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/Logic/Function/Basic.lean
|
Function.Injective2.right
|
[] |
[
935,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
934,
11
] |
Mathlib/Algebra/Periodic.lean
|
Function.Periodic.const_sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.87566\nf g : α → β\nc c₁ c₂ x✝ : α\ninst✝ : AddCommGroup α\nh : Periodic f c\na x : α\n⊢ (fun x => f (a - x)) (x + c) = (fun x => f (a - x)) x",
"tactic": "simp only [← sub_sub, h.sub_eq]"
}
] |
[
193,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.closure_sUnion
|
[] |
[
1059,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1058,
1
] |
Mathlib/CategoryTheory/Monoidal/Free/Basic.lean
|
CategoryTheory.FreeMonoidalCategory.mk_ρ_hom
|
[] |
[
216,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Antiperiodic.add_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.167848\nf g : α → β\nc c₁ c₂ x✝ : α\ninst✝¹ : AddCommSemigroup α\ninst✝ : Neg β\nh : Antiperiodic f c\na x : α\n⊢ (fun x => f (x + a)) (x + c) = -(fun x => f (x + a)) x",
"tactic": "simpa only [add_right_comm] using h (x + a)"
}
] |
[
441,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
439,
1
] |
Mathlib/Data/Polynomial/CancelLeads.lean
|
Polynomial.dvd_cancelLeads_of_dvd_of_dvd
|
[] |
[
86,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Combinatorics/Partition.lean
|
Nat.Partition.ofComposition_surj
|
[
{
"state_after": "case mk\nα : Type ?u.1285\nn : ℕ\nb : Multiset ℕ\nhb₁ : ∀ {i : ℕ}, i ∈ b → 0 < i\nhb₂ : sum b = n\n⊢ ∃ a, ofComposition n a = { parts := b, parts_pos := hb₁, parts_sum := hb₂ }",
"state_before": "α : Type ?u.1285\nn : ℕ\n⊢ Function.Surjective (ofComposition n)",
"tactic": "rintro ⟨b, hb₁, hb₂⟩"
},
{
"state_after": "case mk.intro\nα : Type ?u.1285\nn : ℕ\nb : List ℕ\nhb₁ : ∀ {i : ℕ}, i ∈ Quotient.mk (List.isSetoid ℕ) b → 0 < i\nhb₂ : sum (Quotient.mk (List.isSetoid ℕ) b) = n\n⊢ ∃ a, ofComposition n a = { parts := Quotient.mk (List.isSetoid ℕ) b, parts_pos := hb₁, parts_sum := hb₂ }",
"state_before": "case mk\nα : Type ?u.1285\nn : ℕ\nb : Multiset ℕ\nhb₁ : ∀ {i : ℕ}, i ∈ b → 0 < i\nhb₂ : sum b = n\n⊢ ∃ a, ofComposition n a = { parts := b, parts_pos := hb₁, parts_sum := hb₂ }",
"tactic": "rcases Quotient.exists_rep b with ⟨b, rfl⟩"
},
{
"state_after": "case mk.intro\nα : Type ?u.1285\nn : ℕ\nb : List ℕ\nhb₁ : ∀ {i : ℕ}, i ∈ Quotient.mk (List.isSetoid ℕ) b → 0 < i\nhb₂ : sum (Quotient.mk (List.isSetoid ℕ) b) = n\n⊢ List.sum b = n",
"state_before": "case mk.intro\nα : Type ?u.1285\nn : ℕ\nb : List ℕ\nhb₁ : ∀ {i : ℕ}, i ∈ Quotient.mk (List.isSetoid ℕ) b → 0 < i\nhb₂ : sum (Quotient.mk (List.isSetoid ℕ) b) = n\n⊢ ∃ a, ofComposition n a = { parts := Quotient.mk (List.isSetoid ℕ) b, parts_pos := hb₁, parts_sum := hb₂ }",
"tactic": "refine' ⟨⟨b, fun {i} hi => hb₁ hi, _⟩, Partition.ext _ _ rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.intro\nα : Type ?u.1285\nn : ℕ\nb : List ℕ\nhb₁ : ∀ {i : ℕ}, i ∈ Quotient.mk (List.isSetoid ℕ) b → 0 < i\nhb₂ : sum (Quotient.mk (List.isSetoid ℕ) b) = n\n⊢ List.sum b = n",
"tactic": "simpa using hb₂"
}
] |
[
85,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
Real.neg_sqrt_lt_of_sq_lt
|
[] |
[
441,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
440,
1
] |
Mathlib/Probability/CondCount.lean
|
ProbabilityTheory.condCount_inter_self
|
[
{
"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\n⊢ ↑↑(condCount s) (s ∩ t) = ↑↑(condCount s) t",
"tactic": "rw [condCount, cond_inter_self _ hs.measurableSet]"
}
] |
[
103,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Data/Set/Sups.lean
|
Set.iUnion_image_sup_right
|
[] |
[
183,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometry.isComplete_image_iff
|
[] |
[
249,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.n_lt_a_pow
|
[
{
"state_after": "a : ℕ\na1 : 1 < a\nn : ℕ\nIH : n < a ^ n\n⊢ n + 1 < a ^ (n + 1)",
"state_before": "a : ℕ\na1 : 1 < a\nn : ℕ\n⊢ n + 1 < a ^ (n + 1)",
"tactic": "have IH := n_lt_a_pow n"
},
{
"state_after": "a : ℕ\na1 : 1 < a\nn : ℕ\nIH : n < a ^ n\nthis : a ^ n + a ^ n ≤ a ^ n * a\n⊢ n + 1 < a ^ (n + 1)",
"state_before": "a : ℕ\na1 : 1 < a\nn : ℕ\nIH : n < a ^ n\n⊢ n + 1 < a ^ (n + 1)",
"tactic": "have : a ^ n + a ^ n ≤ a ^ n * a := by\n rw [← mul_two]\n exact Nat.mul_le_mul_left _ a1"
},
{
"state_after": "a : ℕ\na1 : 1 < a\nn : ℕ\nIH : n < a ^ n\nthis : a ^ n + a ^ n ≤ a ^ n * a\n⊢ n + 1 < a ^ n * a",
"state_before": "a : ℕ\na1 : 1 < a\nn : ℕ\nIH : n < a ^ n\nthis : a ^ n + a ^ n ≤ a ^ n * a\n⊢ n + 1 < a ^ (n + 1)",
"tactic": "simp [_root_.pow_succ']"
},
{
"state_after": "a : ℕ\na1 : 1 < a\nn : ℕ\nIH : n < a ^ n\nthis : a ^ n + a ^ n ≤ a ^ n * a\n⊢ n + 1 < a ^ n + a ^ n",
"state_before": "a : ℕ\na1 : 1 < a\nn : ℕ\nIH : n < a ^ n\nthis : a ^ n + a ^ n ≤ a ^ n * a\n⊢ n + 1 < a ^ n * a",
"tactic": "refine' lt_of_lt_of_le _ this"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\nn : ℕ\nIH : n < a ^ n\nthis : a ^ n + a ^ n ≤ a ^ n * a\n⊢ n + 1 < a ^ n + a ^ n",
"tactic": "exact add_lt_add_of_lt_of_le IH (lt_of_le_of_lt (Nat.zero_le _) IH)"
},
{
"state_after": "a : ℕ\na1 : 1 < a\nn : ℕ\nIH : n < a ^ n\n⊢ a ^ n * 2 ≤ a ^ n * a",
"state_before": "a : ℕ\na1 : 1 < a\nn : ℕ\nIH : n < a ^ n\n⊢ a ^ n + a ^ n ≤ a ^ n * a",
"tactic": "rw [← mul_two]"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\nn : ℕ\nIH : n < a ^ n\n⊢ a ^ n * 2 ≤ a ^ n * a",
"tactic": "exact Nat.mul_le_mul_left _ a1"
}
] |
[
279,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.monomial_pow
|
[
{
"state_after": "case zero\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nr : R\n⊢ ↑(monomial n) r ^ Nat.zero = ↑(monomial (n * Nat.zero)) (r ^ Nat.zero)\n\ncase succ\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nr : R\nk : ℕ\nih : ↑(monomial n) r ^ k = ↑(monomial (n * k)) (r ^ k)\n⊢ ↑(monomial n) r ^ Nat.succ k = ↑(monomial (n * Nat.succ k)) (r ^ Nat.succ k)",
"state_before": "R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nr : R\nk : ℕ\n⊢ ↑(monomial n) r ^ k = ↑(monomial (n * k)) (r ^ k)",
"tactic": "induction' k with k ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nr : R\n⊢ ↑(monomial n) r ^ Nat.zero = ↑(monomial (n * Nat.zero)) (r ^ Nat.zero)",
"tactic": "simp [pow_zero, monomial_zero_one]"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nr : R\nk : ℕ\nih : ↑(monomial n) r ^ k = ↑(monomial (n * k)) (r ^ k)\n⊢ ↑(monomial n) r ^ Nat.succ k = ↑(monomial (n * Nat.succ k)) (r ^ Nat.succ k)",
"tactic": "simp [pow_succ, ih, monomial_mul_monomial, Nat.succ_eq_add_one, mul_add, add_comm]"
}
] |
[
456,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
453,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.norm_eq_of_isometry
|
[] |
[
855,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
854,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
NNRat.coe_div
|
[] |
[
145,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Algebra/Ring/Prod.lean
|
RingHom.coe_prodMap
|
[] |
[
263,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
summable_int_of_summable_nat
|
[] |
[
1033,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1031,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.partiallyWellOrderedOn_union
|
[] |
[
270,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Topology/Order/Basic.lean
|
comap_coe_Iio_nhdsWithin_Iio
|
[] |
[
2506,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2505,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.liftRel_pure_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nca : Computation α\nb : β\n⊢ LiftRel R ca (pure b) ↔ ∃ a, a ∈ ca ∧ R a b",
"tactic": "rw [LiftRel.swap, liftRel_pure_left]"
}
] |
[
1188,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1187,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.biUnion_mono
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.517176\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\nh : ∀ (a : α), a ∈ s → t₁ a ⊆ t₂ a\nthis : ∀ (b : β) (a : α), a ∈ s → b ∈ t₁ a → ∃ a, a ∈ s ∧ b ∈ t₂ a\n⊢ Finset.biUnion s t₁ ⊆ Finset.biUnion s t₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.517176\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\nh : ∀ (a : α), a ∈ s → t₁ a ⊆ t₂ a\n⊢ Finset.biUnion s t₁ ⊆ Finset.biUnion s t₂",
"tactic": "have : ∀ b a, a ∈ s → b ∈ t₁ a → ∃ a : α, a ∈ s ∧ b ∈ t₂ a := fun b a ha hb =>\n ⟨a, ha, Finset.mem_of_subset (h a ha) hb⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.517176\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\nh : ∀ (a : α), a ∈ s → t₁ a ⊆ t₂ a\nthis : ∀ (b : β) (a : α), a ∈ s → b ∈ t₁ a → ∃ a, a ∈ s ∧ b ∈ t₂ a\n⊢ Finset.biUnion s t₁ ⊆ Finset.biUnion s t₂",
"tactic": "simpa only [subset_iff, mem_biUnion, exists_imp, and_imp, exists_prop]"
}
] |
[
3609,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3606,
1
] |
Mathlib/Data/List/Pairwise.lean
|
List.pwFilter_cons_of_pos
|
[] |
[
363,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.mul_lt_mul'
|
[] |
[
1194,
95
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1192,
11
] |
Mathlib/Data/Set/Basic.lean
|
Set.singleton_union
|
[] |
[
1326,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1325,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
inv_le_iff_one_le_mul'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c d : α\n⊢ a * a⁻¹ ≤ a * b ↔ 1 ≤ a * b",
"tactic": "rw [mul_inv_self]"
}
] |
[
130,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.subset_one_iff_eq
|
[] |
[
134,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.degree_add_div
|
[
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nhpq : degree q ≤ degree p\nthis : degree (p % q) < degree (q * (p / q))\n⊢ degree q + degree (p / q) = degree p",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nhpq : degree q ≤ degree p\n⊢ degree q + degree (p / q) = degree p",
"tactic": "have : degree (p % q) < degree (q * (p / q)) :=\n calc\n degree (p % q) < degree q := EuclideanDomain.mod_lt _ hq0\n _ ≤ _ := degree_le_mul_left _ (mt (div_eq_zero_iff hq0).1 (not_lt_of_ge hpq))"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nhpq : degree q ≤ degree p\nthis : degree (p % q) < degree (q * (p / q))\n⊢ degree q + degree (p / q) = degree p",
"tactic": "conv_rhs =>\n rw [← EuclideanDomain.div_add_mod p q, degree_add_eq_left_of_degree_lt this, degree_mul]"
