file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Nat/Basic.lean
|
Nat.mul_dvd_of_dvd_div
|
[
{
"state_after": "no goals",
"state_before": "m n k a b c : ℕ\nhab : c ∣ b\nh : a ∣ b / c\nh1 : ∃ d, b / c = a * d\nd : ℕ\nhd : b / c = a * d\nh3 : b = a * d * c\n⊢ b = c * a * d",
"tactic": "rwa [mul_comm, ←mul_assoc] at h3"
}
] |
[
774,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
769,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.subtype_injective
|
[] |
[
802,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.congr_const
|
[] |
[
323,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
SubmonoidClass.coe_finset_prod
|
[] |
[
66,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Algebra/CharP/Two.lean
|
CharTwo.bit0_apply_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nι : Type ?u.2610\ninst✝¹ : Semiring R\ninst✝ : CharP R 2\nx : R\n⊢ bit0 x = 0",
"tactic": "simp"
}
] |
[
46,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toReal_min
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.801740\nβ : Type ?u.801743\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nhr : a ≠ ⊤\nhp : b ≠ ⊤\nh : a ≤ b\n⊢ ENNReal.toReal (min a b) = min (ENNReal.toReal a) (ENNReal.toReal b)",
"tactic": "simp only [h, (ENNReal.toReal_le_toReal hr hp).2 h, min_eq_left]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.801740\nβ : Type ?u.801743\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nhr : a ≠ ⊤\nhp : b ≠ ⊤\nh : b ≤ a\n⊢ ENNReal.toReal (min a b) = min (ENNReal.toReal a) (ENNReal.toReal b)",
"tactic": "simp only [h, (ENNReal.toReal_le_toReal hp hr).2 h, min_eq_right]"
}
] |
[
2056,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2053,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.mul_le_mul_iff_left
|
[] |
[
825,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
824,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.map_div_atTop_eq_nat
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.345892\nι' : Type ?u.345895\nα : Type ?u.345898\nβ : Type ?u.345901\nγ : Type ?u.345904\nk : ℕ\nhk : 0 < k\na b : ℕ\nx✝ : b ≥ 1\n⊢ a / k ≤ b ↔ a ≤ (fun b => b * k + (k - 1)) b",
"tactic": "simp only [← Nat.lt_succ_iff, Nat.div_lt_iff_lt_mul hk, Nat.succ_eq_add_one,\nadd_assoc, tsub_add_cancel_of_le (Nat.one_le_iff_ne_zero.2 hk.ne'), add_mul, one_mul]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.345892\nι' : Type ?u.345895\nα : Type ?u.345898\nβ : Type ?u.345901\nγ : Type ?u.345904\nk : ℕ\nhk : 0 < k\nb : ℕ\nx✝ : b ≥ 1\n⊢ b = b * k / k",
"tactic": "rw [Nat.mul_div_cancel b hk]"
}
] |
[
1671,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1663,
1
] |
Mathlib/Init/Data/Ordering/Lemmas.lean
|
cmpUsing_eq_lt
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nlt : α → α → Prop\ninst✝ : DecidableRel lt\na b : α\n⊢ (cmpUsing lt a b = Ordering.lt) = lt a b",
"tactic": "simp"
}
] |
[
51,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Algebra/Tropical/Basic.lean
|
Tropical.untrop_monotone
|
[] |
[
221,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
LinearIndependent.set_finite_of_isNoetherian
|
[] |
[
314,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.IsCycleOn.extendDomain
|
[
{
"state_after": "case intro.intro.intro.intro\nι : Type ?u.2071598\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\ns t : Set α\na✝ b✝ x y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\nh : IsCycleOn g s\na : α\nha : a ∈ s\nb : α\nhb : b ∈ s\n⊢ SameCycle (Perm.extendDomain g f) ((Subtype.val ∘ ↑f) a) ((Subtype.val ∘ ↑f) b)",
"state_before": "ι : Type ?u.2071598\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\ns t : Set α\na b x y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\nh : IsCycleOn g s\n⊢ ∀ ⦃x : β⦄, x ∈ Subtype.val ∘ ↑f '' s → ∀ ⦃y : β⦄, y ∈ Subtype.val ∘ ↑f '' s → SameCycle (Perm.extendDomain g f) x y",
"tactic": "rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nι : Type ?u.2071598\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\ns t : Set α\na✝ b✝ x y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\nh : IsCycleOn g s\na : α\nha : a ∈ s\nb : α\nhb : b ∈ s\n⊢ SameCycle (Perm.extendDomain g f) ((Subtype.val ∘ ↑f) a) ((Subtype.val ∘ ↑f) b)",
"tactic": "exact (h.2 ha hb).extendDomain"
}
] |
[
941,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
937,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean
|
CategoryTheory.Limits.map_π_preserves_coequalizer_inv
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nG : C ⥤ D\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\nw : f ≫ h = g ≫ h\ninst✝² : HasCoequalizer f g\ninst✝¹ : HasCoequalizer (G.map f) (G.map g)\ninst✝ : PreservesColimit (parallelPair f g) G\n⊢ G.map (coequalizer.π f g) ≫ (PreservesCoequalizer.iso G f g).inv = coequalizer.π (G.map f) (G.map g)",
"tactic": "rw [← ι_comp_coequalizerComparison_assoc, ← PreservesCoequalizer.iso_hom, Iso.hom_inv_id,\n comp_id]"
}
] |
[
209,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Order/Concept.lean
|
Concept.fst_injective
|
[] |
[
207,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Combinatorics/Derangements/Finite.lean
|
card_derangements_eq_numDerangements
|
[
{
"state_after": "α✝ : Type ?u.21453\ninst✝³ : DecidableEq α✝\ninst✝² : Fintype α✝\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\n⊢ card ↑(derangements (Fin (card α))) = numDerangements (card α)",
"state_before": "α✝ : Type ?u.21453\ninst✝³ : DecidableEq α✝\ninst✝² : Fintype α✝\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\n⊢ card ↑(derangements α) = numDerangements (card α)",
"tactic": "rw [← card_derangements_invariant (card_fin _)]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.21453\ninst✝³ : DecidableEq α✝\ninst✝² : Fintype α✝\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\n⊢ card ↑(derangements (Fin (card α))) = numDerangements (card α)",
"tactic": "exact card_derangements_fin_eq_numDerangements"
}
] |
[
116,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Std/Data/Rat/Lemmas.lean
|
Rat.normalize_mul_right
|
[
{
"state_after": "d : Nat\nn : Int\na : Nat\nd0 : d ≠ 0\na0 : a ≠ 0\n⊢ normalize (n * ↑a) (d * a) = normalize (↑a * n) (a * d)",
"state_before": "d : Nat\nn : Int\na : Nat\nd0 : d ≠ 0\na0 : a ≠ 0\n⊢ normalize (n * ↑a) (d * a) = normalize n d",
"tactic": "rw [← normalize_mul_left (d0 := d0) a0]"
},
{
"state_after": "no goals",
"state_before": "d : Nat\nn : Int\na : Nat\nd0 : d ≠ 0\na0 : a ≠ 0\n⊢ normalize (n * ↑a) (d * a) = normalize (↑a * n) (a * d)",
"tactic": "congr 1 <;> [apply Int.mul_comm; apply Nat.mul_comm]"
}
] |
[
44,
96
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
42,
1
] |
Mathlib/RingTheory/Localization/AtPrime.lean
|
Localization.localRingHom_id
|
[] |
[
257,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.union_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nl t₁ t₂ : List α\nh : t₁ ~ t₂\n⊢ List.union l t₁ ~ List.union l t₂",
"tactic": "induction l <;> simp [*, Perm.insert]"
}
] |
[
999,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
998,
1
] |
Mathlib/Data/Int/GCD.lean
|
Nat.gcdB_zero_left
|
[
{
"state_after": "s : ℕ\n⊢ (xgcd 0 s).snd = 1",
"state_before": "s : ℕ\n⊢ gcdB 0 s = 1",
"tactic": "unfold gcdB"
},
{
"state_after": "no goals",
"state_before": "s : ℕ\n⊢ (xgcd 0 s).snd = 1",
"tactic": "rw [xgcd, xgcd_zero_left]"
}
] |
[
92,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.add_pos_of_pos_of_nonneg
|
[] |
[
826,
52
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
825,
11
] |
Mathlib/Order/Heyting/Basic.lean
|
sup_compl_le_himp
|
[] |
[
807,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
806,
1
] |
Mathlib/Algebra/Order/Pointwise.lean
|
sSup_inv
|
[
{
"state_after": "α : Type u_1\ninst✝³ : CompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns✝ t s : Set α\n⊢ (⨆ (a : α) (_ : a ∈ s), a⁻¹) = (sInf s)⁻¹",
"state_before": "α : Type u_1\ninst✝³ : CompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns✝ t s : Set α\n⊢ sSup s⁻¹ = (sInf s)⁻¹",
"tactic": "rw [← image_inv, sSup_image]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : CompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns✝ t s : Set α\n⊢ (⨆ (a : α) (_ : a ∈ s), a⁻¹) = (sInf s)⁻¹",
"tactic": "exact ((OrderIso.inv α).map_sInf _).symm"
}
] |
[
66,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.WellFoundedOn.union
|
[
{
"state_after": "ι : Type ?u.29184\nα : Type u_1\nβ : Type ?u.29190\nr r' : α → α → Prop\ninst✝ : IsStrictOrder α r\ns t : Set α\nhs : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ s\nht : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ t\n⊢ ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ s ∪ t",
