B)) • adjugate (A ⬝ B) =\n (Ring.inverse (det B) • adjugate B) ⬝ (Ring.inverse (det A) • adjugate A)",
"state_before": "l : Type ?u.349525\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B✝ A B : Matrix n n α\n⊢ (A ⬝ B)⁻¹ = B⁻¹ ⬝ A⁻¹",
"tactic": "simp only [inv_def]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.349525\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B✝ A B : Matrix n n α\n⊢ Ring.inverse (det (A ⬝ B)) • adjugate (A ⬝ B) =\n (Ring.inverse (det B) • adjugate B) ⬝ (Ring.inverse (det A) • adjugate A)",
"tactic": "rw [Matrix.smul_mul, Matrix.mul_smul, smul_smul, det_mul, adjugate_mul_distrib,\n Ring.mul_inverse_rev]"
}
] |
[
591,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
588,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Memℒp.toLp_sub
|
[] |
[
150,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.find_mono
|
[] |
[
854,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
852,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
HasFPowerSeriesOnBall.eventually_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.469661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f 0 x r\n⊢ ∀ᶠ (z : E) in 𝓝 x, f z = 0",
"tactic": "filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero"
}
] |
[
509,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
Submodule.finiteDimensional_of_le
|
[] |
[
709,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
704,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Set.toFinset_nonempty
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.81654\nγ : Type ?u.81657\ns✝ t s : Set α\ninst✝ : Fintype ↑s\n⊢ Finset.Nonempty (toFinset s) ↔ Set.Nonempty s",
"tactic": "rw [← Finset.coe_nonempty, coe_toFinset]"
}
] |
[
643,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
642,
1
] |
Mathlib/Order/Minimal.lean
|
maximals_of_symm
|
[] |
[
111,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.adjoin_eq_range_algebraMap_adjoin
|
[] |
[
303,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean
|
MeasureTheory.measure_lt_top_of_isCompact_of_isMulLeftInvariant'
|
[] |
[
603,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
600,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.erase_val
|
[] |
[
1855,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1854,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
GaloisConnection.u_csInf'
|
[] |
[
1305,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1303,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieModuleHom.coe_smul
|
[] |
[
938,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
937,
1
] |
Mathlib/Order/PFilter.lean
|
Order.PFilter.directed
|
[] |
[
93,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Data/Sum/Order.lean
|
Sum.not_inr_lt_inl
|
[] |
[
168,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
coe_subset_nonunits
|
[] |
[
851,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
850,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.toReal_mem_Ioc
|
[] |
[
575,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/Computability/PartrecCode.lean
|
Nat.Partrec.Code.encode_lt_pair
|
[
{
"state_after": "cf cg : Code\n⊢ encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 ∧\n encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4",
"state_before": "cf cg : Code\n⊢ encode cf < encode (pair cf cg) ∧ encode cg < encode (pair cf cg)",
"tactic": "simp [encodeCode_eq, encodeCode]"
},
{
"state_after": "cf cg : Code\nthis : 1 * Nat.pair (encodeCode cf) (encodeCode cg) ≤ 2 * 2 * Nat.pair (encodeCode cf) (encodeCode cg)\n⊢ encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 ∧\n encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4",
"state_before": "cf cg : Code\n⊢ encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 ∧\n encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4",
"tactic": "have := Nat.mul_le_mul_right (Nat.pair cf.encodeCode cg.encodeCode) (by decide : 1 ≤ 2 * 2)"
},
{
"state_after": "cf cg : Code\nthis : Nat.pair (encodeCode cf) (encodeCode cg) ≤ 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg))\n⊢ encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 ∧\n encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4",
"state_before": "cf cg : Code\nthis : 1 * Nat.pair (encodeCode cf) (encodeCode cg) ≤ 2 * 2 * Nat.pair (encodeCode cf) (encodeCode cg)\n⊢ encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 ∧\n encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4",
"tactic": "rw [one_mul, mul_assoc] at this"
},
{
"state_after": "cf cg : Code\nthis✝ : Nat.pair (encodeCode cf) (encodeCode cg) ≤ 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg))\nthis : Nat.pair (encodeCode cf) (encodeCode cg) < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4\n⊢ encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 ∧\n encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4",
"state_before": "cf cg : Code\nthis : Nat.pair (encodeCode cf) (encodeCode cg) ≤ 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg))\n⊢ encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 ∧\n encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4",
"tactic": "have := lt_of_le_of_lt this (lt_add_of_pos_right _ (by decide : 0 < 4))"
},
{
"state_after": "no goals",
"state_before": "cf cg : Code\nthis✝ : Nat.pair (encodeCode cf) (encodeCode cg) ≤ 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg))\nthis : Nat.pair (encodeCode cf) (encodeCode cg) < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4\n⊢ encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 ∧\n encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4",
"tactic": "exact ⟨lt_of_le_of_lt (Nat.left_le_pair _ _) this, lt_of_le_of_lt (Nat.right_le_pair _ _) this⟩"
},
{
"state_after": "no goals",
"state_before": "cf cg : Code\n⊢ 1 ≤ 2 * 2",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "cf cg : Code\nthis : Nat.pair (encodeCode cf) (encodeCode cg) ≤ 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg))\n⊢ 0 < 4",
"tactic": "decide"
}
] |
[
211,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Infinite.nonempty
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nh : Set.Infinite ∅\n⊢ False",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nh : Set.Infinite s\n⊢ s ≠ ∅",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nh : Set.Infinite ∅\n⊢ False",
"tactic": "exact h finite_empty"
}
] |
[
833,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
830,
11
] |
Mathlib/Topology/Instances/AddCircle.lean
|
AddCircle.coe_eq_coe_iff_of_mem_Ico
|
[
{
"state_after": "𝕜 : Type u_1\nB : Type ?u.79444\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ico a (a + p)\nhy : y ∈ Ico a (a + p)\nh : ↑x = ↑y\n⊢ x = y",
"state_before": "𝕜 : Type u_1\nB : Type ?u.79444\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ico a (a + p)\nhy : y ∈ Ico a (a + p)\n⊢ ↑x = ↑y ↔ x = y",
"tactic": "refine' ⟨fun h => _, by tauto⟩"
},
{
"state_after": "𝕜 : Type u_1\nB : Type ?u.79444\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ico a (a + p)\nhy : y ∈ Ico a (a + p)\nh : ↑x = ↑y\n⊢ { val := x, property := hx } = { val := y, property := hy }",
"state_before": "𝕜 : Type u_1\nB : Type ?u.79444\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ico a (a + p)\nhy : y ∈ Ico a (a + p)\nh : ↑x = ↑y\n⊢ x = y",
"tactic": "suffices (⟨x, hx⟩ : Ico a (a + p)) = ⟨y, hy⟩ by exact Subtype.mk.inj this"
},
{
"state_after": "𝕜 : Type u_1\nB : Type ?u.79444\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ico a (a + p)\nhy : y ∈ Ico a (a + p)\nh : ↑(equivIco p a) ↑x = ↑(equivIco p a) ↑y\n⊢ { val := x, property := hx } = { val := y, property := hy }",
"state_before": "𝕜 : Type u_1\nB : Type ?u.79444\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ico a (a + p)\nhy : y ∈ Ico a (a + p)\nh : ↑x = ↑y\n⊢ { val := x, property := hx } = { val := y, property := hy }",
"tactic": "apply_fun equivIco p a at h"
},
{
"state_after": "𝕜 : Type u_1\nB : Type ?u.79444\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ico a (a + p)\nhy : y ∈ Ico a (a + p)\nh : ↑(equivIco p a) ↑x = ↑(equivIco p a) ↑y\n⊢ Equiv.toFun (equivIco p a) (Equiv.invFun (equivIco p a) { val := x, property := hx }) =\n Equiv.toFun (equivIco p a) (Equiv.invFun (equivIco p a) { val := y, property := hy })",
