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/Int/Bitwise.lean
|
Int.bit_negSucc
|
[
{
"state_after": "b : Bool\nn : ℕ\n⊢ (2 * -[n+1] + bif b then 1 else 0) = -[2 * n + bif !b then 1 else 0+1]",
"state_before": "b : Bool\nn : ℕ\n⊢ bit b -[n+1] = -[Nat.bit (!b) n+1]",
"tactic": "rw [bit_val, Nat.bit_val]"
},
{
"state_after": "no goals",
"state_before": "b : Bool\nn : ℕ\n⊢ (2 * -[n+1] + bif b then 1 else 0) = -[2 * n + bif !b then 1 else 0+1]",
"tactic": "cases b <;> rfl"
}
] |
[
163,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.map_id
|
[] |
[
463,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
Bool.inf_eq_band
|
[] |
[
795,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
794,
1
] |
Mathlib/Order/Partition/Finpartition.lean
|
Finpartition.parts_bot
|
[] |
[
497,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.toAddSubsemigroup_closure
|
[] |
[
110,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.assoc_tmul
|
[] |
[
710,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
708,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.val_injective
|
[
{
"state_after": "case zero\ninst✝ : NeZero Nat.zero\n⊢ Injective val\n\ncase succ\nn✝ : ℕ\ninst✝ : NeZero (Nat.succ n✝)\n⊢ Injective val",
"state_before": "n : ℕ\ninst✝ : NeZero n\n⊢ Injective val",
"tactic": "cases n"
},
{
"state_after": "case succ\nn✝ : ℕ\ninst✝ : NeZero (Nat.succ n✝)\na b : ZMod (Nat.succ n✝)\nh : val a = val b\n⊢ a = b",
"state_before": "case succ\nn✝ : ℕ\ninst✝ : NeZero (Nat.succ n✝)\n⊢ Injective val",
"tactic": "intro a b h"
},
{
"state_after": "case succ\nn✝ : ℕ\ninst✝ : NeZero (Nat.succ n✝)\na b : ZMod (Nat.succ n✝)\nh : val a = val b\n⊢ a = b",
"state_before": "case succ\nn✝ : ℕ\ninst✝ : NeZero (Nat.succ n✝)\na b : ZMod (Nat.succ n✝)\nh : val a = val b\n⊢ a = b",
"tactic": "dsimp [ZMod]"
},
{
"state_after": "case succ.h\nn✝ : ℕ\ninst✝ : NeZero (Nat.succ n✝)\na b : ZMod (Nat.succ n✝)\nh : val a = val b\n⊢ ↑a = ↑b",
"state_before": "case succ\nn✝ : ℕ\ninst✝ : NeZero (Nat.succ n✝)\na b : ZMod (Nat.succ n✝)\nh : val a = val b\n⊢ a = b",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case succ.h\nn✝ : ℕ\ninst✝ : NeZero (Nat.succ n✝)\na b : ZMod (Nat.succ n✝)\nh : val a = val b\n⊢ ↑a = ↑b",
"tactic": "exact h"
},
{
"state_after": "no goals",
"state_before": "case zero\ninst✝ : NeZero Nat.zero\n⊢ Injective val",
"tactic": "cases NeZero.ne 0 rfl"
}
] |
[
585,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
579,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
RingHom.ext_iff
|
[] |
[
541,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
1
] |
Mathlib/Logic/Relation.lean
|
Relation.reflexive_reflTransGen
|
[] |
[
545,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
545,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_union
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1027026\nγ : Type ?u.1027029\nδ : Type ?u.1027032\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nA B : Set α\nhB : MeasurableSet B\nhAB : Disjoint A B\n⊢ (∫⁻ (a : α) in A ∪ B, f a ∂μ) = (∫⁻ (a : α) in A, f a ∂μ) + ∫⁻ (a : α) in B, f a ∂μ",
"tactic": "rw [restrict_union hAB hB, lintegral_add_measure]"
}
] |
[
1229,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1227,
1
] |
Mathlib/Data/Vector/Mem.lean
|
Vector.mem_map_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nn : ℕ\na a' : α\nb : β\nv : Vector α n\nf : α → β\n⊢ b ∈ toList (map f v) ↔ ∃ a, a ∈ toList v ∧ f a = b",
"tactic": "rw [Vector.toList_map, List.mem_map]"
}
] |
[
81,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.mul_empty
|
[] |
[
363,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/Topology/Algebra/OpenSubgroup.lean
|
OpenSubgroup.isClopen
|
[] |
[
185,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioi_subset_Ici_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.72950\ninst✝¹ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\ninst✝ : DenselyOrdered α\nh : Ioi b ⊆ Ici a\n⊢ a ≤ b",
"state_before": "α : Type u_1\nβ : Type ?u.72950\ninst✝¹ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\ninst✝ : DenselyOrdered α\n⊢ Ioi b ⊆ Ici a ↔ a ≤ b",
"tactic": "refine' ⟨fun h => _, fun h => Ioi_subset_Ici h⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.72950\ninst✝¹ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\ninst✝ : DenselyOrdered α\nh : Ioi b ⊆ Ici a\nba : ¬a ≤ b\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.72950\ninst✝¹ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\ninst✝ : DenselyOrdered α\nh : Ioi b ⊆ Ici a\n⊢ a ≤ b",
"tactic": "by_contra ba"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.72950\ninst✝¹ : LinearOrder α\na a₁ a₂ b b₁ b₂ c✝ d : α\ninst✝ : DenselyOrdered α\nh : Ioi b ⊆ Ici a\nba : ¬a ≤ b\nc : α\nbc : b < c\nca : c < a\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.72950\ninst✝¹ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\ninst✝ : DenselyOrdered α\nh : Ioi b ⊆ Ici a\nba : ¬a ≤ b\n⊢ False",
"tactic": "obtain ⟨c, bc, ca⟩ : ∃ c, b < c ∧ c < a := exists_between (not_le.mp ba)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.72950\ninst✝¹ : LinearOrder α\na a₁ a₂ b b₁ b₂ c✝ d : α\ninst✝ : DenselyOrdered α\nh : Ioi b ⊆ Ici a\nba : ¬a ≤ b\nc : α\nbc : b < c\nca : c < a\n⊢ False",
"tactic": "exact lt_irrefl _ (ca.trans_le (h bc))"
}
] |
[
1181,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1177,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.diagonal_updateRow_single
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.1233103\nm : Type ?u.1233106\nn : Type u_1\no : Type ?u.1233112\nm' : o → Type ?u.1233117\nn' : o → Type ?u.1233122\nR : Type ?u.1233125\nS : Type ?u.1233128\nα : Type v\nβ : Type w\nγ : Type ?u.1233135\nM : Matrix m n α\ni✝ : m\nj : n\nb : n → α\nc : m → α\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nv : n → α\ni : n\nx : α\n⊢ updateRow (diagonal v) i (Pi.single i x) = diagonal (Function.update v i x)",
"tactic": "rw [← diagonal_transpose, updateRow_transpose, diagonal_updateColumn_single, diagonal_transpose]"
}
] |
[
2868,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2866,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.dist_map_zero_lt_of_semiconjBy
|
[] |
[
537,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.floor_le_sub_one_iff
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.123144\nα : Type u_1\nβ : Type ?u.123150\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\n⊢ ⌊a⌋ ≤ z - 1 ↔ a < ↑z",
"tactic": "rw [← floor_lt, le_sub_one_iff]"
}
] |
[
668,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
668,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
eq_of_one_div_eq_one_div
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.27621\nG : Type ?u.27624\ninst✝ : DivisionMonoid α\na b c : α\nh : 1 / a = 1 / b\n⊢ a = b",
"tactic": "rw [← one_div_one_div a, h, one_div_one_div]"
}
] |
[
455,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.add_domain
|
[] |
[
471,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.one_succAbove_zero
|
[
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\n⊢ ↑(succAbove 1) 0 = 0",
"tactic": "rfl"
}
] |
[
2231,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2230,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.mk_le_mk
|
[] |
[
98,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Analysis/BoxIntegral/Box/Basic.lean
|
BoxIntegral.Box.monotone_face
|
[] |
[
413,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
412,
1
] |
Mathlib/ModelTheory/Substructures.lean
|
FirstOrder.Language.Substructure.map_iSup_comap_of_surjective
|
[] |
[
647,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
645,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean
|
CategoryTheory.Limits.Multicofork.ofSigmaCofork_ι_app_left
|
[] |
[
657,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
654,
1
] |
Mathlib/Topology/Bornology/Basic.lean
|
Bornology.ext_iff_isBounded
|
[
{
"state_after": "case h_cobounded.a\nι : Type ?u.6300\nα : Type u_1\nβ : Type ?u.6306\nt t' : Bornology α\nh : ∀ (s : Set α), IsBounded s ↔ IsBounded s\ns : Set α\n⊢ s ∈ cobounded α ↔ s ∈ cobounded α",
"state_before": "ι : Type ?u.6300\nα : Type u_1\nβ : Type ?u.6306\nt t' : Bornology α\nh : ∀ (s : Set α), IsBounded s ↔ IsBounded s\n⊢ t = t'",
"tactic": "ext s"
},
{
"state_after": "no goals",
"state_before": "case h_cobounded.a\nι : Type ?u.6300\nα : Type u_1\nβ : Type ?u.6306\nt t' : Bornology α\nh : ∀ (s : Set α), IsBounded s ↔ IsBounded s\ns : Set α\n⊢ s ∈ cobounded α ↔ s ∈ cobounded α",
"tactic": "simpa [@isBounded_def _ t, isBounded_def, compl_compl] using h (sᶜ)"
}
] |
[
242,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
lp.sum_rpow_le_of_tendsto
|
[
