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/Analysis/NormedSpace/BoundedLinearMaps.lean
|
ContinuousLinearMap.isBoundedLinearMap_comp_left
|
[] |
[
459,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
457,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.StronglyMeasurable.const_smul
|
[] |
[
467,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
465,
11
] |
Mathlib/Control/Traversable/Lemmas.lean
|
Traversable.naturality_pf
|
[
{
"state_after": "case h\nt : Type u → Type u\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nα β γ : Type u\ng : α → F β\nh : β → G γ\nf✝ : β → γ\nx : t β\nη : ApplicativeTransformation F G\nf : α → F β\nx✝ : t α\n⊢ traverse ((fun {α} => ApplicativeTransformation.app η α) ∘ f) x✝ =\n ((fun {α} => ApplicativeTransformation.app η α) ∘ traverse f) x✝",
"state_before": "t : Type u → Type u\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nα β γ : Type u\ng : α → F β\nh : β → G γ\nf✝ : β → γ\nx : t β\nη : ApplicativeTransformation F G\nf : α → F β\n⊢ traverse ((fun {α} => ApplicativeTransformation.app η α) ∘ f) =\n (fun {α} => ApplicativeTransformation.app η α) ∘ traverse f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nt : Type u → Type u\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nα β γ : Type u\ng : α → F β\nh : β → G γ\nf✝ : β → γ\nx : t β\nη : ApplicativeTransformation F G\nf : α → F β\nx✝ : t α\n⊢ traverse ((fun {α} => ApplicativeTransformation.app η α) ∘ f) x✝ =\n ((fun {α} => ApplicativeTransformation.app η α) ∘ traverse f) x✝",
"tactic": "rw [comp_apply, naturality]"
}
] |
[
149,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
expSeries_summable_of_mem_ball'
|
[] |
[
225,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
TrivSqZeroExt.fst_comp_inr
|
[] |
[
156,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
InfHom.coe_id
|
[] |
[
575,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.closedBall_subset_closedBall
|
[] |
[
599,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
598,
1
] |
Mathlib/CategoryTheory/Monoidal/Opposite.lean
|
CategoryTheory.mop_id_unmop
|
[] |
[
150,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.add_one_pos
|
[
{
"state_after": "case zero\nm : ℕ\ni : Fin (zero + 1)\nh : i < last zero\n⊢ 0 < i + 1\n\ncase succ\nm n✝ : ℕ\ni : Fin (Nat.succ n✝ + 1)\nh : i < last (Nat.succ n✝)\n⊢ 0 < i + 1",
"state_before": "n m : ℕ\ni : Fin (n + 1)\nh : i < last n\n⊢ 0 < i + 1",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\nm : ℕ\ni : Fin (zero + 1)\nh : i < last zero\n⊢ 0 < i + 1",
"tactic": "exact absurd h (Nat.not_lt_zero _)"
},
{
"state_after": "case succ\nm n✝ : ℕ\ni : Fin (Nat.succ n✝ + 1)\nh✝ : ↑i < Nat.succ n✝\nh : ↑i + 1 < Nat.succ n✝ + 1\n⊢ 0 < i + 1",
"state_before": "case succ\nm n✝ : ℕ\ni : Fin (Nat.succ n✝ + 1)\nh : i < last (Nat.succ n✝)\n⊢ 0 < i + 1",
"tactic": "rw [lt_iff_val_lt_val, val_last, ← add_lt_add_iff_right 1] at h"
},
{
"state_after": "case succ\nm n✝ : ℕ\ni : Fin (Nat.succ n✝ + 1)\nh✝ : ↑i < Nat.succ n✝\nh : ↑i + 1 < Nat.succ n✝ + 1\n⊢ 0 < ↑i + 1",
"state_before": "case succ\nm n✝ : ℕ\ni : Fin (Nat.succ n✝ + 1)\nh✝ : ↑i < Nat.succ n✝\nh : ↑i + 1 < Nat.succ n✝ + 1\n⊢ 0 < i + 1",
"tactic": "rw [lt_iff_val_lt_val, val_add, val_zero, val_one, Nat.mod_eq_of_lt h]"
},
{
"state_after": "no goals",
"state_before": "case succ\nm n✝ : ℕ\ni : Fin (Nat.succ n✝ + 1)\nh✝ : ↑i < Nat.succ n✝\nh : ↑i + 1 < Nat.succ n✝ + 1\n⊢ 0 < ↑i + 1",
"tactic": "exact Nat.zero_lt_succ _"
}
] |
[
835,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
830,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
Module.DualBases.lc_coeffs
|
[
{
"state_after": "R : Type u_2\nM : Type u_1\nι : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\nm : M\n⊢ ∀ (i : ι), ↑(ε i) (lc e (coeffs h m) - m) = 0",
"state_before": "R : Type u_2\nM : Type u_1\nι : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\nm : M\n⊢ lc e (coeffs h m) = m",
"tactic": "refine' eq_of_sub_eq_zero (h.Total _)"
},
{
"state_after": "R : Type u_2\nM : Type u_1\nι : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\nm : M\ni : ι\n⊢ ↑(ε i) (lc e (coeffs h m) - m) = 0",
"state_before": "R : Type u_2\nM : Type u_1\nι : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\nm : M\n⊢ ∀ (i : ι), ↑(ε i) (lc e (coeffs h m) - m) = 0",
"tactic": "intro i"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nM : Type u_1\nι : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\nm : M\ni : ι\n⊢ ↑(ε i) (lc e (coeffs h m) - m) = 0",
"tactic": "simp [LinearMap.map_sub, h.dual_lc, sub_eq_zero]"
}
] |
[
722,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
719,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.rotate_eq_drop_append_take
|
[
{
"state_after": "α : Type u\nl : List α\nn : ℕ\n⊢ n ≤ length l → rotate' l n = drop n l ++ take n l",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ n ≤ length l → rotate l n = drop n l ++ take n l",
"tactic": "rw [rotate_eq_rotate']"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ n ≤ length l → rotate' l n = drop n l ++ take n l",
"tactic": "exact rotate'_eq_drop_append_take"
}
] |
[
144,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Std/Data/Int/DivMod.lean
|
Int.dvd_zero
|
[] |
[
593,
75
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
593,
11
] |
Mathlib/Data/List/Sort.lean
|
List.sorted_mergeSort
|
[
{
"state_after": "no goals",
"state_before": "α : Type uu\nr : α → α → Prop\ninst✝² : DecidableRel r\ninst✝¹ : IsTotal α r\ninst✝ : IsTrans α r\n⊢ Sorted r (mergeSort r [])",
"tactic": "simp [mergeSort]"
},
{
"state_after": "no goals",
"state_before": "α : Type uu\nr : α → α → Prop\ninst✝² : DecidableRel r\ninst✝¹ : IsTotal α r\ninst✝ : IsTrans α r\na : α\n⊢ Sorted r (mergeSort r [a])",
"tactic": "simp [mergeSort]"
},
{
"state_after": "case mk\nα : Type uu\nr : α → α → Prop\ninst✝² : DecidableRel r\ninst✝¹ : IsTotal α r\ninst✝ : IsTrans α r\na b : α\nl l₁ l₂ : List α\ne : split (a :: b :: l) = (l₁, l₂)\n⊢ Sorted r (mergeSort r (a :: b :: l))",
"state_before": "α : Type uu\nr : α → α → Prop\ninst✝² : DecidableRel r\ninst✝¹ : IsTotal α r\ninst✝ : IsTrans α r\na b : α\nl : List α\n⊢ Sorted r (mergeSort r (a :: b :: l))",
"tactic": "cases' e : split (a :: b :: l) with l₁ l₂"
},
{
"state_after": "case mk.intro\nα : Type uu\nr : α → α → Prop\ninst✝² : DecidableRel r\ninst✝¹ : IsTotal α r\ninst✝ : IsTrans α r\na b : α\nl l₁ l₂ : List α\ne : split (a :: b :: l) = (l₁, l₂)\nh₁ : length l₁ < length (a :: b :: l)\nh₂ : length l₂ < length (a :: b :: l)\n⊢ Sorted r (mergeSort r (a :: b :: l))",
"state_before": "case mk\nα : Type uu\nr : α → α → Prop\ninst✝² : DecidableRel r\ninst✝¹ : IsTotal α r\ninst✝ : IsTrans α r\na b : α\nl l₁ l₂ : List α\ne : split (a :: b :: l) = (l₁, l₂)\n⊢ Sorted r (mergeSort r (a :: b :: l))",
"tactic": "cases' length_split_lt e with h₁ h₂"
},
{
"state_after": "case mk.intro\nα : Type uu\nr : α → α → Prop\ninst✝² : DecidableRel r\ninst✝¹ : IsTotal α r\ninst✝ : IsTrans α r\na b : α\nl l₁ l₂ : List α\ne : split (a :: b :: l) = (l₁, l₂)\nh₁ : length l₁ < length (a :: b :: l)\nh₂ : length l₂ < length (a :: b :: l)\n⊢ Sorted r (merge r (mergeSort r l₁) (mergeSort r l₂))",
"state_before": "case mk.intro\nα : Type uu\nr : α → α → Prop\ninst✝² : DecidableRel r\ninst✝¹ : IsTotal α r\ninst✝ : IsTrans α r\na b : α\nl l₁ l₂ : List α\ne : split (a :: b :: l) = (l₁, l₂)\nh₁ : length l₁ < length (a :: b :: l)\nh₂ : length l₂ < length (a :: b :: l)\n⊢ Sorted r (mergeSort r (a :: b :: l))",
"tactic": "rw [mergeSort_cons_cons r e]"
},
{
"state_after": "no goals",
"state_before": "case mk.intro\nα : Type uu\nr : α → α → Prop\ninst✝² : DecidableRel r\ninst✝¹ : IsTotal α r\ninst✝ : IsTrans α r\na b : α\nl l₁ l₂ : List α\ne : split (a :: b :: l) = (l₁, l₂)\nh₁ : length l₁ < length (a :: b :: l)\nh₂ : length l₂ < length (a :: b :: l)\n⊢ Sorted r (merge r (mergeSort r l₁) (mergeSort r l₂))",
"tactic": "exact (sorted_mergeSort l₁).merge (sorted_mergeSort l₂)"
}
] |
[
450,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/RingTheory/PowerSeries/WellKnown.lean
|
PowerSeries.coeff_sin_bit1
|
[
{
"state_after": "no goals",
"state_before": "A : Type u_1\nA' : Type ?u.354098\ninst✝⁷ : Ring A\ninst✝⁶ : Ring A'\ninst✝⁵ : Algebra ℚ A\ninst✝⁴ : Algebra ℚ A'\ninst✝³ : Ring A\ninst✝² : Ring A'\ninst✝¹ : Algebra ℚ A\ninst✝ : Algebra ℚ A'\nn : ℕ\nf : A →+* A'\n⊢ ↑(coeff A (bit1 n)) (sin A) = (-1) ^ n * ↑(coeff A (bit1 n)) (exp A)",
"tactic": "rw [sin, coeff_mk, if_neg n.not_even_bit1, Nat.bit1_div_two, ← mul_one_div, map_mul, map_pow,\n map_neg, map_one, coeff_exp]"
}
] |
[
116,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Data/List/Join.lean
|
List.join_filter_isEmpty_eq_false
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1507\ninst✝ : DecidablePred fun l => isEmpty l = false\nL : List (List α)\n⊢ join (filter (fun l => decide (isEmpty l = false)) ([] :: L)) = join ([] :: L)",
"tactic": "simp [join_filter_isEmpty_eq_false (L := L), isEmpty_iff_eq_nil]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1507\ninst✝ : DecidablePred fun l => isEmpty l = false\na : α\nl : List α\nL : List (List α)\ncons_not_empty : isEmpty (a :: l) = false\n⊢ join (filter (fun l => decide (isEmpty l = false)) ((a :: l) :: L)) = join ((a :: l) :: L)",
"state_before": "α : Type u_1\nβ : Type ?u.1507\ninst✝ : DecidablePred fun l => isEmpty l = false\na : α\nl : List α\nL : List (List α)\n⊢ join (filter (fun l => decide (isEmpty l = false)) ((a :: l) :: L)) = join ((a :: l) :: L)",
"tactic": "have cons_not_empty : isEmpty (a :: l) = false := rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1507\ninst✝ : DecidablePred fun l => isEmpty l = false\na : α\nl : List α\nL : List (List α)\ncons_not_empty : isEmpty (a :: l) = false\n⊢ join (filter (fun l => decide (isEmpty l = false)) ((a :: l) :: L)) = join ((a :: l) :: L)",
"tactic": "simp [join_filter_isEmpty_eq_false (L := L), cons_not_empty]"
}
] |
[
59,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Icc_union_Ici_eq_Ici
|
[] |
[
1339,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1338,
1
] |
Mathlib/Algebra/Hom/Aut.lean
|
MulAut.one_def
|
[] |
[