"state_before": "𝕜 : Type u_1\nB : Type ?u.79444\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ico a (a + p)\nhy : y ∈ Ico a (a + p)\nh : ↑(equivIco p a) ↑x = ↑(equivIco p a) ↑y\n⊢ { val := x, property := hx } = { val := y, property := hy }",
"tactic": "rw [← (equivIco p a).right_inv ⟨x, hx⟩, ← (equivIco p a).right_inv ⟨y, hy⟩]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nB : Type ?u.79444\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ico a (a + p)\nhy : y ∈ Ico a (a + p)\nh : ↑(equivIco p a) ↑x = ↑(equivIco p a) ↑y\n⊢ Equiv.toFun (equivIco p a) (Equiv.invFun (equivIco p a) { val := x, property := hx }) =\n Equiv.toFun (equivIco p a) (Equiv.invFun (equivIco p a) { val := y, property := hy })",
"tactic": "exact h"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nB : Type ?u.79444\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ico a (a + p)\nhy : y ∈ Ico a (a + p)\n⊢ x = y → ↑x = ↑y",
"tactic": "tauto"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nB : Type ?u.79444\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ico a (a + p)\nhy : y ∈ Ico a (a + p)\nh : ↑x = ↑y\nthis : { val := x, property := hx } = { val := y, property := hy }\n⊢ x = y",
"tactic": "exact Subtype.mk.inj this"
}
] |
[
244,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Std/Data/List/Lemmas.lean
|
List.length_pos_iff_exists_mem
|
[] |
[
45,
64
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
44,
1
] |
Mathlib/Order/Chain.lean
|
IsMaxChain.not_superChain
|
[] |
[
146,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
QuadraticForm.coeFn_smul
|
[] |
[
386,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/Algebra/ContinuedFractions/Translations.lean
|
GeneralizedContinuedFraction.terminatedAt_iff_s_none
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ng : GeneralizedContinuedFraction α\nn : ℕ\n⊢ TerminatedAt g n ↔ Stream'.Seq.get? g.s n = none",
"tactic": "rfl"
}
] |
[
40,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.preimage_comap_zeroLocus
|
[] |
[
621,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
620,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.sub_add_min_cancel
|
[
{
"state_after": "no goals",
"state_before": "n m : Nat\n⊢ n - m + min n m = n",
"tactic": "rw [sub_eq_sub_min, Nat.sub_add_cancel (Nat.min_le_left n m)]"
}
] |
[
226,
64
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
225,
19
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
Collinear.mem_affineSpan_of_mem_of_ne
|
[
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nh : ∃ v, ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\np₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\n⊢ p₃ ∈ affineSpan k {p₁, p₂}",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\nh : Collinear k s\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\n⊢ p₃ ∈ affineSpan k {p₁, p₂}",
"tactic": "rw [collinear_iff_of_mem hp₁] at h"
},
{
"state_after": "case intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\n⊢ p₃ ∈ affineSpan k {p₁, p₂}",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nh : ∃ v, ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\np₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\n⊢ p₃ ∈ affineSpan k {p₁, p₂}",
"tactic": "rcases h with ⟨v, h⟩"
},
{
"state_after": "case intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ p₃ : P\nhp₁ : p₁ ∈ s\nhp₃ : p₃ ∈ s\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\nr₂ : k\nhp₂ : r₂ • v +ᵥ p₁ ∈ s\nhp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁\n⊢ p₃ ∈ affineSpan k {p₁, r₂ • v +ᵥ p₁}",
"state_before": "case intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\n⊢ p₃ ∈ affineSpan k {p₁, p₂}",
"tactic": "rcases h p₂ hp₂ with ⟨r₂, rfl⟩"
},
{
"state_after": "case intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\nr₂ : k\nhp₂ : r₂ • v +ᵥ p₁ ∈ s\nhp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁\nr₃ : k\nhp₃ : r₃ • v +ᵥ p₁ ∈ s\n⊢ r₃ • v +ᵥ p₁ ∈ affineSpan k {p₁, r₂ • v +ᵥ p₁}",
"state_before": "case intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ p₃ : P\nhp₁ : p₁ ∈ s\nhp₃ : p₃ ∈ s\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\nr₂ : k\nhp₂ : r₂ • v +ᵥ p₁ ∈ s\nhp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁\n⊢ p₃ ∈ affineSpan k {p₁, r₂ • v +ᵥ p₁}",
"tactic": "rcases h p₃ hp₃ with ⟨r₃, rfl⟩"
},
{
"state_after": "case intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\nr₂ : k\nhp₂ : r₂ • v +ᵥ p₁ ∈ s\nhp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁\nr₃ : k\nhp₃ : r₃ • v +ᵥ p₁ ∈ s\n⊢ ∃ r, r • (r₂ • v +ᵥ p₁ -ᵥ p₁) = r₃ • v",
"state_before": "case intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\nr₂ : k\nhp₂ : r₂ • v +ᵥ p₁ ∈ s\nhp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁\nr₃ : k\nhp₃ : r₃ • v +ᵥ p₁ ∈ s\n⊢ r₃ • v +ᵥ p₁ ∈ affineSpan k {p₁, r₂ • v +ᵥ p₁}",
"tactic": "rw [vadd_left_mem_affineSpan_pair]"
},
{
"state_after": "case intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\nr₂ : k\nhp₂ : r₂ • v +ᵥ p₁ ∈ s\nhp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁\nr₃ : k\nhp₃ : r₃ • v +ᵥ p₁ ∈ s\n⊢ (r₃ / r₂) • (r₂ • v +ᵥ p₁ -ᵥ p₁) = r₃ • v",
"state_before": "case intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\nr₂ : k\nhp₂ : r₂ • v +ᵥ p₁ ∈ s\nhp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁\nr₃ : k\nhp₃ : r₃ • v +ᵥ p₁ ∈ s\n⊢ ∃ r, r • (r₂ • v +ᵥ p₁ -ᵥ p₁) = r₃ • v",
"tactic": "refine' ⟨r₃ / r₂, _⟩"
},
{
"state_after": "case intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\nr₂ : k\nhp₂ : r₂ • v +ᵥ p₁ ∈ s\nhp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁\nr₃ : k\nhp₃ : r₃ • v +ᵥ p₁ ∈ s\nh₂ : r₂ ≠ 0\n⊢ (r₃ / r₂) • (r₂ • v +ᵥ p₁ -ᵥ p₁) = r₃ • v",
"state_before": "case intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\nr₂ : k\nhp₂ : r₂ • v +ᵥ p₁ ∈ s\nhp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁\nr₃ : k\nhp₃ : r₃ • v +ᵥ p₁ ∈ s\n⊢ (r₃ / r₂) • (r₂ • v +ᵥ p₁ -ᵥ p₁) = r₃ • v",
"tactic": "have h₂ : r₂ ≠ 0 := by\n rintro rfl\n simp at hp₁p₂"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\nr₂ : k\nhp₂ : r₂ • v +ᵥ p₁ ∈ s\nhp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁\nr₃ : k\nhp₃ : r₃ • v +ᵥ p₁ ∈ s\nh₂ : r₂ ≠ 0\n⊢ (r₃ / r₂) • (r₂ • v +ᵥ p₁ -ᵥ p₁) = r₃ • v",
"tactic": "simp [smul_smul, h₂]"
},
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\nr₃ : k\nhp₃ : r₃ • v +ᵥ p₁ ∈ s\nhp₂ : 0 • v +ᵥ p₁ ∈ s\nhp₁p₂ : p₁ ≠ 0 • v +ᵥ p₁\n⊢ False",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\nr₂ : k\nhp₂ : r₂ • v +ᵥ p₁ ∈ s\nhp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁\nr₃ : k\nhp₃ : r₃ • v +ᵥ p₁ ∈ s\n⊢ r₂ ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.310342\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\nv : V\nh : ∀ (p : P), p ∈ s → ∃ r, p = r • v +ᵥ p₁\nr₃ : k\nhp₃ : r₃ • v +ᵥ p₁ ∈ s\nhp₂ : 0 • v +ᵥ p₁ ∈ s\nhp₁p₂ : p₁ ≠ 0 • v +ᵥ p₁\n⊢ False",
"tactic": "simp at hp₁p₂"
}
] |
[
525,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
514,
1
] |
Mathlib/CategoryTheory/Over.lean
|
CategoryTheory.Over.map_map_left
|
[] |
[
186,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
ContinuousLinearMap.exists_right_inverse_of_surjective
|
[] |
[
531,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/RingTheory/FiniteType.lean
|
MonoidAlgebra.mvPolynomial_aeval_of_surjective_of_closure
|
[
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nf : MonoidAlgebra R M\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = f",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\n⊢ Surjective ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s)",
"tactic": "intro f"
},
{
