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/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.bddAbove_iff_small
|
[] |
[
1384,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1382,
1
] |
Mathlib/GroupTheory/GroupAction/Defs.lean
|
IsScalarTower.of_smul_one_mul
|
[
{
"state_after": "no goals",
"state_before": "M✝ : Type ?u.29021\nN✝ : Type ?u.29024\nG : Type ?u.29027\nA : Type ?u.29030\nB : Type ?u.29033\nα : Type ?u.29036\nβ : Type ?u.29039\nγ : Type ?u.29042\nδ : Type ?u.29045\nM : Type u_1\nN : Type u_2\ninst✝¹ : Monoid N\ninst✝ : SMul M N\nh : ∀ (x : M) (y : N), x • 1 * y = x • y\nx : M\ny z : N\n⊢ (x • y) • z = x • y • z",
"tactic": "rw [← h, smul_eq_mul, mul_assoc, h, smul_eq_mul]"
}
] |
[
662,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
660,
1
] |
Mathlib/Algebra/Parity.lean
|
even_neg_two
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.101304\nα : Type u_1\nβ : Type ?u.101310\nR : Type ?u.101313\ninst✝ : Ring α\na b : α\nn : ℕ\n⊢ Even (-2)",
"tactic": "simp only [even_neg, even_two]"
}
] |
[
420,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/Analysis/NormedSpace/MStructure.lean
|
IsLprojection.le_def
|
[] |
[
222,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Set.finite_Iic
|
[] |
[
701,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
700,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
Ring.eq_mul_inverse_iff_mul_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.8485\nM₀ : Type u_1\nG₀ : Type ?u.8491\nM₀' : Type ?u.8494\nG₀' : Type ?u.8497\nF : Type ?u.8500\nF' : Type ?u.8503\ninst✝ : MonoidWithZero M₀\nx y z : M₀\nh : IsUnit z\nh1 : x = y * inverse z\n⊢ x * z = y",
"tactic": "rw [h1, inverse_mul_cancel_right _ _ h]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.8485\nM₀ : Type u_1\nG₀ : Type ?u.8491\nM₀' : Type ?u.8494\nG₀' : Type ?u.8497\nF : Type ?u.8500\nF' : Type ?u.8503\ninst✝ : MonoidWithZero M₀\nx y z : M₀\nh : IsUnit z\nh1 : x * z = y\n⊢ x = y * inverse z",
"tactic": "rw [← h1, mul_inverse_cancel_right _ _ h]"
}
] |
[
139,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.derivWithin_integral_of_tendsto_ae_left
|
[] |
[
968,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
963,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.univ_pi_update_univ
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.153466\ns✝ s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni✝ : ι\ninst✝ : DecidableEq ι\ni : ι\ns : Set (α i)\n⊢ pi univ (update (fun j => univ) i s) = eval i ⁻¹' s",
"tactic": "rw [univ_pi_update i (fun j => (univ : Set (α j))) s fun j t => t, pi_univ, inter_univ, preimage]"
}
] |
[
795,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
793,
1
] |
Mathlib/GroupTheory/Sylow.lean
|
Sylow.card_eq_multiplicity
|
[
{
"state_after": "case intro\nG : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Sylow p G\nn : ℕ\nheq : Fintype.card { x // x ∈ ↑P } = p ^ n\n⊢ Fintype.card { x // x ∈ ↑P } = p ^ ↑(Nat.factorization (Fintype.card G)) p",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Sylow p G\n⊢ Fintype.card { x // x ∈ ↑P } = p ^ ↑(Nat.factorization (Fintype.card G)) p",
"tactic": "obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'"
},
{
"state_after": "case intro\nG : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Sylow p G\nn : ℕ\nheq : Fintype.card { x // x ∈ ↑P } = p ^ n\n⊢ Fintype.card { x // x ∈ ↑P } ∣ p ^ ↑(Nat.factorization (Fintype.card G)) p",
"state_before": "case intro\nG : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Sylow p G\nn : ℕ\nheq : Fintype.card { x // x ∈ ↑P } = p ^ n\n⊢ Fintype.card { x // x ∈ ↑P } = p ^ ↑(Nat.factorization (Fintype.card G)) p",
"tactic": "refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))"
},
{
"state_after": "case intro\nG : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Sylow p G\nn : ℕ\nheq : Fintype.card { x // x ∈ ↑P } = p ^ n\n⊢ Fintype.card { x // x ∈ ↑P } ∣ Fintype.card G",
"state_before": "case intro\nG : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Sylow p G\nn : ℕ\nheq : Fintype.card { x // x ∈ ↑P } = p ^ n\n⊢ Fintype.card { x // x ∈ ↑P } ∣ p ^ ↑(Nat.factorization (Fintype.card G)) p",
"tactic": "rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]"
},
{
"state_after": "no goals",
"state_before": "case intro\nG : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Sylow p G\nn : ℕ\nheq : Fintype.card { x // x ∈ ↑P } = p ^ n\n⊢ Fintype.card { x // x ∈ ↑P } ∣ Fintype.card G",
"tactic": "exact P.1.card_subgroup_dvd_card"
}
] |
[
696,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
691,
1
] |
Mathlib/RingTheory/Congruence.lean
|
RingCon.coe_nat_cast
|
[] |
[
312,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/CategoryTheory/Equivalence.lean
|
CategoryTheory.Equivalence.cancel_unit_right
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\ne : C ≌ D\nX Y : C\nf f' : X ⟶ Y\n⊢ f ≫ (unit e).app Y = f' ≫ (unit e).app Y ↔ f = f'",
"tactic": "simp only [cancel_mono]"
}
] |
[
387,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.disjoint_supported_supported
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u_2\nN : Type ?u.113774\nP : Type ?u.113777\nR : Type u_3\nS : Type ?u.113783\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\ns t : Set α\nh : Disjoint s t\n⊢ supported M R s ⊓ supported M R t = ⊥",