84,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.abs_exp_sub_one_sub_id_le
|
[
{
"state_after": "x : ℝ\nhx : abs' x ≤ 1\n⊢ abs' (exp x - 1 - x) ≤ abs' x ^ 2",
"state_before": "x : ℝ\nhx : abs' x ≤ 1\n⊢ abs' (exp x - 1 - x) ≤ x ^ 2",
"tactic": "rw [← _root_.sq_abs]"
},
{
"state_after": "x : ℝ\nhx : abs' x ≤ 1\nthis : ↑Complex.abs ↑x ≤ 1\n⊢ abs' (exp x - 1 - x) ≤ abs' x ^ 2",
"state_before": "x : ℝ\nhx : abs' x ≤ 1\n⊢ abs' (exp x - 1 - x) ≤ abs' x ^ 2",
"tactic": "have : Complex.abs x ≤ 1 := by exact_mod_cast hx"
},
{
"state_after": "x : ℝ\nhx : abs' x ≤ 1\nthis✝ : ↑Complex.abs ↑x ≤ 1\nthis : ↑Complex.abs (Complex.exp ↑x - 1 - ↑x) ≤ ↑Complex.abs ↑x ^ 2\n⊢ abs' (exp x - 1 - x) ≤ abs' x ^ 2",
"state_before": "x : ℝ\nhx : abs' x ≤ 1\nthis : ↑Complex.abs ↑x ≤ 1\n⊢ abs' (exp x - 1 - x) ≤ abs' x ^ 2",
"tactic": "have := Complex.abs_exp_sub_one_sub_id_le this"
},
{
"state_after": "x : ℝ\nhx : abs' x ≤ 1\nthis✝ : ↑Complex.abs ↑x ≤ 1\nthis : abs' (exp x - 1 - x) ≤ abs' x ^ 2\n⊢ abs' (exp x - 1 - x) ≤ abs' x ^ 2",
"state_before": "x : ℝ\nhx : abs' x ≤ 1\nthis✝ : ↑Complex.abs ↑x ≤ 1\nthis : ↑Complex.abs (Complex.exp ↑x - 1 - ↑x) ≤ ↑Complex.abs ↑x ^ 2\n⊢ abs' (exp x - 1 - x) ≤ abs' x ^ 2",
"tactic": "rw [← ofReal_one, ← ofReal_exp, ← ofReal_sub, ← ofReal_sub, abs_ofReal, abs_ofReal] at this"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : abs' x ≤ 1\nthis✝ : ↑Complex.abs ↑x ≤ 1\nthis : abs' (exp x - 1 - x) ≤ abs' x ^ 2\n⊢ abs' (exp x - 1 - x) ≤ abs' x ^ 2",
"tactic": "exact this"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : abs' x ≤ 1\n⊢ ↑Complex.abs ↑x ≤ 1",
"tactic": "exact_mod_cast hx"
}
] |
[
1740,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1733,
1
] |
Mathlib/Data/Prod/Basic.lean
|
Function.Bijective.Prod_map
|
[] |
[
326,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
norm_sub_sq
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2406393\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ‖x - y‖ ^ 2 = ‖x‖ ^ 2 - 2 * ↑re (inner x y) + ‖y‖ ^ 2",
"tactic": "rw [sub_eq_add_neg, @norm_add_sq 𝕜 _ _ _ _ x (-y), norm_neg, inner_neg_right, map_neg, mul_neg,\n sub_eq_add_neg]"
}
] |
[
1066,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1064,
1
] |
Mathlib/Control/Bitraversable/Lemmas.lean
|
Bitraversable.tfst_eq_fst_id
|
[
{
"state_after": "no goals",
"state_before": "t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ✝ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : IsLawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα α' β : Type u\nf : α → α'\nx : t α β\n⊢ tfst (pure ∘ f) x = pure (fst f x)",
"tactic": "apply bitraverse_eq_bimap_id"
}
] |
[
118,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Combinatorics/DoubleCounting.lean
|
Finset.coe_bipartiteAbove
|
[] |
[
70,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
continuousWithinAt_of_locally_uniform_approx_of_continuousWithinAt
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\n⊢ ContinuousWithinAt f s x",
"tactic": "refine Uniform.continuousWithinAt_iff'_left.2 fun u₀ hu₀ => ?_"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"tactic": "obtain ⟨u₁, h₁, u₁₀⟩ : ∃ u ∈ 𝓤 β, u ○ u ⊆ u₀ := comp_mem_uniformity_sets hu₀"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\nu₂ : Set (β × β)\nh₂ : u₂ ∈ 𝓤 β\nhsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂\nu₂₁ : u₂ ○ u₂ ⊆ u₁\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"tactic": "obtain ⟨u₂, h₂, hsymm, u₂₁⟩ : ∃ u ∈ 𝓤 β, (∀ {a b}, (a, b) ∈ u → (b, a) ∈ u) ∧ u ○ u ⊆ u₁ :=\n comp_symm_of_uniformity h₁"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF✝ : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\nu₂ : Set (β × β)\nh₂ : u₂ ∈ 𝓤 β\nhsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂\nu₂₁ : u₂ ○ u₂ ⊆ u₁\nt : Set α\ntx : t ∈ 𝓝[s] x\nF : α → β\nhFc : ContinuousWithinAt F s x\nhF : ∀ (y : α), y ∈ t → (f y, F y) ∈ u₂\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\nu₂ : Set (β × β)\nh₂ : u₂ ∈ 𝓤 β\nhsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂\nu₂₁ : u₂ ○ u₂ ⊆ u₁\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"tactic": "rcases L u₂ h₂ with ⟨t, tx, F, hFc, hF⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF✝ : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\nu₂ : Set (β × β)\nh₂ : u₂ ∈ 𝓤 β\nhsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂\nu₂₁ : u₂ ○ u₂ ⊆ u₁\nt : Set α\ntx : t ∈ 𝓝[s] x\nF : α → β\nhFc : ContinuousWithinAt F s x\nhF : ∀ (y : α), y ∈ t → (f y, F y) ∈ u₂\nA : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F y) ∈ u₂\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF✝ : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\nu₂ : Set (β × β)\nh₂ : u₂ ∈ 𝓤 β\nhsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂\nu₂₁ : u₂ ○ u₂ ⊆ u₁\nt : Set α\ntx : t ∈ 𝓝[s] x\nF : α → β\nhFc : ContinuousWithinAt F s x\nhF : ∀ (y : α), y ∈ t → (f y, F y) ∈ u₂\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"tactic": "have A : ∀ᶠ y in 𝓝[s] x, (f y, F y) ∈ u₂ := Eventually.mono tx hF"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF✝ : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\nu₂ : Set (β × β)\nh₂ : u₂ ∈ 𝓤 β\nhsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂\nu₂₁ : u₂ ○ u₂ ⊆ u₁\nt : Set α\ntx : t ∈ 𝓝[s] x\nF : α → β\nhFc : ContinuousWithinAt F s x\nhF : ∀ (y : α), y ∈ t → (f y, F y) ∈ u₂\nA : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F y) ∈ u₂\nB : ∀ᶠ (y : α) in 𝓝[s] x, (F y, F x) ∈ u₂\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF✝ : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\nu₂ : Set (β × β)\nh₂ : u₂ ∈ 𝓤 β\nhsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂\nu₂₁ : u₂ ○ u₂ ⊆ u₁\nt : Set α\ntx : t ∈ 𝓝[s] x\nF : α → β\nhFc : ContinuousWithinAt F s x\nhF : ∀ (y : α), y ∈ t → (f y, F y) ∈ u₂\nA : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F y) ∈ u₂\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"tactic": "have B : ∀ᶠ y in 𝓝[s] x, (F y, F x) ∈ u₂ := Uniform.continuousWithinAt_iff'_left.1 hFc h₂"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF✝ : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\nu₂ : Set (β × β)\nh₂ : u₂ ∈ 𝓤 β\nhsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂\nu₂₁ : u₂ ○ u₂ ⊆ u₁\nt : Set α\ntx : t ∈ 𝓝[s] x\nF : α → β\nhFc : ContinuousWithinAt F s x\nhF : ∀ (y : α), y ∈ t → (f y, F y) ∈ u₂\nA : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F y) ∈ u₂\nB : ∀ᶠ (y : α) in 𝓝[s] x, (F y, F x) ∈ u₂\nC : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F x) ∈ u₁\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF✝ : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\nu₂ : Set (β × β)\nh₂ : u₂ ∈ 𝓤 β\nhsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂\nu₂₁ : u₂ ○ u₂ ⊆ u₁\nt : Set α\ntx : t ∈ 𝓝[s] x\nF : α → β\nhFc : ContinuousWithinAt F s x\nhF : ∀ (y : α), y ∈ t → (f y, F y) ∈ u₂\nA : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F y) ∈ u₂\nB : ∀ᶠ (y : α) in 𝓝[s] x, (F y, F x) ∈ u₂\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"tactic": "have C : ∀ᶠ y in 𝓝[s] x, (f y, F x) ∈ u₁ :=\n (A.and B).mono fun y hy => u₂₁ (prod_mk_mem_compRel hy.1 hy.2)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF✝ : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\nu₂ : Set (β × β)\nh₂ : u₂ ∈ 𝓤 β\nhsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂\nu₂₁ : u₂ ○ u₂ ⊆ u₁\nt : Set α\ntx : t ∈ 𝓝[s] x\nF : α → β\nhFc : ContinuousWithinAt F s x\nhF : ∀ (y : α), y ∈ t → (f y, F y) ∈ u₂\nA : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F y) ∈ u₂\nB : ∀ᶠ (y : α) in 𝓝[s] x, (F y, F x) ∈ u₂\nC : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F x) ∈ u₁\nthis : (F x, f x) ∈ u₁\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF✝ : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\nu₂ : Set (β × β)\nh₂ : u₂ ∈ 𝓤 β\nhsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂\nu₂₁ : u₂ ○ u₂ ⊆ u₁\nt : Set α\ntx : t ∈ 𝓝[s] x\nF : α → β\nhFc : ContinuousWithinAt F s x\nhF : ∀ (y : α), y ∈ t → (f y, F y) ∈ u₂\nA : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F y) ∈ u₂\nB : ∀ᶠ (y : α) in 𝓝[s] x, (F y, F x) ∈ u₂\nC : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F x) ∈ u₁\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"tactic": "have : (F x, f x) ∈ u₁ :=\n u₂₁ (prod_mk_mem_compRel (refl_mem_uniformity h₂) (hsymm (A.self_of_nhdsWithin hx)))"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF✝ : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nhx : x ∈ s\nL : ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∃ t, t ∈ 𝓝[s] x ∧ ∃ F, ContinuousWithinAt F s x ∧ ∀ (y : α), y ∈ t → (f y, F y) ∈ u\nu₀ : Set (β × β)\nhu₀ : u₀ ∈ 𝓤 β\nu₁ : Set (β × β)\nh₁ : u₁ ∈ 𝓤 β\nu₁₀ : u₁ ○ u₁ ⊆ u₀\nu₂ : Set (β × β)\nh₂ : u₂ ∈ 𝓤 β\nhsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂\nu₂₁ : u₂ ○ u₂ ⊆ u₁\nt : Set α\ntx : t ∈ 𝓝[s] x\nF : α → β\nhFc : ContinuousWithinAt F s x\nhF : ∀ (y : α), y ∈ t → (f y, F y) ∈ u₂\nA : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F y) ∈ u₂\nB : ∀ᶠ (y : α) in 𝓝[s] x, (F y, F x) ∈ u₂\nC : ∀ᶠ (y : α) in 𝓝[s] x, (f y, F x) ∈ u₁\nthis : (F x, f x) ∈ u₁\n⊢ u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)",
"tactic": "exact C.mono fun y hy => u₁₀ (prod_mk_mem_compRel hy this)"
}
] |
[
827,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
813,
1
] |
Mathlib/Tactic/IntervalCases.lean
|
Mathlib.Tactic.IntervalCases.of_le_left
|
[] |
[
138,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Topology/Basic.lean
|
map_mem_closure
|
[] |
[
1774,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1772,
1
] |
Mathlib/Topology/Instances/Irrational.lean
|
Irrational.eventually_forall_le_dist_cast_rat_of_den_le
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : Irrational x\nn : ℕ\nε : ℝ\nH : ∀ (k : ℕ), k ≤ n → ∀ (m : ℤ), ε ≤ dist x (↑m / ↑k)\nr : ℚ\nhr : r.den ≤ n\n⊢ ε ≤ dist x ↑r",
"tactic": "simpa only [Rat.cast_def] using H r.den hr r.num"
}
] |
[
101,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.ite_diff_self
|
[] |
[
2262,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2261,
1
] |
Mathlib/Topology/Separation.lean