"state_after": "case hM\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) m\n\ncase hadd\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nf g : MonoidAlgebra R M\nihf : ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = f\nihg : ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = g\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = f + g\n\ncase hsmul\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nr : R\nf : MonoidAlgebra R M\nih : ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = f\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = r • f",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nf : MonoidAlgebra R M\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = f",
"tactic": "induction' f using induction_on with m f g ihf ihg r f ih"
},
{
"state_after": "case hM\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) m",
"state_before": "case hM\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) m",
"tactic": "have : m ∈ closure S := hS.symm ▸ mem_top _"
},
{
"state_after": "case hM.refine'_1\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm✝ : M\nthis : m✝ ∈ closure S\nm : M\nhm : m ∈ S\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) m\n\ncase hM.refine'_2\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) 1\n\ncase hM.refine'_3\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\n⊢ ∀ (x y : M),\n (∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) x) →\n (∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) y) →\n ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) (x * y)",
"state_before": "case hM\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) m",
"tactic": "refine' closure_induction this (fun m hm => _) _ _"
},
{
"state_after": "no goals",
"state_before": "case hM.refine'_1\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm✝ : M\nthis : m✝ ∈ closure S\nm : M\nhm : m ∈ S\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) m",
"tactic": "exact ⟨MvPolynomial.X ⟨m, hm⟩, MvPolynomial.aeval_X _ _⟩"
},
{
"state_after": "no goals",
"state_before": "case hM.refine'_2\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) 1",
"tactic": "exact ⟨1, AlgHom.map_one _⟩"
},
{
"state_after": "case hM.refine'_3.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\nm₁ m₂ : M\nP₁ : MvPolynomial (↑S) R\nhP₁ : ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) P₁ = ↑(of R M) m₁\nP₂ : MvPolynomial (↑S) R\nhP₂ : ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) P₂ = ↑(of R M) m₂\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) (m₁ * m₂)",
"state_before": "case hM.refine'_3\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\n⊢ ∀ (x y : M),\n (∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) x) →\n (∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) y) →\n ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) (x * y)",
"tactic": "rintro m₁ m₂ ⟨P₁, hP₁⟩ ⟨P₂, hP₂⟩"
},
{
"state_after": "no goals",
"state_before": "case hM.refine'_3.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\nm₁ m₂ : M\nP₁ : MvPolynomial (↑S) R\nhP₁ : ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) P₁ = ↑(of R M) m₁\nP₂ : MvPolynomial (↑S) R\nhP₂ : ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) P₂ = ↑(of R M) m₂\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(of R M) (m₁ * m₂)",
"tactic": "exact\n ⟨P₁ * P₂, by\n rw [AlgHom.map_mul, hP₁, hP₂, of_apply, of_apply, of_apply, single_mul_single, one_mul]⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\nm₁ m₂ : M\nP₁ : MvPolynomial (↑S) R\nhP₁ : ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) P₁ = ↑(of R M) m₁\nP₂ : MvPolynomial (↑S) R\nhP₂ : ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) P₂ = ↑(of R M) m₂\n⊢ ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) (P₁ * P₂) = ↑(of R M) (m₁ * m₂)",
"tactic": "rw [AlgHom.map_mul, hP₁, hP₂, of_apply, of_apply, of_apply, single_mul_single, one_mul]"
},
{
"state_after": "case hadd.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\ng : MonoidAlgebra R M\nihg : ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = g\nP : MvPolynomial (↑S) R\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) P + g",
"state_before": "case hadd\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nf g : MonoidAlgebra R M\nihf : ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = f\nihg : ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = g\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = f + g",
"tactic": "rcases ihf with ⟨P, rfl⟩"
},
{
"state_after": "case hadd.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nP Q : MvPolynomial (↑S) R\n⊢ ∃ a,\n ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a =\n ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) P + ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) Q",
"state_before": "case hadd.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\ng : MonoidAlgebra R M\nihg : ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = g\nP : MvPolynomial (↑S) R\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) P + g",
"tactic": "rcases ihg with ⟨Q, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case hadd.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nP Q : MvPolynomial (↑S) R\n⊢ ∃ a,\n ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a =\n ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) P + ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) Q",
"tactic": "exact ⟨P + Q, AlgHom.map_add _ _ _⟩"
},
{
"state_after": "case hsmul.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nr : R\nP : MvPolynomial (↑S) R\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = r • ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) P",
"state_before": "case hsmul\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nr : R\nf : MonoidAlgebra R M\nih : ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = f\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = r • f",
"tactic": "rcases ih with ⟨P, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case hsmul.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nr : R\nP : MvPolynomial (↑S) R\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) a = r • ↑(MvPolynomial.aeval fun s => ↑(of R M) ↑s) P",
"tactic": "exact ⟨r • P, AlgHom.map_smul _ _ _⟩"
}
] |
[
597,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/Analysis/Complex/PhragmenLindelof.lean
|
PhragmenLindelof.quadrant_II
|
[
{
"state_after": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nz : ℂ\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhB : ∃ c, c < 2 ∧ ∃ B, f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nhz_re : z.re ≤ 0\nhz_im : 0 ≤ z.im\n⊢ ∃ z', z' * I = z\n\ncase intro\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhB : ∃ c, c < 2 ∧ ∃ B, f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : (z * I).re ≤ 0\nhz_im : 0 ≤ (z * I).im\n⊢ ‖f (z * I)‖ ≤ C",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nz : ℂ\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhB : ∃ c, c < 2 ∧ ∃ B, f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nhz_re : z.re ≤ 0\nhz_im : 0 ≤ z.im\n⊢ ‖f z‖ ≤ C",
"tactic": "obtain ⟨z, rfl⟩ : ∃ z', z' * I = z"
},
{
"state_after": "case intro\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhB : ∃ c, c < 2 ∧ ∃ B, f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : (z * I).re ≤ 0\nhz_im : 0 ≤ (z * I).im\n⊢ ‖f (z * I)‖ ≤ C",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nz : ℂ\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhB : ∃ c, c < 2 ∧ ∃ B, f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nhz_re : z.re ≤ 0\nhz_im : 0 ≤ z.im\n⊢ ∃ z', z' * I = z\n\ncase intro\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhB : ∃ c, c < 2 ∧ ∃ B, f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : (z * I).re ≤ 0\nhz_im : 0 ≤ (z * I).im\n⊢ ‖f (z * I)‖ ≤ C",
"tactic": "exact ⟨z / I, div_mul_cancel _ I_ne_zero⟩"
},
{
"state_after": "case intro\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhB : ∃ c, c < 2 ∧ ∃ B, f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.im\nhz_im : 0 ≤ z.re\n⊢ ‖f (z * I)‖ ≤ C",
"state_before": "case intro\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhB : ∃ c, c < 2 ∧ ∃ B, f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : (z * I).re ≤ 0\nhz_im : 0 ≤ (z * I).im\n⊢ ‖f (z * I)‖ ≤ C",
"tactic": "simp only [mul_I_re, mul_I_im, neg_nonpos] at hz_re hz_im"
},
{
