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 |
|---|---|---|---|---|---|---|
src/lean/Init/Data/Nat/Basic.lean
|
Nat.eq_of_mul_eq_mul_left
|
[] |
[
466,
71
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
464,
11
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
AddLECancellable.tsub_lt_tsub_iff_left_of_le
|
[] |
[
463,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
11
] |
Mathlib/CategoryTheory/Preadditive/Basic.lean
|
CategoryTheory.Preadditive.hasKernel_of_hasEqualizer
|
[] |
[
362,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/Data/Nat/Cast/Basic.lean
|
Nat.cast_max
|
[] |
[
179,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConvexOn.lt_left_of_right_lt'
|
[] |
[
781,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
772,
1
] |
Mathlib/Data/Part.lean
|
Part.sdiff_mem_sdiff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.95752\nγ : Type ?u.95755\ninst✝ : SDiff α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1, a_1 ∈ a ∧ ∃ a, a ∈ b ∧ a_1 \\ a = ma \\ mb",
"state_before": "α : Type u_1\nβ : Type ?u.95752\nγ : Type ?u.95755\ninst✝ : SDiff α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ma \\ mb ∈ a \\ b",
"tactic": "simp [sdiff_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.95752\nγ : Type ?u.95755\ninst✝ : SDiff α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1, a_1 ∈ a ∧ ∃ a, a ∈ b ∧ a_1 \\ a = ma \\ mb",
"tactic": "aesop"
}
] |
[
863,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
862,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.weightedVSub_map
|
[] |
[
310,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
308,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.coe_ne_coe
|
[] |
[
44,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/GroupTheory/GroupAction/Defs.lean
|
comp_smul_left
|
[] |
[
496,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/Data/List/Count.lean
|
List.count_singleton
|
[
{
"state_after": "α : Type u_1\nl : List α\ninst✝ : DecidableEq α\na : α\n⊢ (if a = a then succ (count a []) else count a []) = 1",
"state_before": "α : Type u_1\nl : List α\ninst✝ : DecidableEq α\na : α\n⊢ count a [a] = 1",
"tactic": "rw [count_cons]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\ninst✝ : DecidableEq α\na : α\n⊢ (if a = a then succ (count a []) else count a []) = 1",
"tactic": "simp"
}
] |
[
223,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/Data/List/ToFinsupp.lean
|
List.toFinsupp_apply_lt
|
[] |
[
73,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
Real.deriv_rpow_const
|
[] |
[
364,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.lt_inv_iff_mul_lt
|
[
{
"state_after": "no goals",
"state_before": "r p : ℝ≥0\nh : p ≠ 0\n⊢ r < p⁻¹ ↔ r * p < 1",
"tactic": "rw [← mul_lt_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h, mul_comm]"
}
] |
[
796,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
795,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
NNReal.sqrt_le_sqrt_iff
|
[] |
[
56,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Data/PEquiv.lean
|
PEquiv.symm_refl
|
[] |
[
134,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_eq_foldr
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1814\nα : Type ?u.1817\nβ : Type ?u.1820\nγ : Type ?u.1823\ninst✝ : CommMonoid α\ns✝ t : Multiset α\na : α\nm : Multiset ι\nf g : ι → α\ns : Multiset α\nx y z : α\n⊢ (fun x x_1 => x * x_1) x ((fun x x_1 => x * x_1) y z) = (fun x x_1 => x * x_1) y ((fun x x_1 => x * x_1) x z)",
"tactic": "simp [mul_left_comm]"
}
] |
[
55,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Control/Traversable/Instances.lean
|
List.comp_traverse
|
[
{
"state_after": "no goals",
"state_before": "F G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα : Type u_1\nβ γ : Type u\nf : β → F γ\ng : α → G β\nx : List α\n⊢ List.traverse (Comp.mk ∘ (fun x x_1 => x <$> x_1) f ∘ g) x = Comp.mk (List.traverse f <$> List.traverse g x)",
"tactic": "induction x <;> simp! [*, functor_norm] <;> rfl"
}
] |
[
86,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
11
] |
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
VitaliFamily.limRatioMeas_measurable
|
[] |
[
435,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Algebra/Module/LocalizedModule.lean
|
LocalizedModule.zero_smul'
|
[
{
"state_after": "case h\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\ns : { x // x ∈ S }\n⊢ 0 • mk m s = 0",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : LocalizedModule S M\n⊢ 0 • x = 0",
"tactic": "induction' x using LocalizedModule.induction_on with m s"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\ns : { x // x ∈ S }\n⊢ 0 • mk m s = 0",
"tactic": "rw [← Localization.mk_zero s, mk_smul_mk, zero_smul, zero_mk]"
}
] |
[
385,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
9
] |
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.aevalTower_comp_toAlgHom
|
[] |
[
430,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
Mathlib/RingTheory/NonZeroDivisors.lean
|
map_le_nonZeroDivisors_of_injective
|
[
{
"state_after": "case inl\nM : Type u_3\nM' : Type u_1\nM₁ : Type ?u.57271\nR : Type ?u.57274\nR' : Type ?u.57277\nF : Type u_2\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M'\ninst✝ : MonoidWithZeroHomClass F M M'\nf : F\nhf : Function.Injective ↑f\nS : Submonoid M\nhS : S ≤ M⁰\nh✝ : Subsingleton M\n⊢ Submonoid.map f S ≤ M'⁰\n\ncase inr\nM : Type u_3\nM' : Type u_1\nM₁ : Type ?u.57271\nR : Type ?u.57274\nR' : Type ?u.57277\nF : Type u_2\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M'\ninst✝ : MonoidWithZeroHomClass F M M'\nf : F\nhf : Function.Injective ↑f\nS : Submonoid M\nhS : S ≤ M⁰\nh✝ : Nontrivial M\n⊢ Submonoid.map f S ≤ M'⁰",
"state_before": "M : Type u_3\nM' : Type u_1\nM₁ : Type ?u.57271\nR : Type ?u.57274\nR' : Type ?u.57277\nF : Type u_2\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M'\ninst✝ : MonoidWithZeroHomClass F M M'\nf : F\nhf : Function.Injective ↑f\nS : Submonoid M\nhS : S ≤ M⁰\n⊢ Submonoid.map f S ≤ M'⁰",
"tactic": "cases subsingleton_or_nontrivial M"
},
{
"state_after": "no goals",
"state_before": "case inl\nM : Type u_3\nM' : Type u_1\nM₁ : Type ?u.57271\nR : Type ?u.57274\nR' : Type ?u.57277\nF : Type u_2\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M'\ninst✝ : MonoidWithZeroHomClass F M M'\nf : F\nhf : Function.Injective ↑f\nS : Submonoid M\nhS : S ≤ M⁰\nh✝ : Subsingleton M\n⊢ Submonoid.map f S ≤ M'⁰",
"tactic": "simp [Subsingleton.elim S ⊥]"
},
{
"state_after": "no goals",
"state_before": "case inr\nM : Type u_3\nM' : Type u_1\nM₁ : Type ?u.57271\nR : Type ?u.57274\nR' : Type ?u.57277\nF : Type u_2\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M'\ninst✝ : MonoidWithZeroHomClass F M M'\nf : F\nhf : Function.Injective ↑f\nS : Submonoid M\nhS : S ≤ M⁰\nh✝ : Nontrivial M\n⊢ Submonoid.map f S ≤ M'⁰",
"tactic": "exact le_nonZeroDivisors_of_noZeroDivisors fun h ↦\n let ⟨x, hx, hx0⟩ := h\n zero_ne_one (hS (hf (hx0.trans (map_zero f).symm) ▸ hx : 0 ∈ S) 1 (mul_zero 1)).symm"
}
] |
[
157,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
mul_zero_eq_const
|
[] |
[
78,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/NumberTheory/Padics/Hensel.lean
|
a_soln_is_unique
|
[
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nha : Polynomial.eval a F = 0\nz' : ℤ_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : ‖z' - a‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2\n⊢ 0 = Polynomial.eval (z' - a + a) F",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nha : Polynomial.eval a F = 0\nz' : ℤ_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : ‖z' - a‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2\n⊢ 0 = Polynomial.eval (a + (z' - a)) F",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nha : Polynomial.eval a F = 0\nz' : ℤ_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : ‖z' - a‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2\n⊢ 0 = Polynomial.eval (z' - a + a) F",