"tactic": "rw [← supported_inter, disjoint_iff_inter_eq_empty.1 h, supported_empty]"
}
] |
[
315,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
|
IsSimpleGroup.prime_card
|
[
{
"state_after": "α : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\n⊢ Nat.Prime (Fintype.card α)",
"state_before": "α : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\n⊢ Nat.Prime (Fintype.card α)",
"tactic": "have h0 : 0 < Fintype.card α := Fintype.card_pos_iff.2 (by infer_instance)"
},
{
"state_after": "case intro\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ Nat.Prime (Fintype.card α)",
"state_before": "α : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\n⊢ Nat.Prime (Fintype.card α)",
"tactic": "obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)"
},
{
"state_after": "case intro\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ 2 ≤ Fintype.card α ∧ ∀ (m : ℕ), m ∣ Fintype.card α → m = 1 ∨ m = Fintype.card α",
"state_before": "case intro\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ Nat.Prime (Fintype.card α)",
"tactic": "rw [Nat.prime_def_lt'']"
},
{
"state_after": "case intro\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\n⊢ n = 1 ∨ n = Fintype.card α",
"state_before": "case intro\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ 2 ≤ Fintype.card α ∧ ∀ (m : ℕ), m ∣ Fintype.card α → m = 1 ∨ m = Fintype.card α",
"tactic": "refine' ⟨Fintype.one_lt_card_iff_nontrivial.2 inferInstance, fun n hn => _⟩"
},
{
"state_after": "case intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\n⊢ Subgroup.zpowers (g ^ n) = ⊤ → n = 1\n\ncase intro.refine'_2\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\n⊢ Subgroup.zpowers (g ^ n) = ⊥ → n = Fintype.card α",
"state_before": "case intro\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\n⊢ n = 1 ∨ n = Fintype.card α",
"tactic": "refine' (IsSimpleOrder.eq_bot_or_eq_top (Subgroup.zpowers (g ^ n))).symm.imp _ _"
},
{
"state_after": "no goals",
"state_before": "α : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\n⊢ Nonempty α",
"tactic": "infer_instance"
},
{
"state_after": "case intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊤\n⊢ n = 1",
"state_before": "case intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\n⊢ Subgroup.zpowers (g ^ n) = ⊤ → n = 1",
"tactic": "intro h"
},
{
"state_after": "case intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊤\nhgo : orderOf (g ^ n) = orderOf g / Nat.gcd (orderOf g) n\n⊢ n = 1",
"state_before": "case intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊤\n⊢ n = 1",
"tactic": "have hgo := orderOf_pow (n := n) g"
},
{
"state_after": "case intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊤\nhgo✝ : orderOf (g ^ n) = Fintype.card α / n\nhgo : Fintype.card α = Fintype.card α * n\n⊢ n = 1\n\ncase intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊤\nhgo : orderOf (g ^ n) = Fintype.card α / n\n⊢ ∀ (x : α), x ∈ Subgroup.zpowers (g ^ n)",
"state_before": "case intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊤\nhgo : orderOf (g ^ n) = orderOf g / Nat.gcd (orderOf g) n\n⊢ n = 1",
"tactic": "rw [orderOf_eq_card_of_forall_mem_zpowers hg, Nat.gcd_eq_right_iff_dvd.1 hn,\n orderOf_eq_card_of_forall_mem_zpowers, eq_comm,\n Nat.div_eq_iff_eq_mul_left (Nat.pos_of_dvd_of_pos hn h0) hn] at hgo"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊤\nhgo✝ : orderOf (g ^ n) = Fintype.card α / n\nhgo : Fintype.card α = Fintype.card α * n\n⊢ n = 1",
"tactic": "exact (mul_left_cancel₀ (ne_of_gt h0) ((mul_one (Fintype.card α)).trans hgo)).symm"
},
{
"state_after": "case intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊤\nhgo : orderOf (g ^ n) = Fintype.card α / n\nx : α\n⊢ x ∈ Subgroup.zpowers (g ^ n)",
"state_before": "case intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊤\nhgo : orderOf (g ^ n) = Fintype.card α / n\n⊢ ∀ (x : α), x ∈ Subgroup.zpowers (g ^ n)",
"tactic": "intro x"
},
{
"state_after": "case intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊤\nhgo : orderOf (g ^ n) = Fintype.card α / n\nx : α\n⊢ x ∈ ⊤",
"state_before": "case intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊤\nhgo : orderOf (g ^ n) = Fintype.card α / n\nx : α\n⊢ x ∈ Subgroup.zpowers (g ^ n)",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_1\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊤\nhgo : orderOf (g ^ n) = Fintype.card α / n\nx : α\n⊢ x ∈ ⊤",
"tactic": "exact Subgroup.mem_top _"
},
{
"state_after": "case intro.refine'_2\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊥\n⊢ n = Fintype.card α",
"state_before": "case intro.refine'_2\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\n⊢ Subgroup.zpowers (g ^ n) = ⊥ → n = Fintype.card α",
"tactic": "intro h"
},
{
"state_after": "case intro.refine'_2\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊥\n⊢ Fintype.card α ≤ n",
"state_before": "case intro.refine'_2\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊥\n⊢ n = Fintype.card α",
"tactic": "apply le_antisymm (Nat.le_of_dvd h0 hn)"
},
{
"state_after": "case intro.refine'_2\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊥\n⊢ orderOf g ≤ n",
"state_before": "case intro.refine'_2\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊥\n⊢ Fintype.card α ≤ n",
"tactic": "rw [← orderOf_eq_card_of_forall_mem_zpowers hg]"
},
{
"state_after": "case intro.refine'_2\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊥\n⊢ g ^ n = 1",