}
] |
[
260,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/MeasureTheory/Integral/CircleIntegral.lean
|
sum_cauchyPowerSeries_eq_integral
|
[] |
[
629,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
626,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_const_mul''
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.935279\nγ : Type ?u.935282\nδ : Type ?u.935285\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : AEMeasurable f\nA : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk f hf a ∂μ\n⊢ (∫⁻ (a : α), r * f a ∂μ) = r * ∫⁻ (a : α), f a ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.935279\nγ : Type ?u.935282\nδ : Type ?u.935285\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : AEMeasurable f\n⊢ (∫⁻ (a : α), r * f a ∂μ) = r * ∫⁻ (a : α), f a ∂μ",
"tactic": "have A : (∫⁻ a, f a ∂μ) = ∫⁻ a, hf.mk f a ∂μ := lintegral_congr_ae hf.ae_eq_mk"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.935279\nγ : Type ?u.935282\nδ : Type ?u.935285\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : AEMeasurable f\nA : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk f hf a ∂μ\nB : (∫⁻ (a : α), r * f a ∂μ) = ∫⁻ (a : α), r * AEMeasurable.mk f hf a ∂μ\n⊢ (∫⁻ (a : α), r * f a ∂μ) = r * ∫⁻ (a : α), f a ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.935279\nγ : Type ?u.935282\nδ : Type ?u.935285\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : AEMeasurable f\nA : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk f hf a ∂μ\n⊢ (∫⁻ (a : α), r * f a ∂μ) = r * ∫⁻ (a : α), f a ∂μ",
"tactic": "have B : (∫⁻ a, r * f a ∂μ) = ∫⁻ a, r * hf.mk f a ∂μ :=\n lintegral_congr_ae (EventuallyEq.fun_comp hf.ae_eq_mk _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.935279\nγ : Type ?u.935282\nδ : Type ?u.935285\nm : MeasurableSpace α\nμ ν : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : AEMeasurable f\nA : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk f hf a ∂μ\nB : (∫⁻ (a : α), r * f a ∂μ) = ∫⁻ (a : α), r * AEMeasurable.mk f hf a ∂μ\n⊢ (∫⁻ (a : α), r * f a ∂μ) = r * ∫⁻ (a : α), f a ∂μ",
"tactic": "rw [A, B, lintegral_const_mul _ hf.measurable_mk]"
}
] |
[
704,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
699,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_const
|
[
{
"state_after": "case inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ < ⊤\n⊢ (∫ (x : α), c ∂μ) = ENNReal.toReal (↑↑μ univ) • c\n\ncase inr\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ = ⊤\n⊢ (∫ (x : α), c ∂μ) = ENNReal.toReal (↑↑μ univ) • c",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\n⊢ (∫ (x : α), c ∂μ) = ENNReal.toReal (↑↑μ univ) • c",
"tactic": "cases' (@le_top _ _ _ (μ univ)).lt_or_eq with hμ hμ"
},
{
"state_after": "case inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ < ⊤\nthis : IsFiniteMeasure μ\n⊢ (∫ (x : α), c ∂μ) = ENNReal.toReal (↑↑μ univ) • c",
"state_before": "case inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ < ⊤\n⊢ (∫ (x : α), c ∂μ) = ENNReal.toReal (↑↑μ univ) • c",
"tactic": "haveI : IsFiniteMeasure μ := ⟨hμ⟩"
},
{
"state_after": "case inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ < ⊤\nthis : IsFiniteMeasure μ\n⊢ (if hf : Integrable fun x => c then ↑L1.integralCLM (Integrable.toL1 (fun x => c) hf) else 0) =\n ENNReal.toReal (↑↑μ univ) • c",
"state_before": "case inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ < ⊤\nthis : IsFiniteMeasure μ\n⊢ (∫ (x : α), c ∂μ) = ENNReal.toReal (↑↑μ univ) • c",
"tactic": "simp only [integral, L1.integral]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ < ⊤\nthis : IsFiniteMeasure μ\n⊢ (if hf : Integrable fun x => c then ↑L1.integralCLM (Integrable.toL1 (fun x => c) hf) else 0) =\n ENNReal.toReal (↑↑μ univ) • c",
"tactic": "exact setToFun_const (dominatedFinMeasAdditive_weightedSMul _) _"
},
{
"state_after": "case pos\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ = ⊤\nhc : c = 0\n⊢ (∫ (x : α), c ∂μ) = ENNReal.toReal (↑↑μ univ) • c\n\ncase neg\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ = ⊤\nhc : ¬c = 0\n⊢ (∫ (x : α), c ∂μ) = ENNReal.toReal (↑↑μ univ) • c",
"state_before": "case inr\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ = ⊤\n⊢ (∫ (x : α), c ∂μ) = ENNReal.toReal (↑↑μ univ) • c",
"tactic": "by_cases hc : c = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ = ⊤\nhc : c = 0\n⊢ (∫ (x : α), c ∂μ) = ENNReal.toReal (↑↑μ univ) • c",
"tactic": "simp [hc, integral_zero]"
},
{
"state_after": "case neg\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ = ⊤\nhc : ¬c = 0\nthis : ¬Integrable fun x => c\n⊢ (∫ (x : α), c ∂μ) = ENNReal.toReal (↑↑μ univ) • c",
"state_before": "case neg\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ = ⊤\nhc : ¬c = 0\n⊢ (∫ (x : α), c ∂μ) = ENNReal.toReal (↑↑μ univ) • c",
"tactic": "have : ¬Integrable (fun _ : α => c) μ := by\n simp only [integrable_const_iff, not_or]\n exact ⟨hc, hμ.not_lt⟩"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ = ⊤\nhc : ¬c = 0\nthis : ¬Integrable fun x => c\n⊢ (∫ (x : α), c ∂μ) = ENNReal.toReal (↑↑μ univ) • c",
"tactic": "simp [integral_undef, *]"
},
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ = ⊤\nhc : ¬c = 0\n⊢ ¬c = 0 ∧ ¬↑↑μ univ < ⊤",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ = ⊤\nhc : ¬c = 0\n⊢ ¬Integrable fun x => c",
"tactic": "simp only [integrable_const_iff, not_or]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1256042\n𝕜 : Type ?u.1256045\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1258736\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nc : E\nhμ : ↑↑μ univ = ⊤\nhc : ¬c = 0\n⊢ ¬c = 0 ∧ ¬↑↑μ univ < ⊤",
"tactic": "exact ⟨hc, hμ.not_lt⟩"
}
] |
[
1371,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1361,
1
] |
Mathlib/Order/BoundedOrder.lean
|
lt_top_iff_ne_top
|
[] |
[
175,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/Data/Nat/ModEq.lean
|
Dvd.dvd.zero_modEq_nat
|
[] |
[
87,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiff_one_iff_deriv
|
[] |
[
2156,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2155,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.copy_eq
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.23151\nδ : Type ?u.23154\ne✝ : LocalEquiv α β\ne' : LocalEquiv β γ\ne : LocalEquiv α β\n⊢ copy e ↑e (_ : ↑e = ↑e) ↑(LocalEquiv.symm e) (_ : ↑(LocalEquiv.symm e) = ↑(LocalEquiv.symm e)) e.source\n (_ : e.source = e.source) e.target (_ : e.target = e.target) =\n e",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.23151\nδ : Type ?u.23154\ne✝ : LocalEquiv α β\ne' : LocalEquiv β γ\ne : LocalEquiv α β\nf : α → β\nhf : ↑e = f\ng : β → α\nhg : ↑(LocalEquiv.symm e) = g\ns : Set α\nhs : e.source = s\nt : Set β\nht : e.target = t\n⊢ copy e f hf g hg s hs t ht = e",
"tactic": "substs f g s t"
},
{
"state_after": "case mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.23151\nδ : Type ?u.23154\ne : LocalEquiv α β\ne' : LocalEquiv β γ\ntoFun✝ : α → β\ninvFun✝ : β → α\nsource✝ : Set α\ntarget✝ : Set β\nmap_source'✝ : ∀ ⦃x : α⦄, x ∈ source✝ → toFun✝ x ∈ target✝\nmap_target'✝ : ∀ ⦃x : β⦄, x ∈ target✝ → invFun✝ x ∈ source✝\nleft_inv'✝ : ∀ ⦃x : α⦄, x ∈ source✝ → invFun✝ (toFun✝ x) = x\nright_inv'✝ : ∀ ⦃x : β⦄, x ∈ target✝ → toFun✝ (invFun✝ x) = x\n⊢ copy\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }\n ↑{ toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }\n (_ :\n ↑{ toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ } =\n ↑{ toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ })\n ↑(LocalEquiv.symm\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ })\n (_ :\n ↑(LocalEquiv.symm\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }) =\n ↑(LocalEquiv.symm\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }))\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }.source\n (_ :\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }.source =\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }.source)\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }.target\n (_ :\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }.target =\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }.target) =\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.23151\nδ : Type ?u.23154\ne✝ : LocalEquiv α β\ne' : LocalEquiv β γ\ne : LocalEquiv α β\n⊢ copy e ↑e (_ : ↑e = ↑e) ↑(LocalEquiv.symm e) (_ : ↑(LocalEquiv.symm e) = ↑(LocalEquiv.symm e)) e.source\n (_ : e.source = e.source) e.target (_ : e.target = e.target) =\n e",
"tactic": "cases e"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.23151\nδ : Type ?u.23154\ne : LocalEquiv α β\ne' : LocalEquiv β γ\ntoFun✝ : α → β\ninvFun✝ : β → α\nsource✝ : Set α\ntarget✝ : Set β\nmap_source'✝ : ∀ ⦃x : α⦄, x ∈ source✝ → toFun✝ x ∈ target✝\nmap_target'✝ : ∀ ⦃x : β⦄, x ∈ target✝ → invFun✝ x ∈ source✝\nleft_inv'✝ : ∀ ⦃x : α⦄, x ∈ source✝ → invFun✝ (toFun✝ x) = x\nright_inv'✝ : ∀ ⦃x : β⦄, x ∈ target✝ → toFun✝ (invFun✝ x) = x\n⊢ copy\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }\n ↑{ toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }\n (_ :\n ↑{ toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ } =\n ↑{ toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ })\n ↑(LocalEquiv.symm\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ })\n (_ :\n ↑(LocalEquiv.symm\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }) =\n ↑(LocalEquiv.symm\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }))\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }.source\n (_ :\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }.source =\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }.source)\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }.target\n (_ :\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }.target =\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }.target) =\n { toFun := toFun✝, invFun := invFun✝, source := source✝, target := target✝, map_source' := map_source'✝,\n map_target' := map_target'✝, left_inv' := left_inv'✝, right_inv' := right_inv'✝ }",
"tactic": "rfl"
}
] |
[
308,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
303,
1
] |
Mathlib/Topology/Sets/Opens.lean
|
TopologicalSpace.Opens.mem_iSup
|
[
{
"state_after": "ι✝ : Type ?u.22254\nα : Type u_2\nβ : Type ?u.22260\nγ : Type ?u.22263\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Sort u_1\nx : α\ns : ι → Opens α\n⊢ x ∈ ↑(iSup s) ↔ ∃ i, x ∈ s i",