"tactic": "simp [hz']"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nha : Polynomial.eval a F = 0\nz' : ℤ_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : ‖z' - a‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2\n⊢ Polynomial.eval (a + h) F = Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2",
"tactic": "rw [hq, ha, zero_add]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nha : Polynomial.eval a F = 0\nz' : ℤ_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : ‖z' - a‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2\n⊢ Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2 =\n (Polynomial.eval a (↑Polynomial.derivative F) + q * h) * h",
"tactic": "rw [sq, right_distrib, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nha : Polynomial.eval a F = 0\nz' : ℤ_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : ‖z' - a‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2\nthis✝ : (Polynomial.eval a (↑Polynomial.derivative F) + q * h) * h = 0\nhne : ¬h = 0\nthis : Polynomial.eval a (↑Polynomial.derivative F) + q * h = 0\n⊢ Polynomial.eval a (↑Polynomial.derivative F) = -q * h",
"tactic": "simpa using eq_neg_of_add_eq_zero_left this"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nha : Polynomial.eval a F = 0\nz' : ℤ_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : ‖z' - a‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2\nthis✝¹ : (Polynomial.eval a (↑Polynomial.derivative F) + q * h) * h = 0\nhne : ¬h = 0\nthis✝ : Polynomial.eval a (↑Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval a (↑Polynomial.derivative F) = -q * h\n⊢ ‖Polynomial.eval a (↑Polynomial.derivative F)‖ = ‖q‖ * ‖h‖",
"tactic": "simp [this]"
},
{
"state_after": "case h\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nha : Polynomial.eval a F = 0\nz' : ℤ_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : ‖z' - a‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2\nthis✝¹ : (Polynomial.eval a (↑Polynomial.derivative F) + q * h) * h = 0\nhne : ¬h = 0\nthis✝ : Polynomial.eval a (↑Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval a (↑Polynomial.derivative F) = -q * h\n⊢ ‖q‖ ≤ 1",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nha : Polynomial.eval a F = 0\nz' : ℤ_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : ‖z' - a‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2\nthis✝¹ : (Polynomial.eval a (↑Polynomial.derivative F) + q * h) * h = 0\nhne : ¬h = 0\nthis✝ : Polynomial.eval a (↑Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval a (↑Polynomial.derivative F) = -q * h\n⊢ ‖q‖ * ‖h‖ ≤ 1 * ‖h‖",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nha : Polynomial.eval a F = 0\nz' : ℤ_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : ‖z' - a‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2\nthis✝¹ : (Polynomial.eval a (↑Polynomial.derivative F) + q * h) * h = 0\nhne : ¬h = 0\nthis✝ : Polynomial.eval a (↑Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval a (↑Polynomial.derivative F) = -q * h\n⊢ ‖q‖ ≤ 1",
"tactic": "apply PadicInt.norm_le_one"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nha : Polynomial.eval a F = 0\nz' : ℤ_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : ‖z' - a‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2\nthis✝¹ : (Polynomial.eval a (↑Polynomial.derivative F) + q * h) * h = 0\nhne : ¬h = 0\nthis✝ : Polynomial.eval a (↑Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval a (↑Polynomial.derivative F) = -q * h\n⊢ 1 * ‖h‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nha : Polynomial.eval a F = 0\nz' : ℤ_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : ‖z' - a‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (↑Polynomial.derivative F) * h + q * h ^ 2\nthis✝ : (Polynomial.eval a (↑Polynomial.derivative F) + q * h) * h = 0\nthis : h = 0\n⊢ z' - a = 0",
"tactic": "rw [← this]"
}
] |
[
485,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
463,
9
] |
Mathlib/Analysis/NormedSpace/Basic.lean
|
frontier_closedBall'
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.308974\nβ : Type ?u.308977\nγ : Type ?u.308980\nι : Type ?u.308983\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.309076\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\n⊢ frontier (closedBall x r) = sphere x r",
"tactic": "rw [frontier, closure_closedBall, interior_closedBall' x r, closedBall_diff_ball]"
}
] |
[
392,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/Combinatorics/Quiver/Basic.lean
|
Prefunctor.comp_assoc
|
[] |
[
133,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.hasFiniteIntegral_const
|
[] |
[
182,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/CategoryTheory/Functor/FullyFaithful.lean
|
CategoryTheory.Functor.preimageIso_mapIso
|
[
{
"state_after": "case w\nC : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nX Y Z : C\nf : X ≅ Y\n⊢ (preimageIso F (F.mapIso f)).hom = f.hom",
"state_before": "C : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nX Y Z : C\nf : X ≅ Y\n⊢ preimageIso F (F.mapIso f) = f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nF : C ⥤ D\ninst✝¹ : Full F\ninst✝ : Faithful F\nX Y Z : C\nf : X ≅ Y\n⊢ (preimageIso F (F.mapIso f)).hom = f.hom",
"tactic": "simp"
}
] |
[
157,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
convex_iff_segment_subset
|
[] |
[
65,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.le_smul_iff
|
[] |
[
1055,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1054,
1
] |
Mathlib/MeasureTheory/Decomposition/Jordan.lean
|
MeasureTheory.JordanDecomposition.real_smul_negPart_neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.15046\ninst✝ : MeasurableSpace α\nj : JordanDecomposition α\nr : ℝ\nhr : r < 0\n⊢ (r • j).negPart = Real.toNNReal (-r) • j.posPart",
"tactic": "rw [real_smul_def, ← smul_posPart, if_neg (not_le.2 hr), neg_negPart]"
}
] |
[
168,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Data/Bool/Basic.lean
|
Bool.coe_decide
|
[] |
[
46,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Data/Fin/Tuple/Sort.lean
|
Tuple.antitone_pair_of_not_sorted
|
[] |
[
183,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Algebra/AddTorsor.lean
|
vsub_eq_sub
|
[] |
[
78,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.coeSucc_eq_succ
|
[
{
"state_after": "case zero\nm : ℕ\na : Fin zero\n⊢ ↑castSucc a + 1 = succ a\n\ncase succ\nm n✝ : ℕ\na : Fin (Nat.succ n✝)\n⊢ ↑castSucc a + 1 = succ a",
"state_before": "n m : ℕ\na : Fin n\n⊢ ↑castSucc a + 1 = succ a",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\nm : ℕ\na : Fin zero\n⊢ ↑castSucc a + 1 = succ a",
"tactic": "exact @finZeroElim (fun _ => _) a"
},
{
"state_after": "no goals",
"state_before": "case succ\nm n✝ : ℕ\na : Fin (Nat.succ n✝)\n⊢ ↑castSucc a + 1 = succ a",
"tactic": "simp [a.is_lt, eq_iff_veq, add_def, Nat.mod_eq_of_lt]"
}
] |
[
1312,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1309,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
measurable_iff_comap_le
|
[] |
[
214,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Algebra/Tropical/Basic.lean
|
Tropical.add_eq_zero_iff
|
[
{
"state_after": "R : Type u\ninst✝ : LinearOrder R\na b : Tropical (WithTop R)\n⊢ a = 0 ∧ a ≤ b ∨ b = 0 ∧ b ≤ a ↔ a = 0 ∧ b = 0",
"state_before": "R : Type u\ninst✝ : LinearOrder R\na b : Tropical (WithTop R)\n⊢ a + b = 0 ↔ a = 0 ∧ b = 0",
"tactic": "rw [add_eq_iff]"
},
{
"state_after": "case mp\nR : Type u\ninst✝ : LinearOrder R\na b : Tropical (WithTop R)\n⊢ a = 0 ∧ a ≤ b ∨ b = 0 ∧ b ≤ a → a = 0 ∧ b = 0\n\ncase mpr\nR : Type u\ninst✝ : LinearOrder R\na b : Tropical (WithTop R)\n⊢ a = 0 ∧ b = 0 → a = 0 ∧ a ≤ b ∨ b = 0 ∧ b ≤ a",
"state_before": "R : Type u\ninst✝ : LinearOrder R\na b : Tropical (WithTop R)\n⊢ a = 0 ∧ a ≤ b ∨ b = 0 ∧ b ≤ a ↔ a = 0 ∧ b = 0",
"tactic": "constructor"
},
{
"state_after": "case mp.inl.intro\nR : Type u\ninst✝ : LinearOrder R\nb : Tropical (WithTop R)\nh : 0 ≤ b\n⊢ 0 = 0 ∧ b = 0\n\ncase mp.inr.intro\nR : Type u\ninst✝ : LinearOrder R\na : Tropical (WithTop R)\nh : 0 ≤ a\n⊢ a = 0 ∧ 0 = 0",