"state_before": "ι : Type ?u.29184\nα : Type u_1\nβ : Type ?u.29190\nr r' : α → α → Prop\ninst✝ : IsStrictOrder α r\ns t : Set α\nhs : WellFoundedOn s r\nht : WellFoundedOn t r\n⊢ WellFoundedOn (s ∪ t) r",
"tactic": "rw [wellFoundedOn_iff_no_descending_seq] at *"
},
{
"state_after": "ι : Type ?u.29184\nα : Type u_1\nβ : Type ?u.29190\nr r' : α → α → Prop\ninst✝ : IsStrictOrder α r\ns t : Set α\nhs : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ s\nht : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ t\nf : (fun x x_1 => x > x_1) ↪r r\nhf : ∀ (n : ℕ), ↑f n ∈ s ∪ t\n⊢ False",
"state_before": "ι : Type ?u.29184\nα : Type u_1\nβ : Type ?u.29190\nr r' : α → α → Prop\ninst✝ : IsStrictOrder α r\ns t : Set α\nhs : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ s\nht : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ t\n⊢ ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ s ∪ t",
"tactic": "rintro f hf"
},
{
"state_after": "case intro.inl\nι : Type ?u.29184\nα : Type u_1\nβ : Type ?u.29190\nr r' : α → α → Prop\ninst✝ : IsStrictOrder α r\ns t : Set α\nhs : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ s\nht : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ t\nf : (fun x x_1 => x > x_1) ↪r r\nhf : ∀ (n : ℕ), ↑f n ∈ s ∪ t\ng : ℕ ↪o ℕ\nhg : ∀ (n : ℕ), ↑f (↑g n) ∈ s\n⊢ False\n\ncase intro.inr\nι : Type ?u.29184\nα : Type u_1\nβ : Type ?u.29190\nr r' : α → α → Prop\ninst✝ : IsStrictOrder α r\ns t : Set α\nhs : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ s\nht : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ t\nf : (fun x x_1 => x > x_1) ↪r r\nhf : ∀ (n : ℕ), ↑f n ∈ s ∪ t\ng : ℕ ↪o ℕ\nhg : ∀ (n : ℕ), ↑f (↑g n) ∈ t\n⊢ False",
"state_before": "ι : Type ?u.29184\nα : Type u_1\nβ : Type ?u.29190\nr r' : α → α → Prop\ninst✝ : IsStrictOrder α r\ns t : Set α\nhs : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ s\nht : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ t\nf : (fun x x_1 => x > x_1) ↪r r\nhf : ∀ (n : ℕ), ↑f n ∈ s ∪ t\n⊢ False",
"tactic": "rcases Nat.exists_subseq_of_forall_mem_union f hf with ⟨g, hg | hg⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.inl\nι : Type ?u.29184\nα : Type u_1\nβ : Type ?u.29190\nr r' : α → α → Prop\ninst✝ : IsStrictOrder α r\ns t : Set α\nhs : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ s\nht : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ t\nf : (fun x x_1 => x > x_1) ↪r r\nhf : ∀ (n : ℕ), ↑f n ∈ s ∪ t\ng : ℕ ↪o ℕ\nhg : ∀ (n : ℕ), ↑f (↑g n) ∈ s\n⊢ False\n\ncase intro.inr\nι : Type ?u.29184\nα : Type u_1\nβ : Type ?u.29190\nr r' : α → α → Prop\ninst✝ : IsStrictOrder α r\ns t : Set α\nhs : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ s\nht : ∀ (f : (fun x x_1 => x > x_1) ↪r r), ¬∀ (n : ℕ), ↑f n ∈ t\nf : (fun x x_1 => x > x_1) ↪r r\nhf : ∀ (n : ℕ), ↑f n ∈ s ∪ t\ng : ℕ ↪o ℕ\nhg : ∀ (n : ℕ), ↑f (↑g n) ∈ t\n⊢ False",
"tactic": "exacts [hs (g.dual.ltEmbedding.trans f) hg, ht (g.dual.ltEmbedding.trans f) hg]"
}
] |
[
166,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Analysis/NormedSpace/LpEquiv.lean
|
coe_addEquiv_lpBcf
|
[] |
[
158,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.neg_def
|
[] |
[
848,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
847,
11
] |
Mathlib/SetTheory/Cardinal/Continuum.lean
|
Cardinal.beth_one
|
[
{
"state_after": "no goals",
"state_before": "⊢ beth 1 = 𝔠",
"tactic": "simpa using beth_succ 0"
}
] |
[
87,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Data/List/Nodup.lean
|
List.get_indexOf
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\ninst✝ : DecidableEq α\nl : List α\nH : Nodup l\ni : Fin (length l)\n⊢ get l { val := indexOf (get l i) l, isLt := (_ : indexOf (get l i) l < length l) } = get l i",
"tactic": "simp"
}
] |
[
172,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.bind
|
[] |
[
822,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
820,
1
] |
Mathlib/Algebra/Order/Kleene.lean
|
mul_kstar_le_kstar
|
[] |
[
207,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean
|
HasFDerivWithinAt.clog
|
[] |
[
121,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
WeakDual.coeFn_continuous
|
[] |
[
280,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/RingTheory/PowerBasis.lean
|
PowerBasis.dim_ne_zero
|
[] |
[
123,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.indicator_mul
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.118109\nι : Type ?u.118112\nM : Type u_2\nN : Type ?u.118118\ninst✝ : MulZeroClass M\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf g : α → M\nx✝ : α\n⊢ indicator s (fun a => f a * g a) x✝ = indicator s f x✝ * indicator s g x✝",
"state_before": "α : Type u_1\nβ : Type ?u.118109\nι : Type ?u.118112\nM : Type u_2\nN : Type ?u.118118\ninst✝ : MulZeroClass M\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf g : α → M\n⊢ (indicator s fun a => f a * g a) = fun a => indicator s f a * indicator s g a",
"tactic": "funext"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.118109\nι : Type ?u.118112\nM : Type u_2\nN : Type ?u.118118\ninst✝ : MulZeroClass M\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf g : α → M\nx✝ : α\n⊢ (if x✝ ∈ s then f x✝ * g x✝ else 0) = (if x✝ ∈ s then f x✝ else 0) * if x✝ ∈ s then g x✝ else 0",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.118109\nι : Type ?u.118112\nM : Type u_2\nN : Type ?u.118118\ninst✝ : MulZeroClass M\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf g : α → M\nx✝ : α\n⊢ indicator s (fun a => f a * g a) x✝ = indicator s f x✝ * indicator s g x✝",
"tactic": "simp only [indicator]"
},
{
"state_after": "case h.inl\nα : Type u_1\nβ : Type ?u.118109\nι : Type ?u.118112\nM : Type u_2\nN : Type ?u.118118\ninst✝ : MulZeroClass M\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf g : α → M\nx✝ : α\nh✝ : x✝ ∈ s\n⊢ f x✝ * g x✝ = f x✝ * g x✝\n\ncase h.inr\nα : Type u_1\nβ : Type ?u.118109\nι : Type ?u.118112\nM : Type u_2\nN : Type ?u.118118\ninst✝ : MulZeroClass M\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf g : α → M\nx✝ : α\nh✝ : ¬x✝ ∈ s\n⊢ 0 = 0 * 0",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.118109\nι : Type ?u.118112\nM : Type u_2\nN : Type ?u.118118\ninst✝ : MulZeroClass M\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf g : α → M\nx✝ : α\n⊢ (if x✝ ∈ s then f x✝ * g x✝ else 0) = (if x✝ ∈ s then f x✝ else 0) * if x✝ ∈ s then g x✝ else 0",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "case h.inr\nα : Type u_1\nβ : Type ?u.118109\nι : Type ?u.118112\nM : Type u_2\nN : Type ?u.118118\ninst✝ : MulZeroClass M\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf g : α → M\nx✝ : α\nh✝ : ¬x✝ ∈ s\n⊢ 0 = 0 * 0",
"tactic": "rw [mul_zero]"
},
{
"state_after": "no goals",
"state_before": "case h.inl\nα : Type u_1\nβ : Type ?u.118109\nι : Type ?u.118112\nM : Type u_2\nN : Type ?u.118118\ninst✝ : MulZeroClass M\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf g : α → M\nx✝ : α\nh✝ : x✝ ∈ s\n⊢ f x✝ * g x✝ = f x✝ * g x✝",
"tactic": "rfl"
}
] |
[
684,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
678,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.const_sub
|
[] |
[
342,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.mapsTo_emetric_ball
|
[] |
[
152,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.mem_zero
|
[] |
[
452,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
451,
1
] |
Mathlib/CategoryTheory/Abelian/Opposite.lean
|
CategoryTheory.factorThruImage_comp_imageUnopOp_inv
|
[
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\nA B : Cᵒᵖ\ng : A ⟶ B\n⊢ factorThruImage g ≫ (imageUnopOp g).inv = (image.ι g.unop).op",
"tactic": "rw [Iso.comp_inv_eq, image_ι_op_comp_imageUnopOp_hom]"
}
] |
[
190,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean
|
AlgebraicGeometry.StructureSheaf.const_self
|
[] |
[
362,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
|
Matrix.SpecialLinearGroup.coe_inv
|
[] |
[
141,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Init/Data/Subtype/Basic.lean
|
Subtype.exists_of_subtype
|
[] |
[
25,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
24,
1
] |
Mathlib/Order/Max.lean
|
IsBot.fst
|
[] |
[
419,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
419,
1
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
TrivSqZeroExt.ext
|
[] |
[
111,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Topology/Maps.lean
|
Inducing.closure_eq_preimage_closure_image