"state_before": "case intro.refine'_2\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊥\n⊢ orderOf g ≤ n",
"tactic": "apply orderOf_le_of_pow_eq_one (Nat.pos_of_dvd_of_pos hn h0)"
},
{
"state_after": "case intro.refine'_2\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊥\n⊢ g ^ n ∈ Subgroup.zpowers (g ^ n)",
"state_before": "case intro.refine'_2\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊥\n⊢ g ^ n = 1",
"tactic": "rw [← Subgroup.mem_bot, ← h]"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_2\nα : Type u\na : α\ninst✝² : CommGroup α\ninst✝¹ : IsSimpleGroup α\ninst✝ : Fintype α\nh0 : 0 < Fintype.card α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : n ∣ Fintype.card α\nh : Subgroup.zpowers (g ^ n) = ⊥\n⊢ g ^ n ∈ Subgroup.zpowers (g ^ n)",
"tactic": "exact Subgroup.mem_zpowers _"
}
] |
[
539,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
519,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.isPell_one
|
[
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\n⊢ az a * az a - ↑(Pell.d a1) * 1 * 1 = 1",
"tactic": "simp [dz_val]"
}
] |
[
227,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
hasDerivWithinAt_Ioi_iff_Ici
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\ninst✝ : PartialOrder 𝕜\n⊢ HasDerivWithinAt f f' (Ioi x) x ↔ HasDerivWithinAt f f' (Ici x) x",
"tactic": "rw [← Ici_diff_left, hasDerivWithinAt_diff_singleton]"
}
] |
[
329,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean
|
GroupSeminorm.apply_one
|
[] |
[
637,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
636,
1
] |
Mathlib/Data/Rat/Lemmas.lean
|
Rat.exists
|
[
{
"state_after": "case h.e'_1\np : ℚ → Prop\nx✝ : ∃ r, p r\nr : ℚ\nhr : p r\n⊢ ↑r.num / ↑↑r.den = r",
"state_before": "p : ℚ → Prop\nx✝ : ∃ r, p r\nr : ℚ\nhr : p r\n⊢ p (↑r.num / ↑↑r.den)",
"tactic": "convert hr"
},
{
"state_after": "no goals",
"state_before": "case h.e'_1\np : ℚ → Prop\nx✝ : ∃ r, p r\nr : ℚ\nhr : p r\n⊢ ↑r.num / ↑↑r.den = r",
"tactic": "convert num_div_den r"
}
] |
[
314,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
11
] |
Mathlib/Topology/Semicontinuous.lean
|
upperSemicontinuousOn_iInf
|
[] |
[
1043,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1041,
1
] |
Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean
|
Complex.measurable_im
|
[] |
[
83,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Order/Concept.lean
|
Concept.sInf_snd
|
[] |
[
369,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
LinearMap.isAlt_iff_eq_neg_flip
|
[
{
"state_after": "case mp\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : IsAlt B\n⊢ B = -flip B\n\ncase mpr\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : B = -flip B\n⊢ IsAlt B",
"state_before": "R : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\n⊢ IsAlt B ↔ B = -flip B",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mpr\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : B = -flip B\nx : M₁\n⊢ ↑(↑B x) x = 0",
"state_before": "case mpr\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : B = -flip B\n⊢ IsAlt B",
"tactic": "intro x"
},
{
"state_after": "case mpr\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : B = -flip B\nx : M₁\nh' : ↑(↑B x) x = ↑(↑(-flip B) x) x := congr_fun₂ h x x\n⊢ ↑(↑B x) x = 0",
"state_before": "case mpr\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : B = -flip B\nx : M₁\n⊢ ↑(↑B x) x = 0",
"tactic": "let h' := congr_fun₂ h x x"
},
{
"state_after": "case mpr\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : B = -flip B\nx : M₁\nh' : ↑(↑B x) x + ↑(↑B x) x = 0\n⊢ ↑(↑B x) x = 0",
"state_before": "case mpr\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : B = -flip B\nx : M₁\nh' : ↑(↑B x) x = ↑(↑(-flip B) x) x := congr_fun₂ h x x\n⊢ ↑(↑B x) x = 0",
"tactic": "simp only [neg_apply, flip_apply, ← add_eq_zero_iff_eq_neg] at h'"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : B = -flip B\nx : M₁\nh' : ↑(↑B x) x + ↑(↑B x) x = 0\n⊢ ↑(↑B x) x = 0",
"tactic": "exact add_self_eq_zero.mp h'"
},
{
"state_after": "case mp.h.h\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : IsAlt B\nx✝¹ x✝ : M₁\n⊢ ↑(↑B x✝¹) x✝ = ↑(↑(-flip B) x✝¹) x✝",
"state_before": "case mp\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : IsAlt B\n⊢ B = -flip B",
"tactic": "ext"
},
{
"state_after": "case mp.h.h\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : IsAlt B\nx✝¹ x✝ : M₁\n⊢ ↑(↑B x✝¹) x✝ = -↑(↑B x✝) x✝¹",
"state_before": "case mp.h.h\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : IsAlt B\nx✝¹ x✝ : M₁\n⊢ ↑(↑B x✝¹) x✝ = ↑(↑(-flip B) x✝¹) x✝",
"tactic": "simp_rw [neg_apply, flip_apply]"
},
{
"state_after": "no goals",
"state_before": "case mp.h.h\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.197485\nR₃ : Type ?u.197488\nM : Type ?u.197491\nM₁ : Type u_3\nM₂ : Type ?u.197497\nMₗ₁ : Type ?u.197500\nMₗ₁' : Type ?u.197503\nMₗ₂ : Type ?u.197506\nMₗ₂' : Type ?u.197509\nK : Type ?u.197512\nK₁ : Type ?u.197515\nK₂ : Type ?u.197518\nV : Type ?u.197521\nV₁ : Type ?u.197524\nV₂ : Type ?u.197527\nn : Type ?u.197530\ninst✝⁵ : CommRing R\ninst✝⁴ : CommSemiring R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB✝ : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nB : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R\nh : IsAlt B\nx✝¹ x✝ : M₁\n⊢ ↑(↑B x✝¹) x✝ = -↑(↑B x✝) x✝¹",
