file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Order/Filter/Germ.lean
|
Filter.Germ.coe_nonneg
|
[] |
[
666,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
665,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.empty_subset
|
[] |
[
573,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
572,
1
] |
Mathlib/Data/Finsupp/Multiset.lean
|
Multiset.toFinsupp_support
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.58480\nι : Type ?u.58483\ninst✝ : DecidableEq α\ns : Multiset α\na✝ : α\n⊢ a✝ ∈ (↑toFinsupp s).support ↔ a✝ ∈ toFinset s",
"state_before": "α : Type u_1\nβ : Type ?u.58480\nι : Type ?u.58483\ninst✝ : DecidableEq α\ns : Multiset α\n⊢ (↑toFinsupp s).support = toFinset s",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.58480\nι : Type ?u.58483\ninst✝ : DecidableEq α\ns : Multiset α\na✝ : α\n⊢ a✝ ∈ (↑toFinsupp s).support ↔ a✝ ∈ toFinset s",
"tactic": "simp [toFinsupp]"
}
] |
[
153,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
lp.norm_le_of_forall_le
|
[
{
"state_after": "case inl\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nC : ℝ\nhC : 0 ≤ C\nhCf : ∀ (i : α), ‖↑f i‖ ≤ C\nh✝ : IsEmpty α\n⊢ ‖f‖ ≤ C\n\ncase inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nC : ℝ\nhC : 0 ≤ C\nhCf : ∀ (i : α), ‖↑f i‖ ≤ C\nh✝ : Nonempty α\n⊢ ‖f‖ ≤ C",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nC : ℝ\nhC : 0 ≤ C\nhCf : ∀ (i : α), ‖↑f i‖ ≤ C\n⊢ ‖f‖ ≤ C",
"tactic": "cases isEmpty_or_nonempty α"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nC : ℝ\nhC : 0 ≤ C\nhCf : ∀ (i : α), ‖↑f i‖ ≤ C\nh✝ : IsEmpty α\n⊢ ‖f‖ ≤ C",
"tactic": "simpa [eq_zero' f] using hC"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\nC : ℝ\nhC : 0 ≤ C\nhCf : ∀ (i : α), ‖↑f i‖ ≤ C\nh✝ : Nonempty α\n⊢ ‖f‖ ≤ C",
"tactic": "exact norm_le_of_forall_le' C hCf"
}
] |
[
587,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Mathlib/Topology/Maps.lean
|
isOpenMap_iff_interior
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.197449\nδ : Type ?u.197452\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nhs : ∀ (s : Set α), f '' interior s ⊆ interior (f '' s)\nu : Set α\nhu : IsOpen u\n⊢ f '' u = f '' interior u",
"tactic": "rw [hu.interior_eq]"
}
] |
[
456,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
450,
1
] |
Mathlib/Data/PFun.lean
|
PFun.mem_prodMap
|
[
{
"state_after": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type u_4\nε : Type ?u.63784\nι : Type ?u.63787\nf✝ : α →. β\nf : α →. γ\ng : β →. δ\nx : α × β\ny : γ × δ\n⊢ y ∈ prodMap f g x ↔ ∃ hp hq, Part.get (f x.fst) hp = y.fst ∧ Part.get (g x.snd) hq = y.snd\n\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type u_4\nε : Type ?u.63784\nι : Type ?u.63787\nf✝ : α →. β\nf : α →. γ\ng : β →. δ\nx : α × β\ny : γ × δ\n⊢ (∃ hp hq, Part.get (f x.fst) hp = y.fst ∧ Part.get (g x.snd) hq = y.snd) ↔ y.fst ∈ f x.fst ∧ y.snd ∈ g x.snd",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type u_4\nε : Type ?u.63784\nι : Type ?u.63787\nf✝ : α →. β\nf : α →. γ\ng : β →. δ\nx : α × β\ny : γ × δ\n⊢ y ∈ prodMap f g x ↔ y.fst ∈ f x.fst ∧ y.snd ∈ g x.snd",
"tactic": "trans ∃ hp hq, (f x.1).get hp = y.1 ∧ (g x.2).get hq = y.2"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type u_4\nε : Type ?u.63784\nι : Type ?u.63787\nf✝ : α →. β\nf : α →. γ\ng : β →. δ\nx : α × β\ny : γ × δ\n⊢ y ∈ prodMap f g x ↔ ∃ hp hq, Part.get (f x.fst) hp = y.fst ∧ Part.get (g x.snd) hq = y.snd",
"tactic": "simp only [prodMap, Part.mem_mk_iff, And.exists, Prod.ext_iff]"
}
] |
[
688,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
684,
1
] |
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
|
ProjectiveSpectrum.mem_vanishingIdeal
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\nf : A\n⊢ f ∈ vanishingIdeal t ↔ ∀ (x : ProjectiveSpectrum 𝒜), x ∈ t → f ∈ x.asHomogeneousIdeal",
"tactic": "rw [← SetLike.mem_coe, coe_vanishingIdeal, Set.mem_setOf_eq]"
}
] |
[
115,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.sin_coe
|
[] |
[
312,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.image₂_subset_iff_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.19045\nβ : Type u_3\nβ' : Type ?u.19051\nγ : Type u_1\nγ' : Type ?u.19057\nδ : Type ?u.19060\nδ' : Type ?u.19063\nε : Type ?u.19066\nε' : Type ?u.19069\nζ : Type ?u.19072\nζ' : Type ?u.19075\nν : Type ?u.19078\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\n⊢ image₂ f s t ⊆ u ↔ ∀ (a : α), a ∈ s → image (fun b => f a b) t ⊆ u",
"tactic": "simp_rw [image₂_subset_iff, image_subset_iff]"
}
] |
[
113,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Topology/LocalExtr.lean
|
Filter.EventuallyEq.isLocalMaxOn_iff
|
[] |
[
565,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
563,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.isFiniteMeasure_map
|
[
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.617578\nδ : Type ?u.617581\nι : Type ?u.617584\nR : Type ?u.617587\nR' : Type ?u.617590\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → β\nhf : AEMeasurable f\n⊢ IsFiniteMeasure (map f μ)\n\ncase neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.617578\nδ : Type ?u.617581\nι : Type ?u.617584\nR : Type ?u.617587\nR' : Type ?u.617590\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → β\nhf : ¬AEMeasurable f\n⊢ IsFiniteMeasure (map f μ)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.617578\nδ : Type ?u.617581\nι : Type ?u.617584\nR : Type ?u.617587\nR' : Type ?u.617590\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → β\n⊢ IsFiniteMeasure (map f μ)",
"tactic": "by_cases hf : AEMeasurable f μ"
},
{
"state_after": "case pos.measure_univ_lt_top\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.617578\nδ : Type ?u.617581\nι : Type ?u.617584\nR : Type ?u.617587\nR' : Type ?u.617590\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → β\nhf : AEMeasurable f\n⊢ ↑↑(map f μ) univ < ⊤",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.617578\nδ : Type ?u.617581\nι : Type ?u.617584\nR : Type ?u.617587\nR' : Type ?u.617590\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → β\nhf : AEMeasurable f\n⊢ IsFiniteMeasure (map f μ)",
"tactic": "constructor"
},
{
"state_after": "case pos.measure_univ_lt_top\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.617578\nδ : Type ?u.617581\nι : Type ?u.617584\nR : Type ?u.617587\nR' : Type ?u.617590\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → β\nhf : AEMeasurable f\n⊢ ↑↑μ (f ⁻¹' univ) < ⊤",
"state_before": "case pos.measure_univ_lt_top\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.617578\nδ : Type ?u.617581\nι : Type ?u.617584\nR : Type ?u.617587\nR' : Type ?u.617590\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → β\nhf : AEMeasurable f\n⊢ ↑↑(map f μ) univ < ⊤",
"tactic": "rw [map_apply_of_aemeasurable hf MeasurableSet.univ]"
},
{
"state_after": "no goals",
"state_before": "case pos.measure_univ_lt_top\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.617578\nδ : Type ?u.617581\nι : Type ?u.617584\nR : Type ?u.617587\nR' : Type ?u.617590\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → β\nhf : AEMeasurable f\n⊢ ↑↑μ (f ⁻¹' univ) < ⊤",
"tactic": "exact measure_lt_top μ _"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.617578\nδ : Type ?u.617581\nι : Type ?u.617584\nR : Type ?u.617587\nR' : Type ?u.617590\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → β\nhf : ¬AEMeasurable f\n⊢ IsFiniteMeasure 0",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.617578\nδ : Type ?u.617581\nι : Type ?u.617584\nR : Type ?u.617587\nR' : Type ?u.617590\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → β\nhf : ¬AEMeasurable f\n⊢ IsFiniteMeasure (map f μ)",
"tactic": "rw [map_of_not_aemeasurable hf]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.617578\nδ : Type ?u.617581\nι : Type ?u.617584\nR : Type ?u.617587\nR' : Type ?u.617590\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → β\nhf : ¬AEMeasurable f\n⊢ IsFiniteMeasure 0",
"tactic": "exact MeasureTheory.isFiniteMeasureZero"
}
] |
[
3133,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3126,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
Ideal.IsHomogeneous.toIdeal_homogeneousHull_eq_self
|
[
{
"state_after": "ι : Type u_1\nσ : Type u_2\nR : Type ?u.212520\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\n⊢ toIdeal (homogeneousHull 𝒜 I) ≤ I",
"state_before": "ι : Type u_1\nσ : Type u_2\nR : Type ?u.212520\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\n⊢ toIdeal (homogeneousHull 𝒜 I) = I",
"tactic": "apply le_antisymm _ (Ideal.le_toIdeal_homogeneousHull _ _)"
},
{
"state_after": "ι : Type u_1\nσ : Type u_2\nR : Type ?u.212520\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\n⊢ {r | ∃ i x, ↑(↑(↑(decompose 𝒜) ↑x) i) = r} ⊆ ↑I",
"state_before": "ι : Type u_1\nσ : Type u_2\nR : Type ?u.212520\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\n⊢ toIdeal (homogeneousHull 𝒜 I) ≤ I",
"tactic": "apply Ideal.span_le.2"
},
{
"state_after": "case intro.intro\nι : Type u_1\nσ : Type u_2\nR : Type ?u.212520\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\ni : ι\nx : { x // x ∈ I }\n⊢ ↑(↑(↑(decompose 𝒜) ↑x) i) ∈ ↑I",
"state_before": "ι : Type u_1\nσ : Type u_2\nR : Type ?u.212520\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\n⊢ {r | ∃ i x, ↑(↑(↑(decompose 𝒜) ↑x) i) = r} ⊆ ↑I",
"tactic": "rintro _ ⟨i, x, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nι : Type u_1\nσ : Type u_2\nR : Type ?u.212520\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\ni : ι\nx : { x // x ∈ I }\n⊢ ↑(↑(↑(decompose 𝒜) ↑x) i) ∈ ↑I",
"tactic": "exact h _ x.prop"
}
] |
[
573,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
568,
1
] |
Mathlib/Analysis/Convex/Normed.lean
|
bounded_convexHull
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.42300\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns✝ t s : Set E\n⊢ Metric.Bounded (↑(convexHull ℝ).toOrderHom s) ↔ Metric.Bounded s",
"tactic": "simp only [Metric.bounded_iff_ediam_ne_top, convexHull_ediam]"
}
] |
[
123,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_extend_by_one
|
[] |
[
1037,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1035,
1
] |
Mathlib/FieldTheory/SplittingField/IsSplittingField.lean
|
Polynomial.IsSplittingField.adjoin_roots
|
[] |
[
69,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
Module.End_isUnit_apply_inv_apply_of_isUnit
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : End R M\nh : IsUnit f\nx : M\n⊢ ↑(f * (IsUnit.unit h).inv) x = x",
"tactic": "simp"
}
] |
[
646,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/Algebra/Ring/Basic.lean
|
AddMonoidHom.coe_mulRight
|
[] |
[
85,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.cons_injective_of_injective
|
[
{
"state_after": "case refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\n⊢ ∀ ⦃a₂ : Fin (n + 1)⦄, cons x₀ x 0 = cons x₀ x a₂ → 0 = a₂\n\ncase refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\n⊢ ∀ (i : Fin n) ⦃a₂ : Fin (n + 1)⦄, cons x₀ x (succ i) = cons x₀ x a₂ → succ i = a₂",
"state_before": "m n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\n⊢ Injective (cons x₀ x)",
"tactic": "refine' Fin.cases _ _"
},
{
"state_after": "case refine'_1.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\n⊢ cons x₀ x 0 = cons x₀ x 0 → 0 = 0\n\ncase refine'_1.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\n⊢ ∀ (i : Fin n), cons x₀ x 0 = cons x₀ x (succ i) → 0 = succ i",
"state_before": "case refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\n⊢ ∀ ⦃a₂ : Fin (n + 1)⦄, cons x₀ x 0 = cons x₀ x a₂ → 0 = a₂",
"tactic": "refine' Fin.cases _ _"
},
{
"state_after": "case refine'_1.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\na✝ : cons x₀ x 0 = cons x₀ x 0\n⊢ 0 = 0",
"state_before": "case refine'_1.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\n⊢ cons x₀ x 0 = cons x₀ x 0 → 0 = 0",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\na✝ : cons x₀ x 0 = cons x₀ x 0\n⊢ 0 = 0",
"tactic": "rfl"
},
{
"state_after": "case refine'_1.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\nj : Fin n\nh : cons x₀ x 0 = cons x₀ x (succ j)\n⊢ 0 = succ j",
"state_before": "case refine'_1.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\n⊢ ∀ (i : Fin n), cons x₀ x 0 = cons x₀ x (succ i) → 0 = succ i",
"tactic": "intro j h"
},
{
"state_after": "case refine'_1.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\nj : Fin n\nh : x₀ = x j\n⊢ 0 = succ j",
"state_before": "case refine'_1.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\nj : Fin n\nh : cons x₀ x 0 = cons x₀ x (succ j)\n⊢ 0 = succ j",
"tactic": "rw [cons_zero, cons_succ] at h"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\nj : Fin n\nh : x₀ = x j\n⊢ 0 = succ j",
"tactic": "exact hx₀.elim ⟨_, h.symm⟩"
},
{
"state_after": "case refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni : Fin n\n⊢ ∀ ⦃a₂ : Fin (n + 1)⦄, cons x₀ x (succ i) = cons x₀ x a₂ → succ i = a₂",