"state_after": "case intro.intro.intro.intro\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.im\nhz_im : 0 ≤ z.re\nH : MapsTo (fun x => x * I) (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Ioi 0)\nc : ℝ\nhc : c < 2\nB : ℝ\nhO : f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\n⊢ ‖(f ∘ fun x => x * I) z‖ ≤ C",
"state_before": "case intro\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhB : ∃ c, c < 2 ∧ ∃ B, f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.im\nhz_im : 0 ≤ z.re\nH : MapsTo (fun x => x * I) (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Ioi 0)\n⊢ ‖(f ∘ fun x => x * I) z‖ ≤ C",
"tactic": "rcases hB with ⟨c, hc, B, hO⟩"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.im\nhz_im : 0 ≤ z.re\nH : MapsTo (fun x => x * I) (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Ioi 0)\nc : ℝ\nhc : c < 2\nB : ℝ\nhO : f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\n⊢ (f ∘ fun x => x * I) =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\n\ncase intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.im\nhz_im : 0 ≤ z.re\nH : MapsTo (fun x => x * I) (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Ioi 0)\nc : ℝ\nhc : c < 2\nB : ℝ\nhO : f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nx : ℝ\nhx : 0 ≤ x\n⊢ ‖(f ∘ fun x => x * I) (↑x * I)‖ ≤ C",
"state_before": "case intro.intro.intro.intro\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.im\nhz_im : 0 ≤ z.re\nH : MapsTo (fun x => x * I) (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Ioi 0)\nc : ℝ\nhc : c < 2\nB : ℝ\nhO : f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\n⊢ ‖(f ∘ fun x => x * I) z‖ ≤ C",
"tactic": "refine' quadrant_I (hd.comp (differentiable_id.mul_const _).diffContOnCl H) ⟨c, hc, B, ?_⟩ him\n (fun x hx => _) hz_im hz_re"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhB : ∃ c, c < 2 ∧ ∃ B, f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.im\nhz_im : 0 ≤ z.re\nw : ℂ\nhw : w ∈ Ioi 0 ×ℂ Ioi 0\n⊢ (fun x => x * I) w ∈ Iio 0 ×ℂ Ioi 0",
"tactic": "simpa only [mem_reProdIm, mul_I_re, mul_I_im, neg_lt_zero, mem_Iio] using hw.symm"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.im\nhz_im : 0 ≤ z.re\nH : MapsTo (fun x => x * I) (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Ioi 0)\nc : ℝ\nhc : c < 2\nB : ℝ\nhO : f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nx : ℝ\nhx : 0 ≤ x\n⊢ ‖f ↑(-x)‖ ≤ C",
"state_before": "case intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.im\nhz_im : 0 ≤ z.re\nH : MapsTo (fun x => x * I) (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Ioi 0)\nc : ℝ\nhc : c < 2\nB : ℝ\nhO : f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nx : ℝ\nhx : 0 ≤ x\n⊢ ‖(f ∘ fun x => x * I) (↑x * I)‖ ≤ C",
"tactic": "rw [comp_apply, mul_assoc, I_mul_I, mul_neg_one, ← ofReal_neg]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf g : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhre : ∀ (x : ℝ), x ≤ 0 → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.im\nhz_im : 0 ≤ z.re\nH : MapsTo (fun x => x * I) (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Ioi 0)\nc : ℝ\nhc : c < 2\nB : ℝ\nhO : f =O[comap (↑Complex.abs) atTop ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * ↑Complex.abs z ^ c)\nx : ℝ\nhx : 0 ≤ x\n⊢ ‖f ↑(-x)‖ ≤ C",
"tactic": "exact hre _ (neg_nonpos.2 hx)"
}
] |
[
488,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/RingTheory/Valuation/Quotient.lean
|
AddValuation.comap_supp
|
[] |
[
117,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.zero_re
|
[] |
[
141,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/RingTheory/Localization/InvSubmonoid.lean
|
IsLocalization.surj''
|
[
{
"state_after": "case intro.mk\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.160993\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nz : S\nr : R\nm : { x // x ∈ M }\ne : z * ↑(algebraMap R S) ↑(r, m).snd = ↑(algebraMap R S) r\n⊢ ∃ r m, z = r • ↑(↑(toInvSubmonoid M S) m)",
"state_before": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.160993\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nz : S\n⊢ ∃ r m, z = r • ↑(↑(toInvSubmonoid M S) m)",
"tactic": "rcases IsLocalization.surj M z with ⟨⟨r, m⟩, e : z * _ = algebraMap R S r⟩"
},
{
"state_after": "case intro.mk\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.160993\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nz : S\nr : R\nm : { x // x ∈ M }\ne : z * ↑(algebraMap R S) ↑(r, m).snd = ↑(algebraMap R S) r\n⊢ z = r • ↑(↑(toInvSubmonoid M S) m)",
"state_before": "case intro.mk\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.160993\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nz : S\nr : R\nm : { x // x ∈ M }\ne : z * ↑(algebraMap R S) ↑(r, m).snd = ↑(algebraMap R S) r\n⊢ ∃ r m, z = r • ↑(↑(toInvSubmonoid M S) m)",
"tactic": "refine' ⟨r, m, _⟩"
},
{
"state_after": "case intro.mk\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.160993\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nz : S\nr : R\nm : { x // x ∈ M }\ne : z * ↑(algebraMap R S) ↑(r, m).snd = ↑(algebraMap R S) r\n⊢ z = z * (↑(algebraMap R S) ↑(r, m).snd * ↑(↑(toInvSubmonoid M S) m))",
"state_before": "case intro.mk\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.160993\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nz : S\nr : R\nm : { x // x ∈ M }\ne : z * ↑(algebraMap R S) ↑(r, m).snd = ↑(algebraMap R S) r\n⊢ z = r • ↑(↑(toInvSubmonoid M S) m)",
"tactic": "rw [Algebra.smul_def, ← e, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case intro.mk\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.160993\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nz : S\nr : R\nm : { x // x ∈ M }\ne : z * ↑(algebraMap R S) ↑(r, m).snd = ↑(algebraMap R S) r\n⊢ z = z * (↑(algebraMap R S) ↑(r, m).snd * ↑(↑(toInvSubmonoid M S) m))",
"tactic": "simp"
}
] |
[
97,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.span_X_isPrime
|
[
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ Ideal.span {X} = RingHom.ker (constantCoeff R)",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ Ideal.IsPrime (Ideal.span {X})",
"tactic": "suffices Ideal.span ({X} : Set (PowerSeries R)) = RingHom.ker (constantCoeff R) by\n rw [this]\n exact RingHom.ker_isPrime _"
},
{
"state_after": "case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ ∀ (x : PowerSeries R), x ∈ Ideal.span {X} ↔ x ∈ RingHom.ker (constantCoeff R)",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ Ideal.span {X} = RingHom.ker (constantCoeff R)",
"tactic": "apply Ideal.ext"
},
{
"state_after": "case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nφ : PowerSeries R\n⊢ φ ∈ Ideal.span {X} ↔ φ ∈ RingHom.ker (constantCoeff R)",
"state_before": "case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ ∀ (x : PowerSeries R), x ∈ Ideal.span {X} ↔ x ∈ RingHom.ker (constantCoeff R)",
"tactic": "intro φ"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nφ : PowerSeries R\n⊢ φ ∈ Ideal.span {X} ↔ φ ∈ RingHom.ker (constantCoeff R)",
"tactic": "rw [RingHom.mem_ker, Ideal.mem_span_singleton, X_dvd_iff]"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nthis : Ideal.span {X} = RingHom.ker (constantCoeff R)\n⊢ Ideal.IsPrime (RingHom.ker (constantCoeff R))",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nthis : Ideal.span {X} = RingHom.ker (constantCoeff R)\n⊢ Ideal.IsPrime (Ideal.span {X})",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nthis : Ideal.span {X} = RingHom.ker (constantCoeff R)\n⊢ Ideal.IsPrime (RingHom.ker (constantCoeff R))",
"tactic": "exact RingHom.ker_isPrime _"
}
] |
[
2044,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2038,
1
] |
Std/Logic.lean
|
Decidable.iff_iff_not_or_and_or_not
|
[
{
"state_after": "a b : Prop\ninst✝¹ : Decidable a\ninst✝ : Decidable b\n⊢ (a → b) ∧ (b → a) ↔ (¬a ∨ b) ∧ (a ∨ ¬b)",
"state_before": "a b : Prop\ninst✝¹ : Decidable a\ninst✝ : Decidable b\n⊢ (a ↔ b) ↔ (¬a ∨ b) ∧ (a ∨ ¬b)",
"tactic": "rw [iff_iff_implies_and_implies a b]"
},
{
"state_after": "no goals",
"state_before": "a b : Prop\ninst✝¹ : Decidable a\ninst✝ : Decidable b\n⊢ (a → b) ∧ (b → a) ↔ (¬a ∨ b) ∧ (a ∨ ¬b)",