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.5388\nδ : Type ?u.5391\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\nhf : Inducing f\ns : Set α\nx : α\n⊢ x ∈ closure s ↔ x ∈ f ⁻¹' closure (f '' s)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.5388\nδ : Type ?u.5391\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\nhf : Inducing f\ns : Set α\n⊢ closure s = f ⁻¹' closure (f '' s)",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.5388\nδ : Type ?u.5391\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\nhf : Inducing f\ns : Set α\nx : α\n⊢ x ∈ closure s ↔ x ∈ f ⁻¹' closure (f '' s)",
"tactic": "rw [Set.mem_preimage, ← closure_induced, hf.induced]"
}
] |
[
158,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Topology/Instances/AddCircle.lean
|
AddCircle.liftIoc_coe_apply
|
[
{
"state_after": "𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nf : 𝕜 → B\nx : 𝕜\nhx : x ∈ Ioc a (a + p)\nthis : ↑(equivIoc p a) ↑x = { val := x, property := hx }\n⊢ liftIoc p a f ↑x = f x",
"state_before": "𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nf : 𝕜 → B\nx : 𝕜\nhx : x ∈ Ioc a (a + p)\n⊢ liftIoc p a f ↑x = f x",
"tactic": "have : (equivIoc p a) x = ⟨x, hx⟩ := by\n rw [Equiv.apply_eq_iff_eq_symm_apply]\n rfl"
},
{
"state_after": "𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nf : 𝕜 → B\nx : 𝕜\nhx : x ∈ Ioc a (a + p)\nthis : ↑(equivIoc p a) ↑x = { val := x, property := hx }\n⊢ restrict (Ioc a (a + p)) f { val := x, property := hx } = f x",
"state_before": "𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nf : 𝕜 → B\nx : 𝕜\nhx : x ∈ Ioc a (a + p)\nthis : ↑(equivIoc p a) ↑x = { val := x, property := hx }\n⊢ liftIoc p a f ↑x = f x",
"tactic": "rw [liftIoc, comp_apply, this]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nf : 𝕜 → B\nx : 𝕜\nhx : x ∈ Ioc a (a + p)\nthis : ↑(equivIoc p a) ↑x = { val := x, property := hx }\n⊢ restrict (Ioc a (a + p)) f { val := x, property := hx } = f x",
"tactic": "rfl"
},
{
"state_after": "𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nf : 𝕜 → B\nx : 𝕜\nhx : x ∈ Ioc a (a + p)\n⊢ ↑x = ↑(equivIoc p a).symm { val := x, property := hx }",
"state_before": "𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nf : 𝕜 → B\nx : 𝕜\nhx : x ∈ Ioc a (a + p)\n⊢ ↑(equivIoc p a) ↑x = { val := x, property := hx }",
"tactic": "rw [Equiv.apply_eq_iff_eq_symm_apply]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nf : 𝕜 → B\nx : 𝕜\nhx : x ∈ Ioc a (a + p)\n⊢ ↑x = ↑(equivIoc p a).symm { val := x, property := hx }",
"tactic": "rfl"
}
] |
[
262,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/Order/CompleteLattice.lean
|
disjoint_sSup_right
|
[] |
[
1955,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1953,
1
] |
Mathlib/GroupTheory/Perm/Fin.lean
|
Fin.cycleRange_zero
|
[
{
"state_after": "case H.h\nn : ℕ\nj : Fin (Nat.succ n)\n⊢ ↑(↑(cycleRange 0) j) = ↑(↑1 j)",
"state_before": "n : ℕ\n⊢ cycleRange 0 = 1",
"tactic": "ext j"
},
{
"state_after": "case H.h.refine'_1\nn : ℕ\nj : Fin (Nat.succ n)\n⊢ ↑(↑(cycleRange 0) 0) = ↑(↑1 0)\n\ncase H.h.refine'_2\nn : ℕ\nj✝ : Fin (Nat.succ n)\nj : Fin n\n⊢ ↑(↑(cycleRange 0) (succ j)) = ↑(↑1 (succ j))",
"state_before": "case H.h\nn : ℕ\nj : Fin (Nat.succ n)\n⊢ ↑(↑(cycleRange 0) j) = ↑(↑1 j)",
"tactic": "refine' Fin.cases _ (fun j => _) j"
},
{
"state_after": "no goals",
"state_before": "case H.h.refine'_1\nn : ℕ\nj : Fin (Nat.succ n)\n⊢ ↑(↑(cycleRange 0) 0) = ↑(↑1 0)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case H.h.refine'_2\nn : ℕ\nj✝ : Fin (Nat.succ n)\nj : Fin n\n⊢ ↑(↑(cycleRange 0) (succ j)) = ↑(↑1 (succ j))",
"tactic": "rw [cycleRange_of_gt (Fin.succ_pos j), one_apply]"
}
] |
[
232,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Data/SetLike/Basic.lean
|
SetLike.mem_coe
|
[] |
[
164,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Order/Max.lean
|
IsMin.eq_of_ge
|
[] |
[
390,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
11
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.toSubsemiring_injective
|
[] |
[
256,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.floor_le_ceil
|
[] |
[
1224,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1223,
1
] |
Mathlib/LinearAlgebra/Multilinear/TensorProduct.lean
|
MultilinearMap.domCoprod_domDomCongr_sumCongr
|
[] |
[
96,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
OrderIso.map_csSup
|
[] |
[
1326,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1324,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.set_biUnion_insert_update
|
[] |
[
2069,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2067,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.sin_sq
|
[
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ sin x ^ 2 = 1 - cos x ^ 2",
"tactic": "rw [← sin_sq_add_cos_sq x, add_sub_cancel]"
}
] |
[
1048,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1048,
1
] |
Mathlib/Analysis/SpecialFunctions/Exp.lean
|
Real.isBigO_exp_comp_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nx y z : ℝ\nl : Filter α\nf : α → ℝ\n⊢ ((fun x => exp (f x)) =O[l] fun x => 1) ↔ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f",
"tactic": "simp only [isBigO_one_iff, norm_eq_abs, abs_exp, isBoundedUnder_le_exp_comp]"
}
] |
[
409,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/CategoryTheory/Groupoid/VertexGroup.lean
|
CategoryTheory.Groupoid.vertexGroup.inv_eq_inv
|
[] |
[
59,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.subtypeRestr_symm_trans_subtypeRestr
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\n⊢ LocalHomeomorph.trans\n (LocalHomeomorph.trans (LocalHomeomorph.symm f) (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)))\n (LocalHomeomorph.trans (Opens.localHomeomorphSubtypeCoe s) f') ≈\n LocalHomeomorph.restr (LocalHomeomorph.trans (LocalHomeomorph.symm f) f')\n (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\n⊢ LocalHomeomorph.trans (LocalHomeomorph.symm (subtypeRestr f s)) (subtypeRestr f' s) ≈\n LocalHomeomorph.restr (LocalHomeomorph.trans (LocalHomeomorph.symm f) f')\n (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)",
"tactic": "simp only [subtypeRestr_def, trans_symm_eq_symm_trans_symm]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\n⊢ LocalHomeomorph.trans\n (LocalHomeomorph.trans (LocalHomeomorph.symm f) (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)))\n (LocalHomeomorph.trans (Opens.localHomeomorphSubtypeCoe s) f') ≈\n LocalHomeomorph.restr (LocalHomeomorph.trans (LocalHomeomorph.symm f) f')\n (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\n⊢ LocalHomeomorph.trans\n (LocalHomeomorph.trans (LocalHomeomorph.symm f) (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)))\n (LocalHomeomorph.trans (Opens.localHomeomorphSubtypeCoe s) f') ≈\n LocalHomeomorph.restr (LocalHomeomorph.trans (LocalHomeomorph.symm f) f')\n (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)",
"tactic": "have openness₁ : IsOpen (f.target ∩ f.symm ⁻¹' s) := f.preimage_open_of_open_symm s.2"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\n⊢ LocalHomeomorph.trans\n (LocalHomeomorph.trans\n (LocalHomeomorph.trans (LocalHomeomorph.symm f) (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)))\n (Opens.localHomeomorphSubtypeCoe s))\n f' ≈\n LocalHomeomorph.trans\n (LocalHomeomorph.trans (ofSet (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) openness₁) (LocalHomeomorph.symm f))\n f'",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\n⊢ LocalHomeomorph.trans\n (LocalHomeomorph.trans (LocalHomeomorph.symm f) (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)))\n (LocalHomeomorph.trans (Opens.localHomeomorphSubtypeCoe s) f') ≈\n LocalHomeomorph.restr (LocalHomeomorph.trans (LocalHomeomorph.symm f) f')\n (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)",
"tactic": "rw [← ofSet_trans _ openness₁, ← trans_assoc, ← trans_assoc]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\n⊢ LocalHomeomorph.trans\n (LocalHomeomorph.trans (LocalHomeomorph.symm f) (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)))\n (Opens.localHomeomorphSubtypeCoe s) ≈\n LocalHomeomorph.trans (ofSet (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) openness₁) (LocalHomeomorph.symm f)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\n⊢ LocalHomeomorph.trans\n (LocalHomeomorph.trans\n (LocalHomeomorph.trans (LocalHomeomorph.symm f) (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)))\n (Opens.localHomeomorphSubtypeCoe s))\n f' ≈\n LocalHomeomorph.trans\n (LocalHomeomorph.trans (ofSet (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) openness₁) (LocalHomeomorph.symm f))\n f'",
