file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/ENat/Basic.lean
|
ENat.coe_mul
|
[] |
[
91,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.mul_eq_map₂
|
[] |
[
565,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
564,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
MonoidHom.mker_prod_map
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_3\nN : Type u_4\nP : Type ?u.142244\ninst✝⁵ : MulOneClass M\ninst✝⁴ : MulOneClass N\ninst✝³ : MulOneClass P\nS : Submonoid M\nA : Type ?u.142265\ninst✝² : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type ?u.142289\nmc : MonoidHomClass F M N\nM' : Type u_1\nN' : Type u_2\ninst✝¹ : MulOneClass M'\ninst✝ : MulOneClass N'\nf : M →* N\ng : M' →* N'\n⊢ mker (prodMap f g) = prod (mker f) (mker g)",
"tactic": "rw [← comap_bot', ← comap_bot', ← comap_bot', ← prod_map_comap_prod', bot_prod_bot]"
}
] |
[
1227,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1225,
1
] |
Mathlib/CategoryTheory/Limits/ExactFunctor.lean
|
CategoryTheory.LeftExactFunctor.forget_obj_of
|
[] |
[
246,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.mem_inter
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.176256\nγ : Type ?u.176259\ninst✝ : DecidableEq α\ns t u : Multiset α\na b : α\nx✝ : a ∈ s ∧ a ∈ t\nh₁ : a ∈ s\nh₂ : a ∈ t\n⊢ a ∈ a ::ₘ erase s a ∩ erase t a",
"state_before": "α : Type u_1\nβ : Type ?u.176256\nγ : Type ?u.176259\ninst✝ : DecidableEq α\ns t u : Multiset α\na b : α\nx✝ : a ∈ s ∧ a ∈ t\nh₁ : a ∈ s\nh₂ : a ∈ t\n⊢ a ∈ s ∩ t",
"tactic": "rw [← cons_erase h₁, cons_inter_of_pos _ h₂]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.176256\nγ : Type ?u.176259\ninst✝ : DecidableEq α\ns t u : Multiset α\na b : α\nx✝ : a ∈ s ∧ a ∈ t\nh₁ : a ∈ s\nh₂ : a ∈ t\n⊢ a ∈ a ::ₘ erase s a ∩ erase t a",
"tactic": "apply mem_cons_self"
}
] |
[
1794,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1792,
1
] |
Mathlib/Algebra/Order/WithZero.lean
|
inv_mul_lt_of_lt_mul₀
|
[
{
"state_after": "α : Type u_1\na b c d x y z : α\ninst✝ : LinearOrderedCommGroupWithZero α\nh : x < z * y\n⊢ x * y⁻¹ < z",
"state_before": "α : Type u_1\na b c d x y z : α\ninst✝ : LinearOrderedCommGroupWithZero α\nh : x < y * z\n⊢ y⁻¹ * x < z",
"tactic": "rw [mul_comm] at *"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b c d x y z : α\ninst✝ : LinearOrderedCommGroupWithZero α\nh : x < z * y\n⊢ x * y⁻¹ < z",
"tactic": "exact mul_inv_lt_of_lt_mul₀ h"
}
] |
[
200,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.lipschitz_comp
|
[
{
"state_after": "case h\nF : Type ?u.376927\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : PseudoMetricSpace γ\nf✝ g✝ : α →ᵇ β\nx✝ : α\nC✝ : ℝ\nG : β → γ\nC : ℝ≥0\nH : LipschitzWith C G\nf g : α →ᵇ β\nx : α\n⊢ dist (↑f x) (↑g x) ≤ dist f g",
"state_before": "F : Type ?u.376927\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : PseudoMetricSpace γ\nf✝ g✝ : α →ᵇ β\nx✝ : α\nC✝ : ℝ\nG : β → γ\nC : ℝ≥0\nH : LipschitzWith C G\nf g : α →ᵇ β\nx : α\n⊢ ↑C * dist (↑f x) (↑g x) ≤ ↑C * dist f g",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type ?u.376927\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : PseudoMetricSpace γ\nf✝ g✝ : α →ᵇ β\nx✝ : α\nC✝ : ℝ\nG : β → γ\nC : ℝ≥0\nH : LipschitzWith C G\nf g : α →ᵇ β\nx : α\n⊢ dist (↑f x) (↑g x) ≤ dist f g",
"tactic": "apply dist_coe_le_dist"
}
] |
[
430,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
|
CategoryTheory.Equalizer.Sieve.w
|
[
{
"state_after": "case w\nC : Type u₁\ninst✝ : Category C\nP : Cᵒᵖ ⥤ Type (max v₁ u₁)\nX : C\nR : Presieve X\nS : Sieve X\n⊢ ∀ (j : Discrete ((Y : C) × (Z : C) × (_ : Z ⟶ Y) × { f' // S.arrows f' })),\n (forkMap P S.arrows ≫ firstMap P S) ≫ limit.π (Discrete.functor fun f => P.obj f.snd.fst.op) j =\n (forkMap P S.arrows ≫ secondMap P S) ≫ limit.π (Discrete.functor fun f => P.obj f.snd.fst.op) j",
"state_before": "C : Type u₁\ninst✝ : Category C\nP : Cᵒᵖ ⥤ Type (max v₁ u₁)\nX : C\nR : Presieve X\nS : Sieve X\n⊢ forkMap P S.arrows ≫ firstMap P S = forkMap P S.arrows ≫ secondMap P S",
"tactic": "apply limit.hom_ext"
},
{
"state_after": "case w.mk.mk.mk.mk.mk\nC : Type u₁\ninst✝ : Category C\nP : Cᵒᵖ ⥤ Type (max v₁ u₁)\nX : C\nR : Presieve X\nS : Sieve X\nY Z : C\ng : Z ⟶ Y\nf : Y ⟶ X\nhf : S.arrows f\n⊢ (forkMap P S.arrows ≫ firstMap P S) ≫\n limit.π (Discrete.functor fun f => P.obj f.snd.fst.op)\n { as := { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } } =\n (forkMap P S.arrows ≫ secondMap P S) ≫\n limit.π (Discrete.functor fun f => P.obj f.snd.fst.op)\n { as := { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } }",
"state_before": "case w\nC : Type u₁\ninst✝ : Category C\nP : Cᵒᵖ ⥤ Type (max v₁ u₁)\nX : C\nR : Presieve X\nS : Sieve X\n⊢ ∀ (j : Discrete ((Y : C) × (Z : C) × (_ : Z ⟶ Y) × { f' // S.arrows f' })),\n (forkMap P S.arrows ≫ firstMap P S) ≫ limit.π (Discrete.functor fun f => P.obj f.snd.fst.op) j =\n (forkMap P S.arrows ≫ secondMap P S) ≫ limit.π (Discrete.functor fun f => P.obj f.snd.fst.op) j",
"tactic": "rintro ⟨Y, Z, g, f, hf⟩"
},
{
"state_after": "no goals",
"state_before": "case w.mk.mk.mk.mk.mk\nC : Type u₁\ninst✝ : Category C\nP : Cᵒᵖ ⥤ Type (max v₁ u₁)\nX : C\nR : Presieve X\nS : Sieve X\nY Z : C\ng : Z ⟶ Y\nf : Y ⟶ X\nhf : S.arrows f\n⊢ (forkMap P S.arrows ≫ firstMap P S) ≫\n limit.π (Discrete.functor fun f => P.obj f.snd.fst.op)\n { as := { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } } =\n (forkMap P S.arrows ≫ secondMap P S) ≫\n limit.π (Discrete.functor fun f => P.obj f.snd.fst.op)\n { as := { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } }",
"tactic": "simp [firstMap, secondMap, forkMap]"
}
] |
[
890,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
887,
1
] |
Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean
|
SimpleGraph.ComponentCompl.nonempty
|
[] |
[
117,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
11
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.erase_zero
|
[
{
"state_after": "R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ Finsupp.erase n 0 = 0",
"state_before": "R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ (erase n 0).toFinsupp = 0.toFinsupp",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ Finsupp.erase n 0 = 0",
"tactic": "exact Finsupp.erase_zero n"
}
] |
[
1058,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1056,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.exists_finset_coe
|
[
{
"state_after": "case intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\na✝ : Fintype ↑s\n⊢ ∃ s', ↑s' = s",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nh : Set.Finite s\n⊢ ∃ s', ↑s' = s",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\na✝ : Fintype ↑s\n⊢ ∃ s', ↑s' = s",
"tactic": "exact ⟨s.toFinset, s.coe_toFinset⟩"
}
] |
[
123,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.isMultiplicative_zeta
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.536824\n⊢ ↑ζ 1 = 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.536824\n⊢ ∀ {m n : ℕ}, m ≠ 0 → n ≠ 0 → coprime m n → ↑ζ (m * n) = ↑ζ m * ↑ζ n",
"tactic": "simp (config := { contextual := true })"
}
] |
[
820,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
819,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_finset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1733020\nγ : Type ?u.1733023\nδ : Type ?u.1733026\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSingletonClass α\ns : Finset α\nf : α → ℝ≥0∞\n⊢ (∫⁻ (x : α) in ↑s, f x ∂μ) = ∑ x in s, f x * ↑↑μ {x}",
"tactic": "simp only [lintegral_countable _ s.countable_toSet, ← Finset.tsum_subtype']"
}
] |
[
1479,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1477,
1
] |
Mathlib/Computability/Reduce.lean
|
OneOneEquiv.congr_right
|
[] |
[
284,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
PseudoEpimorphism.ext
|
[] |
[
133,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Std/Data/RBMap/Alter.lean
|
Std.RBNode.modify_eq_alter
|
[
{
"state_after": "α : Type u_1\ncut : α → Ordering\nf : α → α\nt : RBNode α\n⊢ (match zoom cut t Path.root with\n | (nil, snd) => t\n | (node c a x b, path) => Path.fill path (node c a (f x) b)) =\n match zoom cut t Path.root with\n | (nil, path) =>\n match Option.map f none with\n | none => t\n | some y => Path.insertNew path y\n | (node c a x b, path) =>\n match Option.map f (some x) with\n | none => Path.del path (append a b) c\n | some y => Path.fill path (node c a y b)",
"state_before": "α : Type u_1\ncut : α → Ordering\nf : α → α\nt : RBNode α\n⊢ modify cut f t = alter cut (Option.map f) t",
"tactic": "simp [modify, alter]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ncut : α → Ordering\nf : α → α\nt : RBNode α\n⊢ (match zoom cut t Path.root with\n | (nil, snd) => t\n | (node c a x b, path) => Path.fill path (node c a (f x) b)) =\n match zoom cut t Path.root with\n | (nil, path) =>\n match Option.map f none with\n | none => t\n | some y => Path.insertNew path y\n | (node c a x b, path) =>\n match Option.map f (some x) with\n | none => Path.del path (append a b) c\n | some y => Path.fill path (node c a y b)",
"tactic": "split <;> simp [Option.map]"
}
] |
[
380,
52
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
379,
1
] |
Std/Data/List/Lemmas.lean
|
List.nil_suffix
|
[] |
[
1574,
64
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1574,
1
] |
Mathlib/MeasureTheory/Function/EssSup.lean
|
essSup_smul_measure
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : ℝ≥0∞\nhc : c ≠ 0\n⊢ limsup f (Measure.ae (c • μ)) = limsup f (Measure.ae μ)",
"state_before": "α : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : ℝ≥0∞\nhc : c ≠ 0\n⊢ essSup f (c • μ) = essSup f μ",
"tactic": "simp_rw [essSup]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : ℝ≥0∞\nhc : c ≠ 0\nh_smul : Measure.ae (c • μ) = Measure.ae μ\n⊢ limsup f (Measure.ae (c • μ)) = limsup f (Measure.ae μ)\n\ncase h_smul\nα : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : ℝ≥0∞\nhc : c ≠ 0\n⊢ Measure.ae (c • μ) = Measure.ae μ",
"state_before": "α : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : ℝ≥0∞\nhc : c ≠ 0\n⊢ limsup f (Measure.ae (c • μ)) = limsup f (Measure.ae μ)",
"tactic": "suffices h_smul : (c • μ).ae = μ.ae"
},
{
"state_after": "case h_smul.a\nα : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : ℝ≥0∞\nhc : c ≠ 0\ns✝ : Set α\n⊢ s✝ ∈ Measure.ae (c • μ) ↔ s✝ ∈ Measure.ae μ",