"tactic": "simp only [imp_iff_not_or, Or.comm]"
}
] |
[
594,
76
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
592,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.mapsTo_sInter
|
[] |
[
1450,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1449,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.smul_mem_smul_finset_iff₀
|
[] |
[
2036,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2035,
1
] |
Mathlib/Data/MvPolynomial/PDeriv.lean
|
MvPolynomial.pderiv_monomial_single
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nσ : Type v\na a' a₁ a₂ : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni : σ\nn : ℕ\n⊢ ↑(pderiv i) (↑(monomial (single i n)) a) = ↑(monomial (single i (n - 1))) (a * ↑n)",
"tactic": "simp"
}
] |
[
117,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
mul_zpow
|
[] |
[
387,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Topology/Algebra/GroupCompletion.lean
|
AddMonoidHom.continuous_extension
|
[] |
[
265,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.infinitePos_add_infinitePos
|
[] |
[
564,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
562,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.valid_nil
|
[] |
[
1064,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1063,
1
] |
Mathlib/Order/ModularLattice.lean
|
inf_covby_of_covby_sup_of_covby_sup_right
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsWeakLowerModularLattice α\na b : α\n⊢ a ⋖ b ⊔ a → b ⋖ b ⊔ a → b ⊓ a ⋖ b",
"state_before": "α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsWeakLowerModularLattice α\na b : α\n⊢ a ⋖ a ⊔ b → b ⋖ a ⊔ b → a ⊓ b ⋖ b",
"tactic": "rw [sup_comm, inf_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsWeakLowerModularLattice α\na b : α\n⊢ a ⋖ b ⊔ a → b ⋖ b ⊔ a → b ⊓ a ⋖ b",
"tactic": "exact fun ha hb => inf_covby_of_covby_sup_of_covby_sup_left hb ha"
}
] |
[
131,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Data/PFun.lean
|
PFun.fix_fwd_eq
|
[
{
"state_after": "case H\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\n⊢ b ∈ fix f a ↔ b ∈ fix f a'",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\n⊢ fix f a = fix f a'",
"tactic": "ext b"
},
{
"state_after": "case H.mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\n⊢ b ∈ fix f a → b ∈ fix f a'\n\ncase H.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\n⊢ b ∈ fix f a' → b ∈ fix f a",
"state_before": "case H\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\n⊢ b ∈ fix f a ↔ b ∈ fix f a'",
"tactic": "constructor"
},
{
"state_after": "case H.mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\nh : b ∈ fix f a\n⊢ b ∈ fix f a'",
"state_before": "case H.mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\n⊢ b ∈ fix f a → b ∈ fix f a'",
"tactic": "intro h"
},
{
"state_after": "case H.mp.inr.intro.intro.refl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\nh : b ∈ fix f a\nh' : Sum.inr a' ∈ f a\ne' : b ∈ fix f a'\n⊢ b ∈ fix f a'",
"state_before": "case H.mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\nh : b ∈ fix f a\n⊢ b ∈ fix f a'",
"tactic": "obtain h' | ⟨a, h', e'⟩ := mem_fix_iff.1 h <;> cases Part.mem_unique ha' h'"
},
{
"state_after": "no goals",
"state_before": "case H.mp.inr.intro.intro.refl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\nh : b ∈ fix f a\nh' : Sum.inr a' ∈ f a\ne' : b ∈ fix f a'\n⊢ b ∈ fix f a'",
"tactic": "exact e'"
},
{
"state_after": "case H.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\nh : b ∈ fix f a'\n⊢ b ∈ fix f a",
"state_before": "case H.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\n⊢ b ∈ fix f a' → b ∈ fix f a",
"tactic": "intro h"
},
{
"state_after": "case H.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\nh : b ∈ fix f a'\n⊢ Sum.inl b ∈ f a ∨ ∃ a', Sum.inr a' ∈ f a ∧ b ∈ fix f a'",
"state_before": "case H.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\nh : b ∈ fix f a'\n⊢ b ∈ fix f a",
"tactic": "rw [PFun.mem_fix_iff]"
},
{
"state_after": "no goals",
"state_before": "case H.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.29936\nδ : Type ?u.29939\nε : Type ?u.29942\nι : Type ?u.29945\nf : α →. β ⊕ α\na a' : α\nha' : Sum.inr a' ∈ f a\nb : β\nh : b ∈ fix f a'\n⊢ Sum.inl b ∈ f a ∨ ∃ a', Sum.inr a' ∈ f a ∧ b ∈ fix f a'",
"tactic": "exact Or.inr ⟨a', ha', h⟩"
}
] |
[
322,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/LinearAlgebra/StdBasis.lean
|
LinearMap.ker_stdBasis
|
[] |
[
86,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEStronglyMeasurable.mul
|
[] |
[
1290,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1287,
11
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.le_add_sub
|
[] |
[
542,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
541,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.rpow_def_of_neg
|
[
{
"state_after": "x : ℝ\nhx : x < 0\ny : ℝ\n⊢ (Complex.exp (Complex.log ↑x * ↑y)).re = exp (log x * y) * cos (y * π)\n\ncase hnc\nx : ℝ\nhx : x < 0\ny : ℝ\n⊢ ¬↑x = 0",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\n⊢ x ^ y = exp (log x * y) * cos (y * π)",
"tactic": "rw [rpow_def, Complex.cpow_def, if_neg]"
},
{
"state_after": "x : ℝ\nhx : x < 0\ny : ℝ\nthis : Complex.log ↑x * ↑y = ↑(log (-x) * y) + ↑(y * π) * Complex.I\n⊢ (Complex.exp (Complex.log ↑x * ↑y)).re = exp (log x * y) * cos (y * π)\n\ncase hnc\nx : ℝ\nhx : x < 0\ny : ℝ\n⊢ ¬↑x = 0",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\n⊢ (Complex.exp (Complex.log ↑x * ↑y)).re = exp (log x * y) * cos (y * π)\n\ncase hnc\nx : ℝ\nhx : x < 0\ny : ℝ\n⊢ ¬↑x = 0",
"tactic": "have : Complex.log x * y = ↑(log (-x) * y) + ↑(y * π) * Complex.I := by\n simp only [Complex.log, abs_of_neg hx, Complex.arg_ofReal_of_neg hx, Complex.abs_ofReal,\n Complex.ofReal_mul]\n ring"
},
{
"state_after": "x : ℝ\nhx : x < 0\ny : ℝ\n⊢ (↑(log (-x)) + ↑π * Complex.I) * ↑y = ↑(log (-x)) * ↑y + ↑y * ↑π * Complex.I",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\n⊢ Complex.log ↑x * ↑y = ↑(log (-x) * y) + ↑(y * π) * Complex.I",
"tactic": "simp only [Complex.log, abs_of_neg hx, Complex.arg_ofReal_of_neg hx, Complex.abs_ofReal,\n Complex.ofReal_mul]"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\n⊢ (↑(log (-x)) + ↑π * Complex.I) * ↑y = ↑(log (-x)) * ↑y + ↑y * ↑π * Complex.I",
"tactic": "ring"
},
{
"state_after": "x : ℝ\nhx : x < 0\ny : ℝ\nthis : Complex.log ↑x * ↑y = ↑(log (-x) * y) + ↑(y * π) * Complex.I\n⊢ exp (log x * y) * cos (y * π) + ((↑(exp (log x * y) * sin (y * π))).re * 0 - 0 * Complex.I.im) =\n exp (log x * y) * cos (y * π)",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\nthis : Complex.log ↑x * ↑y = ↑(log (-x) * y) + ↑(y * π) * Complex.I\n⊢ (Complex.exp (Complex.log ↑x * ↑y)).re = exp (log x * y) * cos (y * π)",
"tactic": "rw [this, Complex.exp_add_mul_I, ← Complex.ofReal_exp, ← Complex.ofReal_cos, ←\n Complex.ofReal_sin, mul_add, ← Complex.ofReal_mul, ← mul_assoc, ← Complex.ofReal_mul,\n Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.ofReal_im,\n Real.log_neg_eq_log]"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\nthis : Complex.log ↑x * ↑y = ↑(log (-x) * y) + ↑(y * π) * Complex.I\n⊢ exp (log x * y) * cos (y * π) + ((↑(exp (log x * y) * sin (y * π))).re * 0 - 0 * Complex.I.im) =\n exp (log x * y) * cos (y * π)",
"tactic": "ring"
},
{
"state_after": "case hnc\nx : ℝ\nhx : x < 0\ny : ℝ\n⊢ ¬x = 0",
"state_before": "case hnc\nx : ℝ\nhx : x < 0\ny : ℝ\n⊢ ¬↑x = 0",
"tactic": "rw [Complex.ofReal_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case hnc\nx : ℝ\nhx : x < 0\ny : ℝ\n⊢ ¬x = 0",
"tactic": "exact ne_of_lt hx"
}
] |
[
87,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/GroupTheory/Nilpotent.lean
|
is_ascending_rev_series_of_is_descending
|
[
{
"state_after": "case intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\n⊢ IsAscendingCentralSeries fun m => H (n - m)",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nhdesc : IsDescendingCentralSeries H\n⊢ IsAscendingCentralSeries fun m => H (n - m)",
"tactic": "cases' hdesc with h0 hH"
},
{
"state_after": "case intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ (fun m => H (n - m)) (m + 1)\ng : G\n⊢ x * g * x⁻¹ * g⁻¹ ∈ (fun m => H (n - m)) m",