"tactic": "refine' EqOnSource.trans' _ (eqOnSource_refl _)"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\nsets_identity :\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) =\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s\n⊢ LocalHomeomorph.trans\n (LocalHomeomorph.trans (LocalHomeomorph.symm f) (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)))\n (Opens.localHomeomorphSubtypeCoe s) ≈\n LocalHomeomorph.trans (ofSet (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) openness₁) (LocalHomeomorph.symm f)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\n⊢ LocalHomeomorph.trans\n (LocalHomeomorph.trans (LocalHomeomorph.symm f) (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)))\n (Opens.localHomeomorphSubtypeCoe s) ≈\n LocalHomeomorph.trans (ofSet (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) openness₁) (LocalHomeomorph.symm f)",
"tactic": "have sets_identity : f.symm.source ∩ (f.target ∩ f.symm ⁻¹' s) = f.symm.source ∩ f.symm ⁻¹' s :=\n by mfld_set_tac"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\nsets_identity :\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) =\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s\nopenness₂ : IsOpen ↑s\n⊢ LocalHomeomorph.trans\n (LocalHomeomorph.trans (LocalHomeomorph.symm f) (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)))\n (Opens.localHomeomorphSubtypeCoe s) ≈\n LocalHomeomorph.trans (ofSet (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) openness₁) (LocalHomeomorph.symm f)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\nsets_identity :\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) =\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s\n⊢ LocalHomeomorph.trans\n (LocalHomeomorph.trans (LocalHomeomorph.symm f) (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)))\n (Opens.localHomeomorphSubtypeCoe s) ≈\n LocalHomeomorph.trans (ofSet (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) openness₁) (LocalHomeomorph.symm f)",
"tactic": "have openness₂ : IsOpen (s : Set α) := s.2"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\nsets_identity :\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) =\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s\nopenness₂ : IsOpen ↑s\n⊢ LocalHomeomorph.trans (LocalHomeomorph.symm f)\n (LocalHomeomorph.trans (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s))\n (Opens.localHomeomorphSubtypeCoe s)) ≈\n LocalHomeomorph.trans (LocalHomeomorph.symm f) (ofSet (↑s) openness₂)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\nsets_identity :\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) =\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s\nopenness₂ : IsOpen ↑s\n⊢ LocalHomeomorph.trans\n (LocalHomeomorph.trans (LocalHomeomorph.symm f) (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)))\n (Opens.localHomeomorphSubtypeCoe s) ≈\n LocalHomeomorph.trans (ofSet (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) openness₁) (LocalHomeomorph.symm f)",
"tactic": "rw [ofSet_trans', sets_identity, ← trans_of_set' _ openness₂, trans_assoc]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\nsets_identity :\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) =\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s\nopenness₂ : IsOpen ↑s\n⊢ LocalHomeomorph.trans (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)) (Opens.localHomeomorphSubtypeCoe s) ≈\n ofSet (↑s) openness₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\nsets_identity :\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) =\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s\nopenness₂ : IsOpen ↑s\n⊢ LocalHomeomorph.trans (LocalHomeomorph.symm f)\n (LocalHomeomorph.trans (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s))\n (Opens.localHomeomorphSubtypeCoe s)) ≈\n LocalHomeomorph.trans (LocalHomeomorph.symm f) (ofSet (↑s) openness₂)",
"tactic": "refine' EqOnSource.trans' (eqOnSource_refl _) _"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\nsets_identity :\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) =\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s\nopenness₂ : IsOpen ↑s\n⊢ ofSet (Opens.localHomeomorphSubtypeCoe s).toLocalEquiv.target\n (_ : IsOpen (Opens.localHomeomorphSubtypeCoe s).toLocalEquiv.target) ≈\n ofSet (↑s) openness₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\nsets_identity :\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) =\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s\nopenness₂ : IsOpen ↑s\n⊢ LocalHomeomorph.trans (LocalHomeomorph.symm (Opens.localHomeomorphSubtypeCoe s)) (Opens.localHomeomorphSubtypeCoe s) ≈\n ofSet (↑s) openness₂",
"tactic": "refine' Setoid.trans (trans_symm_self s.localHomeomorphSubtypeCoe) _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\nsets_identity :\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) =\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s\nopenness₂ : IsOpen ↑s\n⊢ ofSet (Opens.localHomeomorphSubtypeCoe s).toLocalEquiv.target\n (_ : IsOpen (Opens.localHomeomorphSubtypeCoe s).toLocalEquiv.target) ≈\n ofSet (↑s) openness₂",
"tactic": "simp only [mfld_simps, Setoid.refl]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.116202\nδ : Type ?u.116205\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ne : LocalHomeomorph α β\ns : Opens α\ninst✝ : Nonempty { x // x ∈ s }\nf f' : LocalHomeomorph α β\nopenness₁ : IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s)\n⊢ (LocalHomeomorph.symm f).toLocalEquiv.source ∩ (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s) =\n (LocalHomeomorph.symm f).toLocalEquiv.source ∩ ↑(LocalHomeomorph.symm f) ⁻¹' ↑s",
"tactic": "mfld_set_tac"
}
] |
[
1399,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1384,
1
] |
Mathlib/RingTheory/Finiteness.lean
|
Submodule.fg_finset_sup
|
[] |
[
190,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.tendsto_lift'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Sort ?u.28305\nf f₁ f₂ : Filter α\nh h₁ h₂ : Set α → Set β\nm : γ → β\nl : Filter γ\n⊢ Tendsto m l (Filter.lift' f h) ↔ ∀ (s : Set α), s ∈ f → ∀ᶠ (a : γ) in l, m a ∈ h s",
"tactic": "simp only [Filter.lift', tendsto_lift, tendsto_principal, comp]"
}
] |
[
255,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.le_congr
|
[] |
[
802,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.rootSet_C_mul_X_pow
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na✝ b : R\nn✝ : ℕ\ninst✝³ : Field R\np q : R[X]\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nn : ℕ\nhn : n ≠ 0\na : R\nha : a ≠ 0\n⊢ rootSet (↑C a * X ^ n) S = {0}",
"tactic": "rw [C_mul_X_pow_eq_monomial, rootSet_monomial hn ha]"
}
] |
[
376,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Ultrafilter.comap_pure
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.20593\nf g : Ultrafilter α\ns t : Set α\np q : α → Prop\nm : α → β\na : α\ninj : Injective m\nlarge : range m ∈ pure (m a)\n⊢ 𝓟 (m ⁻¹' {m a}) = ↑(pure a)",
"tactic": "rw [coe_pure, ← principal_singleton, ← image_singleton, preimage_image_eq _ inj]"
}
] |
[
306,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.Memℒp.ae_eq
|
[] |
[
532,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
531,
1
] |
Mathlib/Computability/Ackermann.lean
|
max_ack_right
|
[] |
[
174,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.pow_multiset_sum_mem_span_pow
|
[
{
"state_after": "case empty\nα : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\n⊢ Multiset.sum 0 ^ (↑Multiset.card 0 * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) 0))\n\ncase cons\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n⊢ Multiset.sum (a ::ₘ s) ^ (↑Multiset.card (a ::ₘ s) * n + 1) ∈\n span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) (a ::ₘ s)))",
"state_before": "α : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nI : Ideal α\ns : Multiset α\nn : ℕ\n⊢ Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))",
"tactic": "induction' s using Multiset.induction_on with a s hs"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n⊢ ∑ m in Finset.range ((↑Multiset.card s + 1) * n + 1 + 1),\n a ^ m * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - m) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) m) ∈\n span (insert (a ^ (n + 1)) ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)))",
"state_before": "case cons\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n⊢ Multiset.sum (a ::ₘ s) ^ (↑Multiset.card (a ::ₘ s) * n + 1) ∈\n span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) (a ::ₘ s)))",