"state_before": "case h_smul\nα : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : ℝ≥0∞\nhc : c ≠ 0\n⊢ Measure.ae (c • μ) = Measure.ae μ",
"tactic": "ext1"
},
{
"state_after": "case h_smul.a\nα : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : ℝ≥0∞\nhc : c ≠ 0\ns✝ : Set α\n⊢ ↑↑(c • μ) (s✝ᶜ) = 0 ↔ ↑↑μ (s✝ᶜ) = 0",
"state_before": "case h_smul.a\nα : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : ℝ≥0∞\nhc : c ≠ 0\ns✝ : Set α\n⊢ s✝ ∈ Measure.ae (c • μ) ↔ s✝ ∈ Measure.ae μ",
"tactic": "simp_rw [mem_ae_iff]"
},
{
"state_after": "no goals",
"state_before": "case h_smul.a\nα : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : ℝ≥0∞\nhc : c ≠ 0\ns✝ : Set α\n⊢ ↑↑(c • μ) (s✝ᶜ) = 0 ↔ ↑↑μ (s✝ᶜ) = 0",
"tactic": "simp [hc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : ℝ≥0∞\nhc : c ≠ 0\nh_smul : Measure.ae (c • μ) = Measure.ae μ\n⊢ limsup f (Measure.ae (c • μ)) = limsup f (Measure.ae μ)",
"tactic": "rw [h_smul]"
}
] |
[
219,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Std/Data/List/Lemmas.lean
|
List.get?_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\n⊢ get? l 0 = head? l",
"tactic": "cases l <;> rfl"
}
] |
[
542,
74
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
542,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
ZNum.zneg_succ
|
[
{
"state_after": "case neg\nα : Type ?u.658662\na✝ : PosNum\n⊢ -succ (neg a✝) = pred (-neg a✝)",
"state_before": "α : Type ?u.658662\nn : ZNum\n⊢ -succ n = pred (-n)",
"tactic": "cases n <;> try { rfl }"
},
{
"state_after": "case neg\nα : Type ?u.658662\na✝ : PosNum\n⊢ Num.toZNum (pred' a✝) = pred (-neg a✝)",
"state_before": "case neg\nα : Type ?u.658662\na✝ : PosNum\n⊢ -succ (neg a✝) = pred (-neg a✝)",
"tactic": "rw [succ, Num.zneg_toZNumNeg]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type ?u.658662\na✝ : PosNum\n⊢ Num.toZNum (pred' a✝) = pred (-neg a✝)",
"tactic": "rfl"
}
] |
[
1089,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1088,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
differentiableAt_const
|
[] |
[
1079,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1078,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.prod_mono
|
[] |
[
233,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.indicatorConstLp_coeFn_mem
|
[] |
[
741,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
740,
1
] |
Mathlib/SetTheory/Ordinal/Exponential.lean
|
Ordinal.log_opow_mul_add
|
[
{
"state_after": "b u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\n⊢ log b (b ^ u * v + w) = u",
"state_before": "b u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\n⊢ log b (b ^ u * v + w) = u",
"tactic": "have hne' := (opow_mul_add_pos (zero_lt_one.trans hb).ne' u hv w).ne'"
},
{
"state_after": "b u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\nhne : ¬log b (b ^ u * v + w) = u\n⊢ False",
"state_before": "b u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\n⊢ log b (b ^ u * v + w) = u",
"tactic": "by_contra' hne"
},
{
"state_after": "case inl\nb u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\nhne : ¬log b (b ^ u * v + w) = u\nh : log b (b ^ u * v + w) < u\n⊢ False\n\ncase inr\nb u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\nhne : ¬log b (b ^ u * v + w) = u\nh : log b (b ^ u * v + w) > u\n⊢ False",
"state_before": "b u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\nhne : ¬log b (b ^ u * v + w) = u\n⊢ False",
"tactic": "cases' lt_or_gt_of_ne hne with h h"
},
{
"state_after": "case inl\nb u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\nhne : ¬log b (b ^ u * v + w) = u\nh : b ^ u * v + w < b ^ u\n⊢ False",
"state_before": "case inl\nb u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\nhne : ¬log b (b ^ u * v + w) = u\nh : log b (b ^ u * v + w) < u\n⊢ False",
"tactic": "rw [← lt_opow_iff_log_lt hb hne'] at h"
},
{
"state_after": "no goals",
"state_before": "case inl\nb u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\nhne : ¬log b (b ^ u * v + w) = u\nh : b ^ u * v + w < b ^ u\n⊢ False",
"tactic": "exact h.not_le ((le_mul_left _ (Ordinal.pos_iff_ne_zero.2 hv)).trans (le_add_right _ _))"
},
{
"state_after": "case inr\nb u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\nhne : ¬log b (b ^ u * v + w) = u\nh : u < log b (b ^ u * v + w)\n⊢ False",
"state_before": "case inr\nb u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\nhne : ¬log b (b ^ u * v + w) = u\nh : log b (b ^ u * v + w) > u\n⊢ False",
"tactic": "conv at h => change u < log b (b ^ u * v + w)"
},
{
"state_after": "case inr\nb u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\nhne : ¬log b (b ^ u * v + w) = u\nh✝ : u < log b (b ^ u * v + w)\nh : b ^ succ u ≤ b ^ u * v + w\n⊢ False",
"state_before": "case inr\nb u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\nhne : ¬log b (b ^ u * v + w) = u\nh : u < log b (b ^ u * v + w)\n⊢ False",
"tactic": "rw [← succ_le_iff, ← opow_le_iff_le_log hb hne'] at h"
},
{
"state_after": "no goals",
"state_before": "case inr\nb u v w : Ordinal\nhb : 1 < b\nhv : v ≠ 0\nhvb : v < b\nhw : w < b ^ u\nhne' : b ^ u * v + w ≠ 0\nhne : ¬log b (b ^ u * v + w) = u\nh✝ : u < log b (b ^ u * v + w)\nh : b ^ succ u ≤ b ^ u * v + w\n⊢ False",
"tactic": "exact (not_lt_of_le h) (opow_mul_add_lt_opow_succ hvb hw)"
}
] |
[
422,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
413,
1
] |
Mathlib/Data/Prod/Basic.lean
|
Prod.swap_leftInverse
|
[] |
[
188,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.limitRecOn_limit
|
[
{
"state_after": "α : Type ?u.97233\nβ : Type ?u.97236\nγ : Type ?u.97239\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nC : Ordinal → Sort u_2\no : Ordinal\nH₁ : C 0\nH₂ : (o : Ordinal) → C o → C (succ o)\nH₃ : (o : Ordinal) → IsLimit o → ((o' : Ordinal) → o' < o → C o') → C o\nh : IsLimit o\n⊢ (H₃ o (_ : o ≠ 0 ∧ ∀ (a : Ordinal), a < o → succ a < o) fun y x =>\n WellFounded.fix lt_wf\n (fun o IH =>\n if o0 : o = 0 then Eq.mpr (_ : C o = C 0) H₁\n else\n if h : ∃ a, o = succ a then\n Eq.mpr (_ : C o = C (succ (pred o))) (H₂ (pred o) (IH (pred o) (_ : pred o < o)))\n else H₃ o (_ : o ≠ 0 ∧ ∀ (a : Ordinal), a < o → succ a < o) IH)\n y) =\n H₃ o h fun x _h => limitRecOn x H₁ H₂ H₃",
"state_before": "α : Type ?u.97233\nβ : Type ?u.97236\nγ : Type ?u.97239\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nC : Ordinal → Sort u_2\no : Ordinal\nH₁ : C 0\nH₂ : (o : Ordinal) → C o → C (succ o)\nH₃ : (o : Ordinal) → IsLimit o → ((o' : Ordinal) → o' < o → C o') → C o\nh : IsLimit o\n⊢ limitRecOn o H₁ H₂ H₃ = H₃ o h fun x _h => limitRecOn x H₁ H₂ H₃",
"tactic": "rw [limitRecOn, lt_wf.fix_eq, dif_neg h.1, dif_neg (not_succ_of_isLimit h)]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.97233\nβ : Type ?u.97236\nγ : Type ?u.97239\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nC : Ordinal → Sort u_2\no : Ordinal\nH₁ : C 0\nH₂ : (o : Ordinal) → C o → C (succ o)\nH₃ : (o : Ordinal) → IsLimit o → ((o' : Ordinal) → o' < o → C o') → C o\nh : IsLimit o\n⊢ (H₃ o (_ : o ≠ 0 ∧ ∀ (a : Ordinal), a < o → succ a < o) fun y x =>\n WellFounded.fix lt_wf\n (fun o IH =>\n if o0 : o = 0 then Eq.mpr (_ : C o = C 0) H₁\n else\n if h : ∃ a, o = succ a then\n Eq.mpr (_ : C o = C (succ (pred o))) (H₂ (pred o) (IH (pred o) (_ : pred o < o)))\n else H₃ o (_ : o ≠ 0 ∧ ∀ (a : Ordinal), a < o → succ a < o) IH)\n y) =\n H₃ o h fun x _h => limitRecOn x H₁ H₂ H₃",
"tactic": "rfl"
}
] |
[
341,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.mem_map
|
[
{
"state_after": "G : Type u_1\nG' : Type ?u.242723\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.242732\ninst✝² : AddGroup A\nH K✝ : Subgroup G\nk : Set G\nN : Type u_2\ninst✝¹ : Group N\nP : Type ?u.242759\ninst✝ : Group P\nf : G →* N\nK : Subgroup G\ny : N\n⊢ (∃ x x_1, ↑f x = y) ↔ ∃ x, x ∈ K ∧ ↑f x = y",
"state_before": "G : Type u_1\nG' : Type ?u.242723\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.242732\ninst✝² : AddGroup A\nH K✝ : Subgroup G\nk : Set G\nN : Type u_2\ninst✝¹ : Group N\nP : Type ?u.242759\ninst✝ : Group P\nf : G →* N\nK : Subgroup G\ny : N\n⊢ y ∈ map f K ↔ ∃ x, x ∈ K ∧ ↑f x = y",
"tactic": "erw [mem_image_iff_bex]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.242723\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.242732\ninst✝² : AddGroup A\nH K✝ : Subgroup G\nk : Set G\nN : Type u_2\ninst✝¹ : Group N\nP : Type ?u.242759\ninst✝ : Group P\nf : G →* N\nK : Subgroup G\ny : N\n⊢ (∃ x x_1, ↑f x = y) ↔ ∃ x, x ∈ K ∧ ↑f x = y",
"tactic": "simp"
}
] |
[
1414,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1413,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisConnection.u_unique
|
[] |
[
186,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.coe_eq_coe
|
[] |
[
131,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
|
deriv_arctan
|
[] |
[
157,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
AEMeasurable.div
|
[] |
[
353,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
351,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.coe_injective
|
[] |
[
159,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/RingTheory/EisensteinCriterion.lean
|
Polynomial.EisensteinCriterionAux.isUnit_of_natDegree_eq_zero_of_isPrimitive
|
[
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhu : IsPrimitive (p * q)\nhpm : natDegree p = 0\n⊢ IsUnit (coeff p 0)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhu : IsPrimitive (p * q)\nhpm : natDegree p = 0\n⊢ IsUnit p",
"tactic": "rw [eq_C_of_degree_le_zero (natDegree_eq_zero_iff_degree_le_zero.1 hpm), isUnit_C]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhu : IsPrimitive (p * q)\nhpm : natDegree p = 0\n⊢ ↑C (coeff p 0) ∣ p * q",
"state_before": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhu : IsPrimitive (p * q)\nhpm : natDegree p = 0\n⊢ IsUnit (coeff p 0)",
"tactic": "refine' hu _ _"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhu : IsPrimitive (p * q)\nhpm : natDegree p = 0\n⊢ p ∣ p * q",
"state_before": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhu : IsPrimitive (p * q)\nhpm : natDegree p = 0\n⊢ ↑C (coeff p 0) ∣ p * q",
"tactic": "rw [← eq_C_of_degree_le_zero (natDegree_eq_zero_iff_degree_le_zero.1 hpm)]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhu : IsPrimitive (p * q)\nhpm : natDegree p = 0\n⊢ p ∣ p * q",
"tactic": "exact dvd_mul_right _ _"
}
] |
[
83,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Topology/MetricSpace/IsometricSMul.lean