"state_before": "ι✝ : Type ?u.22254\nα : Type u_2\nβ : Type ?u.22260\nγ : Type ?u.22263\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Sort u_1\nx : α\ns : ι → Opens α\n⊢ x ∈ iSup s ↔ ∃ i, x ∈ s i",
"tactic": "rw [← SetLike.mem_coe]"
},
{
"state_after": "no goals",
"state_before": "ι✝ : Type ?u.22254\nα : Type u_2\nβ : Type ?u.22260\nγ : Type ?u.22263\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Sort u_1\nx : α\ns : ι → Opens α\n⊢ x ∈ ↑(iSup s) ↔ ∃ i, x ∈ s i",
"tactic": "simp"
}
] |
[
248,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
Associates.factors_one
|
[
{
"state_after": "case h\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\n⊢ FactorSet.prod (factors 1) = FactorSet.prod 0",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\n⊢ factors 1 = 0",
"tactic": "apply eq_of_prod_eq_prod"
},
{
"state_after": "case h\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\n⊢ 1 = FactorSet.prod 0",
"state_before": "case h\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\n⊢ FactorSet.prod (factors 1) = FactorSet.prod 0",
"tactic": "rw [Associates.factors_prod]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\n⊢ 1 = FactorSet.prod 0",
"tactic": "exact Multiset.prod_zero"
}
] |
[
1792,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1789,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.span_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na b : α\n⊢ span 0 = ⊥",
"tactic": "rw [← Set.singleton_zero, span_singleton_eq_bot]"
}
] |
[
206,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Analysis/Convex/Quasiconvex.lean
|
Antitone.quasilinearOn
|
[] |
[
229,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean
|
Ordinal.CNF_zero
|
[] |
[
85,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.le_ofDual_iff
|
[] |
[
998,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
996,
1
] |
Mathlib/Algebra/Hom/Centroid.lean
|
CentroidHom.add_apply
|
[] |
[
330,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
1
] |
Mathlib/Data/Set/Image.lean
|
Disjoint.of_image
|
[] |
[
1598,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1596,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.TM1to1.trTape_mk'
|
[
{
"state_after": "no goals",
"state_before": "Γ : Type u_1\ninst✝² : Inhabited Γ\nΛ : Type ?u.363898\ninst✝¹ : Inhabited Λ\nσ : Type ?u.363904\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nL R : ListBlank Γ\n⊢ trTape enc0 (Tape.mk' L R) = trTape' enc0 L R",
"tactic": "simp only [trTape, Tape.mk'_left, Tape.mk'_right₀]"
}
] |
[
1751,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1750,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.cof_le_card
|
[
{
"state_after": "α : Type ?u.21105\nr : α → α → Prop\no : Ordinal\n⊢ sInf {a | ∃ ι f, lsub f = o ∧ (#ι) = a} ≤ card o",
"state_before": "α : Type ?u.21105\nr : α → α → Prop\no : Ordinal\n⊢ cof o ≤ card o",
"tactic": "rw [cof_eq_sInf_lsub]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.21105\nr : α → α → Prop\no : Ordinal\n⊢ sInf {a | ∃ ι f, lsub f = o ∧ (#ι) = a} ≤ card o",
"tactic": "exact csInf_le' card_mem_cof"
}
] |
[
290,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/Order/OmegaCompletePartialOrder.lean
|
OmegaCompletePartialOrder.ContinuousHom.forall_forall_merge'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nα' : Type ?u.68385\nβ : Type v\nβ' : Type ?u.68390\nγ : Type ?u.68393\nφ : Type ?u.68396\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β\ninst✝³ : OmegaCompletePartialOrder γ\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\nc₀ : Chain (α →𝒄 β)\nc₁ : Chain α\nz : β\n⊢ (∀ (j i : ℕ), ↑(↑c₀ i).toOrderHom (↑c₁ j) ≤ z) ↔ ∀ (i : ℕ), ↑(↑c₀ i).toOrderHom (↑c₁ i) ≤ z",
"tactic": "rw [forall_swap, forall_forall_merge]"
}
] |
[
784,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
782,
1
] |
Mathlib/LinearAlgebra/Determinant.lean
|
Matrix.det_comm
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.18410\ninst✝¹⁰ : CommRing R\nM✝ : Type ?u.18416\ninst✝⁹ : AddCommGroup M✝\ninst✝⁸ : Module R M✝\nM' : Type ?u.19001\ninst✝⁷ : AddCommGroup M'\ninst✝⁶ : Module R M'\nι : Type ?u.19543\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Fintype ι\ne : Basis ι R M✝\nA : Type u_2\ninst✝³ : CommRing A\nm : Type ?u.20027\nn : Type u_1\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nM N : Matrix n n A\n⊢ det (M ⬝ N) = det (N ⬝ M)",
"tactic": "rw [det_mul, det_mul, mul_comm]"
}
] |
[
83,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.Iso.cancel_iso_hom_left
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX✝ Y✝ Z✝ X Y Z : C\nf : X ≅ Y\ng g' : Y ⟶ Z\n⊢ f.hom ≫ g = f.hom ≫ g' ↔ g = g'",
"tactic": "simp only [cancel_epi]"
}
] |
[
538,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
536,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.smul_eval
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf x : σ → R\np : MvPolynomial σ R\ns : R\n⊢ ↑(eval x) (s • p) = s * ↑(eval x) p",
"tactic": "rw [smul_eq_C_mul, (eval x).map_mul, eval_C]"
}
] |
[
1153,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1152,
1
] |
Mathlib/Analysis/Convex/Integral.lean
|
ConvexOn.set_average_mem_epigraph
|
[
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.1095696\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\nhg : ConvexOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed s\nh0 : ↑↑μ t ≠ 0\nht : ↑↑μ t ≠ ⊤\nhfs : ∀ᵐ (x : α) ∂Measure.restrict μ t, f x ∈ s\nhfi : IntegrableOn f t\nhgi : IntegrableOn (g ∘ f) t\nthis : Fact (↑↑μ t < ⊤)\n⊢ (⨍ (x : α) in t, f x ∂μ, ⨍ (x : α) in t, g (f x) ∂μ) ∈ {p | p.fst ∈ s ∧ g p.fst ≤ p.snd}",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1095696\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\nhg : ConvexOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed s\nh0 : ↑↑μ t ≠ 0\nht : ↑↑μ t ≠ ⊤\nhfs : ∀ᵐ (x : α) ∂Measure.restrict μ t, f x ∈ s\nhfi : IntegrableOn f t\nhgi : IntegrableOn (g ∘ f) t\n⊢ (⨍ (x : α) in t, f x ∂μ, ⨍ (x : α) in t, g (f x) ∂μ) ∈ {p | p.fst ∈ s ∧ g p.fst ≤ p.snd}",
"tactic": "haveI : Fact (μ t < ∞) := ⟨ht.lt_top⟩"
},
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.1095696\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\nhg : ConvexOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed s\nh0 : ↑↑μ t ≠ 0\nht : ↑↑μ t ≠ ⊤\nhfs : ∀ᵐ (x : α) ∂Measure.restrict μ t, f x ∈ s\nhfi : IntegrableOn f t\nhgi : IntegrableOn (g ∘ f) t\nthis : Fact (↑↑μ t < ⊤)\n⊢ Measure.restrict μ t ≠ 0",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1095696\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\nhg : ConvexOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed s\nh0 : ↑↑μ t ≠ 0\nht : ↑↑μ t ≠ ⊤\nhfs : ∀ᵐ (x : α) ∂Measure.restrict μ t, f x ∈ s\nhfi : IntegrableOn f t\nhgi : IntegrableOn (g ∘ f) t\nthis : Fact (↑↑μ t < ⊤)\n⊢ (⨍ (x : α) in t, f x ∂μ, ⨍ (x : α) in t, g (f x) ∂μ) ∈ {p | p.fst ∈ s ∧ g p.fst ≤ p.snd}",
"tactic": "refine' hg.average_mem_epigraph hgc hsc _ hfs hfi hgi"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1095696\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\nhg : ConvexOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed s\nh0 : ↑↑μ t ≠ 0\nht : ↑↑μ t ≠ ⊤\nhfs : ∀ᵐ (x : α) ∂Measure.restrict μ t, f x ∈ s\nhfi : IntegrableOn f t\nhgi : IntegrableOn (g ∘ f) t\nthis : Fact (↑↑μ t < ⊤)\n⊢ Measure.restrict μ t ≠ 0",
"tactic": "rwa [Ne.def, restrict_eq_zero]"
}
] |
[
167,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.mapRange.linearEquiv_symm
|
[] |
[
877,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
875,
1
] |
Std/Logic.lean
|
not_not_not
|
[] |
[
24,
69
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
24,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
HasFDerivWithinAt.cpow
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : E → ℂ\nf' g' : E →L[ℂ] ℂ\nx : E\ns : Set E\nc : ℂ\nhf : HasFDerivWithinAt f f' s x\nhg : HasFDerivWithinAt g g' s x\nh0 : 0 < (f x).re ∨ (f x).im ≠ 0\n⊢ HasFDerivWithinAt (fun x => f x ^ g x) ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Complex.log (f x)) • g') s x",
"tactic": "convert\n (@Complex.hasFDerivAt_cpow ((fun x => (f x, g x)) x) h0).comp_hasFDerivWithinAt x (hf.prod hg)"
}
] |
[
108,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/RingTheory/ChainOfDivisors.lean
|
map_prime_of_factor_orderIso
|
[
{
"state_after": "M : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ Irreducible ↑(↑d { val := p, property := (_ : p ∣ m) })",
"state_before": "M : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ Prime ↑(↑d { val := p, property := (_ : p ∣ m) })",
"tactic": "rw [← irreducible_iff_prime]"
},
{
"state_after": "case refine'_1\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ ↑(↑d { val := p, property := (_ : p ∣ m) }) ≠ ⊥\n\ncase refine'_2\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\n⊢ b = ⊥",
"state_before": "M : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ Irreducible ↑(↑d { val := p, property := (_ : p ∣ m) })",
"tactic": "refine' (Associates.isAtom_iff <| ne_zero_of_dvd_ne_zero hn (d ⟨p, _⟩).prop).mp ⟨_, fun b hb => _⟩"
},
{
"state_after": "case refine'_1\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ ¬p = 1",
"state_before": "case refine'_1\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ ↑(↑d { val := p, property := (_ : p ∣ m) }) ≠ ⊥",
"tactic": "rw [Ne.def, ← Associates.isUnit_iff_eq_bot, Associates.isUnit_iff_eq_one,\n coe_factor_orderIso_map_eq_one_iff _ d]"
},
{
"state_after": "case refine'_1\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n ≠ 0\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nhp : 1 ∈ normalizedFactors m\n⊢ False",
"state_before": "case refine'_1\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ ¬p = 1",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n ≠ 0\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nhp : 1 ∈ normalizedFactors m\n⊢ False",
"tactic": "exact (prime_of_normalized_factor 1 hp).not_unit isUnit_one"
},
{
"state_after": "case refine'_2.intro\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhx : ↑d x = { val := b, property := (_ : b ≤ n) }\n⊢ b = ⊥",
"state_before": "case refine'_2\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\n⊢ b = ⊥",