"tactic": "simp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons,\n Multiset.card_cons, add_pow]"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n⊢ ∀ (c : ℕ),\n c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1) →\n a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) ∈\n span (insert (a ^ (n + 1)) ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)))",
"state_before": "case cons\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n⊢ ∑ m in Finset.range ((↑Multiset.card s + 1) * n + 1 + 1),\n a ^ m * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - m) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) m) ∈\n span (insert (a ^ (n + 1)) ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)))",
"tactic": "refine' Submodule.sum_mem _ _"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) ∈\n span (insert (a ^ (n + 1)) ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)))",
"state_before": "case cons\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n⊢ ∀ (c : ℕ),\n c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1) →\n a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) ∈\n span (insert (a ^ (n + 1)) ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)))",
"tactic": "intro c _hc"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\n⊢ ∃ a_1 z,\n z ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) ∧\n a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n a_1 * a ^ (n + 1) + z",
"state_before": "case cons\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) ∈\n span (insert (a ^ (n + 1)) ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)))",
"tactic": "rw [mem_span_insert]"
},
{
"state_after": "case pos\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : n + 1 ≤ c\n⊢ ∃ a_1 z,\n z ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) ∧\n a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n a_1 * a ^ (n + 1) + z\n\ncase neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : ¬n + 1 ≤ c\n⊢ ∃ a_1 z,\n z ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) ∧\n a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n a_1 * a ^ (n + 1) + z",
"state_before": "case cons\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\n⊢ ∃ a_1 z,\n z ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) ∧\n a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n a_1 * a ^ (n + 1) + z",
"tactic": "by_cases h : n + 1 ≤ c"
},
{
"state_after": "no goals",
"state_before": "case empty\nα : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\n⊢ Multiset.sum 0 ^ (↑Multiset.card 0 * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) 0))",
"tactic": "simp"
},
{
"state_after": "case pos\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : n + 1 ≤ c\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n a ^ (c - (n + 1)) * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) *\n ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) *\n a ^ (n + 1) +\n 0",
"state_before": "case pos\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : n + 1 ≤ c\n⊢ ∃ a_1 z,\n z ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) ∧\n a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n a_1 * a ^ (n + 1) + z",
"tactic": "refine' ⟨a ^ (c - (n + 1)) * s.sum ^ ((Multiset.card s + 1) * n + 1 - c) *\n ((Multiset.card s + 1) * n + 1).choose c, 0, Submodule.zero_mem _, _⟩"
},
{
"state_after": "case pos\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : n + 1 ≤ c\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n a ^ (n + 1) *\n (a ^ (c - (n + 1)) * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) *\n ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c)) +\n 0",
"state_before": "case pos\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : n + 1 ≤ c\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n a ^ (c - (n + 1)) * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) *\n ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) *\n a ^ (n + 1) +\n 0",
"tactic": "rw [mul_comm _ (a ^ (n + 1))]"
},
{
"state_after": "case pos\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : n + 1 ≤ c\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n a ^ (n + 1) * a ^ (c - (n + 1)) * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) *\n ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) +\n 0",
"state_before": "case pos\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : n + 1 ≤ c\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n a ^ (n + 1) *\n (a ^ (c - (n + 1)) * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) *\n ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c)) +\n 0",
"tactic": "simp_rw [← mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : n + 1 ≤ c\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n a ^ (n + 1) * a ^ (c - (n + 1)) * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) *\n ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) +\n 0",
"tactic": "rw [← pow_add, add_zero, add_tsub_cancel_of_le h]"
},
{
"state_after": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : ¬n + 1 ≤ c\n⊢ ∃ z,\n z ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) ∧\n a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n 0 * a ^ (n + 1) + z",
"state_before": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : ¬n + 1 ≤ c\n⊢ ∃ a_1 z,\n z ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) ∧\n a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n a_1 * a ^ (n + 1) + z",
"tactic": "use 0"
},
{
"state_after": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : ¬n + 1 ≤ c\n⊢ ∃ z,\n z ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) ∧\n a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n z",
"state_before": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : ¬n + 1 ≤ c\n⊢ ∃ z,\n z ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) ∧\n a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n 0 * a ^ (n + 1) + z",
"tactic": "simp_rw [zero_mul, zero_add]"
},
{
"state_after": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : ¬n + 1 ≤ c\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) ∈\n span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))",
"state_before": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : ¬n + 1 ≤ c\n⊢ ∃ z,\n z ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) ∧\n a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) =\n z",
"tactic": "refine' ⟨_, _, rfl⟩"
},
{
"state_after": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : c ≤ n\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) ∈\n span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))",
"state_before": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : ¬n + 1 ≤ c\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) ∈\n span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))",
"tactic": "replace h : c ≤ n := Nat.lt_succ_iff.mp (not_le.mp h)"
},
{
"state_after": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : c ≤ n\nthis : (↑Multiset.card s + 1) * n + 1 - c = ↑Multiset.card s * n + 1 + (n - c)\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) ∈\n span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))",
"state_before": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : c ≤ n\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) ∈\n span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))",
"tactic": "have : (Multiset.card s + 1) * n + 1 - c = Multiset.card s * n + 1 + (n - c) := by\n rw [add_mul, one_mul, add_assoc, add_comm n 1, ← add_assoc, add_tsub_assoc_of_le h]"
},
{
"state_after": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : c ≤ n\nthis : (↑Multiset.card s + 1) * n + 1 - c = ↑Multiset.card s * n + 1 + (n - c)\n⊢ a ^ c * (Multiset.sum s ^ (↑Multiset.card s * n + 1) * Multiset.sum s ^ (n - c)) *\n ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) ∈\n span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))",
"state_before": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : c ≤ n\nthis : (↑Multiset.card s + 1) * n + 1 - c = ↑Multiset.card s * n + 1 + (n - c)\n⊢ a ^ c * Multiset.sum s ^ ((↑Multiset.card s + 1) * n + 1 - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) ∈\n span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))",
"tactic": "rw [this, pow_add]"
},
{
"state_after": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : c ≤ n\nthis : (↑Multiset.card s + 1) * n + 1 - c = ↑Multiset.card s * n + 1 + (n - c)\n⊢ a ^ c * Multiset.sum s ^ (n - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) *\n Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈\n span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))",