"state_before": "case refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\n⊢ ∀ (i : Fin n) ⦃a₂ : Fin (n + 1)⦄, cons x₀ x (succ i) = cons x₀ x a₂ → succ i = a₂",
"tactic": "intro i"
},
{
"state_after": "case refine'_2.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni : Fin n\n⊢ cons x₀ x (succ i) = cons x₀ x 0 → succ i = 0\n\ncase refine'_2.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni : Fin n\n⊢ ∀ (i_1 : Fin n), cons x₀ x (succ i) = cons x₀ x (succ i_1) → succ i = succ i_1",
"state_before": "case refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni : Fin n\n⊢ ∀ ⦃a₂ : Fin (n + 1)⦄, cons x₀ x (succ i) = cons x₀ x a₂ → succ i = a₂",
"tactic": "refine' Fin.cases _ _"
},
{
"state_after": "case refine'_2.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni : Fin n\nh : cons x₀ x (succ i) = cons x₀ x 0\n⊢ succ i = 0",
"state_before": "case refine'_2.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni : Fin n\n⊢ cons x₀ x (succ i) = cons x₀ x 0 → succ i = 0",
"tactic": "intro h"
},
{
"state_after": "case refine'_2.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni : Fin n\nh : x i = x₀\n⊢ succ i = 0",
"state_before": "case refine'_2.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni : Fin n\nh : cons x₀ x (succ i) = cons x₀ x 0\n⊢ succ i = 0",
"tactic": "rw [cons_zero, cons_succ] at h"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni : Fin n\nh : x i = x₀\n⊢ succ i = 0",
"tactic": "exact hx₀.elim ⟨_, h⟩"
},
{
"state_after": "case refine'_2.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni j : Fin n\nh : cons x₀ x (succ i) = cons x₀ x (succ j)\n⊢ succ i = succ j",
"state_before": "case refine'_2.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni : Fin n\n⊢ ∀ (i_1 : Fin n), cons x₀ x (succ i) = cons x₀ x (succ i_1) → succ i = succ i_1",
"tactic": "intro j h"
},
{
"state_after": "case refine'_2.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni j : Fin n\nh : x i = x j\n⊢ succ i = succ j",
"state_before": "case refine'_2.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni j : Fin n\nh : cons x₀ x (succ i) = cons x₀ x (succ j)\n⊢ succ i = succ j",
"tactic": "rw [cons_succ, cons_succ] at h"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx✝ : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\nα : Type u_1\nx₀ : α\nx : Fin n → α\nhx₀ : ¬x₀ ∈ Set.range x\nhx : Injective x\ni j : Fin n\nh : x i = x j\n⊢ succ i = succ j",
"tactic": "exact congr_arg _ (hx h)"
}
] |
[
187,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/RingTheory/PrincipalIdealDomain.lean
|
Irreducible.dvd_iff_not_coprime
|
[] |
[
448,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
1
] |
Mathlib/Algebra/Invertible.lean
|
invOf_eq_inv
|
[] |
[
384,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.isCaratheodory_sum
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\n⊢ ∑ i in Finset.range 0, ↑m (t ∩ s i) = ↑m (t ∩ ⋃ (i : ℕ) (_ : i < 0), s i)",
"tactic": "simp [Nat.not_lt_zero, m.empty]"
},
{
"state_after": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nn : ℕ\n⊢ (s n ∩ ⋃ (k : ℕ) (_ : k < n), s k) ⊆ ∅",
"state_before": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nn : ℕ\n⊢ ∑ i in Finset.range (Nat.succ n), ↑m (t ∩ s i) = ↑m (t ∩ ⋃ (i : ℕ) (_ : i < Nat.succ n), s i)",
"tactic": "rw [biUnion_lt_succ, Finset.sum_range_succ, Set.union_comm, isCaratheodory_sum h hd,\n m.measure_inter_union _ (h n), add_comm]"
},
{
"state_after": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nn : ℕ\na : α\n⊢ (a ∈ s n ∩ ⋃ (k : ℕ) (_ : k < n), s k) → a ∈ ∅",
"state_before": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nn : ℕ\n⊢ (s n ∩ ⋃ (k : ℕ) (_ : k < n), s k) ⊆ ∅",
"tactic": "intro a"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nm : OuterMeasure α\ns✝ s₁ s₂ : Set α\ns : ℕ → Set α\nh : ∀ (i : ℕ), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set α\nn : ℕ\na : α\n⊢ (a ∈ s n ∩ ⋃ (k : ℕ) (_ : k < n), s k) → a ∈ ∅",
"tactic": "simpa using fun (h₁ : a ∈ s n) i (hi : i < n) h₂ => (hd (ne_of_gt hi)).le_bot ⟨h₁, h₂⟩"
}
] |
[
1003,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
995,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.lift_of_comp
|
[] |
[
528,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
526,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
inner_add_left
|
[] |
[
461,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
460,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
IsComplete.nonempty_iInter_of_nonempty_biInter
|
[
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\n⊢ Set.Nonempty (⋂ (n : ℕ), s n)",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\n⊢ Set.Nonempty (⋂ (n : ℕ), s n)",
"tactic": "let u N := (h N).some"
},
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\n⊢ Set.Nonempty (⋂ (n : ℕ), s n)",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\n⊢ Set.Nonempty (⋂ (n : ℕ), s n)",
"tactic": "have I : ∀ n N, n ≤ N → u N ∈ s n := by\n intro n N hn\n apply mem_of_subset_of_mem _ (h N).choose_spec\n intro x hx\n simp only [mem_iInter] at hx\n exact hx n hn"
},
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\nthis : CauchySeq u\n⊢ Set.Nonempty (⋂ (n : ℕ), s n)",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\n⊢ Set.Nonempty (⋂ (n : ℕ), s n)",
"tactic": "have : CauchySeq u := by\n apply cauchySeq_of_le_tendsto_0 _ _ h'\n intro m n N hm hn\n exact dist_le_diam_of_mem (h's N) (I _ _ hm) (I _ _ hn)"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx✝ y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\nthis : CauchySeq u\nx : α\nxlim : Tendsto (fun n => u n) atTop (𝓝 x)\n⊢ Set.Nonempty (⋂ (n : ℕ), s n)",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\nthis : CauchySeq u\n⊢ Set.Nonempty (⋂ (n : ℕ), s n)",
"tactic": "obtain ⟨x, -, xlim⟩ : ∃ x ∈ s 0, Tendsto (fun n : ℕ => u n) atTop (𝓝 x) :=\n cauchySeq_tendsto_of_isComplete h0 (fun n => I 0 n (zero_le _)) this"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx✝ y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\nthis : CauchySeq u\nx : α\nxlim : Tendsto (fun n => u n) atTop (𝓝 x)\nn : ℕ\n⊢ x ∈ s n",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx✝ y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\nthis : CauchySeq u\nx : α\nxlim : Tendsto (fun n => u n) atTop (𝓝 x)\n⊢ Set.Nonempty (⋂ (n : ℕ), s n)",
"tactic": "refine' ⟨x, mem_iInter.2 fun n => _⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx✝ y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\nthis : CauchySeq u\nx : α\nxlim : Tendsto (fun n => u n) atTop (𝓝 x)\nn : ℕ\n⊢ ∀ᶠ (x : ℕ) in atTop, u x ∈ s n",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx✝ y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\nthis : CauchySeq u\nx : α\nxlim : Tendsto (fun n => u n) atTop (𝓝 x)\nn : ℕ\n⊢ x ∈ s n",
"tactic": "apply (hs n).mem_of_tendsto xlim"
},
{
"state_after": "case h\nα : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx✝ y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\nthis : CauchySeq u\nx : α\nxlim : Tendsto (fun n => u n) atTop (𝓝 x)\nn p : ℕ\nhp : p ∈ Ici n\n⊢ Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ p), s n)) ∈ s n",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx✝ y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\nthis : CauchySeq u\nx : α\nxlim : Tendsto (fun n => u n) atTop (𝓝 x)\nn : ℕ\n⊢ ∀ᶠ (x : ℕ) in atTop, u x ∈ s n",
"tactic": "filter_upwards [Ici_mem_atTop n]with p hp"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx✝ y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\nthis : CauchySeq u\nx : α\nxlim : Tendsto (fun n => u n) atTop (𝓝 x)\nn p : ℕ\nhp : p ∈ Ici n\n⊢ Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ p), s n)) ∈ s n",
"tactic": "exact I n p hp"
},
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nn N : ℕ\nhn : n ≤ N\n⊢ u N ∈ s n",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\n⊢ ∀ (n N : ℕ), n ≤ N → u N ∈ s n",
"tactic": "intro n N hn"
},
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nn N : ℕ\nhn : n ≤ N\n⊢ (⋂ (n : ℕ) (_ : n ≤ N), s n) ⊆ s n",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nn N : ℕ\nhn : n ≤ N\n⊢ u N ∈ s n",
"tactic": "apply mem_of_subset_of_mem _ (h N).choose_spec"
},
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx✝ y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nn N : ℕ\nhn : n ≤ N\nx : α\nhx : x ∈ ⋂ (n : ℕ) (_ : n ≤ N), s n\n⊢ x ∈ s n",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nn N : ℕ\nhn : n ≤ N\n⊢ (⋂ (n : ℕ) (_ : n ≤ N), s n) ⊆ s n",
"tactic": "intro x hx"
},
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx✝ y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nn N : ℕ\nhn : n ≤ N\nx : α\nhx : ∀ (i : ℕ), i ≤ N → x ∈ s i\n⊢ x ∈ s n",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx✝ y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nn N : ℕ\nhn : n ≤ N\nx : α\nhx : x ∈ ⋂ (n : ℕ) (_ : n ≤ N), s n\n⊢ x ∈ s n",
"tactic": "simp only [mem_iInter] at hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx✝ y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nn N : ℕ\nhn : n ≤ N\nx : α\nhx : ∀ (i : ℕ), i ≤ N → x ∈ s i\n⊢ x ∈ s n",
"tactic": "exact hx n hn"
},
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\n⊢ ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) ≤ diam (s N)",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\n⊢ CauchySeq u",
"tactic": "apply cauchySeq_of_le_tendsto_0 _ _ h'"
},
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\nm n N : ℕ\nhm : N ≤ m\nhn : N ≤ n\n⊢ dist (u m) (u n) ≤ diam (s N)",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\n⊢ ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) ≤ diam (s N)",
"tactic": "intro m n N hm hn"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.540098\nι : Type ?u.540101\ninst✝ : PseudoMetricSpace α\ns✝ : Set α\nx y z : α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed (s n)\nh's : ∀ (n : ℕ), Bounded (s n)\nh : ∀ (N : ℕ), Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)\nu : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ (n : ℕ) (_ : n ≤ N), s n))\nI : ∀ (n N : ℕ), n ≤ N → u N ∈ s n\nm n N : ℕ\nhm : N ≤ m\nhn : N ≤ n\n⊢ dist (u m) (u n) ≤ diam (s N)",
"tactic": "exact dist_le_diam_of_mem (h's N) (I _ _ hm) (I _ _ hn)"
}
] |
[
2755,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2735,
1
] |
Mathlib/Data/MvPolynomial/Equiv.lean
|
MvPolynomial.natDegree_finSuccEquiv
|
[
{
"state_after": "case pos\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1366622\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nc : f = 0\n⊢ natDegree (↑(finSuccEquiv R n) f) = degreeOf 0 f\n\ncase neg\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1366622\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nc : ¬f = 0\n⊢ natDegree (↑(finSuccEquiv R n) f) = degreeOf 0 f",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1366622\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\n⊢ natDegree (↑(finSuccEquiv R n) f) = degreeOf 0 f",
"tactic": "by_cases c : f = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1366622\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nc : f = 0\n⊢ natDegree (↑(finSuccEquiv R n) f) = degreeOf 0 f",
"tactic": "rw [c, (finSuccEquiv R n).map_zero, Polynomial.natDegree_zero, degreeOf_zero]"
},
{
"state_after": "case neg\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1366622\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nc : ¬f = 0\n⊢ WithBot.unbot' 0 ↑(degreeOf 0 f) = degreeOf 0 f",
"state_before": "case neg\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1366622\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nc : ¬f = 0\n⊢ natDegree (↑(finSuccEquiv R n) f) = degreeOf 0 f",
"tactic": "rw [Polynomial.natDegree, degree_finSuccEquiv (by simpa only [Ne.def] )]"
},
{
"state_after": "case neg\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1366622\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nc : ¬f = 0\n⊢ ↑(degreeOf 0 f) = degreeOf 0 f",
"state_before": "case neg\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1366622\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nc : ¬f = 0\n⊢ WithBot.unbot' 0 ↑(degreeOf 0 f) = degreeOf 0 f",
"tactic": "erw [WithBot.unbot'_coe]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1366622\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nc : ¬f = 0\n⊢ ↑(degreeOf 0 f) = degreeOf 0 f",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1366622\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nc : ¬f = 0\n⊢ f ≠ 0",
"tactic": "simpa only [Ne.def]"
}
] |
[
494,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.exp_neg
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ ↑(exp (-x)) = ↑(exp x)⁻¹",
"tactic": "simp [exp_neg]"
}
] |
[
1169,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1168,
8
] |
Mathlib/InformationTheory/Hamming.lean
|
hammingNorm_pos_iff
|
[] |
[
203,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.range_const
|
[] |
[