"tactic": "exact (h.neg _ _).symm"
}
] |
[
305,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Data/Rel.lean
|
Rel.mem_image
|
[] |
[
135,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
dist_nndist
|
[] |
[
309,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
IsCompact.exists_infEdist_eq_edist
|
[
{
"state_after": "ι : Sort ?u.27190\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nhs : IsCompact s\nhne : Set.Nonempty s\nx : α\nA : Continuous fun y => edist x y\n⊢ ∃ y, y ∈ s ∧ infEdist x s = edist x y",
"state_before": "ι : Sort ?u.27190\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nhs : IsCompact s\nhne : Set.Nonempty s\nx : α\n⊢ ∃ y, y ∈ s ∧ infEdist x s = edist x y",
"tactic": "have A : Continuous fun y => edist x y := continuous_const.edist continuous_id"
},
{
"state_after": "case intro.intro\nι : Sort ?u.27190\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y✝ : α\ns t : Set α\nΦ : α → β\nhs : IsCompact s\nhne : Set.Nonempty s\nx : α\nA : Continuous fun y => edist x y\ny : α\nys : y ∈ s\nhy : ∀ (z : α), z ∈ s → edist x y ≤ edist x z\n⊢ ∃ y, y ∈ s ∧ infEdist x s = edist x y",
"state_before": "ι : Sort ?u.27190\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nhs : IsCompact s\nhne : Set.Nonempty s\nx : α\nA : Continuous fun y => edist x y\n⊢ ∃ y, y ∈ s ∧ infEdist x s = edist x y",
"tactic": "obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, ∀ z, z ∈ s → edist x y ≤ edist x z :=\n hs.exists_forall_le hne A.continuousOn"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nι : Sort ?u.27190\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y✝ : α\ns t : Set α\nΦ : α → β\nhs : IsCompact s\nhne : Set.Nonempty s\nx : α\nA : Continuous fun y => edist x y\ny : α\nys : y ∈ s\nhy : ∀ (z : α), z ∈ s → edist x y ≤ edist x z\n⊢ ∃ y, y ∈ s ∧ infEdist x s = edist x y",
"tactic": "exact ⟨y, ys, le_antisymm (infEdist_le_edist_of_mem ys) (by rwa [le_infEdist])⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.27190\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y✝ : α\ns t : Set α\nΦ : α → β\nhs : IsCompact s\nhne : Set.Nonempty s\nx : α\nA : Continuous fun y => edist x y\ny : α\nys : y ∈ s\nhy : ∀ (z : α), z ∈ s → edist x y ≤ edist x z\n⊢ edist x y ≤ infEdist x s",
"tactic": "rwa [le_infEdist]"
}
] |
[
237,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.commute_cast
|
[] |
[
78,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
|
CategoryTheory.leftAdjointMate_comp
|
[
{
"state_after": "C : Type u₁\ninst✝³ : Category C\ninst✝² : MonoidalCategory C\nX Y Z : C\ninst✝¹ : HasLeftDual X\ninst✝ : HasLeftDual Y\nf : X ⟶ Y\ng : ᘁX ⟶ Z\n⊢ ((λ_ ᘁY).inv ≫ (η_ (ᘁX) X ⊗ 𝟙 ᘁY) ≫ ((𝟙 ᘁX ⊗ f) ⊗ 𝟙 ᘁY) ≫ (α_ (ᘁX) Y ᘁY).hom ≫ (𝟙 ᘁX ⊗ ε_ (ᘁY) Y) ≫ (ρ_ ᘁX).hom) ≫ g =\n (λ_ ᘁY).inv ≫ (η_ (ᘁX) X ⊗ 𝟙 ᘁY) ≫ ((g ⊗ f) ⊗ 𝟙 ᘁY) ≫ (α_ Z Y ᘁY).hom ≫ (𝟙 Z ⊗ ε_ (ᘁY) Y) ≫ (ρ_ Z).hom",
"state_before": "C : Type u₁\ninst✝³ : Category C\ninst✝² : MonoidalCategory C\nX Y Z : C\ninst✝¹ : HasLeftDual X\ninst✝ : HasLeftDual Y\nf : X ⟶ Y\ng : ᘁX ⟶ Z\n⊢ (ᘁf) ≫ g = (λ_ ᘁY).inv ≫ (η_ (ᘁX) X ⊗ 𝟙 ᘁY) ≫ ((g ⊗ f) ⊗ 𝟙 ᘁY) ≫ (α_ Z Y ᘁY).hom ≫ (𝟙 Z ⊗ ε_ (ᘁY) Y) ≫ (ρ_ Z).hom",
"tactic": "dsimp only [leftAdjointMate]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\ninst✝² : MonoidalCategory C\nX Y Z : C\ninst✝¹ : HasLeftDual X\ninst✝ : HasLeftDual Y\nf : X ⟶ Y\ng : ᘁX ⟶ Z\n⊢ ((λ_ ᘁY).inv ≫ (η_ (ᘁX) X ⊗ 𝟙 ᘁY) ≫ ((𝟙 ᘁX ⊗ f) ⊗ 𝟙 ᘁY) ≫ (α_ (ᘁX) Y ᘁY).hom ≫ (𝟙 ᘁX ⊗ ε_ (ᘁY) Y) ≫ (ρ_ ᘁX).hom) ≫ g =\n (λ_ ᘁY).inv ≫ (η_ (ᘁX) X ⊗ 𝟙 ᘁY) ≫ ((g ⊗ f) ⊗ 𝟙 ᘁY) ≫ (α_ Z Y ᘁY).hom ≫ (𝟙 Z ⊗ ε_ (ᘁY) Y) ≫ (ρ_ Z).hom",
"tactic": "rw [Category.assoc, Category.assoc, associator_naturality_assoc, associator_naturality_assoc, ←\n id_tensor_comp_tensor_id _ g, Category.assoc, Category.assoc, Category.assoc, Category.assoc,\n tensor_id_comp_id_tensor_assoc, ← rightUnitor_naturality, id_tensor_comp_tensor_id_assoc]"
}
] |
[
216,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.even_add_one
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ Even (n + 1) ↔ ¬Even n",
"tactic": "simp [even_add]"
}
] |
[
125,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Analysis/Convex/Gauge.lean
|
gauge_empty
|
[
{
"state_after": "case h\n𝕜 : Type ?u.33039\nE : Type u_1\nF : Type ?u.33045\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx✝ : E\n⊢ gauge ∅ x✝ = OfNat.ofNat 0 x✝",
"state_before": "𝕜 : Type ?u.33039\nE : Type u_1\nF : Type ?u.33045\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\n⊢ gauge ∅ = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type ?u.33039\nE : Type u_1\nF : Type ?u.33045\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx✝ : E\n⊢ gauge ∅ x✝ = OfNat.ofNat 0 x✝",
"tactic": "simp only [gauge_def', Real.sInf_empty, mem_empty_iff_false, Pi.zero_apply, sep_false]"
}
] |
[
123,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.measure_zero_iff_ae_nmem
|
[] |
[
394,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
QuadraticForm.discr_smul
|
[
{
"state_after": "no goals",
"state_before": "S : Type ?u.614314\nR : Type ?u.614317\nR₁ : Type u_1\nM : Type ?u.614323\nn : Type w\ninst✝⁵ : Fintype n\ninst✝⁴ : CommRing R₁\ninst✝³ : DecidableEq n\ninst✝² : Invertible 2\nm : Type w\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\nQ : QuadraticForm R₁ (n → R₁)\na : R₁\n⊢ discr (a • Q) = a ^ Fintype.card n * discr Q",