"tactic": "obtain ⟨x, hx⟩ :=\n d.surjective ⟨b, le_trans (le_of_lt hb) (d ⟨p, dvd_of_mem_normalizedFactors hp⟩).prop⟩"
},
{
"state_after": "case refine'_2.intro\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhb : ↑(↑d x) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d x = { val := b, property := (_ : b ≤ n) }\n⊢ b = ⊥",
"state_before": "case refine'_2.intro\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhx : ↑d x = { val := b, property := (_ : b ≤ n) }\n⊢ b = ⊥",
"tactic": "rw [← Subtype.coe_mk b _, ← hx] at hb"
},
{
"state_after": "case refine'_2.intro\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhb : ↑(↑d x) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d x = { val := b, property := (_ : b ≤ n) }\nthis : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\n⊢ b = ⊥",
"state_before": "case refine'_2.intro\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhb : ↑(↑d x) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d x = { val := b, property := (_ : b ≤ n) }\n⊢ b = ⊥",
"tactic": "letI : OrderBot { l : Associates M // l ≤ m } := Subtype.orderBot bot_le"
},
{
"state_after": "case refine'_2.intro\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhb : ↑(↑d x) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d x = { val := b, property := (_ : b ≤ n) }\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\n⊢ b = ⊥",
"state_before": "case refine'_2.intro\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhb : ↑(↑d x) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d x = { val := b, property := (_ : b ≤ n) }\nthis : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\n⊢ b = ⊥",
"tactic": "letI : OrderBot { l : Associates N // l ≤ n } := Subtype.orderBot bot_le"
},
{
"state_after": "case refine'_2.intro\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhb : ↑(↑d x) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d x = { val := b, property := (_ : b ≤ n) }\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\n⊢ x = ⊥",
"state_before": "case refine'_2.intro\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhb : ↑(↑d x) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d x = { val := b, property := (_ : b ≤ n) }\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\n⊢ b = ⊥",
"tactic": "suffices x = ⊥ by\n rw [this, OrderIso.map_bot d] at hx\n refine' (Subtype.mk_eq_bot_iff _ _).mp hx.symm\n simp"
},
{
"state_after": "case refine'_2.intro.mk\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\na : Associates M\nha : a ∈ Set.Iic m\nhb : ↑(↑d { val := a, property := ha }) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d { val := a, property := ha } = { val := b, property := (_ : b ≤ n) }\n⊢ { val := a, property := ha } = ⊥",
"state_before": "case refine'_2.intro\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhb : ↑(↑d x) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d x = { val := b, property := (_ : b ≤ n) }\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\n⊢ x = ⊥",
"tactic": "obtain ⟨a, ha⟩ := x"
},
{
"state_after": "case refine'_2.intro.mk\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\na : Associates M\nha : a ∈ Set.Iic m\nhb : ↑(↑d { val := a, property := ha }) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d { val := a, property := ha } = { val := b, property := (_ : b ≤ n) }\n⊢ a = ⊥\n\ncase refine'_2.intro.mk.hbot\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\na : Associates M\nha : a ∈ Set.Iic m\nhb : ↑(↑d { val := a, property := ha }) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d { val := a, property := ha } = { val := b, property := (_ : b ≤ n) }\n⊢ ⊥ ∈ Set.Iic m",
"state_before": "case refine'_2.intro.mk\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\na : Associates M\nha : a ∈ Set.Iic m\nhb : ↑(↑d { val := a, property := ha }) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d { val := a, property := ha } = { val := b, property := (_ : b ≤ n) }\n⊢ { val := a, property := ha } = ⊥",
"tactic": "rw [Subtype.mk_eq_bot_iff]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.mk.hbot\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\na : Associates M\nha : a ∈ Set.Iic m\nhb : ↑(↑d { val := a, property := ha }) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d { val := a, property := ha } = { val := b, property := (_ : b ≤ n) }\n⊢ ⊥ ∈ Set.Iic m",
"tactic": "simp"
},
{
"state_after": "M : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhb : ↑(↑d x) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ⊥ = { val := b, property := (_ : b ≤ n) }\nthis✝¹ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis✝ : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\nthis : x = ⊥\n⊢ b = ⊥",
"state_before": "M : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhb : ↑(↑d x) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d x = { val := b, property := (_ : b ≤ n) }\nthis✝¹ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis✝ : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\nthis : x = ⊥\n⊢ b = ⊥",
"tactic": "rw [this, OrderIso.map_bot d] at hx"
},
{
"state_after": "M : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhb : ↑(↑d x) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ⊥ = { val := b, property := (_ : b ≤ n) }\nthis✝¹ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis✝ : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\nthis : x = ⊥\n⊢ ⊥ ∈ Set.Iic n",
"state_before": "M : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhb : ↑(↑d x) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ⊥ = { val := b, property := (_ : b ≤ n) }\nthis✝¹ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis✝ : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\nthis : x = ⊥\n⊢ b = ⊥",
"tactic": "refine' (Subtype.mk_eq_bot_iff _ _).mp hx.symm"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nx : ↑(Set.Iic m)\nhb : ↑(↑d x) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ⊥ = { val := b, property := (_ : b ≤ n) }\nthis✝¹ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis✝ : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\nthis : x = ⊥\n⊢ ⊥ ∈ Set.Iic n",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.mk\nM : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝³ : CancelCommMonoidWithZero N\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nb : Associates N\nhb✝ : b < ↑(↑d { val := p, property := (_ : p ∣ m) })\nthis✝ : OrderBot { l // l ≤ m } := Subtype.orderBot (_ : ⊥ ≤ m)\nthis : OrderBot { l // l ≤ n } := Subtype.orderBot (_ : ⊥ ≤ n)\na : Associates M\nha : a ∈ Set.Iic m\nhb : ↑(↑d { val := a, property := ha }) < ↑(↑d { val := p, property := (_ : p ∣ m) })\nhx : ↑d { val := a, property := ha } = { val := b, property := (_ : b ≤ n) }\n⊢ a = ⊥",
"tactic": "exact\n ((Associates.isAtom_iff <| Prime.ne_zero <| prime_of_normalized_factor p hp).mpr <|\n irreducible_of_normalized_factor p hp).right\n a (Subtype.mk_lt_mk.mp <| d.lt_iff_lt.mp hb)"
}
] |
[
311,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
LinearMap.rTensor_add
|
[] |
[
1043,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1042,
1
] |
Std/Data/List/Lemmas.lean
|
List.head_eq_of_cons_eq
|
[] |
[
24,
81
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
24,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.antitoneOn_iff_antitone
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α → β\n⊢ AntitoneOn f s ↔ Antitone fun a => f ↑a",
"tactic": "simp [Antitone, AntitoneOn]"
}
] |
[
2658,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2656,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.mem_dlookup_kunion
|
[
{
"state_after": "case nil\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nl₂ : List (Sigma β)\n⊢ b ∈ dlookup a (kunion [] l₂) ↔ b ∈ dlookup a [] ∨ ¬a ∈ keys [] ∧ b ∈ dlookup a l₂\n\ncase cons\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nhead✝ : Sigma β\ntail✝ : List (Sigma β)\ntail_ih✝ :\n ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\n⊢ b ∈ dlookup a (kunion (head✝ :: tail✝) l₂) ↔\n b ∈ dlookup a (head✝ :: tail✝) ∨ ¬a ∈ keys (head✝ :: tail✝) ∧ b ∈ dlookup a l₂",
"state_before": "α : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nl₁ l₂ : List (Sigma β)\n⊢ b ∈ dlookup a (kunion l₁ l₂) ↔ b ∈ dlookup a l₁ ∨ ¬a ∈ keys l₁ ∧ b ∈ dlookup a l₂",
"tactic": "induction l₁ generalizing l₂"
},
{
"state_after": "case cons\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nhead✝ : Sigma β\ntail✝ : List (Sigma β)\ntail_ih✝ :\n ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\n⊢ b ∈ dlookup a (kunion (head✝ :: tail✝) l₂) ↔\n b ∈ dlookup a (head✝ :: tail✝) ∨ ¬a ∈ keys (head✝ :: tail✝) ∧ b ∈ dlookup a l₂",
"state_before": "case nil\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nl₂ : List (Sigma β)\n⊢ b ∈ dlookup a (kunion [] l₂) ↔ b ∈ dlookup a [] ∨ ¬a ∈ keys [] ∧ b ∈ dlookup a l₂\n\ncase cons\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nhead✝ : Sigma β\ntail✝ : List (Sigma β)\ntail_ih✝ :\n ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\n⊢ b ∈ dlookup a (kunion (head✝ :: tail✝) l₂) ↔\n b ∈ dlookup a (head✝ :: tail✝) ∨ ¬a ∈ keys (head✝ :: tail✝) ∧ b ∈ dlookup a l₂",
"tactic": "case nil => simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nl₂ : List (Sigma β)\n⊢ b ∈ dlookup a (kunion [] l₂) ↔ b ∈ dlookup a [] ∨ ¬a ∈ keys [] ∧ b ∈ dlookup a l₂",
"tactic": "simp"
},
{
"state_after": "case mk\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\ntail✝ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\na' : α\nsnd✝ : β a'\n⊢ b ∈ dlookup a (kunion ({ fst := a', snd := snd✝ } :: tail✝) l₂) ↔\n b ∈ dlookup a ({ fst := a', snd := snd✝ } :: tail✝) ∨\n ¬a ∈ keys ({ fst := a', snd := snd✝ } :: tail✝) ∧ b ∈ dlookup a l₂",
"state_before": "α : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\ns : Sigma β\ntail✝ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\n⊢ b ∈ dlookup a (kunion (s :: tail✝) l₂) ↔ b ∈ dlookup a (s :: tail✝) ∨ ¬a ∈ keys (s :: tail✝) ∧ b ∈ dlookup a l₂",
"tactic": "cases' s with a'"
},
{