"state_before": "case intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\n⊢ IsAscendingCentralSeries fun m => H (n - m)",
"tactic": "refine' ⟨hn, fun x m hx g => _⟩"
},
{
"state_after": "case intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\n⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)",
"state_before": "case intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ (fun m => H (n - m)) (m + 1)\ng : G\n⊢ x * g * x⁻¹ * g⁻¹ ∈ (fun m => H (n - m)) m",
"tactic": "dsimp only at hx⊢"
},
{
"state_after": "case pos\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\nhm : n ≤ m\n⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)\n\ncase neg\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\nhm : ¬n ≤ m\n⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)",
"state_before": "case intro\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\n⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)",
"tactic": "by_cases hm : n ≤ m"
},
{
"state_after": "case pos\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\nhm : n ≤ m\nhnm : n - m = 0\n⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)",
"state_before": "case pos\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\nhm : n ≤ m\n⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)",
"tactic": "have hnm : n - m = 0 := tsub_eq_zero_iff_le.mpr hm"
},
{
"state_after": "case pos\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\nhm : n ≤ m\nhnm : n - m = 0\n⊢ x * g * x⁻¹ * g⁻¹ ∈ ⊤",
"state_before": "case pos\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\nhm : n ≤ m\nhnm : n - m = 0\n⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)",
"tactic": "rw [hnm, h0]"
},
{
"state_after": "no goals",
"state_before": "case pos\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\nhm : n ≤ m\nhnm : n - m = 0\n⊢ x * g * x⁻¹ * g⁻¹ ∈ ⊤",
"tactic": "exact mem_top _"
},
{
"state_after": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\nhm : m < n\n⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)",
"state_before": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\nhm : ¬n ≤ m\n⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)",
"tactic": "push_neg at hm"
},
{
"state_after": "case h.e'_5\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\nhm : m < n\n⊢ H (n - m) = H (n - (m + 1) + 1)",
"state_before": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\nhm : m < n\n⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)",
"tactic": "convert hH x _ hx g using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5\nG : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)\nx : G\nm : ℕ\nhx : x ∈ H (n - (m + 1))\ng : G\nhm : m < n\n⊢ H (n - m) = H (n - (m + 1) + 1)",
"tactic": "rw [tsub_add_eq_add_tsub (Nat.succ_le_of_lt hm), Nat.succ_sub_succ]"
}
] |
[
260,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_zero'
|
[] |
[
852,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
851,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
sbtw_const_vsub_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.124810\nP : Type u_3\nP' : Type ?u.124816\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z p : P\n⊢ Sbtw R (p -ᵥ x) (p -ᵥ y) (p -ᵥ z) ↔ Sbtw R x y z",
"tactic": "rw [Sbtw, Sbtw, wbtw_const_vsub_iff, (vsub_right_injective p).ne_iff,\n (vsub_right_injective p).ne_iff]"
}
] |
[
228,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
union_lowerBounds_subset_lowerBounds_inter
|
[] |
[
379,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
StrictAnti.prod_map
|
[
{
"state_after": "ι : Type ?u.48700\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nπ : ι → Type ?u.48714\nr : α → α → Prop\ninst✝³ : PartialOrder α\ninst✝² : PartialOrder β\ninst✝¹ : Preorder γ\ninst✝ : Preorder δ\nf : α → γ\ng : β → δ\nhf : StrictAnti f\nhg : StrictAnti g\na b : α × β\n⊢ a.fst < b.fst ∧ a.snd ≤ b.snd ∨ a.fst ≤ b.fst ∧ a.snd < b.snd →\n (Prod.map f g b).fst < (Prod.map f g a).fst ∧ (Prod.map f g b).snd ≤ (Prod.map f g a).snd ∨\n (Prod.map f g b).fst ≤ (Prod.map f g a).fst ∧ (Prod.map f g b).snd < (Prod.map f g a).snd",
"state_before": "ι : Type ?u.48700\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nπ : ι → Type ?u.48714\nr : α → α → Prop\ninst✝³ : PartialOrder α\ninst✝² : PartialOrder β\ninst✝¹ : Preorder γ\ninst✝ : Preorder δ\nf : α → γ\ng : β → δ\nhf : StrictAnti f\nhg : StrictAnti g\na b : α × β\n⊢ a < b → Prod.map f g b < Prod.map f g a",
"tactic": "simp only [Prod.lt_iff]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.48700\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nπ : ι → Type ?u.48714\nr : α → α → Prop\ninst✝³ : PartialOrder α\ninst✝² : PartialOrder β\ninst✝¹ : Preorder γ\ninst✝ : Preorder δ\nf : α → γ\ng : β → δ\nhf : StrictAnti f\nhg : StrictAnti g\na b : α × β\n⊢ a.fst < b.fst ∧ a.snd ≤ b.snd ∨ a.fst ≤ b.fst ∧ a.snd < b.snd →\n (Prod.map f g b).fst < (Prod.map f g a).fst ∧ (Prod.map f g b).snd ≤ (Prod.map f g a).snd ∨\n (Prod.map f g b).fst ≤ (Prod.map f g a).fst ∧ (Prod.map f g b).snd < (Prod.map f g a).snd",
"tactic": "exact Or.imp (And.imp hf.imp hg.antitone.imp) (And.imp hf.antitone.imp hg.imp)"
}
] |
[
1179,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1176,
1
] |
Mathlib/AlgebraicTopology/SimplicialObject.lean
|
CategoryTheory.SimplicialObject.augment_hom_zero
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nX✝ X : SimplicialObject C\nX₀ : C\nf : X.obj [0].op ⟶ X₀\nw : ∀ (i : SimplexCategory) (g₁ g₂ : [0] ⟶ i), X.map g₁.op ≫ f = X.map g₂.op ≫ f\n⊢ X.map (SimplexCategory.const [0] 0).op ≫ f = f",
"state_before": "C : Type u\ninst✝ : Category C\nX✝ X : SimplicialObject C\nX₀ : C\nf : X.obj [0].op ⟶ X₀\nw : ∀ (i : SimplexCategory) (g₁ g₂ : [0] ⟶ i), X.map g₁.op ≫ f = X.map g₂.op ≫ f\n⊢ (augment X X₀ f w).hom.app [0].op = f",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX✝ X : SimplicialObject C\nX₀ : C\nf : X.obj [0].op ⟶ X₀\nw : ∀ (i : SimplexCategory) (g₁ g₂ : [0] ⟶ i), X.map g₁.op ≫ f = X.map g₂.op ≫ f\n⊢ X.map (SimplexCategory.const [0] 0).op ≫ f = f",
"tactic": "rw [SimplexCategory.hom_zero_zero ([0].const 0), op_id, X.map_id, Category.id_comp]"
}
] |
[
406,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/Data/Finset/Sigma.lean
|
Finset.sigma_eq_biUnion
|
[
{
"state_after": "case a.mk\nι : Type u_2\nα : ι → Type u_1\nβ : Type ?u.9726\ns✝ s₁ s₂ : Finset ι\nt✝ t₁ t₂ : (i : ι) → Finset (α i)\ninst✝ : DecidableEq ((i : ι) × α i)\ns : Finset ι\nt : (i : ι) → Finset (α i)\nx : ι\ny : α x\n⊢ { fst := x, snd := y } ∈ Finset.sigma s t ↔\n { fst := x, snd := y } ∈ Finset.biUnion s fun i => map (Embedding.sigmaMk i) (t i)",
"state_before": "ι : Type u_2\nα : ι → Type u_1\nβ : Type ?u.9726\ns✝ s₁ s₂ : Finset ι\nt✝ t₁ t₂ : (i : ι) → Finset (α i)\ninst✝ : DecidableEq ((i : ι) × α i)\ns : Finset ι\nt : (i : ι) → Finset (α i)\n⊢ Finset.sigma s t = Finset.biUnion s fun i => map (Embedding.sigmaMk i) (t i)",
"tactic": "ext ⟨x, y⟩"
},
{
"state_after": "no goals",
"state_before": "case a.mk\nι : Type u_2\nα : ι → Type u_1\nβ : Type ?u.9726\ns✝ s₁ s₂ : Finset ι\nt✝ t₁ t₂ : (i : ι) → Finset (α i)\ninst✝ : DecidableEq ((i : ι) × α i)\ns : Finset ι\nt : (i : ι) → Finset (α i)\nx : ι\ny : α x\n⊢ { fst := x, snd := y } ∈ Finset.sigma s t ↔\n { fst := x, snd := y } ∈ Finset.biUnion s fun i => map (Embedding.sigmaMk i) (t i)",
"tactic": "simp [and_left_comm]"
}
] |
[
97,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
Convex.starConvex_iff
|
[] |
[
167,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.EqOnSource.symm'
|
[] |
[
947,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
946,
1
] |
Mathlib/Order/Max.lean
|
not_isBot
|
[] |
[
216,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Data/Set/Pairwise/Lattice.lean
|
Pairwise.subset_of_biUnion_subset_biUnion
|
[] |
[
128,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Init/Algebra/Classes.lean
|
irrefl
|
[] |
[
258,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean
|
GroupSeminorm.zero_apply
|
[] |
[
264,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.mulIndicator_finset_prod