|
EMetric.preimage_mul_left_ball
|
[] |
[
305,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
303,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
pow_boole
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.22144\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\nP : Prop\ninst✝ : Decidable P\na : M\n⊢ (a ^ if P then 1 else 0) = if P then a else 1",
"tactic": "simp"
}
] |
[
155,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/LinearAlgebra/Matrix/Hermitian.lean
|
Matrix.IsHermitian.apply
|
[] |
[
61,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Topology/Order/Basic.lean
|
countable_setOf_covby_right
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"tactic": "nontriviality α"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"tactic": "let s := { x : α | ∃ y, x ⋖ y }"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\nthis : ∀ (x : α), x ∈ s → ∃ y, x ⋖ y\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"tactic": "have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\nthis : ∀ (x : α), x ∈ s → ∃ y, x ⋖ y\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"tactic": "choose! y hy using this"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"tactic": "have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\nH : ∀ (a : Set α), IsOpen a → Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\n⊢ Set.Countable {x | ∃ y, x ⋖ y}\n\ncase H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\n⊢ ∀ (a : Set α), IsOpen a → Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"tactic": "suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\n⊢ Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\n⊢ ∀ (a : Set α), IsOpen a → Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}",
"tactic": "intro a ha"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nH : Set.Countable {x | (x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a) ∧ ¬IsBot x}\n⊢ Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\n\ncase H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\n⊢ Set.Countable {x | (x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a) ∧ ¬IsBot x}",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\n⊢ Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}",
"tactic": "suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\n⊢ Set.Countable {x | x ∈ {x | ∃ y, x ⋖ y} ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\n⊢ Set.Countable {x | (x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a) ∧ ¬IsBot x}",
"tactic": "simp only [and_assoc]"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\n⊢ Set.Countable {x | x ∈ {x | ∃ y, x ⋖ y} ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\n⊢ Set.Countable {x | x ∈ {x | ∃ y, x ⋖ y} ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}",
"tactic": "let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nthis : ∀ (x : α), x ∈ t → ∃ z, z < x ∧ Ioc z x ⊆ a\n⊢ Set.Countable {x | x ∈ {x | ∃ y, x ⋖ y} ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\n⊢ Set.Countable {x | x ∈ {x | ∃ y, x ⋖ y} ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}",
"tactic": "have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\n⊢ Set.Countable {x | x ∈ {x | ∃ y, x ⋖ y} ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nthis : ∀ (x : α), x ∈ t → ∃ z, z < x ∧ Ioc z x ⊆ a\n⊢ Set.Countable {x | x ∈ {x | ∃ y, x ⋖ y} ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}",
"tactic": "choose! z hz h'z using this"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nthis : PairwiseDisjoint t fun x => Ioc (z x) x\nx : α\nhx : x ∈ t\n⊢ IsOpen (Ioc (z x) x)",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nthis : PairwiseDisjoint t fun x => Ioc (z x) x\n⊢ Set.Countable {x | x ∈ {x | ∃ y, x ⋖ y} ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}",
"tactic": "refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nthis : PairwiseDisjoint t fun x => Ioc (z x) x\nx : α\nhx : x ∈ t\nH : Ioc (z x) x = Ioo (z x) (y x)\n⊢ IsOpen (Ioc (z x) x)\n\ncase H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nthis : PairwiseDisjoint t fun x => Ioc (z x) x\nx : α\nhx : x ∈ t\n⊢ Ioc (z x) x = Ioo (z x) (y x)",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nthis : PairwiseDisjoint t fun x => Ioc (z x) x\nx : α\nhx : x ∈ t\n⊢ IsOpen (Ioc (z x) x)",
"tactic": "suffices H : Ioc (z x) x = Ioo (z x) (y x)"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nthis : PairwiseDisjoint t fun x => Ioc (z x) x\nx : α\nhx : x ∈ t\n⊢ Ioc (z x) x = Ioo (z x) (y x)",
"tactic": "exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\nH : ∀ (a : Set α), IsOpen a → Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\nthis : s ⊆ ⋃ (a : Set α) (_ : a ∈ countableBasis α), {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\nH : ∀ (a : Set α), IsOpen a → Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"tactic": "have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\nH : ∀ (a : Set α), IsOpen a → Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\nthis : s ⊆ ⋃ (a : Set α) (_ : a ∈ countableBasis α), {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\n⊢ Set.Countable (⋃ (a : Set α) (_ : a ∈ countableBasis α), {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a})",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\nH : ∀ (a : Set α), IsOpen a → Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\nthis : s ⊆ ⋃ (a : Set α) (_ : a ∈ countableBasis α), {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\n⊢ Set.Countable {x | ∃ y, x ⋖ y}",
"tactic": "refine Set.Countable.mono this ?_"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\nH : ∀ (a : Set α), IsOpen a → Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\nthis : s ⊆ ⋃ (a : Set α) (_ : a ∈ countableBasis α), {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\na : Set α\nha : a ∈ countableBasis α\n⊢ IsOpen a",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\nH : ∀ (a : Set α), IsOpen a → Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\nthis : s ⊆ ⋃ (a : Set α) (_ : a ∈ countableBasis α), {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\n⊢ Set.Countable (⋃ (a : Set α) (_ : a ∈ countableBasis α), {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a})",
"tactic": "refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\nH : ∀ (a : Set α), IsOpen a → Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\nthis : s ⊆ ⋃ (a : Set α) (_ : a ∈ countableBasis α), {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\na : Set α\nha : a ∈ countableBasis α\n⊢ IsOpen a",
"tactic": "exact isOpen_of_mem_countableBasis ha"
},
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\nH : ∀ (a : Set α), IsOpen a → Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\nx : α\nhx : x ∈ s\na : Set α\nab : a ∈ countableBasis α\nxa : x ∈ a\nya : ¬y x ∈ a\n⊢ x ∈ ⋃ (a : Set α) (_ : a ∈ countableBasis α), {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\nH : ∀ (a : Set α), IsOpen a → Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\nx : α\nhx : x ∈ s\n⊢ x ∈ ⋃ (a : Set α) (_ : a ∈ countableBasis α), {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}",
"tactic": "rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\nH : ∀ (a : Set α), IsOpen a → Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}\nx : α\nhx : x ∈ s\na : Set α\nab : a ∈ countableBasis α\nxa : x ∈ a\nya : ¬y x ∈ a\n⊢ x ∈ ⋃ (a : Set α) (_ : a ∈ countableBasis α), {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}",
"tactic": "exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nH : Set.Countable {x | (x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a) ∧ ¬IsBot x}\n⊢ Set.Countable {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a}",
"tactic": "exact H.of_diff (subsingleton_isBot α).countable"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nx : α\nhx : x ∈ t\n⊢ ∃ z, z < x ∧ Ioc z x ⊆ a",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\n⊢ ∀ (x : α), x ∈ t → ∃ z, z < x ∧ Ioc z x ⊆ a",
"tactic": "intro x hx"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nx : α\nhx : x ∈ t\n⊢ ∃ l, l < x",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nx : α\nhx : x ∈ t\n⊢ ∃ z, z < x ∧ Ioc z x ⊆ a",
"tactic": "apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nx : α\nhx : x ∈ t\n⊢ ∃ l, l < x",
"tactic": "simpa only [IsBot, not_forall, not_le] using hx.right.right.right"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x < x'\n⊢ (Disjoint on fun x => Ioc (z x) x) x x'\n\ncase inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x' < x\n⊢ (Disjoint on fun x => Ioc (z x) x) x x'",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\n⊢ (Disjoint on fun x => Ioc (z x) x) x x'",
"tactic": "rcases hxx'.lt_or_lt with (h' | h')"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x < x'\nu : α\nux : u ∈ (fun x => Ioc (z x) x) x\nux' : u ∈ (fun x => Ioc (z x) x) x'\n⊢ y x ∈ a",
"state_before": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x < x'\n⊢ (Disjoint on fun x => Ioc (z x) x) x x'",
"tactic": "refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x < x'\nu : α\nux : u ∈ (fun x => Ioc (z x) x) x\nux' : u ∈ (fun x => Ioc (z x) x) x'\n⊢ y x ≤ x'",
"state_before": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x < x'\nu : α\nux : u ∈ (fun x => Ioc (z x) x) x\nux' : u ∈ (fun x => Ioc (z x) x) x'\n⊢ y x ∈ a",
"tactic": "refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x < x'\nu : α\nux : u ∈ (fun x => Ioc (z x) x) x\nux' : u ∈ (fun x => Ioc (z x) x) x'\nH : x' < y x\n⊢ False",
"state_before": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x < x'\nu : α\nux : u ∈ (fun x => Ioc (z x) x) x\nux' : u ∈ (fun x => Ioc (z x) x) x'\n⊢ y x ≤ x'",
"tactic": "by_contra' H"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x < x'\nu : α\nux : u ∈ (fun x => Ioc (z x) x) x\nux' : u ∈ (fun x => Ioc (z x) x) x'\nH : x' < y x\n⊢ False",
"tactic": "exact False.elim (lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h'))"
},
{
"state_after": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x' < x\nu : α\nux : u ∈ (fun x => Ioc (z x) x) x\nux' : u ∈ (fun x => Ioc (z x) x) x'\n⊢ y x' ∈ a",
"state_before": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x' < x\n⊢ (Disjoint on fun x => Ioc (z x) x) x x'",
"tactic": "refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _"
},
{