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\n⊢ ∑ i in s, ‖f i‖ ^ ENNReal.toReal p ≤ C ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\n⊢ ∑ i in s, ‖f i‖ ^ ENNReal.toReal p ≤ C ^ ENNReal.toReal p",
"tactic": "have hp' : p ≠ 0 := (zero_lt_one.trans_le _i.elim).ne'"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\n⊢ ∑ i in s, ‖f i‖ ^ ENNReal.toReal p ≤ C ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\n⊢ ∑ i in s, ‖f i‖ ^ ENNReal.toReal p ≤ C ^ ENNReal.toReal p",
"tactic": "have hp'' : 0 < p.toReal := ENNReal.toReal_pos hp' hp"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\n⊢ ∑ i in s, ‖f i‖ ^ ENNReal.toReal p ≤ C ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\n⊢ ∑ i in s, ‖f i‖ ^ ENNReal.toReal p ≤ C ^ ENNReal.toReal p",
"tactic": "let G : (∀ a, E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ p.toReal"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\nhG : Continuous G\n⊢ ∑ i in s, ‖f i‖ ^ ENNReal.toReal p ≤ C ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\n⊢ ∑ i in s, ‖f i‖ ^ ENNReal.toReal p ≤ C ^ ENNReal.toReal p",
"tactic": "have hG : Continuous G := by\n refine' continuous_finset_sum s _\n intro a _\n have : Continuous fun f : ∀ a, E a => f a := continuous_apply a\n exact this.norm.rpow_const fun _ => Or.inr hp''.le"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\nhG : Continuous G\n⊢ ∀ᶠ (c : ι) in l, (G ∘ id fun i => ↑(F i)) c ≤ C ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\nhG : Continuous G\n⊢ ∑ i in s, ‖f i‖ ^ ENNReal.toReal p ≤ C ^ ENNReal.toReal p",
"tactic": "refine' le_of_tendsto (hG.continuousAt.tendsto.comp hf) _"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\nhG : Continuous G\n⊢ ∀ (x : ι), ‖F x‖ ≤ C → (G ∘ id fun i => ↑(F i)) x ≤ C ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\nhG : Continuous G\n⊢ ∀ᶠ (c : ι) in l, (G ∘ id fun i => ↑(F i)) c ≤ C ^ ENNReal.toReal p",
"tactic": "refine' hCF.mono _"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\nhG : Continuous G\nk : ι\nhCFk : ‖F k‖ ≤ C\n⊢ (G ∘ id fun i => ↑(F i)) k ≤ C ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\nhG : Continuous G\n⊢ ∀ (x : ι), ‖F x‖ ≤ C → (G ∘ id fun i => ↑(F i)) x ≤ C ^ ENNReal.toReal p",
"tactic": "intro k hCFk"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\nhG : Continuous G\nk : ι\nhCFk : ‖F k‖ ≤ C\n⊢ ‖F k‖ ^ ENNReal.toReal p ≤ C ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\nhG : Continuous G\nk : ι\nhCFk : ‖F k‖ ≤ C\n⊢ (G ∘ id fun i => ↑(F i)) k ≤ C ^ ENNReal.toReal p",
"tactic": "refine' (lp.sum_rpow_le_norm_rpow hp'' (F k) s).trans _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\nhG : Continuous G\nk : ι\nhCFk : ‖F k‖ ≤ C\n⊢ ‖F k‖ ^ ENNReal.toReal p ≤ C ^ ENNReal.toReal p",
"tactic": "exact Real.rpow_le_rpow (norm_nonneg _) hCFk hp''.le"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\n⊢ ∀ (i : α), i ∈ s → Continuous fun f => ‖f i‖ ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\n⊢ Continuous G",
"tactic": "refine' continuous_finset_sum s _"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\na : α\na✝ : a ∈ s\n⊢ Continuous fun f => ‖f a‖ ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\n⊢ ∀ (i : α), i ∈ s → Continuous fun f => ‖f i‖ ^ ENNReal.toReal p",
"tactic": "intro a _"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\na : α\na✝ : a ∈ s\nthis : Continuous fun f => f a\n⊢ Continuous fun f => ‖f a‖ ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\na : α\na✝ : a ∈ s\n⊢ Continuous fun f => ‖f a‖ ^ ENNReal.toReal p",
"tactic": "have : Continuous fun f : ∀ a, E a => f a := continuous_apply a"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nhp : p ≠ ⊤\nC : ℝ\nF : ι → { x // x ∈ lp E p }\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : α) → E a) → ℝ := fun f => ∑ a in s, ‖f a‖ ^ ENNReal.toReal p\na : α\na✝ : a ∈ s\nthis : Continuous fun f => f a\n⊢ Continuous fun f => ‖f a‖ ^ ENNReal.toReal p",
"tactic": "exact this.norm.rpow_const fun _ => Or.inr hp''.le"
}
] |
[
1156,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1141,
1
] |
Mathlib/Topology/UniformSpace/Completion.lean
|
UniformSpace.Completion.uniformContinuous_completionSeparationQuotientEquiv
|
[] |
[
661,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
659,
1
] |
Mathlib/Analysis/SpecialFunctions/Exp.lean
|
Real.tendsto_exp_atTop
|
[
{
"state_after": "α : Type ?u.20985\nx y z : ℝ\nl : Filter α\nA : Tendsto (fun x => x + 1) atTop atTop\n⊢ Tendsto exp atTop atTop",
"state_before": "α : Type ?u.20985\nx y z : ℝ\nl : Filter α\n⊢ Tendsto exp atTop atTop",
"tactic": "have A : Tendsto (fun x : ℝ => x + 1) atTop atTop :=\n tendsto_atTop_add_const_right atTop 1 tendsto_id"
},
{
"state_after": "α : Type ?u.20985\nx y z : ℝ\nl : Filter α\nA : Tendsto (fun x => x + 1) atTop atTop\nB : ∀ᶠ (x : ℝ) in atTop, x + 1 ≤ exp x\n⊢ Tendsto exp atTop atTop",
"state_before": "α : Type ?u.20985\nx y z : ℝ\nl : Filter α\nA : Tendsto (fun x => x + 1) atTop atTop\n⊢ Tendsto exp atTop atTop",
"tactic": "have B : ∀ᶠ x in atTop, x + 1 ≤ exp x := eventually_atTop.2 ⟨0, fun x _ => add_one_le_exp x⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.20985\nx y z : ℝ\nl : Filter α\nA : Tendsto (fun x => x + 1) atTop atTop\nB : ∀ᶠ (x : ℝ) in atTop, x + 1 ≤ exp x\n⊢ Tendsto exp atTop atTop",
"tactic": "exact tendsto_atTop_mono' atTop B A"
}
] |
[
180,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/CategoryTheory/Idempotents/Karoubi.lean
|
CategoryTheory.Idempotents.Karoubi.comp_p
|
[
{
"state_after": "no goals",
"state_before": "C : Type u_1\ninst✝ : Category C\nP Q : Karoubi C\nf : Hom P Q\n⊢ f.f ≫ Q.p = f.f",
"tactic": "rw [f.comm, assoc, assoc, Q.idem]"
}
] |
[
93,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Analysis/Convex/StrictConvexBetween.lean
|
Wbtw.dist_le_max_dist
|
[
{
"state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhp₁ : p₂ = p₁\n⊢ dist p₂ p ≤ max (dist p₁ p) (dist p₃ p)\n\ncase neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhp₁ : ¬p₂ = p₁\n⊢ dist p₂ p ≤ max (dist p₁ p) (dist p₃ p)",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃\n⊢ dist p₂ p ≤ max (dist p₁ p) (dist p₃ p)",
"tactic": "by_cases hp₁ : p₂ = p₁"
},
{
"state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhp₁ : ¬p₂ = p₁\nhp₃ : p₂ = p₃\n⊢ dist p₂ p ≤ max (dist p₁ p) (dist p₃ p)\n\ncase neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhp₁ : ¬p₂ = p₁\nhp₃ : ¬p₂ = p₃\n⊢ dist p₂ p ≤ max (dist p₁ p) (dist p₃ p)",
"state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhp₁ : ¬p₂ = p₁\n⊢ dist p₂ p ≤ max (dist p₁ p) (dist p₃ p)",
"tactic": "by_cases hp₃ : p₂ = p₃"
},
{
"state_after": "case neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhp₁ : ¬p₂ = p₁\nhp₃ : ¬p₂ = p₃\nhs : Sbtw ℝ p₁ p₂ p₃\n⊢ dist p₂ p ≤ max (dist p₁ p) (dist p₃ p)",
"state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhp₁ : ¬p₂ = p₁\nhp₃ : ¬p₂ = p₃\n⊢ dist p₂ p ≤ max (dist p₁ p) (dist p₃ p)",
"tactic": "have hs : Sbtw ℝ p₁ p₂ p₃ := ⟨h, hp₁, hp₃⟩"
},
{
"state_after": "no goals",
"state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhp₁ : ¬p₂ = p₁\nhp₃ : ¬p₂ = p₃\nhs : Sbtw ℝ p₁ p₂ p₃\n⊢ dist p₂ p ≤ max (dist p₁ p) (dist p₃ p)",
"tactic": "exact (hs.dist_lt_max_dist _).le"
},
{
"state_after": "no goals",
"state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhp₁ : p₂ = p₁\n⊢ dist p₂ p ≤ max (dist p₁ p) (dist p₃ p)",
"tactic": "simp [hp₁]"
},
{
"state_after": "no goals",
"state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhp₁ : ¬p₂ = p₁\nhp₃ : p₂ = p₃\n⊢ dist p₂ p ≤ max (dist p₁ p) (dist p₃ p)",
"tactic": "simp [hp₃]"
}
] |
[
49,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/MeasureTheory/Integral/Average.lean
|
MeasureTheory.average_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.232029\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nh : NeBot (ae μ)\nc : E\n⊢ (⨍ (x : α), c ∂μ) = c",
"tactic": "simp only [average_eq, integral_const, Measure.restrict_apply, MeasurableSet.univ, one_smul,\n univ_inter, smul_smul, ← ENNReal.toReal_inv, ← ENNReal.toReal_mul, ENNReal.inv_mul_cancel,\n measure_ne_top μ univ, Ne.def, measure_univ_eq_zero, ae_neBot.1 h, not_false_iff,\n ENNReal.one_toReal]"
}
] |
[
200,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Std/Logic.lean
|
imp_iff_right
|
[] |
[
113,
86
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
113,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.IsCycle.isConj_iff
|
[
{
"state_after": "ι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nhσ : IsCycle σ\nhτ : IsCycle τ\nh : IsConj σ τ\n⊢ card (support σ) = card (support τ)",
"state_before": "ι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nhσ : IsCycle σ\nhτ : IsCycle τ\n⊢ IsConj σ τ → card (support σ) = card (support τ)",