"state_after": "case pos\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\ntail✝ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\na' : α\nsnd✝ : β a'\nh₁ : a = a'\n⊢ b ∈ dlookup a (kunion ({ fst := a', snd := snd✝ } :: tail✝) l₂) ↔\n b ∈ dlookup a ({ fst := a', snd := snd✝ } :: tail✝) ∨\n ¬a ∈ keys ({ fst := a', snd := snd✝ } :: tail✝) ∧ b ∈ dlookup a l₂\n\ncase neg\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\ntail✝ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\na' : α\nsnd✝ : β a'\nh₁ : ¬a = a'\n⊢ b ∈ dlookup a (kunion ({ fst := a', snd := snd✝ } :: tail✝) l₂) ↔\n b ∈ dlookup a ({ fst := a', snd := snd✝ } :: tail✝) ∨\n ¬a ∈ keys ({ fst := a', snd := snd✝ } :: tail✝) ∧ b ∈ dlookup a l₂",
"state_before": "case mk\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\ntail✝ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\na' : α\nsnd✝ : β a'\n⊢ b ∈ dlookup a (kunion ({ fst := a', snd := snd✝ } :: tail✝) l₂) ↔\n b ∈ dlookup a ({ fst := a', snd := snd✝ } :: tail✝) ∨\n ¬a ∈ keys ({ fst := a', snd := snd✝ } :: tail✝) ∧ b ∈ dlookup a l₂",
"tactic": "by_cases h₁ : a = a'"
},
{
"state_after": "case pos\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\ntail✝ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\nsnd✝ : β a\n⊢ b ∈ dlookup a (kunion ({ fst := a, snd := snd✝ } :: tail✝) l₂) ↔\n b ∈ dlookup a ({ fst := a, snd := snd✝ } :: tail✝) ∨\n ¬a ∈ keys ({ fst := a, snd := snd✝ } :: tail✝) ∧ b ∈ dlookup a l₂",
"state_before": "case pos\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\ntail✝ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\na' : α\nsnd✝ : β a'\nh₁ : a = a'\n⊢ b ∈ dlookup a (kunion ({ fst := a', snd := snd✝ } :: tail✝) l₂) ↔\n b ∈ dlookup a ({ fst := a', snd := snd✝ } :: tail✝) ∨\n ¬a ∈ keys ({ fst := a', snd := snd✝ } :: tail✝) ∧ b ∈ dlookup a l₂",
"tactic": "subst h₁"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\ntail✝ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\nsnd✝ : β a\n⊢ b ∈ dlookup a (kunion ({ fst := a, snd := snd✝ } :: tail✝) l₂) ↔\n b ∈ dlookup a ({ fst := a, snd := snd✝ } :: tail✝) ∨\n ¬a ∈ keys ({ fst := a, snd := snd✝ } :: tail✝) ∧ b ∈ dlookup a l₂",
"tactic": "simp"
},
{
"state_after": "case neg\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\ntail✝ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\na' : α\nsnd✝ : β a'\nh₁ : ¬a = a'\nh₂ : b ∈ dlookup a (kunion tail✝ (kerase a' l₂)) ↔\n b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a (kerase a' l₂) :=\n ih\n⊢ b ∈ dlookup a (kunion ({ fst := a', snd := snd✝ } :: tail✝) l₂) ↔\n b ∈ dlookup a ({ fst := a', snd := snd✝ } :: tail✝) ∨\n ¬a ∈ keys ({ fst := a', snd := snd✝ } :: tail✝) ∧ b ∈ dlookup a l₂",
"state_before": "case neg\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\ntail✝ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\na' : α\nsnd✝ : β a'\nh₁ : ¬a = a'\n⊢ b ∈ dlookup a (kunion ({ fst := a', snd := snd✝ } :: tail✝) l₂) ↔\n b ∈ dlookup a ({ fst := a', snd := snd✝ } :: tail✝) ∨\n ¬a ∈ keys ({ fst := a', snd := snd✝ } :: tail✝) ∧ b ∈ dlookup a l₂",
"tactic": "let h₂ := @ih (kerase a' l₂)"
},
{
"state_after": "case neg\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\ntail✝ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\na' : α\nsnd✝ : β a'\nh₁ : ¬a = a'\nh₂ :\n dlookup a (kunion tail✝ (kerase a' l₂)) = some b ↔ dlookup a tail✝ = some b ∨ ¬a ∈ keys tail✝ ∧ dlookup a l₂ = some b\n⊢ b ∈ dlookup a (kunion ({ fst := a', snd := snd✝ } :: tail✝) l₂) ↔\n b ∈ dlookup a ({ fst := a', snd := snd✝ } :: tail✝) ∨\n ¬a ∈ keys ({ fst := a', snd := snd✝ } :: tail✝) ∧ b ∈ dlookup a l₂",
"state_before": "case neg\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\ntail✝ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\na' : α\nsnd✝ : β a'\nh₁ : ¬a = a'\nh₂ : b ∈ dlookup a (kunion tail✝ (kerase a' l₂)) ↔\n b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a (kerase a' l₂) :=\n ih\n⊢ b ∈ dlookup a (kunion ({ fst := a', snd := snd✝ } :: tail✝) l₂) ↔\n b ∈ dlookup a ({ fst := a', snd := snd✝ } :: tail✝) ∨\n ¬a ∈ keys ({ fst := a', snd := snd✝ } :: tail✝) ∧ b ∈ dlookup a l₂",
"tactic": "simp [h₁] at h₂"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\ntail✝ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, b ∈ dlookup a (kunion tail✝ l₂) ↔ b ∈ dlookup a tail✝ ∨ ¬a ∈ keys tail✝ ∧ b ∈ dlookup a l₂\nl₂ : List (Sigma β)\na' : α\nsnd✝ : β a'\nh₁ : ¬a = a'\nh₂ :\n dlookup a (kunion tail✝ (kerase a' l₂)) = some b ↔ dlookup a tail✝ = some b ∨ ¬a ∈ keys tail✝ ∧ dlookup a l₂ = some b\n⊢ b ∈ dlookup a (kunion ({ fst := a', snd := snd✝ } :: tail✝) l₂) ↔\n b ∈ dlookup a ({ fst := a', snd := snd✝ } :: tail✝) ∨\n ¬a ∈ keys ({ fst := a', snd := snd✝ } :: tail✝) ∧ b ∈ dlookup a l₂",
"tactic": "simp [h₁, h₂]"
}
] |
[
778,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
767,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
|
UniformFun.postcomp_uniformInducing
|
[
{
"state_after": "case comap_uniformity\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.40608\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : UniformInducing f\n⊢ comap (fun x => ((↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun) x.fst, (↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun) x.snd))\n (𝓤 (α →ᵤ β)) =\n 𝓤 (α →ᵤ γ)",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.40608\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : UniformInducing f\n⊢ UniformInducing (↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun)",
"tactic": "constructor"
},
{
"state_after": "case comap_uniformity\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.40608\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : comap (Prod.map f f) (𝓤 β) = 𝓤 γ\n⊢ comap (fun x => ((↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun) x.fst, (↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun) x.snd))\n (𝓤 (α →ᵤ β)) =\n 𝓤 (α →ᵤ γ)",
"state_before": "case comap_uniformity\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.40608\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : UniformInducing f\n⊢ comap (fun x => ((↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun) x.fst, (↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun) x.snd))\n (𝓤 (α →ᵤ β)) =\n 𝓤 (α →ᵤ γ)",
"tactic": "replace hf : (𝓤 β).comap (Prod.map f f) = _ := hf.comap_uniformity"
},
{
"state_after": "case comap_uniformity\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.40608\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : 𝓤 γ = 𝓤 γ\n⊢ 𝓤 (α →ᵤ γ) = 𝓤 (α →ᵤ γ)",
"state_before": "case comap_uniformity\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.40608\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : comap (Prod.map f f) (𝓤 β) = 𝓤 γ\n⊢ comap (Prod.map (↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun) (↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun)) (𝓤 (α →ᵤ β)) = 𝓤 (α →ᵤ γ)",
"tactic": "rw [← uniformity_comap] at hf⊢"
},
{
"state_after": "case comap_uniformity.h.e_2.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.40608\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : 𝓤 γ = 𝓤 γ\n⊢ UniformSpace.comap (↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun) (uniformSpace α β) = uniformSpace α γ",
"state_before": "case comap_uniformity\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.40608\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : 𝓤 γ = 𝓤 γ\n⊢ 𝓤 (α →ᵤ γ) = 𝓤 (α →ᵤ γ)",
"tactic": "congr"
},
{
"state_after": "case comap_uniformity.h.e_2.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.40608\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : 𝓤 γ = 𝓤 γ\n⊢ UniformSpace.comap (↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun) (uniformSpace α β) =\n UniformSpace.comap (fun x => f ∘ x) (uniformSpace α β)",
"state_before": "case comap_uniformity.h.e_2.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.40608\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : 𝓤 γ = 𝓤 γ\n⊢ UniformSpace.comap (↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun) (uniformSpace α β) = uniformSpace α γ",
"tactic": "rw [← uniformSpace_eq hf, UniformFun.comap_eq]"
},
{
"state_after": "no goals",
"state_before": "case comap_uniformity.h.e_2.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.40608\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : γ → β\nhf : 𝓤 γ = 𝓤 γ\n⊢ UniformSpace.comap (↑ofFun ∘ (fun x => f ∘ x) ∘ ↑toFun) (uniformSpace α β) =\n UniformSpace.comap (fun x => f ∘ x) (uniformSpace α β)",
"tactic": "rfl"
}
] |
[
440,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
431,
11
] |
Mathlib/Topology/Sets/Order.lean
|
ClopenUpperSet.coe_bot
|
[] |
[
106,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.infinitePos_add_not_infiniteNeg
|
[
{
"state_after": "x y : ℝ*\nhip : InfinitePos x\nhnin : ¬InfiniteNeg y\nr : ℝ\n⊢ ↑r < x + y",
"state_before": "x y : ℝ*\n⊢ InfinitePos x → ¬InfiniteNeg y → InfinitePos (x + y)",
"tactic": "intro hip hnin r"
},
{
"state_after": "case intro\nx y : ℝ*\nhip : InfinitePos x\nhnin : ¬InfiniteNeg y\nr r₂ : ℝ\nhr₂ : ¬y < ↑r₂\n⊢ ↑r < x + y",
"state_before": "x y : ℝ*\nhip : InfinitePos x\nhnin : ¬InfiniteNeg y\nr : ℝ\n⊢ ↑r < x + y",
"tactic": "cases' not_forall.mp hnin with r₂ hr₂"
},
{
"state_after": "case h.e'_3\nx y : ℝ*\nhip : InfinitePos x\nhnin : ¬InfiniteNeg y\nr r₂ : ℝ\nhr₂ : ¬y < ↑r₂\n⊢ ↑r = ↑(r + -r₂) + ↑r₂",
"state_before": "case intro\nx y : ℝ*\nhip : InfinitePos x\nhnin : ¬InfiniteNeg y\nr r₂ : ℝ\nhr₂ : ¬y < ↑r₂\n⊢ ↑r < x + y",
"tactic": "convert add_lt_add_of_lt_of_le (hip (r + -r₂)) (not_lt.mp hr₂) using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nx y : ℝ*\nhip : InfinitePos x\nhnin : ¬InfiniteNeg y\nr r₂ : ℝ\nhr₂ : ¬y < ↑r₂\n⊢ ↑r = ↑(r + -r₂) + ↑r₂",
"tactic": "simp"
}
] |
[
543,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
538,
1
] |
Mathlib/LinearAlgebra/Dfinsupp.lean
|
Dfinsupp.mapRange.linearMap_comp
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_5\nR : Type u_1\nS : Type ?u.148536\nM : ι → Type ?u.148541\nN : Type ?u.148544\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_4\nβ₁ : ι → Type u_2\nβ₂ : ι → Type u_3\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)\nf : (i : ι) → β₁ i →ₗ[R] β₂ i\nf₂ : (i : ι) → β i →ₗ[R] β₁ i\n⊢ ∀ (i : ι), ((fun i x => ↑(f i) x) i ∘ (fun i x => ↑(f₂ i) x) i) 0 = 0",
"tactic": "simp"
}
] |
[
218,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Data/Holor.lean
|
Holor.cprank_upper_bound
|
[] |
[
413,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
div_lt_comm
|
[] |
[
1002,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1001,