"state_before": "case mp\nR : Type u\ninst✝ : LinearOrder R\na b : Tropical (WithTop R)\n⊢ a = 0 ∧ a ≤ b ∨ b = 0 ∧ b ≤ a → a = 0 ∧ b = 0",
"tactic": "rintro (⟨rfl, h⟩ | ⟨rfl, h⟩)"
},
{
"state_after": "no goals",
"state_before": "case mp.inl.intro\nR : Type u\ninst✝ : LinearOrder R\nb : Tropical (WithTop R)\nh : 0 ≤ b\n⊢ 0 = 0 ∧ b = 0",
"tactic": "exact ⟨rfl, le_antisymm (le_zero _) h⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.inr.intro\nR : Type u\ninst✝ : LinearOrder R\na : Tropical (WithTop R)\nh : 0 ≤ a\n⊢ a = 0 ∧ 0 = 0",
"tactic": "exact ⟨le_antisymm (le_zero _) h, rfl⟩"
},
{
"state_after": "case mpr.intro\nR : Type u\ninst✝ : LinearOrder R\n⊢ 0 = 0 ∧ 0 ≤ 0 ∨ 0 = 0 ∧ 0 ≤ 0",
"state_before": "case mpr\nR : Type u\ninst✝ : LinearOrder R\na b : Tropical (WithTop R)\n⊢ a = 0 ∧ b = 0 → a = 0 ∧ a ≤ b ∨ b = 0 ∧ b ≤ a",
"tactic": "rintro ⟨rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nR : Type u\ninst✝ : LinearOrder R\n⊢ 0 = 0 ∧ 0 ≤ 0 ∨ 0 = 0 ∧ 0 ≤ 0",
"tactic": "simp"
}
] |
[
370,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.image_add_left_Ioo
|
[
{
"state_after": "ι : Type ?u.242100\nα : Type u_1\ninst✝³ : OrderedCancelAddCommMonoid α\ninst✝² : ExistsAddOfLE α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b c : α\n⊢ image ((fun x x_1 => x + x_1) c) (Ioo a b) = image (↑(addLeftEmbedding c)) (Ioo a b)",
"state_before": "ι : Type ?u.242100\nα : Type u_1\ninst✝³ : OrderedCancelAddCommMonoid α\ninst✝² : ExistsAddOfLE α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b c : α\n⊢ image ((fun x x_1 => x + x_1) c) (Ioo a b) = Ioo (c + a) (c + b)",
"tactic": "rw [← map_add_left_Ioo, map_eq_image]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.242100\nα : Type u_1\ninst✝³ : OrderedCancelAddCommMonoid α\ninst✝² : ExistsAddOfLE α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b c : α\n⊢ image ((fun x x_1 => x + x_1) c) (Ioo a b) = image (↑(addLeftEmbedding c)) (Ioo a b)",
"tactic": "rfl"
}
] |
[
1119,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1117,
1
] |
Mathlib/Order/Bounded.lean
|
Set.unbounded_le_inter_not_le
|
[
{
"state_after": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : SemilatticeSup α\na : α\n⊢ Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | ¬b ≤ a}) ↔ Bounded (fun x x_1 => x ≤ x_1) s",
"state_before": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : SemilatticeSup α\na : α\n⊢ Unbounded (fun x x_1 => x ≤ x_1) (s ∩ {b | ¬b ≤ a}) ↔ Unbounded (fun x x_1 => x ≤ x_1) s",
"tactic": "rw [← not_bounded_iff, ← not_bounded_iff, not_iff_not]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : SemilatticeSup α\na : α\n⊢ Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | ¬b ≤ a}) ↔ Bounded (fun x x_1 => x ≤ x_1) s",
"tactic": "exact bounded_le_inter_not_le a"
}
] |
[
327,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
Summable.update
|
[] |
[
850,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
848,
1
] |
Mathlib/Algebra/LinearRecurrence.lean
|
LinearRecurrence.is_sol_iff_mem_solSpace
|
[] |
[
140,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean
|
DoubleCentralizer.zero_snd
|
[] |
[
269,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/NumberTheory/Divisors.lean
|
Nat.divisorsAntidiagonal_zero
|
[
{
"state_after": "case a\nn : ℕ\na✝ : ℕ × ℕ\n⊢ a✝ ∈ divisorsAntidiagonal 0 ↔ a✝ ∈ ∅",
"state_before": "n : ℕ\n⊢ divisorsAntidiagonal 0 = ∅",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nn : ℕ\na✝ : ℕ × ℕ\n⊢ a✝ ∈ divisorsAntidiagonal 0 ↔ a✝ ∈ ∅",
"tactic": "simp"
}
] |
[
204,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
MeasurableSet.image_inclusion'
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.62091\nγ : Type ?u.62094\nδ : Type ?u.62097\nδ' : Type ?u.62100\nι : Sort uι\ns✝ t✝ u✝ : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ns t : Set α\nh : s ⊆ t\nhs : MeasurableSet (Subtype.val ⁻¹' s)\nu : Set α\nhu : MeasurableSet u\n⊢ MeasurableSet (inclusion h '' (Subtype.val ⁻¹' u))",
"state_before": "α : Type u_1\nβ : Type ?u.62091\nγ : Type ?u.62094\nδ : Type ?u.62097\nδ' : Type ?u.62100\nι : Sort uι\ns✝ t✝ u✝ : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ns t : Set α\nh : s ⊆ t\nu : Set ↑s\nhs : MeasurableSet (Subtype.val ⁻¹' s)\nhu : MeasurableSet u\n⊢ MeasurableSet (inclusion h '' u)",
"tactic": "rcases hu with ⟨u, hu, rfl⟩"
},
{
"state_after": "case h.e'_3\nα : Type u_1\nβ : Type ?u.62091\nγ : Type ?u.62094\nδ : Type ?u.62097\nδ' : Type ?u.62100\nι : Sort uι\ns✝ t✝ u✝ : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ns t : Set α\nh : s ⊆ t\nhs : MeasurableSet (Subtype.val ⁻¹' s)\nu : Set α\nhu : MeasurableSet u\n⊢ inclusion h '' (Subtype.val ⁻¹' u) = Subtype.val ⁻¹' u ∩ Subtype.val ⁻¹' s",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.62091\nγ : Type ?u.62094\nδ : Type ?u.62097\nδ' : Type ?u.62100\nι : Sort uι\ns✝ t✝ u✝ : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ns t : Set α\nh : s ⊆ t\nhs : MeasurableSet (Subtype.val ⁻¹' s)\nu : Set α\nhu : MeasurableSet u\n⊢ MeasurableSet (inclusion h '' (Subtype.val ⁻¹' u))",
"tactic": "convert (measurable_subtype_coe hu).inter hs"
},
{
"state_after": "case h.e'_3.h.mk\nα : Type u_1\nβ : Type ?u.62091\nγ : Type ?u.62094\nδ : Type ?u.62097\nδ' : Type ?u.62100\nι : Sort uι\ns✝ t✝ u✝ : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ns t : Set α\nh : s ⊆ t\nhs : MeasurableSet (Subtype.val ⁻¹' s)\nu : Set α\nhu : MeasurableSet u\nx : α\nhx : x ∈ t\n⊢ { val := x, property := hx } ∈ inclusion h '' (Subtype.val ⁻¹' u) ↔\n { val := x, property := hx } ∈ Subtype.val ⁻¹' u ∩ Subtype.val ⁻¹' s",
"state_before": "case h.e'_3\nα : Type u_1\nβ : Type ?u.62091\nγ : Type ?u.62094\nδ : Type ?u.62097\nδ' : Type ?u.62100\nι : Sort uι\ns✝ t✝ u✝ : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ns t : Set α\nh : s ⊆ t\nhs : MeasurableSet (Subtype.val ⁻¹' s)\nu : Set α\nhu : MeasurableSet u\n⊢ inclusion h '' (Subtype.val ⁻¹' u) = Subtype.val ⁻¹' u ∩ Subtype.val ⁻¹' s",
"tactic": "ext ⟨x, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.mk\nα : Type u_1\nβ : Type ?u.62091\nγ : Type ?u.62094\nδ : Type ?u.62097\nδ' : Type ?u.62100\nι : Sort uι\ns✝ t✝ u✝ : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ns t : Set α\nh : s ⊆ t\nhs : MeasurableSet (Subtype.val ⁻¹' s)\nu : Set α\nhu : MeasurableSet u\nx : α\nhx : x ∈ t\n⊢ { val := x, property := hx } ∈ inclusion h '' (Subtype.val ⁻¹' u) ↔\n { val := x, property := hx } ∈ Subtype.val ⁻¹' u ∩ Subtype.val ⁻¹' s",
"tactic": "simpa [@and_comm _ (_ = x)] using and_comm"
}
] |
[
588,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
582,
1
] |
Mathlib/Data/Multiset/Sort.lean
|
Multiset.length_sort
|
[] |
[
58,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
OrthogonalFamily.inner_sum
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3657440\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl₁ l₂ : (i : ι) → G i\ns : Finset ι\nem✝ : (a : Prop) → Decidable a\n⊢ inner (∑ i in s, ↑(V i) (l₁ i)) (∑ j in s, ↑(V j) (l₂ j)) = ∑ i in s, inner (l₁ i) (l₂ i)",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3657440\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl₁ l₂ : (i : ι) → G i\ns : Finset ι\n⊢ inner (∑ i in s, ↑(V i) (l₁ i)) (∑ j in s, ↑(V j) (l₂ j)) = ∑ i in s, inner (l₁ i) (l₂ i)",
"tactic": "classical!"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3657440\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl₁ l₂ : (i : ι) → G i\ns : Finset ι\nem✝ : (a : Prop) → Decidable a\n⊢ inner (∑ i in s, ↑(V i) (l₁ i)) (∑ j in s, ↑(V j) (l₂ j)) = ∑ i in s, inner (l₁ i) (l₂ i)",