|
[] |
[
637,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
635,
1
] |
src/lean/Init/SimpLemmas.lean
|
ite_false
|
[] |
[
78,
74
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
78,
9
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
Submonoid.coe_iSup_of_directed
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_2\nA : Type ?u.34491\nB : Type ?u.34494\ninst✝¹ : MulOneClass M\nι : Sort u_1\ninst✝ : Nonempty ι\nS : ι → Submonoid M\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : M\n⊢ x ∈ ↑(⨆ (i : ι), S i) ↔ x ∈ ⋃ (i : ι), ↑(S i)",
"tactic": "simp [mem_iSup_of_directed hS]"
}
] |
[
217,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Topology/Algebra/Order/IntermediateValue.lean
|
IsPreconnected.Icc_subset
|
[
{
"state_after": "no goals",
"state_before": "X : Type u\nα : Type v\ninst✝³ : TopologicalSpace X\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\ns : Set α\nhs : IsPreconnected s\na b : α\nha : a ∈ s\nhb : b ∈ s\n⊢ Icc a b ⊆ s",
"tactic": "simpa only [image_id] using hs.intermediate_value ha hb continuousOn_id"
}
] |
[
219,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/NumberTheory/FermatPsp.lean
|
FermatPsp.probablePrime_iff_modEq
|
[
{
"state_after": "n b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\n⊢ ProbablePrime n b ↔ b ^ (n - 1) ≡ 1 [MOD n]",
"state_before": "n b : ℕ\nh : 1 ≤ b\n⊢ ProbablePrime n b ↔ b ^ (n - 1) ≡ 1 [MOD n]",
"tactic": "have : 1 ≤ b ^ (n - 1) := one_le_pow_of_one_le h (n - 1)"
},
{
"state_after": "n b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\n⊢ ProbablePrime n b ↔ 1 ≡ b ^ (n - 1) [MOD n]",
"state_before": "n b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\n⊢ ProbablePrime n b ↔ b ^ (n - 1) ≡ 1 [MOD n]",
"tactic": "rw [Nat.ModEq.comm]"
},
{
"state_after": "case mp\nn b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\n⊢ ProbablePrime n b → 1 ≡ b ^ (n - 1) [MOD n]\n\ncase mpr\nn b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\n⊢ 1 ≡ b ^ (n - 1) [MOD n] → ProbablePrime n b",
"state_before": "n b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\n⊢ ProbablePrime n b ↔ 1 ≡ b ^ (n - 1) [MOD n]",
"tactic": "constructor"
},
{
"state_after": "case mp\nn b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\nh₁ : ProbablePrime n b\n⊢ 1 ≡ b ^ (n - 1) [MOD n]",
"state_before": "case mp\nn b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\n⊢ ProbablePrime n b → 1 ≡ b ^ (n - 1) [MOD n]",
"tactic": "intro h₁"
},
{
"state_after": "case mp.a\nn b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\nh₁ : ProbablePrime n b\n⊢ ↑n ∣ ↑(b ^ (n - 1)) - ↑1",
"state_before": "case mp\nn b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\nh₁ : ProbablePrime n b\n⊢ 1 ≡ b ^ (n - 1) [MOD n]",
"tactic": "apply Nat.modEq_of_dvd"
},
{
"state_after": "no goals",
"state_before": "case mp.a\nn b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\nh₁ : ProbablePrime n b\n⊢ ↑n ∣ ↑(b ^ (n - 1)) - ↑1",
"tactic": "exact_mod_cast h₁"
},
{
"state_after": "case mpr\nn b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\nh₁ : 1 ≡ b ^ (n - 1) [MOD n]\n⊢ ProbablePrime n b",
"state_before": "case mpr\nn b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\n⊢ 1 ≡ b ^ (n - 1) [MOD n] → ProbablePrime n b",
"tactic": "intro h₁"
},
{
"state_after": "no goals",
"state_before": "case mpr\nn b : ℕ\nh : 1 ≤ b\nthis : 1 ≤ b ^ (n - 1)\nh₁ : 1 ≡ b ^ (n - 1) [MOD n]\n⊢ ProbablePrime n b",
"tactic": "exact_mod_cast Nat.ModEq.dvd h₁"
}
] |
[
117,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.add_vecMul
|
[
{
"state_after": "case h\nl : Type ?u.857786\nm : Type u_1\nn : Type u_2\no : Type ?u.857795\nm' : o → Type ?u.857800\nn' : o → Type ?u.857805\nR : Type ?u.857808\nS : Type ?u.857811\nα : Type v\nβ : Type w\nγ : Type ?u.857818\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype m\nA : Matrix m n α\nx y : m → α\nx✝ : n\n⊢ vecMul (x + y) A x✝ = (vecMul x A + vecMul y A) x✝",
"state_before": "l : Type ?u.857786\nm : Type u_1\nn : Type u_2\no : Type ?u.857795\nm' : o → Type ?u.857800\nn' : o → Type ?u.857805\nR : Type ?u.857808\nS : Type ?u.857811\nα : Type v\nβ : Type w\nγ : Type ?u.857818\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype m\nA : Matrix m n α\nx y : m → α\n⊢ vecMul (x + y) A = vecMul x A + vecMul y A",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nl : Type ?u.857786\nm : Type u_1\nn : Type u_2\no : Type ?u.857795\nm' : o → Type ?u.857800\nn' : o → Type ?u.857805\nR : Type ?u.857808\nS : Type ?u.857811\nα : Type v\nβ : Type w\nγ : Type ?u.857818\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype m\nA : Matrix m n α\nx y : m → α\nx✝ : n\n⊢ vecMul (x + y) A x✝ = (vecMul x A + vecMul y A) x✝",
"tactic": "apply add_dotProduct"
}
] |
[
1769,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1766,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.coprod_zero_right
|
[] |
[
243,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.disjoint_comm
|
[] |
[
57,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Order/Circular.lean
|
sbtw_of_btw_not_btw
|
[] |
[
211,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Algebra/Category/GroupCat/EpiMono.lean
|
GroupCat.SurjectiveOfEpiAuxs.fromCoset_ne_of_nin_range
|
[
{
"state_after": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ¬b ∈ MonoidHom.range f\nr :\n fromCoset\n { val := b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) } =\n fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }\n⊢ False",
"state_before": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ¬b ∈ MonoidHom.range f\n⊢ fromCoset\n { val := b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) } ≠\n fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }",
"tactic": "intro r"
},
{
"state_after": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ¬b ∈ MonoidHom.range f\nr : b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier = (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\n⊢ False",
"state_before": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ¬b ∈ MonoidHom.range f\nr :\n fromCoset\n { val := b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) } =\n fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }\n⊢ False",
"tactic": "simp only [fromCoset.injEq, Subtype.mk.injEq] at r"
},
{
"state_after": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ¬b ∈ MonoidHom.range f\nr : b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier = (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\nb' : ↑B := b\n⊢ False",
"state_before": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ¬b ∈ MonoidHom.range f\nr : b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier = (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\n⊢ False",
"tactic": "let b' : B.α := b"
},
{
"state_after": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ¬b ∈ MonoidHom.range f\nb' : ↑B := b\nr : b' *l ↑(MonoidHom.range f) = ↑(MonoidHom.range f)\n⊢ False",
"state_before": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ¬b ∈ MonoidHom.range f\nr : b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier = (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\nb' : ↑B := b\n⊢ False",
"tactic": "change b' *l f.range = f.range at r"
},
{
"state_after": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ¬b ∈ MonoidHom.range f\nb' : ↑B := b\nr : b' *l ↑(MonoidHom.range f) = 1 *l ↑(MonoidHom.range f)\n⊢ False",
"state_before": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ¬b ∈ MonoidHom.range f\nb' : ↑B := b\nr : b' *l ↑(MonoidHom.range f) = ↑(MonoidHom.range f)\n⊢ False",
"tactic": "nth_rw 2 [show (f.range : Set B.α) = 1 *l f.range from (one_leftCoset _).symm] at r"
},
{
"state_after": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ¬b ∈ MonoidHom.range f\nb' : ↑B := b\nr : b'⁻¹ ∈ MonoidHom.range f\n⊢ False",
"state_before": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ¬b ∈ MonoidHom.range f\nb' : ↑B := b\nr : b' *l ↑(MonoidHom.range f) = 1 *l ↑(MonoidHom.range f)\n⊢ False",
"tactic": "rw [leftCoset_eq_iff, mul_one] at r"
},
{