"state_before": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : c ≤ n\nthis : (↑Multiset.card s + 1) * n + 1 - c = ↑Multiset.card s * n + 1 + (n - c)\n⊢ a ^ c * (Multiset.sum s ^ (↑Multiset.card s * n + 1) * Multiset.sum s ^ (n - c)) *\n ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) ∈\n span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))",
"tactic": "simp_rw [mul_assoc, mul_comm (s.sum ^ (Multiset.card s * n + 1)), ← mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : c ≤ n\nthis : (↑Multiset.card s + 1) * n + 1 - c = ↑Multiset.card s * n + 1 + (n - c)\n⊢ a ^ c * Multiset.sum s ^ (n - c) * ↑(Nat.choose ((↑Multiset.card s + 1) * n + 1) c) *\n Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈\n span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))",
"tactic": "exact mul_mem_left _ _ hs"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\na✝ b : α\ninst✝ : CommSemiring α\nI : Ideal α\nn : ℕ\na : α\ns : Multiset α\nhs : Multiset.sum s ^ (↑Multiset.card s * n + 1) ∈ span ↑(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : ℕ\n_hc : c ∈ Finset.range ((↑Multiset.card s + 1) * n + 1 + 1)\nh : c ≤ n\n⊢ (↑Multiset.card s + 1) * n + 1 - c = ↑Multiset.card s * n + 1 + (n - c)",
"tactic": "rw [add_mul, one_mul, add_assoc, add_comm n 1, ← add_assoc, add_tsub_assoc_of_le h]"
}
] |
[
611,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
588,
1
] |
Mathlib/MeasureTheory/Measure/Portmanteau.lean
|
MeasureTheory.tendsto_lintegral_thickenedIndicator_of_isClosed
|
[
{
"state_after": "Ω✝ : Type ?u.24613\ninst✝⁴ : MeasurableSpace Ω✝\nΩ : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nF_closed : IsClosed F\nδs : ℕ → ℝ\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\n⊢ Tendsto (fun n => ↑(thickenedIndicator (_ : 0 < δs n) F)) atTop (𝓝 (indicator F fun x => 1))",
"state_before": "Ω✝ : Type ?u.24613\ninst✝⁴ : MeasurableSpace Ω✝\nΩ : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nF_closed : IsClosed F\nδs : ℕ → ℝ\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\n⊢ Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(thickenedIndicator (_ : 0 < δs n) F) ω) ∂μ) atTop (𝓝 (↑↑μ F))",
"tactic": "apply measure_of_cont_bdd_of_tendsto_indicator μ F_closed.measurableSet\n (fun n => thickenedIndicator (δs_pos n) F) fun n ω => thickenedIndicator_le_one (δs_pos n) F ω"
},
{
"state_after": "Ω✝ : Type ?u.24613\ninst✝⁴ : MeasurableSpace Ω✝\nΩ : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nF_closed : IsClosed F\nδs : ℕ → ℝ\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey : Tendsto (fun n => ↑(thickenedIndicator (_ : 0 < δs n) F)) atTop (𝓝 (indicator (closure F) fun x => 1))\n⊢ Tendsto (fun n => ↑(thickenedIndicator (_ : 0 < δs n) F)) atTop (𝓝 (indicator F fun x => 1))",
"state_before": "Ω✝ : Type ?u.24613\ninst✝⁴ : MeasurableSpace Ω✝\nΩ : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nF_closed : IsClosed F\nδs : ℕ → ℝ\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\n⊢ Tendsto (fun n => ↑(thickenedIndicator (_ : 0 < δs n) F)) atTop (𝓝 (indicator F fun x => 1))",
"tactic": "have key := thickenedIndicator_tendsto_indicator_closure δs_pos δs_lim F"
},
{
"state_after": "no goals",
"state_before": "Ω✝ : Type ?u.24613\ninst✝⁴ : MeasurableSpace Ω✝\nΩ : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nF_closed : IsClosed F\nδs : ℕ → ℝ\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nkey : Tendsto (fun n => ↑(thickenedIndicator (_ : 0 < δs n) F)) atTop (𝓝 (indicator (closure F) fun x => 1))\n⊢ Tendsto (fun n => ↑(thickenedIndicator (_ : 0 < δs n) F)) atTop (𝓝 (indicator F fun x => 1))",
"tactic": "rwa [F_closed.closure_eq] at key"
}
] |
[
333,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
|
AffineEquiv.coe_toAffineMap
|
[] |
[
124,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/RingTheory/Subsemiring/Pointwise.lean
|
Subsemiring.mem_smul_pointwise_iff_exists
|
[] |
[
77,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.ite_inter_compl_self
|
[
{
"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 t s s' : Set α\n⊢ Set.ite t s s' ∩ tᶜ = s' ∩ tᶜ",
"tactic": "rw [← ite_compl, ite_inter_self]"
}
] |
[
2257,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2256,
1
] |
Mathlib/CategoryTheory/Limits/IsLimit.lean
|
CategoryTheory.Limits.IsColimit.coconePointUniqueUpToIso_inv_desc
|
[
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nr s t : Cocone F\nP : IsColimit s\nQ : IsColimit t\n⊢ ∀ (j : J), t.ι.app j ≫ (coconePointUniqueUpToIso P Q).inv ≫ desc P r = r.ι.app j",
"tactic": "simp"
}
] |
[
683,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
681,
1
] |
Mathlib/Combinatorics/Quiver/Cast.lean
|
Quiver.Path.cast_heq
|
[
{
"state_after": "U : Type u_1\ninst✝ : Quiver U\nu v u' v' : U\nhu : u = u'\nhv : v = v'\np : Path u v\n⊢ HEq (_root_.cast (_ : Path u v = Path u' v') p) p",
"state_before": "U : Type u_1\ninst✝ : Quiver U\nu v u' v' : U\nhu : u = u'\nhv : v = v'\np : Path u v\n⊢ HEq (cast hu hv p) p",
"tactic": "rw [Path.cast_eq_cast]"
},
{
"state_after": "no goals",
"state_before": "U : Type u_1\ninst✝ : Quiver U\nu v u' v' : U\nhu : u = u'\nhv : v = v'\np : Path u v\n⊢ HEq (_root_.cast (_ : Path u v = Path u' v') p) p",
"tactic": "exact _root_.cast_heq _ _"
}
] |
[
118,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Analysis/NormedSpace/LpEquiv.lean
|
Memℓp.all
|
[
{
"state_after": "case inl\nα : Type u_1\nE : α → Type u_2\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : Finite α\nf : (i : α) → E i\n⊢ Memℓp f 0\n\ncase inr.inl\nα : Type u_1\nE : α → Type u_2\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : Finite α\nf : (i : α) → E i\n⊢ Memℓp f ⊤\n\ncase inr.inr\nα : Type u_1\nE : α → Type u_2\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\ninst✝ : Finite α\nf : (i : α) → E i\n_h : 0 < ENNReal.toReal p\n⊢ Memℓp f p",
"state_before": "α : Type u_1\nE : α → Type u_2\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\ninst✝ : Finite α\nf : (i : α) → E i\n⊢ Memℓp f p",
"tactic": "rcases p.trichotomy with (rfl | rfl | _h)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nE : α → Type u_2\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : Finite α\nf : (i : α) → E i\n⊢ Memℓp f 0",
"tactic": "exact memℓp_zero_iff.mpr { i : α | f i ≠ 0 }.toFinite"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\nE : α → Type u_2\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : Finite α\nf : (i : α) → E i\n⊢ Memℓp f ⊤",
"tactic": "exact memℓp_infty_iff.mpr (Set.Finite.bddAbove (Set.range fun i : α => ‖f i‖).toFinite)"
},
{
"state_after": "case inr.inr.intro\nα : Type u_1\nE : α → Type u_2\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\ninst✝ : Finite α\nf : (i : α) → E i\n_h : 0 < ENNReal.toReal p\nval✝ : Fintype α\n⊢ Memℓp f p",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\ninst✝ : Finite α\nf : (i : α) → E i\n_h : 0 < ENNReal.toReal p\n⊢ Memℓp f p",
"tactic": "cases nonempty_fintype α"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.intro\nα : Type u_1\nE : α → Type u_2\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\ninst✝ : Finite α\nf : (i : α) → E i\n_h : 0 < ENNReal.toReal p\nval✝ : Fintype α\n⊢ Memℓp f p",
"tactic": "exact memℓp_gen ⟨Finset.univ.sum _, hasSum_fintype _⟩"
}
] |
[
57,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Data/Rat/Floor.lean
|
Rat.ceil_cast
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : ℚ\n⊢ ⌈↑x⌉ = ⌈x⌉",
"tactic": "rw [← neg_inj, ← floor_neg, ← floor_neg, ← Rat.cast_neg, Rat.floor_cast]"
}
] |
[
79,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/SetTheory/Cardinal/Continuum.lean
|
Cardinal.continuum_lt_lift
|
[
{
"state_after": "no goals",
"state_before": "c : Cardinal\n⊢ 𝔠 < lift c ↔ 𝔠 < c",
"tactic": "rw [← lift_continuum.{u,v}, lift_lt]"
}
] |
[
64,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Algebra/Field/Basic.lean
|
same_add_div
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1565\nK : Type ?u.1568\ninst✝ : DivisionSemiring α\na b c d : α\nh : b ≠ 0\n⊢ (b + a) / b = 1 + a / b",
"tactic": "rw [← div_self h, add_div]"
}
] |
[
40,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.liminf_le_liminf
|
[] |
[
526,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffWithinAt.fderivWithin