"tactic": "calc\n ⟪∑ i in s, V i (l₁ i), ∑ j in s, V j (l₂ j)⟫ = ∑ j in s, ∑ i in s, ⟪V i (l₁ i), V j (l₂ j)⟫ :=\n by simp only [_root_.sum_inner, _root_.inner_sum]\n _ = ∑ j in s, ∑ i in s, ite (i = j) ⟪V i (l₁ i), V j (l₂ j)⟫ 0 := by\n congr with i\n congr with j\n apply hV.eq_ite\n _ = ∑ i in s, ⟪l₁ i, l₂ i⟫ := by\n simp only [Finset.sum_ite_of_true, Finset.sum_ite_eq', LinearIsometry.inner_map_map,\n imp_self, imp_true_iff]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3657440\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl₁ l₂ : (i : ι) → G i\ns : Finset ι\nem✝ : (a : Prop) → Decidable a\n⊢ inner (∑ i in s, ↑(V i) (l₁ i)) (∑ j in s, ↑(V j) (l₂ j)) = ∑ j in s, ∑ i in s, inner (↑(V i) (l₁ i)) (↑(V j) (l₂ j))",
"tactic": "simp only [_root_.sum_inner, _root_.inner_sum]"
},
{
"state_after": "case e_f.h\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3657440\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl₁ l₂ : (i : ι) → G i\ns : Finset ι\nem✝ : (a : Prop) → Decidable a\ni : ι\n⊢ ∑ i_1 in s, inner (↑(V i_1) (l₁ i_1)) (↑(V i) (l₂ i)) =\n ∑ i_1 in s, if i_1 = i then inner (↑(V i_1) (l₁ i_1)) (↑(V i) (l₂ i)) else 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3657440\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl₁ l₂ : (i : ι) → G i\ns : Finset ι\nem✝ : (a : Prop) → Decidable a\n⊢ ∑ j in s, ∑ i in s, inner (↑(V i) (l₁ i)) (↑(V j) (l₂ j)) =\n ∑ j in s, ∑ i in s, if i = j then inner (↑(V i) (l₁ i)) (↑(V j) (l₂ j)) else 0",
"tactic": "congr with i"
},
{
"state_after": "case e_f.h.e_f.h\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3657440\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl₁ l₂ : (i : ι) → G i\ns : Finset ι\nem✝ : (a : Prop) → Decidable a\ni j : ι\n⊢ inner (↑(V j) (l₁ j)) (↑(V i) (l₂ i)) = if j = i then inner (↑(V j) (l₁ j)) (↑(V i) (l₂ i)) else 0",
"state_before": "case e_f.h\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3657440\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl₁ l₂ : (i : ι) → G i\ns : Finset ι\nem✝ : (a : Prop) → Decidable a\ni : ι\n⊢ ∑ i_1 in s, inner (↑(V i_1) (l₁ i_1)) (↑(V i) (l₂ i)) =\n ∑ i_1 in s, if i_1 = i then inner (↑(V i_1) (l₁ i_1)) (↑(V i) (l₂ i)) else 0",
"tactic": "congr with j"
},
{
"state_after": "no goals",
"state_before": "case e_f.h.e_f.h\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3657440\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl₁ l₂ : (i : ι) → G i\ns : Finset ι\nem✝ : (a : Prop) → Decidable a\ni j : ι\n⊢ inner (↑(V j) (l₁ j)) (↑(V i) (l₂ i)) = if j = i then inner (↑(V j) (l₁ j)) (↑(V i) (l₂ i)) else 0",
"tactic": "apply hV.eq_ite"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.3657440\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nl₁ l₂ : (i : ι) → G i\ns : Finset ι\nem✝ : (a : Prop) → Decidable a\n⊢ (∑ j in s, ∑ i in s, if i = j then inner (↑(V i) (l₁ i)) (↑(V j) (l₂ j)) else 0) = ∑ i in s, inner (l₁ i) (l₂ i)",
"tactic": "simp only [Finset.sum_ite_of_true, Finset.sum_ite_eq', LinearIsometry.inner_map_map,\n imp_self, imp_true_iff]"
}
] |
[
2051,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2039,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measure_add_measure_compl
|
[] |
[
157,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.span_mono
|
[] |
[
153,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
LinearEquiv.mapMatrix_trans
|
[] |
[
1514,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1512,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.pairwise_iff
|
[
{
"state_after": "case nil\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝² : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝ l₂✝¹ l₁ l₂✝ : List α\np✝ : l₁ ~ l₂✝\nl₂ : List α\np : [] ~ l₂\n⊢ Pairwise R l₂\n\ncase cons\nα : Type uu\nβ : Type vv\nl₁✝² l₂✝² : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝¹ l₂✝¹ l₁✝ l₂✝ : List α\np✝ : l₁✝ ~ l₂✝\na : α\nl₁ : List α\nh : ∀ (a' : α), a' ∈ l₁ → R a a'\na✝ : Pairwise R l₁\nIH : ∀ (l₂ : List α), l₁ ~ l₂ → Pairwise R l₂\nl₂ : List α\np : a :: l₁ ~ l₂\n⊢ Pairwise R l₂",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝ l₂✝ l₁ l₂ : List α\np : l₁ ~ l₂\nd : Pairwise R l₁\n⊢ Pairwise R l₂",
"tactic": "induction' d with a l₁ h _ IH generalizing l₂"
},
{
"state_after": "case nil\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝² : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝ l₂✝¹ l₁ l₂✝ : List α\np✝ : l₁ ~ l₂✝\nl₂ : List α\np : [] ~ l₂\n⊢ Pairwise R []",
"state_before": "case nil\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝² : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝ l₂✝¹ l₁ l₂✝ : List α\np✝ : l₁ ~ l₂✝\nl₂ : List α\np : [] ~ l₂\n⊢ Pairwise R l₂",
"tactic": "rw [← p.nil_eq]"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝² : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝ l₂✝¹ l₁ l₂✝ : List α\np✝ : l₁ ~ l₂✝\nl₂ : List α\np : [] ~ l₂\n⊢ Pairwise R []",
"tactic": "constructor"
},
{
"state_after": "case cons\nα : Type uu\nβ : Type vv\nl₁✝² l₂✝² : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝¹ l₂✝¹ l₁✝ l₂✝ : List α\np✝ : l₁✝ ~ l₂✝\na : α\nl₁ : List α\nh : ∀ (a' : α), a' ∈ l₁ → R a a'\na✝ : Pairwise R l₁\nIH : ∀ (l₂ : List α), l₁ ~ l₂ → Pairwise R l₂\nl₂ : List α\np : a :: l₁ ~ l₂\nthis : a ∈ l₂\n⊢ Pairwise R l₂",
"state_before": "case cons\nα : Type uu\nβ : Type vv\nl₁✝² l₂✝² : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝¹ l₂✝¹ l₁✝ l₂✝ : List α\np✝ : l₁✝ ~ l₂✝\na : α\nl₁ : List α\nh : ∀ (a' : α), a' ∈ l₁ → R a a'\na✝ : Pairwise R l₁\nIH : ∀ (l₂ : List α), l₁ ~ l₂ → Pairwise R l₂\nl₂ : List α\np : a :: l₁ ~ l₂\n⊢ Pairwise R l₂",
"tactic": "have : a ∈ l₂ := p.subset (mem_cons_self _ _)"
},
{
"state_after": "case cons.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝² l₂✝¹ : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝¹ l₂✝ l₁✝ l₂ : List α\np✝ : l₁✝ ~ l₂\na : α\nl₁ : List α\nh : ∀ (a' : α), a' ∈ l₁ → R a a'\na✝ : Pairwise R l₁\nIH : ∀ (l₂ : List α), l₁ ~ l₂ → Pairwise R l₂\ns₂ t₂ : List α\np : a :: l₁ ~ s₂ ++ a :: t₂\nthis : a ∈ s₂ ++ a :: t₂\n⊢ Pairwise R (s₂ ++ a :: t₂)",
"state_before": "case cons\nα : Type uu\nβ : Type vv\nl₁✝² l₂✝² : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝¹ l₂✝¹ l₁✝ l₂✝ : List α\np✝ : l₁✝ ~ l₂✝\na : α\nl₁ : List α\nh : ∀ (a' : α), a' ∈ l₁ → R a a'\na✝ : Pairwise R l₁\nIH : ∀ (l₂ : List α), l₁ ~ l₂ → Pairwise R l₂\nl₂ : List α\np : a :: l₁ ~ l₂\nthis : a ∈ l₂\n⊢ Pairwise R l₂",
"tactic": "rcases mem_split this with ⟨s₂, t₂, rfl⟩"
},
{
"state_after": "case cons.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝² l₂✝¹ : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝¹ l₂✝ l₁✝ l₂ : List α\np✝ : l₁✝ ~ l₂\na : α\nl₁ : List α\nh : ∀ (a' : α), a' ∈ l₁ → R a a'\na✝ : Pairwise R l₁\nIH : ∀ (l₂ : List α), l₁ ~ l₂ → Pairwise R l₂\ns₂ t₂ : List α\np : a :: l₁ ~ s₂ ++ a :: t₂\nthis : a ∈ s₂ ++ a :: t₂\np' : l₁ ~ s₂ ++ t₂\n⊢ Pairwise R (s₂ ++ a :: t₂)",
"state_before": "case cons.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝² l₂✝¹ : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝¹ l₂✝ l₁✝ l₂ : List α\np✝ : l₁✝ ~ l₂\na : α\nl₁ : List α\nh : ∀ (a' : α), a' ∈ l₁ → R a a'\na✝ : Pairwise R l₁\nIH : ∀ (l₂ : List α), l₁ ~ l₂ → Pairwise R l₂\ns₂ t₂ : List α\np : a :: l₁ ~ s₂ ++ a :: t₂\nthis : a ∈ s₂ ++ a :: t₂\n⊢ Pairwise R (s₂ ++ a :: t₂)",
"tactic": "have p' := (p.trans perm_middle).cons_inv"
},
{
"state_after": "case cons.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝² l₂✝¹ : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝¹ l₂✝ l₁✝ l₂ : List α\np✝ : l₁✝ ~ l₂\na : α\nl₁ : List α\nh : ∀ (a' : α), a' ∈ l₁ → R a a'\na✝ : Pairwise R l₁\nIH : ∀ (l₂ : List α), l₁ ~ l₂ → Pairwise R l₂\ns₂ t₂ : List α\np : a :: l₁ ~ s₂ ++ a :: t₂\nthis : a ∈ s₂ ++ a :: t₂\np' : l₁ ~ s₂ ++ t₂\nb : α\nm : b ∈ s₂ ++ t₂\n⊢ R a b",
"state_before": "case cons.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝² l₂✝¹ : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝¹ l₂✝ l₁✝ l₂ : List α\np✝ : l₁✝ ~ l₂\na : α\nl₁ : List α\nh : ∀ (a' : α), a' ∈ l₁ → R a a'\na✝ : Pairwise R l₁\nIH : ∀ (l₂ : List α), l₁ ~ l₂ → Pairwise R l₂\ns₂ t₂ : List α\np : a :: l₁ ~ s₂ ++ a :: t₂\nthis : a ∈ s₂ ++ a :: t₂\np' : l₁ ~ s₂ ++ t₂\n⊢ Pairwise R (s₂ ++ a :: t₂)",
"tactic": "refine' (pairwise_middle S).2 (pairwise_cons.2 ⟨fun b m => _, IH _ p'⟩)"
},
{