1002,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
999,
1
] |
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.eq_reflection_of_eq_subspace
|
[
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s }\ninst✝² : CompleteSpace { x // x ∈ direction s }\np : P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\n⊢ ↑(reflection s) p = ↑(reflection s) p",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns s' : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s }\ninst✝² : Nonempty { x // x ∈ s' }\ninst✝¹ : CompleteSpace { x // x ∈ direction s }\ninst✝ : CompleteSpace { x // x ∈ direction s' }\nh : s = s'\np : P\n⊢ ↑(reflection s) p = ↑(reflection s') p",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝³ : Nonempty { x // x ∈ s }\ninst✝² : CompleteSpace { x // x ∈ direction s }\np : P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\n⊢ ↑(reflection s) p = ↑(reflection s) p",
"tactic": "rfl"
}
] |
[
552,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
1
] |
Mathlib/Combinatorics/Quiver/Path.lean
|
Prefunctor.mapPath_comp
|
[
{
"state_after": "V : Type u₁\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF : V ⥤q W\na b : V\np : Path a b\nc b✝ : V\nq : Path b b✝\ne : b✝ ⟶ c\n⊢ Path.cons (mapPath F (Path.comp p q)) (F.map e) = Path.cons (Path.comp (mapPath F p) (mapPath F q)) (F.map e)",
"state_before": "V : Type u₁\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF : V ⥤q W\na b : V\np : Path a b\nc b✝ : V\nq : Path b b✝\ne : b✝ ⟶ c\n⊢ mapPath F (Path.comp p (Path.cons q e)) = Path.comp (mapPath F p) (mapPath F (Path.cons q e))",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "V : Type u₁\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF : V ⥤q W\na b : V\np : Path a b\nc b✝ : V\nq : Path b b✝\ne : b✝ ⟶ c\n⊢ Path.cons (mapPath F (Path.comp p q)) (F.map e) = Path.cons (Path.comp (mapPath F p) (mapPath F q)) (F.map e)",
"tactic": "rw [mapPath_comp p q]"
}
] |
[
234,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Std/Logic.lean
|
forall_prop_of_true
|
[] |
[
488,
28
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
487,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.Monic.natDegree_pow
|
[
{
"state_after": "case zero\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\n⊢ natDegree (p ^ Nat.zero) = Nat.zero * natDegree p\n\ncase succ\nR : Type u\nS : Type v\na b : R\nm n✝ : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\nn : ℕ\nhn : natDegree (p ^ n) = n * natDegree p\n⊢ natDegree (p ^ Nat.succ n) = Nat.succ n * natDegree p",
"state_before": "R : Type u\nS : Type v\na b : R\nm n✝ : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\nn : ℕ\n⊢ natDegree (p ^ n) = n * natDegree p",
"tactic": "induction' n with n hn"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\n⊢ natDegree (p ^ Nat.zero) = Nat.zero * natDegree p",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\nS : Type v\na b : R\nm n✝ : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\nn : ℕ\nhn : natDegree (p ^ n) = n * natDegree p\n⊢ natDegree (p ^ Nat.succ n) = Nat.succ n * natDegree p",
"tactic": "rw [pow_succ, hp.natDegree_mul (hp.pow n), hn, Nat.succ_mul, add_comm]"
}
] |
[
236,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.diff_mem
|
[] |
[
168,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigO_of_div_tendsto_nhds
|
[] |
[
2010,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2007,
1
] |
Mathlib/CategoryTheory/LiftingProperties/Adjunction.lean
|
CategoryTheory.CommSq.left_adjoint_hasLift_iff
|
[
{
"state_after": "C : Type u_4\nD : Type u_1\ninst✝¹ : Category C\ninst✝ : Category D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : A ⟶ F.obj X\nv : B ⟶ F.obj Y\nsq : CommSq u i (F.map p) v\nadj : G ⊣ F\n⊢ Nonempty\n (LiftStruct\n (_ : CommSq (↑(Adjunction.homEquiv adj A X).symm u) (G.map i) p (↑(Adjunction.homEquiv adj B Y).symm v))) ↔\n Nonempty (LiftStruct sq)",
"state_before": "C : Type u_4\nD : Type u_1\ninst✝¹ : Category C\ninst✝ : Category D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : A ⟶ F.obj X\nv : B ⟶ F.obj Y\nsq : CommSq u i (F.map p) v\nadj : G ⊣ F\n⊢ HasLift (_ : CommSq (↑(Adjunction.homEquiv adj A X).symm u) (G.map i) p (↑(Adjunction.homEquiv adj B Y).symm v)) ↔\n HasLift sq",
"tactic": "simp only [HasLift.iff]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_4\nD : Type u_1\ninst✝¹ : Category C\ninst✝ : Category D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : A ⟶ F.obj X\nv : B ⟶ F.obj Y\nsq : CommSq u i (F.map p) v\nadj : G ⊣ F\n⊢ Nonempty\n (LiftStruct\n (_ : CommSq (↑(Adjunction.homEquiv adj A X).symm u) (G.map i) p (↑(Adjunction.homEquiv adj B Y).symm v))) ↔\n Nonempty (LiftStruct sq)",
"tactic": "exact Equiv.nonempty_congr (sq.leftAdjointLiftStructEquiv adj).symm"
}
] |
[
118,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Order/SymmDiff.lean
|
bihimp_himp_eq_inf
|
[] |
[
302,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.closure_ball_subset_closedBall
|
[] |
[
1882,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1881,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_cond_eq_prod_of_cond_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\n⊢ (∏ᶠ (i : α) (_ : p i), f i) = ∏ i in t, f i",
"state_before": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\n⊢ (∏ᶠ (i : α) (_ : p i), f i) = ∏ i in t, f i",
"tactic": "set s := { x | p x }"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\nthis : mulSupport (mulIndicator s f) ⊆ ↑t\n⊢ (∏ᶠ (i : α) (_ : p i), f i) = ∏ i in t, f i",
"state_before": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\n⊢ (∏ᶠ (i : α) (_ : p i), f i) = ∏ i in t, f i",
"tactic": "have : mulSupport (s.mulIndicator f) ⊆ t := by\n rw [Set.mulSupport_mulIndicator]\n intro x hx\n exact (h hx.2).1 hx.1"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\nthis : mulSupport (mulIndicator s f) ⊆ ↑t\n⊢ ∏ i in t, mulIndicator s f i = ∏ i in t, f i",
"state_before": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\nthis : mulSupport (mulIndicator s f) ⊆ ↑t\n⊢ (∏ᶠ (i : α) (_ : p i), f i) = ∏ i in t, f i",
"tactic": "erw [finprod_mem_def, finprod_eq_prod_of_mulSupport_subset _ this]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\nthis : mulSupport (mulIndicator s f) ⊆ ↑t\nx : α\nhx : x ∈ t\nhxs : ¬x ∈ s\n⊢ f x = 1",
"state_before": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\nthis : mulSupport (mulIndicator s f) ⊆ ↑t\n⊢ ∏ i in t, mulIndicator s f i = ∏ i in t, f i",
"tactic": "refine' Finset.prod_congr rfl fun x hx => mulIndicator_apply_eq_self.2 fun hxs => _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\nthis : mulSupport (mulIndicator s f) ⊆ ↑t\nx : α\nhx : x ∈ t\nhxs : f x ≠ 1\n⊢ x ∈ {x | p x}",
"state_before": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\nthis : mulSupport (mulIndicator s f) ⊆ ↑t\nx : α\nhx : x ∈ t\nhxs : ¬x ∈ s\n⊢ f x = 1",
"tactic": "contrapose! hxs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\nthis : mulSupport (mulIndicator s f) ⊆ ↑t\nx : α\nhx : x ∈ t\nhxs : f x ≠ 1\n⊢ x ∈ {x | p x}",
"tactic": "exact (h hxs).2 hx"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\n⊢ s ∩ mulSupport f ⊆ ↑t",
"state_before": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\n⊢ mulSupport (mulIndicator s f) ⊆ ↑t",
"tactic": "rw [Set.mulSupport_mulIndicator]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\nx : α\nhx : x ∈ s ∩ mulSupport f\n⊢ x ∈ ↑t",
"state_before": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\n⊢ s ∩ mulSupport f ⊆ ↑t",
"tactic": "intro x hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.188274\nι : Type ?u.188277\nG : Type ?u.188280\nM : Type u_2\nN : Type ?u.188286\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\np : α → Prop\nt : Finset α\nh : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)\ns : Set α := {x | p x}\nx : α\nhx : x ∈ s ∩ mulSupport f\n⊢ x ∈ ↑t",
"tactic": "exact (h hx.2).1 hx.1"
}
] |
[
452,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.range_castSucc
|
[] |
[
1323,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1321,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.image_const_add_uIcc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na b c d : α\n⊢ (fun x => a + x) '' [[b, c]] = [[a + b, a + c]]",
"tactic": "simp [add_comm]"
}
] |
[
456,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
456,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
MeasurableEmbedding.comp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.379755\nδ' : Type ?u.379758\nι : Sort uι\ns✝ t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\ns : Set α\nhs : MeasurableSet s\n⊢ MeasurableSet (g ∘ f '' s)",
"tactic": "rwa [image_comp, hg.measurableSet_image, hf.measurableSet_image]"
}
] |
[
1094,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1091,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.lift'_principal
|
[] |
[
320,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/Order/SuccPred/LinearLocallyFinite.lean
|
iterate_succ_toZ
|
[
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i : ι\nhi : i0 ≤ i\n⊢ (succ^[Nat.find (_ : ∃ n, (succ^[n]) i0 = i)]) i0 = i",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i : ι\nhi : i0 ≤ i\n⊢ (succ^[Int.toNat (toZ i0 i)]) i0 = i",
"tactic": "rw [toZ_of_ge hi, Int.toNat_coe_nat]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i : ι\nhi : i0 ≤ i\n⊢ (succ^[Nat.find (_ : ∃ n, (succ^[n]) i0 = i)]) i0 = i",
"tactic": "exact Nat.find_spec (exists_succ_iterate_of_le hi)"
}
] |
[
218,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
div_div_cancel_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.70321\nβ : Type ?u.70324\nG : Type u_1\ninst✝ : CommGroup G\na✝ b✝ c d a b : G\n⊢ a / b / a = b⁻¹",
"tactic": "simp"
}
] |
[
948,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
948,
1
] |
Mathlib/GroupTheory/GroupAction/Defs.lean
|
SMul.comp.smulCommClass'
|
[] |
[
398,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.IsPeriodicPt.iterate_mod_apply
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3547\nf fa : α → α\nfb : β → β\nx y : α\nm✝ n : ℕ\nh : IsPeriodicPt f n x\nm : ℕ\n⊢ (f^[m % n]) x = (f^[m]) x",
"tactic": "conv_rhs => rw [← Nat.mod_add_div m n, iterate_add_apply, (h.mul_const _).eq]"
}
] |
[
161,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
uniformity_eq_comap_nhds_one'
|
[] |
[
581,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/Algebra/Regular/Basic.lean
|
IsRightRegular.subsingleton
|
[] |
[
199,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/MeasureTheory/Group/FundamentalDomain.lean
|
MeasureTheory.IsFundamentalDomain.integrableOn_iff
|
[] |
[
399,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
11
] |
Mathlib/Data/PEquiv.lean
|
PEquiv.trans_assoc
|
[] |
[
147,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Order/SymmDiff.lean
|
sdiff_symmDiff_left
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.62901\nα : Type u_1\nβ : Type ?u.62907\nπ : ι → Type ?u.62912\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\n⊢ a \\ a ∆ b = a ⊓ b",
"tactic": "simp [sdiff_symmDiff]"
}
] |
[
433,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Data/Bitvec/Lemmas.lean
|
Bitvec.addLsb_div_two
|
[
{
"state_after": "case true\nx : ℕ\n⊢ x + 1 / 2 = x",
"state_before": "x : ℕ\nb : Bool\n⊢ addLsb x b / 2 = x",
"tactic": "cases b <;>\n simp only [Nat.add_mul_div_left, addLsb, ← two_mul, add_comm, Nat.succ_pos',\n Nat.mul_div_right, gt_iff_lt, zero_add, cond]"
},
{
"state_after": "no goals",
"state_before": "case true\nx : ℕ\n⊢ x + 1 / 2 = x",
"tactic": "norm_num"
}
] |
[
119,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Data/List/Permutation.lean
|
List.mem_permutationsAux2'
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.19875\nt : α\nts ys l : List α\n⊢ l ∈ (permutationsAux2 t ts [] ys fun x => [] ++ x).snd ↔ ∃ l₁ l₂, l₂ ≠ [] ∧ ys = l₁ ++ l₂ ∧ l = l₁ ++ t :: l₂ ++ ts",
"state_before": "α : Type u_1\nβ : Type ?u.19875\nt : α\nts ys l : List α\n⊢ l ∈ (permutationsAux2 t ts [] ys id).snd ↔ ∃ l₁ l₂, l₂ ≠ [] ∧ ys = l₁ ++ l₂ ∧ l = l₁ ++ t :: l₂ ++ ts",
"tactic": "rw [show @id (List α) = ([] ++ .) by funext _; rfl]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.19875\nt : α\nts ys l : List α\n⊢ l ∈ (permutationsAux2 t ts [] ys fun x => [] ++ x).snd ↔ ∃ l₁ l₂, l₂ ≠ [] ∧ ys = l₁ ++ l₂ ∧ l = l₁ ++ t :: l₂ ++ ts",
"tactic": "apply mem_permutationsAux2"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.19875\nt : α\nts ys l x✝ : List α\n⊢ id x✝ = [] ++ x✝",
"state_before": "α : Type u_1\nβ : Type ?u.19875\nt : α\nts ys l : List α\n⊢ id = fun x => [] ++ x",
"tactic": "funext _"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.19875\nt : α\nts ys l x✝ : List α\n⊢ id x✝ = [] ++ x✝",
"tactic": "rfl"
}
] |
[
167,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Topology/Algebra/Valuation.lean
|
Valued.cauchy_iff
|
[
{