"tactic": "intro h"
},
{
"state_after": "case intro\nι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : IsCycle σ\nπ : Perm α\nhτ : IsCycle (π * σ * π⁻¹)\nh : IsConj σ (π * σ * π⁻¹)\n⊢ card (support σ) = card (support (π * σ * π⁻¹))",
"state_before": "ι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nhσ : IsCycle σ\nhτ : IsCycle τ\nh : IsConj σ τ\n⊢ card (support σ) = card (support τ)",
"tactic": "obtain ⟨π, rfl⟩ := (_root_.isConj_iff).1 h"
},
{
"state_after": "case intro.refine'_1\nι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : IsCycle σ\nπ : Perm α\nhτ : IsCycle (π * σ * π⁻¹)\nh : IsConj σ (π * σ * π⁻¹)\nx✝ : α\nha : x✝ ∈ support σ\n⊢ (fun a x => ↑π a) x✝ ha ∈ support (π * σ * π⁻¹)\n\ncase intro.refine'_2\nι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : IsCycle σ\nπ : Perm α\nhτ : IsCycle (π * σ * π⁻¹)\nh : IsConj σ (π * σ * π⁻¹)\nb : α\nhb : b ∈ support (π * σ * π⁻¹)\n⊢ ∃ a ha, (fun a x => ↑π a) a ha = b",
"state_before": "case intro\nι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : IsCycle σ\nπ : Perm α\nhτ : IsCycle (π * σ * π⁻¹)\nh : IsConj σ (π * σ * π⁻¹)\n⊢ card (support σ) = card (support (π * σ * π⁻¹))",
"tactic": "refine' Finset.card_congr (fun a _ => π a) (fun _ ha => _) (fun _ _ _ _ ab => π.injective ab)\n fun b hb => _"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_1\nι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : IsCycle σ\nπ : Perm α\nhτ : IsCycle (π * σ * π⁻¹)\nh : IsConj σ (π * σ * π⁻¹)\nx✝ : α\nha : x✝ ∈ support σ\n⊢ (fun a x => ↑π a) x✝ ha ∈ support (π * σ * π⁻¹)",
"tactic": "simp [mem_support.1 ha]"
},
{
"state_after": "case intro.refine'_2\nι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : IsCycle σ\nπ : Perm α\nhτ : IsCycle (π * σ * π⁻¹)\nh : IsConj σ (π * σ * π⁻¹)\nb : α\nhb : b ∈ support (π * σ * π⁻¹)\n⊢ ↑π⁻¹ b ∈ support σ",
"state_before": "case intro.refine'_2\nι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : IsCycle σ\nπ : Perm α\nhτ : IsCycle (π * σ * π⁻¹)\nh : IsConj σ (π * σ * π⁻¹)\nb : α\nhb : b ∈ support (π * σ * π⁻¹)\n⊢ ∃ a ha, (fun a x => ↑π a) a ha = b",
"tactic": "refine' ⟨π⁻¹ b, ⟨_, π.apply_inv_self b⟩⟩"
},
{
"state_after": "case intro.refine'_2\nι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : IsCycle σ\nπ : Perm α\nhτ : IsCycle (π * σ * π⁻¹)\nh : IsConj σ (π * σ * π⁻¹)\nb : α\nhb : ¬↑π⁻¹ b ∈ support σ\n⊢ ¬b ∈ support (π * σ * π⁻¹)",
"state_before": "case intro.refine'_2\nι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : IsCycle σ\nπ : Perm α\nhτ : IsCycle (π * σ * π⁻¹)\nh : IsConj σ (π * σ * π⁻¹)\nb : α\nhb : b ∈ support (π * σ * π⁻¹)\n⊢ ↑π⁻¹ b ∈ support σ",
"tactic": "contrapose! hb"
},
{
"state_after": "case intro.refine'_2\nι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : IsCycle σ\nπ : Perm α\nhτ : IsCycle (π * σ * π⁻¹)\nh : IsConj σ (π * σ * π⁻¹)\nb : α\nhb : ↑σ (↑π⁻¹ b) = ↑π⁻¹ b\n⊢ ¬b ∈ support (π * σ * π⁻¹)",
"state_before": "case intro.refine'_2\nι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : IsCycle σ\nπ : Perm α\nhτ : IsCycle (π * σ * π⁻¹)\nh : IsConj σ (π * σ * π⁻¹)\nb : α\nhb : ¬↑π⁻¹ b ∈ support σ\n⊢ ¬b ∈ support (π * σ * π⁻¹)",
"tactic": "rw [mem_support, Classical.not_not] at hb"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_2\nι : Type ?u.3035426\nα : Type u_1\nβ : Type ?u.3035432\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : IsCycle σ\nπ : Perm α\nhτ : IsCycle (π * σ * π⁻¹)\nh : IsConj σ (π * σ * π⁻¹)\nb : α\nhb : ↑σ (↑π⁻¹ b) = ↑π⁻¹ b\n⊢ ¬b ∈ support (π * σ * π⁻¹)",
"tactic": "rw [mem_support, Classical.not_not, Perm.mul_apply, Perm.mul_apply, hb, Perm.apply_inv_self]"
}
] |
[
1755,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1743,
1
] |
Mathlib/CategoryTheory/Monoidal/NaturalTransformation.lean
|
CategoryTheory.MonoidalNatIso.ofComponents.hom_app
|
[] |
[
169,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Topology/LocallyFinite.lean
|
LocallyFinite.isClosed_iUnion
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.8039\nα : Type ?u.8042\nX : Type u_2\nY : Type ?u.8048\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf g : ι → Set X\nhf : LocallyFinite f\nhc : ∀ (i : ι), IsClosed (f i)\n⊢ IsClosed (⋃ (i : ι), f i)",
"tactic": "simp only [← closure_eq_iff_isClosed, hf.closure_iUnion, (hc _).closure_eq]"
}
] |
[
139,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Algebra/Ring/Semiconj.lean
|
SemiconjBy.neg_one_left
|
[] |
[
82,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/GroupTheory/Index.lean
|
Subgroup.card_eq_one
|
[] |
[
504,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
1
] |
Mathlib/Algebra/Star/Basic.lean
|
star_zsmul
|
[] |
[
311,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
310,
1
] |
Mathlib/Topology/Algebra/UniformMulAction.lean
|
uniformContinuousConstSMul_of_continuousConstSMul
|
[] |
[
70,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Control/Functor/Multivariate.lean
|
MvFunctor.id_map'
|
[] |
[
115,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Order/SuccPred/IntervalSucc.lean
|
Monotone.pairwise_disjoint_on_Ioo_pred
|
[] |
[
96,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.isCountablyGenerated_of_seq
|
[
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.110343\nγ : Type ?u.110346\nι : Type ?u.110349\nι' : Sort ?u.110352\nx : ℕ → Set α\n⊢ IsCountablyGenerated (⨅ (i : ℕ), 𝓟 (x i))",
"state_before": "α : Type u_1\nβ : Type ?u.110343\nγ : Type ?u.110346\nι : Type ?u.110349\nι' : Sort ?u.110352\nf : Filter α\nh : ∃ x, f = ⨅ (i : ℕ), 𝓟 (x i)\n⊢ IsCountablyGenerated f",
"tactic": "rcases h with ⟨x, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.110343\nγ : Type ?u.110346\nι : Type ?u.110349\nι' : Sort ?u.110352\nx : ℕ → Set α\n⊢ IsCountablyGenerated (⨅ (i : ℕ), 𝓟 (x i))",
"tactic": "apply isCountablyGenerated_seq"
}
] |
[
1157,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1154,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
|
CategoryTheory.Limits.eq_zero_of_mono_cokernel
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝¹ : HasCokernel f\ninst✝ : Mono (cokernel.π f)\n⊢ f ≫ cokernel.π f = 0 ≫ cokernel.π f",
"tactic": "simp"
}
] |
[
796,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
795,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
AEMeasurable.isLUB
|
[
{
"state_after": "case pos\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : μ = 0\n⊢ AEMeasurable g\n\ncase neg\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\n⊢ AEMeasurable g",
"state_before": "α : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\n⊢ AEMeasurable g",
"tactic": "by_cases hμ : μ = 0"
},
{
"state_after": "case neg\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\n⊢ AEMeasurable g",
"state_before": "case neg\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\n⊢ AEMeasurable g",
"tactic": "have : μ.ae.NeBot := by simpa [neBot_iff]"
},
{
"state_after": "case pos\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : Nonempty ι\n⊢ AEMeasurable g\n\ncase neg\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\n⊢ AEMeasurable g",
"state_before": "case neg\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\n⊢ AEMeasurable g",
"tactic": "by_cases hι : Nonempty ι"
},
{
"state_after": "case neg\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\n⊢ ∃ x, g =ᵐ[μ] fun x_1 => g x",
"state_before": "case neg\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\n⊢ AEMeasurable g",
"tactic": "suffices ∃ x, g =ᵐ[μ] fun _ => g x by\n exact ⟨fun _ => g this.choose, measurable_const, this.choose_spec⟩"
},
{
"state_after": "case neg\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\nh_empty : ∀ (x : δ), {a | ∃ i, f i x = a} = ∅\n⊢ ∃ x, g =ᵐ[μ] fun x_1 => g x",
"state_before": "case neg\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\n⊢ ∃ x, g =ᵐ[μ] fun x_1 => g x",
"tactic": "have h_empty : ∀ x, { a : α | ∃ i : ι, f i x = a } = ∅ := by\n intro x\n ext1 y\n rw [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff]\n exact fun hi => hι (nonempty_of_exists hi)"
},
{
"state_after": "case neg\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\nh_empty : ∀ (x : δ), {a | ∃ i, f i x = a} = ∅\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB ∅ (g b)\n⊢ ∃ x, g =ᵐ[μ] fun x_1 => g x",
"state_before": "case neg\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\nh_empty : ∀ (x : δ), {a | ∃ i, f i x = a} = ∅\n⊢ ∃ x, g =ᵐ[μ] fun x_1 => g x",
"tactic": "simp_rw [h_empty] at hg"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\nh_empty : ∀ (x : δ), {a | ∃ i, f i x = a} = ∅\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB ∅ (g b)\n⊢ ∃ x, g =ᵐ[μ] fun x_1 => g x",
"tactic": "exact ⟨hg.exists.choose, hg.mono fun y hy => IsLUB.unique hy hg.exists.choose_spec⟩"