"state_after": "no goals",
"state_before": "case cons.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝² l₂✝¹ : List α\nR : α → α → Prop\nS : Symmetric R\nl₁✝¹ l₂✝ l₁✝ l₂ : List α\np✝ : l₁✝ ~ l₂\na : α\nl₁ : List α\nh : ∀ (a' : α), a' ∈ l₁ → R a a'\na✝ : Pairwise R l₁\nIH : ∀ (l₂ : List α), l₁ ~ l₂ → Pairwise R l₂\ns₂ t₂ : List α\np : a :: l₁ ~ s₂ ++ a :: t₂\nthis : a ∈ s₂ ++ a :: t₂\np' : l₁ ~ s₂ ++ t₂\nb : α\nm : b ∈ s₂ ++ t₂\n⊢ R a b",
"tactic": "exact h _ (p'.symm.subset m)"
}
] |
[
1053,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1041,
1
] |
Mathlib/Data/Real/Irrational.lean
|
Irrational.nat_div
|
[] |
[
454,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
453,
1
] |
Mathlib/Data/Polynomial/Splits.lean
|
Polynomial.eq_prod_roots_of_splits
|
[
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni✝ : K →+* L\np : K[X]\ni : K →+* L\nhsplit : Splits i p\n⊢ map i p = ↑C (leadingCoeff (map i p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (map i p)))",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni✝ : K →+* L\np : K[X]\ni : K →+* L\nhsplit : Splits i p\n⊢ map i p = ↑C (↑i (leadingCoeff p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (map i p)))",
"tactic": "rw [← leadingCoeff_map]"
},
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni✝ : K →+* L\np : K[X]\ni : K →+* L\nhsplit : Splits i p\n⊢ ↑C (leadingCoeff (map i p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (map i p))) = map i p",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni✝ : K →+* L\np : K[X]\ni : K →+* L\nhsplit : Splits i p\n⊢ map i p = ↑C (leadingCoeff (map i p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (map i p)))",
"tactic": "symm"
},
{
"state_after": "case hroots\nF : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni✝ : K →+* L\np : K[X]\ni : K →+* L\nhsplit : Splits i p\n⊢ ↑Multiset.card (roots (map i p)) = natDegree (map i p)",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni✝ : K →+* L\np : K[X]\ni : K →+* L\nhsplit : Splits i p\n⊢ ↑C (leadingCoeff (map i p)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots (map i p))) = map i p",
"tactic": "apply C_leadingCoeff_mul_prod_multiset_X_sub_C"
},
{
"state_after": "case hroots\nF : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni✝ : K →+* L\np : K[X]\ni : K →+* L\nhsplit : Splits i p\n⊢ ↑Multiset.card (roots (map i p)) = natDegree p",
"state_before": "case hroots\nF : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni✝ : K →+* L\np : K[X]\ni : K →+* L\nhsplit : Splits i p\n⊢ ↑Multiset.card (roots (map i p)) = natDegree (map i p)",
"tactic": "rw [natDegree_map]"
},
{
"state_after": "no goals",
"state_before": "case hroots\nF : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni✝ : K →+* L\np : K[X]\ni : K →+* L\nhsplit : Splits i p\n⊢ ↑Multiset.card (roots (map i p)) = natDegree p",
"tactic": "exact (natDegree_eq_card_roots hsplit).symm"
}
] |
[
355,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
351,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
LipschitzOnWith.norm_div_le_of_le
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.186854\n𝕜 : Type ?u.186857\nα : Type ?u.186860\nι : Type ?u.186863\nκ : Type ?u.186866\nE : Type u_1\nF : Type u_2\nG : Type ?u.186875\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : E → F\nC : ℝ≥0\nh : LipschitzOnWith C f s\nha : a ∈ s\nhb : b ∈ s\nhr : ‖a / b‖ ≤ r\n⊢ ↑C * ‖a / b‖ ≤ ↑C * r",
"tactic": "gcongr"
}
] |
[
815,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
813,
1
] |
Std/Data/Int/DivMod.lean
|
Int.mul_ediv_cancel_of_emod_eq_zero
|
[
{
"state_after": "a b : Int\nH : a % b = 0\nthis : a % b + b * (a / b) = a\n⊢ b * (a / b) = a",
"state_before": "a b : Int\nH : a % b = 0\n⊢ b * (a / b) = a",
"tactic": "have := emod_add_ediv a b"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nH : a % b = 0\nthis : a % b + b * (a / b) = a\n⊢ b * (a / b) = a",
"tactic": "rwa [H, Int.zero_add] at this"
}
] |
[
586,
59
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
585,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
Isometry.mapsTo_sphere
|
[] |
[
262,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Data/Int/Basic.lean
|
Int.natAbs_surjective
|
[] |
[
200,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/MeasureTheory/Measure/Content.lean
|
MeasureTheory.Content.outerMeasure_le
|
[] |
[
268,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/LinearAlgebra/PiTensorProduct.lean
|
PiTensorProduct.add_tprodCoeff'
|
[] |
[
165,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
EuclideanSpace.dist_eq
|
[] |
[
133,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Data/PNat/Find.lean
|
PNat.find_spec
|
[] |
[
63,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
11
] |
src/lean/Init/Data/List/Basic.lean
|
List.append_assoc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nas bs cs : List α\n⊢ as ++ bs ++ cs = as ++ (bs ++ cs)",
"tactic": "induction as with\n| nil => rfl\n| cons a as ih => simp [ih]"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nβ : Type v\nγ : Type w\nbs cs : List α\n⊢ nil ++ bs ++ cs = nil ++ (bs ++ cs)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nβ : Type v\nγ : Type w\nbs cs : List α\na : α\nas : List α\nih : as ++ bs ++ cs = as ++ (bs ++ cs)\n⊢ a :: as ++ bs ++ cs = a :: as ++ (bs ++ cs)",
"tactic": "simp [ih]"
}
] |
[
106,
30
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
103,
1
] |
Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
|
IsAddCyclic.card_orderOf_eq_totient
|
[
{
"state_after": "case mk.intro\nα✝ : Type u\na : α✝\ninst✝³ : Group α✝\nα : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : IsAddCyclic α\ninst✝ : Fintype α\nd : ℕ\nhd : d ∣ Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ AddSubgroup.zmultiples g\n⊢ card (filter (fun a => addOrderOf a = d) univ) = φ d",
"state_before": "α✝ : Type u\na : α✝\ninst✝³ : Group α✝\nα : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : IsAddCyclic α\ninst✝ : Fintype α\nd : ℕ\nhd : d ∣ Fintype.card α\n⊢ card (filter (fun a => addOrderOf a = d) univ) = φ d",
"tactic": "obtain ⟨g, hg⟩ := id ‹IsAddCyclic α›"
},
{
"state_after": "no goals",
"state_before": "case mk.intro\nα✝ : Type u\na : α✝\ninst✝³ : Group α✝\nα : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : IsAddCyclic α\ninst✝ : Fintype α\nd : ℕ\nhd : d ∣ Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ AddSubgroup.zmultiples g\n⊢ card (filter (fun a => addOrderOf a = d) univ) = φ d",
"tactic": "apply @IsCyclic.card_orderOf_eq_totient (Multiplicative α) _ ⟨⟨g, hg⟩⟩ (_) _ hd"
}
] |
[
426,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
422,
1
] |
Mathlib/CategoryTheory/Abelian/Pseudoelements.lean
|
CategoryTheory.Abelian.Pseudoelement.epi_of_pseudo_surjective
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\n⊢ Epi f",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\n⊢ Function.Surjective (pseudoApply f) → Epi f",
"tactic": "intro h"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\n⊢ Epi f",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\n⊢ Epi f",
"tactic": "have ⟨pbar, hpbar⟩ := h (𝟙 Q)"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhp : Quotient.mk (setoid P) p = pbar\n⊢ Epi f",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\n⊢ Epi f",
"tactic": "have ⟨p, hp⟩ := Quotient.exists_rep pbar"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhp : Quotient.mk (setoid P) p = pbar\nthis : Quotient.mk (setoid Q) (Over.mk (p.hom ≫ f)) = Quotient.mk (setoid Q) (Over.mk (𝟙 Q))\n⊢ Epi f",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhp : Quotient.mk (setoid P) p = pbar\n⊢ Epi f",
"tactic": "have : ⟦(p.hom ≫ f : Over Q)⟧ = ⟦↑(𝟙 Q)⟧ := by\n rw [← hp] at hpbar\n exact hpbar"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhp : Quotient.mk (setoid P) p = pbar\nthis : Quotient.mk (setoid Q) (Over.mk (p.hom ≫ f)) = Quotient.mk (setoid Q) (Over.mk (𝟙 Q))\nR : C\nx : R ⟶ (Over.mk (p.hom ≫ f)).left\ny : R ⟶ (Over.mk (𝟙 Q)).left\nw✝ : Epi x\ney : Epi y\ncomm : x ≫ (Over.mk (p.hom ≫ f)).hom = y ≫ (Over.mk (𝟙 Q)).hom\n⊢ Epi f",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhp : Quotient.mk (setoid P) p = pbar\nthis : Quotient.mk (setoid Q) (Over.mk (p.hom ≫ f)) = Quotient.mk (setoid Q) (Over.mk (𝟙 Q))\n⊢ Epi f",