"state_after": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\n⊢ (Filter.NeBot F ∧\n ∀ (U : Set R),\n U ∈ RingFilterBasis.toAddGroupFilterBasis → ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ U) ↔\n Filter.NeBot F ∧ ∀ (γ : Γ₀ˣ), ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ",
"state_before": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\n⊢ Cauchy F ↔ Filter.NeBot F ∧ ∀ (γ : Γ₀ˣ), ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ",
"tactic": "rw [toUniformSpace_eq, AddGroupFilterBasis.cauchy_iff]"
},
{
"state_after": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\n⊢ (∀ (U : Set R),\n U ∈ RingFilterBasis.toAddGroupFilterBasis → ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ U) ↔\n ∀ (γ : Γ₀ˣ), ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ",
"state_before": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\n⊢ (Filter.NeBot F ∧\n ∀ (U : Set R),\n U ∈ RingFilterBasis.toAddGroupFilterBasis → ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ U) ↔\n Filter.NeBot F ∧ ∀ (γ : Γ₀ˣ), ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ",
"tactic": "apply and_congr Iff.rfl"
},
{
"state_after": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\n⊢ (∀ (U : Set R), (∃ i, U = ↑(ltAddSubgroup v i)) → ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ U) ↔\n ∀ (γ : Γ₀ˣ), ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ",
"state_before": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\n⊢ (∀ (U : Set R),\n U ∈ RingFilterBasis.toAddGroupFilterBasis → ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ U) ↔\n ∀ (γ : Γ₀ˣ), ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ",
"tactic": "simp_rw [Valued.v.subgroups_basis.mem_addGroupFilterBasis_iff]"
},
{
"state_after": "case mp\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\n⊢ (∀ (U : Set R), (∃ i, U = ↑(ltAddSubgroup v i)) → ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ U) →\n ∀ (γ : Γ₀ˣ), ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ\n\ncase mpr\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\n⊢ (∀ (γ : Γ₀ˣ), ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ) →\n ∀ (U : Set R), (∃ i, U = ↑(ltAddSubgroup v i)) → ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ U",
"state_before": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\n⊢ (∀ (U : Set R), (∃ i, U = ↑(ltAddSubgroup v i)) → ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ U) ↔\n ∀ (γ : Γ₀ˣ), ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\nh : ∀ (U : Set R), (∃ i, U = ↑(ltAddSubgroup v i)) → ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ U\nγ : Γ₀ˣ\n⊢ ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ",
"state_before": "case mp\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\n⊢ (∀ (U : Set R), (∃ i, U = ↑(ltAddSubgroup v i)) → ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ U) →\n ∀ (γ : Γ₀ˣ), ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ",
"tactic": "intro h γ"
},
{
"state_after": "no goals",
"state_before": "case mp\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\nh : ∀ (U : Set R), (∃ i, U = ↑(ltAddSubgroup v i)) → ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ U\nγ : Γ₀ˣ\n⊢ ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ",
"tactic": "exact h _ (Valued.v.subgroups_basis.mem_addGroupFilterBasis _)"
},
{
"state_after": "case mpr.intro\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\nh : ∀ (γ : Γ₀ˣ), ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ\nγ : Γ₀ˣ\n⊢ ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ ↑(ltAddSubgroup v γ)",
"state_before": "case mpr\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\n⊢ (∀ (γ : Γ₀ˣ), ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ) →\n ∀ (U : Set R), (∃ i, U = ↑(ltAddSubgroup v i)) → ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ U",
"tactic": "rintro h - ⟨γ, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nF : Filter R\nh : ∀ (γ : Γ₀ˣ), ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → ↑v (y - x) < ↑γ\nγ : Γ₀ˣ\n⊢ ∃ M, M ∈ F ∧ ∀ (x : R), x ∈ M → ∀ (y : R), y ∈ M → y - x ∈ ↑(ltAddSubgroup v γ)",
"tactic": "exact h γ"
}
] |
[
170,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/ModelTheory/Satisfiability.lean
|
FirstOrder.Language.BoundedFormula.induction_on_all_ex
|
[
{
"state_after": "L : Language\nT : Theory L\nα : Type w\nn : ℕ\nP : {m : ℕ} → BoundedFormula L α m → Prop\nφ : BoundedFormula L α n\nhqf : ∀ {m : ℕ} {ψ : BoundedFormula L α m}, IsQF ψ → P ψ\nhall : ∀ {m : ℕ} {ψ : BoundedFormula L α (m + 1)}, P ψ → P (∀'ψ)\nhex : ∀ {m : ℕ} {φ : BoundedFormula L α (m + 1)}, P φ → P (∃'φ)\nhse : ∀ {m : ℕ} {φ₁ φ₂ : BoundedFormula L α m}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)\nh' : ∀ {m : ℕ} {φ : BoundedFormula L α m}, IsPrenex φ → P φ\n⊢ P φ\n\ncase h'\nL : Language\nT : Theory L\nα : Type w\nn : ℕ\nP : {m : ℕ} → BoundedFormula L α m → Prop\nφ : BoundedFormula L α n\nhqf : ∀ {m : ℕ} {ψ : BoundedFormula L α m}, IsQF ψ → P ψ\nhall : ∀ {m : ℕ} {ψ : BoundedFormula L α (m + 1)}, P ψ → P (∀'ψ)\nhex : ∀ {m : ℕ} {φ : BoundedFormula L α (m + 1)}, P φ → P (∃'φ)\nhse : ∀ {m : ℕ} {φ₁ φ₂ : BoundedFormula L α m}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)\n⊢ ∀ {m : ℕ} {φ : BoundedFormula L α m}, IsPrenex φ → P φ",
"state_before": "L : Language\nT : Theory L\nα : Type w\nn : ℕ\nP : {m : ℕ} → BoundedFormula L α m → Prop\nφ : BoundedFormula L α n\nhqf : ∀ {m : ℕ} {ψ : BoundedFormula L α m}, IsQF ψ → P ψ\nhall : ∀ {m : ℕ} {ψ : BoundedFormula L α (m + 1)}, P ψ → P (∀'ψ)\nhex : ∀ {m : ℕ} {φ : BoundedFormula L α (m + 1)}, P φ → P (∃'φ)\nhse : ∀ {m : ℕ} {φ₁ φ₂ : BoundedFormula L α m}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)\n⊢ P φ",
"tactic": "suffices h' : ∀ {m} {φ : L.BoundedFormula α m}, φ.IsPrenex → P φ"
},
{
"state_after": "case h'\nL : Language\nT : Theory L\nα : Type w\nn : ℕ\nP : {m : ℕ} → BoundedFormula L α m → Prop\nφ✝ : BoundedFormula L α n\nhqf : ∀ {m : ℕ} {ψ : BoundedFormula L α m}, IsQF ψ → P ψ\nhall : ∀ {m : ℕ} {ψ : BoundedFormula L α (m + 1)}, P ψ → P (∀'ψ)\nhex : ∀ {m : ℕ} {φ : BoundedFormula L α (m + 1)}, P φ → P (∃'φ)\nhse : ∀ {m : ℕ} {φ₁ φ₂ : BoundedFormula L α m}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)\nm : ℕ\nφ : BoundedFormula L α m\nhφ : IsPrenex φ\n⊢ P φ",
"state_before": "case h'\nL : Language\nT : Theory L\nα : Type w\nn : ℕ\nP : {m : ℕ} → BoundedFormula L α m → Prop\nφ : BoundedFormula L α n\nhqf : ∀ {m : ℕ} {ψ : BoundedFormula L α m}, IsQF ψ → P ψ\nhall : ∀ {m : ℕ} {ψ : BoundedFormula L α (m + 1)}, P ψ → P (∀'ψ)\nhex : ∀ {m : ℕ} {φ : BoundedFormula L α (m + 1)}, P φ → P (∃'φ)\nhse : ∀ {m : ℕ} {φ₁ φ₂ : BoundedFormula L α m}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)\n⊢ ∀ {m : ℕ} {φ : BoundedFormula L α m}, IsPrenex φ → P φ",
"tactic": "intro m φ hφ"
},
{
"state_after": "case h'.of_isQF\nL : Language\nT : Theory L\nα : Type w\nn : ℕ\nP : {m : ℕ} → BoundedFormula L α m → Prop\nφ✝¹ : BoundedFormula L α n\nhqf : ∀ {m : ℕ} {ψ : BoundedFormula L α m}, IsQF ψ → P ψ\nhall : ∀ {m : ℕ} {ψ : BoundedFormula L α (m + 1)}, P ψ → P (∀'ψ)\nhex : ∀ {m : ℕ} {φ : BoundedFormula L α (m + 1)}, P φ → P (∃'φ)\nhse : ∀ {m : ℕ} {φ₁ φ₂ : BoundedFormula L α m}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)\nm : ℕ\nφ : BoundedFormula L α m\nn✝ : ℕ\nφ✝ : BoundedFormula L α n✝\nhφ : IsQF φ✝\n⊢ P φ✝\n\ncase h'.all\nL : Language\nT : Theory L\nα : Type w\nn : ℕ\nP : {m : ℕ} → BoundedFormula L α m → Prop\nφ✝¹ : BoundedFormula L α n\nhqf : ∀ {m : ℕ} {ψ : BoundedFormula L α m}, IsQF ψ → P ψ\nhall : ∀ {m : ℕ} {ψ : BoundedFormula L α (m + 1)}, P ψ → P (∀'ψ)\nhex : ∀ {m : ℕ} {φ : BoundedFormula L α (m + 1)}, P φ → P (∃'φ)\nhse : ∀ {m : ℕ} {φ₁ φ₂ : BoundedFormula L α m}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)\nm : ℕ\nφ : BoundedFormula L α m\nn✝ : ℕ\nφ✝ : BoundedFormula L α (n✝ + 1)\nh✝ : IsPrenex φ✝\nhφ : P φ✝\n⊢ P (∀'φ✝)\n\ncase h'.ex\nL : Language\nT : Theory L\nα : Type w\nn : ℕ\nP : {m : ℕ} → BoundedFormula L α m → Prop\nφ✝¹ : BoundedFormula L α n\nhqf : ∀ {m : ℕ} {ψ : BoundedFormula L α m}, IsQF ψ → P ψ\nhall : ∀ {m : ℕ} {ψ : BoundedFormula L α (m + 1)}, P ψ → P (∀'ψ)\nhex : ∀ {m : ℕ} {φ : BoundedFormula L α (m + 1)}, P φ → P (∃'φ)\nhse : ∀ {m : ℕ} {φ₁ φ₂ : BoundedFormula L α m}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)\nm : ℕ\nφ : BoundedFormula L α m\nn✝ : ℕ\nφ✝ : BoundedFormula L α (n✝ + 1)\nh✝ : IsPrenex φ✝\nhφ : P φ✝\n⊢ P (∃'φ✝)",
"state_before": "case h'\nL : Language\nT : Theory L\nα : Type w\nn : ℕ\nP : {m : ℕ} → BoundedFormula L α m → Prop\nφ✝ : BoundedFormula L α n\nhqf : ∀ {m : ℕ} {ψ : BoundedFormula L α m}, IsQF ψ → P ψ\nhall : ∀ {m : ℕ} {ψ : BoundedFormula L α (m + 1)}, P ψ → P (∀'ψ)\nhex : ∀ {m : ℕ} {φ : BoundedFormula L α (m + 1)}, P φ → P (∃'φ)\nhse : ∀ {m : ℕ} {φ₁ φ₂ : BoundedFormula L α m}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)\nm : ℕ\nφ : BoundedFormula L α m\nhφ : IsPrenex φ\n⊢ P φ",
"tactic": "induction' hφ with _ _ hφ _ _ _ hφ _ _ _ hφ"
},
{
"state_after": "no goals",
"state_before": "L : Language\nT : Theory L\nα : Type w\nn : ℕ\nP : {m : ℕ} → BoundedFormula L α m → Prop\nφ : BoundedFormula L α n\nhqf : ∀ {m : ℕ} {ψ : BoundedFormula L α m}, IsQF ψ → P ψ\nhall : ∀ {m : ℕ} {ψ : BoundedFormula L α (m + 1)}, P ψ → P (∀'ψ)\nhex : ∀ {m : ℕ} {φ : BoundedFormula L α (m + 1)}, P φ → P (∃'φ)\nhse : ∀ {m : ℕ} {φ₁ φ₂ : BoundedFormula L α m}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)\nh' : ∀ {m : ℕ} {φ : BoundedFormula L α m}, IsPrenex φ → P φ\n⊢ P φ",
"tactic": "exact (hse φ.semanticallyEquivalent_toPrenex).2 (h' φ.toPrenex_isPrenex)"
},
{
"state_after": "no goals",
"state_before": "case h'.of_isQF\nL : Language\nT : Theory L\nα : Type w\nn : ℕ\nP : {m : ℕ} → BoundedFormula L α m → Prop\nφ✝¹ : BoundedFormula L α n\nhqf : ∀ {m : ℕ} {ψ : BoundedFormula L α m}, IsQF ψ → P ψ\nhall : ∀ {m : ℕ} {ψ : BoundedFormula L α (m + 1)}, P ψ → P (∀'ψ)\nhex : ∀ {m : ℕ} {φ : BoundedFormula L α (m + 1)}, P φ → P (∃'φ)\nhse : ∀ {m : ℕ} {φ₁ φ₂ : BoundedFormula L α m}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)\nm : ℕ\nφ : BoundedFormula L α m\nn✝ : ℕ\nφ✝ : BoundedFormula L α n✝\nhφ : IsQF φ✝\n⊢ P φ✝",
"tactic": "exact hqf hφ"
},
{
"state_after": "no goals",
"state_before": "case h'.all\nL : Language\nT : Theory L\nα : Type w\nn : ℕ\nP : {m : ℕ} → BoundedFormula L α m → Prop\nφ✝¹ : BoundedFormula L α n\nhqf : ∀ {m : ℕ} {ψ : BoundedFormula L α m}, IsQF ψ → P ψ\nhall : ∀ {m : ℕ} {ψ : BoundedFormula L α (m + 1)}, P ψ → P (∀'ψ)\nhex : ∀ {m : ℕ} {φ : BoundedFormula L α (m + 1)}, P φ → P (∃'φ)\nhse : ∀ {m : ℕ} {φ₁ φ₂ : BoundedFormula L α m}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)\nm : ℕ\nφ : BoundedFormula L α m\nn✝ : ℕ\nφ✝ : BoundedFormula L α (n✝ + 1)\nh✝ : IsPrenex φ✝\nhφ : P φ✝\n⊢ P (∀'φ✝)",
"tactic": "exact hall hφ"
},
{
"state_after": "no goals",
"state_before": "case h'.ex\nL : Language\nT : Theory L\nα : Type w\nn : ℕ\nP : {m : ℕ} → BoundedFormula L α m → Prop\nφ✝¹ : BoundedFormula L α n\nhqf : ∀ {m : ℕ} {ψ : BoundedFormula L α m}, IsQF ψ → P ψ\nhall : ∀ {m : ℕ} {ψ : BoundedFormula L α (m + 1)}, P ψ → P (∀'ψ)\nhex : ∀ {m : ℕ} {φ : BoundedFormula L α (m + 1)}, P φ → P (∃'φ)\nhse : ∀ {m : ℕ} {φ₁ φ₂ : BoundedFormula L α m}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)\nm : ℕ\nφ : BoundedFormula L α m\nn✝ : ℕ\nφ✝ : BoundedFormula L α (n✝ + 1)\nh✝ : IsPrenex φ✝\nhφ : P φ✝\n⊢ P (∃'φ✝)",
"tactic": "exact hex hφ"
}
] |
[
623,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
610,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.exists_degree_lt
|
[
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.474475\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : Fintype σ\nf : MvPolynomial σ R\nn : ℕ\nd : σ →₀ ℕ\nhd : d ∈ support f\nh : ∀ (i : σ), n ≤ ↑d i\n⊢ n * Fintype.card σ ≤ totalDegree f",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.474475\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : Fintype σ\nf : MvPolynomial σ R\nn : ℕ\nh : totalDegree f < n * Fintype.card σ\nd : σ →₀ ℕ\nhd : d ∈ support f\n⊢ ∃ i, ↑d i < n",
"tactic": "contrapose! h"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.474475\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : Fintype σ\nf : MvPolynomial σ R\nn : ℕ\nd : σ →₀ ℕ\nhd : d ∈ support f\nh : ∀ (i : σ), n ≤ ↑d i\n⊢ n * Fintype.card σ ≤ totalDegree f",
"tactic": "calc\n n * Fintype.card σ = ∑ _s : σ, n := by\n rw [Finset.sum_const, Nat.nsmul_eq_mul, mul_comm, Finset.card_univ]\n _ ≤ ∑ s, d s := (Finset.sum_le_sum fun s _ => h s)\n _ ≤ d.sum fun _ e => e := by\n rw [Finsupp.sum_fintype]\n intros\n rfl\n _ ≤ f.totalDegree := le_totalDegree hd"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.474475\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : Fintype σ\nf : MvPolynomial σ R\nn : ℕ\nd : σ →₀ ℕ\nhd : d ∈ support f\nh : ∀ (i : σ), n ≤ ↑d i\n⊢ n * Fintype.card σ = ∑ _s : σ, n",
"tactic": "rw [Finset.sum_const, Nat.nsmul_eq_mul, mul_comm, Finset.card_univ]"
},
{