},
{
"state_after": "case pos\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : μ = 0\n⊢ AEMeasurable g",
"state_before": "case pos\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : μ = 0\n⊢ AEMeasurable g",
"tactic": "rw [hμ]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : μ = 0\n⊢ AEMeasurable g",
"tactic": "exact aemeasurable_zero_measure"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\n⊢ NeBot (Measure.ae μ)",
"tactic": "simpa [neBot_iff]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : Nonempty ι\n⊢ AEMeasurable g",
"tactic": "exact AEMeasurable.is_lub_of_nonempty hι hf hg"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis✝ : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\nthis : ∃ x, g =ᵐ[μ] fun x_1 => g x\n⊢ AEMeasurable g",
"tactic": "exact ⟨fun _ => g this.choose, measurable_const, this.choose_spec⟩"
},
{
"state_after": "α : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\nx : δ\n⊢ {a | ∃ i, f i x = a} = ∅",
"state_before": "α : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\n⊢ ∀ (x : δ), {a | ∃ i, f i x = a} = ∅",
"tactic": "intro x"
},
{
"state_after": "case h\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\nx : δ\ny : α\n⊢ y ∈ {a | ∃ i, f i x = a} ↔ y ∈ ∅",
"state_before": "α : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\nx : δ\n⊢ {a | ∃ i, f i x = a} = ∅",
"tactic": "ext1 y"
},
{
"state_after": "case h\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\nx : δ\ny : α\n⊢ ¬∃ i, f i x = y",
"state_before": "case h\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\nx : δ\ny : α\n⊢ y ∈ {a | ∃ i, f i x = a} ↔ y ∈ ∅",
"tactic": "rw [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_3\nβ : Type ?u.1154773\nγ : Type ?u.1154776\nγ₂ : Type ?u.1154779\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\nμ : MeasureTheory.Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasurable (f i)\nhg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)\nhμ : ¬μ = 0\nthis : NeBot (Measure.ae μ)\nhι : ¬Nonempty ι\nx : δ\ny : α\n⊢ ¬∃ i, f i x = y",
"tactic": "exact fun hi => hι (nonempty_of_exists hi)"
}
] |
[
1128,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1111,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.biInter_mem
|
[] |
[
824,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
822,
11
] |
Std/Data/Int/Lemmas.lean
|
Int.mul_neg_one
|
[
{
"state_after": "no goals",
"state_before": "a : Int\n⊢ a * -1 = -a",
"tactic": "rw [Int.mul_neg, Int.mul_one]"
}
] |
[
526,
90
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
526,
11
] |
Mathlib/Analysis/InnerProductSpace/Positive.lean
|
ContinuousLinearMap.isPositive_iff_complex
|
[
{
"state_after": "𝕜 : Type ?u.285201\nE : Type ?u.285204\nF : Type ?u.285207\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace 𝕜 E\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : CompleteSpace E\ninst✝³ : CompleteSpace F\nE' : Type u_1\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℂ E'\ninst✝ : CompleteSpace E'\nT : E' →L[ℂ] E'\n⊢ ((∀ (v : E'), ↑(↑re (inner (↑↑T v) v)) = inner (↑↑T v) v) ∧ ∀ (x : E'), 0 ≤ reApplyInnerSelf T x) ↔\n (∀ (x : E'), ↑(↑re (inner (↑T x) x)) = inner (↑T x) x) ∧ ∀ (x : E'), 0 ≤ ↑re (inner (↑T x) x)",
"state_before": "𝕜 : Type ?u.285201\nE : Type ?u.285204\nF : Type ?u.285207\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace 𝕜 E\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : CompleteSpace E\ninst✝³ : CompleteSpace F\nE' : Type u_1\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℂ E'\ninst✝ : CompleteSpace E'\nT : E' →L[ℂ] E'\n⊢ IsPositive T ↔ ∀ (x : E'), ↑(↑re (inner (↑T x) x)) = inner (↑T x) x ∧ 0 ≤ ↑re (inner (↑T x) x)",
"tactic": "simp_rw [IsPositive, forall_and, isSelfAdjoint_iff_isSymmetric,\n LinearMap.isSymmetric_iff_inner_map_self_real, conj_eq_iff_re]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.285201\nE : Type ?u.285204\nF : Type ?u.285207\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace 𝕜 E\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : CompleteSpace E\ninst✝³ : CompleteSpace F\nE' : Type u_1\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace ℂ E'\ninst✝ : CompleteSpace E'\nT : E' →L[ℂ] E'\n⊢ ((∀ (v : E'), ↑(↑re (inner (↑↑T v) v)) = inner (↑↑T v) v) ∧ ∀ (x : E'), 0 ≤ reApplyInnerSelf T x) ↔\n (∀ (x : E'), ↑(↑re (inner (↑T x) x)) = inner (↑T x) x) ∧ ∀ (x : E'), 0 ≤ ↑re (inner (↑T x) x)",
"tactic": "rfl"
}
] |
[
129,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousLinearMap.norm_compContinuousMultilinearMap_le
|
[] |
[
955,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
950,
1
] |
Mathlib/Topology/Algebra/Module/CharacterSpace.lean
|
WeakDual.CharacterSpace.union_zero_isClosed
|
[
{
"state_after": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : CommSemiring 𝕜\ninst✝⁷ : TopologicalSpace 𝕜\ninst✝⁶ : ContinuousAdd 𝕜\ninst✝⁵ : ContinuousConstSMul 𝕜 𝕜\ninst✝⁴ : NonUnitalNonAssocSemiring A\ninst✝³ : TopologicalSpace A\ninst✝² : Module 𝕜 A\ninst✝¹ : T2Space 𝕜\ninst✝ : ContinuousMul 𝕜\n⊢ IsClosed (⋂ (i : A) (i_1 : A), {x | ↑x (i * i_1) = ↑x i * ↑x i_1})",
"state_before": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : CommSemiring 𝕜\ninst✝⁷ : TopologicalSpace 𝕜\ninst✝⁶ : ContinuousAdd 𝕜\ninst✝⁵ : ContinuousConstSMul 𝕜 𝕜\ninst✝⁴ : NonUnitalNonAssocSemiring A\ninst✝³ : TopologicalSpace A\ninst✝² : Module 𝕜 A\ninst✝¹ : T2Space 𝕜\ninst✝ : ContinuousMul 𝕜\n⊢ IsClosed (characterSpace 𝕜 A ∪ {0})",
"tactic": "simp only [union_zero, Set.setOf_forall]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : CommSemiring 𝕜\ninst✝⁷ : TopologicalSpace 𝕜\ninst✝⁶ : ContinuousAdd 𝕜\ninst✝⁵ : ContinuousConstSMul 𝕜 𝕜\ninst✝⁴ : NonUnitalNonAssocSemiring A\ninst✝³ : TopologicalSpace A\ninst✝² : Module 𝕜 A\ninst✝¹ : T2Space 𝕜\ninst✝ : ContinuousMul 𝕜\n⊢ IsClosed (⋂ (i : A) (i_1 : A), {x | ↑x (i * i_1) = ↑x i * ↑x i_1})",
"tactic": "exact\n isClosed_iInter fun x =>\n isClosed_iInter fun y =>\n isClosed_eq (eval_continuous _) <| (eval_continuous _).mul (eval_continuous _)"
}
] |
[
134,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Logic/Basic.lean
|
not_ball_of_bex_not
|
[] |
[
1083,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1082,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Order.lean
|
le_hasSum'
|
[] |
[
202,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.aleph0_le_mul_iff'
|
[
{
"state_after": "α β : Type u\na b : Cardinal\nthis : ∀ {a : Cardinal}, ℵ₀ ≤ a → a ≠ 0\n⊢ ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ ℵ₀ ≤ b ∨ ℵ₀ ≤ a ∧ b ≠ 0",
"state_before": "α β : Type u\na b : Cardinal\n⊢ ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ ℵ₀ ≤ b ∨ ℵ₀ ≤ a ∧ b ≠ 0",
"tactic": "have : ∀ {a : Cardinal.{u}}, ℵ₀ ≤ a → a ≠ 0 := fun a => ne_bot_of_le_ne_bot aleph0_ne_zero a"
},
{
"state_after": "α β : Type u\na b : Cardinal\nthis : ∀ {a : Cardinal}, ℵ₀ ≤ a → a ≠ 0\n⊢ b ≠ 0 ∧ ℵ₀ ≤ a ∨ a ≠ 0 ∧ ℵ₀ ≤ b ↔ a ≠ 0 ∧ ℵ₀ ≤ b ∨ ℵ₀ ≤ a ∧ b ≠ 0",
"state_before": "α β : Type u\na b : Cardinal\nthis : ∀ {a : Cardinal}, ℵ₀ ≤ a → a ≠ 0\n⊢ ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ ℵ₀ ≤ b ∨ ℵ₀ ≤ a ∧ b ≠ 0",
"tactic": "simp only [aleph0_le_mul_iff, and_or_left, and_iff_right_of_imp this, @and_left_comm (a ≠ 0)]"
},
{
"state_after": "no goals",
"state_before": "α β : Type u\na b : Cardinal\nthis : ∀ {a : Cardinal}, ℵ₀ ≤ a → a ≠ 0\n⊢ b ≠ 0 ∧ ℵ₀ ≤ a ∨ a ≠ 0 ∧ ℵ₀ ≤ b ↔ a ≠ 0 ∧ ℵ₀ ≤ b ∨ ℵ₀ ≤ a ∧ b ≠ 0",
"tactic": "simp only [and_comm, or_comm]"
}
] |
[
1586,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1583,
1
] |
Mathlib/Data/List/Indexes.lean
|
List.oldMapIdx_append
|
[
{
"state_after": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl : List α\ne : α\n⊢ List.oldMapIdx f (l ++ [e]) = List.oldMapIdx f l ++ [f (length l) e]",
"state_before": "α : Type u\nβ : Type v\n⊢ ∀ (f : ℕ → α → β) (l : List α) (e : α), List.oldMapIdx f (l ++ [e]) = List.oldMapIdx f l ++ [f (length l) e]",
"tactic": "intros f l e"
},
{
"state_after": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl : List α\ne : α\n⊢ List.oldMapIdxCore f 0 (l ++ [e]) = List.oldMapIdxCore f 0 l ++ [f (length l) e]",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl : List α\ne : α\n⊢ List.oldMapIdx f (l ++ [e]) = List.oldMapIdx f l ++ [f (length l) e]",
"tactic": "unfold List.oldMapIdx"
},
{
"state_after": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl : List α\ne : α\n⊢ List.oldMapIdxCore f 0 l ++ List.oldMapIdxCore f (0 + length l) [e] = List.oldMapIdxCore f 0 l ++ [f (length l) e]",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl : List α\ne : α\n⊢ List.oldMapIdxCore f 0 (l ++ [e]) = List.oldMapIdxCore f 0 l ++ [f (length l) e]",
"tactic": "rw [List.oldMapIdxCore_append f 0 l [e]]"
},
{