"state_after": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x' < x\nu : α\nux : u ∈ (fun x => Ioc (z x) x) x\nux' : u ∈ (fun x => Ioc (z x) x) x'\n⊢ y x' ≤ x",
"state_before": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x' < x\nu : α\nux : u ∈ (fun x => Ioc (z x) x) x\nux' : u ∈ (fun x => Ioc (z x) x) x'\n⊢ y x' ∈ a",
"tactic": "refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩"
},
{
"state_after": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x' < x\nu : α\nux : u ∈ (fun x => Ioc (z x) x) x\nux' : u ∈ (fun x => Ioc (z x) x) x'\nH : x < y x'\n⊢ False",
"state_before": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x' < x\nu : α\nux : u ∈ (fun x => Ioc (z x) x) x\nux' : u ∈ (fun x => Ioc (z x) x) x'\n⊢ y x' ≤ x",
"tactic": "by_contra' H"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nx : α\nxt : x ∈ t\nx' : α\nx't : x' ∈ t\nhxx' : x ≠ x'\nh' : x' < x\nu : α\nux : u ∈ (fun x => Ioc (z x) x) x\nux' : u ∈ (fun x => Ioc (z x) x) x'\nH : x < y x'\n⊢ False",
"tactic": "exact False.elim (lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h'))"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nthis : PairwiseDisjoint t fun x => Ioc (z x) x\nx : α\nhx : x ∈ t\nH : Ioc (z x) x = Ioo (z x) (y x)\n⊢ IsOpen (Ioo (z x) (y x))",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nthis : PairwiseDisjoint t fun x => Ioc (z x) x\nx : α\nhx : x ∈ t\nH : Ioc (z x) x = Ioo (z x) (y x)\n⊢ IsOpen (Ioc (z x) x)",
"tactic": "rw [H]"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\n✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ (x : α), x ∈ s → x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen a\nt : Set α := {x | x ∈ s ∧ x ∈ a ∧ ¬y x ∈ a ∧ ¬IsBot x}\nz : α → α\nhz : ∀ (x : α), x ∈ t → z x < x\nh'z : ∀ (x : α), x ∈ t → Ioc (z x) x ⊆ a\nthis : PairwiseDisjoint t fun x => Ioc (z x) x\nx : α\nhx : x ∈ t\nH : Ioc (z x) x = Ioo (z x) (y x)\n⊢ IsOpen (Ioo (z x) (y x))",
"tactic": "exact isOpen_Ioo"
}
] |
[
1411,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1373,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.lipschitz_infNndist_pt
|
[] |
[
669,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
668,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.Compacts.map_id
|
[] |
[
146,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.map_iSup
|
[] |
[
1477,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1476,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.ae_le_of_ae_lt
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.100460\nγ : Type ?u.100463\nδ : Type ?u.100466\nι : Type ?u.100469\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nf g : α → ℝ≥0∞\nh : ∀ᵐ (x : α) ∂μ, f x < g x\n⊢ ↑↑μ {a | ¬f a ≤ g a} = 0",
"state_before": "α : Type u_1\nβ : Type ?u.100460\nγ : Type ?u.100463\nδ : Type ?u.100466\nι : Type ?u.100469\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nf g : α → ℝ≥0∞\nh : ∀ᵐ (x : α) ∂μ, f x < g x\n⊢ f ≤ᵐ[μ] g",
"tactic": "rw [Filter.EventuallyLE, ae_iff]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.100460\nγ : Type ?u.100463\nδ : Type ?u.100466\nι : Type ?u.100469\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nf g : α → ℝ≥0∞\nh : ↑↑μ {a | ¬f a < g a} = 0\n⊢ ↑↑μ {a | ¬f a ≤ g a} = 0",
"state_before": "α : Type u_1\nβ : Type ?u.100460\nγ : Type ?u.100463\nδ : Type ?u.100466\nι : Type ?u.100469\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nf g : α → ℝ≥0∞\nh : ∀ᵐ (x : α) ∂μ, f x < g x\n⊢ ↑↑μ {a | ¬f a ≤ g a} = 0",
"tactic": "rw [ae_iff] at h"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.100460\nγ : Type ?u.100463\nδ : Type ?u.100466\nι : Type ?u.100469\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nf g : α → ℝ≥0∞\nh : ↑↑μ {a | ¬f a < g a} = 0\nx : α\nhx : x ∈ {a | ¬f a ≤ g a}\n⊢ x ∈ {a | ¬f a < g a}",
"state_before": "α : Type u_1\nβ : Type ?u.100460\nγ : Type ?u.100463\nδ : Type ?u.100466\nι : Type ?u.100469\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nf g : α → ℝ≥0∞\nh : ↑↑μ {a | ¬f a < g a} = 0\n⊢ ↑↑μ {a | ¬f a ≤ g a} = 0",
"tactic": "refine' measure_mono_null (fun x hx => _) h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.100460\nγ : Type ?u.100463\nδ : Type ?u.100466\nι : Type ?u.100469\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nf g : α → ℝ≥0∞\nh : ↑↑μ {a | ¬f a < g a} = 0\nx : α\nhx : x ∈ {a | ¬f a ≤ g a}\n⊢ x ∈ {a | ¬f a < g a}",
"tactic": "exact not_lt.2 (le_of_lt (not_le.1 hx))"
}
] |
[
438,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Algebra/AddTorsor.lean
|
vadd_vsub_assoc
|
[
{
"state_after": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\ng : G\np1 p2 : P\n⊢ g +ᵥ p1 -ᵥ p2 +ᵥ p2 = g + (p1 -ᵥ p2) +ᵥ p2",
"state_before": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\ng : G\np1 p2 : P\n⊢ g +ᵥ p1 -ᵥ p2 = g + (p1 -ᵥ p2)",
"tactic": "apply vadd_right_cancel p2"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\ng : G\np1 p2 : P\n⊢ g +ᵥ p1 -ᵥ p2 +ᵥ p2 = g + (p1 -ᵥ p2) +ᵥ p2",
"tactic": "rw [vsub_vadd, add_vadd, vsub_vadd]"
}
] |
[
122,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Limits.lean
|
CategoryTheory.lift_comp_preservesLimitsIso_hom
|
[
{
"state_after": "case w\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\nJ : Type w\ninst✝³ : Category J\nF : J ⥤ C\ninst✝² : PreservesLimit F G\ninst✝¹ : HasLimit F\ninst✝ : HasLimit (F ⋙ G)\nt : Cone F\nj✝ : J\n⊢ (G.map (limit.lift F t) ≫ (preservesLimitIso G F).hom) ≫ limit.π (F ⋙ G) j✝ =\n limit.lift (F ⋙ G) (G.mapCone t) ≫ limit.π (F ⋙ G) j✝",
"state_before": "C : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\nJ : Type w\ninst✝³ : Category J\nF : J ⥤ C\ninst✝² : PreservesLimit F G\ninst✝¹ : HasLimit F\ninst✝ : HasLimit (F ⋙ G)\nt : Cone F\n⊢ G.map (limit.lift F t) ≫ (preservesLimitIso G F).hom = limit.lift (F ⋙ G) (G.mapCone t)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u₁\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nG : C ⥤ D\nJ : Type w\ninst✝³ : Category J\nF : J ⥤ C\ninst✝² : PreservesLimit F G\ninst✝¹ : HasLimit F\ninst✝ : HasLimit (F ⋙ G)\nt : Cone F\nj✝ : J\n⊢ (G.map (limit.lift F t) ≫ (preservesLimitIso G F).hom) ≫ limit.π (F ⋙ G) j✝ =\n limit.lift (F ⋙ G) (G.mapCone t) ≫ limit.π (F ⋙ G) j✝",
"tactic": "simp [← G.map_comp]"
}
] |
[
80,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
|
CategoryTheory.Limits.Trident.IsLimit.homIso_natural
|
[] |
[
402,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
398,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.LiftRel.swap
|
[
{
"state_after": "case inl\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type u_2\nr r₁ r₂ : α → γ → Prop\ns s₁ s₂ : β → δ → Prop\na : α\nb : β\nc : γ\nd : δ\na✝¹ : α\nc✝ : γ\na✝ : r a✝¹ c✝\n⊢ LiftRel s r (swap (inl a✝¹)) (swap (inl c✝))\n\ncase inr\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type u_2\nr r₁ r₂ : α → γ → Prop\ns s₁ s₂ : β → δ → Prop\na : α\nb : β\nc : γ\nd : δ\nb✝ : β\nd✝ : δ\na✝ : s b✝ d✝\n⊢ LiftRel s r (swap (inr b✝)) (swap (inr d✝))",
"state_before": "α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type u_2\nr r₁ r₂ : α → γ → Prop\ns s₁ s₂ : β → δ → Prop\na : α\nb : β\nc : γ\nd : δ\nx : α ⊕ β\ny : γ ⊕ δ\nh : LiftRel r s x y\n⊢ LiftRel s r (swap x) (swap y)",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type u_2\nr r₁ r₂ : α → γ → Prop\ns s₁ s₂ : β → δ → Prop\na : α\nb : β\nc : γ\nd : δ\na✝¹ : α\nc✝ : γ\na✝ : r a✝¹ c✝\n⊢ LiftRel s r (swap (inl a✝¹)) (swap (inl c✝))",
"tactic": "exact LiftRel.inr ‹_›"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type u_2\nr r₁ r₂ : α → γ → Prop\ns s₁ s₂ : β → δ → Prop\na : α\nb : β\nc : γ\nd : δ\nb✝ : β\nd✝ : δ\na✝ : s b✝ d✝\n⊢ LiftRel s r (swap (inr b✝)) (swap (inr d✝))",
"tactic": "exact LiftRel.inl ‹_›"
}
] |
[
447,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
444,
11
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.compl_top
|
[] |
[
901,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
900,
11
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
HomogeneousIdeal.coe_top
|
[] |
[
345,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Data/List/Basic.lean
|
List.drop_sizeOf_le
|
[
{
"state_after": "case nil\nι : Type ?u.208871\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nn : ℕ\n⊢ sizeOf (drop n []) ≤ sizeOf []\n\ncase cons\nι : Type ?u.208871\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nhead✝ : α\ntail✝ : List α\nlih : ∀ (n : ℕ), sizeOf (drop n tail✝) ≤ sizeOf tail✝\nn : ℕ\n⊢ sizeOf (drop n (head✝ :: tail✝)) ≤ sizeOf (head✝ :: tail✝)",
"state_before": "ι : Type ?u.208871\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nl : List α\n⊢ ∀ (n : ℕ), sizeOf (drop n l) ≤ sizeOf l",
"tactic": "induction' l with _ _ lih <;> intro n"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.208871\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nn : ℕ\n⊢ sizeOf (drop n []) ≤ sizeOf []",
"tactic": "rw [drop_nil]"
},
{
"state_after": "case cons.zero\nι : Type ?u.208871\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nhead✝ : α\ntail✝ : List α\nlih : ∀ (n : ℕ), sizeOf (drop n tail✝) ≤ sizeOf tail✝\n⊢ sizeOf (drop zero (head✝ :: tail✝)) ≤ sizeOf (head✝ :: tail✝)\n\ncase cons.succ\nι : Type ?u.208871\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nhead✝ : α\ntail✝ : List α\nlih : ∀ (n : ℕ), sizeOf (drop n tail✝) ≤ sizeOf tail✝\nn : ℕ\nn_ih✝ : sizeOf (drop n (head✝ :: tail✝)) ≤ sizeOf (head✝ :: tail✝)\n⊢ sizeOf (drop (succ n) (head✝ :: tail✝)) ≤ sizeOf (head✝ :: tail✝)",
"state_before": "case cons\nι : Type ?u.208871\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nhead✝ : α\ntail✝ : List α\nlih : ∀ (n : ℕ), sizeOf (drop n tail✝) ≤ sizeOf tail✝\nn : ℕ\n⊢ sizeOf (drop n (head✝ :: tail✝)) ≤ sizeOf (head✝ :: tail✝)",
"tactic": "induction' n with n"
},
{
"state_after": "no goals",
"state_before": "case cons.zero\nι : Type ?u.208871\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nhead✝ : α\ntail✝ : List α\nlih : ∀ (n : ℕ), sizeOf (drop n tail✝) ≤ sizeOf tail✝\n⊢ sizeOf (drop zero (head✝ :: tail✝)) ≤ sizeOf (head✝ :: tail✝)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case cons.succ\nι : Type ?u.208871\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : SizeOf α\nhead✝ : α\ntail✝ : List α\nlih : ∀ (n : ℕ), sizeOf (drop n tail✝) ≤ sizeOf tail✝\nn : ℕ\nn_ih✝ : sizeOf (drop n (head✝ :: tail✝)) ≤ sizeOf (head✝ :: tail✝)\n⊢ sizeOf (drop (succ n) (head✝ :: tail✝)) ≤ sizeOf (head✝ :: tail✝)",