|
SeparatedNhds.disjoint_closure_left
|
[] |
[
143,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
ZFSet.mem_funs
|
[
{
"state_after": "no goals",
"state_before": "x y f : ZFSet\n⊢ f ∈ funs x y ↔ IsFunc x y f",
"tactic": "simp [funs, IsFunc]"
}
] |
[
1364,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1364,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
lie_lie
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nt : R\nx y z : L\nm n : M\n⊢ ⁅⁅x, y⁆, m⁆ = ⁅x, ⁅y, m⁆⁆ - ⁅y, ⁅x, m⁆⁆",
"tactic": "rw [leibniz_lie, add_sub_cancel]"
}
] |
[
212,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/CategoryTheory/Abelian/Pseudoelements.lean
|
CategoryTheory.Abelian.Pseudoelement.zero_morphism_ext
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : ∀ (a : Pseudoelement P), pseudoApply f a = 0\n⊢ 𝟙 P ≫ f = 0",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : ∀ (a : Pseudoelement P), pseudoApply f a = 0\n⊢ f = 0",
"tactic": "rw [← Category.id_comp f]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : ∀ (a : Pseudoelement P), pseudoApply f a = 0\n⊢ 𝟙 P ≫ f = 0",
"tactic": "exact (pseudoZero_iff (𝟙 P ≫ f : Over Q)).1 (h (𝟙 P))"
}
] |
[
286,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/Order/OmegaCompletePartialOrder.lean
|
OmegaCompletePartialOrder.ContinuousHom.bind_continuous'
|
[
{
"state_after": "α : Type u\nα' : Type ?u.58987\nβ✝ : Type v\nβ' : Type ?u.58992\nγ✝ : Type ?u.58995\nφ : Type ?u.58998\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β✝\ninst✝³ : OmegaCompletePartialOrder γ✝\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\nβ γ : Type v\nf : α → Part β\ng : α → β → Part γ\nhf : Monotone f\nhf' : Continuous { toFun := f, monotone' := hf }\nhg : Monotone g\nhg' : Continuous { toFun := g, monotone' := hg }\nc : Chain α\n⊢ ↑(OrderHom.bind { toFun := f, monotone' := hf } { toFun := g, monotone' := hg }) (ωSup c) =\n ωSup (Chain.map c (OrderHom.bind { toFun := f, monotone' := hf } { toFun := g, monotone' := hg }))",
"state_before": "α : Type u\nα' : Type ?u.58987\nβ✝ : Type v\nβ' : Type ?u.58992\nγ✝ : Type ?u.58995\nφ : Type ?u.58998\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β✝\ninst✝³ : OmegaCompletePartialOrder γ✝\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\nβ γ : Type v\nf : α → Part β\ng : α → β → Part γ\nhf : Monotone f\nhf' : Continuous { toFun := f, monotone' := hf }\nhg : Monotone g\nhg' : Continuous { toFun := g, monotone' := hg }\n⊢ Continuous (OrderHom.bind { toFun := f, monotone' := hf } { toFun := g, monotone' := hg })",
"tactic": "intro c"
},
{
"state_after": "α : Type u\nα' : Type ?u.58987\nβ✝ : Type v\nβ' : Type ?u.58992\nγ✝ : Type ?u.58995\nφ : Type ?u.58998\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β✝\ninst✝³ : OmegaCompletePartialOrder γ✝\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\nβ γ : Type v\nf : α → Part β\ng : α → β → Part γ\nhf : Monotone f\nhf' : Continuous { toFun := f, monotone' := hf }\nhg : Monotone g\nhg' : Continuous { toFun := g, monotone' := hg }\nc : Chain α\n⊢ ↑(OrderHom.bind { toFun := f, monotone' := hf } { toFun := g, monotone' := hg }) (ωSup c) =\n ↑{ toFun := f, monotone' := hf } (ωSup c) >>= ↑{ toFun := g, monotone' := hg } (ωSup c)",
"state_before": "α : Type u\nα' : Type ?u.58987\nβ✝ : Type v\nβ' : Type ?u.58992\nγ✝ : Type ?u.58995\nφ : Type ?u.58998\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β✝\ninst✝³ : OmegaCompletePartialOrder γ✝\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\nβ γ : Type v\nf : α → Part β\ng : α → β → Part γ\nhf : Monotone f\nhf' : Continuous { toFun := f, monotone' := hf }\nhg : Monotone g\nhg' : Continuous { toFun := g, monotone' := hg }\nc : Chain α\n⊢ ↑(OrderHom.bind { toFun := f, monotone' := hf } { toFun := g, monotone' := hg }) (ωSup c) =\n ωSup (Chain.map c (OrderHom.bind { toFun := f, monotone' := hf } { toFun := g, monotone' := hg }))",
"tactic": "rw [ωSup_bind, ← hf', ← hg']"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nα' : Type ?u.58987\nβ✝ : Type v\nβ' : Type ?u.58992\nγ✝ : Type ?u.58995\nφ : Type ?u.58998\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β✝\ninst✝³ : OmegaCompletePartialOrder γ✝\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\nβ γ : Type v\nf : α → Part β\ng : α → β → Part γ\nhf : Monotone f\nhf' : Continuous { toFun := f, monotone' := hf }\nhg : Monotone g\nhg' : Continuous { toFun := g, monotone' := hg }\nc : Chain α\n⊢ ↑(OrderHom.bind { toFun := f, monotone' := hf } { toFun := g, monotone' := hg }) (ωSup c) =\n ↑{ toFun := f, monotone' := hf } (ωSup c) >>= ↑{ toFun := g, monotone' := hg } (ωSup c)",
"tactic": "rfl"
}
] |
[
659,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
655,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
dist_mul_mul_le_of_le
|
[] |
[
1436,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1434,
1
] |
Mathlib/Topology/MetricSpace/Antilipschitz.lean
|
AntilipschitzWith.properSpace
|
[
{
"state_after": "α✝ : Type ?u.96684\nβ : Type u_2\nγ : Type ?u.96690\ninst✝³ : PseudoMetricSpace α✝\ninst✝² : PseudoMetricSpace β\nK✝ : ℝ≥0\nf✝ : α✝ → β\nα : Type u_1\ninst✝¹ : MetricSpace α\nK : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K f\nf_cont : Continuous f\nhf : Function.Surjective f\nx₀ : β\nr : ℝ\n⊢ IsCompact (closedBall x₀ r)",
"state_before": "α✝ : Type ?u.96684\nβ : Type u_2\nγ : Type ?u.96690\ninst✝³ : PseudoMetricSpace α✝\ninst✝² : PseudoMetricSpace β\nK✝ : ℝ≥0\nf✝ : α✝ → β\nα : Type u_1\ninst✝¹ : MetricSpace α\nK : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K f\nf_cont : Continuous f\nhf : Function.Surjective f\n⊢ ProperSpace β",
"tactic": "refine ⟨fun x₀ r => ?_⟩"
},
{
"state_after": "α✝ : Type ?u.96684\nβ : Type u_2\nγ : Type ?u.96690\ninst✝³ : PseudoMetricSpace α✝\ninst✝² : PseudoMetricSpace β\nK✝¹ : ℝ≥0\nf✝ : α✝ → β\nα : Type u_1\ninst✝¹ : MetricSpace α\nK✝ : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K✝ f\nf_cont : Continuous f\nhf : Function.Surjective f\nx₀ : β\nr : ℝ\nK : Set α := f ⁻¹' closedBall x₀ r\n⊢ IsCompact (closedBall x₀ r)",
"state_before": "α✝ : Type ?u.96684\nβ : Type u_2\nγ : Type ?u.96690\ninst✝³ : PseudoMetricSpace α✝\ninst✝² : PseudoMetricSpace β\nK✝ : ℝ≥0\nf✝ : α✝ → β\nα : Type u_1\ninst✝¹ : MetricSpace α\nK : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K f\nf_cont : Continuous f\nhf : Function.Surjective f\nx₀ : β\nr : ℝ\n⊢ IsCompact (closedBall x₀ r)",
"tactic": "let K := f ⁻¹' closedBall x₀ r"
},
{
"state_after": "α✝ : Type ?u.96684\nβ : Type u_2\nγ : Type ?u.96690\ninst✝³ : PseudoMetricSpace α✝\ninst✝² : PseudoMetricSpace β\nK✝¹ : ℝ≥0\nf✝ : α✝ → β\nα : Type u_1\ninst✝¹ : MetricSpace α\nK✝ : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K✝ f\nf_cont : Continuous f\nhf : Function.Surjective f\nx₀ : β\nr : ℝ\nK : Set α := f ⁻¹' closedBall x₀ r\nA : IsClosed K\n⊢ IsCompact (closedBall x₀ r)",
"state_before": "α✝ : Type ?u.96684\nβ : Type u_2\nγ : Type ?u.96690\ninst✝³ : PseudoMetricSpace α✝\ninst✝² : PseudoMetricSpace β\nK✝¹ : ℝ≥0\nf✝ : α✝ → β\nα : Type u_1\ninst✝¹ : MetricSpace α\nK✝ : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K✝ f\nf_cont : Continuous f\nhf : Function.Surjective f\nx₀ : β\nr : ℝ\nK : Set α := f ⁻¹' closedBall x₀ r\n⊢ IsCompact (closedBall x₀ r)",
"tactic": "have A : IsClosed K := isClosed_ball.preimage f_cont"
},
{
"state_after": "α✝ : Type ?u.96684\nβ : Type u_2\nγ : Type ?u.96690\ninst✝³ : PseudoMetricSpace α✝\ninst✝² : PseudoMetricSpace β\nK✝¹ : ℝ≥0\nf✝ : α✝ → β\nα : Type u_1\ninst✝¹ : MetricSpace α\nK✝ : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K✝ f\nf_cont : Continuous f\nhf : Function.Surjective f\nx₀ : β\nr : ℝ\nK : Set α := f ⁻¹' closedBall x₀ r\nA : IsClosed K\nB : Metric.Bounded K\n⊢ IsCompact (closedBall x₀ r)",
"state_before": "α✝ : Type ?u.96684\nβ : Type u_2\nγ : Type ?u.96690\ninst✝³ : PseudoMetricSpace α✝\ninst✝² : PseudoMetricSpace β\nK✝¹ : ℝ≥0\nf✝ : α✝ → β\nα : Type u_1\ninst✝¹ : MetricSpace α\nK✝ : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K✝ f\nf_cont : Continuous f\nhf : Function.Surjective f\nx₀ : β\nr : ℝ\nK : Set α := f ⁻¹' closedBall x₀ r\nA : IsClosed K\n⊢ IsCompact (closedBall x₀ r)",
"tactic": "have B : Bounded K := hK.bounded_preimage bounded_closedBall"
},
{
"state_after": "α✝ : Type ?u.96684\nβ : Type u_2\nγ : Type ?u.96690\ninst✝³ : PseudoMetricSpace α✝\ninst✝² : PseudoMetricSpace β\nK✝¹ : ℝ≥0\nf✝ : α✝ → β\nα : Type u_1\ninst✝¹ : MetricSpace α\nK✝ : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K✝ f\nf_cont : Continuous f\nhf : Function.Surjective f\nx₀ : β\nr : ℝ\nK : Set α := f ⁻¹' closedBall x₀ r\nA : IsClosed K\nB : Metric.Bounded K\nthis : IsCompact K\n⊢ IsCompact (closedBall x₀ r)",
"state_before": "α✝ : Type ?u.96684\nβ : Type u_2\nγ : Type ?u.96690\ninst✝³ : PseudoMetricSpace α✝\ninst✝² : PseudoMetricSpace β\nK✝¹ : ℝ≥0\nf✝ : α✝ → β\nα : Type u_1\ninst✝¹ : MetricSpace α\nK✝ : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K✝ f\nf_cont : Continuous f\nhf : Function.Surjective f\nx₀ : β\nr : ℝ\nK : Set α := f ⁻¹' closedBall x₀ r\nA : IsClosed K\nB : Metric.Bounded K\n⊢ IsCompact (closedBall x₀ r)",
"tactic": "have : IsCompact K := isCompact_iff_isClosed_bounded.2 ⟨A, B⟩"
},
{
"state_after": "case h.e'_3\nα✝ : Type ?u.96684\nβ : Type u_2\nγ : Type ?u.96690\ninst✝³ : PseudoMetricSpace α✝\ninst✝² : PseudoMetricSpace β\nK✝¹ : ℝ≥0\nf✝ : α✝ → β\nα : Type u_1\ninst✝¹ : MetricSpace α\nK✝ : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K✝ f\nf_cont : Continuous f\nhf : Function.Surjective f\nx₀ : β\nr : ℝ\nK : Set α := f ⁻¹' closedBall x₀ r\nA : IsClosed K\nB : Metric.Bounded K\nthis : IsCompact K\n⊢ closedBall x₀ r = f '' K",
"state_before": "α✝ : Type ?u.96684\nβ : Type u_2\nγ : Type ?u.96690\ninst✝³ : PseudoMetricSpace α✝\ninst✝² : PseudoMetricSpace β\nK✝¹ : ℝ≥0\nf✝ : α✝ → β\nα : Type u_1\ninst✝¹ : MetricSpace α\nK✝ : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K✝ f\nf_cont : Continuous f\nhf : Function.Surjective f\nx₀ : β\nr : ℝ\nK : Set α := f ⁻¹' closedBall x₀ r\nA : IsClosed K\nB : Metric.Bounded K\nthis : IsCompact K\n⊢ IsCompact (closedBall x₀ r)",