"state_after": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl : List α\ne : α\n⊢ List.oldMapIdxCore f (length l) [e] = [f (length l) e]",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl : List α\ne : α\n⊢ List.oldMapIdxCore f 0 l ++ List.oldMapIdxCore f (0 + length l) [e] = List.oldMapIdxCore f 0 l ++ [f (length l) e]",
"tactic": "simp only [zero_add, append_cancel_left_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nl : List α\ne : α\n⊢ List.oldMapIdxCore f (length l) [e] = [f (length l) e]",
"tactic": "rfl"
}
] |
[
102,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
11
] |
Mathlib/Algebra/Algebra/Unitization.lean
|
Unitization.inr_injective
|
[] |
[
138,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Data/Nat/Fib.lean
|
Nat.fib_lt_fib_succ
|
[
{
"state_after": "case intro\nn : ℕ\nhn : 2 ≤ 2 + n\n⊢ fib (2 + n) < fib (2 + n + 1)",
"state_before": "n : ℕ\nhn : 2 ≤ n\n⊢ fib n < fib (n + 1)",
"tactic": "rcases exists_add_of_le hn with ⟨n, rfl⟩"
},
{
"state_after": "case intro\nn : ℕ\nhn : 2 ≤ 2 + n\n⊢ 0 < fib (n + 1)",
"state_before": "case intro\nn : ℕ\nhn : 2 ≤ 2 + n\n⊢ fib (2 + n) < fib (2 + n + 1)",
"tactic": "rw [← tsub_pos_iff_lt, add_comm 2, fib_add_two_sub_fib_add_one]"
},
{
"state_after": "no goals",
"state_before": "case intro\nn : ℕ\nhn : 2 ≤ 2 + n\n⊢ 0 < fib (n + 1)",
"tactic": "apply fib_pos (succ_pos n)"
}
] |
[
118,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Data/List/Basic.lean
|
List.get_map_rev
|
[] |
[
1297,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1296,
1
] |
Mathlib/RingTheory/Coprime/Basic.lean
|
IsCoprime.mul_dvd
|
[
{
"state_after": "case intro.intro\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ z",
"state_before": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nH : IsCoprime x y\nH1 : x ∣ z\nH2 : y ∣ z\n⊢ x * y ∣ z",
"tactic": "obtain ⟨a, b, h⟩ := H"
},
{
"state_after": "case intro.intro\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ z * (a * x) + z * (b * y)",
"state_before": "case intro.intro\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ z",
"tactic": "rw [← mul_one z, ← h, mul_add]"
},
{
"state_after": "case intro.intro.h₁\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ z * (a * x)\n\ncase intro.intro.h₂\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ z * (b * y)",
"state_before": "case intro.intro\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ z * (a * x) + z * (b * y)",
"tactic": "apply dvd_add"
},
{
"state_after": "case intro.intro.h₁\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ a * (x * z)",
"state_before": "case intro.intro.h₁\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ z * (a * x)",
"tactic": "rw [mul_comm z, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.h₁\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ a * (x * z)",
"tactic": "exact (mul_dvd_mul_left _ H2).mul_left _"
},
{
"state_after": "case intro.intro.h₂\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ z * y * b",
"state_before": "case intro.intro.h₂\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ z * (b * y)",
"tactic": "rw [mul_comm b, ← mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.h₂\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ z * y * b",
"tactic": "exact (mul_dvd_mul_right H1 _).mul_right _"
}
] |
[
126,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Data/Multiset/Nodup.lean
|
Multiset.nodup_of_le
|
[] |
[
66,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Topology/Algebra/Valuation.lean
|
Valued.hasBasis_nhds_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\n⊢ Filter.HasBasis (𝓝 0) (fun x => True) fun γ => {x | ↑v x < ↑γ}",
"tactic": "simp [Filter.hasBasis_iff, is_topological_valuation]"
}
] |
[
122,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.image_sub_const_Ioo
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => x - a) '' Ioo b c = Ioo (b - a) (c - a)",
"tactic": "simp [sub_eq_neg_add]"
}
] |
[
406,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter_true
|
[] |
[
675,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
674,
1
] |
Std/Control/ForInStep/Lemmas.lean
|
ForInStep.bindList_nil
|
[] |
[
21,
56
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
20,
9
] |
Mathlib/Data/Fintype/Basic.lean
|
Set.toFinset_univ
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.91847\nγ : Type ?u.91850\ns t : Set α\ninst✝¹ : Fintype α\ninst✝ : Fintype ↑univ\na✝ : α\n⊢ a✝ ∈ toFinset univ ↔ a✝ ∈ Finset.univ",
"state_before": "α : Type u_1\nβ : Type ?u.91847\nγ : Type ?u.91850\ns t : Set α\ninst✝¹ : Fintype α\ninst✝ : Fintype ↑univ\n⊢ toFinset univ = Finset.univ",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.91847\nγ : Type ?u.91850\ns t : Set α\ninst✝¹ : Fintype α\ninst✝ : Fintype ↑univ\na✝ : α\n⊢ a✝ ∈ toFinset univ ↔ a✝ ∈ Finset.univ",
"tactic": "simp"
}
] |
[
743,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
740,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.mem_of_max
|
[
{
"state_after": "case empty\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\n⊢ ∀ {a : α}, Finset.max ∅ = ↑a → a ∈ ∅\n\ncase insert\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\n⊢ ∀ {a : α}, Finset.max (insert b s) = ↑a → a ∈ insert b s",
"state_before": "F : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\ns : Finset α\n⊢ ∀ {a : α}, Finset.max s = ↑a → a ∈ s",
"tactic": "induction' s using Finset.induction_on with b s _ ih"
},
{
"state_after": "case empty\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\na✝ : α\nH : Finset.max ∅ = ↑a✝\n⊢ a✝ ∈ ∅",
"state_before": "case empty\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\n⊢ ∀ {a : α}, Finset.max ∅ = ↑a → a ∈ ∅",
"tactic": "intro _ H"
},
{
"state_after": "no goals",
"state_before": "case empty\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\na✝ : α\nH : Finset.max ∅ = ↑a✝\n⊢ a✝ ∈ ∅",
"tactic": "cases H"
},
{
"state_after": "case insert\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\na : α\nh : Finset.max (insert b s) = ↑a\n⊢ a ∈ insert b s",
"state_before": "case insert\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\n⊢ ∀ {a : α}, Finset.max (insert b s) = ↑a → a ∈ insert b s",
"tactic": "intro a h"
},
{
"state_after": "case pos\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\na : α\nh : Finset.max (insert b s) = ↑a\np : b = a\n⊢ a ∈ insert b s\n\ncase neg\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\na : α\nh : Finset.max (insert b s) = ↑a\np : ¬b = a\n⊢ a ∈ insert b s",
"state_before": "case insert\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\na : α\nh : Finset.max (insert b s) = ↑a\n⊢ a ∈ insert b s",
"tactic": "by_cases p : b = a"
},
{
"state_after": "case pos.refl\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\na : α\nh : Finset.max (insert b s) = ↑b\n⊢ b ∈ insert b s",
"state_before": "case pos\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\na : α\nh : Finset.max (insert b s) = ↑a\np : b = a\n⊢ a ∈ insert b s",
"tactic": "induction p"
},
{
"state_after": "no goals",
"state_before": "case pos.refl\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\na : α\nh : Finset.max (insert b s) = ↑b\n⊢ b ∈ insert b s",
"tactic": "exact mem_insert_self b s"
},
{
"state_after": "case neg.inl\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\na : α\nh : ↑b = ↑a\np : ¬b = a\nq : max (↑b) (Finset.max s) = ↑b\n⊢ a ∈ insert b s\n\ncase neg.inr\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\na : α\nh : Finset.max s = ↑a\np : ¬b = a\nq : max (↑b) (Finset.max s) = Finset.max s\n⊢ a ∈ insert b s",
"state_before": "case neg\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\na : α\nh : Finset.max (insert b s) = ↑a\np : ¬b = a\n⊢ a ∈ insert b s",
"tactic": "cases' max_choice (↑b) s.max with q q <;> rw [max_insert, q] at h"
},
{
"state_after": "case neg.inl.refl\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\nq : max (↑b) (Finset.max s) = ↑b\np : ¬b = b\n⊢ b ∈ insert b s",
"state_before": "case neg.inl\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\na : α\nh : ↑b = ↑a\np : ¬b = a\nq : max (↑b) (Finset.max s) = ↑b\n⊢ a ∈ insert b s",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case neg.inl.refl\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\nq : max (↑b) (Finset.max s) = ↑b\np : ¬b = b\n⊢ b ∈ insert b s",
"tactic": "cases p rfl"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\nF : Type ?u.314698\nα : Type u_1\nβ : Type ?u.314704\nγ : Type ?u.314707\nι : Type ?u.314710\nκ : Type ?u.314713\ninst✝ : LinearOrder α\nb : α\ns : Finset α\na✝ : ¬b ∈ s\nih : ∀ {a : α}, Finset.max s = ↑a → a ∈ s\na : α\nh : Finset.max s = ↑a\np : ¬b = a\nq : max (↑b) (Finset.max s) = Finset.max s\n⊢ a ∈ insert b s",
"tactic": "exact mem_insert_of_mem (ih h)"