"tactic": "simp only [discr, toMatrix'_smul, Matrix.det_smul]"
}
] |
[
1035,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1034,
1
] |
Mathlib/Topology/Sheaves/Stalks.lean
|
TopCat.Presheaf.stalk_open_algebraMap
|
[] |
[
649,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
647,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.localization_away_comap_range
|
[
{
"state_after": "R : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\n⊢ {p | Disjoint ↑(Submonoid.powers r) ↑p.asIdeal} = ↑(basicOpen r)",
"state_before": "R : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\n⊢ Set.range ↑(comap (algebraMap R S)) = ↑(basicOpen r)",
"tactic": "rw [localization_comap_range S (Submonoid.powers r)]"
},
{
"state_after": "case h\nR : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nx : PrimeSpectrum R\n⊢ x ∈ {p | Disjoint ↑(Submonoid.powers r) ↑p.asIdeal} ↔ x ∈ ↑(basicOpen r)",
"state_before": "R : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\n⊢ {p | Disjoint ↑(Submonoid.powers r) ↑p.asIdeal} = ↑(basicOpen r)",
"tactic": "ext x"
},
{
"state_after": "case h\nR : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nx : PrimeSpectrum R\n⊢ ↑(Submonoid.powers r) ⊓ ↑x.asIdeal ≤ ⊥ ↔ ¬r ∈ x.asIdeal",
"state_before": "case h\nR : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nx : PrimeSpectrum R\n⊢ x ∈ {p | Disjoint ↑(Submonoid.powers r) ↑p.asIdeal} ↔ x ∈ ↑(basicOpen r)",
"tactic": "simp only [mem_zeroLocus, basicOpen_eq_zeroLocus_compl, SetLike.mem_coe, Set.mem_setOf_eq,\n Set.singleton_subset_iff, Set.mem_compl_iff, disjoint_iff_inf_le]"
},
{
"state_after": "case h.mp\nR : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nx : PrimeSpectrum R\n⊢ ↑(Submonoid.powers r) ⊓ ↑x.asIdeal ≤ ⊥ → ¬r ∈ x.asIdeal\n\ncase h.mpr\nR : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nx : PrimeSpectrum R\n⊢ ¬r ∈ x.asIdeal → ↑(Submonoid.powers r) ⊓ ↑x.asIdeal ≤ ⊥",
"state_before": "case h\nR : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nx : PrimeSpectrum R\n⊢ ↑(Submonoid.powers r) ⊓ ↑x.asIdeal ≤ ⊥ ↔ ¬r ∈ x.asIdeal",
"tactic": "constructor"
},
{
"state_after": "case h.mp\nR : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nx : PrimeSpectrum R\nh₁ : ↑(Submonoid.powers r) ⊓ ↑x.asIdeal ≤ ⊥\nh₂ : r ∈ x.asIdeal\n⊢ False",
"state_before": "case h.mp\nR : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nx : PrimeSpectrum R\n⊢ ↑(Submonoid.powers r) ⊓ ↑x.asIdeal ≤ ⊥ → ¬r ∈ x.asIdeal",
"tactic": "intro h₁ h₂"
},
{
"state_after": "no goals",
"state_before": "case h.mp\nR : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nx : PrimeSpectrum R\nh₁ : ↑(Submonoid.powers r) ⊓ ↑x.asIdeal ≤ ⊥\nh₂ : r ∈ x.asIdeal\n⊢ False",
"tactic": "exact h₁ ⟨Submonoid.mem_powers r, h₂⟩"
},
{
"state_after": "case h.mpr.intro.intro\nR : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nx : PrimeSpectrum R\nh₁ : ¬r ∈ x.asIdeal\nn : ℕ\nh₃ : (fun x x_1 => x ^ x_1) r n ∈ ↑x.asIdeal\n⊢ (fun x x_1 => x ^ x_1) r n ∈ ⊥",
"state_before": "case h.mpr\nR : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nx : PrimeSpectrum R\n⊢ ¬r ∈ x.asIdeal → ↑(Submonoid.powers r) ⊓ ↑x.asIdeal ≤ ⊥",
"tactic": "rintro h₁ _ ⟨⟨n, rfl⟩, h₃⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro.intro\nR : Type u\nS✝ : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nx : PrimeSpectrum R\nh₁ : ¬r ∈ x.asIdeal\nn : ℕ\nh₃ : (fun x x_1 => x ^ x_1) r n ∈ ↑x.asIdeal\n⊢ (fun x x_1 => x ^ x_1) r n ∈ ⊥",
"tactic": "exact h₁ (x.2.mem_of_pow_mem _ h₃)"
}
] |
[
887,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
877,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
Algebra.algebraMap_ofSubring
|
[] |
[
537,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/CategoryTheory/Subobject/Basic.lean
|
CategoryTheory.Subobject.ofMkLEMk_refl
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\nB A₁ : C\nf : A₁ ⟶ B\ninst✝ : Mono f\n⊢ ofMkLEMk f f (_ : mk f ≤ mk f) ≫ f = 𝟙 A₁ ≫ f",
"state_before": "C : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\nB A₁ : C\nf : A₁ ⟶ B\ninst✝ : Mono f\n⊢ ofMkLEMk f f (_ : mk f ≤ mk f) = 𝟙 A₁",
"tactic": "apply (cancel_mono f).mp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\nB A₁ : C\nf : A₁ ⟶ B\ninst✝ : Mono f\n⊢ ofMkLEMk f f (_ : mk f ≤ mk f) ≫ f = 𝟙 A₁ ≫ f",
"tactic": "simp"
}
] |
[
461,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
459,
1
] |
Mathlib/Topology/LocalExtr.lean
|
isLocalMaxOn_const
|
[] |
[
195,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/SetTheory/Ordinal/Exponential.lean
|
Ordinal.log_le_self
|
[
{
"state_after": "no goals",
"state_before": "b x : Ordinal\nhx : x = 0\n⊢ log b x ≤ x",
"tactic": "simp only [hx, log_zero_right, Ordinal.zero_le]"
},
{
"state_after": "no goals",
"state_before": "b x : Ordinal\nhx : ¬x = 0\nhb : ¬1 < b\n⊢ log b x ≤ x",
"tactic": "simp only [log_of_not_one_lt_left hb, Ordinal.zero_le]"
}
] |
[
371,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.isCoseparator_prod_of_isCoseparator_right
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nG H : C\ninst✝ : HasBinaryProduct G H\nhH : IsCoseparator H\n⊢ {H} ⊆ {G, H}",
"tactic": "simp"
}
] |
[
609,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
607,
1
] |
Mathlib/CategoryTheory/Limits/ExactFunctor.lean
|
CategoryTheory.RightExactFunctor.ofExact_map
|
[] |
[
163,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