"tactic": "convert this.image f_cont"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nα✝ : Type ?u.96684\nβ : Type u_2\nγ : Type ?u.96690\ninst✝³ : PseudoMetricSpace α✝\ninst✝² : PseudoMetricSpace β\nK✝¹ : ℝ≥0\nf✝ : α✝ → β\nα : Type u_1\ninst✝¹ : MetricSpace α\nK✝ : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K✝ f\nf_cont : Continuous f\nhf : Function.Surjective f\nx₀ : β\nr : ℝ\nK : Set α := f ⁻¹' closedBall x₀ r\nA : IsClosed K\nB : Metric.Bounded K\nthis : IsCompact K\n⊢ closedBall x₀ r = f '' K",
"tactic": "exact (hf.image_preimage _).symm"
}
] |
[
249,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
11
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
StrictAnti.const_mul_of_neg
|
[] |
[
746,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
744,
1
] |
Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
|
TopCat.embedding_pullback_to_prod
|
[] |
[
185,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Std/Data/Rat/Lemmas.lean
|
Rat.inv_zero
|
[
{
"state_after": "⊢ (if h : 0.num < 0 then mk' (-↑0.den) (Int.natAbs 0.num)\n else if h : 0.num > 0 then mk' (↑0.den) (Int.natAbs 0.num) else 0) =\n 0",
"state_before": "⊢ Rat.inv 0 = 0",
"tactic": "unfold Rat.inv"
},
{
"state_after": "no goals",
"state_before": "⊢ (if h : 0.num < 0 then mk' (-↑0.den) (Int.natAbs 0.num)\n else if h : 0.num > 0 then mk' (↑0.den) (Int.natAbs 0.num) else 0) =\n 0",
"tactic": "simp"
}
] |
[
297,
7
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
295,
19
] |
Mathlib/Analysis/BoxIntegral/Basic.lean
|
BoxIntegral.integralSum_sub_partitions
|
[
{
"state_after": "ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ₁ π₂ : TaggedPrepartition I\nh₁ : IsPartition π₁\nh₂ : IsPartition π₂\n⊢ ∑ J in (infPrepartition π₁ π₂.toPrepartition).toPrepartition.boxes,\n ↑(↑vol J) (f (tag (infPrepartition π₁ π₂.toPrepartition) J)) -\n ∑ J in (infPrepartition π₂ π₁.toPrepartition).toPrepartition.boxes,\n ↑(↑vol J) (f (tag (infPrepartition π₂ π₁.toPrepartition) J)) =\n ∑ x in (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes, ↑(↑vol x) (f (tag (infPrepartition π₁ π₂.toPrepartition) x)) -\n ∑ x in (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes, ↑(↑vol x) (f (tag (infPrepartition π₂ π₁.toPrepartition) x))",
"state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ₁ π₂ : TaggedPrepartition I\nh₁ : IsPartition π₁\nh₂ : IsPartition π₂\n⊢ integralSum f vol π₁ - integralSum f vol π₂ =\n ∑ J in (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n (↑(↑vol J) (f (tag (infPrepartition π₁ π₂.toPrepartition) J)) -\n ↑(↑vol J) (f (tag (infPrepartition π₂ π₁.toPrepartition) J)))",
"tactic": "rw [← integralSum_inf_partition f vol π₁ h₂, ← integralSum_inf_partition f vol π₂ h₁,\n integralSum, integralSum, Finset.sum_sub_distrib]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ₁ π₂ : TaggedPrepartition I\nh₁ : IsPartition π₁\nh₂ : IsPartition π₂\n⊢ ∑ J in (infPrepartition π₁ π₂.toPrepartition).toPrepartition.boxes,\n ↑(↑vol J) (f (tag (infPrepartition π₁ π₂.toPrepartition) J)) -\n ∑ J in (infPrepartition π₂ π₁.toPrepartition).toPrepartition.boxes,\n ↑(↑vol J) (f (tag (infPrepartition π₂ π₁.toPrepartition) J)) =\n ∑ x in (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes, ↑(↑vol x) (f (tag (infPrepartition π₁ π₂.toPrepartition) x)) -\n ∑ x in (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes, ↑(↑vol x) (f (tag (infPrepartition π₂ π₁.toPrepartition) x))",
"tactic": "simp only [infPrepartition_toPrepartition, inf_comm]"
}
] |
[
125,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMinOn.neg
|
[] |
[
409,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
408,
8
] |
Mathlib/Order/Heyting/Regular.lean
|
Heyting.isRegular_of_boolean
|
[] |
[
260,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
FormalMultilinearSeries.le_changeOriginSeries_radius
|
[] |
[
1238,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1236,
1
] |
Mathlib/Order/Heyting/Hom.lean
|
HeytingHom.id_comp
|
[] |
[
355,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
mul_self_nonneg
|
[] |
[
1097,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1096,
1
] |
Mathlib/Topology/Basic.lean
|
isOpen_empty
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\n⊢ IsOpen (⋃₀ ∅)",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\n⊢ IsOpen ∅",
"tactic": "rw [← sUnion_empty]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\n⊢ IsOpen (⋃₀ ∅)",
"tactic": "exact isOpen_sUnion fun a => False.elim"
}
] |
[
160,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
9
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.op_norm_zero
|
[
{
"state_after": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.403828\nE : Type u_1\nEₗ : Type ?u.403834\nF : Type u_2\nFₗ : Type ?u.403840\nG : Type ?u.403843\nGₗ : Type ?u.403846\n𝓕 : Type ?u.403849\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx x✝ : E\n⊢ ‖↑0 x✝‖ = 0",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.403828\nE : Type u_1\nEₗ : Type ?u.403834\nF : Type u_2\nFₗ : Type ?u.403840\nG : Type ?u.403843\nGₗ : Type ?u.403846\n𝓕 : Type ?u.403849\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx x✝ : E\n⊢ ‖↑0 x✝‖ = 0 * ‖x✝‖",
"tactic": "rw [MulZeroClass.zero_mul]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.403828\nE : Type u_1\nEₗ : Type ?u.403834\nF : Type u_2\nFₗ : Type ?u.403840\nG : Type ?u.403843\nGₗ : Type ?u.403846\n𝓕 : Type ?u.403849\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx x✝ : E\n⊢ ‖↑0 x✝‖ = 0",
"tactic": "exact norm_zero"
}
] |
[
281,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.erase_sub
|
[] |
[
937,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
935,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_one
|
[] |
[
202,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.continuousOn_sub
|
[
{
"state_after": "α : Type ?u.107946\nβ : Type ?u.107949\nγ : Type ?u.107952\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ ∀ (x : ℝ≥0∞ × ℝ≥0∞), x ∈ {p | p ≠ (⊤, ⊤)} → ContinuousWithinAt (fun p => p.fst - p.snd) {p | p ≠ (⊤, ⊤)} x",
"state_before": "α : Type ?u.107946\nβ : Type ?u.107949\nγ : Type ?u.107952\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ ContinuousOn (fun p => p.fst - p.snd) {p | p ≠ (⊤, ⊤)}",
"tactic": "rw [ContinuousOn]"
},
{
"state_after": "case mk\nα : Type ?u.107946\nβ : Type ?u.107949\nγ : Type ?u.107952\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx✝ y✝ z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nx y : ℝ≥0∞\nhp : (x, y) ∈ {p | p ≠ (⊤, ⊤)}\n⊢ ContinuousWithinAt (fun p => p.fst - p.snd) {p | p ≠ (⊤, ⊤)} (x, y)",
"state_before": "α : Type ?u.107946\nβ : Type ?u.107949\nγ : Type ?u.107952\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ ∀ (x : ℝ≥0∞ × ℝ≥0∞), x ∈ {p | p ≠ (⊤, ⊤)} → ContinuousWithinAt (fun p => p.fst - p.snd) {p | p ≠ (⊤, ⊤)} x",
"tactic": "rintro ⟨x, y⟩ hp"
},
{
"state_after": "case mk\nα : Type ?u.107946\nβ : Type ?u.107949\nγ : Type ?u.107952\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx✝ y✝ z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nx y : ℝ≥0∞\nhp : ¬(x = ⊤ ∧ y = ⊤)\n⊢ ContinuousWithinAt (fun p => p.fst - p.snd) {p | p ≠ (⊤, ⊤)} (x, y)",
"state_before": "case mk\nα : Type ?u.107946\nβ : Type ?u.107949\nγ : Type ?u.107952\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx✝ y✝ z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nx y : ℝ≥0∞\nhp : (x, y) ∈ {p | p ≠ (⊤, ⊤)}\n⊢ ContinuousWithinAt (fun p => p.fst - p.snd) {p | p ≠ (⊤, ⊤)} (x, y)",
"tactic": "simp only [Ne.def, Set.mem_setOf_eq, Prod.mk.inj_iff] at hp"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Type ?u.107946\nβ : Type ?u.107949\nγ : Type ?u.107952\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx✝ y✝ z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nx y : ℝ≥0∞\nhp : ¬(x = ⊤ ∧ y = ⊤)\n⊢ ContinuousWithinAt (fun p => p.fst - p.snd) {p | p ≠ (⊤, ⊤)} (x, y)",
"tactic": "refine' tendsto_nhdsWithin_of_tendsto_nhds (tendsto_sub (not_and_or.mp hp))"
}
] |
[
445,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
440,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.symm_map_nhds_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.31160\nδ : Type ?u.31163\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nx : α\nhx : x ∈ e.source\n⊢ 𝓝 (↑(LocalHomeomorph.symm e) (↑e x)) = 𝓝 x",
"tactic": "rw [e.left_inv hx]"
}
] |
[
372,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean
|
Real.differentiableWithinAt_arccos_Iic
|
[] |
[
171,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometry.diam_image
|
[] |
[
198,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.isUnit_leadingCoeff_mul_left_eq_zero_iff
|
[
{
"state_after": "case mp\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\nq : R[X]\n⊢ q * p = 0 → q = 0\n\ncase mpr\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\nq : R[X]\n⊢ q = 0 → q * p = 0",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\nq : R[X]\n⊢ q * p = 0 ↔ q = 0",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\nq : R[X]\nhp : q * p = 0\n⊢ q = 0",
"state_before": "case mp\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\nq : R[X]\n⊢ q * p = 0 → q = 0",
"tactic": "intro hp"
},
{
"state_after": "case mp\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\nq : R[X]\nhp : q * p * ↑C ↑(IsUnit.unit h)⁻¹ = 0\n⊢ q = 0",
"state_before": "case mp\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\nq : R[X]\nhp : (fun x => x * ↑C ↑(IsUnit.unit h)⁻¹) (q * p) = (fun x => x * ↑C ↑(IsUnit.unit h)⁻¹) 0\n⊢ q = 0",
"tactic": "simp only [zero_mul] at hp"
},
{
"state_after": "case mp.h\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\nq : R[X]\nhp : q * (p * ↑C ↑(IsUnit.unit h)⁻¹) = 0\n⊢ Monic (p * ↑C ↑(IsUnit.unit h)⁻¹)",
"state_before": "case mp\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\nq : R[X]\nhp : q * p * ↑C ↑(IsUnit.unit h)⁻¹ = 0\n⊢ q = 0",
"tactic": "rwa [mul_assoc, Monic.mul_left_eq_zero_iff] at hp"
},
{
"state_after": "case mp.h\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\nq : R[X]\nhp : q * (p * ↑C ↑(IsUnit.unit h)⁻¹) = 0\n⊢ leadingCoeff p * ↑(IsUnit.unit h)⁻¹ = 1",