}
] |
[
1207,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1197,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
stronglyMeasurable_iff_measurable
|
[] |
[
638,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
636,
1
] |
Mathlib/Data/Polynomial/Coeff.lean
|
Polynomial.coeff_monomial_mul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r✝ p : R[X]\nn d : ℕ\nr : R\n⊢ coeff (↑(monomial n) r * p) (d + n) = r * coeff p d",
"tactic": "rw [← C_mul_X_pow_eq_monomial, mul_assoc, coeff_C_mul, X_pow_mul, coeff_mul_X_pow]"
}
] |
[
280,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/Data/Nat/Cast/Basic.lean
|
Nat.cast_le
|
[] |
[
133,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/GroupTheory/Finiteness.lean
|
Submonoid.FG.map_injective
|
[
{
"state_after": "case intro\nM : Type u_2\nN : Type ?u.23387\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\ne : M →* M'\nhe : Function.Injective ↑e\ns : Finset M'\nhs : closure ↑s = Submonoid.map e P\n⊢ FG P",
"state_before": "M : Type u_2\nN : Type ?u.23387\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\ne : M →* M'\nhe : Function.Injective ↑e\nh : FG (Submonoid.map e P)\n⊢ FG P",
"tactic": "obtain ⟨s, hs⟩ := h"
},
{
"state_after": "case intro\nM : Type u_2\nN : Type ?u.23387\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\ne : M →* M'\nhe : Function.Injective ↑e\ns : Finset M'\nhs : closure ↑s = Submonoid.map e P\n⊢ closure ↑(Finset.preimage s ↑e (_ : Set.InjOn (↑e) (↑e ⁻¹' ↑s))) = P",
"state_before": "case intro\nM : Type u_2\nN : Type ?u.23387\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\ne : M →* M'\nhe : Function.Injective ↑e\ns : Finset M'\nhs : closure ↑s = Submonoid.map e P\n⊢ FG P",
"tactic": "use s.preimage e (he.injOn _)"
},
{
"state_after": "case intro.a\nM : Type u_2\nN : Type ?u.23387\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\ne : M →* M'\nhe : Function.Injective ↑e\ns : Finset M'\nhs : closure ↑s = Submonoid.map e P\n⊢ Submonoid.map e (closure ↑(Finset.preimage s ↑e (_ : Set.InjOn (↑e) (↑e ⁻¹' ↑s)))) = Submonoid.map e P",
"state_before": "case intro\nM : Type u_2\nN : Type ?u.23387\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\ne : M →* M'\nhe : Function.Injective ↑e\ns : Finset M'\nhs : closure ↑s = Submonoid.map e P\n⊢ closure ↑(Finset.preimage s ↑e (_ : Set.InjOn (↑e) (↑e ⁻¹' ↑s))) = P",
"tactic": "apply Submonoid.map_injective_of_injective he"
},
{
"state_after": "case intro.a\nM : Type u_2\nN : Type ?u.23387\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\ne : M →* M'\nhe : Function.Injective ↑e\ns : Finset M'\nhs : closure ↑s = Submonoid.map e P\n⊢ closure (↑e '' (↑e ⁻¹' ↑s)) = closure ↑s",
"state_before": "case intro.a\nM : Type u_2\nN : Type ?u.23387\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\ne : M →* M'\nhe : Function.Injective ↑e\ns : Finset M'\nhs : closure ↑s = Submonoid.map e P\n⊢ Submonoid.map e (closure ↑(Finset.preimage s ↑e (_ : Set.InjOn (↑e) (↑e ⁻¹' ↑s)))) = Submonoid.map e P",
"tactic": "rw [← hs, MonoidHom.map_mclosure e, Finset.coe_preimage]"
},
{
"state_after": "case intro.a.e_s\nM : Type u_2\nN : Type ?u.23387\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\ne : M →* M'\nhe : Function.Injective ↑e\ns : Finset M'\nhs : closure ↑s = Submonoid.map e P\n⊢ ↑e '' (↑e ⁻¹' ↑s) = ↑s",
"state_before": "case intro.a\nM : Type u_2\nN : Type ?u.23387\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\ne : M →* M'\nhe : Function.Injective ↑e\ns : Finset M'\nhs : closure ↑s = Submonoid.map e P\n⊢ closure (↑e '' (↑e ⁻¹' ↑s)) = closure ↑s",
"tactic": "congr"
},
{
"state_after": "case intro.a.e_s\nM : Type u_2\nN : Type ?u.23387\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\ne : M →* M'\nhe : Function.Injective ↑e\ns : Finset M'\nhs : closure ↑s = Submonoid.map e P\n⊢ Submonoid.map e P ≤ Submonoid.map e ⊤",
"state_before": "case intro.a.e_s\nM : Type u_2\nN : Type ?u.23387\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\ne : M →* M'\nhe : Function.Injective ↑e\ns : Finset M'\nhs : closure ↑s = Submonoid.map e P\n⊢ ↑e '' (↑e ⁻¹' ↑s) = ↑s",
"tactic": "rw [Set.image_preimage_eq_iff, ← MonoidHom.coe_mrange e, ← Submonoid.closure_le, hs,\n MonoidHom.mrange_eq_map e]"
},
{
"state_after": "no goals",
"state_before": "case intro.a.e_s\nM : Type u_2\nN : Type ?u.23387\ninst✝² : Monoid M\ninst✝¹ : AddMonoid N\nM' : Type u_1\ninst✝ : Monoid M'\nP : Submonoid M\ne : M →* M'\nhe : Function.Injective ↑e\ns : Finset M'\nhs : closure ↑s = Submonoid.map e P\n⊢ Submonoid.map e P ≤ Submonoid.map e ⊤",
"tactic": "exact Submonoid.monotone_map le_top"
}
] |
[
164,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
zpow_mono_right
|
[
{
"state_after": "α : Type u_1\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✝ : OrderedCommGroup α\nm✝ n✝ : ℤ\na b : α\nha : 1 ≤ a\nm n : ℤ\nh : m ≤ n\n⊢ a ^ (m + (n - m)) = a ^ n",
"state_before": "α : Type u_1\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✝ : OrderedCommGroup α\nm✝ n✝ : ℤ\na b : α\nha : 1 ≤ a\nm n : ℤ\nh : m ≤ n\n⊢ a ^ m * a ^ (n - m) = a ^ n",
"tactic": "rw [← zpow_add]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\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✝ : OrderedCommGroup α\nm✝ n✝ : ℤ\na b : α\nha : 1 ≤ a\nm n : ℤ\nh : m ≤ n\n⊢ a ^ (m + (n - m)) = a ^ n",
"tactic": "simp"
}
] |
[
343,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/Data/Real/Pi/Bounds.lean
|
Real.pi_lower_bound_start
|
[
{
"state_after": "n : ℕ\na : ℝ\nh : sqrtTwoAddSeries (↑0 / ↑1) n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2\n⊢ a ≤ 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n)",
"state_before": "n : ℕ\na : ℝ\nh : sqrtTwoAddSeries (↑0 / ↑1) n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2\n⊢ a < π",
"tactic": "refine' lt_of_le_of_lt _ (pi_gt_sqrtTwoAddSeries n)"
},
{
"state_after": "n : ℕ\na : ℝ\nh : sqrtTwoAddSeries (↑0 / ↑1) n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2\n⊢ a ≤ sqrt (2 - sqrtTwoAddSeries 0 n) * 2 ^ (n + 1)",
"state_before": "n : ℕ\na : ℝ\nh : sqrtTwoAddSeries (↑0 / ↑1) n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2\n⊢ a ≤ 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n)",
"tactic": "rw [mul_comm]"
},
{
"state_after": "n : ℕ\na : ℝ\nh : sqrtTwoAddSeries (↑0 / ↑1) n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2\n⊢ (a / 2 ^ (n + 1)) ^ 2 ≤ 2 - sqrtTwoAddSeries 0 n",
"state_before": "n : ℕ\na : ℝ\nh : sqrtTwoAddSeries (↑0 / ↑1) n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2\n⊢ a ≤ sqrt (2 - sqrtTwoAddSeries 0 n) * 2 ^ (n + 1)",
"tactic": "refine' (div_le_iff (pow_pos (by norm_num) _ : (0 : ℝ) < _)).mp (le_sqrt_of_sq_le _)"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\na : ℝ\nh : sqrtTwoAddSeries (↑0 / ↑1) n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2\n⊢ (a / 2 ^ (n + 1)) ^ 2 ≤ 2 - sqrtTwoAddSeries 0 n",
"tactic": "rwa [le_sub_comm, show (0 : ℝ) = (0 : ℕ) / (1 : ℕ) by rw [Nat.cast_zero, zero_div]]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\na : ℝ\nh : sqrtTwoAddSeries (↑0 / ↑1) n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2\n⊢ 0 < 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\na : ℝ\nh : sqrtTwoAddSeries (↑0 / ↑1) n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2\n⊢ 0 = ↑0 / ↑1",
"tactic": "rw [Nat.cast_zero, zero_div]"
}
] |
[
80,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Analysis/Convex/Hull.lean
|
convexHull_min
|
[] |
[
76,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
abs_eq_max_neg
|
[] |
[
45,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_strict_mono_of_ae_le_of_ae_lt_on
|
[] |
[
949,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
945,
1
] |
Mathlib/Topology/Instances/Nat.lean
|
Nat.closedBall_eq_Icc
|
[
{
"state_after": "case inl\nx : ℕ\nr : ℝ\nhr : 0 ≤ r\n⊢ closedBall x r = Icc ⌈↑x - r⌉₊ ⌊↑x + r⌋₊\n\ncase inr\nx : ℕ\nr : ℝ\nhr : r < 0\n⊢ closedBall x r = Icc ⌈↑x - r⌉₊ ⌊↑x + r⌋₊",
"state_before": "x : ℕ\nr : ℝ\n⊢ closedBall x r = Icc ⌈↑x - r⌉₊ ⌊↑x + r⌋₊",
"tactic": "rcases le_or_lt 0 r with (hr | hr)"
},
{
"state_after": "case inl\nx : ℕ\nr : ℝ\nhr : 0 ≤ r\n⊢ 0 ≤ ↑x + r",
"state_before": "case inl\nx : ℕ\nr : ℝ\nhr : 0 ≤ r\n⊢ closedBall x r = Icc ⌈↑x - r⌉₊ ⌊↑x + r⌋₊",
"tactic": "rw [← preimage_closedBall, Real.closedBall_eq_Icc, preimage_Icc]"
},
{
"state_after": "no goals",
"state_before": "case inl\nx : ℕ\nr : ℝ\nhr : 0 ≤ r\n⊢ 0 ≤ ↑x + r",
"tactic": "exact add_nonneg (cast_nonneg x) hr"
},
{
"state_after": "case inr\nx : ℕ\nr : ℝ\nhr : r < 0\n⊢ ⌊↑x + r⌋₊ < ⌈↑x - r⌉₊",
"state_before": "case inr\nx : ℕ\nr : ℝ\nhr : r < 0\n⊢ closedBall x r = Icc ⌈↑x - r⌉₊ ⌊↑x + r⌋₊",
"tactic": "rw [closedBall_eq_empty.2 hr, Icc_eq_empty_of_lt]"