"tactic": "exact Trans.trans (lih _) le_add_self"
}
] |
[
2197,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2192,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.bounded_Ioc
|
[] |
[
2550,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2549,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.isComplement_top_singleton
|
[] |
[
115,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.filter_eq'
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.420411\nγ : Type ?u.420414\ninst✝ : DecidableEq α\ns : Multiset α\nb : α\nl : List α\n⊢ ↑(List.filter (fun b_1 => decide (b_1 = b)) l) = replicate (List.count b l) b",
"state_before": "α : Type u_1\nβ : Type ?u.420411\nγ : Type ?u.420414\ninst✝ : DecidableEq α\ns : Multiset α\nb : α\nl : List α\n⊢ filter (fun x => x = b) (Quotient.mk (isSetoid α) l) = replicate (count b (Quotient.mk (isSetoid α) l)) b",
"tactic": "simp only [quot_mk_to_coe, coe_filter, mem_coe, coe_count]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.420411\nγ : Type ?u.420414\ninst✝ : DecidableEq α\ns : Multiset α\nb : α\nl : List α\n⊢ ↑(List.filter (fun b_1 => decide (b_1 = b)) l) = replicate (List.count b l) b",
"tactic": "rw [List.filter_eq' l b, coe_replicate]"
}
] |
[
2574,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2571,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
csInf_upper_bounds_eq_csSup
|
[] |
[
595,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
593,
1
] |
Mathlib/RingTheory/Algebraic.lean
|
isAlgebraic_algebraMap
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.65885\nA : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\ninst✝ : Nontrivial R\nx : R\n⊢ ↑(aeval (↑(algebraMap R A) x)) (X - ↑C x) = 0",
"tactic": "rw [_root_.map_sub, aeval_X, aeval_C, sub_self]"
}
] |
[
111,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
src/lean/Init/SimpLemmas.lean
|
Eq.mpr_prop
|
[] |
[
62,
72
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
62,
1
] |
Mathlib/Computability/Partrec.lean
|
Partrec.map_encode_iff
|
[] |
[
491,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
490,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.ae_le_set_union
|
[] |
[
466,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
464,
1
] |
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
Matrix.toEuclideanLin_eq_toLin
|
[] |
[
1002,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
999,
1
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.Quotient.eq
|
[] |
[
86,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
11
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.compContinuous_apply
|
[] |
[
384,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/Combinatorics/SimpleGraph/Coloring.lean
|
SimpleGraph.cliqueFree_of_chromaticNumber_lt
|
[] |
[
459,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
457,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.orderIsoIicCoe_symm_apply_coe
|
[] |
[
1804,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1802,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
PosNum.cmp_to_nat
|
[
{
"state_after": "α : Type ?u.145219\na b : PosNum\nthis : Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a)\n⊢ Ordering.casesOn (cmp (bit0 a) (bit0 b)) (↑(bit0 a) < ↑(bit0 b)) (bit0 a = bit0 b) (↑(bit0 b) < ↑(bit0 a))",
"state_before": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.casesOn (cmp (bit0 a) (bit0 b)) (↑(bit0 a) < ↑(bit0 b)) (bit0 a = bit0 b) (↑(bit0 b) < ↑(bit0 a))",
"tactic": "have := cmp_to_nat a b"
},
{
"state_after": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a) →\n Ordering.casesOn (cmp (bit0 a) (bit0 b)) (↑(bit0 a) < ↑(bit0 b)) (bit0 a = bit0 b) (↑(bit0 b) < ↑(bit0 a))",
"state_before": "α : Type ?u.145219\na b : PosNum\nthis : Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a)\n⊢ Ordering.casesOn (cmp (bit0 a) (bit0 b)) (↑(bit0 a) < ↑(bit0 b)) (bit0 a = bit0 b) (↑(bit0 b) < ↑(bit0 a))",
"tactic": "revert this"
},
{
"state_after": "case lt\nα : Type ?u.145219\na b : PosNum\nthis : ↑a < ↑b\n⊢ _root_.bit0 ↑a < _root_.bit0 ↑b\n\ncase eq\nα : Type ?u.145219\na b : PosNum\nthis : a = b\n⊢ bit0 a = bit0 b\n\ncase gt\nα : Type ?u.145219\na b : PosNum\nthis : ↑b < ↑a\n⊢ _root_.bit0 ↑b < _root_.bit0 ↑a",
"state_before": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a) →\n Ordering.casesOn (cmp (bit0 a) (bit0 b)) (↑(bit0 a) < ↑(bit0 b)) (bit0 a = bit0 b) (↑(bit0 b) < ↑(bit0 a))",
"tactic": "cases cmp a b <;> dsimp <;> intro this"
},
{
"state_after": "no goals",
"state_before": "case lt\nα : Type ?u.145219\na b : PosNum\nthis : ↑a < ↑b\n⊢ _root_.bit0 ↑a < _root_.bit0 ↑b",
"tactic": "exact add_lt_add this this"
},
{
"state_after": "no goals",
"state_before": "case eq\nα : Type ?u.145219\na b : PosNum\nthis : a = b\n⊢ bit0 a = bit0 b",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "case gt\nα : Type ?u.145219\na b : PosNum\nthis : ↑b < ↑a\n⊢ _root_.bit0 ↑b < _root_.bit0 ↑a",
"tactic": "exact add_lt_add this this"
},
{
"state_after": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.rec (_root_.bit0 ↑a < _root_.bit1 ↑b) (bit0 a = bit1 b) (_root_.bit1 ↑b < _root_.bit0 ↑a)\n (Ordering.rec Ordering.lt Ordering.lt Ordering.gt (cmp a b))",
"state_before": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.casesOn (cmp (bit0 a) (bit1 b)) (↑(bit0 a) < ↑(bit1 b)) (bit0 a = bit1 b) (↑(bit1 b) < ↑(bit0 a))",
"tactic": "dsimp [cmp]"
},
{
"state_after": "α : Type ?u.145219\na b : PosNum\nthis : Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a)\n⊢ Ordering.rec (_root_.bit0 ↑a < _root_.bit1 ↑b) (bit0 a = bit1 b) (_root_.bit1 ↑b < _root_.bit0 ↑a)\n (Ordering.rec Ordering.lt Ordering.lt Ordering.gt (cmp a b))",
"state_before": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.rec (_root_.bit0 ↑a < _root_.bit1 ↑b) (bit0 a = bit1 b) (_root_.bit1 ↑b < _root_.bit0 ↑a)\n (Ordering.rec Ordering.lt Ordering.lt Ordering.gt (cmp a b))",
"tactic": "have := cmp_to_nat a b"
},
{
"state_after": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a) →\n Ordering.rec (_root_.bit0 ↑a < _root_.bit1 ↑b) (bit0 a = bit1 b) (_root_.bit1 ↑b < _root_.bit0 ↑a)\n (Ordering.rec Ordering.lt Ordering.lt Ordering.gt (cmp a b))",
"state_before": "α : Type ?u.145219\na b : PosNum\nthis : Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a)\n⊢ Ordering.rec (_root_.bit0 ↑a < _root_.bit1 ↑b) (bit0 a = bit1 b) (_root_.bit1 ↑b < _root_.bit0 ↑a)\n (Ordering.rec Ordering.lt Ordering.lt Ordering.gt (cmp a b))",
"tactic": "revert this"
},
{
"state_after": "case lt\nα : Type ?u.145219\na b : PosNum\nthis : ↑a < ↑b\n⊢ _root_.bit0 ↑a < _root_.bit1 ↑b\n\ncase eq\nα : Type ?u.145219\na b : PosNum\nthis : a = b\n⊢ _root_.bit0 ↑a < _root_.bit1 ↑b\n\ncase gt\nα : Type ?u.145219\na b : PosNum\nthis : ↑b < ↑a\n⊢ _root_.bit1 ↑b < _root_.bit0 ↑a",
"state_before": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a) →\n Ordering.rec (_root_.bit0 ↑a < _root_.bit1 ↑b) (bit0 a = bit1 b) (_root_.bit1 ↑b < _root_.bit0 ↑a)\n (Ordering.rec Ordering.lt Ordering.lt Ordering.gt (cmp a b))",
"tactic": "cases cmp a b <;> dsimp <;> intro this"
},
{
"state_after": "no goals",
"state_before": "case lt\nα : Type ?u.145219\na b : PosNum\nthis : ↑a < ↑b\n⊢ _root_.bit0 ↑a < _root_.bit1 ↑b",
"tactic": "exact Nat.le_succ_of_le (add_lt_add this this)"
},
{
"state_after": "case eq\nα : Type ?u.145219\na b : PosNum\nthis : a = b\n⊢ _root_.bit0 ↑b < _root_.bit1 ↑b",
"state_before": "case eq\nα : Type ?u.145219\na b : PosNum\nthis : a = b\n⊢ _root_.bit0 ↑a < _root_.bit1 ↑b",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "case eq\nα : Type ?u.145219\na b : PosNum\nthis : a = b\n⊢ _root_.bit0 ↑b < _root_.bit1 ↑b",
"tactic": "apply Nat.lt_succ_self"
},
{
"state_after": "no goals",
"state_before": "case gt\nα : Type ?u.145219\na b : PosNum\nthis : ↑b < ↑a\n⊢ _root_.bit1 ↑b < _root_.bit0 ↑a",
"tactic": "exact cmp_to_nat_lemma this"
},
{
"state_after": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.rec (_root_.bit1 ↑a < _root_.bit0 ↑b) (bit1 a = bit0 b) (_root_.bit0 ↑b < _root_.bit1 ↑a)\n (Ordering.rec Ordering.lt Ordering.gt Ordering.gt (cmp a b))",
"state_before": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.casesOn (cmp (bit1 a) (bit0 b)) (↑(bit1 a) < ↑(bit0 b)) (bit1 a = bit0 b) (↑(bit0 b) < ↑(bit1 a))",
"tactic": "dsimp [cmp]"
},
{
"state_after": "α : Type ?u.145219\na b : PosNum\nthis : Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a)\n⊢ Ordering.rec (_root_.bit1 ↑a < _root_.bit0 ↑b) (bit1 a = bit0 b) (_root_.bit0 ↑b < _root_.bit1 ↑a)\n (Ordering.rec Ordering.lt Ordering.gt Ordering.gt (cmp a b))",
"state_before": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.rec (_root_.bit1 ↑a < _root_.bit0 ↑b) (bit1 a = bit0 b) (_root_.bit0 ↑b < _root_.bit1 ↑a)\n (Ordering.rec Ordering.lt Ordering.gt Ordering.gt (cmp a b))",
"tactic": "have := cmp_to_nat a b"
},
{
"state_after": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a) →\n Ordering.rec (_root_.bit1 ↑a < _root_.bit0 ↑b) (bit1 a = bit0 b) (_root_.bit0 ↑b < _root_.bit1 ↑a)\n (Ordering.rec Ordering.lt Ordering.gt Ordering.gt (cmp a b))",
"state_before": "α : Type ?u.145219\na b : PosNum\nthis : Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a)\n⊢ Ordering.rec (_root_.bit1 ↑a < _root_.bit0 ↑b) (bit1 a = bit0 b) (_root_.bit0 ↑b < _root_.bit1 ↑a)\n (Ordering.rec Ordering.lt Ordering.gt Ordering.gt (cmp a b))",
"tactic": "revert this"
},
{
"state_after": "case lt\nα : Type ?u.145219\na b : PosNum\nthis : ↑a < ↑b\n⊢ _root_.bit1 ↑a < _root_.bit0 ↑b\n\ncase eq\nα : Type ?u.145219\na b : PosNum\nthis : a = b\n⊢ _root_.bit0 ↑b < _root_.bit1 ↑a\n\ncase gt\nα : Type ?u.145219\na b : PosNum\nthis : ↑b < ↑a\n⊢ _root_.bit0 ↑b < _root_.bit1 ↑a",
"state_before": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a) →\n Ordering.rec (_root_.bit1 ↑a < _root_.bit0 ↑b) (bit1 a = bit0 b) (_root_.bit0 ↑b < _root_.bit1 ↑a)\n (Ordering.rec Ordering.lt Ordering.gt Ordering.gt (cmp a b))",
"tactic": "cases cmp a b <;> dsimp <;> intro this"
},
{
"state_after": "no goals",
"state_before": "case lt\nα : Type ?u.145219\na b : PosNum\nthis : ↑a < ↑b\n⊢ _root_.bit1 ↑a < _root_.bit0 ↑b",
"tactic": "exact cmp_to_nat_lemma this"
},
{
"state_after": "case eq\nα : Type ?u.145219\na b : PosNum\nthis : a = b\n⊢ _root_.bit0 ↑b < _root_.bit1 ↑b",