"state_before": "case mp.h\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\nq : R[X]\nhp : q * (p * ↑C ↑(IsUnit.unit h)⁻¹) = 0\n⊢ Monic (p * ↑C ↑(IsUnit.unit h)⁻¹)",
"tactic": "refine' monic_mul_C_of_leadingCoeff_mul_eq_one _"
},
{
"state_after": "no goals",
"state_before": "case mp.h\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\nq : R[X]\nhp : q * (p * ↑C ↑(IsUnit.unit h)⁻¹) = 0\n⊢ leadingCoeff p * ↑(IsUnit.unit h)⁻¹ = 1",
"tactic": "simp [Units.mul_inv_eq_iff_eq_mul, IsUnit.unit_spec]"
},
{
"state_after": "case mpr\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\n⊢ 0 * p = 0",
"state_before": "case mpr\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\nq : R[X]\n⊢ q = 0 → q * p = 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : IsUnit (leadingCoeff p)\n⊢ 0 * p = 0",
"tactic": "rw [zero_mul]"
}
] |
[
550,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.prod_coe
|
[] |
[
953,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
951,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.vars_monomial_single
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.263576\nr✝ : R\ne✝ : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf : R →+* S\ni : σ\ne : ℕ\nr : R\nhe : e ≠ 0\nhr : r ≠ 0\n⊢ vars (↑(monomial (Finsupp.single i e)) r) = {i}",
"tactic": "rw [vars_monomial hr, Finsupp.support_single_ne_zero _ he]"
}
] |
[
467,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
465,
1
] |
Mathlib/Topology/MetricSpace/MetricSeparated.lean
|
IsMetricSeparated.subset_compl_right
|
[] |
[
64,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
AddMonoidHom.coe_addMonoidHom_mkRingHomOfMulSelfOfTwoNeZero
|
[
{
"state_after": "case h\nF : Type ?u.127699\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127708\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : CommRing β\nf : β →+ α\nh : ∀ (x : β), ↑f (x * x) = ↑f x * ↑f x\nh_two : 2 ≠ 0\nh_one : ↑f 1 = 1\n⊢ ∀ (x : β), ↑↑(mkRingHomOfMulSelfOfTwoNeZero f h h_two h_one) x = ↑f x",
"state_before": "F : Type ?u.127699\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127708\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : CommRing β\nf : β →+ α\nh : ∀ (x : β), ↑f (x * x) = ↑f x * ↑f x\nh_two : 2 ≠ 0\nh_one : ↑f 1 = 1\n⊢ ↑(mkRingHomOfMulSelfOfTwoNeZero f h h_two h_one) = f",
"tactic": "apply AddMonoidHom.ext"
},
{
"state_after": "case h\nF : Type ?u.127699\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127708\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : CommRing β\nf : β →+ α\nh : ∀ (x : β), ↑f (x * x) = ↑f x * ↑f x\nh_two : 2 ≠ 0\nh_one : ↑f 1 = 1\nx✝ : β\n⊢ ↑↑(mkRingHomOfMulSelfOfTwoNeZero f h h_two h_one) x✝ = ↑f x✝",
"state_before": "case h\nF : Type ?u.127699\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127708\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : CommRing β\nf : β →+ α\nh : ∀ (x : β), ↑f (x * x) = ↑f x * ↑f x\nh_two : 2 ≠ 0\nh_one : ↑f 1 = 1\n⊢ ∀ (x : β), ↑↑(mkRingHomOfMulSelfOfTwoNeZero f h h_two h_one) x = ↑f x",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type ?u.127699\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127708\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : CommRing β\nf : β →+ α\nh : ∀ (x : β), ↑f (x * x) = ↑f x * ↑f x\nh_two : 2 ≠ 0\nh_one : ↑f 1 = 1\nx✝ : β\n⊢ ↑↑(mkRingHomOfMulSelfOfTwoNeZero f h h_two h_one) x✝ = ↑f x✝",
"tactic": "rfl"
}
] |
[
805,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/Data/ENat/Basic.lean
|
ENat.sub_top
|
[] |
[
153,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Order/Cover.lean
|
AntisymmRel.trans_covby
|
[] |
[
314,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/Probability/CondCount.lean
|
ProbabilityTheory.condCount_eq_one_of
|
[
{
"state_after": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\nht : s ⊆ t\nthis : IsProbabilityMeasure (condCount s)\n⊢ ↑↑(condCount s) t = 1",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\nht : s ⊆ t\n⊢ ↑↑(condCount s) t = 1",
"tactic": "haveI := condCount_isProbabilityMeasure hs hs'"
},
{
"state_after": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\nht : s ⊆ t\nthis : IsProbabilityMeasure (condCount s)\n⊢ ¬↑↑(condCount s) t < 1",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\nht : s ⊆ t\nthis : IsProbabilityMeasure (condCount s)\n⊢ ↑↑(condCount s) t = 1",
"tactic": "refine' eq_of_le_of_not_lt prob_le_one _"
},
{
"state_after": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\nht : s ⊆ t\nthis : IsProbabilityMeasure (condCount s)\n⊢ ↑↑(condCount s) s ≤ ↑↑(condCount s) t",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\nht : s ⊆ t\nthis : IsProbabilityMeasure (condCount s)\n⊢ ¬↑↑(condCount s) t < 1",
"tactic": "rw [not_lt, ← condCount_self hs hs']"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\nht : s ⊆ t\nthis : IsProbabilityMeasure (condCount s)\n⊢ ↑↑(condCount s) s ≤ ↑↑(condCount s) t",
"tactic": "exact measure_mono ht"
}
] |
[
117,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Data/Finset/Preimage.lean
|
Finset.sigma_preimage_mk
|
[
{
"state_after": "case a\nα : Type u\nβ✝ : Type v\nι : Sort w\nγ : Type x\nβ : α → Type u_1\ninst✝ : DecidableEq α\ns : Finset ((a : α) × β a)\nt : Finset α\nx : (i : α) × β i\n⊢ (x ∈ Finset.sigma t fun a => preimage s (Sigma.mk a) (_ : InjOn (Sigma.mk a) (Sigma.mk a ⁻¹' ↑s))) ↔\n x ∈ filter (fun a => a.fst ∈ t) s",
"state_before": "α : Type u\nβ✝ : Type v\nι : Sort w\nγ : Type x\nβ : α → Type u_1\ninst✝ : DecidableEq α\ns : Finset ((a : α) × β a)\nt : Finset α\n⊢ (Finset.sigma t fun a => preimage s (Sigma.mk a) (_ : InjOn (Sigma.mk a) (Sigma.mk a ⁻¹' ↑s))) =\n filter (fun a => a.fst ∈ t) s",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ✝ : Type v\nι : Sort w\nγ : Type x\nβ : α → Type u_1\ninst✝ : DecidableEq α\ns : Finset ((a : α) × β a)\nt : Finset α\nx : (i : α) × β i\n⊢ (x ∈ Finset.sigma t fun a => preimage s (Sigma.mk a) (_ : InjOn (Sigma.mk a) (Sigma.mk a ⁻¹' ↑s))) ↔\n x ∈ filter (fun a => a.fst ∈ t) s",
"tactic": "simp [and_comm]"
}
] |
[
127,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_nonneg
|
[] |
[
286,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.sum_le_iSup
|
[
{
"state_after": "α β ι : Type u\nf : ι → Cardinal\n⊢ sum f ≤ lift (#ι) * iSup f",
"state_before": "α β ι : Type u\nf : ι → Cardinal\n⊢ sum f ≤ (#ι) * iSup f",
"tactic": "rw [← lift_id (#ι)]"
},
{
"state_after": "no goals",
"state_before": "α β ι : Type u\nf : ι → Cardinal\n⊢ sum f ≤ lift (#ι) * iSup f",
"tactic": "exact sum_le_iSup_lift f"
}
] |
[
999,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
997,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
|
Polynomial.trailingDegree_monomial
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nha : a ≠ 0\n⊢ ↑n = ↑n",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nha : a ≠ 0\n⊢ trailingDegree (↑(monomial n) a) = ↑n",
"tactic": "rw [trailingDegree, support_monomial n ha, min_singleton]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nha : a ≠ 0\n⊢ ↑n = ↑n",
"tactic": "rfl"
}
] |
[
204,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/CategoryTheory/Limits/ExactFunctor.lean
|
CategoryTheory.ExactFunctor.forget_obj
|
[] |
[
181,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Analysis/SpecialFunctions/Arsinh.lean
|
Real.sinh_arsinh
|
[
{
"state_after": "x✝ y x : ℝ\n⊢ (x + sqrt (1 + x ^ 2) - (-x + sqrt (1 + x ^ 2))) / 2 = x",
"state_before": "x✝ y x : ℝ\n⊢ sinh (arsinh x) = x",
"tactic": "rw [sinh_eq, ← arsinh_neg, exp_arsinh, exp_arsinh, neg_sq]"
},
{
"state_after": "no goals",
"state_before": "x✝ y x : ℝ\n⊢ (x + sqrt (1 + x ^ 2) - (-x + sqrt (1 + x ^ 2))) / 2 = x",
"tactic": "field_simp"
}
] |
[
82,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Topology/Separation.lean
|
compl_singleton_mem_nhds_iff
|
[] |
[
618,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
617,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.Bounded.ediam_ne_top
|
[] |
[
2653,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2652,
1
] |
Mathlib/Data/Nat/Count.lean
|
Nat.count_zero
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ → Prop\ninst✝ : DecidablePred p\n⊢ count p 0 = 0",
"tactic": "rw [count, List.range_zero, List.countp, List.countp.go]"
}
] |
[
42,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
isClosed_of_mem_irreducibleComponents
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.184023\nπ : ι → Type ?u.184028\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s : Set α\nH : s ∈ irreducibleComponents α\n⊢ s = closure s",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.184023\nπ : ι → Type ?u.184028\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s : Set α\nH : s ∈ irreducibleComponents α\n⊢ IsClosed s",
"tactic": "rw [← closure_eq_iff_isClosed, eq_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.184023\nπ : ι → Type ?u.184028\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s : Set α\nH : s ∈ irreducibleComponents α\n⊢ s = closure s",
"tactic": "exact subset_closure.antisymm (H.2 H.1.closure subset_closure)"
}
] |
[
1769,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1766,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
hasFDerivWithinAt_insert
|
[
{
"state_after": "case inl\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.177830\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.177925\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\n⊢ HasFDerivWithinAt f f' (insert x s) x ↔ HasFDerivWithinAt f f' s x\n\ncase inr\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.177830\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.177925\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\ny : E\nh : x ≠ y\n⊢ HasFDerivWithinAt f f' (insert y s) x ↔ HasFDerivWithinAt f f' s x",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.177830\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.177925\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\ny : E\n⊢ HasFDerivWithinAt f f' (insert y s) x ↔ HasFDerivWithinAt f f' s x",
"tactic": "rcases eq_or_ne x y with (rfl | h)"
},
{
"state_after": "case inr\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.177830\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.177925\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\ny : E\nh : x ≠ y\nhf : HasFDerivWithinAt f f' s x\n⊢ s ∈ 𝓝[insert y s] x",