"tactic": "have ⟨R, x, y, _, ey, comm⟩ := Quotient.exact this"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhp : Quotient.mk (setoid P) p = pbar\nthis : Quotient.mk (setoid Q) (Over.mk (p.hom ≫ f)) = Quotient.mk (setoid Q) (Over.mk (𝟙 Q))\nR : C\nx : R ⟶ (Over.mk (p.hom ≫ f)).left\ny : R ⟶ (Over.mk (𝟙 Q)).left\nw✝ : Epi x\ney : Epi y\ncomm : x ≫ (Over.mk (p.hom ≫ f)).hom = y ≫ (Over.mk (𝟙 Q)).hom\n⊢ (x ≫ p.hom) ≫ f = y",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhp : Quotient.mk (setoid P) p = pbar\nthis : Quotient.mk (setoid Q) (Over.mk (p.hom ≫ f)) = Quotient.mk (setoid Q) (Over.mk (𝟙 Q))\nR : C\nx : R ⟶ (Over.mk (p.hom ≫ f)).left\ny : R ⟶ (Over.mk (𝟙 Q)).left\nw✝ : Epi x\ney : Epi y\ncomm : x ≫ (Over.mk (p.hom ≫ f)).hom = y ≫ (Over.mk (𝟙 Q)).hom\n⊢ Epi f",
"tactic": "apply @epi_of_epi_fac _ _ _ _ _ (x ≫ p.hom) f y ey"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhp : Quotient.mk (setoid P) p = pbar\nthis : Quotient.mk (setoid Q) (Over.mk (p.hom ≫ f)) = Quotient.mk (setoid Q) (Over.mk (𝟙 Q))\nR : C\nx : R ⟶ (Over.mk (p.hom ≫ f)).left\ny : R ⟶ (Over.mk (𝟙 Q)).left\nw✝ : Epi x\ney : Epi y\ncomm : x ≫ p.hom ≫ f = y ≫ 𝟙 Q\n⊢ (x ≫ p.hom) ≫ f = y",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhp : Quotient.mk (setoid P) p = pbar\nthis : Quotient.mk (setoid Q) (Over.mk (p.hom ≫ f)) = Quotient.mk (setoid Q) (Over.mk (𝟙 Q))\nR : C\nx : R ⟶ (Over.mk (p.hom ≫ f)).left\ny : R ⟶ (Over.mk (𝟙 Q)).left\nw✝ : Epi x\ney : Epi y\ncomm : x ≫ (Over.mk (p.hom ≫ f)).hom = y ≫ (Over.mk (𝟙 Q)).hom\n⊢ (x ≫ p.hom) ≫ f = y",
"tactic": "dsimp at comm"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhp : Quotient.mk (setoid P) p = pbar\nthis : Quotient.mk (setoid Q) (Over.mk (p.hom ≫ f)) = Quotient.mk (setoid Q) (Over.mk (𝟙 Q))\nR : C\nx : R ⟶ (Over.mk (p.hom ≫ f)).left\ny : R ⟶ (Over.mk (𝟙 Q)).left\nw✝ : Epi x\ney : Epi y\ncomm : x ≫ p.hom ≫ f = y ≫ 𝟙 Q\n⊢ y ≫ 𝟙 Q = y",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhp : Quotient.mk (setoid P) p = pbar\nthis : Quotient.mk (setoid Q) (Over.mk (p.hom ≫ f)) = Quotient.mk (setoid Q) (Over.mk (𝟙 Q))\nR : C\nx : R ⟶ (Over.mk (p.hom ≫ f)).left\ny : R ⟶ (Over.mk (𝟙 Q)).left\nw✝ : Epi x\ney : Epi y\ncomm : x ≫ p.hom ≫ f = y ≫ 𝟙 Q\n⊢ (x ≫ p.hom) ≫ f = y",
"tactic": "rw [Category.assoc, comm]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhp : Quotient.mk (setoid P) p = pbar\nthis : Quotient.mk (setoid Q) (Over.mk (p.hom ≫ f)) = Quotient.mk (setoid Q) (Over.mk (𝟙 Q))\nR : C\nx : R ⟶ (Over.mk (p.hom ≫ f)).left\ny : R ⟶ (Over.mk (𝟙 Q)).left\nw✝ : Epi x\ney : Epi y\ncomm : x ≫ p.hom ≫ f = y ≫ 𝟙 Q\n⊢ y ≫ 𝟙 Q = y",
"tactic": "apply Category.comp_id"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar✝ : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhpbar : pseudoApply f (Quotient.mk (setoid P) p) = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\nhp : Quotient.mk (setoid P) p = pbar\n⊢ Quotient.mk (?m.445621 hp) (Over.mk (p.hom ≫ f)) = Quotient.mk (?m.445621 hp) (Over.mk (𝟙 Q))",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhp : Quotient.mk (setoid P) p = pbar\n⊢ Quotient.mk ?m.415411 (Over.mk (p.hom ≫ f)) = Quotient.mk ?m.415411 (Over.mk (𝟙 Q))",
"tactic": "rw [← hp] at hpbar"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : Function.Surjective (pseudoApply f)\npbar : Pseudoelement P\nhpbar✝ : pseudoApply f pbar = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\np : Over P\nhpbar : pseudoApply f (Quotient.mk (setoid P) p) = Quot.mk (PseudoEqual Q) (Over.mk (𝟙 Q))\nhp : Quotient.mk (setoid P) p = pbar\n⊢ Quotient.mk (?m.445621 hp) (Over.mk (p.hom ≫ f)) = Quotient.mk (?m.445621 hp) (Over.mk (𝟙 Q))",
"tactic": "exact hpbar"
}
] |
[
355,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.conjTranspose_zero
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.955922\nm : Type u_1\nn : Type u_2\no : Type ?u.955931\nm' : o → Type ?u.955936\nn' : o → Type ?u.955941\nR : Type ?u.955944\nS : Type ?u.955947\nα : Type v\nβ : Type w\nγ : Type ?u.955954\ninst✝¹ : AddMonoid α\ninst✝ : StarAddMonoid α\n⊢ ∀ (i : n) (j : m), 0ᴴ i j = OfNat.ofNat 0 i j",
"tactic": "simp"
}
] |
[
2140,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2139,
1
] |
Mathlib/Algebra/Lie/Semisimple.lean
|
LieAlgebra.isSemisimple_iff_no_solvable_ideals
|
[] |
[
69,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
Associates.count_some
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : DecidableEq (Associates α)\np : Associates α\nhp : Irreducible p\ns : Multiset { a // Irreducible a }\n⊢ dite (Irreducible p) (fun hp => bcount { val := p, property := hp }) (fun hp => 0) (some s) =\n Multiset.count { val := p, property := hp } s",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : DecidableEq (Associates α)\np : Associates α\nhp : Irreducible p\ns : Multiset { a // Irreducible a }\n⊢ count p (some s) = Multiset.count { val := p, property := hp } s",
"tactic": "dsimp only [count]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : DecidableEq (Associates α)\np : Associates α\nhp : Irreducible p\ns : Multiset { a // Irreducible a }\n⊢ bcount { val := p, property := hp } (some s) = Multiset.count { val := p, property := hp } s",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : DecidableEq (Associates α)\np : Associates α\nhp : Irreducible p\ns : Multiset { a // Irreducible a }\n⊢ dite (Irreducible p) (fun hp => bcount { val := p, property := hp }) (fun hp => 0) (some s) =\n Multiset.count { val := p, property := hp } s",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : DecidableEq (Associates α)\np : Associates α\nhp : Irreducible p\ns : Multiset { a // Irreducible a }\n⊢ bcount { val := p, property := hp } (some s) = Multiset.count { val := p, property := hp } s",
"tactic": "rfl"
}
] |
[
1296,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1292,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.isRight_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.8798\nδ : Type ?u.8801\nx y : α ⊕ β\n⊢ isRight x = true ↔ ∃ y, x = inr y",
"tactic": "cases x <;> simp"
}
] |
[
150,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.finset_sum_apply'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nι : Type u_1\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nI : Finset ι\nκ : ι → { x // x ∈ kernel α β }\na : α\ns : Set β\n⊢ ↑↑(↑(∑ i in I, κ i) a) s = ∑ i in I, ↑↑(↑(κ i) a) s",
"tactic": "rw [finset_sum_apply, Measure.finset_sum_apply]"
}
] |
[
111,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Data/Set/Pointwise/Finite.lean
|
Set.Finite.mul
|
[] |
[
53,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.diff_diff
|
[] |
[
1916,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1915,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.nextCoeff_X_add_C
|
[
{
"state_after": "R : Type u\nS : Type v\na b c✝ d : R\nn m : ℕ\ninst✝² : Nontrivial R\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nc : S\n✝ : Nontrivial S\n⊢ nextCoeff (X + ↑C c) = c",
"state_before": "R : Type u\nS : Type v\na b c✝ d : R\nn m : ℕ\ninst✝² : Nontrivial R\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nc : S\n⊢ nextCoeff (X + ↑C c) = c",
"tactic": "nontriviality S"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c✝ d : R\nn m : ℕ\ninst✝² : Nontrivial R\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nc : S\n✝ : Nontrivial S\n⊢ nextCoeff (X + ↑C c) = c",
"tactic": "simp [nextCoeff_of_pos_natDegree]"
}
] |
[
1375,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1373,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Finite.Preorder.wellFounded_lt
|
[] |
[
915,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