"state_after": "case h\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.474475\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : Fintype σ\nf : MvPolynomial σ R\nn : ℕ\nd : σ →₀ ℕ\nhd : d ∈ support f\nh : ∀ (i : σ), n ≤ ↑d i\n⊢ σ → 0 = 0",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.474475\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : Fintype σ\nf : MvPolynomial σ R\nn : ℕ\nd : σ →₀ ℕ\nhd : d ∈ support f\nh : ∀ (i : σ), n ≤ ↑d i\n⊢ ∑ s : σ, ↑d s ≤ sum d fun x e => e",
"tactic": "rw [Finsupp.sum_fintype]"
},
{
"state_after": "case h\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.474475\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : Fintype σ\nf : MvPolynomial σ R\nn : ℕ\nd : σ →₀ ℕ\nhd : d ∈ support f\nh : ∀ (i : σ), n ≤ ↑d i\ni✝ : σ\n⊢ 0 = 0",
"state_before": "case h\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.474475\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : Fintype σ\nf : MvPolynomial σ R\nn : ℕ\nd : σ →₀ ℕ\nhd : d ∈ support f\nh : ∀ (i : σ), n ≤ ↑d i\n⊢ σ → 0 = 0",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.474475\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : Fintype σ\nf : MvPolynomial σ R\nn : ℕ\nd : σ →₀ ℕ\nhd : d ∈ support f\nh : ∀ (i : σ), n ≤ ↑d i\ni✝ : σ\n⊢ 0 = 0",
"tactic": "rfl"
}
] |
[
769,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
758,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.factorization_gcd
|
[
{
"state_after": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\n⊢ factorization (gcd a b) = factorization a ⊓ factorization b",
"state_before": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\n⊢ factorization (gcd a b) = factorization a ⊓ factorization b",
"tactic": "let dfac := a.factorization ⊓ b.factorization"
},
{
"state_after": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\n⊢ factorization (gcd a b) = factorization a ⊓ factorization b",
"state_before": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\n⊢ factorization (gcd a b) = factorization a ⊓ factorization b",
"tactic": "let d := dfac.prod Nat.pow"
},
{
"state_after": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\n⊢ factorization (gcd a b) = factorization a ⊓ factorization b",
"state_before": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\n⊢ factorization (gcd a b) = factorization a ⊓ factorization b",
"tactic": "have dfac_prime : ∀ p : ℕ, p ∈ dfac.support → Prime p := by\n intro p hp\n have : p ∈ a.factors ∧ p ∈ b.factors := by simpa using hp\n exact prime_of_mem_factors this.1"
},
{
"state_after": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\n⊢ factorization (gcd a b) = factorization a ⊓ factorization b",
"state_before": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\n⊢ factorization (gcd a b) = factorization a ⊓ factorization b",
"tactic": "have h1 : d.factorization = dfac := prod_pow_factorization_eq_self dfac_prime"
},
{
"state_after": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ factorization (gcd a b) = factorization a ⊓ factorization b",
"state_before": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\n⊢ factorization (gcd a b) = factorization a ⊓ factorization b",
"tactic": "have hd_pos : d ≠ 0 := (factorizationEquiv.invFun ⟨dfac, dfac_prime⟩).2.ne'"
},
{
"state_after": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ d = gcd a b",
"state_before": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ factorization (gcd a b) = factorization a ⊓ factorization b",
"tactic": "suffices d = gcd a b by rwa [← this]"
},
{
"state_after": "case hda\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ d ∣ a\n\ncase hdb\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ d ∣ b\n\ncase hd\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ ∀ (e : ℕ), e ∣ a → e ∣ b → e ∣ d",
"state_before": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ d = gcd a b",
"tactic": "apply gcd_greatest"
},
{
"state_after": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\np : ℕ\nhp : p ∈ dfac.support\n⊢ Prime p",
"state_before": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\n⊢ ∀ (p : ℕ), p ∈ dfac.support → Prime p",
"tactic": "intro p hp"
},
{
"state_after": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\np : ℕ\nhp : p ∈ dfac.support\nthis : p ∈ factors a ∧ p ∈ factors b\n⊢ Prime p",
"state_before": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\np : ℕ\nhp : p ∈ dfac.support\n⊢ Prime p",
"tactic": "have : p ∈ a.factors ∧ p ∈ b.factors := by simpa using hp"
},
{
"state_after": "no goals",
"state_before": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\np : ℕ\nhp : p ∈ dfac.support\nthis : p ∈ factors a ∧ p ∈ factors b\n⊢ Prime p",
"tactic": "exact prime_of_mem_factors this.1"
},
{
"state_after": "no goals",
"state_before": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\np : ℕ\nhp : p ∈ dfac.support\n⊢ p ∈ factors a ∧ p ∈ factors b",
"tactic": "simpa using hp"
},
{
"state_after": "no goals",
"state_before": "a b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\nthis : d = gcd a b\n⊢ factorization (gcd a b) = factorization a ⊓ factorization b",
"tactic": "rwa [← this]"
},
{
"state_after": "case hda\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ dfac ≤ factorization a",
"state_before": "case hda\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ d ∣ a",
"tactic": "rw [← factorization_le_iff_dvd hd_pos ha_pos, h1]"
},
{
"state_after": "no goals",
"state_before": "case hda\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ dfac ≤ factorization a",
"tactic": "exact inf_le_left"
},
{
"state_after": "case hdb\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ dfac ≤ factorization b",
"state_before": "case hdb\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ d ∣ b",
"tactic": "rw [← factorization_le_iff_dvd hd_pos hb_pos, h1]"
},
{
"state_after": "no goals",
"state_before": "case hdb\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ dfac ≤ factorization b",
"tactic": "exact inf_le_right"
},
{
"state_after": "case hd\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\ne : ℕ\nhea : e ∣ a\nheb : e ∣ b\n⊢ e ∣ d",
"state_before": "case hd\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\n⊢ ∀ (e : ℕ), e ∣ a → e ∣ b → e ∣ d",
"tactic": "intro e hea heb"
},
{
"state_after": "case hd.inl\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\nhea : 0 ∣ a\nheb : 0 ∣ b\n⊢ 0 ∣ d\n\ncase hd.inr\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\ne : ℕ\nhea : e ∣ a\nheb : e ∣ b\nhe_pos : e ≠ 0\n⊢ e ∣ d",
"state_before": "case hd\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\ne : ℕ\nhea : e ∣ a\nheb : e ∣ b\n⊢ e ∣ d",
"tactic": "rcases Decidable.eq_or_ne e 0 with (rfl | he_pos)"
},
{
"state_after": "case hd.inr\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\ne : ℕ\nhea : e ∣ a\nheb : e ∣ b\nhe_pos : e ≠ 0\nhea' : factorization e ≤ factorization a\n⊢ e ∣ d",
"state_before": "case hd.inr\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\ne : ℕ\nhea : e ∣ a\nheb : e ∣ b\nhe_pos : e ≠ 0\n⊢ e ∣ d",
"tactic": "have hea' := (factorization_le_iff_dvd he_pos ha_pos).mpr hea"
},
{
"state_after": "case hd.inr\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\ne : ℕ\nhea : e ∣ a\nheb : e ∣ b\nhe_pos : e ≠ 0\nhea' : factorization e ≤ factorization a\nheb' : factorization e ≤ factorization b\n⊢ e ∣ d",
"state_before": "case hd.inr\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\ne : ℕ\nhea : e ∣ a\nheb : e ∣ b\nhe_pos : e ≠ 0\nhea' : factorization e ≤ factorization a\n⊢ e ∣ d",
"tactic": "have heb' := (factorization_le_iff_dvd he_pos hb_pos).mpr heb"
},
{
"state_after": "no goals",
"state_before": "case hd.inr\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\ne : ℕ\nhea : e ∣ a\nheb : e ∣ b\nhe_pos : e ≠ 0\nhea' : factorization e ≤ factorization a\nheb' : factorization e ≤ factorization b\n⊢ e ∣ d",
"tactic": "simp [← factorization_le_iff_dvd he_pos hd_pos, h1, hea', heb']"
},
{
"state_after": "case hd.inl\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\nheb : 0 ∣ b\nhea : a = 0\n⊢ 0 ∣ d",
"state_before": "case hd.inl\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\nhea : 0 ∣ a\nheb : 0 ∣ b\n⊢ 0 ∣ d",
"tactic": "simp only [zero_dvd_iff] at hea"
},
{
"state_after": "no goals",
"state_before": "case hd.inl\na b : ℕ\nha_pos : a ≠ 0\nhb_pos : b ≠ 0\ndfac : ℕ →₀ ℕ := factorization a ⊓ factorization b\nd : ℕ := Finsupp.prod dfac Nat.pow\ndfac_prime : ∀ (p : ℕ), p ∈ dfac.support → Prime p\nh1 : factorization d = dfac\nhd_pos : d ≠ 0\nheb : 0 ∣ b\nhea : a = 0\n⊢ 0 ∣ d",
"tactic": "contradiction"
}
] |
[
689,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
667,
1
] |
Mathlib/Algebra/Module/Torsion.lean
|
Submodule.torsion_isTorsion
|
[] |
[
661,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
660,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.refl_coe
|
[] |
[
619,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
618,
1
] |
Mathlib/Analysis/Convex/Quasiconvex.lean
|
quasilinearOn_iff_monotoneOn_or_antitoneOn
|
[] |
[
250,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
WithTop.sum_lt_top_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nα : Type ?u.191723\nβ : Type ?u.191726\nM : Type u_1\nN : Type ?u.191732\nG : Type ?u.191735\nk : Type ?u.191738\nR : Type ?u.191741\ninst✝¹ : AddCommMonoid M\ninst✝ : LT M\ns : Finset ι\nf : ι → WithTop M\n⊢ ∑ i in s, f i < ⊤ ↔ ∀ (i : ι), i ∈ s → f i < ⊤",
"tactic": "simp only [WithTop.lt_top_iff_ne_top, ne_eq, sum_eq_top_iff, not_exists, not_and]"
}
] |
[
727,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
725,
1
] |
Mathlib/Topology/Order/Basic.lean
|
right_nhdsWithin_Ioo_neBot
|
[] |
[
2442,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2441,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
pow_bit1_neg
|
[
{
"state_after": "β : Type ?u.235419\nA : Type ?u.235422\nG : Type ?u.235425\nM : Type ?u.235428\nR : Type u_1\ninst✝ : StrictOrderedRing R\na : R\nha : a < 0\nn : ℕ\n⊢ a * a ^ bit0 n < 0",
"state_before": "β : Type ?u.235419\nA : Type ?u.235422\nG : Type ?u.235425\nM : Type ?u.235428\nR : Type u_1\ninst✝ : StrictOrderedRing R\na : R\nha : a < 0\nn : ℕ\n⊢ a ^ bit1 n < 0",
"tactic": "rw [bit1, pow_succ]"
},
{
"state_after": "no goals",
"state_before": "β : Type ?u.235419\nA : Type ?u.235422\nG : Type ?u.235425\nM : Type ?u.235428\nR : Type u_1\ninst✝ : StrictOrderedRing R\na : R\nha : a < 0\nn : ℕ\n⊢ a * a ^ bit0 n < 0",
"tactic": "exact mul_neg_of_neg_of_pos ha (pow_bit0_pos_of_neg ha n)"
}
] |
[
551,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
549,
1
] |
Mathlib/Analysis/Calculus/Deriv/Mul.lean
|
HasDerivWithinAt.clm_comp
|
[
{
"state_after": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nG : Type u_1\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →L[𝕜] G\nc' : F →L[𝕜] G\nd : 𝕜 → E →L[𝕜] F\nd' : E →L[𝕜] F\nu : 𝕜 → F\nu' : F\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\nthis :\n HasDerivWithinAt (fun y => comp (c y) (d y))\n (↑(comp (↑(compL 𝕜 E F G) (c x)) (smulRight 1 d') +\n comp (↑(ContinuousLinearMap.flip (compL 𝕜 E F G)) (d x)) (smulRight 1 c'))\n 1)\n s x\n⊢ HasDerivWithinAt (fun y => comp (c y) (d y)) (comp c' (d x) + comp (c x) d') s x",
"state_before": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nG : Type u_1\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →L[𝕜] G\nc' : F →L[𝕜] G\nd : 𝕜 → E →L[𝕜] F\nd' : E →L[𝕜] F\nu : 𝕜 → F\nu' : F\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\n⊢ HasDerivWithinAt (fun y => comp (c y) (d y)) (comp c' (d x) + comp (c x) d') s x",
"tactic": "have := (hc.hasFDerivWithinAt.clm_comp hd.hasFDerivWithinAt).hasDerivWithinAt"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nG : Type u_1\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →L[𝕜] G\nc' : F →L[𝕜] G\nd : 𝕜 → E →L[𝕜] F\nd' : E →L[𝕜] F\nu : 𝕜 → F\nu' : F\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\nthis :\n HasDerivWithinAt (fun y => comp (c y) (d y))\n (↑(comp (↑(compL 𝕜 E F G) (c x)) (smulRight 1 d') +\n comp (↑(ContinuousLinearMap.flip (compL 𝕜 E F G)) (d x)) (smulRight 1 c'))\n 1)\n s x\n⊢ HasDerivWithinAt (fun y => comp (c y) (d y)) (comp c' (d x) + comp (c x) d') s x",
"tactic": "rwa [add_apply, comp_apply, comp_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,\n one_smul, add_comm] at this"
}
] |
[
362,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/Algebra/Order/Sub/WithTop.lean
|
WithTop.map_sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝³ : Sub α\ninst✝² : Zero α\ninst✝¹ : Sub β\ninst✝ : Zero β\nf : α → β\nh : ∀ (x y : α), f (x - y) = f x - f y\nh₀ : f 0 = 0\nx✝ : WithTop α\n⊢ map f (x✝ - ⊤) = map f x✝ - map f ⊤",
"tactic": "simp only [h₀, sub_top, WithTop.map_zero, coe_zero, map_top]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝³ : Sub α\ninst✝² : Zero α\ninst✝¹ : Sub β\ninst✝ : Zero β\nf : α → β\nh : ∀ (x y : α), f (x - y) = f x - f y\nh₀ : f 0 = 0\nx y : α\n⊢ map f (↑x - ↑y) = map f ↑x - map f ↑y",
"tactic": "simp only [← coe_sub, map_coe, h]"
}
] |
[