1
] |
Mathlib/GroupTheory/Perm/Basic.lean
|
Equiv.Perm.subtypePerm_mul
|
[] |
[
382,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Data/Finset/NatAntidiagonal.lean
|
Finset.Nat.mem_antidiagonal
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\nx : ℕ × ℕ\n⊢ x ∈ antidiagonal n ↔ x.fst + x.snd = n",
"tactic": "rw [antidiagonal, mem_def, Multiset.Nat.mem_antidiagonal]"
}
] |
[
40,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Data/Set/Equitable.lean
|
Finset.EquitableOn.le_add_one
|
[] |
[
124,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/LinearAlgebra/Dfinsupp.lean
|
Dfinsupp.lhom_ext'
|
[] |
[
76,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.coeff_X_pow
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nm n : ℕ\n⊢ ↑(coeff R m) (X ^ n) = if m = n then 1 else 0",
"tactic": "rw [X_pow_eq, coeff_monomial]"
}
] |
[
1457,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1456,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.cast_commute
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.2807\nι : Type ?u.2810\nα : Type u_1\nβ : Type ?u.2816\ninst✝ : DivisionRing α\nr : ℚ\na : α\n⊢ Commute (↑r) a",
"tactic": "simpa only [cast_def] using (r.1.cast_commute a).div_left (r.2.cast_commute a)"
}
] |
[
70,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.foldr_reverse
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nl : List α\nf : α → β → β\nb : β\n⊢ foldl (fun y x => f x y) b (reverse (reverse l)) = foldl (fun x y => f y x) b l",
"tactic": "simp"
}
] |
[
190,
43
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
188,
9
] |
Mathlib/Data/Set/Basic.lean
|
Set.diff_inter
|
[] |
[
1955,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1954,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.adjoin_one
|
[] |
[
692,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
691,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.restrict_self
|
[] |
[
522,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
Mathlib/Combinatorics/Quiver/Symmetric.lean
|
Quiver.Path.reverse_comp
|
[
{
"state_after": "case nil\nU : Type ?u.5751\nV : Type u_1\nW : Type ?u.5757\ninst✝³ : Quiver U\ninst✝² : Quiver V\ninst✝¹ : Quiver W\ninst✝ : HasReverse V\na b c : V\np : Path a b\n⊢ reverse (comp p nil) = comp (reverse nil) (reverse p)\n\ncase cons\nU : Type ?u.5751\nV : Type u_1\nW : Type ?u.5757\ninst✝³ : Quiver U\ninst✝² : Quiver V\ninst✝¹ : Quiver W\ninst✝ : HasReverse V\na b c : V\np : Path a b\nb✝ c✝ : V\na✝¹ : Path b b✝\na✝ : b✝ ⟶ c✝\nh : reverse (comp p a✝¹) = comp (reverse a✝¹) (reverse p)\n⊢ reverse (comp p (cons a✝¹ a✝)) = comp (reverse (cons a✝¹ a✝)) (reverse p)",
"state_before": "U : Type ?u.5751\nV : Type u_1\nW : Type ?u.5757\ninst✝³ : Quiver U\ninst✝² : Quiver V\ninst✝¹ : Quiver W\ninst✝ : HasReverse V\na b c : V\np : Path a b\nq : Path b c\n⊢ reverse (comp p q) = comp (reverse q) (reverse p)",
"tactic": "induction' q with _ _ _ _ h"
},
{
"state_after": "no goals",
"state_before": "case nil\nU : Type ?u.5751\nV : Type u_1\nW : Type ?u.5757\ninst✝³ : Quiver U\ninst✝² : Quiver V\ninst✝¹ : Quiver W\ninst✝ : HasReverse V\na b c : V\np : Path a b\n⊢ reverse (comp p nil) = comp (reverse nil) (reverse p)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nU : Type ?u.5751\nV : Type u_1\nW : Type ?u.5757\ninst✝³ : Quiver U\ninst✝² : Quiver V\ninst✝¹ : Quiver W\ninst✝ : HasReverse V\na b c : V\np : Path a b\nb✝ c✝ : V\na✝¹ : Path b b✝\na✝ : b✝ ⟶ c✝\nh : reverse (comp p a✝¹) = comp (reverse a✝¹) (reverse p)\n⊢ reverse (comp p (cons a✝¹ a✝)) = comp (reverse (cons a✝¹ a✝)) (reverse p)",
"tactic": "simp [h]"
}
] |
[
159,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_const_mul_atTop_iff_neg
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.239844\nι' : Type ?u.239847\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.239856\ninst✝¹ : LinearOrderedField α\nl : Filter β\nf : β → α\nr : α\ninst✝ : NeBot l\nh : Tendsto f l atBot\n⊢ Tendsto (fun x => r * f x) l atTop ↔ r < 0",
"tactic": "simp [tendsto_const_mul_atTop_iff, h, h.not_tendsto disjoint_atBot_atTop]"
}
] |
[
1152,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1150,
1
] |
Mathlib/CategoryTheory/Limits/Types.lean
|
CategoryTheory.Limits.Types.Colimit.ι_desc_apply'
|
[] |
[
334,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
Pi.himp_def
|
[] |
[
162,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Analysis/Quaternion.lean
|
Quaternion.coeComplex_imK
|
[] |
[
132,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.mem_range_iff_mem_finset_range_of_mod_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.102484\nγ : Type ?u.102487\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns : Finset α\nt : Finset β\na✝ : α\nb c : β\ninst✝ : DecidableEq α\nf : ℤ → α\na : α\nn : ℕ\nhn : 0 < n\nh : ∀ (i : ℤ), f (i % ↑n) = f i\nthis : (∃ i, f (i % ↑n) = a) ↔ ∃ i, i < n ∧ f ↑i = a\n⊢ a ∈ Set.range f ↔ a ∈ image (fun i => f ↑i) (range n)",
"tactic": "simpa [h]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.102484\nγ : Type ?u.102487\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns : Finset α\nt : Finset β\na✝ : α\nb c : β\ninst✝ : DecidableEq α\nf : ℤ → α\na : α\nn : ℕ\nhn : 0 < n\nh : ∀ (i : ℤ), f (i % ↑n) = f i\nhn' : 0 < ↑n\nx✝ : ∃ i, f (i % ↑n) = a\ni : ℤ\nhi : f (i % ↑n) = a\nthis : 0 ≤ i % ↑n\n⊢ i % ↑n < ↑n ∧ f (i % ↑n) = a",
"state_before": "α : Type u_1\nβ : Type ?u.102484\nγ : Type ?u.102487\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns : Finset α\nt : Finset β\na✝ : α\nb c : β\ninst✝ : DecidableEq α\nf : ℤ → α\na : α\nn : ℕ\nhn : 0 < n\nh : ∀ (i : ℤ), f (i % ↑n) = f i\nhn' : 0 < ↑n\nx✝ : ∃ i, f (i % ↑n) = a\ni : ℤ\nhi : f (i % ↑n) = a\nthis : 0 ≤ i % ↑n\n⊢ Int.toNat (i % ↑n) < n ∧ f ↑(Int.toNat (i % ↑n)) = a",
"tactic": "rw [← Int.ofNat_lt, Int.toNat_of_nonneg this]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.102484\nγ : Type ?u.102487\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns : Finset α\nt : Finset β\na✝ : α\nb c : β\ninst✝ : DecidableEq α\nf : ℤ → α\na : α\nn : ℕ\nhn : 0 < n\nh : ∀ (i : ℤ), f (i % ↑n) = f i\nhn' : 0 < ↑n\nx✝ : ∃ i, f (i % ↑n) = a\ni : ℤ\nhi : f (i % ↑n) = a\nthis : 0 ≤ i % ↑n\n⊢ i % ↑n < ↑n ∧ f (i % ↑n) = a",
"tactic": "exact ⟨Int.emod_lt_of_pos i hn', hi⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.102484\nγ : Type ?u.102487\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns : Finset α\nt : Finset β\na✝ : α\nb c : β\ninst✝ : DecidableEq α\nf : ℤ → α\na : α\nn : ℕ\nhn : 0 < n\nh : ∀ (i : ℤ), f (i % ↑n) = f i\nhn' : 0 < ↑n\nx✝ : ∃ i, i < n ∧ f ↑i = a\ni : ℕ\nhi : i < n\nha : f ↑i = a\n⊢ f (↑i % ↑n) = a",
"tactic": "rw [Int.emod_eq_of_lt (Int.ofNat_zero_le _) (Int.ofNat_lt_ofNat_of_lt hi), ha]"
}
] |
[
576,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
565,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean
|
PiLp.edist_eq_of_L2
|
[
{
"state_after": "no goals",
"state_before": "p : ℝ≥0∞\n𝕜 : Type ?u.293461\n𝕜' : Type ?u.293464\nι : Type u_2\nα : ι → Type ?u.293472\nβ✝ : ι → Type ?u.293477\ninst✝² : Fintype ι\ninst✝¹ : Fact (1 ≤ p)\nβ : ι → Type u_1\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx y : PiLp 2 β\n⊢ edist x y = (∑ i : ι, edist (x i) (y i) ^ 2) ^ (1 / 2)",
"tactic": "simp [PiLp.edist_eq_sum]"
}
] |
[
618,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
617,
1
] |
Mathlib/Algebra/Module/LocalizedModule.lean
|
LocalizedModule.lift_mk
|
[] |
[
657,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
654,
1
] |
Mathlib/CategoryTheory/Limits/Lattice.lean
|
CategoryTheory.Limits.CompleteLattice.pullback_eq_inf
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nJ : Type w\ninst✝³ : SmallCategory J\ninst✝² : FinCategory J\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderTop α\nx y z : α\nf : x ⟶ z\ng : y ⟶ z\n⊢ limit (cospan f g) = Finset.inf Finset.univ (cospan f g).toPrefunctor.obj",
"tactic": "rw [finite_limit_eq_finset_univ_inf]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nJ : Type w\ninst✝³ : SmallCategory J\ninst✝² : FinCategory J\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderTop α\nx y z : α\nf : x ⟶ z\ng : y ⟶ z\n⊢ z ⊓ (x ⊓ (y ⊓ ⊤)) = z ⊓ (x ⊓ y)",
"tactic": "rw [inf_top_eq]"
}
] |
[
165,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.count_apply_finite
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.373940\nγ : Type ?u.373943\nδ : Type ?u.373946\nι : Type ?u.373949\nR : Type ?u.373952\nR' : Type ?u.373955\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\ns : Set α\nhs : Set.Finite s\n⊢ ↑↑count s = ↑(Finset.card (Finite.toFinset hs))",
"tactic": "rw [← count_apply_finset, Finite.coe_toFinset]"
}
] |
[
2226,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2225,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.sUnion_insert
|
[] |
[
1140,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1139,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieSubmodule.coe_injective
|
[] |
[
357,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
PosNum.cast_le
|
[
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedSemiring α\nm n : PosNum\n⊢ ¬↑n < ↑m ↔ m ≤ n",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedSemiring α\nm n : PosNum\n⊢ ↑m ≤ ↑n ↔ m ≤ n",
"tactic": "rw [← not_lt]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedSemiring α\nm n : PosNum\n⊢ ¬↑n < ↑m ↔ m ≤ n",
"tactic": "exact not_congr cast_lt"
}
] |
[
707,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
706,
1
] |
Mathlib/GroupTheory/GroupAction/Group.lean
|
smul_eq_zero_iff_eq'
|
[] |
[
288,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/Order/Hom/Basic.lean
|
disjoint_map_orderIso_iff
|
[] |
[
1226,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1223,
1
] |
Mathlib/Dynamics/Flow.lean
|
isInvariant_iff_image
|
[
{
"state_after": "no goals",
"state_before": "τ : Type u_1\nα : Type u_2\nϕ : τ → α → α\ns : Set α\n⊢ IsInvariant ϕ s ↔ ∀ (t : τ), ϕ t '' s ⊆ s",
"tactic": "simp_rw [IsInvariant, mapsTo']"
}
] |
[
55,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Data/List/Basic.lean
|
List.get_pmap
|
[
{