linearIndependent_subtype_range
|
[] |
[
267,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/CategoryTheory/Monoidal/Category.lean
|
CategoryTheory.MonoidalCategory.tensor_left_iff
|
[
{
"state_after": "no goals",
"state_before": "C✝ : Type u\n𝒞 : Category C✝\ninst✝² : MonoidalCategory C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nU V W X✝ Y✝ Z X Y : C\nf g : X ⟶ Y\n⊢ 𝟙 (𝟙_ C) ⊗ f = 𝟙 (𝟙_ C) ⊗ g ↔ f = g",
"tactic": "simp"
}
] |
[
271,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.of_add_of
|
[] |
[
328,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Order/GameAdd.lean
|
Prod.GameAdd.induction
|
[] |
[
143,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEFinStronglyMeasurable.exists_set_sigmaFinite
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.549973\nι : Type ?u.549976\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\nf g✝ : α → β\ninst✝¹ : Zero β\ninst✝ : T2Space β\ng : α → β\nhg : FinStronglyMeasurable g μ\nhfg : f =ᶠ[ae μ] g\n⊢ ∃ t, MeasurableSet t ∧ f =ᶠ[ae (Measure.restrict μ (tᶜ))] 0 ∧ SigmaFinite (Measure.restrict μ t)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.549973\nι : Type ?u.549976\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\nf g : α → β\ninst✝¹ : Zero β\ninst✝ : T2Space β\nhf : AEFinStronglyMeasurable f μ\n⊢ ∃ t, MeasurableSet t ∧ f =ᶠ[ae (Measure.restrict μ (tᶜ))] 0 ∧ SigmaFinite (Measure.restrict μ t)",
"tactic": "rcases hf with ⟨g, hg, hfg⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.549973\nι : Type ?u.549976\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\nf g✝ : α → β\ninst✝¹ : Zero β\ninst✝ : T2Space β\ng : α → β\nhg : FinStronglyMeasurable g μ\nhfg : f =ᶠ[ae μ] g\nt : Set α\nht : MeasurableSet t\nhgt_zero : ∀ (x : α), x ∈ tᶜ → g x = 0\nhtμ : SigmaFinite (Measure.restrict μ t)\n⊢ ∃ t, MeasurableSet t ∧ f =ᶠ[ae (Measure.restrict μ (tᶜ))] 0 ∧ SigmaFinite (Measure.restrict μ t)",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.549973\nι : Type ?u.549976\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\nf g✝ : α → β\ninst✝¹ : Zero β\ninst✝ : T2Space β\ng : α → β\nhg : FinStronglyMeasurable g μ\nhfg : f =ᶠ[ae μ] g\n⊢ ∃ t, MeasurableSet t ∧ f =ᶠ[ae (Measure.restrict μ (tᶜ))] 0 ∧ SigmaFinite (Measure.restrict μ t)",
"tactic": "obtain ⟨t, ht, hgt_zero, htμ⟩ := hg.exists_set_sigmaFinite"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.549973\nι : Type ?u.549976\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\nf g✝ : α → β\ninst✝¹ : Zero β\ninst✝ : T2Space β\ng : α → β\nhg : FinStronglyMeasurable g μ\nhfg : f =ᶠ[ae μ] g\nt : Set α\nht : MeasurableSet t\nhgt_zero : ∀ (x : α), x ∈ tᶜ → g x = 0\nhtμ : SigmaFinite (Measure.restrict μ t)\n⊢ f =ᶠ[ae (Measure.restrict μ (tᶜ))] 0",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.549973\nι : Type ?u.549976\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\nf g✝ : α → β\ninst✝¹ : Zero β\ninst✝ : T2Space β\ng : α → β\nhg : FinStronglyMeasurable g μ\nhfg : f =ᶠ[ae μ] g\nt : Set α\nht : MeasurableSet t\nhgt_zero : ∀ (x : α), x ∈ tᶜ → g x = 0\nhtμ : SigmaFinite (Measure.restrict μ t)\n⊢ ∃ t, MeasurableSet t ∧ f =ᶠ[ae (Measure.restrict μ (tᶜ))] 0 ∧ SigmaFinite (Measure.restrict μ t)",
"tactic": "refine' ⟨t, ht, _, htμ⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.549973\nι : Type ?u.549976\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\nf g✝ : α → β\ninst✝¹ : Zero β\ninst✝ : T2Space β\ng : α → β\nhg : FinStronglyMeasurable g μ\nhfg : f =ᶠ[ae μ] g\nt : Set α\nht : MeasurableSet t\nhgt_zero : ∀ (x : α), x ∈ tᶜ → g x = 0\nhtμ : SigmaFinite (Measure.restrict μ t)\n⊢ (fun x => g x) =ᶠ[ae (Measure.restrict μ (tᶜ))] 0",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.549973\nι : Type ?u.549976\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\nf g✝ : α → β\ninst✝¹ : Zero β\ninst✝ : T2Space β\ng : α → β\nhg : FinStronglyMeasurable g μ\nhfg : f =ᶠ[ae μ] g\nt : Set α\nht : MeasurableSet t\nhgt_zero : ∀ (x : α), x ∈ tᶜ → g x = 0\nhtμ : SigmaFinite (Measure.restrict μ t)\n⊢ f =ᶠ[ae (Measure.restrict μ (tᶜ))] 0",
"tactic": "refine' EventuallyEq.trans (ae_restrict_of_ae hfg) _"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.549973\nι : Type ?u.549976\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\nf g✝ : α → β\ninst✝¹ : Zero β\ninst✝ : T2Space β\ng : α → β\nhg : FinStronglyMeasurable g μ\nhfg : f =ᶠ[ae μ] g\nt : Set α\nht : MeasurableSet t\nhgt_zero : ∀ (x : α), x ∈ tᶜ → g x = 0\nhtμ : SigmaFinite (Measure.restrict μ t)\n⊢ ∀ᵐ (x : α) ∂μ, x ∈ tᶜ → g x = OfNat.ofNat 0 x",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.549973\nι : Type ?u.549976\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\nf g✝ : α → β\ninst✝¹ : Zero β\ninst✝ : T2Space β\ng : α → β\nhg : FinStronglyMeasurable g μ\nhfg : f =ᶠ[ae μ] g\nt : Set α\nht : MeasurableSet t\nhgt_zero : ∀ (x : α), x ∈ tᶜ → g x = 0\nhtμ : SigmaFinite (Measure.restrict μ t)\n⊢ (fun x => g x) =ᶠ[ae (Measure.restrict μ (tᶜ))] 0",
"tactic": "rw [EventuallyEq, ae_restrict_iff' ht.compl]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.549973\nι : Type ?u.549976\ninst✝³ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\nf g✝ : α → β\ninst✝¹ : Zero β\ninst✝ : T2Space β\ng : α → β\nhg : FinStronglyMeasurable g μ\nhfg : f =ᶠ[ae μ] g\nt : Set α\nht : MeasurableSet t\nhgt_zero : ∀ (x : α), x ∈ tᶜ → g x = 0\nhtμ : SigmaFinite (Measure.restrict μ t)\n⊢ ∀ᵐ (x : α) ∂μ, x ∈ tᶜ → g x = OfNat.ofNat 0 x",