914,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.powerlt_zero
|
[] |
[
2324,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2321,
1
] |
Mathlib/Algebra/Module/PointwisePi.lean
|
smul_univ_pi
|
[
{
"state_after": "case refine'_1\nK : Type u_1\nι : Type u_3\nR : ι → Type u_2\ninst✝ : (i : ι) → SMul K (R i)\nr : K\nt : (i : ι) → Set (R i)\nx : (i : ι) → R i\nh : x ∈ pi univ (r • t)\ni : ι\nx✝ : i ∈ univ\n⊢ (fun i => Classical.choose (_ : x i ∈ (r • t) i)) i ∈ t i\n\ncase refine'_2\nK : Type u_1\nι : Type u_3\nR : ι → Type u_2\ninst✝ : (i : ι) → SMul K (R i)\nr : K\nt : (i : ι) → Set (R i)\nx : (i : ι) → R i\nh : x ∈ pi univ (r • t)\ni : ι\n⊢ (fun x => r • x) (fun i => Classical.choose (_ : x i ∈ (r • t) i)) i = x i",
"state_before": "K : Type u_1\nι : Type u_3\nR : ι → Type u_2\ninst✝ : (i : ι) → SMul K (R i)\nr : K\nt : (i : ι) → Set (R i)\nx : (i : ι) → R i\nh : x ∈ pi univ (r • t)\n⊢ x ∈ r • pi univ t",
"tactic": "refine' ⟨fun i ↦ Classical.choose (h i <| Set.mem_univ _), fun i _ ↦ _, funext fun i ↦ _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nK : Type u_1\nι : Type u_3\nR : ι → Type u_2\ninst✝ : (i : ι) → SMul K (R i)\nr : K\nt : (i : ι) → Set (R i)\nx : (i : ι) → R i\nh : x ∈ pi univ (r • t)\ni : ι\nx✝ : i ∈ univ\n⊢ (fun i => Classical.choose (_ : x i ∈ (r • t) i)) i ∈ t i",
"tactic": "exact (Classical.choose_spec (h i <| Set.mem_univ i)).left"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nK : Type u_1\nι : Type u_3\nR : ι → Type u_2\ninst✝ : (i : ι) → SMul K (R i)\nr : K\nt : (i : ι) → Set (R i)\nx : (i : ι) → R i\nh : x ∈ pi univ (r • t)\ni : ι\n⊢ (fun x => r • x) (fun i => Classical.choose (_ : x i ∈ (r • t) i)) i = x i",
"tactic": "exact (Classical.choose_spec (h i <| Set.mem_univ i)).right"
}
] |
[
46,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.exists
|
[] |
[
62,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Analysis/Convex/Topology.lean
|
Convex.closure_subset_interior_image_homothety_of_one_lt
|
[] |
[
329,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.smul_empty
|
[] |
[
117,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
fderiv_zero_of_not_differentiableAt
|
[
{
"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.41308\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.41403\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 : ¬DifferentiableAt 𝕜 f x\nthis : ¬∃ f', HasFDerivAt f f' x\n⊢ fderiv 𝕜 f x = 0",
"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.41308\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.41403\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 : ¬DifferentiableAt 𝕜 f x\n⊢ fderiv 𝕜 f x = 0",
"tactic": "have : ¬∃ f', HasFDerivAt f f' x := 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.41308\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.41403\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 : ¬DifferentiableAt 𝕜 f x\nthis : ¬∃ f', HasFDerivAt f f' x\n⊢ fderiv 𝕜 f x = 0",
"tactic": "simp [fderiv, this]"
}
] |
[
227,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.op_norm_mul_apply_le
|
[] |
[
1118,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1117,
1
] |
Mathlib/Data/Nat/GCD/Basic.lean
|
Nat.coprime_self_add_left
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ coprime (m + n) m ↔ coprime n m",
"tactic": "rw [coprime, coprime, gcd_self_add_left]"
}
] |
[
149,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
iteratedFDerivWithin_inter
|
[] |
[
948,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
946,
1
] |
Mathlib/Order/JordanHolder.lean
|
CompositionSeries.lt_succ
|
[] |
[
171,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.prod_iInf_left
|
[
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.11121\nδ : Type ?u.11124\nι : Sort u_1\ns : Set α\nt : Set β\nf✝ : Filter α\ng✝ : Filter β\ninst✝ : Nonempty ι\nf : ι → Filter α\ng : Filter β\n⊢ Filter.prod (⨅ (i : ι), f i) g = ⨅ (i : ι), Filter.prod (f i) g",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.11121\nδ : Type ?u.11124\nι : Sort u_1\ns : Set α\nt : Set β\nf✝ : Filter α\ng✝ : Filter β\ninst✝ : Nonempty ι\nf : ι → Filter α\ng : Filter β\n⊢ (⨅ (i : ι), f i) ×ˢ g = ⨅ (i : ι), f i ×ˢ g",
"tactic": "dsimp only [SProd.sprod]"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.11121\nδ : Type ?u.11124\nι : Sort u_1\ns : Set α\nt : Set β\nf✝ : Filter α\ng✝ : Filter β\ninst✝ : Nonempty ι\nf : ι → Filter α\ng : Filter β\n⊢ (⨅ (x : ι), comap Prod.fst (f x) ⊓ comap Prod.snd g) = ⨅ (i : ι), Filter.prod (f i) g",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.11121\nδ : Type ?u.11124\nι : Sort u_1\ns : Set α\nt : Set β\nf✝ : Filter α\ng✝ : Filter β\ninst✝ : Nonempty ι\nf : ι → Filter α\ng : Filter β\n⊢ Filter.prod (⨅ (i : ι), f i) g = ⨅ (i : ι), Filter.prod (f i) g",
"tactic": "rw [Filter.prod, comap_iInf, iInf_inf]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.11121\nδ : Type ?u.11124\nι : Sort u_1\ns : Set α\nt : Set β\nf✝ : Filter α\ng✝ : Filter β\ninst✝ : Nonempty ι\nf : ι → Filter α\ng : Filter β\n⊢ (⨅ (x : ι), comap Prod.fst (f x) ⊓ comap Prod.snd g) = ⨅ (i : ι), Filter.prod (f i) g",
"tactic": "simp only [Filter.prod, eq_self_iff_true]"
}
] |
[
220,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/LinearAlgebra/Dfinsupp.lean
|
Dfinsupp.mapRange.linearMap_id
|
[
{
"state_after": "case h.h.h\nι : Type u_1\nR : Type u_3\nS : Type ?u.140242\nM : ι → Type ?u.140247\nN : Type ?u.140250\ndec_ι : DecidableEq ι\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : ι) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → Module R (M i)\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nβ : ι → Type ?u.140337\nβ₁ : ι → Type ?u.140342\nβ₂ : ι → Type u_2\ninst✝⁵ : (i : ι) → AddCommMonoid (β i)\ninst✝⁴ : (i : ι) → AddCommMonoid (β₁ i)\ninst✝³ : (i : ι) → AddCommMonoid (β₂ i)\ninst✝² : (i : ι) → Module R (β i)\ninst✝¹ : (i : ι) → Module R (β₁ i)\ninst✝ : (i : ι) → Module R (β₂ i)\ni✝¹ : ι\nx✝ : β₂ i✝¹\ni✝ : ι\n⊢ ↑(↑(LinearMap.comp (linearMap fun i => LinearMap.id) (lsingle i✝¹)) x✝) i✝ =\n ↑(↑(LinearMap.comp LinearMap.id (lsingle i✝¹)) x✝) i✝",
"state_before": "ι : Type u_1\nR : Type u_3\nS : Type ?u.140242\nM : ι → Type ?u.140247\nN : Type ?u.140250\ndec_ι : DecidableEq ι\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : ι) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → Module R (M i)\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nβ : ι → Type ?u.140337\nβ₁ : ι → Type ?u.140342\nβ₂ : ι → Type u_2\ninst✝⁵ : (i : ι) → AddCommMonoid (β i)\ninst✝⁴ : (i : ι) → AddCommMonoid (β₁ i)\ninst✝³ : (i : ι) → AddCommMonoid (β₂ i)\ninst✝² : (i : ι) → Module R (β i)\ninst✝¹ : (i : ι) → Module R (β₁ i)\ninst✝ : (i : ι) → Module R (β₂ i)\n⊢ (linearMap fun i => LinearMap.id) = LinearMap.id",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nι : Type u_1\nR : Type u_3\nS : Type ?u.140242\nM : ι → Type ?u.140247\nN : Type ?u.140250\ndec_ι : DecidableEq ι\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : ι) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → Module R (M i)\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nβ : ι → Type ?u.140337\nβ₁ : ι → Type ?u.140342\nβ₂ : ι → Type u_2\ninst✝⁵ : (i : ι) → AddCommMonoid (β i)\ninst✝⁴ : (i : ι) → AddCommMonoid (β₁ i)\ninst✝³ : (i : ι) → AddCommMonoid (β₂ i)\ninst✝² : (i : ι) → Module R (β i)\ninst✝¹ : (i : ι) → Module R (β₁ i)\ninst✝ : (i : ι) → Module R (β₂ i)\ni✝¹ : ι\nx✝ : β₂ i✝¹\ni✝ : ι\n⊢ ↑(↑(LinearMap.comp (linearMap fun i => LinearMap.id) (lsingle i✝¹)) x✝) i✝ =\n ↑(↑(LinearMap.comp LinearMap.id (lsingle i✝¹)) x✝) i✝",
"tactic": "simp [linearMap]"
}
] |
[
211,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.lineMap_apply_module'
|
[] |
[
522,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
src/lean/Init/Data/String/Basic.lean
|
String.utf8PrevAux_lt_of_pos
|
[
{
"state_after": "c : Char\ncs : List Char\ni p : Pos\nh : p ≠ 0\n⊢ (ite (i + c = p) i (utf8PrevAux cs (i + c) p)).byteIdx < p.byteIdx",