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1064062\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : UniqueDiffOn 𝕜 t\nhmn : m + 1 ≤ n\nhx₀ : x₀ ∈ s\nhst : s ⊆ g ⁻¹' t\n⊢ ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1064062\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : UniqueDiffOn 𝕜 t\nhmn : m + 1 ≤ n\nhx₀ : x₀ ∈ s\nhst : s ⊆ g ⁻¹' t\n⊢ ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"tactic": "rw [← insert_eq_self.mpr hx₀] at hf"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1064062\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : UniqueDiffOn 𝕜 t\nhmn : m + 1 ≤ n\nhx₀ : x₀ ∈ s\nhst : s ⊆ g ⁻¹' t\n⊢ ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1064062\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : UniqueDiffOn 𝕜 t\nhmn : m + 1 ≤ n\nhx₀ : x₀ ∈ s\nhst : s ⊆ g ⁻¹' t\n⊢ ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"tactic": "refine' hf.fderivWithin' hg _ hmn hst"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1064062\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : UniqueDiffOn 𝕜 t\nhmn : m + 1 ≤ n\nhx₀ : x₀ ∈ s\nhst : s ⊆ g ⁻¹' t\n⊢ ∀ᶠ (x : E) in 𝓝[s] x₀, UniqueDiffWithinAt 𝕜 t (g x)",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1064062\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : UniqueDiffOn 𝕜 t\nhmn : m + 1 ≤ n\nhx₀ : x₀ ∈ s\nhst : s ⊆ g ⁻¹' t\n⊢ ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)",
"tactic": "rw [insert_eq_self.mpr hx₀]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1064062\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : UniqueDiffOn 𝕜 t\nhmn : m + 1 ≤ n\nhx₀ : x₀ ∈ s\nhst : s ⊆ g ⁻¹' t\n⊢ ∀ᶠ (x : E) in 𝓝[s] x₀, UniqueDiffWithinAt 𝕜 t (g x)",
"tactic": "exact eventually_of_mem self_mem_nhdsWithin fun x hx => ht _ (hst hx)"
}
] |
[
1018,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1011,
11
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.mem_sup_left
|
[] |
[
408,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/Probability/ConditionalProbability.lean
|
ProbabilityTheory.cond_cond_eq_cond_inter
|
[] |
[
141,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Analysis/LocallyConvex/Basic.lean
|
balanced_iInter₂
|
[] |
[
205,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Algebra/Homology/Augment.lean
|
ChainComplex.augment_d_one_zero
|
[] |
[
90,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_eq_finset_prod_of_mulSupport_subset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.180289\nι : Type ?u.180292\nG : Type ?u.180295\nM : Type u_2\nN : Type ?u.180301\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\ns : Finset α\nh : mulSupport f ⊆ ↑s\n⊢ Finite.toFinset (_ : Set.Finite (mulSupport f)) ⊆ s",
"tactic": "simpa [← Finset.coe_subset, Set.coe_toFinset]"
}
] |
[
408,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/Data/Set/Sups.lean
|
Set.sups_sups_sups_comm
|
[] |
[
211,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
|
MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone
|
[] |
[
527,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
525,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.normSq_pos
|
[
{
"state_after": "K : Type u_1\nE : Type ?u.4450828\ninst✝ : IsROrC K\nz : K\n⊢ 0 ≤ ↑normSq z ∧ ¬↑normSq z = 0 ↔ z ≠ 0",
"state_before": "K : Type u_1\nE : Type ?u.4450828\ninst✝ : IsROrC K\nz : K\n⊢ 0 < ↑normSq z ↔ z ≠ 0",
"tactic": "rw [lt_iff_le_and_ne, Ne, eq_comm]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.4450828\ninst✝ : IsROrC K\nz : K\n⊢ 0 ≤ ↑normSq z ∧ ¬↑normSq z = 0 ↔ z ≠ 0",
"tactic": "simp [normSq_nonneg]"
}
] |
[
494,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.uniformEmbedding_iff'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.553589\nι : Type ?u.553592\ninst✝² : PseudoMetricSpace α\nγ : Type w\ninst✝¹ : MetricSpace γ\nx : γ\ns : Set γ\ninst✝ : MetricSpace β\nf : γ → β\n⊢ UniformEmbedding f ↔\n (∀ (ε : ℝ), ε > 0 → ∃ δ, δ > 0 ∧ ∀ {a b : γ}, dist a b < δ → dist (f a) (f b) < ε) ∧\n ∀ (δ : ℝ), δ > 0 → ∃ ε, ε > 0 ∧ ∀ {a b : γ}, dist (f a) (f b) < ε → dist a b < δ",
"tactic": "rw [uniformEmbedding_iff_uniformInducing, uniformInducing_iff, uniformContinuous_iff]"
}
] |
[
2927,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2923,
1
] |
Mathlib/CategoryTheory/FintypeCat.lean
|
FintypeCat.Skeleton.incl_mk_nat_card
|
[
{
"state_after": "case h.e'_2\nn : ℕ\n⊢ Fintype.card ↑(incl.obj (mk n)) = Finset.card Finset.univ",
"state_before": "n : ℕ\n⊢ Fintype.card ↑(incl.obj (mk n)) = n",
"tactic": "convert Finset.card_fin n"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nn : ℕ\n⊢ Fintype.card ↑(incl.obj (mk n)) = Finset.card Finset.univ",
"tactic": "apply Fintype.ofEquiv_card"
}
] |
[
207,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
src/lean/Init/Classical.lean
|
Classical.axiomOfChoice
|
[] |
[
108,
34
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
107,
1
] |
Mathlib/Tactic/Ring/Basic.lean
|
Mathlib.Tactic.Ring.cast_pos
|
[
{
"state_after": "no goals",
"state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na : R\nn : ℕ\ne : a = ↑n\n⊢ a = Nat.rawCast n + 0",
"tactic": "simp [e]"
}
] |
[
832,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
831,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.untop_coe
|
[] |
[
782,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
781,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.mem_direction_iff_eq_vsub_left
|
[
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\nv : V\n⊢ v ∈ (fun x x_1 => x -ᵥ x_1) p '' ↑s ↔ ∃ p2, p2 ∈ s ∧ v = p -ᵥ p2",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\nv : V\n⊢ v ∈ direction s ↔ ∃ p2, p2 ∈ s ∧ v = p -ᵥ p2",
"tactic": "rw [← SetLike.mem_coe, coe_direction_eq_vsub_set_left hp]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\nv : V\n⊢ v ∈ (fun x x_1 => x -ᵥ x_1) p '' ↑s ↔ ∃ p2, p2 ∈ s ∧ v = p -ᵥ p2",
"tactic": "exact ⟨fun ⟨p2, hp2, hv⟩ => ⟨p2, hp2, hv.symm⟩, fun ⟨p2, hp2, hv⟩ => ⟨p2, hp2, hv.symm⟩⟩"
}
] |
[
331,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.infinitePos_omega
|
[] |
[
807,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
806,
1
] |
Mathlib/CategoryTheory/Category/Basic.lean
|
CategoryTheory.mono_of_mono
|
[
{
"state_after": "case right_cancellation\nC : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝ X Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝ : Mono (f ≫ g)\n⊢ ∀ {Z : C} (g h : Z ⟶ X), g ≫ f = h ≫ f → g = h",
"state_before": "C : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝ X Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝ : Mono (f ≫ g)\n⊢ Mono f",
"tactic": "constructor"
},
{
"state_after": "case right_cancellation\nC : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝¹ X Y Z✝ : C\nf : X ⟶ Y\ng : Y ⟶ Z✝\ninst✝ : Mono (f ≫ g)\nZ : C\na b : Z ⟶ X\nw : a ≫ f = b ≫ f\n⊢ a = b",
"state_before": "case right_cancellation\nC : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝ X Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝ : Mono (f ≫ g)\n⊢ ∀ {Z : C} (g h : Z ⟶ X), g ≫ f = h ≫ f → g = h",
"tactic": "intro Z a b w"
},
{
"state_after": "case right_cancellation\nC : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝¹ X Y Z✝ : C\nf : X ⟶ Y\ng : Y ⟶ Z✝\ninst✝ : Mono (f ≫ g)\nZ : C\na b : Z ⟶ X\nw : (fun k => k ≫ g) (a ≫ f) = (fun k => k ≫ g) (b ≫ f)\n⊢ a = b",
"state_before": "case right_cancellation\nC : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝¹ X Y Z✝ : C\nf : X ⟶ Y\ng : Y ⟶ Z✝\ninst✝ : Mono (f ≫ g)\nZ : C\na b : Z ⟶ X\nw : a ≫ f = b ≫ f\n⊢ a = b",
"tactic": "replace w := congr_arg (fun k => k ≫ g) w"
},
{
"state_after": "case right_cancellation\nC : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝¹ X Y Z✝ : C\nf : X ⟶ Y\ng : Y ⟶ Z✝\ninst✝ : Mono (f ≫ g)\nZ : C\na b : Z ⟶ X\nw : (a ≫ f) ≫ g = (b ≫ f) ≫ g\n⊢ a = b",
"state_before": "case right_cancellation\nC : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝¹ X Y Z✝ : C\nf : X ⟶ Y\ng : Y ⟶ Z✝\ninst✝ : Mono (f ≫ g)\nZ : C\na b : Z ⟶ X\nw : (fun k => k ≫ g) (a ≫ f) = (fun k => k ≫ g) (b ≫ f)\n⊢ a = b",
"tactic": "dsimp at w"
},
{
"state_after": "case right_cancellation\nC : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝¹ X Y Z✝ : C\nf : X ⟶ Y\ng : Y ⟶ Z✝\ninst✝ : Mono (f ≫ g)\nZ : C\na b : Z ⟶ X\nw : a ≫ f ≫ g = b ≫ f ≫ g\n⊢ a = b",
"state_before": "case right_cancellation\nC : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝¹ X Y Z✝ : C\nf : X ⟶ Y\ng : Y ⟶ Z✝\ninst✝ : Mono (f ≫ g)\nZ : C\na b : Z ⟶ X\nw : (a ≫ f) ≫ g = (b ≫ f) ≫ g\n⊢ a = b",
"tactic": "rw [Category.assoc, Category.assoc] at w"
},
{
"state_after": "no goals",
"state_before": "case right_cancellation\nC : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝¹ X Y Z✝ : C\nf : X ⟶ Y\ng : Y ⟶ Z✝\ninst✝ : Mono (f ≫ g)\nZ : C\na b : Z ⟶ X\nw : a ≫ f ≫ g = b ≫ f ≫ g\n⊢ a = b",
"tactic": "exact (cancel_mono _).1 w"