},
{
"state_after": "no goals",
"state_before": "case inr\nx : ℕ\nr : ℝ\nhr : r < 0\n⊢ ⌊↑x + r⌋₊ < ⌈↑x - r⌉₊",
"tactic": "calc ⌊(x : ℝ) + r⌋₊ ≤ ⌊(x : ℝ)⌋₊ := floor_mono <| by linarith\n_ < ⌈↑x - r⌉₊ := by\n rw [floor_coe, Nat.lt_ceil]\n linarith"
},
{
"state_after": "no goals",
"state_before": "x : ℕ\nr : ℝ\nhr : r < 0\n⊢ ↑x + r ≤ ↑x",
"tactic": "linarith"
},
{
"state_after": "x : ℕ\nr : ℝ\nhr : r < 0\n⊢ ↑x < ↑x - r",
"state_before": "x : ℕ\nr : ℝ\nhr : r < 0\n⊢ ⌊↑x⌋₊ < ⌈↑x - r⌉₊",
"tactic": "rw [floor_coe, Nat.lt_ceil]"
},
{
"state_after": "no goals",
"state_before": "x : ℕ\nr : ℝ\nhr : r < 0\n⊢ ↑x < ↑x - r",
"tactic": "linarith"
}
] |
[
66,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
measurable_quot_mk
|
[] |
[
504,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
1
] |
Mathlib/Data/Real/NNReal.lean
|
Real.toNNReal_mul
|
[
{
"state_after": "no goals",
"state_before": "p q : ℝ\nhp : 0 ≤ p\n⊢ ↑(toNNReal (p * q)) = ↑(toNNReal p * toNNReal q)",
"tactic": "simp [mul_max_of_nonneg, hp]"
}
] |
[
704,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
702,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Split.lean
|
BoxIntegral.Box.splitUpper_eq_bot
|
[
{
"state_after": "ι : Type u_1\nM : Type ?u.13653\nn : ℕ\nI : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\ni : ι\nx : ℝ\n⊢ (upper I i ≤ max x (lower I i) ∨ ∃ x x_1, upper I x ≤ lower I x) ↔ upper I i ≤ x",
"state_before": "ι : Type u_1\nM : Type ?u.13653\nn : ℕ\nI : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\ni : ι\nx : ℝ\n⊢ splitUpper I i x = ⊥ ↔ upper I i ≤ x",
"tactic": "rw [splitUpper, mk'_eq_bot, exists_update_iff I.lower fun j y => I.upper j ≤ y]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nM : Type ?u.13653\nn : ℕ\nI : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\ni : ι\nx : ℝ\n⊢ (upper I i ≤ max x (lower I i) ∨ ∃ x x_1, upper I x ≤ lower I x) ↔ upper I i ≤ x",
"tactic": "simp [(I.lower_lt_upper _).not_le]"
}
] |
[
123,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Conformal.lean
|
InnerProductGeometry.IsConformalMap.preserves_angle
|
[
{
"state_after": "case intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : InnerProductSpace ℝ F\nu v : E\nc : ℝ\nhc : c ≠ 0\nli : E →ₗᵢ[ℝ] F\n⊢ angle (↑(c • LinearIsometry.toContinuousLinearMap li) u) (↑(c • LinearIsometry.toContinuousLinearMap li) v) =\n angle u v",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : InnerProductSpace ℝ F\nf' : E →L[ℝ] F\nh : IsConformalMap f'\nu v : E\n⊢ angle (↑f' u) (↑f' v) = angle u v",
"tactic": "obtain ⟨c, hc, li, rfl⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : InnerProductSpace ℝ F\nu v : E\nc : ℝ\nhc : c ≠ 0\nli : E →ₗᵢ[ℝ] F\n⊢ angle (↑(c • LinearIsometry.toContinuousLinearMap li) u) (↑(c • LinearIsometry.toContinuousLinearMap li) v) =\n angle u v",
"tactic": "exact (angle_smul_smul hc _ _).trans (li.angle_map _ _)"
}
] |
[
33,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
30,
1
] |
Mathlib/Logic/Function/Conjugate.lean
|
Function.Semiconj.commute
|
[] |
[
93,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Logic/Basic.lean
|
not_bex
|
[] |
[
1079,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1079,
1
] |
Mathlib/Analysis/Normed/Group/AddTorsor.lean
|
LipschitzWith.vsub
|
[] |
[
258,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Order/Heyting/Boundary.lean
|
Coheyting.boundary_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ ∂ ⊤ = ⊥",
"tactic": "rw [boundary, hnot_top, inf_bot_eq]"
}
] |
[
66,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Topology/Basic.lean
|
IsClosed.closure_subset_iff
|
[] |
[
450,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffAt.add
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1726488\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf g : E → F\nhf : ContDiffWithinAt 𝕜 n f univ x\nhg : ContDiffWithinAt 𝕜 n g univ x\n⊢ ContDiffWithinAt 𝕜 n (fun x => f x + g x) univ x",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1726488\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf g : E → F\nhf : ContDiffAt 𝕜 n f x\nhg : ContDiffAt 𝕜 n g x\n⊢ ContDiffAt 𝕜 n (fun x => f x + g x) x",
"tactic": "rw [← contDiffWithinAt_univ] at *"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1726488\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf g : E → F\nhf : ContDiffWithinAt 𝕜 n f univ x\nhg : ContDiffWithinAt 𝕜 n g univ x\n⊢ ContDiffWithinAt 𝕜 n (fun x => f x + g x) univ x",
"tactic": "exact hf.add hg"
}
] |
[
1204,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1202,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.subtypeRestr_def
|
[] |
[
1368,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1367,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.Tendsto.frequently_map
|
[] |
[
2850,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2847,
1
] |
Mathlib/RingTheory/Nilpotent.lean
|
isRadical_iff_span_singleton
|
[
{
"state_after": "R S : Type u\nx y : R\ninst✝ : CommSemiring R\n⊢ (∀ (n : ℕ) (x : R), x ^ n ∈ Ideal.span {y} → x ∈ Ideal.span {y}) ↔ Ideal.IsRadical (Ideal.span {y})",
"state_before": "R S : Type u\nx y : R\ninst✝ : CommSemiring R\n⊢ IsRadical y ↔ Ideal.IsRadical (Ideal.span {y})",
"tactic": "simp_rw [IsRadical, ← Ideal.mem_span_singleton]"
},
{
"state_after": "no goals",
"state_before": "R S : Type u\nx y : R\ninst✝ : CommSemiring R\n⊢ (∀ (n : ℕ) (x : R), x ^ n ∈ Ideal.span {y} → x ∈ Ideal.span {y}) ↔ Ideal.IsRadical (Ideal.span {y})",
"tactic": "exact forall_swap.trans (forall_congr' fun r => exists_imp.symm)"
}
] |
[
124,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
mul_eq_of_eq_mul_inv
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.52587\nβ : Type ?u.52590\nG : Type u_1\ninst✝ : Group G\na b c d : G\nh : a = c * b⁻¹\n⊢ a * b = c",
"tactic": "simp [h]"
}
] |
[
663,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
663,
1
] |
Mathlib/Order/CompleteLatticeIntervals.lean
|
sInf_within_of_ordConnected
|
[
{
"state_after": "case intro\nα : Type u_1\ns✝ : Set α\ninst✝ : ConditionallyCompleteLinearOrder α\ns : Set α\nhs : OrdConnected s\nt : Set ↑s\nh_bdd : BddBelow t\nc : ↑s\nhct : c ∈ t\n⊢ sInf (Subtype.val '' t) ∈ s",
"state_before": "α : Type u_1\ns✝ : Set α\ninst✝ : ConditionallyCompleteLinearOrder α\ns : Set α\nhs : OrdConnected s\nt : Set ↑s\nht : Set.Nonempty t\nh_bdd : BddBelow t\n⊢ sInf (Subtype.val '' t) ∈ s",
"tactic": "obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht"
},
{
"state_after": "case intro.intro\nα : Type u_1\ns✝ : Set α\ninst✝ : ConditionallyCompleteLinearOrder α\ns : Set α\nhs : OrdConnected s\nt : Set ↑s\nc : ↑s\nhct : c ∈ t\nB : ↑s\nhB : B ∈ lowerBounds t\n⊢ sInf (Subtype.val '' t) ∈ s",
"state_before": "case intro\nα : Type u_1\ns✝ : Set α\ninst✝ : ConditionallyCompleteLinearOrder α\ns : Set α\nhs : OrdConnected s\nt : Set ↑s\nh_bdd : BddBelow t\nc : ↑s\nhct : c ∈ t\n⊢ sInf (Subtype.val '' t) ∈ s",
"tactic": "obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd"
},
{
"state_after": "case intro.intro.refine'_1\nα : Type u_1\ns✝ : Set α\ninst✝ : ConditionallyCompleteLinearOrder α\ns : Set α\nhs : OrdConnected s\nt : Set ↑s\nc : ↑s\nhct : c ∈ t\nB : ↑s\nhB : B ∈ lowerBounds t\n⊢ ↑B ≤ sInf (Subtype.val '' t)\n\ncase intro.intro.refine'_2\nα : Type u_1\ns✝ : Set α\ninst✝ : ConditionallyCompleteLinearOrder α\ns : Set α\nhs : OrdConnected s\nt : Set ↑s\nc : ↑s\nhct : c ∈ t\nB : ↑s\nhB : B ∈ lowerBounds t\n⊢ sInf (Subtype.val '' t) ≤ ↑c",
"state_before": "case intro.intro\nα : Type u_1\ns✝ : Set α\ninst✝ : ConditionallyCompleteLinearOrder α\ns : Set α\nhs : OrdConnected s\nt : Set ↑s\nc : ↑s\nhct : c ∈ t\nB : ↑s\nhB : B ∈ lowerBounds t\n⊢ sInf (Subtype.val '' t) ∈ s",
"tactic": "refine' hs.out B.2 c.2 ⟨_, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1\nα : Type u_1\ns✝ : Set α\ninst✝ : ConditionallyCompleteLinearOrder α\ns : Set α\nhs : OrdConnected s\nt : Set ↑s\nc : ↑s\nhct : c ∈ t\nB : ↑s\nhB : B ∈ lowerBounds t\n⊢ ↑B ≤ sInf (Subtype.val '' t)",
"tactic": "exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_2\nα : Type u_1\ns✝ : Set α\ninst✝ : ConditionallyCompleteLinearOrder α\ns : Set α\nhs : OrdConnected s\nt : Set ↑s\nc : ↑s\nhct : c ∈ t\nB : ↑s\nhB : B ∈ lowerBounds t\n⊢ sInf (Subtype.val '' t) ≤ ↑c",
"tactic": "exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩"
}
] |
[
154,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/AlgebraicTopology/SimplexCategory.lean
|
SimplexCategory.eq_σ_of_epi
|
[
{
"state_after": "case intro.intro\nn : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\ni : Fin (n + 1)\nθ' : [n] ⟶ [n]\nh : θ = σ i ≫ θ'\n⊢ ∃ i, θ = σ i",
"state_before": "n : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\n⊢ ∃ i, θ = σ i",