"state_before": "c : Char\ncs : List Char\ni p : Pos\nh : p ≠ 0\n⊢ (utf8PrevAux (c :: cs) i p).byteIdx < p.byteIdx",
"tactic": "simp [utf8PrevAux]"
},
{
"state_after": "case hneg\nc : Char\ncs : List Char\ni p : Pos\nh : p ≠ 0\nh' : ¬i + c = p\n⊢ (utf8PrevAux cs (i + c) p).byteIdx < p.byteIdx",
"state_before": "case hpos\nc : Char\ncs : List Char\ni p : Pos\nh : p ≠ 0\nh' : i + c = p\n⊢ i.byteIdx < p.byteIdx\n\ncase hneg\nc : Char\ncs : List Char\ni p : Pos\nh : p ≠ 0\nh' : ¬i + c = p\n⊢ (utf8PrevAux cs (i + c) p).byteIdx < p.byteIdx",
"tactic": "next => exact h' ▸ Nat.add_lt_add_left (one_le_csize _) _"
},
{
"state_after": "no goals",
"state_before": "case hneg\nc : Char\ncs : List Char\ni p : Pos\nh : p ≠ 0\nh' : ¬i + c = p\n⊢ (utf8PrevAux cs (i + c) p).byteIdx < p.byteIdx",
"tactic": "next => exact utf8PrevAux_lt_of_pos _ _ _ h"
},
{
"state_after": "no goals",
"state_before": "c : Char\ncs : List Char\ni p : Pos\nh : p ≠ 0\nh' : i + c = p\n⊢ i.byteIdx < p.byteIdx",
"tactic": "exact h' ▸ Nat.add_lt_add_left (one_le_csize _) _"
},
{
"state_after": "no goals",
"state_before": "c : Char\ncs : List Char\ni p : Pos\nh : p ≠ 0\nh' : ¬i + c = p\n⊢ (utf8PrevAux cs (i + c) p).byteIdx < p.byteIdx",
"tactic": "exact utf8PrevAux_lt_of_pos _ _ _ h"
}
] |
[
149,
48
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
140,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.sum_apply
|
[
{
"state_after": "no goals",
"state_before": "R₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type ?u.493091\ninst✝¹⁷ : Semiring R₁\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nM₁ : Type u_4\ninst✝¹⁴ : TopologicalSpace M₁\ninst✝¹³ : AddCommMonoid M₁\nM'₁ : Type ?u.493178\ninst✝¹² : TopologicalSpace M'₁\ninst✝¹¹ : AddCommMonoid M'₁\nM₂ : Type u_5\ninst✝¹⁰ : TopologicalSpace M₂\ninst✝⁹ : AddCommMonoid M₂\nM₃ : Type ?u.493196\ninst✝⁸ : TopologicalSpace M₃\ninst✝⁷ : AddCommMonoid M₃\nM₄ : Type ?u.493205\ninst✝⁶ : TopologicalSpace M₄\ninst✝⁵ : AddCommMonoid M₄\ninst✝⁴ : Module R₁ M₁\ninst✝³ : Module R₁ M'₁\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\ninst✝ : ContinuousAdd M₂\nι : Type u_1\nt : Finset ι\nf : ι → M₁ →SL[σ₁₂] M₂\nb : M₁\n⊢ ↑(∑ d in t, f d) b = ∑ d in t, ↑(f d) b",
"tactic": "simp only [coe_sum', Finset.sum_apply]"
}
] |
[
763,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
762,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.map_sub
|
[] |
[
360,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
11
] |
Mathlib/Data/Nat/Factors.lean
|
Nat.factors_sublist_of_dvd
|
[
{
"state_after": "case intro\nn a : ℕ\nh' : n * a ≠ 0\n⊢ factors n <+ factors (n * a)",
"state_before": "n k : ℕ\nh : n ∣ k\nh' : k ≠ 0\n⊢ factors n <+ factors k",
"tactic": "obtain ⟨a, rfl⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case intro\nn a : ℕ\nh' : n * a ≠ 0\n⊢ factors n <+ factors (n * a)",
"tactic": "exact factors_sublist_right (right_ne_zero_of_mul h')"
}
] |
[
230,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Algebra/Quandle.lean
|
Rack.PreEnvelGroupRel'.rel
|
[] |
[
640,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
639,
1
] |
Mathlib/Data/Int/Interval.lean
|
Int.card_Ioo
|
[
{
"state_after": "a b : ℤ\n⊢ card (map (Function.Embedding.trans Nat.castEmbedding (addLeftEmbedding (a + 1))) (range (toNat (b - a - 1)))) =\n toNat (b - a - 1)",
"state_before": "a b : ℤ\n⊢ card (Ioo a b) = toNat (b - a - 1)",
"tactic": "change (Finset.map _ _).card = _"
},
{
"state_after": "no goals",
"state_before": "a b : ℤ\n⊢ card (map (Function.Embedding.trans Nat.castEmbedding (addLeftEmbedding (a + 1))) (range (toNat (b - a - 1)))) =\n toNat (b - a - 1)",
"tactic": "rw [Finset.card_map, Finset.card_range]"
}
] |
[
127,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.sub_re
|
[] |
[
927,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
927,
9
] |
Mathlib/ModelTheory/Definability.lean
|
FirstOrder.Language.DefinableSet.coe_top
|
[] |
[
344,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/Topology/Maps.lean
|
closedEmbedding_of_embedding_closed
|
[] |
[
677,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
675,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
lowerBounds_Ico
|
[] |
[
732,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
731,
1
] |
Mathlib/NumberTheory/RamificationInertia.lean
|
Ideal.ramificationIdx_spec
|
[
{
"state_after": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\n⊢ ramificationIdx f p P = n",
"state_before": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\n⊢ ramificationIdx f p P = n",
"tactic": "let Q : ℕ → Prop := fun m => ∀ k : ℕ, map f p ≤ P ^ k → k ≤ m"
},
{
"state_after": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nthis : Q n\n⊢ ramificationIdx f p P = n",
"state_before": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\n⊢ ramificationIdx f p P = n",
"tactic": "have : Q n := by\n intro k hk\n refine le_of_not_lt fun hnk => ?_\n exact hgt (hk.trans (Ideal.pow_le_pow hnk))"
},
{
"state_after": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nthis : Q n\n⊢ Nat.find (_ : ∃ n, ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ n) = n",
"state_before": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nthis : Q n\n⊢ ramificationIdx f p P = n",
"tactic": "rw [ramificationIdx_eq_find ⟨n, this⟩]"
},
{
"state_after": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nthis : Q n\nh : Nat.find (_ : ∃ n, ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ n) < n\n⊢ False",
"state_before": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nthis : Q n\n⊢ Nat.find (_ : ∃ n, ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ n) = n",
"tactic": "refine le_antisymm (Nat.find_min' _ this) (le_of_not_gt fun h : Nat.find _ < n => ?_)"
},
{
"state_after": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nthis : Q n\nh : Nat.find (_ : ∃ n, ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ n) < n\nthis' : Q (Nat.find (_ : ∃ n, Q n))\n⊢ False",
"state_before": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nthis : Q n\nh : Nat.find (_ : ∃ n, ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ n) < n\n⊢ False",
"tactic": "obtain this' := Nat.find_spec ⟨n, this⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nthis : Q n\nh : Nat.find (_ : ∃ n, ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ n) < n\nthis' : Q (Nat.find (_ : ∃ n, Q n))\n⊢ False",
"tactic": "exact h.not_le (this' _ hle)"
},
{
"state_after": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nk : ℕ\nhk : map f p ≤ P ^ k\n⊢ k ≤ n",
"state_before": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\n⊢ Q n",
"tactic": "intro k hk"
},
{
"state_after": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nk : ℕ\nhk : map f p ≤ P ^ k\nhnk : n < k\n⊢ False",
"state_before": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nk : ℕ\nhk : map f p ≤ P ^ k\n⊢ k ≤ n",
"tactic": "refine le_of_not_lt fun hnk => ?_"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommRing R\nS : Type v\ninst✝ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nk : ℕ\nhk : map f p ≤ P ^ k\nhnk : n < k\n⊢ False",
"tactic": "exact hgt (hk.trans (Ideal.pow_le_pow hnk))"
}
] |
[
98,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Computability/Reduce.lean
|
ManyOneEquiv.le_congr_left
|
[] |
[
245,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/CategoryTheory/Skeletal.lean
|
CategoryTheory.Functor.eq_of_iso
|
[] |
[
66,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.C_one
|
[] |
[
973,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
972,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.MapsTo.congr
|
[] |
[
412,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
411,
1
] |
Mathlib/NumberTheory/BernoulliPolynomials.lean
|
Polynomial.bernoulli_zero
|
[
{
"state_after": "no goals",
"state_before": "⊢ bernoulli 0 = 1",
"tactic": "simp [bernoulli]"
}
] |
[
76,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
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