"state_before": "case eq\nα : Type ?u.145219\na b : PosNum\nthis : a = b\n⊢ _root_.bit0 ↑b < _root_.bit1 ↑a",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "case eq\nα : Type ?u.145219\na b : PosNum\nthis : a = b\n⊢ _root_.bit0 ↑b < _root_.bit1 ↑b",
"tactic": "apply Nat.lt_succ_self"
},
{
"state_after": "no goals",
"state_before": "case gt\nα : Type ?u.145219\na b : PosNum\nthis : ↑b < ↑a\n⊢ _root_.bit0 ↑b < _root_.bit1 ↑a",
"tactic": "exact Nat.le_succ_of_le (add_lt_add this this)"
},
{
"state_after": "α : Type ?u.145219\na b : PosNum\nthis : Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a)\n⊢ Ordering.casesOn (cmp (bit1 a) (bit1 b)) (↑(bit1 a) < ↑(bit1 b)) (bit1 a = bit1 b) (↑(bit1 b) < ↑(bit1 a))",
"state_before": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.casesOn (cmp (bit1 a) (bit1 b)) (↑(bit1 a) < ↑(bit1 b)) (bit1 a = bit1 b) (↑(bit1 b) < ↑(bit1 a))",
"tactic": "have := cmp_to_nat a b"
},
{
"state_after": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a) →\n Ordering.casesOn (cmp (bit1 a) (bit1 b)) (↑(bit1 a) < ↑(bit1 b)) (bit1 a = bit1 b) (↑(bit1 b) < ↑(bit1 a))",
"state_before": "α : Type ?u.145219\na b : PosNum\nthis : Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a)\n⊢ Ordering.casesOn (cmp (bit1 a) (bit1 b)) (↑(bit1 a) < ↑(bit1 b)) (bit1 a = bit1 b) (↑(bit1 b) < ↑(bit1 a))",
"tactic": "revert this"
},
{
"state_after": "case lt\nα : Type ?u.145219\na b : PosNum\nthis : ↑a < ↑b\n⊢ _root_.bit1 ↑a < _root_.bit1 ↑b\n\ncase eq\nα : Type ?u.145219\na b : PosNum\nthis : a = b\n⊢ bit1 a = bit1 b\n\ncase gt\nα : Type ?u.145219\na b : PosNum\nthis : ↑b < ↑a\n⊢ _root_.bit1 ↑b < _root_.bit1 ↑a",
"state_before": "α : Type ?u.145219\na b : PosNum\n⊢ Ordering.casesOn (cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a) →\n Ordering.casesOn (cmp (bit1 a) (bit1 b)) (↑(bit1 a) < ↑(bit1 b)) (bit1 a = bit1 b) (↑(bit1 b) < ↑(bit1 a))",
"tactic": "cases cmp a b <;> dsimp <;> intro this"
},
{
"state_after": "no goals",
"state_before": "case lt\nα : Type ?u.145219\na b : PosNum\nthis : ↑a < ↑b\n⊢ _root_.bit1 ↑a < _root_.bit1 ↑b",
"tactic": "exact Nat.succ_lt_succ (add_lt_add this this)"
},
{
"state_after": "no goals",
"state_before": "case eq\nα : Type ?u.145219\na b : PosNum\nthis : a = b\n⊢ bit1 a = bit1 b",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "case gt\nα : Type ?u.145219\na b : PosNum\nthis : ↑b < ↑a\n⊢ _root_.bit1 ↑b < _root_.bit1 ↑a",
"tactic": "exact Nat.succ_lt_succ (add_lt_add this this)"
}
] |
[
185,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
StrictMonoOn.exists_deriv_lt_slope_aux
|
[
{
"state_after": "E : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"state_before": "E : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"tactic": "have A : DifferentiableOn ℝ f (Ioo x y) := fun w wmem =>\n (differentiableAt_of_deriv_ne_zero (h w wmem)).differentiableWithinAt"
},
{
"state_after": "E : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a = (f y - f x) / (y - x)\n\ncase intro.intro.intro\nE : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"state_before": "E : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"tactic": "obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x)"
},
{
"state_after": "case intro.intro.intro\nE : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"state_before": "E : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a = (f y - f x) / (y - x)\n\ncase intro.intro.intro\nE : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"tactic": "exact exists_deriv_eq_slope f hxy hf A"
},
{
"state_after": "case intro.intro.intro.intro.intro\nE : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : ℝ\nhxb : x < b\nhba : b < a\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"state_before": "case intro.intro.intro\nE : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"tactic": "rcases nonempty_Ioo.2 hxa with ⟨b, ⟨hxb, hba⟩⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nE : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : ℝ\nhxb : x < b\nhba : b < a\n⊢ deriv f b < (f y - f x) / (y - x)",
"state_before": "case intro.intro.intro.intro.intro\nE : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : ℝ\nhxb : x < b\nhba : b < a\n⊢ ∃ a, a ∈ Ioo x y ∧ deriv f a < (f y - f x) / (y - x)",
"tactic": "refine' ⟨b, ⟨hxb, hba.trans hay⟩, _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nE : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : ℝ\nhxb : x < b\nhba : b < a\n⊢ deriv f b < deriv f a",
"state_before": "case intro.intro.intro.intro.intro\nE : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : ℝ\nhxb : x < b\nhba : b < a\n⊢ deriv f b < (f y - f x) / (y - x)",
"tactic": "rw [← ha]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nE : Type ?u.437432\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.437528\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ (w : ℝ), w ∈ Ioo x y → deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : ℝ\nhxb : x < b\nhba : b < a\n⊢ deriv f b < deriv f a",
"tactic": "exact hf'_mono ⟨hxb, hba.trans hay⟩ ⟨hxa, hay⟩ hba"
}
] |
[
1064,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1054,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
mul_self_zpow
|
[
{
"state_after": "α : Type ?u.125817\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\nb : G\nm : ℤ\n⊢ b ^ 1 * b ^ m = b ^ (m + 1)",
"state_before": "α : Type ?u.125817\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\nb : G\nm : ℤ\n⊢ b * b ^ m = b ^ (m + 1)",
"tactic": "conv_lhs =>\n congr\n rw [← zpow_one b]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.125817\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\nb : G\nm : ℤ\n⊢ b ^ 1 * b ^ m = b ^ (m + 1)",
"tactic": "rw [← zpow_add, add_comm]"
}
] |
[
229,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Order/SymmDiff.lean
|
bihimp_right_comm
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.81353\nα : Type u_1\nβ : Type ?u.81359\nπ : ι → Type ?u.81364\ninst✝ : BooleanAlgebra α\na b c d : α\n⊢ a ⇔ b ⇔ c = a ⇔ c ⇔ b",
"tactic": "simp_rw [bihimp_assoc, bihimp_comm]"
}
] |
[
637,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
637,
1
] |
Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean
|
Polynomial.exists_approx_polynomial_aux
|
[
{
"state_after": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ degree (A i₁ - A i₀) < ↑(natDegree b - d)",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ degree (A i₁ - A i₀) < ↑(natDegree b - d)",
"tactic": "have hb : b ≠ 0 := by\n rintro rfl\n specialize hA 0\n rw [degree_zero] at hA\n exact not_lt_of_le bot_le hA"
},
{
"state_after": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ degree (A i₁ - A i₀) < ↑(natDegree b - d)",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ degree (A i₁ - A i₀) < ↑(natDegree b - d)",
"tactic": "set f : Fin m.succ → Fin d → Fq := fun i j => (A i).coeff (natDegree b - j.succ)"
},
{
"state_after": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ degree (A i₁ - A i₀) < ↑(natDegree b - d)",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ degree (A i₁ - A i₀) < ↑(natDegree b - d)",
"tactic": "have : Fintype.card (Fin d → Fq) < Fintype.card (Fin m.succ) := by\n simpa using lt_of_le_of_lt hm (Nat.lt_succ_self m)"
},
{
"state_after": "case intro.intro.intro\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ degree (A i₁ - A i₀) < ↑(natDegree b - d)",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ degree (A i₁ - A i₀) < ↑(natDegree b - d)",
"tactic": "obtain ⟨i₀, i₁, i_ne, i_eq⟩ := Fintype.exists_ne_map_eq_of_card_lt f this"
},
{
"state_after": "case intro.intro.intro\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\n⊢ degree (A i₁ - A i₀) < ↑(natDegree b - d)",
"state_before": "case intro.intro.intro\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ degree (A i₁ - A i₀) < ↑(natDegree b - d)",
"tactic": "use i₀, i₁, i_ne"
},
{
"state_after": "case intro.intro.intro\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\n⊢ coeff (A i₁ - A i₀) j = 0",
"state_before": "case intro.intro.intro\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\n⊢ degree (A i₁ - A i₀) < ↑(natDegree b - d)",
"tactic": "refine' (degree_lt_iff_coeff_zero _ _).mpr fun j hj => _"
},
{
"state_after": "case pos\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj : degree b ≤ ↑j\n⊢ coeff (A i₁ - A i₀) j = 0\n\ncase neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj : ¬degree b ≤ ↑j\n⊢ coeff (A i₁ - A i₀) j = 0",
"state_before": "case intro.intro.intro\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\n⊢ coeff (A i₁ - A i₀) j = 0",
"tactic": "by_cases hbj : degree b ≤ j"
},
{
"state_after": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj : ¬degree b ≤ ↑j\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"state_before": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj : ¬degree b ≤ ↑j\n⊢ coeff (A i₁ - A i₀) j = 0",
"tactic": "rw [coeff_sub, sub_eq_zero]"
},
{
"state_after": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj : ↑j < ↑(natDegree b)\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"state_before": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj : ¬degree b ≤ ↑j\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"tactic": "rw [not_le, degree_eq_natDegree hb] at hbj"
},
{
"state_after": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"state_before": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj : ↑j < ↑(natDegree b)\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"tactic": "have hbj : j < natDegree b := (@WithBot.coe_lt_coe _ _ _).mp hbj"
},
{
"state_after": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis✝ : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj✝ : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\nhj : natDegree b - Nat.succ j < d\nthis : j = natDegree b - Nat.succ (natDegree b - Nat.succ j)\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"state_before": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj✝ : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\nhj : natDegree b - Nat.succ j < d\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"tactic": "have : j = b.natDegree - (natDegree b - j.succ).succ := by\n rw [← Nat.succ_sub hbj, Nat.succ_sub_succ, tsub_tsub_cancel_of_le hbj.le]"
},
{
"state_after": "no goals",