"state_after": "case nil\nι : Type ?u.337556\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nh✝ : ∀ (a : α), a ∈ l → p a\nn✝ : ℕ\nhn✝ : n✝ < length (pmap f l h✝)\nh : ∀ (a : α), a ∈ [] → p a\nn : ℕ\nhn : n < length (pmap f [] h)\n⊢ get (pmap f [] h) { val := n, isLt := hn } =\n f (get [] { val := n, isLt := (_ : n < length []) }) (_ : p (get [] { val := n, isLt := (_ : n < length []) }))\n\ncase cons\nι : Type ?u.337556\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nh✝ : ∀ (a : α), a ∈ l → p a\nn✝ : ℕ\nhn✝ : n✝ < length (pmap f l h✝)\nhd : α\ntl : List α\nhl :\n ∀ (h : ∀ (a : α), a ∈ tl → p a) {n : ℕ} (hn : n < length (pmap f tl h)),\n get (pmap f tl h) { val := n, isLt := hn } =\n f (get tl { val := n, isLt := (_ : n < length tl) }) (_ : p (get tl { val := n, isLt := (_ : n < length tl) }))\nh : ∀ (a : α), a ∈ hd :: tl → p a\nn : ℕ\nhn : n < length (pmap f (hd :: tl) h)\n⊢ get (pmap f (hd :: tl) h) { val := n, isLt := hn } =\n f (get (hd :: tl) { val := n, isLt := (_ : n < length (hd :: tl)) })\n (_ : p (get (hd :: tl) { val := n, isLt := (_ : n < length (hd :: tl)) }))",
"state_before": "ι : Type ?u.337556\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nh : ∀ (a : α), a ∈ l → p a\nn : ℕ\nhn : n < length (pmap f l h)\n⊢ get (pmap f l h) { val := n, isLt := hn } =\n f (get l { val := n, isLt := (_ : n < length l) }) (_ : p (get l { val := n, isLt := (_ : n < length l) }))",
"tactic": "induction' l with hd tl hl generalizing n"
},
{
"state_after": "case nil\nι : Type ?u.337556\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nh✝ : ∀ (a : α), a ∈ l → p a\nn✝ : ℕ\nhn✝ : n✝ < length (pmap f l h✝)\nh : ∀ (a : α), a ∈ [] → p a\nn : ℕ\nhn : n < 0\n⊢ get (pmap f [] h) { val := n, isLt := hn } =\n f (get [] { val := n, isLt := (_ : n < length []) }) (_ : p (get [] { val := n, isLt := (_ : n < length []) }))",
"state_before": "case nil\nι : Type ?u.337556\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nh✝ : ∀ (a : α), a ∈ l → p a\nn✝ : ℕ\nhn✝ : n✝ < length (pmap f l h✝)\nh : ∀ (a : α), a ∈ [] → p a\nn : ℕ\nhn : n < length (pmap f [] h)\n⊢ get (pmap f [] h) { val := n, isLt := hn } =\n f (get [] { val := n, isLt := (_ : n < length []) }) (_ : p (get [] { val := n, isLt := (_ : n < length []) }))",
"tactic": "simp only [length, pmap] at hn"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.337556\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nh✝ : ∀ (a : α), a ∈ l → p a\nn✝ : ℕ\nhn✝ : n✝ < length (pmap f l h✝)\nh : ∀ (a : α), a ∈ [] → p a\nn : ℕ\nhn : n < 0\n⊢ get (pmap f [] h) { val := n, isLt := hn } =\n f (get [] { val := n, isLt := (_ : n < length []) }) (_ : p (get [] { val := n, isLt := (_ : n < length []) }))",
"tactic": "exact absurd hn (not_lt_of_le n.zero_le)"
},
{
"state_after": "case cons.zero\nι : Type ?u.337556\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nh✝ : ∀ (a : α), a ∈ l → p a\nn : ℕ\nhn✝ : n < length (pmap f l h✝)\nhd : α\ntl : List α\nhl :\n ∀ (h : ∀ (a : α), a ∈ tl → p a) {n : ℕ} (hn : n < length (pmap f tl h)),\n get (pmap f tl h) { val := n, isLt := hn } =\n f (get tl { val := n, isLt := (_ : n < length tl) }) (_ : p (get tl { val := n, isLt := (_ : n < length tl) }))\nh : ∀ (a : α), a ∈ hd :: tl → p a\nhn : zero < length (pmap f (hd :: tl) h)\n⊢ get (pmap f (hd :: tl) h) { val := zero, isLt := hn } =\n f (get (hd :: tl) { val := zero, isLt := (_ : zero < length (hd :: tl)) })\n (_ : p (get (hd :: tl) { val := zero, isLt := (_ : zero < length (hd :: tl)) }))\n\ncase cons.succ\nι : Type ?u.337556\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nh✝ : ∀ (a : α), a ∈ l → p a\nn : ℕ\nhn✝ : n < length (pmap f l h✝)\nhd : α\ntl : List α\nhl :\n ∀ (h : ∀ (a : α), a ∈ tl → p a) {n : ℕ} (hn : n < length (pmap f tl h)),\n get (pmap f tl h) { val := n, isLt := hn } =\n f (get tl { val := n, isLt := (_ : n < length tl) }) (_ : p (get tl { val := n, isLt := (_ : n < length tl) }))\nh : ∀ (a : α), a ∈ hd :: tl → p a\nn✝ : ℕ\nhn : succ n✝ < length (pmap f (hd :: tl) h)\n⊢ get (pmap f (hd :: tl) h) { val := succ n✝, isLt := hn } =\n f (get (hd :: tl) { val := succ n✝, isLt := (_ : succ n✝ < length (hd :: tl)) })\n (_ : p (get (hd :: tl) { val := succ n✝, isLt := (_ : succ n✝ < length (hd :: tl)) }))",
"state_before": "case cons\nι : Type ?u.337556\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nh✝ : ∀ (a : α), a ∈ l → p a\nn✝ : ℕ\nhn✝ : n✝ < length (pmap f l h✝)\nhd : α\ntl : List α\nhl :\n ∀ (h : ∀ (a : α), a ∈ tl → p a) {n : ℕ} (hn : n < length (pmap f tl h)),\n get (pmap f tl h) { val := n, isLt := hn } =\n f (get tl { val := n, isLt := (_ : n < length tl) }) (_ : p (get tl { val := n, isLt := (_ : n < length tl) }))\nh : ∀ (a : α), a ∈ hd :: tl → p a\nn : ℕ\nhn : n < length (pmap f (hd :: tl) h)\n⊢ get (pmap f (hd :: tl) h) { val := n, isLt := hn } =\n f (get (hd :: tl) { val := n, isLt := (_ : n < length (hd :: tl)) })\n (_ : p (get (hd :: tl) { val := n, isLt := (_ : n < length (hd :: tl)) }))",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case cons.zero\nι : Type ?u.337556\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nh✝ : ∀ (a : α), a ∈ l → p a\nn : ℕ\nhn✝ : n < length (pmap f l h✝)\nhd : α\ntl : List α\nhl :\n ∀ (h : ∀ (a : α), a ∈ tl → p a) {n : ℕ} (hn : n < length (pmap f tl h)),\n get (pmap f tl h) { val := n, isLt := hn } =\n f (get tl { val := n, isLt := (_ : n < length tl) }) (_ : p (get tl { val := n, isLt := (_ : n < length tl) }))\nh : ∀ (a : α), a ∈ hd :: tl → p a\nhn : zero < length (pmap f (hd :: tl) h)\n⊢ get (pmap f (hd :: tl) h) { val := zero, isLt := hn } =\n f (get (hd :: tl) { val := zero, isLt := (_ : zero < length (hd :: tl)) })\n (_ : p (get (hd :: tl) { val := zero, isLt := (_ : zero < length (hd :: tl)) }))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons.succ\nι : Type ?u.337556\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nh✝ : ∀ (a : α), a ∈ l → p a\nn : ℕ\nhn✝ : n < length (pmap f l h✝)\nhd : α\ntl : List α\nhl :\n ∀ (h : ∀ (a : α), a ∈ tl → p a) {n : ℕ} (hn : n < length (pmap f tl h)),\n get (pmap f tl h) { val := n, isLt := hn } =\n f (get tl { val := n, isLt := (_ : n < length tl) }) (_ : p (get tl { val := n, isLt := (_ : n < length tl) }))\nh : ∀ (a : α), a ∈ hd :: tl → p a\nn✝ : ℕ\nhn : succ n✝ < length (pmap f (hd :: tl) h)\n⊢ get (pmap f (hd :: tl) h) { val := succ n✝, isLt := hn } =\n f (get (hd :: tl) { val := succ n✝, isLt := (_ : succ n✝ < length (hd :: tl)) })\n (_ : p (get (hd :: tl) { val := succ n✝, isLt := (_ : succ n✝ < length (hd :: tl)) }))",
"tactic": "simp [hl]"
}
] |
[
3198,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3188,
1
] |
Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
|
LiouvilleWith.nat_add_iff
|
[
{
"state_after": "no goals",
"state_before": "p q x y : ℝ\nr : ℚ\nm : ℤ\nn : ℕ\n⊢ LiouvilleWith p (↑n + x) ↔ LiouvilleWith p x",
"tactic": "rw [add_comm, add_nat_iff]"
}
] |
[
229,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
eventually_nhds_nhdsWithin
|
[] |
[
49,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean
|
Asymptotics.IsLittleO.sum_range
|
[
{
"state_after": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\n⊢ (fun n => ∑ i in range n, f i) =o[atTop] fun n => ∑ i in range n, g i",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\n⊢ (fun n => ∑ i in range n, f i) =o[atTop] fun n => ∑ i in range n, g i",
"tactic": "have A : ∀ i, ‖g i‖ = g i := fun i => Real.norm_of_nonneg (hg i)"
},
{
"state_after": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\n⊢ (fun n => ∑ i in range n, f i) =o[atTop] fun n => ∑ i in range n, g i",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\n⊢ (fun n => ∑ i in range n, f i) =o[atTop] fun n => ∑ i in range n, g i",
"tactic": "have B : ∀ n, ‖∑ i in range n, g i‖ = ∑ i in range n, g i := fun n => by\n rwa [Real.norm_eq_abs, abs_sum_of_nonneg']"
},
{
"state_after": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\n⊢ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (x : ℕ) in atTop, ‖∑ i in range x, f i‖ ≤ ε * ‖∑ i in range x, g i‖",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\n⊢ (fun n => ∑ i in range n, f i) =o[atTop] fun n => ∑ i in range n, g i",
"tactic": "apply isLittleO_iff.2 fun ε εpos => _"
},
{
"state_after": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖∑ i in range x, f i‖ ≤ ε * ‖∑ i in range x, g i‖",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\n⊢ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (x : ℕ) in atTop, ‖∑ i in range x, f i‖ ≤ ε * ‖∑ i in range x, g i‖",
"tactic": "intro ε εpos"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖∑ i in range x, f i‖ ≤ ε * ‖∑ i in range x, g i‖",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖∑ i in range x, f i‖ ≤ ε * ‖∑ i in range x, g i‖",
"tactic": "obtain ⟨N, hN⟩ : ∃ N : ℕ, ∀ b : ℕ, N ≤ b → ‖f b‖ ≤ ε / 2 * g b := by\n simpa only [A, eventually_atTop] using isLittleO_iff.mp h (half_pos εpos)"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖∑ i in range x, f i‖ ≤ ε * ‖∑ i in range x, g i‖",
"state_before": "case intro\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖∑ i in range x, f i‖ ≤ ε * ‖∑ i in range x, g i‖",
"tactic": "have : (fun _ : ℕ => ∑ i in range N, f i) =o[atTop] fun n : ℕ => ∑ i in range n, g i := by\n apply isLittleO_const_left.2\n exact Or.inr (h'g.congr fun n => (B n).symm)"
},
{
"state_after": "case h\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ ‖∑ i in range n, f i‖ ≤ ε * ‖∑ i in range n, g i‖",
"state_before": "case intro\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖∑ i in range x, f i‖ ≤ ε * ‖∑ i in range x, g i‖",
"tactic": "filter_upwards [isLittleO_iff.1 this (half_pos εpos), Ici_mem_atTop N] with n hn Nn"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nn : ℕ\n⊢ ‖∑ i in range n, g i‖ = ∑ i in range n, g i",
"tactic": "rwa [Real.norm_eq_abs, abs_sum_of_nonneg']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\n⊢ ∃ N, ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b",
"tactic": "simpa only [A, eventually_atTop] using isLittleO_iff.mp h (half_pos εpos)"
},
{
"state_after": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\n⊢ ∑ i in range N, f i = 0 ∨ Tendsto (norm ∘ fun n => ∑ i in range n, g i) atTop atTop",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\n⊢ (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i",
"tactic": "apply isLittleO_const_left.2"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\n⊢ ∑ i in range N, f i = 0 ∨ Tendsto (norm ∘ fun n => ∑ i in range n, g i) atTop atTop",