"state_before": "case inr\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.177830\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.177925\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\ny : E\nh : x ≠ y\n⊢ HasFDerivWithinAt f f' (insert y s) x ↔ HasFDerivWithinAt f f' s x",
"tactic": "refine' ⟨fun h => h.mono <| subset_insert y s, fun hf => hf.mono_of_mem _⟩"
},
{
"state_after": "no goals",
"state_before": "case inr\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.177830\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.177925\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\ny : E\nh : x ≠ y\nhf : HasFDerivWithinAt f f' s x\n⊢ s ∈ 𝓝[insert y s] x",
"tactic": "simp_rw [nhdsWithin_insert_of_ne h, self_mem_nhdsWithin]"
},
{
"state_after": "case inl\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.177830\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.177925\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\n⊢ ((fun x' => f x' - f x - ↑f' (x' - x)) =o[𝓝[insert x s] x] fun x' => x' - x) ↔\n (fun x' => f x' - f x - ↑f' (x' - x)) =o[𝓝[s] x] fun x' => x' - x",
"state_before": "case inl\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.177830\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.177925\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\n⊢ HasFDerivWithinAt f f' (insert x s) x ↔ HasFDerivWithinAt f f' s x",
"tactic": "simp_rw [HasFDerivWithinAt, HasFDerivAtFilter]"
},
{
"state_after": "case inl.h\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.177830\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.177925\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\n⊢ f x - f x - ↑f' (x - x) = 0",
"state_before": "case inl\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.177830\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.177925\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\n⊢ ((fun x' => f x' - f x - ↑f' (x' - x)) =o[𝓝[insert x s] x] fun x' => x' - x) ↔\n (fun x' => f x' - f x - ↑f' (x' - x)) =o[𝓝[s] x] fun x' => x' - x",
"tactic": "apply Asymptotics.isLittleO_insert"
},
{
"state_after": "no goals",
"state_before": "case inl.h\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.177830\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.177925\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\n⊢ f x - f x - ↑f' (x - x) = 0",
"tactic": "simp only [sub_self, map_zero]"
}
] |
[
401,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
mem_vadd_const_affineSegment
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\nV : Type u_2\nV' : Type ?u.49755\nP : Type u_1\nP' : Type ?u.49761\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z : V\np : P\n⊢ z +ᵥ p ∈ affineSegment R (x +ᵥ p) (y +ᵥ p) ↔ z ∈ affineSegment R x y",
"tactic": "rw [← affineSegment_vadd_const_image, (vadd_right_injective p).mem_set_image]"
}
] |
[
127,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.le_induction
|
[
{
"state_after": "case refl\nm✝ n k m : ℕ\nP : (n : ℕ) → m ≤ n → Prop\nbase : P m (_ : m ≤ m)\nsucc : ∀ (n : ℕ) (hn : m ≤ n), P n hn → P (n + 1) (_ : m ≤ n + 1)\n⊢ P m (_ : Nat.le m m)\n\ncase step\nm✝ n k m : ℕ\nP : (n : ℕ) → m ≤ n → Prop\nbase : P m (_ : m ≤ m)\nsucc : ∀ (n : ℕ) (hn : m ≤ n), P n hn → P (n + 1) (_ : m ≤ n + 1)\n⊢ ∀ {m_1 : ℕ} (a : Nat.le m m_1), P m_1 a → P (Nat.succ m_1) (_ : Nat.le m (Nat.succ m_1))",
"state_before": "m✝ n k m : ℕ\nP : (n : ℕ) → m ≤ n → Prop\nbase : P m (_ : m ≤ m)\nsucc : ∀ (n : ℕ) (hn : m ≤ n), P n hn → P (n + 1) (_ : m ≤ n + 1)\n⊢ ∀ (n : ℕ) (hn : m ≤ n), P n hn",
"tactic": "apply Nat.le.rec"
},
{
"state_after": "no goals",
"state_before": "case refl\nm✝ n k m : ℕ\nP : (n : ℕ) → m ≤ n → Prop\nbase : P m (_ : m ≤ m)\nsucc : ∀ (n : ℕ) (hn : m ≤ n), P n hn → P (n + 1) (_ : m ≤ n + 1)\n⊢ P m (_ : Nat.le m m)",
"tactic": "exact base"
},
{
"state_after": "case step\nm✝ n✝ k m : ℕ\nP : (n : ℕ) → m ≤ n → Prop\nbase : P m (_ : m ≤ m)\nsucc : ∀ (n : ℕ) (hn : m ≤ n), P n hn → P (n + 1) (_ : m ≤ n + 1)\nn : ℕ\nhn : Nat.le m n\n⊢ P n hn → P (Nat.succ n) (_ : Nat.le m (Nat.succ n))",
"state_before": "case step\nm✝ n k m : ℕ\nP : (n : ℕ) → m ≤ n → Prop\nbase : P m (_ : m ≤ m)\nsucc : ∀ (n : ℕ) (hn : m ≤ n), P n hn → P (n + 1) (_ : m ≤ n + 1)\n⊢ ∀ {m_1 : ℕ} (a : Nat.le m m_1), P m_1 a → P (Nat.succ m_1) (_ : Nat.le m (Nat.succ m_1))",
"tactic": "intros n hn"
},
{
"state_after": "no goals",
"state_before": "case step\nm✝ n✝ k m : ℕ\nP : (n : ℕ) → m ≤ n → Prop\nbase : P m (_ : m ≤ m)\nsucc : ∀ (n : ℕ) (hn : m ≤ n), P n hn → P (n + 1) (_ : m ≤ n + 1)\nn : ℕ\nhn : Nat.le m n\n⊢ P n hn → P (Nat.succ n) (_ : Nat.le m (Nat.succ n))",
"tactic": "apply succ n hn"
}
] |
[
488,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Mathlib/Combinatorics/Colex.lean
|
Colex.eq_iff
|
[] |
[
81,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
Associates.prod_coe
|
[] |
[
1246,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1244,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.restrictScalars_apply
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_3\nι' : Type ?u.1271219\nR : Type u_2\nR₂ : Type ?u.1271225\nK : Type ?u.1271228\nM : Type u_1\nM' : Type ?u.1271234\nM'' : Type ?u.1271237\nV : Type u\nV' : Type ?u.1271242\nS : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Nontrivial S\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Algebra R S\ninst✝³ : Module S M\ninst✝² : Module R M\ninst✝¹ : IsScalarTower R S M\ninst✝ : NoZeroSMulDivisors R S\nb : Basis ι S M\ni : ι\n⊢ ↑(↑(restrictScalars R b) i) = ↑b i",
"tactic": "simp only [Basis.restrictScalars, Basis.span_apply]"
}
] |
[
1661,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1660,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
DifferentiableWithinAt.hasFDerivWithinAt
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.253209\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.253304\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh : DifferentiableWithinAt 𝕜 f s x\n⊢ HasFDerivWithinAt f (if h : ∃ f', HasFDerivWithinAt f f' s x then choose h else 0) s x",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.253209\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.253304\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh : DifferentiableWithinAt 𝕜 f s x\n⊢ HasFDerivWithinAt f (fderivWithin 𝕜 f s x) s x",
"tactic": "dsimp only [fderivWithin]"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.253209\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.253304\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh : ∃ f', HasFDerivWithinAt f f' s x\n⊢ HasFDerivWithinAt f (if h : ∃ f', HasFDerivWithinAt f f' s x then choose h else 0) s x",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.253209\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.253304\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh : DifferentiableWithinAt 𝕜 f s x\n⊢ HasFDerivWithinAt f (if h : ∃ f', HasFDerivWithinAt f f' s x then choose h else 0) s x",
"tactic": "dsimp only [DifferentiableWithinAt] at h"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.253209\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.253304\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh : ∃ f', HasFDerivWithinAt f f' s x\n⊢ HasFDerivWithinAt f (choose h) s x",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.253209\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.253304\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh : ∃ f', HasFDerivWithinAt f f' s x\n⊢ HasFDerivWithinAt f (if h : ∃ f', HasFDerivWithinAt f f' s x then choose h else 0) s x",
"tactic": "rw [dif_pos h]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.253209\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.253304\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh : ∃ f', HasFDerivWithinAt f f' s x\n⊢ HasFDerivWithinAt f (choose h) s x",
"tactic": "exact Classical.choose_spec h"
}
] |
[
513,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
508,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.map_map
|
[] |
[
1280,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1279,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
NNReal.rpow_le_rpow_of_exponent_ge
|
[] |
[
185,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Analysis/SpecialFunctions/CompareExp.lean
|
Complex.IsExpCmpFilter.isLittleO_zpow_mul_exp
|
[
{
"state_after": "no goals",
"state_before": "l : Filter ℂ\nb₁ b₂ : ℝ\nhl : IsExpCmpFilter l\nhb : b₁ < b₂\nm n : ℤ\n⊢ (fun z => z ^ ↑m * exp (↑b₁ * z)) =o[l] fun z => z ^ ↑n * exp (↑b₂ * z)",
"tactic": "simpa only [cpow_int_cast] using hl.isLittleO_cpow_mul_exp hb m n"
}
] |
[
225,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Order.lean
|
hasSum_lt
|
[
{
"state_after": "ι : Type u_1\nκ : Type ?u.38034\nα : Type u_2\ninst✝³ : OrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalAddGroup α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na₁ a₂ : α\ni : ι\nh : f ≤ g\nhi : f i < g i\nhf : HasSum f a₁\nhg : HasSum g a₂\nthis : update f i 0 ≤ update g i 0\n⊢ a₁ < a₂",
"state_before": "ι : Type u_1\nκ : Type ?u.38034\nα : Type u_2\ninst✝³ : OrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalAddGroup α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na₁ a₂ : α\ni : ι\nh : f ≤ g\nhi : f i < g i\nhf : HasSum f a₁\nhg : HasSum g a₂\n⊢ a₁ < a₂",
"tactic": "have : update f i 0 ≤ update g i 0 := update_le_update_iff.mpr ⟨rfl.le, fun i _ => h i⟩"
},
{
"state_after": "ι : Type u_1\nκ : Type ?u.38034\nα : Type u_2\ninst✝³ : OrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalAddGroup α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na₁ a₂ : α\ni : ι\nh : f ≤ g\nhi : f i < g i\nhf : HasSum f a₁\nhg : HasSum g a₂\nthis✝ : update f i 0 ≤ update g i 0\nthis : 0 - f i + a₁ ≤ 0 - g i + a₂\n⊢ a₁ < a₂",
"state_before": "ι : Type u_1\nκ : Type ?u.38034\nα : Type u_2\ninst✝³ : OrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalAddGroup α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na₁ a₂ : α\ni : ι\nh : f ≤ g\nhi : f i < g i\nhf : HasSum f a₁\nhg : HasSum g a₂\nthis : update f i 0 ≤ update g i 0\n⊢ a₁ < a₂",