"state_before": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis✝ : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj✝ : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\nhj : natDegree b - Nat.succ j < d\nthis : j = natDegree b - Nat.succ (natDegree b - Nat.succ j)\n⊢ coeff (A i₁) j = coeff (A i₀) j",
"tactic": "convert congr_fun i_eq.symm ⟨natDegree b - j.succ, hj⟩"
},
{
"state_after": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree 0\n⊢ False",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\n⊢ b ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nA : Fin (Nat.succ m) → Fq[X]\nhA : degree (A 0) < degree 0\n⊢ False",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree 0\n⊢ False",
"tactic": "specialize hA 0"
},
{
"state_after": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nA : Fin (Nat.succ m) → Fq[X]\nhA : degree (A 0) < ⊥\n⊢ False",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nA : Fin (Nat.succ m) → Fq[X]\nhA : degree (A 0) < degree 0\n⊢ False",
"tactic": "rw [degree_zero] at hA"
},
{
"state_after": "no goals",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nA : Fin (Nat.succ m) → Fq[X]\nhA : degree (A 0) < ⊥\n⊢ False",
"tactic": "exact not_lt_of_le bot_le hA"
},
{
"state_after": "no goals",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\n⊢ Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))",
"tactic": "simpa using lt_of_le_of_lt hm (Nat.lt_succ_self m)"
},
{
"state_after": "case pos\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj : degree b ≤ ↑j\n⊢ degree (A i₁ - A i₀) < degree b",
"state_before": "case pos\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj : degree b ≤ ↑j\n⊢ coeff (A i₁ - A i₀) j = 0",
"tactic": "refine' coeff_eq_zero_of_degree_lt (lt_of_lt_of_le _ hbj)"
},
{
"state_after": "no goals",
"state_before": "case pos\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj : degree b ≤ ↑j\n⊢ degree (A i₁ - A i₀) < degree b",
"tactic": "exact lt_of_le_of_lt (degree_sub_le _ _) (max_lt (hA _) (hA _))"
},
{
"state_after": "case pos\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\nhd : natDegree b < d\n⊢ natDegree b - Nat.succ j < d\n\ncase neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\nhd : ¬natDegree b < d\n⊢ natDegree b - Nat.succ j < d",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\n⊢ natDegree b - Nat.succ j < d",
"tactic": "by_cases hd : natDegree b < d"
},
{
"state_after": "no goals",
"state_before": "case pos\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\nhd : natDegree b < d\n⊢ natDegree b - Nat.succ j < d",
"tactic": "exact lt_of_le_of_lt tsub_le_self hd"
},
{
"state_after": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\nhd : d ≤ natDegree b\n⊢ natDegree b - Nat.succ j < d",
"state_before": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\nhd : ¬natDegree b < d\n⊢ natDegree b - Nat.succ j < d",
"tactic": "rw [not_lt] at hd"
},
{
"state_after": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis✝ : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\nhd : d ≤ natDegree b\nthis : natDegree b - d < Nat.succ j\n⊢ natDegree b - Nat.succ j < d",
"state_before": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\nhd : d ≤ natDegree b\n⊢ natDegree b - Nat.succ j < d",
"tactic": "have := lt_of_le_of_lt hj (Nat.lt_succ_self j)"
},
{
"state_after": "no goals",
"state_before": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis✝ : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\nhd : d ≤ natDegree b\nthis : natDegree b - d < Nat.succ j\n⊢ natDegree b - Nat.succ j < d",
"tactic": "rwa [tsub_lt_iff_tsub_lt hd hbj] at this"
},
{
"state_after": "no goals",
"state_before": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin (Nat.succ m) → Fq[X]\nhA : ∀ (i : Fin (Nat.succ m)), degree (A i) < degree b\nhb : b ≠ 0\nf : Fin (Nat.succ m) → Fin d → Fq := fun i j => coeff (A i) (natDegree b - ↑(Fin.succ j))\nthis : Fintype.card (Fin d → Fq) < Fintype.card (Fin (Nat.succ m))\ni₀ i₁ : Fin (Nat.succ m)\ni_ne : i₀ ≠ i₁\ni_eq : f i₀ = f i₁\nj : ℕ\nhj✝ : natDegree b - d ≤ j\nhbj✝ : ↑j < ↑(natDegree b)\nhbj : j < natDegree b\nhj : natDegree b - Nat.succ j < d\n⊢ j = natDegree b - Nat.succ (natDegree b - Nat.succ j)",
"tactic": "rw [← Nat.succ_sub hbj, Nat.succ_sub_succ, tsub_tsub_cancel_of_le hbj.le]"
}
] |
[
101,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/CategoryTheory/Functor/Const.lean
|
CategoryTheory.Functor.const.unop_functor_op_obj_map
|
[] |
[
87,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/LinearAlgebra/Projection.lean
|
LinearMap.IsProj.eq_conj_prod_map'
|
[
{
"state_after": "R : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.451177\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.451693\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.452656\ninst✝² : Semiring S\nM : Type ?u.452662\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] E\nh : IsProj p f\n⊢ comp f ↑(prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f))) =\n comp (↑(prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))) (prodMap id 0)",
"state_before": "R : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.451177\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.451693\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.452656\ninst✝² : Semiring S\nM : Type ?u.452662\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] E\nh : IsProj p f\n⊢ f =\n comp (↑(prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f))))\n (comp (prodMap id 0) ↑(LinearEquiv.symm (prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))))",
"tactic": "rw [← LinearMap.comp_assoc, LinearEquiv.eq_comp_toLinearMap_symm]"
},
{
"state_after": "case hl.h\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.451177\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.451693\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.452656\ninst✝² : Semiring S\nM : Type ?u.452662\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] E\nh : IsProj p f\nx : { x // x ∈ p }\n⊢ ↑(comp (comp f ↑(prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))) (inl R { x // x ∈ p } { x // x ∈ ker f })) x =\n ↑(comp (comp (↑(prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))) (prodMap id 0))\n (inl R { x // x ∈ p } { x // x ∈ ker f }))\n x\n\ncase hr.h\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.451177\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.451693\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.452656\ninst✝² : Semiring S\nM : Type ?u.452662\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] E\nh : IsProj p f\nx : { x // x ∈ ker f }\n⊢ ↑(comp (comp f ↑(prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))) (inr R { x // x ∈ p } { x // x ∈ ker f })) x =\n ↑(comp (comp (↑(prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))) (prodMap id 0))\n (inr R { x // x ∈ p } { x // x ∈ ker f }))\n x",
"state_before": "R : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.451177\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.451693\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.452656\ninst✝² : Semiring S\nM : Type ?u.452662\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] E\nh : IsProj p f\n⊢ comp f ↑(prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f))) =\n comp (↑(prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))) (prodMap id 0)",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case hl.h\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.451177\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.451693\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.452656\ninst✝² : Semiring S\nM : Type ?u.452662\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] E\nh : IsProj p f\nx : { x // x ∈ p }\n⊢ ↑(comp (comp f ↑(prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))) (inl R { x // x ∈ p } { x // x ∈ ker f })) x =\n ↑(comp (comp (↑(prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))) (prodMap id 0))\n (inl R { x // x ∈ p } { x // x ∈ ker f }))\n x",
"tactic": "simp only [coe_prodEquivOfIsCompl, comp_apply, coe_inl, coprod_apply, coeSubtype,\n _root_.map_zero, add_zero, h.map_id x x.2, prodMap_apply, id_apply]"
},
{
"state_after": "no goals",
"state_before": "case hr.h\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.451177\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.451693\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.452656\ninst✝² : Semiring S\nM : Type ?u.452662\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] E\nh : IsProj p f\nx : { x // x ∈ ker f }\n⊢ ↑(comp (comp f ↑(prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))) (inr R { x // x ∈ p } { x // x ∈ ker f })) x =\n ↑(comp (comp (↑(prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))) (prodMap id 0))\n (inr R { x // x ∈ p } { x // x ∈ ker f }))\n x",
"tactic": "simp only [coe_prodEquivOfIsCompl, comp_apply, coe_inr, coprod_apply, _root_.map_zero,\n coeSubtype, zero_add, map_coe_ker, prodMap_apply, zero_apply, add_zero]"
}
] |
[
441,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Topology/ContinuousFunction/Ordered.lean
|
ContinuousMap.le_def
|
[] |
[
60,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Computability/Reduce.lean
|
ComputablePred.computable_of_oneOneReducible
|
[] |
[
147,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.preimage_image_preimage
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.77955\nι : Sort ?u.77958\nι' : Sort ?u.77961\nf✝ : ι → α\ns✝ t : Set α\nf : α → β\ns : Set β\n⊢ f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s",
"tactic": "rw [image_preimage_eq_inter_range, preimage_inter_range]"
}
] |
[
852,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
851,
1
] |
Mathlib/LinearAlgebra/BilinearMap.lean
|
LinearMap.lsmul_apply
|
[] |
[
398,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
398,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
MeasureTheory.Integrable.indicator
|
[] |
[
273,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.zero_imK
|
[] |
[
179,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
9
] |
Mathlib/LinearAlgebra/Matrix/ZPow.lean
|
Matrix.zero_zpow
|
[
{
"state_after": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\nh : ↑n ≠ 0\n⊢ 0 < n",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\nh : ↑n ≠ 0\n⊢ 0 ^ ↑n = 0",
"tactic": "rw [zpow_ofNat, zero_pow]"
},
{
"state_after": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\nh : ↑n ≠ 0\n⊢ n ≠ 0",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\nh : ↑n ≠ 0\n⊢ 0 < n",