"tactic": "exact Or.inr (h'g.congr fun n => (B n).symm)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ ‖∑ i in range n, f i‖ = ‖∑ i in range N, f i + ∑ i in Ico N n, f i‖",
"tactic": "rw [sum_range_add_sum_Ico _ Nn]"
},
{
"state_after": "case bc\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ ∑ i in Ico N n, ε / 2 * g i ≤ ∑ i in range n, ε / 2 * g i",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ ‖∑ i in range N, f i‖ + ∑ i in Ico N n, ε / 2 * g i ≤ ‖∑ i in range N, f i‖ + ∑ i in range n, ε / 2 * g i",
"tactic": "gcongr"
},
{
"state_after": "case bc.h\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ Ico N n ⊆ range n\n\ncase bc.hf\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ ∀ (i : ℕ), i ∈ range n → ¬i ∈ Ico N n → 0 ≤ ε / 2 * g i",
"state_before": "case bc\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ ∑ i in Ico N n, ε / 2 * g i ≤ ∑ i in range n, ε / 2 * g i",
"tactic": "apply sum_le_sum_of_subset_of_nonneg"
},
{
"state_after": "case bc.h\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ Ico N n ⊆ Ico 0 n",
"state_before": "case bc.h\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ Ico N n ⊆ range n",
"tactic": "rw [range_eq_Ico]"
},
{
"state_after": "no goals",
"state_before": "case bc.h\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ Ico N n ⊆ Ico 0 n",
"tactic": "exact Ico_subset_Ico (zero_le _) le_rfl"
},
{
"state_after": "case bc.hf\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\ni : ℕ\na✝¹ : i ∈ range n\na✝ : ¬i ∈ Ico N n\n⊢ 0 ≤ ε / 2 * g i",
"state_before": "case bc.hf\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ ∀ (i : ℕ), i ∈ range n → ¬i ∈ Ico N n → 0 ≤ ε / 2 * g i",
"tactic": "intro i _ _"
},
{
"state_after": "no goals",
"state_before": "case bc.hf\nα : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\ni : ℕ\na✝¹ : i ∈ range n\na✝ : ¬i ∈ Ico N n\n⊢ 0 ≤ ε / 2 * g i",
"tactic": "exact mul_nonneg (half_pos εpos).le (hg i)"
},
{
"state_after": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ ‖∑ i in range N, f i‖ + ε / 2 * ∑ x in range n, g x ≤ ε / 2 * ‖∑ i in range n, g i‖ + ε / 2 * ∑ i in range n, g i",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ ‖∑ i in range N, f i‖ + ∑ i in range n, ε / 2 * g i ≤ ε / 2 * ‖∑ i in range n, g i‖ + ε / 2 * ∑ i in range n, g i",
"tactic": "rw [← mul_sum]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ ‖∑ i in range N, f i‖ + ε / 2 * ∑ x in range n, g x ≤ ε / 2 * ‖∑ i in range n, g i‖ + ε / 2 * ∑ i in range n, g i",
"tactic": "gcongr"
},
{
"state_after": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ ε / 2 * ∑ i in range n, g i + ε / 2 * ∑ i in range n, g i = ε * ∑ i in range n, g i",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ ε / 2 * ‖∑ i in range n, g i‖ + ε / 2 * ∑ i in range n, g i = ε * ‖∑ i in range n, g i‖",
"tactic": "simp only [B]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n => ∑ i in range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i in range n, g i‖ = ∑ i in range n, g i\nε : ℝ\nεpos : 0 < ε\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b\nthis : (fun x => ∑ i in range N, f i) =o[atTop] fun n => ∑ i in range n, g i\nn : ℕ\nhn : ‖∑ i in range N, f i‖ ≤ ε / 2 * ‖∑ i in range n, g i‖\nNn : n ∈ Set.Ici N\n⊢ ε / 2 * ∑ i in range n, g i + ε / 2 * ∑ i in range n, g i = ε * ∑ i in range n, g i",
"tactic": "ring"
}
] |
[
139,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/Topology/UniformSpace/UniformEmbedding.lean
|
totallyBounded_preimage
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\n⊢ ∃ t_1, Set.Finite t_1 ∧ f ⁻¹' s ⊆ ⋃ (y : α) (_ : y ∈ t_1), {x | (x, y) ∈ t}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ 𝓤 α\n⊢ ∃ t_1, Set.Finite t_1 ∧ f ⁻¹' s ⊆ ⋃ (y : α) (_ : y ∈ t_1), {x | (x, y) ∈ t}",
"tactic": "rw [← hf.comap_uniformity] at ht"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\n⊢ ∃ t_1, Set.Finite t_1 ∧ f ⁻¹' s ⊆ ⋃ (y : α) (_ : y ∈ t_1), {x | (x, y) ∈ t}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\n⊢ ∃ t_1, Set.Finite t_1 ∧ f ⁻¹' s ⊆ ⋃ (y : α) (_ : y ∈ t_1), {x | (x, y) ∈ t}",
"tactic": "rcases mem_comap.2 ht with ⟨t', ht', ts⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\nc : Set β\ncs : c ⊆ f '' (f ⁻¹' s)\nhfc : Set.Finite c\nhct : f '' (f ⁻¹' s) ⊆ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\n⊢ ∃ t_1, Set.Finite t_1 ∧ f ⁻¹' s ⊆ ⋃ (y : α) (_ : y ∈ t_1), {x | (x, y) ∈ t}",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\n⊢ ∃ t_1, Set.Finite t_1 ∧ f ⁻¹' s ⊆ ⋃ (y : α) (_ : y ∈ t_1), {x | (x, y) ∈ t}",
"tactic": "rcases totallyBounded_iff_subset.1 (totallyBounded_subset (image_preimage_subset f s) hs) _ ht'\n with ⟨c, cs, hfc, hct⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\nc : Set β\ncs : c ⊆ f '' (f ⁻¹' s)\nhfc : Set.Finite c\nhct : f '' (f ⁻¹' s) ⊆ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\nx : α\nh : x ∈ f ⁻¹' s\n⊢ x ∈ ⋃ (y : α) (_ : y ∈ f ⁻¹' c), {x | (x, y) ∈ t}",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\nc : Set β\ncs : c ⊆ f '' (f ⁻¹' s)\nhfc : Set.Finite c\nhct : f '' (f ⁻¹' s) ⊆ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\n⊢ ∃ t_1, Set.Finite t_1 ∧ f ⁻¹' s ⊆ ⋃ (y : α) (_ : y ∈ t_1), {x | (x, y) ∈ t}",
"tactic": "refine' ⟨f ⁻¹' c, hfc.preimage (hf.inj.injOn _), fun x h => _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\nc : Set β\ncs : c ⊆ f '' (f ⁻¹' s)\nhfc : Set.Finite c\nhct : f '' (f ⁻¹' s) ⊆ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\nx : α\nh : x ∈ f ⁻¹' s\nthis : f x ∈ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\n⊢ x ∈ ⋃ (y : α) (_ : y ∈ f ⁻¹' c), {x | (x, y) ∈ t}",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\nc : Set β\ncs : c ⊆ f '' (f ⁻¹' s)\nhfc : Set.Finite c\nhct : f '' (f ⁻¹' s) ⊆ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\nx : α\nh : x ∈ f ⁻¹' s\n⊢ x ∈ ⋃ (y : α) (_ : y ∈ f ⁻¹' c), {x | (x, y) ∈ t}",
"tactic": "have := hct (mem_image_of_mem f h)"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\nc : Set β\ncs : c ⊆ f '' (f ⁻¹' s)\nhfc : Set.Finite c\nhct : f '' (f ⁻¹' s) ⊆ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\nx : α\nh : x ∈ f ⁻¹' s\nthis : ∃ i, i ∈ c ∧ (f x, i) ∈ t'\n⊢ ∃ i, f i ∈ c ∧ (x, i) ∈ t",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\nc : Set β\ncs : c ⊆ f '' (f ⁻¹' s)\nhfc : Set.Finite c\nhct : f '' (f ⁻¹' s) ⊆ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\nx : α\nh : x ∈ f ⁻¹' s\nthis : f x ∈ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\n⊢ x ∈ ⋃ (y : α) (_ : y ∈ f ⁻¹' c), {x | (x, y) ∈ t}",
"tactic": "simp at this⊢"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\nc : Set β\ncs : c ⊆ f '' (f ⁻¹' s)\nhfc : Set.Finite c\nhct : f '' (f ⁻¹' s) ⊆ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\nx : α\nh : x ∈ f ⁻¹' s\nz : β\nzc : z ∈ c\nzt : (f x, z) ∈ t'\n⊢ ∃ i, f i ∈ c ∧ (x, i) ∈ t",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\nc : Set β\ncs : c ⊆ f '' (f ⁻¹' s)\nhfc : Set.Finite c\nhct : f '' (f ⁻¹' s) ⊆ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\nx : α\nh : x ∈ f ⁻¹' s\nthis : ∃ i, i ∈ c ∧ (f x, i) ∈ t'\n⊢ ∃ i, f i ∈ c ∧ (x, i) ∈ t",
"tactic": "rcases this with ⟨z, zc, zt⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\nc : Set β\ncs : c ⊆ f '' (f ⁻¹' s)\nhfc : Set.Finite c\nhct : f '' (f ⁻¹' s) ⊆ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\nx : α\nh : x ∈ f ⁻¹' s\ny : α\nzc : f y ∈ c\nzt : (f x, f y) ∈ t'\n⊢ ∃ i, f i ∈ c ∧ (x, i) ∈ t",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\nc : Set β\ncs : c ⊆ f '' (f ⁻¹' s)\nhfc : Set.Finite c\nhct : f '' (f ⁻¹' s) ⊆ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\nx : α\nh : x ∈ f ⁻¹' s\nz : β\nzc : z ∈ c\nzt : (f x, z) ∈ t'\n⊢ ∃ i, f i ∈ c ∧ (x, i) ∈ t",
"tactic": "rcases cs zc with ⟨y, -, rfl⟩"
},
{
"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\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set β\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (α × α)\nht : t ∈ comap (fun x => (f x.fst, f x.snd)) (𝓤 β)\nt' : Set (β × β)\nht' : t' ∈ 𝓤 β\nts : (fun x => (f x.fst, f x.snd)) ⁻¹' t' ⊆ t\nc : Set β\ncs : c ⊆ f '' (f ⁻¹' s)\nhfc : Set.Finite c\nhct : f '' (f ⁻¹' s) ⊆ ⋃ (y : β) (_ : y ∈ c), {x | (x, y) ∈ t'}\nx : α\nh : x ∈ f ⁻¹' s\ny : α\nzc : f y ∈ c\nzt : (f x, f y) ∈ t'\n⊢ ∃ i, f i ∈ c ∧ (x, i) ∈ t",
"tactic": "exact ⟨y, zc, ts zt⟩"
}
] |
[
396,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Split.lean
|
BoxIntegral.Prepartition.splitMany_empty
|
[] |
[
250,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.sum_smul_index
|
[
{
"state_after": "case ofFinsupp\nR : Type u\na b✝ : R\nm n : ℕ\ninst✝¹ : Semiring R\np q : R[X]\nS : Type u_1\ninst✝ : AddCommMonoid S\nb : R\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ sum (b • { toFinsupp := toFinsupp✝ }) f = sum { toFinsupp := toFinsupp✝ } fun n a => f n (b * a)",
"state_before": "R : Type u\na b✝ : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q : R[X]\nS : Type u_1\ninst✝ : AddCommMonoid S\np : R[X]\nb : R\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\n⊢ sum (b • p) f = sum p fun n a => f n (b * a)",
"tactic": "rcases p with ⟨⟩"
},
{
"state_after": "no goals",
"state_before": "case ofFinsupp\nR : Type u\na b✝ : R\nm n : ℕ\ninst✝¹ : Semiring R\np q : R[X]\nS : Type u_1\ninst✝ : AddCommMonoid S\nb : R\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ sum (b • { toFinsupp := toFinsupp✝ }) f = sum { toFinsupp := toFinsupp✝ } fun n a => f n (b * a)",
"tactic": "simpa [sum, support, coeff] using Finsupp.sum_smul_index hf"
}
] |
[
1005,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1002,
1
] |
Mathlib/Data/Part.lean
|
Part.right_dom_of_union_dom
|
[] |
[
850,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
850,
1
] |
Mathlib/Data/Rat/Defs.lean
|
Rat.coe_int_inj
|
[] |
[
545,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
544,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
eventually_ne_of_tendsto_norm_atTop'
|
[] |
[
1261,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1259,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.ker_prod_ker_le_ker_coprod
|
[] |
[
1324,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1322,
1
] |
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