"tactic": "exact eventually_of_forall hgt_zero"
}
] |
[
1932,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1925,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.star_self_mem_adjoin_singleton
|
[] |
[
456,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/CategoryTheory/Closed/Cartesian.lean
|
CategoryTheory.CartesianClosed.uncurry_natural_left
|
[] |
[
200,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Analysis/MeanInequalitiesPow.lean
|
NNReal.rpow_add_rpow_le
|
[
{
"state_after": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)\n⊢ (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\n⊢ (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"tactic": "have h_rpow : ∀ a : ℝ≥0, a ^ q = (a ^ p) ^ (q / p) := fun a => by\n rw [← NNReal.rpow_mul, _root_.div_eq_inv_mul, ← mul_assoc, _root_.mul_inv_cancel hp_pos.ne.symm,\n one_mul]"
},
{
"state_after": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)\n⊢ (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"tactic": "have h_rpow_add_rpow_le_add :\n ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p := by\n refine' rpow_add_rpow_le_add (a ^ p) (b ^ p) _\n rwa [one_le_div hp_pos]"
},
{
"state_after": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (p / q) ≤ a ^ p + b ^ p",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"tactic": "rw [h_rpow a, h_rpow b, NNReal.le_rpow_one_div_iff hp_pos, ← NNReal.rpow_mul, mul_comm,\n mul_one_div]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (p / q) ≤ a ^ p + b ^ p",
"tactic": "rwa [one_div_div] at h_rpow_add_rpow_le_add"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na✝ b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\na : ℝ≥0\n⊢ a ^ q = (a ^ p) ^ (q / p)",
"tactic": "rw [← NNReal.rpow_mul, _root_.div_eq_inv_mul, ← mul_assoc, _root_.mul_inv_cancel hp_pos.ne.symm,\n one_mul]"
},
{
"state_after": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)\n⊢ 1 ≤ q / p",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)\n⊢ ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p",
"tactic": "refine' rpow_add_rpow_le_add (a ^ p) (b ^ p) _"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)\n⊢ 1 ≤ q / p",
"tactic": "rwa [one_le_div hp_pos]"
}
] |
[
218,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Order/SymmDiff.lean
|
symmDiff_eq'
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.86870\nα : Type u_1\nβ : Type ?u.86876\nπ : ι → Type ?u.86881\ninst✝ : BooleanAlgebra α\na b c d : α\n⊢ a ∆ b = (a ⊔ b) ⊓ (aᶜ ⊔ bᶜ)",
"tactic": "rw [symmDiff_eq_sup_sdiff_inf, sdiff_eq, compl_inf]"
}
] |
[
721,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
720,
1
] |
Mathlib/Analysis/Calculus/IteratedDeriv.lean
|
iteratedDerivWithin_univ
|
[
{
"state_after": "case h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.3042\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx✝ x : 𝕜\n⊢ iteratedDerivWithin n f univ x = iteratedDeriv n f x",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.3042\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\n⊢ iteratedDerivWithin n f univ = iteratedDeriv n f",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.3042\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx✝ x : 𝕜\n⊢ iteratedDerivWithin n f univ x = iteratedDeriv n f x",
"tactic": "rw [iteratedDerivWithin, iteratedDeriv, iteratedFDerivWithin_univ]"
}
] |
[
76,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_le_one_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.52121\ns t : Finset α\nf : α → β\nn : ℕ\n⊢ (∀ (a : α), a ∈ s → ∀ (b : α), b ∈ s → a = b) ↔ ∀ {a b : α}, a ∈ s → b ∈ s → a = b",
"state_before": "α : Type u_1\nβ : Type ?u.52121\ns t : Finset α\nf : α → β\nn : ℕ\n⊢ card s ≤ 1 ↔ ∀ {a b : α}, a ∈ s → b ∈ s → a = b",
"tactic": "rw [card_le_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.52121\ns t : Finset α\nf : α → β\nn : ℕ\n⊢ (∀ (a : α), a ∈ s → ∀ (b : α), b ∈ s → a = b) ↔ ∀ {a b : α}, a ∈ s → b ∈ s → a = b",
"tactic": "tauto"
}
] |
[
541,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
539,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.normSq_le_zero
|
[] |
[
1284,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1283,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.sin_sub_pi_div_two
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ sin (x - π / 2) = -cos x",
"tactic": "simp [sub_eq_add_neg, sin_add]"
}
] |
[
448,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalExtrOn.on_subset
|
[] |
[
117,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.cos_pos_of_le_one
|
[
{
"state_after": "case h₁.h\nx : ℝ\nhx : abs' x ≤ 1\n⊢ abs' x ^ 4 ≤ 1\n\ncase h₂.h\nx : ℝ\nhx : abs' x ≤ 1\n⊢ x ^ 2 ≤ 1",
"state_before": "x : ℝ\nhx : abs' x ≤ 1\n⊢ abs' x ^ 4 * (5 / 96) + x ^ 2 / 2 ≤ 1 * (5 / 96) + 1 / 2",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h₁.h\nx : ℝ\nhx : abs' x ≤ 1\n⊢ abs' x ^ 4 ≤ 1",
"tactic": "exact pow_le_one _ (abs_nonneg _) hx"
},
{
"state_after": "case h₂.h\nx : ℝ\nhx : abs' x ≤ 1\n⊢ abs' x * abs' x ≤ 1",
"state_before": "case h₂.h\nx : ℝ\nhx : abs' x ≤ 1\n⊢ x ^ 2 ≤ 1",
"tactic": "rw [sq, ← abs_mul_self, abs_mul]"
},
{
"state_after": "no goals",
"state_before": "case h₂.h\nx : ℝ\nhx : abs' x ≤ 1\n⊢ abs' x * abs' x ≤ 1",