"tactic": "have : 0 - f i + a₁ ≤ 0 - g i + a₂ := hasSum_le this (hf.update i 0) (hg.update i 0)"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nκ : Type ?u.38034\nα : Type u_2\ninst✝³ : OrderedAddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalAddGroup α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na₁ a₂ : α\ni : ι\nh : f ≤ g\nhi : f i < g i\nhf : HasSum f a₁\nhg : HasSum g a₂\nthis✝ : update f i 0 ≤ update g i 0\nthis : 0 - f i + a₁ ≤ 0 - g i + a₂\n⊢ a₁ < a₂",
"tactic": "simpa only [zero_sub, add_neg_cancel_left] using add_lt_add_of_lt_of_le hi this"
}
] |
[
167,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
hasFDerivAtFilter_pi
|
[] |
[
404,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
Nat.toAdd_pow
|
[] |
[
1208,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1207,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.eq_comp_toLinearMap_symm
|
[
{
"state_after": "case mp.h\nR : Type ?u.193960\nR₁ : Type u_5\nR₂ : Type u_1\nR₃ : Type u_2\nk : Type ?u.193972\nS : Type ?u.193975\nM : Type ?u.193978\nM₁ : Type u_6\nM₂ : Type u_3\nM₃ : Type u_4\nN₁ : Type ?u.193990\nN₂ : Type ?u.193993\nN₃ : Type ?u.193996\nN₄ : Type ?u.193999\nι : Type ?u.194002\nM₄ : Type ?u.194005\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\nmodule_M : Module R M\nmodule_S_M₂ : Module S M₂\nσ : R →+* S\nσ' : S →+* R\nre₁ : RingHomInvPair σ σ'\nre₂ : RingHomInvPair σ' σ\ne e' : M ≃ₛₗ[σ] M₂\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : Module R₃ M₃\nmodule_N₁ : Module R₁ N₁\nmodule_N₂ : Module R₁ N₂\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ₂₁ : R₂ →+* R₁\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₃ : RingHomInvPair σ₂₃ σ₃₂\ninst✝³ : RingHomInvPair σ₁₃ σ₃₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\nre₃₂ : RingHomInvPair σ₃₂ σ₂₃\ninst✝² : RingHomInvPair σ₃₁ σ₁₃\ne₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂\ne₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃\ninst✝¹ : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃\ninst✝ : RingHomCompTriple σ₃₁ σ₁₂ σ₃₂\nf : M₂ →ₛₗ[σ₂₃] M₃\ng : M₁ →ₛₗ[σ₁₃] M₃\nH : f = LinearMap.comp g ↑(symm e₁₂)\nx✝ : M₁\n⊢ ↑(LinearMap.comp f ↑e₁₂) x✝ = ↑g x✝\n\ncase mpr.h\nR : Type ?u.193960\nR₁ : Type u_5\nR₂ : Type u_1\nR₃ : Type u_2\nk : Type ?u.193972\nS : Type ?u.193975\nM : Type ?u.193978\nM₁ : Type u_6\nM₂ : Type u_3\nM₃ : Type u_4\nN₁ : Type ?u.193990\nN₂ : Type ?u.193993\nN₃ : Type ?u.193996\nN₄ : Type ?u.193999\nι : Type ?u.194002\nM₄ : Type ?u.194005\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\nmodule_M : Module R M\nmodule_S_M₂ : Module S M₂\nσ : R →+* S\nσ' : S →+* R\nre₁ : RingHomInvPair σ σ'\nre₂ : RingHomInvPair σ' σ\ne e' : M ≃ₛₗ[σ] M₂\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : Module R₃ M₃\nmodule_N₁ : Module R₁ N₁\nmodule_N₂ : Module R₁ N₂\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ₂₁ : R₂ →+* R₁\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₃ : RingHomInvPair σ₂₃ σ₃₂\ninst✝³ : RingHomInvPair σ₁₃ σ₃₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\nre₃₂ : RingHomInvPair σ₃₂ σ₂₃\ninst✝² : RingHomInvPair σ₃₁ σ₁₃\ne₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂\ne₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃\ninst✝¹ : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃\ninst✝ : RingHomCompTriple σ₃₁ σ₁₂ σ₃₂\nf : M₂ →ₛₗ[σ₂₃] M₃\ng : M₁ →ₛₗ[σ₁₃] M₃\nH : LinearMap.comp f ↑e₁₂ = g\nx✝ : M₂\n⊢ ↑f x✝ = ↑(LinearMap.comp g ↑(symm e₁₂)) x✝",
"state_before": "R : Type ?u.193960\nR₁ : Type u_5\nR₂ : Type u_1\nR₃ : Type u_2\nk : Type ?u.193972\nS : Type ?u.193975\nM : Type ?u.193978\nM₁ : Type u_6\nM₂ : Type u_3\nM₃ : Type u_4\nN₁ : Type ?u.193990\nN₂ : Type ?u.193993\nN₃ : Type ?u.193996\nN₄ : Type ?u.193999\nι : Type ?u.194002\nM₄ : Type ?u.194005\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\nmodule_M : Module R M\nmodule_S_M₂ : Module S M₂\nσ : R →+* S\nσ' : S →+* R\nre₁ : RingHomInvPair σ σ'\nre₂ : RingHomInvPair σ' σ\ne e' : M ≃ₛₗ[σ] M₂\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : Module R₃ M₃\nmodule_N₁ : Module R₁ N₁\nmodule_N₂ : Module R₁ N₂\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ₂₁ : R₂ →+* R₁\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₃ : RingHomInvPair σ₂₃ σ₃₂\ninst✝³ : RingHomInvPair σ₁₃ σ₃₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\nre₃₂ : RingHomInvPair σ₃₂ σ₂₃\ninst✝² : RingHomInvPair σ₃₁ σ₁₃\ne₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂\ne₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃\ninst✝¹ : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃\ninst✝ : RingHomCompTriple σ₃₁ σ₁₂ σ₃₂\nf : M₂ →ₛₗ[σ₂₃] M₃\ng : M₁ →ₛₗ[σ₁₃] M₃\n⊢ f = LinearMap.comp g ↑(symm e₁₂) ↔ LinearMap.comp f ↑e₁₂ = g",
"tactic": "constructor <;> intro H <;> ext"
},
{
"state_after": "no goals",
"state_before": "case mp.h\nR : Type ?u.193960\nR₁ : Type u_5\nR₂ : Type u_1\nR₃ : Type u_2\nk : Type ?u.193972\nS : Type ?u.193975\nM : Type ?u.193978\nM₁ : Type u_6\nM₂ : Type u_3\nM₃ : Type u_4\nN₁ : Type ?u.193990\nN₂ : Type ?u.193993\nN₃ : Type ?u.193996\nN₄ : Type ?u.193999\nι : Type ?u.194002\nM₄ : Type ?u.194005\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\nmodule_M : Module R M\nmodule_S_M₂ : Module S M₂\nσ : R →+* S\nσ' : S →+* R\nre₁ : RingHomInvPair σ σ'\nre₂ : RingHomInvPair σ' σ\ne e' : M ≃ₛₗ[σ] M₂\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : Module R₃ M₃\nmodule_N₁ : Module R₁ N₁\nmodule_N₂ : Module R₁ N₂\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ₂₁ : R₂ →+* R₁\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₃ : RingHomInvPair σ₂₃ σ₃₂\ninst✝³ : RingHomInvPair σ₁₃ σ₃₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\nre₃₂ : RingHomInvPair σ₃₂ σ₂₃\ninst✝² : RingHomInvPair σ₃₁ σ₁₃\ne₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂\ne₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃\ninst✝¹ : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃\ninst✝ : RingHomCompTriple σ₃₁ σ₁₂ σ₃₂\nf : M₂ →ₛₗ[σ₂₃] M₃\ng : M₁ →ₛₗ[σ₁₃] M₃\nH : f = LinearMap.comp g ↑(symm e₁₂)\nx✝ : M₁\n⊢ ↑(LinearMap.comp f ↑e₁₂) x✝ = ↑g x✝",
"tactic": "simp [H, e₁₂.toEquiv.eq_comp_symm f g]"
},
{
"state_after": "no goals",
"state_before": "case mpr.h\nR : Type ?u.193960\nR₁ : Type u_5\nR₂ : Type u_1\nR₃ : Type u_2\nk : Type ?u.193972\nS : Type ?u.193975\nM : Type ?u.193978\nM₁ : Type u_6\nM₂ : Type u_3\nM₃ : Type u_4\nN₁ : Type ?u.193990\nN₂ : Type ?u.193993\nN₃ : Type ?u.193996\nN₄ : Type ?u.193999\nι : Type ?u.194002\nM₄ : Type ?u.194005\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\nmodule_M : Module R M\nmodule_S_M₂ : Module S M₂\nσ : R →+* S\nσ' : S →+* R\nre₁ : RingHomInvPair σ σ'\nre₂ : RingHomInvPair σ' σ\ne e' : M ≃ₛₗ[σ] M₂\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : Module R₃ M₃\nmodule_N₁ : Module R₁ N₁\nmodule_N₂ : Module R₁ N₂\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ₂₁ : R₂ →+* R₁\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₃ : RingHomInvPair σ₂₃ σ₃₂\ninst✝³ : RingHomInvPair σ₁₃ σ₃₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\nre₃₂ : RingHomInvPair σ₃₂ σ₂₃\ninst✝² : RingHomInvPair σ₃₁ σ₁₃\ne₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂\ne₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃\ninst✝¹ : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃\ninst✝ : RingHomCompTriple σ₃₁ σ₁₂ σ₃₂\nf : M₂ →ₛₗ[σ₂₃] M₃\ng : M₁ →ₛₗ[σ₁₃] M₃\nH : LinearMap.comp f ↑e₁₂ = g\nx✝ : M₂\n⊢ ↑f x✝ = ↑(LinearMap.comp g ↑(symm e₁₂)) x✝",
"tactic": "simp [← H, ← e₁₂.toEquiv.eq_comp_symm f g]"
}
] |
[
436,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.BoundedFormula.IsAtomic.isPrenex
|
[] |
[
764,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
763,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
|
tendsto_of_le_liminf_of_limsup_le
|
[] |
[
227,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Antitone.map_sSup_of_continuousAt
|
[] |
[
2718,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2715,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_eq_zero_iff
|
[
{
"state_after": "case refine'_1\nz : ℂ\nh : arg z = 0\n⊢ 0 ≤ z.re ∧ z.im = 0\n\ncase refine'_2\nz : ℂ\n⊢ 0 ≤ z.re ∧ z.im = 0 → arg z = 0",
"state_before": "z : ℂ\n⊢ arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0",
"tactic": "refine' ⟨fun h => _, _⟩"
},
{
"state_after": "case refine'_1\nz : ℂ\nh : arg z = 0\n⊢ 0 ≤ (↑(↑abs z) * (cos ↑0 + sin ↑0 * I)).re ∧ (↑(↑abs z) * (cos ↑0 + sin ↑0 * I)).im = 0",
"state_before": "case refine'_1\nz : ℂ\nh : arg z = 0\n⊢ 0 ≤ z.re ∧ z.im = 0",
"tactic": "rw [← abs_mul_cos_add_sin_mul_I z, h]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nz : ℂ\nh : arg z = 0\n⊢ 0 ≤ (↑(↑abs z) * (cos ↑0 + sin ↑0 * I)).re ∧ (↑(↑abs z) * (cos ↑0 + sin ↑0 * I)).im = 0",
"tactic": "simp [abs.nonneg]"
},
{
"state_after": "case refine'_2.mk\nx y : ℝ\n⊢ 0 ≤ { re := x, im := y }.re ∧ { re := x, im := y }.im = 0 → arg { re := x, im := y } = 0",
"state_before": "case refine'_2\nz : ℂ\n⊢ 0 ≤ z.re ∧ z.im = 0 → arg z = 0",
"tactic": "cases' z with x y"
},
{
"state_after": "case refine'_2.mk.intro\nx : ℝ\nh : 0 ≤ { re := x, im := 0 }.re\n⊢ arg { re := x, im := 0 } = 0",
"state_before": "case refine'_2.mk\nx y : ℝ\n⊢ 0 ≤ { re := x, im := y }.re ∧ { re := x, im := y }.im = 0 → arg { re := x, im := y } = 0",
"tactic": "rintro ⟨h, rfl : y = 0⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.mk.intro\nx : ℝ\nh : 0 ≤ { re := x, im := 0 }.re\n⊢ arg { re := x, im := 0 } = 0",
"tactic": "exact arg_ofReal_of_nonneg h"
}
] |
[
230,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/Topology/FiberBundle/Constructions.lean
|
Trivialization.prod_symm_apply
|
[] |
[
259,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.aeval_eq_eval₂Hom
|
[] |
[
1440,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1439,
1
] |
Mathlib/Topology/Algebra/Order/Floor.lean
|
ContinuousOn.comp_fract
|
[] |
[
220,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/CategoryTheory/Limits/Creates.lean
|