"tactic": "refine' lt_of_le_of_ne n.zero_le (Ne.symm _)"
},
{
"state_after": "no goals",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\nh : ↑n ≠ 0\n⊢ n ≠ 0",
"tactic": "simpa using h"
},
{
"state_after": "no goals",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\nx✝ : -[n+1] ≠ 0\n⊢ 0 ^ -[n+1] = 0",
"tactic": "simp [zero_pow n.zero_lt_succ]"
}
] |
[
92,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
isClopen_univ
|
[] |
[
1564,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1564,
9
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isLittleO_sup
|
[] |
[
659,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
isUnit_exp_of_mem_ball
|
[] |
[
304,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Deprecated/Subfield.lean
|
isSubfield_iUnion_of_directed
|
[] |
[
158,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/CategoryTheory/IsConnected.lean
|
CategoryTheory.equiv_relation
|
[
{
"state_after": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : IsConnected J\nr : J → J → Prop\nhr : _root_.Equivalence r\nh : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → r j₁ j₂\nz : ∀ (j : J), r (Classical.arbitrary J) j\n⊢ ∀ (j₁ j₂ : J), r j₁ j₂",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : IsConnected J\nr : J → J → Prop\nhr : _root_.Equivalence r\nh : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → r j₁ j₂\n⊢ ∀ (j₁ j₂ : J), r j₁ j₂",
"tactic": "have z : ∀ j : J, r (Classical.arbitrary J) j :=\n induct_on_objects (fun k => r (Classical.arbitrary J) k) (hr.1 (Classical.arbitrary J))\n fun f => ⟨fun t => hr.3 t (h f), fun t => hr.3 t (hr.2 (h f))⟩"
},
{
"state_after": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : IsConnected J\nr : J → J → Prop\nhr : _root_.Equivalence r\nh : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → r j₁ j₂\nz : ∀ (j : J), r (Classical.arbitrary J) j\nj₁✝ j₂✝ : J\n⊢ r j₁✝ j₂✝",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : IsConnected J\nr : J → J → Prop\nhr : _root_.Equivalence r\nh : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → r j₁ j₂\nz : ∀ (j : J), r (Classical.arbitrary J) j\n⊢ ∀ (j₁ j₂ : J), r j₁ j₂",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : IsConnected J\nr : J → J → Prop\nhr : _root_.Equivalence r\nh : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → r j₁ j₂\nz : ∀ (j : J), r (Classical.arbitrary J) j\nj₁✝ j₂✝ : J\n⊢ r j₁✝ j₂✝",
"tactic": "apply hr.3 (hr.2 (z _)) (z _)"
}
] |
[
306,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Order/Ideal.lean
|
Order.Ideal.coe_toLowerSet
|
[] |
[
125,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Data/MvPolynomial/Equiv.lean
|
MvPolynomial.sumToIter_Xr
|
[] |
[
184,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Order/FixedPoints.lean
|
OrderHom.map_le_lfp
|
[] |
[
76,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderHom.le_def
|
[] |
[
309,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
308,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.Lifts.eq_of_le
|
[] |
[
1005,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1004,
1
] |
Mathlib/Algebra/Order/Ring/WithTop.lean
|
WithTop.mul_def
|
[] |
[
41,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Data/Bool/Count.lean
|
List.Chain'.length_sub_one_le_two_mul_count_bool
|
[
{
"state_after": "l : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) l\nb : Bool\n⊢ length l - 1 ≤ if Even (length l) then length l else if (some b == head? l) = true then length l + 1 else length l - 1",
"state_before": "l : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) l\nb : Bool\n⊢ length l - 1 ≤ 2 * count b l",
"tactic": "rw [hl.two_mul_count_bool_eq_ite]"
},
{
"state_after": "no goals",
"state_before": "l : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) l\nb : Bool\n⊢ length l - 1 ≤ if Even (length l) then length l else if (some b == head? l) = true then length l + 1 else length l - 1",
"tactic": "split_ifs <;> simp [le_tsub_add, Nat.le_succ_of_le]"
}
] |
[
126,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
StrictAnti.ite
|
[] |
[
624,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
11
] |
Mathlib/Order/Zorn.lean
|
zorn_subset
|
[] |
[
192,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderIso.trans_refl
|
[
{
"state_after": "case h.h\nF : Type ?u.83017\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.83026\nδ : Type ?u.83029\ninst✝² : LE α\ninst✝¹ : LE β\ninst✝ : LE γ\ne : α ≃o β\nx : α\n⊢ ↑(trans e (refl β)) x = ↑e x",
"state_before": "F : Type ?u.83017\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.83026\nδ : Type ?u.83029\ninst✝² : LE α\ninst✝¹ : LE β\ninst✝ : LE γ\ne : α ≃o β\n⊢ trans e (refl β) = e",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h.h\nF : Type ?u.83017\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.83026\nδ : Type ?u.83029\ninst✝² : LE α\ninst✝¹ : LE β\ninst✝ : LE γ\ne : α ≃o β\nx : α\n⊢ ↑(trans e (refl β)) x = ↑e x",
"tactic": "rfl"
}
] |
[
918,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
916,
1
] |
Mathlib/GroupTheory/Solvable.lean
|
derivedSeries_le_map_derivedSeries
|
[
{
"state_after": "case zero\nG : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\nhf : Function.Surjective ↑f\n⊢ derivedSeries G' Nat.zero ≤ map f (derivedSeries G Nat.zero)\n\ncase succ\nG : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\nhf : Function.Surjective ↑f\nn : ℕ\nih : derivedSeries G' n ≤ map f (derivedSeries G n)\n⊢ derivedSeries G' (Nat.succ n) ≤ map f (derivedSeries G (Nat.succ n))",
"state_before": "G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\nhf : Function.Surjective ↑f\nn : ℕ\n⊢ derivedSeries G' n ≤ map f (derivedSeries G n)",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nG : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\nhf : Function.Surjective ↑f\n⊢ derivedSeries G' Nat.zero ≤ map f (derivedSeries G Nat.zero)",
"tactic": "exact (map_top_of_surjective f hf).ge"
},
{
"state_after": "no goals",
"state_before": "case succ\nG : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\nhf : Function.Surjective ↑f\nn : ℕ\nih : derivedSeries G' n ≤ map f (derivedSeries G n)\n⊢ derivedSeries G' (Nat.succ n) ≤ map f (derivedSeries G (Nat.succ n))",
"tactic": "exact commutator_le_map_commutator ih ih"
}
] |
[
92,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
inv_nonpos
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.6857\nα : Type u_1\nβ : Type ?u.6863\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\n⊢ a⁻¹ ≤ 0 ↔ a ≤ 0",
"tactic": "simp only [← not_lt, inv_pos]"
}
] |
[
72,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Algebra/Order/Invertible.lean
|
invOf_pos
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedSemiring α\na : α\ninst✝ : Invertible a\n⊢ 0 < a * ⅟a",
"tactic": "simp only [mul_invOf_self, zero_lt_one]"
}
] |
[
24,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
22,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
NNRat.coe_multiset_sum
|
[] |
[
266,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.mem_smul_set_iff_inv_smul_mem
|
[] |
[
886,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
885,
1
] |
Mathlib/Topology/Support.lean
|
HasCompactMulSupport.comp_left
|
[] |
[
205,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.L1.ofReal_norm_sub_eq_lintegral
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1306428\nδ : Type ?u.1306431\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ (∫⁻ (x : α), edist (↑↑(f - g) x) 0 ∂μ) = ∫⁻ (x : α), edist (↑↑f x - ↑↑g x) 0 ∂μ",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1306428\nδ : Type ?u.1306431\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ ENNReal.ofReal ‖f - g‖ = ∫⁻ (x : α), ↑‖↑↑f x - ↑↑g x‖₊ ∂μ",
"tactic": "simp_rw [ofReal_norm_eq_lintegral, ← edist_eq_coe_nnnorm]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1306428\nδ : Type ?u.1306431\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ (fun a => edist (↑↑(f - g) a) 0) =ᵐ[μ] fun a => edist (↑↑f a - ↑↑g a) 0",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1306428\nδ : Type ?u.1306431\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ (∫⁻ (x : α), edist (↑↑(f - g) x) 0 ∂μ) = ∫⁻ (x : α), edist (↑↑f x - ↑↑g x) 0 ∂μ",
"tactic": "apply lintegral_congr_ae"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1306428\nδ : Type ?u.1306431\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\na✝ : α\nha : ↑↑(f - g) a✝ = (↑↑f - ↑↑g) a✝\n⊢ edist (↑↑(f - g) a✝) 0 = edist (↑↑f a✝ - ↑↑g a✝) 0",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1306428\nδ : Type ?u.1306431\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ (fun a => edist (↑↑(f - g) a) 0) =ᵐ[μ] fun a => edist (↑↑f a - ↑↑g a) 0",
"tactic": "filter_upwards [Lp.coeFn_sub f g] with _ ha"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1306428\nδ : Type ?u.1306431\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\na✝ : α\nha : ↑↑(f - g) a✝ = (↑↑f - ↑↑g) a✝\n⊢ edist (↑↑(f - g) a✝) 0 = edist (↑↑f a✝ - ↑↑g a✝) 0",
"tactic": "simp only [ha, Pi.sub_apply]"
}
] |
[
1338,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1333,
1
] |
Mathlib/Analysis/Calculus/Deriv/Linear.lean
|
LinearMap.hasStrictDerivAt
|
[] |
[
91,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
11
] |
Mathlib/Algebra/Order/Rearrangement.lean
|
MonovaryOn.sum_comp_perm_smul_le_sum_smul
|
[
{
"state_after": "case h.e'_3\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | ↑σ x ≠ x} ⊆ ↑s\n⊢ ∑ i in s, f (↑σ i) • g i = ∑ i in s, f i • g (↑σ⁻¹ i)",
"state_before": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | ↑σ x ≠ x} ⊆ ↑s\n⊢ ∑ i in s, f (↑σ i) • g i ≤ ∑ i in s, f i • g i",
"tactic": "convert hfg.sum_smul_comp_perm_le_sum_smul\n (show { x | σ⁻¹ x ≠ x } ⊆ s by simp only [set_support_inv_eq, hσ]) using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | ↑σ x ≠ x} ⊆ ↑s\n⊢ ∑ i in s, f (↑σ i) • g i = ∑ i in s, f i • g (↑σ⁻¹ i)",
"tactic": "exact σ.sum_comp' s (fun i j ↦ f i • g j) hσ"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | ↑σ x ≠ x} ⊆ ↑s\n⊢ {x | ↑σ⁻¹ x ≠ x} ⊆ ↑s",
"tactic": "simp only [set_support_inv_eq, hσ]"
}
] |
[
158,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.Lp.simpleFunc.toLp_toSimpleFunc
|
[] |
[
635,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
633,
1
] |
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