}
] |
[
327,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/Algebra/Symmetrized.lean
|
SymAlg.sym_bijective
|
[] |
[
95,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/AlgebraicTopology/SimplicialObject.lean
|
CategoryTheory.SimplicialObject.δ_comp_σ_of_gt
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : ↑Fin.castSucc j < i\n⊢ X.map (SimplexCategory.σ (↑Fin.castSucc j)).op ≫ X.map (SimplexCategory.δ (Fin.succ i)).op =\n X.map (SimplexCategory.δ i).op ≫ X.map (SimplexCategory.σ j).op",
"state_before": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : ↑Fin.castSucc j < i\n⊢ σ X (↑Fin.castSucc j) ≫ δ X (Fin.succ i) = δ X i ≫ σ X j",
"tactic": "dsimp [δ, σ]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : ↑Fin.castSucc j < i\n⊢ X.map (SimplexCategory.σ (↑Fin.castSucc j)).op ≫ X.map (SimplexCategory.δ (Fin.succ i)).op =\n X.map (SimplexCategory.δ i).op ≫ X.map (SimplexCategory.σ j).op",
"tactic": "simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_σ_of_gt H]"
}
] |
[
188,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.div_le_of_le_mul'
|
[] |
[
1594,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1593,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.Chain.nil
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr : α → α → Prop\n⊢ Chain r Cycle.nil",
"tactic": "trivial"
}
] |
[
935,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
935,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.foldrM_reverse
|
[
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝ : Monad m\nl : List α\nf : α → β → m β\nb : β\n⊢ List.foldlM (fun y x => f x y) b (reverse (reverse l)) = List.foldlM (fun x y => f y x) b l",
"tactic": "simp"
}
] |
[
175,
44
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
173,
9
] |
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
|
StructureGroupoid.LocalInvariantProp.liftPropAt_symm_of_mem_maximalAtlas
|
[
{
"state_after": "H : Type u_1\nM : Type u_2\nH' : Type ?u.56211\nM' : Type ?u.56214\nX : Type ?u.56217\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nhG✝ : LocalInvariantProp G G' P\ninst✝ : HasGroupoid M G\nx : H\nhG : LocalInvariantProp G G Q\nhQ : ∀ (y : H), Q id univ y\nhe : e ∈ maximalAtlas M G\nhx : x ∈ e.target\nh : Q (↑e ∘ ↑(LocalHomeomorph.symm e)) univ x\n⊢ LiftPropAt Q (↑(LocalHomeomorph.symm e)) x\n\ncase h\nH : Type u_1\nM : Type u_2\nH' : Type ?u.56211\nM' : Type ?u.56214\nX : Type ?u.56217\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nhG✝ : LocalInvariantProp G G' P\ninst✝ : HasGroupoid M G\nx : H\nhG : LocalInvariantProp G G Q\nhQ : ∀ (y : H), Q id univ y\nhe : e ∈ maximalAtlas M G\nhx : x ∈ e.target\n⊢ Q (↑e ∘ ↑(LocalHomeomorph.symm e)) univ x",
"state_before": "H : Type u_1\nM : Type u_2\nH' : Type ?u.56211\nM' : Type ?u.56214\nX : Type ?u.56217\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nhG✝ : LocalInvariantProp G G' P\ninst✝ : HasGroupoid M G\nx : H\nhG : LocalInvariantProp G G Q\nhQ : ∀ (y : H), Q id univ y\nhe : e ∈ maximalAtlas M G\nhx : x ∈ e.target\n⊢ LiftPropAt Q (↑(LocalHomeomorph.symm e)) x",
"tactic": "suffices h : Q (e ∘ e.symm) univ x"
},
{
"state_after": "no goals",
"state_before": "case h\nH : Type u_1\nM : Type u_2\nH' : Type ?u.56211\nM' : Type ?u.56214\nX : Type ?u.56217\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nhG✝ : LocalInvariantProp G G' P\ninst✝ : HasGroupoid M G\nx : H\nhG : LocalInvariantProp G G Q\nhQ : ∀ (y : H), Q id univ y\nhe : e ∈ maximalAtlas M G\nhx : x ∈ e.target\n⊢ Q (↑e ∘ ↑(LocalHomeomorph.symm e)) univ x",
"tactic": "exact hG.congr' (e.eventually_right_inverse hx) (hQ x)"
},
{
"state_after": "H : Type u_1\nM : Type u_2\nH' : Type ?u.56211\nM' : Type ?u.56214\nX : Type ?u.56217\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nhG✝ : LocalInvariantProp G G' P\ninst✝ : HasGroupoid M G\nx : H\nhG : LocalInvariantProp G G Q\nhQ : ∀ (y : H), Q id univ y\nhe : e ∈ maximalAtlas M G\nhx : x ∈ e.target\nh : Q (↑e ∘ ↑(LocalHomeomorph.symm e)) univ x\nthis : ↑(LocalHomeomorph.symm e) x ∈ e.source\n⊢ LiftPropAt Q (↑(LocalHomeomorph.symm e)) x",
"state_before": "H : Type u_1\nM : Type u_2\nH' : Type ?u.56211\nM' : Type ?u.56214\nX : Type ?u.56217\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nhG✝ : LocalInvariantProp G G' P\ninst✝ : HasGroupoid M G\nx : H\nhG : LocalInvariantProp G G Q\nhQ : ∀ (y : H), Q id univ y\nhe : e ∈ maximalAtlas M G\nhx : x ∈ e.target\nh : Q (↑e ∘ ↑(LocalHomeomorph.symm e)) univ x\n⊢ LiftPropAt Q (↑(LocalHomeomorph.symm e)) x",
"tactic": "have : e.symm x ∈ e.source := by simp only [hx, mfld_simps]"
},
{
"state_after": "H : Type u_1\nM : Type u_2\nH' : Type ?u.56211\nM' : Type ?u.56214\nX : Type ?u.56217\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nhG✝ : LocalInvariantProp G G' P\ninst✝ : HasGroupoid M G\nx : H\nhG : LocalInvariantProp G G Q\nhQ : ∀ (y : H), Q id univ y\nhe : e ∈ maximalAtlas M G\nhx : x ∈ e.target\nh : Q (↑e ∘ ↑(LocalHomeomorph.symm e)) univ x\nthis : ↑(LocalHomeomorph.symm e) x ∈ e.source\n⊢ ContinuousWithinAt (↑(LocalHomeomorph.symm e)) univ x ∧\n Q (↑e ∘ ↑(LocalHomeomorph.symm e) ∘ ↑(LocalHomeomorph.symm (LocalHomeomorph.refl H)))\n (↑(LocalHomeomorph.symm (LocalHomeomorph.refl H)) ⁻¹' univ) (↑(LocalHomeomorph.refl H) x)",
"state_before": "H : Type u_1\nM : Type u_2\nH' : Type ?u.56211\nM' : Type ?u.56214\nX : Type ?u.56217\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nhG✝ : LocalInvariantProp G G' P\ninst✝ : HasGroupoid M G\nx : H\nhG : LocalInvariantProp G G Q\nhQ : ∀ (y : H), Q id univ y\nhe : e ∈ maximalAtlas M G\nhx : x ∈ e.target\nh : Q (↑e ∘ ↑(LocalHomeomorph.symm e)) univ x\nthis : ↑(LocalHomeomorph.symm e) x ∈ e.source\n⊢ LiftPropAt Q (↑(LocalHomeomorph.symm e)) x",
"tactic": "rw [LiftPropAt, hG.liftPropWithinAt_indep_chart G.id_mem_maximalAtlas (mem_univ _) he this]"
},
{
"state_after": "H : Type u_1\nM : Type u_2\nH' : Type ?u.56211\nM' : Type ?u.56214\nX : Type ?u.56217\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nhG✝ : LocalInvariantProp G G' P\ninst✝ : HasGroupoid M G\nx : H\nhG : LocalInvariantProp G G Q\nhQ : ∀ (y : H), Q id univ y\nhe : e ∈ maximalAtlas M G\nhx : x ∈ e.target\nh : Q (↑e ∘ ↑(LocalHomeomorph.symm e)) univ x\nthis : ↑(LocalHomeomorph.symm e) x ∈ e.source\n⊢ Q (↑e ∘ ↑(LocalHomeomorph.symm e) ∘ ↑(LocalHomeomorph.symm (LocalHomeomorph.refl H)))\n (↑(LocalHomeomorph.symm (LocalHomeomorph.refl H)) ⁻¹' univ) (↑(LocalHomeomorph.refl H) x)",
"state_before": "H : Type u_1\nM : Type u_2\nH' : Type ?u.56211\nM' : Type ?u.56214\nX : Type ?u.56217\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nhG✝ : LocalInvariantProp G G' P\ninst✝ : HasGroupoid M G\nx : H\nhG : LocalInvariantProp G G Q\nhQ : ∀ (y : H), Q id univ y\nhe : e ∈ maximalAtlas M G\nhx : x ∈ e.target\nh : Q (↑e ∘ ↑(LocalHomeomorph.symm e)) univ x\nthis : ↑(LocalHomeomorph.symm e) x ∈ e.source\n⊢ ContinuousWithinAt (↑(LocalHomeomorph.symm e)) univ x ∧\n Q (↑e ∘ ↑(LocalHomeomorph.symm e) ∘ ↑(LocalHomeomorph.symm (LocalHomeomorph.refl H)))\n (↑(LocalHomeomorph.symm (LocalHomeomorph.refl H)) ⁻¹' univ) (↑(LocalHomeomorph.refl H) x)",
"tactic": "refine' ⟨(e.symm.continuousAt hx).continuousWithinAt, _⟩"
},
{
"state_after": "no goals",
"state_before": "H : Type u_1\nM : Type u_2\nH' : Type ?u.56211\nM' : Type ?u.56214\nX : Type ?u.56217\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nhG✝ : LocalInvariantProp G G' P\ninst✝ : HasGroupoid M G\nx : H\nhG : LocalInvariantProp G G Q\nhQ : ∀ (y : H), Q id univ y\nhe : e ∈ maximalAtlas M G\nhx : x ∈ e.target\nh : Q (↑e ∘ ↑(LocalHomeomorph.symm e)) univ x\nthis : ↑(LocalHomeomorph.symm e) x ∈ e.source\n⊢ Q (↑e ∘ ↑(LocalHomeomorph.symm e) ∘ ↑(LocalHomeomorph.symm (LocalHomeomorph.refl H)))\n (↑(LocalHomeomorph.symm (LocalHomeomorph.refl H)) ⁻¹' univ) (↑(LocalHomeomorph.refl H) x)",
"tactic": "simp only [h, mfld_simps]"
},
{
"state_after": "no goals",
"state_before": "H : Type u_1\nM : Type u_2\nH' : Type ?u.56211\nM' : Type ?u.56214\nX : Type ?u.56217\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nhG✝ : LocalInvariantProp G G' P\ninst✝ : HasGroupoid M G\nx : H\nhG : LocalInvariantProp G G Q\nhQ : ∀ (y : H), Q id univ y\nhe : e ∈ maximalAtlas M G\nhx : x ∈ e.target\nh : Q (↑e ∘ ↑(LocalHomeomorph.symm e)) univ x\n⊢ ↑(LocalHomeomorph.symm e) x ∈ e.source",
"tactic": "simp only [hx, mfld_simps]"
}
] |
[
501,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
MeasureTheory.FiniteMeasure.toWeakDualBCNN_apply
|
[] |
[
457,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
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