CategoryTheory.hasLimitsOfShape_of_hasLimitsOfShape_createsLimitsOfShape
|
[] |
[
171,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Algebra/Support.lean
|
Function.support_smul_subset_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type ?u.86818\nA : Type u_1\nB : Type u_2\nM : Type ?u.86827\nN : Type ?u.86830\nP : Type ?u.86833\nR : Type ?u.86836\nS : Type ?u.86839\nG : Type ?u.86842\nM₀ : Type ?u.86845\nG₀ : Type ?u.86848\nι : Sort ?u.86851\ninst✝² : AddMonoid A\ninst✝¹ : Monoid B\ninst✝ : DistribMulAction B A\nb : B\nf : α → A\nx : α\nhbf : x ∈ support (b • f)\nhf : f x = 0\n⊢ (b • f) x = 0",
"tactic": "rw [Pi.smul_apply, hf, smul_zero]"
}
] |
[
371,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/RingTheory/Henselian.lean
|
isLocalRingHom_of_le_jacobson_bot
|
[
{
"state_after": "case map_nonunit\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh : I ≤ Ideal.jacobson ⊥\n⊢ ∀ (a : R), IsUnit (↑(Ideal.Quotient.mk I) a) → IsUnit a",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh : I ≤ Ideal.jacobson ⊥\n⊢ IsLocalRingHom (Ideal.Quotient.mk I)",
"tactic": "constructor"
},
{
"state_after": "case map_nonunit\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\n⊢ IsUnit a",
"state_before": "case map_nonunit\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh : I ≤ Ideal.jacobson ⊥\n⊢ ∀ (a : R), IsUnit (↑(Ideal.Quotient.mk I) a) → IsUnit a",
"tactic": "intro a h"
},
{
"state_after": "case map_nonunit\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\nthis : IsUnit (↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a)\n⊢ IsUnit a",
"state_before": "case map_nonunit\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\n⊢ IsUnit a",
"tactic": "have : IsUnit (Ideal.Quotient.mk (Ideal.jacobson ⊥) a) := by\n rw [isUnit_iff_exists_inv] at *\n obtain ⟨b, hb⟩ := h\n obtain ⟨b, rfl⟩ := Ideal.Quotient.mk_surjective b\n use Ideal.Quotient.mk _ b\n rw [← (Ideal.Quotient.mk _).map_one, ← (Ideal.Quotient.mk _).map_mul, Ideal.Quotient.eq] at hb⊢\n exact h hb"
},
{
"state_after": "case map_nonunit.intro.mk\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\ny : R ⧸ Ideal.jacobson ⊥\nh1 : ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a * y = 1\nh2 : y * ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a = 1\n⊢ IsUnit a",
"state_before": "case map_nonunit\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\nthis : IsUnit (↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a)\n⊢ IsUnit a",
"tactic": "obtain ⟨⟨x, y, h1, h2⟩, rfl : x = _⟩ := this"
},
{
"state_after": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\ny : R\nh1 : ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a * ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) y = 1\nh2 : ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) y * ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a = 1\n⊢ IsUnit a",
"state_before": "case map_nonunit.intro.mk\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\ny : R ⧸ Ideal.jacobson ⊥\nh1 : ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a * y = 1\nh2 : y * ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a = 1\n⊢ IsUnit a",
"tactic": "obtain ⟨y, rfl⟩ := Ideal.Quotient.mk_surjective y"
},
{
"state_after": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\ny : R\nh1 : ∀ (y_1 : R), IsUnit ((a * y - 1) * y_1 + 1)\nh2 : ∀ (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\n⊢ IsUnit a",
"state_before": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\ny : R\nh1 : ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a * ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) y = 1\nh2 : ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) y * ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a = 1\n⊢ IsUnit a",
"tactic": "rw [← (Ideal.Quotient.mk _).map_mul, ← (Ideal.Quotient.mk _).map_one, Ideal.Quotient.eq,\n Ideal.mem_jacobson_bot] at h1 h2"
},
{
"state_after": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\ny : R\nh2 : ∀ (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\nh1 : IsUnit ((a * y - 1) * 1 + 1)\n⊢ IsUnit a",
"state_before": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\ny : R\nh1 : ∀ (y_1 : R), IsUnit ((a * y - 1) * y_1 + 1)\nh2 : ∀ (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\n⊢ IsUnit a",
"tactic": "specialize h1 1"
},
{
"state_after": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\ny : R\nh2 : ∀ (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\nh1 : IsUnit a ∧ IsUnit y\n⊢ IsUnit a",
"state_before": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\ny : R\nh2 : ∀ (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\nh1 : IsUnit ((a * y - 1) * 1 + 1)\n⊢ IsUnit a",
"tactic": "simp at h1"
},
{
"state_after": "no goals",
"state_before": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\ny : R\nh2 : ∀ (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\nh1 : IsUnit a ∧ IsUnit y\n⊢ IsUnit a",
"tactic": "exact h1.1"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : ∃ b, ↑(Ideal.Quotient.mk I) a * b = 1\n⊢ ∃ b, ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a * b = 1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : IsUnit (↑(Ideal.Quotient.mk I) a)\n⊢ IsUnit (↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a)",
"tactic": "rw [isUnit_iff_exists_inv] at *"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh : I ≤ Ideal.jacobson ⊥\na : R\nb : R ⧸ I\nhb : ↑(Ideal.Quotient.mk I) a * b = 1\n⊢ ∃ b, ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a * b = 1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ Ideal.jacobson ⊥\na : R\nh : ∃ b, ↑(Ideal.Quotient.mk I) a * b = 1\n⊢ ∃ b, ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a * b = 1",
"tactic": "obtain ⟨b, hb⟩ := h"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh : I ≤ Ideal.jacobson ⊥\na b : R\nhb : ↑(Ideal.Quotient.mk I) a * ↑(Ideal.Quotient.mk I) b = 1\n⊢ ∃ b, ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a * b = 1",
"state_before": "case intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh : I ≤ Ideal.jacobson ⊥\na : R\nb : R ⧸ I\nhb : ↑(Ideal.Quotient.mk I) a * b = 1\n⊢ ∃ b, ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a * b = 1",
"tactic": "obtain ⟨b, rfl⟩ := Ideal.Quotient.mk_surjective b"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh : I ≤ Ideal.jacobson ⊥\na b : R\nhb : ↑(Ideal.Quotient.mk I) a * ↑(Ideal.Quotient.mk I) b = 1\n⊢ ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a * ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) b = 1",
"state_before": "case intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh : I ≤ Ideal.jacobson ⊥\na b : R\nhb : ↑(Ideal.Quotient.mk I) a * ↑(Ideal.Quotient.mk I) b = 1\n⊢ ∃ b, ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a * b = 1",
"tactic": "use Ideal.Quotient.mk _ b"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh : I ≤ Ideal.jacobson ⊥\na b : R\nhb : a * b - 1 ∈ I\n⊢ a * b - 1 ∈ Ideal.jacobson ⊥",
"state_before": "case intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh : I ≤ Ideal.jacobson ⊥\na b : R\nhb : ↑(Ideal.Quotient.mk I) a * ↑(Ideal.Quotient.mk I) b = 1\n⊢ ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) a * ↑(Ideal.Quotient.mk (Ideal.jacobson ⊥)) b = 1",
"tactic": "rw [← (Ideal.Quotient.mk _).map_one, ← (Ideal.Quotient.mk _).map_mul, Ideal.Quotient.eq] at hb⊢"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh : I ≤ Ideal.jacobson ⊥\na b : R\nhb : a * b - 1 ∈ I\n⊢ a * b - 1 ∈ Ideal.jacobson ⊥",
"tactic": "exact h hb"
}
] |
[
85,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
le_div_iff'
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.23085\nα : Type u_1\nβ : Type ?u.23091\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nhc : 0 < c\n⊢ a ≤ b / c ↔ c * a ≤ b",
"tactic": "rw [mul_comm, le_div_iff hc]"
}
] |
[
143,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Algebra/Algebra/Bilinear.lean
|
LinearMap.mul_apply'
|
[] |
[
77,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Data/Set/Intervals/ProjIcc.lean
|
Set.IccExtend_left
|
[] |
[
120,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.coe_injective
|
[] |
[
165,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
11
] |
Mathlib/RingTheory/Derivation/Basic.lean
|
Derivation.coe_neg
|
[] |
[
427,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConcaveOn.convex_hypograph
|
[] |
[
259,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.sum_fintype
|
[
{
"state_after": "case h.h\nα : Type u_2\nβ : Type u_3\nι : Type u_1\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : { x // x ∈ kernel α β }\ninst✝ : Fintype ι\nκ : ι → { x // x ∈ kernel α β }\na : α\ns : Set β\n⊢ MeasurableSet s → ↑↑(↑(kernel.sum κ) a) s = ↑↑(↑(∑ i : ι, κ i) a) s",
"state_before": "α : Type u_2\nβ : Type u_3\nι : Type u_1\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : { x // x ∈ kernel α β }\ninst✝ : Fintype ι\nκ : ι → { x // x ∈ kernel α β }\n⊢ kernel.sum κ = ∑ i : ι, κ i",
"tactic": "ext a s"
},
{
"state_after": "case h.h\nα : Type u_2\nβ : Type u_3\nι : Type u_1\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : { x // x ∈ kernel α β }\ninst✝ : Fintype ι\nκ : ι → { x // x ∈ kernel α β }\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ ↑↑(↑(kernel.sum κ) a) s = ↑↑(↑(∑ i : ι, κ i) a) s",
"state_before": "case h.h\nα : Type u_2\nβ : Type u_3\nι : Type u_1\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : { x // x ∈ kernel α β }\ninst✝ : Fintype ι\nκ : ι → { x // x ∈ kernel α β }\na : α\ns : Set β\n⊢ MeasurableSet s → ↑↑(↑(kernel.sum κ) a) s = ↑↑(↑(∑ i : ι, κ i) a) s",
"tactic": "intro hs"
},
{
"state_after": "no goals",
"state_before": "case h.h\nα : Type u_2\nβ : Type u_3\nι : Type u_1\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : { x // x ∈ kernel α β }\ninst✝ : Fintype ι\nκ : ι → { x // x ∈ kernel α β }\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ ↑↑(↑(kernel.sum κ) a) s = ↑↑(↑(∑ i : ι, κ i) a) s",
"tactic": "simp only [sum_apply' κ a hs, finset_sum_apply' _ κ a s, tsum_fintype]"
}
] |
[
252,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
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