61,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/GroupTheory/Perm/Basic.lean
|
Equiv.Perm.subtypePerm_inv
|
[] |
[
391,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
RingHom.mul_def
|
[] |
[
734,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
733,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.exists_le_of_tendsto_atBot
|
[] |
[
523,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
Std/Data/RBMap/WF.lean
|
Std.RBNode.Balanced.balLeft
|
[
{
"state_after": "α✝ : Type u_1\nl : RBNode α✝\nv : α✝\nr : RBNode α✝\ncr : RBColor\nn : Nat\nhl : RedRed True l n\nhr : Balanced r cr (n + 1)\n⊢ RedRed (cr = red)\n (match l with\n | node red a x b => node red (node black a x b) v r\n | l =>\n match r with\n | node black a y b => balance2 l v (node red a y b)\n | node red (node black a y b) z c => node red (node black l v a) y (balance2 b z (setRed c))\n | r => node red l v r)\n (n + 1)",
"state_before": "α✝ : Type u_1\nl : RBNode α✝\nv : α✝\nr : RBNode α✝\ncr : RBColor\nn : Nat\nhl : RedRed True l n\nhr : Balanced r cr (n + 1)\n⊢ RedRed (cr = red) (balLeft l v r) (n + 1)",
"tactic": "unfold balLeft"
},
{
"state_after": "case h_1\nα✝ : Type u_1\nv : α✝\nr : RBNode α✝\ncr : RBColor\nn : Nat\nhr : Balanced r cr (n + 1)\nl✝ a✝ : RBNode α✝\nx✝ : α✝\nb✝ : RBNode α✝\nhl : RedRed True (node red a✝ x✝ b✝) n\n⊢ RedRed (cr = red) (node red (node black a✝ x✝ b✝) v r) (n + 1)\n\ncase h_2\nα✝ : Type u_1\nl : RBNode α✝\nv : α✝\nr : RBNode α✝\ncr : RBColor\nn : Nat\nhl : RedRed True l n\nhr : Balanced r cr (n + 1)\nl✝ : RBNode α✝\nx✝ : ∀ (a : RBNode α✝) (x : α✝) (b : RBNode α✝), l = node red a x b → False\n⊢ RedRed (cr = red)\n (match r with\n | node black a y b => balance2 l v (node red a y b)\n | node red (node black a y b) z c => node red (node black l v a) y (balance2 b z (setRed c))\n | r => node red l v r)\n (n + 1)",
"state_before": "α✝ : Type u_1\nl : RBNode α✝\nv : α✝\nr : RBNode α✝\ncr : RBColor\nn : Nat\nhl : RedRed True l n\nhr : Balanced r cr (n + 1)\n⊢ RedRed (cr = red)\n (match l with\n | node red a x b => node red (node black a x b) v r\n | l =>\n match r with\n | node black a y b => balance2 l v (node red a y b)\n | node red (node black a y b) z c => node red (node black l v a) y (balance2 b z (setRed c))\n | r => node red l v r)\n (n + 1)",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case h_1\nα✝ : Type u_1\nv : α✝\nr : RBNode α✝\ncr : RBColor\nn : Nat\nhr : Balanced r cr (n + 1)\nl✝ a✝ : RBNode α✝\nx✝ : α✝\nb✝ : RBNode α✝\nhl : RedRed True (node red a✝ x✝ b✝) n\n⊢ RedRed (cr = red) (node red (node black a✝ x✝ b✝) v r) (n + 1)",
"tactic": "next a x b => exact\nlet ⟨ca, cb, ha, hb⟩ := hl.of_red\nmatch cr with\n| red => .redred rfl (.black ha hb) hr\n| black => .balanced (.red (.black ha hb) hr)"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\nv : α✝\nr : RBNode α✝\ncr : RBColor\nn : Nat\nhr : Balanced r cr (n + 1)\nl✝ a : RBNode α✝\nx : α✝\nb : RBNode α✝\nhl : RedRed True (node red a x b) n\n⊢ RedRed (cr = red) (node red (node black a x b) v r) (n + 1)",
"tactic": "exact\nlet ⟨ca, cb, ha, hb⟩ := hl.of_red\nmatch cr with\n| red => .redred rfl (.black ha hb) hr\n| black => .balanced (.red (.black ha hb) hr)"
},
{
"state_after": "no goals",
"state_before": "case h_2\nα✝ : Type u_1\nl : RBNode α✝\nv : α✝\nr : RBNode α✝\ncr : RBColor\nn : Nat\nhl : RedRed True l n\nhr : Balanced r cr (n + 1)\nl✝ : RBNode α✝\nx✝ : ∀ (a : RBNode α✝) (x : α✝) (b : RBNode α✝), l = node red a x b → False\n⊢ RedRed (cr = red)\n (match r with\n | node black a y b => balance2 l v (node red a y b)\n | node red (node black a y b) z c => node red (node black l v a) y (balance2 b z (setRed c))\n | r => node red l v r)\n (n + 1)",
"tactic": "next H => exact match hl with\n| .redred .. => nomatch H _ _ _ rfl\n| .balanced hl => match hr with\n | .black ha hb =>\n let ⟨c, h⟩ := RedRed.balance2 hl (.redred trivial ha hb); .balanced h\n | .red (.black ha hb) (.black hc hd) =>\n let ⟨c, h⟩ := RedRed.balance2 hb (.redred trivial hc hd); .redred rfl (.black hl ha) h"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\nl : RBNode α✝\nv : α✝\nr : RBNode α✝\ncr : RBColor\nn : Nat\nhl : RedRed True l n\nhr : Balanced r cr (n + 1)\nl✝ : RBNode α✝\nH : ∀ (a : RBNode α✝) (x : α✝) (b : RBNode α✝), l = node red a x b → False\n⊢ RedRed (cr = red)\n (match r with\n | node black a y b => balance2 l v (node red a y b)\n | node red (node black a y b) z c => node red (node black l v a) y (balance2 b z (setRed c))\n | r => node red l v r)\n (n + 1)",
"tactic": "exact match hl with\n| .redred .. => nomatch H _ _ _ rfl\n| .balanced hl => match hr with\n| .black ha hb =>\nlet ⟨c, h⟩ := RedRed.balance2 hl (.redred trivial ha hb); .balanced h\n| .red (.black ha hb) (.black hc hd) =>\nlet ⟨c, h⟩ := RedRed.balance2 hb (.redred trivial hc hd); .redred rfl (.black hl ha) h"
}
] |
[
267,
95
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
253,
11
] |
Mathlib/Order/Bounds/Basic.lean
|
BddBelow.insert
|
[] |
[
943,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
941,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
norm_iteratedFDerivWithin_smul_le
|
[] |
[
2489,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2482,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
HasDerivWithinAt.hasDerivAt
|
[] |
[
424,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
422,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieModuleEquiv.toEquiv_mk
|
[] |
[
1009,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1007,
1
] |
Mathlib/NumberTheory/LucasLehmer.lean
|
LucasLehmer.X.ωb_mul_ω
|
[
{
"state_after": "no goals",
"state_before": "q✝ q : ℕ+\n⊢ ωb * ω = 1",
"tactic": "rw [mul_comm, ω_mul_ωb]"
}
] |
[
375,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.multiplicity_eq_factorization
|
[
{
"state_after": "no goals",
"state_before": "n p : ℕ\npp : Prime p\nhn : n ≠ 0\n⊢ multiplicity p n = ↑(↑(factorization n) p)",
"tactic": "simp [factorization, pp, padicValNat_def' pp.ne_one hn.bot_lt]"
}
] |
[
99,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_eq_iff_natDegree_eq
|
[
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhp : p ≠ 0\n⊢ ↑(natDegree p) = ↑n ↔ natDegree p = n",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhp : p ≠ 0\n⊢ degree p = ↑n ↔ natDegree p = n",
"tactic": "rw [degree_eq_natDegree hp]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhp : p ≠ 0\n⊢ ↑(natDegree p) = ↑n ↔ natDegree p = n",
"tactic": "exact WithBot.coe_eq_coe"
}
] |
[
136,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
|
HasStrictFDerivAt.exp
|
[] |
[
289,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean
|
PiLp.norm_sq_eq_of_L2
|
[
{
"state_after": "p : ℝ≥0∞\n𝕜 : Type ?u.285404\n𝕜' : Type ?u.285407\nι : Type u_2\nα : ι → Type ?u.285415\nβ✝ : ι → Type ?u.285420\ninst✝² : Fintype ι\ninst✝¹ : Fact (1 ≤ p)\nβ : ι → Type u_1\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp 2 β\n⊢ ‖x‖₊ ^ 2 = ∑ i : ι, ‖x i‖₊ ^ 2",
"state_before": "p : ℝ≥0∞\n𝕜 : Type ?u.285404\n𝕜' : Type ?u.285407\nι : Type u_2\nα : ι → Type ?u.285415\nβ✝ : ι → Type ?u.285420\ninst✝² : Fintype ι\ninst✝¹ : Fact (1 ≤ p)\nβ : ι → Type u_1\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp 2 β\n⊢ ‖x‖ ^ 2 = ∑ i : ι, ‖x i‖ ^ 2",
"tactic": "suffices ‖x‖₊ ^ 2 = ∑ i : ι, ‖x i‖₊ ^ 2 by\n simpa only [NNReal.coe_sum] using congr_arg ((↑) : ℝ≥0 → ℝ) this"
},
{
"state_after": "no goals",
"state_before": "p : ℝ≥0∞\n𝕜 : Type ?u.285404\n𝕜' : Type ?u.285407\nι : Type u_2\nα : ι → Type ?u.285415\nβ✝ : ι → Type ?u.285420\ninst✝² : Fintype ι\ninst✝¹ : Fact (1 ≤ p)\nβ : ι → Type u_1\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp 2 β\n⊢ ‖x‖₊ ^ 2 = ∑ i : ι, ‖x i‖₊ ^ 2",
"tactic": "rw [nnnorm_eq_of_L2, NNReal.sq_sqrt]"
},
{
"state_after": "no goals",
"state_before": "p : ℝ≥0∞\n𝕜 : Type ?u.285404\n𝕜' : Type ?u.285407\nι : Type u_2\nα : ι → Type ?u.285415\nβ✝ : ι → Type ?u.285420\ninst✝² : Fintype ι\ninst✝¹ : Fact (1 ≤ p)\nβ : ι → Type u_1\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp 2 β\nthis : ‖x‖₊ ^ 2 = ∑ i : ι, ‖x i‖₊ ^ 2\n⊢ ‖x‖ ^ 2 = ∑ i : ι, ‖x i‖ ^ 2",
"tactic": "simpa only [NNReal.coe_sum] using congr_arg ((↑) : ℝ≥0 → ℝ) this"
}
] |
[
600,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
596,
1
] |
Mathlib/Init/Algebra/Classes.lean
|
lt_of_incomp_of_lt
|
[] |
[
406,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
VectorBundleCore.mem_localTriv_target
|
[] |
[
713,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
711,
1
] |
Mathlib/Analysis/NormedSpace/Star/Basic.lean
|
starₗᵢ_apply
|
[] |
[
308,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/Analysis/Normed/Group/AddTorsor.lean
|
dist_vsub_vsub_le
|
[
{
"state_after": "α : Type ?u.24677\nV : Type u_1\nP : Type u_2\nW : Type ?u.24686\nQ : Type ?u.24689\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : NormedAddCommGroup W\ninst✝¹ : MetricSpace Q\ninst✝ : NormedAddTorsor W Q\np₁ p₂ p₃ p₄ : P\n⊢ ‖p₁ -ᵥ p₃ - (p₂ -ᵥ p₄)‖ ≤ ‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖",
"state_before": "α : Type ?u.24677\nV : Type u_1\nP : Type u_2\nW : Type ?u.24686\nQ : Type ?u.24689\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : NormedAddCommGroup W\ninst✝¹ : MetricSpace Q\ninst✝ : NormedAddTorsor W Q\np₁ p₂ p₃ p₄ : P\n⊢ dist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ dist p₁ p₃ + dist p₂ p₄",
"tactic": "rw [dist_eq_norm, vsub_sub_vsub_comm, dist_eq_norm_vsub V, dist_eq_norm_vsub V]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.24677\nV : Type u_1\nP : Type u_2\nW : Type ?u.24686\nQ : Type ?u.24689\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : NormedAddCommGroup W\ninst✝¹ : MetricSpace Q\ninst✝ : NormedAddTorsor W Q\np₁ p₂ p₃ p₄ : P\n⊢ ‖p₁ -ᵥ p₃ - (p₂ -ᵥ p₄)‖ ≤ ‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖",
"tactic": "exact norm_sub_le _ _"
}
] |
[
186,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/ModelTheory/Skolem.lean
|
FirstOrder.Language.Substructure.skolem₁_reduct_isElementary
|
[
{
"state_after": "L : Language\nM : Type w\ninst✝¹ : Nonempty M\ninst✝ : Structure L M\nS : Substructure (Language.sum L (skolem₁ L)) M\n⊢ ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → { x // x ∈ ↑(LHom.substructureReduct LHom.sumInl) S })\n (a : M),\n BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) ↑b)",
"state_before": "L : Language\nM : Type w\ninst✝¹ : Nonempty M\ninst✝ : Structure L M\nS : Substructure (Language.sum L (skolem₁ L)) M\n⊢ IsElementary (↑(LHom.substructureReduct LHom.sumInl) S)",
"tactic": "apply (LHom.sumInl.substructureReduct S).isElementary_of_exists"
},
{
"state_after": "L : Language\nM : Type w\ninst✝¹ : Nonempty M\ninst✝ : Structure L M\nS : Substructure (Language.sum L (skolem₁ L)) M\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nx : Fin n → { x // x ∈ ↑(LHom.substructureReduct LHom.sumInl) S }\na : M\nh : BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a)\n⊢ ∃ b, BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) ↑b)",
"state_before": "L : Language\nM : Type w\ninst✝¹ : Nonempty M\ninst✝ : Structure L M\nS : Substructure (Language.sum L (skolem₁ L)) M\n⊢ ∀ (n : ℕ) (φ : BoundedFormula L Empty (n + 1)) (x : Fin n → { x // x ∈ ↑(LHom.substructureReduct LHom.sumInl) S })\n (a : M),\n BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a) →\n ∃ b, BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) ↑b)",
"tactic": "intro n φ x a h"
},
{
"state_after": "L : Language\nM : Type w\ninst✝¹ : Nonempty M\ninst✝ : Structure L M\nS : Substructure (Language.sum L (skolem₁ L)) M\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nx : Fin n → { x // x ∈ ↑(LHom.substructureReduct LHom.sumInl) S }\na : M\nh : BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a)\nφ' : Functions (Language.sum L (skolem₁ L)) n := LHom.onFunction LHom.sumInr φ\n⊢ ∃ b, BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) ↑b)",
"state_before": "L : Language\nM : Type w\ninst✝¹ : Nonempty M\ninst✝ : Structure L M\nS : Substructure (Language.sum L (skolem₁ L)) M\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nx : Fin n → { x // x ∈ ↑(LHom.substructureReduct LHom.sumInl) S }\na : M\nh : BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a)\n⊢ ∃ b, BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) ↑b)",
"tactic": "let φ' : (L.sum L.skolem₁).Functions n := LHom.sumInr.onFunction φ"
},
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type w\ninst✝¹ : Nonempty M\ninst✝ : Structure L M\nS : Substructure (Language.sum L (skolem₁ L)) M\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nx : Fin n → { x // x ∈ ↑(LHom.substructureReduct LHom.sumInl) S }\na : M\nh : BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a)\nφ' : Functions (Language.sum L (skolem₁ L)) n := LHom.onFunction LHom.sumInr φ\n⊢ ∃ b, BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) ↑b)",
"tactic": "exact\n ⟨⟨funMap φ' ((↑) ∘ x), S.fun_mem (LHom.sumInr.onFunction φ) ((↑) ∘ x) (by\n exact fun i => (x i).2)⟩,\n by exact Classical.epsilon_spec (p := fun a => BoundedFormula.Realize φ default\n (Fin.snoc (Subtype.val ∘ x) a)) ⟨a, h⟩⟩"