"state_after": "no goals",
"state_before": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ¬b ∈ MonoidHom.range f\nb' : ↑B := b\nr : b'⁻¹ ∈ MonoidHom.range f\n⊢ False",
"tactic": "exact hb (inv_inv b ▸ Subgroup.inv_mem _ r)"
}
] |
[
170,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Data/List/Pairwise.lean
|
List.pairwise_singleton
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7483\nR✝ S T : α → α → Prop\na✝ : α\nl : List α\nR : α → α → Prop\na : α\n⊢ Pairwise R [a]",
"tactic": "simp [Pairwise.nil]"
}
] |
[
156,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.inr_map_mul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.152136\nM₆ : Type ?u.152139\nS : Type ?u.152142\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid M₂\ninst✝¹³ : AddCommMonoid M₃\ninst✝¹² : AddCommMonoid M₄\ninst✝¹¹ : AddCommMonoid M₅\ninst✝¹⁰ : AddCommMonoid M₆\ninst✝⁹ : Module R M\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M₃\ninst✝⁶ : Module R M₄\ninst✝⁵ : Module R M₅\ninst✝⁴ : Module R M₆\nf : M →ₗ[R] M₂\nA : Type u_1\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\nB : Type u_2\ninst✝¹ : NonUnitalNonAssocSemiring B\ninst✝ : Module R B\nb₁ b₂ : B\n⊢ (↑(inr R A B) (b₁ * b₂)).fst = (↑(inr R A B) b₁ * ↑(inr R A B) b₂).fst",
"tactic": "simp"
}
] |
[
409,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/Combinatorics/Additive/SalemSpencer.lean
|
MulSalemSpencer.image
|
[
{
"state_after": "case intro.intro.intro.intro.intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\n𝕜 : Type ?u.12908\nE : Type ?u.12911\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ns : Set α\na✝ : α\ninst✝ : MulHomClass F α β\nf : F\nhf : InjOn (↑f) (s * s)\nh : MulSalemSpencer s\na : α\nha : a ∈ s\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nhabc : ↑f a * ↑f b = ↑f c * ↑f c\n⊢ ↑f a = ↑f b",
"state_before": "F : Type u_1\nα : Type u_2\nβ : Type u_3\n𝕜 : Type ?u.12908\nE : Type ?u.12911\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ns : Set α\na : α\ninst✝ : MulHomClass F α β\nf : F\nhf : InjOn (↑f) (s * s)\nh : MulSalemSpencer s\n⊢ MulSalemSpencer (↑f '' s)",
"tactic": "rintro _ _ _ ⟨a, ha, rfl⟩ ⟨b, hb, rfl⟩ ⟨c, hc, rfl⟩ habc"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\n𝕜 : Type ?u.12908\nE : Type ?u.12911\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ns : Set α\na✝ : α\ninst✝ : MulHomClass F α β\nf : F\nhf : InjOn (↑f) (s * s)\nh : MulSalemSpencer s\na : α\nha : a ∈ s\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nhabc : ↑f a * ↑f b = ↑f c * ↑f c\n⊢ ↑f a = ↑f b",
"tactic": "rw [h ha hb hc (hf (mul_mem_mul ha hb) (mul_mem_mul hc hc) <| by rwa [map_mul, map_mul])]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\nα : Type u_2\nβ : Type u_3\n𝕜 : Type ?u.12908\nE : Type ?u.12911\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ns : Set α\na✝ : α\ninst✝ : MulHomClass F α β\nf : F\nhf : InjOn (↑f) (s * s)\nh : MulSalemSpencer s\na : α\nha : a ∈ s\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nhabc : ↑f a * ↑f b = ↑f c * ↑f c\n⊢ ↑f (a * b) = ↑f (c * c)",
"tactic": "rwa [map_mul, map_mul]"
}
] |
[
137,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.apply_mem_support
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nx : α\n⊢ ↑f x ∈ support f ↔ x ∈ support f",
"tactic": "rw [mem_support, mem_support, Ne.def, Ne.def, apply_eq_iff_eq]"
}
] |
[
360,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/Data/Multiset/FinsetOps.lean
|
Multiset.ndinsert_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns t : Multiset α\nx✝ : s ≤ t ∧ a ∈ t\nl : s ≤ t\nm : a ∈ t\nh : a ∈ s\n⊢ ndinsert a s ≤ t",
"tactic": "simp [h, l]"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns t : Multiset α\nx✝ : s ≤ t ∧ a ∈ t\nl : s ≤ t\nm : a ∈ t\nh : ¬a ∈ s\n⊢ s ≤ t",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns t : Multiset α\nx✝ : s ≤ t ∧ a ∈ t\nl : s ≤ t\nm : a ∈ t\nh : ¬a ∈ s\n⊢ ndinsert a s ≤ t",
"tactic": "rw [ndinsert_of_not_mem h, ← cons_erase m, cons_le_cons_iff, ← le_cons_of_not_mem h,\n cons_erase m]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns t : Multiset α\nx✝ : s ≤ t ∧ a ∈ t\nl : s ≤ t\nm : a ∈ t\nh : ¬a ∈ s\n⊢ s ≤ t",
"tactic": "exact l"
}
] |
[
100,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Data/Int/Associated.lean
|
Int.natAbs_eq_iff_associated
|
[
{
"state_after": "a b : ℤ\n⊢ a = b ∨ a = -b ↔ Associated a b",
"state_before": "a b : ℤ\n⊢ natAbs a = natAbs b ↔ Associated a b",
"tactic": "refine' Int.natAbs_eq_natAbs_iff.trans _"
},
{
"state_after": "case mp\na b : ℤ\n⊢ a = b ∨ a = -b → Associated a b\n\ncase mpr\na b : ℤ\n⊢ Associated a b → a = b ∨ a = -b",
"state_before": "a b : ℤ\n⊢ a = b ∨ a = -b ↔ Associated a b",
"tactic": "constructor"
},
{
"state_after": "case mp.inl\na : ℤ\n⊢ Associated a a\n\ncase mp.inr\nb : ℤ\n⊢ Associated (-b) b",
"state_before": "case mp\na b : ℤ\n⊢ a = b ∨ a = -b → Associated a b",
"tactic": "rintro (rfl | rfl)"
},
{
"state_after": "no goals",
"state_before": "case mp.inl\na : ℤ\n⊢ Associated a a",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case mp.inr\nb : ℤ\n⊢ Associated (-b) b",
"tactic": "exact ⟨-1, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "b : ℤ\n⊢ -b * ↑(-1) = b",
"tactic": "simp"
},
{
"state_after": "case mpr.intro\na : ℤ\nu : ℤˣ\n⊢ a = a * ↑u ∨ a = -(a * ↑u)",
"state_before": "case mpr\na b : ℤ\n⊢ Associated a b → a = b ∨ a = -b",
"tactic": "rintro ⟨u, rfl⟩"
},
{
"state_after": "case mpr.intro.inl\na : ℤ\n⊢ a = a * ↑1 ∨ a = -(a * ↑1)\n\ncase mpr.intro.inr\na : ℤ\n⊢ a = a * ↑(-1) ∨ a = -(a * ↑(-1))",
"state_before": "case mpr.intro\na : ℤ\nu : ℤˣ\n⊢ a = a * ↑u ∨ a = -(a * ↑u)",
"tactic": "obtain rfl | rfl := Int.units_eq_one_or u"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.inl\na : ℤ\n⊢ a = a * ↑1 ∨ a = -(a * ↑1)",
"tactic": "exact Or.inl (by simp)"
},
{
"state_after": "no goals",
"state_before": "a : ℤ\n⊢ a = a * ↑1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.inr\na : ℤ\n⊢ a = a * ↑(-1) ∨ a = -(a * ↑(-1))",
"tactic": "exact Or.inr (by simp)"
},
{
"state_after": "no goals",
"state_before": "a : ℤ\n⊢ a = -(a * ↑(-1))",
"tactic": "simp"
}
] |
[
34,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
25,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
Measurable.ennreal_ofReal
|
[] |
[
1799,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1797,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.reindexFinsetRange_self
|
[
{
"state_after": "ι : Type u_2\nι' : Type ?u.421677\nR : Type u_3\nR₂ : Type ?u.421683\nK : Type ?u.421686\nM : Type u_1\nM' : Type ?u.421692\nM'' : Type ?u.421695\nV : Type u\nV' : Type ?u.421700\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq M\ni : ι\nh : optParam (↑b i ∈ Finset.image (↑b) Finset.univ) (_ : ↑b i ∈ Finset.image (↑b) Finset.univ)\n⊢ ↑(↑(Equiv.subtypeEquiv (Equiv.refl M) (_ : ∀ (a : M), a ∈ range ↑b ↔ a ∈ Finset.image (↑b) Finset.univ)).symm\n { val := ↑b i, property := h }) =\n ↑b i",
"state_before": "ι : Type u_2\nι' : Type ?u.421677\nR : Type u_3\nR₂ : Type ?u.421683\nK : Type ?u.421686\nM : Type u_1\nM' : Type ?u.421692\nM'' : Type ?u.421695\nV : Type u\nV' : Type ?u.421700\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq M\ni : ι\nh : optParam (↑b i ∈ Finset.image (↑b) Finset.univ) (_ : ↑b i ∈ Finset.image (↑b) Finset.univ)\n⊢ ↑(reindexFinsetRange b) { val := ↑b i, property := h } = ↑b i",
"tactic": "rw [reindexFinsetRange, reindex_apply, reindexRange_apply]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nι' : Type ?u.421677\nR : Type u_3\nR₂ : Type ?u.421683\nK : Type ?u.421686\nM : Type u_1\nM' : Type ?u.421692\nM'' : Type ?u.421695\nV : Type u\nV' : Type ?u.421700\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq M\ni : ι\nh : optParam (↑b i ∈ Finset.image (↑b) Finset.univ) (_ : ↑b i ∈ Finset.image (↑b) Finset.univ)\n⊢ ↑(↑(Equiv.subtypeEquiv (Equiv.refl M) (_ : ∀ (a : M), a ∈ range ↑b ↔ a ∈ Finset.image (↑b) Finset.univ)).symm\n { val := ↑b i, property := h }) =\n ↑b i",
"tactic": "rfl"
}
] |
[
529,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
526,
1
] |
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