"tactic": "rcases eq_σ_comp_of_not_injective θ (by\n by_contra h\n simpa using le_of_mono (mono_iff_injective.mpr h)) with ⟨i, θ', h⟩"
},
{
"state_after": "case intro.intro\nn : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\ni : Fin (n + 1)\nθ' : [n] ⟶ [n]\nh : θ = σ i ≫ θ'\n⊢ θ = σ i",
"state_before": "case intro.intro\nn : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\ni : Fin (n + 1)\nθ' : [n] ⟶ [n]\nh : θ = σ i ≫ θ'\n⊢ ∃ i, θ = σ i",
"tactic": "use i"
},
{
"state_after": "case intro.intro\nn : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\ni : Fin (n + 1)\nθ' : [n] ⟶ [n]\nh : θ = σ i ≫ θ'\nthis : Epi (σ i ≫ θ')\n⊢ θ = σ i",
"state_before": "case intro.intro\nn : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\ni : Fin (n + 1)\nθ' : [n] ⟶ [n]\nh : θ = σ i ≫ θ'\n⊢ θ = σ i",
"tactic": "haveI : Epi (σ i ≫ θ') := by\n rw [← h]\n infer_instance"
},
{
"state_after": "case intro.intro\nn : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\ni : Fin (n + 1)\nθ' : [n] ⟶ [n]\nh : θ = σ i ≫ θ'\nthis✝ : Epi (σ i ≫ θ')\nthis : Epi θ'\n⊢ θ = σ i",
"state_before": "case intro.intro\nn : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\ni : Fin (n + 1)\nθ' : [n] ⟶ [n]\nh : θ = σ i ≫ θ'\nthis : Epi (σ i ≫ θ')\n⊢ θ = σ i",
"tactic": "haveI := CategoryTheory.epi_of_epi (σ i) θ'"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nn : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\ni : Fin (n + 1)\nθ' : [n] ⟶ [n]\nh : θ = σ i ≫ θ'\nthis✝ : Epi (σ i ≫ θ')\nthis : Epi θ'\n⊢ θ = σ i",
"tactic": "rw [h, eq_id_of_epi θ', Category.comp_id]"
},
{
"state_after": "n : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\nh : Function.Injective ↑(Hom.toOrderHom θ)\n⊢ False",
"state_before": "n : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\n⊢ ¬Function.Injective ↑(Hom.toOrderHom θ)",
"tactic": "by_contra h"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\nh : Function.Injective ↑(Hom.toOrderHom θ)\n⊢ False",
"tactic": "simpa using le_of_mono (mono_iff_injective.mpr h)"
},
{
"state_after": "n : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\ni : Fin (n + 1)\nθ' : [n] ⟶ [n]\nh : θ = σ i ≫ θ'\n⊢ Epi θ",
"state_before": "n : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\ni : Fin (n + 1)\nθ' : [n] ⟶ [n]\nh : θ = σ i ≫ θ'\n⊢ Epi (σ i ≫ θ')",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nθ : [n + 1] ⟶ [n]\ninst✝ : Epi θ\ni : Fin (n + 1)\nθ' : [n] ⟶ [n]\nh : θ = σ i ≫ θ'\n⊢ Epi θ",
"tactic": "infer_instance"
}
] |
[
743,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
734,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.cycleOf_zpow_apply_self
|
[
{
"state_after": "ι : Type ?u.2180524\nα : Type u_1\nβ : Type ?u.2180530\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝ y : α\nf : Perm α\nx : α\nz : ℤ\n⊢ ↑(cycleOf f x ^ z) x = ↑(f ^ z) x",
"state_before": "ι : Type ?u.2180524\nα : Type u_1\nβ : Type ?u.2180530\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝ y : α\nf : Perm α\nx : α\n⊢ ∀ (n : ℤ), ↑(cycleOf f x ^ n) x = ↑(f ^ n) x",
"tactic": "intro z"
},
{
"state_after": "case ofNat\nι : Type ?u.2180524\nα : Type u_1\nβ : Type ?u.2180530\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝ y : α\nf : Perm α\nx : α\nz : ℕ\n⊢ ↑(cycleOf f x ^ Int.ofNat z) x = ↑(f ^ Int.ofNat z) x\n\ncase negSucc\nι : Type ?u.2180524\nα : Type u_1\nβ : Type ?u.2180530\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝ y : α\nf : Perm α\nx : α\nhz : ℕ\n⊢ ↑(cycleOf f x ^ Int.negSucc hz) x = ↑(f ^ Int.negSucc hz) x",
"state_before": "ι : Type ?u.2180524\nα : Type u_1\nβ : Type ?u.2180530\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝ y : α\nf : Perm α\nx : α\nz : ℤ\n⊢ ↑(cycleOf f x ^ z) x = ↑(f ^ z) x",
"tactic": "induction' z with z hz"
},
{
"state_after": "no goals",
"state_before": "case ofNat\nι : Type ?u.2180524\nα : Type u_1\nβ : Type ?u.2180530\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝ y : α\nf : Perm α\nx : α\nz : ℕ\n⊢ ↑(cycleOf f x ^ Int.ofNat z) x = ↑(f ^ Int.ofNat z) x",
"tactic": "exact cycleOf_pow_apply_self f x z"
},
{
"state_after": "no goals",
"state_before": "case negSucc\nι : Type ?u.2180524\nα : Type u_1\nβ : Type ?u.2180530\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g : Perm α\nx✝ y : α\nf : Perm α\nx : α\nhz : ℕ\n⊢ ↑(cycleOf f x ^ Int.negSucc hz) x = ↑(f ^ Int.negSucc hz) x",
"tactic": "rw [zpow_negSucc, ← inv_pow, cycleOf_inv, zpow_negSucc, ← inv_pow, cycleOf_pow_apply_self]"
}
] |
[
997,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
992,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
StrictAntiOn.antitoneOn
|
[] |
[
462,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
11
] |
Mathlib/Algebra/Group/WithOne/Defs.lean
|
WithOne.coe_inv
|
[] |
[
215,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.filter_apply_pos
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : (i : ι) → Zero (β i)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\ni : ι\nh : p i\n⊢ ↑(filter p f) i = ↑f i",
"tactic": "simp only [filter_apply, if_pos h]"
}
] |
[
412,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
411,
1
] |
Mathlib/AlgebraicGeometry/SheafedSpace.lean
|
AlgebraicGeometry.SheafedSpace.Γ_map_op
|
[] |
[
221,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/Data/Finmap.lean
|
Finmap.lookup_singleton_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\nb : β a\n⊢ lookup a (singleton a b) = some b",
"tactic": "rw [singleton, lookup_toFinmap, AList.singleton, AList.lookup, dlookup_cons_eq]"
}
] |
[
308,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/Data/PFunctor/Univariate/Basic.lean
|
PFunctor.iget_map
|
[
{
"state_after": "P : PFunctor\nα✝ β✝ : Type u\ninst✝² : DecidableEq P.A\nα β : Type u\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nx : Obj P α\nf : α → β\ni : IdxCat P\nh : i.fst = x.fst\n⊢ Sigma.snd (f <$> x) (cast (_ : B P i.fst = B P (f <$> x).fst) i.snd) =\n f (Sigma.snd x (cast (_ : B P i.fst = B P x.fst) i.snd))",
"state_before": "P : PFunctor\nα✝ β✝ : Type u\ninst✝² : DecidableEq P.A\nα β : Type u\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nx : Obj P α\nf : α → β\ni : IdxCat P\nh : i.fst = x.fst\n⊢ Obj.iget (f <$> x) i = f (Obj.iget x i)",
"tactic": "simp only [Obj.iget, fst_map, *, dif_pos, eq_self_iff_true]"
},
{
"state_after": "case mk\nP : PFunctor\nα✝ β✝ : Type u\ninst✝² : DecidableEq P.A\nα β : Type u\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nf : α → β\ni : IdxCat P\nfst✝ : P.A\nsnd✝ : B P fst✝ → α\nh : i.fst = { fst := fst✝, snd := snd✝ }.fst\n⊢ Sigma.snd (f <$> { fst := fst✝, snd := snd✝ })\n (cast (_ : B P i.fst = B P (f <$> { fst := fst✝, snd := snd✝ }).fst) i.snd) =\n f (Sigma.snd { fst := fst✝, snd := snd✝ } (cast (_ : B P i.fst = B P { fst := fst✝, snd := snd✝ }.fst) i.snd))",
"state_before": "P : PFunctor\nα✝ β✝ : Type u\ninst✝² : DecidableEq P.A\nα β : Type u\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nx : Obj P α\nf : α → β\ni : IdxCat P\nh : i.fst = x.fst\n⊢ Sigma.snd (f <$> x) (cast (_ : B P i.fst = B P (f <$> x).fst) i.snd) =\n f (Sigma.snd x (cast (_ : B P i.fst = B P x.fst) i.snd))",
"tactic": "cases x"
},
{
"state_after": "no goals",
"state_before": "case mk\nP : PFunctor\nα✝ β✝ : Type u\ninst✝² : DecidableEq P.A\nα β : Type u\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nf : α → β\ni : IdxCat P\nfst✝ : P.A\nsnd✝ : B P fst✝ → α\nh : i.fst = { fst := fst✝, snd := snd✝ }.fst\n⊢ Sigma.snd (f <$> { fst := fst✝, snd := snd✝ })\n (cast (_ : B P i.fst = B P (f <$> { fst := fst✝, snd := snd✝ }).fst) i.snd) =\n f (Sigma.snd { fst := fst✝, snd := snd✝ } (cast (_ : B P i.fst = B P { fst := fst✝, snd := snd✝ }.fst) i.snd))",
"tactic": "rfl"
}
] |
[
151,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
HomologicalComplex.d_comp_eqToHom
|
[
{
"state_after": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j : ι\nrij rij' : ComplexShape.Rel c i j\n⊢ d C i j ≫ eqToHom (_ : X C j = X C j) = d C i j",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j j' : ι\nrij : ComplexShape.Rel c i j\nrij' : ComplexShape.Rel c i j'\n⊢ d C i j' ≫ eqToHom (_ : X C j' = X C j) = d C i j",
"tactic": "obtain rfl := c.next_eq rij rij'"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j : ι\nrij rij' : ComplexShape.Rel c i j\n⊢ d C i j ≫ eqToHom (_ : X C j = X C j) = d C i j",
"tactic": "simp only [eqToHom_refl, comp_id]"
}
] |
[
322,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MonoidHom.map_zpow'
|
[] |
[
1303,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1301,
11
] |
Mathlib/Order/Filter/NAry.lean
|
Filter.NeBot.map₂
|
[] |
[
134,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
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