},
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type w\ninst✝¹ : Nonempty M\ninst✝ : Structure L M\nS : Substructure (Language.sum L (skolem₁ L)) M\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nx : Fin n → { x // x ∈ ↑(LHom.substructureReduct LHom.sumInl) S }\na : M\nh : BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a)\nφ' : Functions (Language.sum L (skolem₁ L)) n := LHom.onFunction LHom.sumInr φ\n⊢ ∀ (i : Fin n), (Subtype.val ∘ x) i ∈ ↑S",
"tactic": "exact fun i => (x i).2"
},
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type w\ninst✝¹ : Nonempty M\ninst✝ : Structure L M\nS : Substructure (Language.sum L (skolem₁ L)) M\nn : ℕ\nφ : BoundedFormula L Empty (n + 1)\nx : Fin n → { x // x ∈ ↑(LHom.substructureReduct LHom.sumInl) S }\na : M\nh : BoundedFormula.Realize φ default (Fin.snoc (Subtype.val ∘ x) a)\nφ' : Functions (Language.sum L (skolem₁ L)) n := LHom.onFunction LHom.sumInr φ\n⊢ BoundedFormula.Realize φ default\n (Fin.snoc (Subtype.val ∘ x)\n ↑{ val := funMap φ' (Subtype.val ∘ x),\n property := (_ : funMap (LHom.onFunction LHom.sumInr φ) (Subtype.val ∘ x) ∈ ↑S) })",
"tactic": "exact Classical.epsilon_spec (p := fun a => BoundedFormula.Realize φ default\n (Fin.snoc (Subtype.val ∘ x) a)) ⟨a, h⟩"
}
] |
[
98,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.ne_of_mem_sphere
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.32596\nι : Type ?u.32599\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\nh : y ∈ sphere x ε\nhε : ε ≠ 0\n⊢ ¬x ∈ sphere x ε",
"tactic": "simpa using hε.symm"
}
] |
[
498,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/CategoryTheory/Equivalence.lean
|
CategoryTheory.Equivalence.unit_inverse_comp
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app (e.inverse.obj Y) ≫\n (e.inverse.map ((counitInv e).app (e.functor.obj (e.inverse.obj Y))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app (e.inverse.obj Y) ≫ e.inverse.map ((counit e).app Y) = 𝟙 (e.inverse.obj Y)",
"tactic": "rw [← id_comp (e.inverse.map _), ← map_id e.inverse, ← counitInv_functor_comp, map_comp]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app (e.inverse.obj Y) ≫\n (e.inverse.map ((counitInv e).app (e.functor.obj (e.inverse.obj Y))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app (e.inverse.obj Y) ≫\n (e.inverse.map ((counitInv e).app (e.functor.obj (e.inverse.obj Y))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "dsimp"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app (e.inverse.obj Y) ≫\n (e.inverse.map ((counitInv e).app (e.functor.obj (e.inverse.obj Y))) ≫\n e.unitIso.hom.app\n (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y))))) ≫\n e.unitIso.inv.app\n (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app (e.inverse.obj Y) ≫\n (e.inverse.map ((counitInv e).app (e.functor.obj (e.inverse.obj Y))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "rw [← Iso.hom_inv_id_assoc (e.unitIso.app _) (e.inverse.map (e.functor.map _)), app_hom, app_inv]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app (e.inverse.obj Y) ≫\n ((((unit e).app (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj Y))) ≫\n (e.functor ⋙ e.inverse).map\n (e.inverse.toPrefunctor.2 ((counitInv e).app (e.functor.obj (e.inverse.obj Y))))) ≫\n e.unitIso.inv.app\n (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app (e.inverse.obj Y) ≫\n (e.inverse.map ((counitInv e).app (e.functor.obj (e.inverse.obj Y))) ≫\n e.unitIso.hom.app\n (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y))))) ≫\n e.unitIso.inv.app\n (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "slice_lhs 2 3 => erw [e.unit.naturality]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (((((unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫ (e.functor ⋙ e.inverse).map ((unit e).1 (e.inverse.obj Y))) ≫\n (e.functor ⋙ e.inverse).map\n (e.inverse.toPrefunctor.2 ((counitInv e).app (e.functor.obj (e.inverse.obj Y))))) ≫\n e.unitIso.inv.app\n (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app (e.inverse.obj Y) ≫\n ((((unit e).app (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj Y))) ≫\n (e.functor ⋙ e.inverse).map\n (e.inverse.toPrefunctor.2 ((counitInv e).app (e.functor.obj (e.inverse.obj Y))))) ≫\n e.unitIso.inv.app\n (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "slice_lhs 1 2 => erw [e.unit.naturality]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n (e.functor ⋙ e.inverse).map ((unit e).1 (e.inverse.obj Y)) ≫\n (e.functor ⋙ e.inverse).map (e.inverse.toPrefunctor.2 ((counitInv e).app (e.functor.obj (e.inverse.obj Y)))) ≫\n (((e.inverse.mapIso (e.counitIso.app (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))).hom ≫\n (e.inverse.mapIso\n (e.counitIso.app (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))).inv ≫\n (unitInv e).app (e.inverse.obj (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (((((unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫ (e.functor ⋙ e.inverse).map ((unit e).1 (e.inverse.obj Y))) ≫\n (e.functor ⋙ e.inverse).map\n (e.inverse.toPrefunctor.2 ((counitInv e).app (e.functor.obj (e.inverse.obj Y))))) ≫\n e.unitIso.inv.app\n (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "slice_lhs 4 4 =>\n rw [← Iso.hom_inv_id_assoc (e.inverse.mapIso (e.counitIso.app _)) (e.unitInv.app _)]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n (e.functor ⋙ e.inverse).map ((unit e).1 (e.inverse.obj Y)) ≫\n (((𝟙 (e.inverse.obj (e.functor.obj (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj Y))))) ≫\n (e.inverse.mapIso\n (e.counitIso.app (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))).inv) ≫\n (unitInv e).app (e.inverse.obj (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n (e.functor ⋙ e.inverse).map ((unit e).1 (e.inverse.obj Y)) ≫\n (e.functor ⋙ e.inverse).map (e.inverse.toPrefunctor.2 ((counitInv e).app (e.functor.obj (e.inverse.obj Y)))) ≫\n (((e.inverse.mapIso (e.counitIso.app (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))).hom ≫\n (e.inverse.mapIso\n (e.counitIso.app (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))).inv ≫\n (unitInv e).app (e.inverse.obj (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "slice_lhs 3 4 =>\n erw [← map_comp e.inverse, e.counit.naturality]\n erw [(e.counitIso.app _).hom_inv_id, map_id]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n (e.functor ⋙ e.inverse).map ((unit e).1 (e.inverse.obj Y)) ≫\n (((e.inverse.mapIso (e.counitIso.app (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))).inv ≫\n (unitInv e).app (e.inverse.obj (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n (e.functor ⋙ e.inverse).map ((unit e).1 (e.inverse.obj Y)) ≫\n (((𝟙 (e.inverse.obj (e.functor.obj (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj Y))))) ≫\n (e.inverse.mapIso\n (e.counitIso.app (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))).inv) ≫\n (unitInv e).app (e.inverse.obj (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "erw [id_comp]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n (((e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n e.inverse.map ((e.inverse ⋙ e.functor).map (e.functor.toPrefunctor.2 ((unit e).1 (e.inverse.obj Y))))) ≫\n (unitInv e).app (e.inverse.obj (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n (e.functor ⋙ e.inverse).map ((unit e).1 (e.inverse.obj Y)) ≫\n (((e.inverse.mapIso (e.counitIso.app (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))).inv ≫\n (unitInv e).app (e.inverse.obj (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "slice_lhs 2 3 => erw [← map_comp e.inverse, e.counitIso.inv.naturality, map_comp]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n (((unitInv e).app (e.inverse.obj (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n (𝟭 C).map (e.inverse.map (e.functor.toPrefunctor.2 ((unit e).1 (e.inverse.obj Y))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n (((e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n e.inverse.map ((e.inverse ⋙ e.functor).map (e.functor.toPrefunctor.2 ((unit e).1 (e.inverse.obj Y))))) ≫\n (unitInv e).app (e.inverse.obj (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "slice_lhs 3 4 => erw [e.unitInv.naturality]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n (unitInv e).app (e.inverse.obj (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n 𝟙 ((e.functor ⋙ e.inverse).obj ((𝟭 C).obj (e.inverse.obj Y))) ≫ e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n (((unitInv e).app (e.inverse.obj (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n (𝟭 C).map (e.inverse.map (e.functor.toPrefunctor.2 ((unit e).1 (e.inverse.obj Y))))) ≫\n e.inverse.map (e.functor.map ((unitInv e).app (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "slice_lhs 4 5 => erw [← map_comp (e.functor ⋙ e.inverse), (e.unitIso.app _).hom_inv_id, map_id]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n (unitInv e).app (e.inverse.obj (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n (unitInv e).app (e.inverse.obj (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n 𝟙 ((e.functor ⋙ e.inverse).obj ((𝟭 C).obj (e.inverse.obj Y))) ≫ e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "erw [id_comp]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n (e.functor ⋙ e.inverse).map (e.inverse.toPrefunctor.2 ((counit e).app Y)) ≫\n (unitInv e).app (e.inverse.toPrefunctor.1 Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n (unitInv e).app (e.inverse.obj (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n e.inverse.map ((counit e).app Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "slice_lhs 3 4 => erw [← e.unitInv.naturality]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n 𝟙 (e.inverse.obj (e.functor.obj ((𝟭 C).obj (e.inverse.obj Y)))) ≫ (unitInv e).app (e.inverse.toPrefunctor.1 Y) =\n 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫\n (e.functor ⋙ e.inverse).map (e.inverse.toPrefunctor.2 ((counit e).app Y)) ≫\n (unitInv e).app (e.inverse.toPrefunctor.1 Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "slice_lhs 2 3 =>\n erw [← map_comp e.inverse, ← e.counitIso.inv.naturality, (e.counitIso.app _).hom_inv_id,\n map_id]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ 𝟙 ((𝟭 C).obj ((𝟭 C).obj (e.inverse.obj Y))) = 𝟙 (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unit e).app ((𝟭 C).obj (e.inverse.obj Y)) ≫\n 𝟙 (e.inverse.obj (e.functor.obj ((𝟭 C).obj (e.inverse.obj Y)))) ≫ (unitInv e).app (e.inverse.toPrefunctor.1 Y) =\n 𝟙 (e.inverse.obj Y)",
"tactic": "erw [id_comp, (e.unitIso.app _).hom_inv_id]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ 𝟙 ((𝟭 C).obj ((𝟭 C).obj (e.inverse.obj Y))) = 𝟙 (e.inverse.obj Y)",
"tactic": "rfl"
}
] |
[
204,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Order/Filter/Partial.lean
|
Filter.mem_pmap
|
[] |
[
219,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean
|
MulChar.IsNontrivial.sum_eq_zero
|
[
{
"state_after": "case intro\nR : Type u\ninst✝⁴ : CommRing R\nR' : Type v\ninst✝³ : CommRing R'\nR'' : Type w\ninst✝² : CommRing R''\ninst✝¹ : Fintype R\ninst✝ : IsDomain R'\nχ : MulChar R R'\nb : Rˣ\nhb : ↑χ ↑b ≠ 1\n⊢ ∑ a : R, ↑χ a = 0",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\nR' : Type v\ninst✝³ : CommRing R'\nR'' : Type w\ninst✝² : CommRing R''\ninst✝¹ : Fintype R\ninst✝ : IsDomain R'\nχ : MulChar R R'\nhχ : IsNontrivial χ\n⊢ ∑ a : R, ↑χ a = 0",
"tactic": "rcases hχ with ⟨b, hb⟩"
},
{
"state_after": "case intro\nR : Type u\ninst✝⁴ : CommRing R\nR' : Type v\ninst✝³ : CommRing R'\nR'' : Type w\ninst✝² : CommRing R''\ninst✝¹ : Fintype R\ninst✝ : IsDomain R'\nχ : MulChar R R'\nb : Rˣ\nhb : ↑χ ↑b ≠ 1\n⊢ ↑χ ↑b * ∑ a : R, ↑χ a = ∑ a : R, ↑χ a",
"state_before": "case intro\nR : Type u\ninst✝⁴ : CommRing R\nR' : Type v\ninst✝³ : CommRing R'\nR'' : Type w\ninst✝² : CommRing R''\ninst✝¹ : Fintype R\ninst✝ : IsDomain R'\nχ : MulChar R R'\nb : Rˣ\nhb : ↑χ ↑b ≠ 1\n⊢ ∑ a : R, ↑χ a = 0",
"tactic": "refine' eq_zero_of_mul_eq_self_left hb _"
},
{
"state_after": "case intro\nR : Type u\ninst✝⁴ : CommRing R\nR' : Type v\ninst✝³ : CommRing R'\nR'' : Type w\ninst✝² : CommRing R''\ninst✝¹ : Fintype R\ninst✝ : IsDomain R'\nχ : MulChar R R'\nb : Rˣ\nhb : ↑χ ↑b ≠ 1\n⊢ ∑ x : R, ↑χ ↑b * ↑χ x = ∑ a : R, ↑χ a",
"state_before": "case intro\nR : Type u\ninst✝⁴ : CommRing R\nR' : Type v\ninst✝³ : CommRing R'\nR'' : Type w\ninst✝² : CommRing R''\ninst✝¹ : Fintype R\ninst✝ : IsDomain R'\nχ : MulChar R R'\nb : Rˣ\nhb : ↑χ ↑b ≠ 1\n⊢ ↑χ ↑b * ∑ a : R, ↑χ a = ∑ a : R, ↑χ a",
"tactic": "simp only [Finset.mul_sum, ← map_mul]"
},
{
"state_after": "case intro\nR : Type u\ninst✝⁴ : CommRing R\nR' : Type v\ninst✝³ : CommRing R'\nR'' : Type w\ninst✝² : CommRing R''\ninst✝¹ : Fintype R\ninst✝ : IsDomain R'\nχ : MulChar R R'\nb : Rˣ\nhb : ↑χ ↑b ≠ 1\nx : R\n⊢ ↑χ ↑b * ↑χ x = ↑χ (↑b * x)",
"state_before": "case intro\nR : Type u\ninst✝⁴ : CommRing R\nR' : Type v\ninst✝³ : CommRing R'\nR'' : Type w\ninst✝² : CommRing R''\ninst✝¹ : Fintype R\ninst✝ : IsDomain R'\nχ : MulChar R R'\nb : Rˣ\nhb : ↑χ ↑b ≠ 1\n⊢ ∑ x : R, ↑χ ↑b * ↑χ x = ∑ a : R, ↑χ a",
"tactic": "refine Fintype.sum_bijective _ (Units.mulLeft_bijective b) _ _ fun x => ?_"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\ninst✝⁴ : CommRing R\nR' : Type v\ninst✝³ : CommRing R'\nR'' : Type w\ninst✝² : CommRing R''\ninst✝¹ : Fintype R\ninst✝ : IsDomain R'\nχ : MulChar R R'\nb : Rˣ\nhb : ↑χ ↑b ≠ 1\nx : R\n⊢ ↑χ ↑b * ↑χ x = ↑χ (↑b * x)",
"tactic": "exact (map_mul χ (b : R) x).symm"
}
] |
[
545,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
538,
1
] |
Mathlib/Data/List/Infix.lean
|
List.insert_nil
|
[] |
[
471,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
470,
1
] |
Mathlib/Order/OrdContinuous.lean
|
RightOrdContinuous.le_iff
|
[] |
[
209,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Topology/GDelta.lean
|
Finset.isGδ_compl
|
[] |
[
158,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/Algebra/Parity.lean
|
isSquare_op_iff
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.311\nα : Type u_1\nβ : Type ?u.317\nR : Type ?u.320\ninst✝ : Mul α\na : α\nx✝ : IsSquare (op a)\nc : αᵐᵒᵖ\nhc : op a = c * c\n⊢ a = unop c * unop c",
"tactic": "rw [← unop_mul, ← hc, unop_op]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.311\nα : Type u_1\nβ : Type ?u.317\nR : Type ?u.320\ninst✝ : Mul α\na : α\nx✝ : IsSquare a\nc : α\nhc : a = c * c\n⊢ IsSquare (op a)",
"tactic": "simp [hc]"
}
] |
[
65,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/GroupTheory/Subgroup/ZPowers.lean
|
Subgroup.zpowers_eq_bot
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝² : Group G\nA : Type ?u.38880\ninst✝¹ : AddGroup A\nN : Type ?u.38886\ninst✝ : Group N\ns : Set G\ng✝ g : G\n⊢ zpowers g = ⊥ ↔ g = 1",
"tactic": "rw [eq_bot_iff, zpowers_le, mem_bot]"
}
] |
[
213,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Topology/DiscreteQuotient.lean
|
DiscreteQuotient.ofLE_map
|
[
{
"state_after": "case mk\nα : Type ?u.32610\nX : Type u_1\nY : Type u_2\nZ : Type ?u.32619\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\ncond : LEComap f A B\nh : B ≤ B'\na : Quotient A.toSetoid\na✝ : X\n⊢ ofLE h (map f cond (Quot.mk Setoid.r a✝)) = map f (_ : LEComap f A B') (Quot.mk Setoid.r a✝)",
"state_before": "α : Type ?u.32610\nX : Type u_1\nY : Type u_2\nZ : Type ?u.32619\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\ncond : LEComap f A B\nh : B ≤ B'\na : Quotient A.toSetoid\n⊢ ofLE h (map f cond a) = map f (_ : LEComap f A B') a",
"tactic": "rcases a with ⟨⟩"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Type ?u.32610\nX : Type u_1\nY : Type u_2\nZ : Type ?u.32619\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\ncond : LEComap f A B\nh : B ≤ B'\na : Quotient A.toSetoid\na✝ : X\n⊢ ofLE h (map f cond (Quot.mk Setoid.r a✝)) = map f (_ : LEComap f A B') (Quot.mk Setoid.r a✝)",
"tactic": "rfl"
}
] |
[
343,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Algebra/Order/SMul.lean
|
lowerBounds_smul_of_pos
|
[] |
[
281,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.comap_equiv_symm
|
[] |
[
2542,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2541,
1
] |
Mathlib/CategoryTheory/Sites/Canonical.lean
|
CategoryTheory.Sheaf.isSheafFor_trans
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\n⊢ Presieve.IsSheafFor P S.arrows",
"state_before": "C : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\n⊢ Presieve.IsSheafFor P S.arrows",
"tactic": "have : (bind R fun Y f _ => S.pullback f : Presieve X) ≤ S := by\n rintro Z f ⟨W, f, g, hg, hf : S _, rfl⟩\n apply hf"
},
{
"state_after": "case hS\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\n⊢ Presieve.IsSheafFor P (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows\n\ncase trans\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,\n S.arrows f →\n Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S)).arrows",
"state_before": "C : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\n⊢ Presieve.IsSheafFor P S.arrows",
"tactic": "apply Presieve.isSheafFor_subsieve_aux P this"
},
{
"state_after": "case hS\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,\n R.arrows f → ∀ ⦃Z : C⦄ (g : Z ⟶ Y), Presieve.IsSeparatedFor P (Sieve.pullback g (Sieve.pullback f S)).arrows\n\ncase trans\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,\n S.arrows f →\n Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S)).arrows",
"state_before": "case hS\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\n⊢ Presieve.IsSheafFor P (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows\n\ncase trans\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,\n S.arrows f →\n Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S)).arrows",
"tactic": "apply isSheafFor_bind _ _ _ hR hS"
},
{
"state_after": "case intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nZ W : C\nf : Z ⟶ W\ng : W ⟶ X\nhg : R.arrows g\nhf : S.arrows (f ≫ g)\n⊢ f ≫ g ∈ S.arrows",
"state_before": "C : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\n⊢ (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows",
"tactic": "rintro Z f ⟨W, f, g, hg, hf : S _, rfl⟩"
},
{
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"state_before": "case intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nZ W : C\nf : Z ⟶ W\ng : W ⟶ X\nhg : R.arrows g\nhf : S.arrows (f ≫ g)\n⊢ f ≫ g ∈ S.arrows",
"tactic": "apply hf"
},
{
"state_after": "case hS\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : R.arrows f\nZ : C\ng : Z ⟶ Y\n⊢ Presieve.IsSeparatedFor P (Sieve.pullback g (Sieve.pullback f S)).arrows",
"state_before": "case hS\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,\n R.arrows f → ∀ ⦃Z : C⦄ (g : Z ⟶ Y), Presieve.IsSeparatedFor P (Sieve.pullback g (Sieve.pullback f S)).arrows",
"tactic": "intro Y f hf Z g"
},
{
"state_after": "case hS\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : R.arrows f\nZ : C\ng : Z ⟶ Y\n⊢ Presieve.IsSeparatedFor P (Sieve.pullback g (Sieve.pullback f S)).arrows",
"state_before": "case hS\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : R.arrows f\nZ : C\ng : Z ⟶ Y\n⊢ Presieve.IsSeparatedFor P (Sieve.pullback g (Sieve.pullback f S)).arrows",
"tactic": "dsimp"
},
{
"state_after": "case hS\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : R.arrows f\nZ : C\ng : Z ⟶ Y\n⊢ Presieve.IsSeparatedFor P (Sieve.pullback (g ≫ f) S).arrows",
"state_before": "case hS\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : R.arrows f\nZ : C\ng : Z ⟶ Y\n⊢ Presieve.IsSeparatedFor P (Sieve.pullback g (Sieve.pullback f S)).arrows",
"tactic": "rw [← pullback_comp]"
},
{
"state_after": "no goals",
"state_before": "case hS\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : R.arrows f\nZ : C\ng : Z ⟶ Y\n⊢ Presieve.IsSeparatedFor P (Sieve.pullback (g ≫ f) S).arrows",
"tactic": "apply (hS (R.downward_closed hf _)).isSeparatedFor"
},
{
"state_after": "case trans\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S)).arrows",
"state_before": "case trans\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,\n S.arrows f →\n Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S)).arrows",
"tactic": "intro Y f hf"
},
{
"state_after": "case trans\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis✝ : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nthis : Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S) = Sieve.pullback f R\n⊢ Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows",
"state_before": "case trans\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis✝ : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nthis : Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S) = Sieve.pullback f R\n⊢ Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S)).arrows",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "case trans\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis✝ : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nthis : Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S) = Sieve.pullback f R\n⊢ Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows",
"tactic": "apply hR' hf"
},
{
"state_after": "case h\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\n⊢ ∀ (f_1 : Z ⟶ Y),\n (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows f_1 ↔\n (Sieve.pullback f R).arrows f_1",
"state_before": "C : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S) = Sieve.pullback f R",
"tactic": "ext Z"
},
{
"state_after": "case h\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\n⊢ (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows g ↔ (Sieve.pullback f R).arrows g",
"state_before": "case h\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\n⊢ ∀ (f_1 : Z ⟶ Y),\n (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows f_1 ↔\n (Sieve.pullback f R).arrows f_1",
"tactic": "intro g"
},
{
"state_after": "case h.mp\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\n⊢ (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows g → (Sieve.pullback f R).arrows g\n\ncase h.mpr\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\n⊢ (Sieve.pullback f R).arrows g → (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows g",
"state_before": "case h\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\n⊢ (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows g ↔ (Sieve.pullback f R).arrows g",
"tactic": "constructor"
},
{
"state_after": "case h.mp.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\nW : C\nk : Z ⟶ W\nl : W ⟶ X\nhl : R.arrows l\nleft✝ : ((fun T k x => Sieve.pullback k S) W l hl).arrows k\ncomm : k ≫ l = g ≫ f\n⊢ (Sieve.pullback f R).arrows g",
"state_before": "case h.mp\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\n⊢ (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows g → (Sieve.pullback f R).arrows g",
"tactic": "rintro ⟨W, k, l, hl, _, comm⟩"
},
{
"state_after": "case h.mp.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\nW : C\nk : Z ⟶ W\nl : W ⟶ X\nhl : R.arrows l\nleft✝ : ((fun T k x => Sieve.pullback k S) W l hl).arrows k\ncomm : k ≫ l = g ≫ f\n⊢ R.arrows (k ≫ l)",
"state_before": "case h.mp.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\nW : C\nk : Z ⟶ W\nl : W ⟶ X\nhl : R.arrows l\nleft✝ : ((fun T k x => Sieve.pullback k S) W l hl).arrows k\ncomm : k ≫ l = g ≫ f\n⊢ (Sieve.pullback f R).arrows g",
"tactic": "rw [pullback_apply, ← comm]"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\nW : C\nk : Z ⟶ W\nl : W ⟶ X\nhl : R.arrows l\nleft✝ : ((fun T k x => Sieve.pullback k S) W l hl).arrows k\ncomm : k ≫ l = g ≫ f\n⊢ R.arrows (k ≫ l)",
"tactic": "simp [hl]"
},
{
"state_after": "case h.mpr\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\na : (Sieve.pullback f R).arrows g\n⊢ (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows g",
"state_before": "case h.mpr\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\n⊢ (Sieve.pullback f R).arrows g → (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows g",
"tactic": "intro a"
},
{
"state_after": "case h.mpr\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\na : (Sieve.pullback f R).arrows g\n⊢ (fun Y f h => ((fun T k x => Sieve.pullback k S) Y f h).arrows) Z (g ≫ f) a (𝟙 Z) ∧ 𝟙 Z ≫ g ≫ f = g ≫ f",
"state_before": "case h.mpr\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\na : (Sieve.pullback f R).arrows g\n⊢ (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows g",
"tactic": "refine' ⟨Z, 𝟙 Z, _, a, _⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mpr\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows ≤ S.arrows\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\ng : Z ⟶ Y\na : (Sieve.pullback f R).arrows g\n⊢ (fun Y f h => ((fun T k x => Sieve.pullback k S) Y f h).arrows) Z (g ≫ f) a (𝟙 Z) ∧ 𝟙 Z ≫ g ≫ f = g ≫ f",
"tactic": "simp [hf]"
}
] |
[
156,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
isClosedMap_div_right
|
[] |
[
1191,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1190,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
|
AffineEquiv.coe_linear
|
[] |
[
134,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.ext_fun_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type ?u.364824\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : { x // x ∈ kernel α β }\nh : κ = η\na : α\nf : β → ℝ≥0∞\nx✝ : Measurable f\n⊢ (∫⁻ (b : β), f b ∂↑κ a) = ∫⁻ (b : β), f b ∂↑η a",
"tactic": "rw [h]"
}
] |
[
205,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_eq_head_iff_eq_getLast
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.671041\ninst✝ : DecidableEq α\nl : List α\nx✝ x y : α\n⊢ ↑(formPerm (y :: l)) x = y ↔ ↑(formPerm (y :: l)) x = ↑(formPerm (y :: l)) (getLast (y :: l) (_ : y :: l ≠ []))",
"tactic": "rw [formPerm_apply_getLast]"
}
] |
[
165,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
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