"tactic": "exact mul_le_one hx (abs_nonneg _) hx"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : abs' x ≤ 1\n⊢ 1 * (5 / 96) + 1 / 2 < 1",
"tactic": "norm_num"
}
] |
[
1878,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1867,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.eq_div_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.310360\nβ : Type ?u.310363\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ 0\nha' : a ≠ ⊤\nh : b = c / a\n⊢ a * b = c",
"tactic": "rw [h, ENNReal.mul_div_cancel' ha ha']"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.310360\nβ : Type ?u.310363\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ 0\nha' : a ≠ ⊤\nh : a * b = c\n⊢ b = c / a",
"tactic": "rw [← h, mul_div_assoc, ENNReal.mul_div_cancel' ha ha']"
}
] |
[
1704,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1702,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.coe_mk'
|
[] |
[
428,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
427,
1
] |
Mathlib/Data/Set/Basic.lean
|
Disjoint.inter_right'
|
[] |
[
2973,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2972,
1
] |
Mathlib/Order/Ideal.lean
|
Order.Ideal.nonempty
|
[] |
[
133,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
11
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.neg_lt_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "x : PGame\n⊢ -x < 0 ↔ 0 < x",
"tactic": "rw [neg_lt_iff, neg_zero]"
}
] |
[
1388,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1388,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Reachable.map
|
[] |
[
1905,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1903,
11
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.inf_product_left
|
[] |
[
432,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.map_mk
|
[] |
[
178,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.Lp_toLp_restrict_smul
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.322244\nE : Type ?u.322247\nF : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\nc : 𝕜\nf : { x // x ∈ Lp F p }\ns : Set α\n⊢ ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) =ᵐ[Measure.restrict μ s] ↑↑(c • Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p))",
"state_before": "α : Type u_1\nβ : Type ?u.322244\nE : Type ?u.322247\nF : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\nc : 𝕜\nf : { x // x ∈ Lp F p }\ns : Set α\n⊢ Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p) = c • Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)",
"tactic": "ext1"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.322244\nE : Type ?u.322247\nF : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\nc : 𝕜\nf : { x // x ∈ Lp F p }\ns : Set α\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s,\n ↑↑(c • f) x = (c • ↑↑f) x →\n ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) x = ↑↑(c • Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.322244\nE : Type ?u.322247\nF : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\nc : 𝕜\nf : { x // x ∈ Lp F p }\ns : Set α\n⊢ ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) =ᵐ[Measure.restrict μ s] ↑↑(c • Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p))",
"tactic": "refine' (ae_restrict_of_ae (Lp.coeFn_smul c f)).mp _"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.322244\nE : Type ?u.322247\nF : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\nc : 𝕜\nf : { x // x ∈ Lp F p }\ns : Set α\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s,\n ↑↑(Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x = ↑↑f x →\n ↑↑(c • f) x = (c • ↑↑f) x →\n ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) x = ↑↑(c • Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.322244\nE : Type ?u.322247\nF : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\nc : 𝕜\nf : { x // x ∈ Lp F p }\ns : Set α\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s,\n ↑↑(c • f) x = (c • ↑↑f) x →\n ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) x = ↑↑(c • Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x",
"tactic": "refine' (Memℒp.coeFn_toLp ((Lp.memℒp f).restrict s)).mp _"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.322244\nE : Type ?u.322247\nF : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\nc : 𝕜\nf : { x // x ∈ Lp F p }\ns : Set α\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s,\n ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) x = ↑↑(c • f) x →\n ↑↑(Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x = ↑↑f x →\n ↑↑(c • f) x = (c • ↑↑f) x →\n ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) x = ↑↑(c • Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.322244\nE : Type ?u.322247\nF : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\nc : 𝕜\nf : { x // x ∈ Lp F p }\ns : Set α\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s,\n ↑↑(Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x = ↑↑f x →\n ↑↑(c • f) x = (c • ↑↑f) x →\n ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) x = ↑↑(c • Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x",
"tactic": "refine' (Memℒp.coeFn_toLp ((Lp.memℒp (c • f)).restrict s)).mp _"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.322244\nE : Type ?u.322247\nF : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\nc : 𝕜\nf : { x // x ∈ Lp F p }\ns : Set α\nx : α\nhx1 : ↑↑(c • Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x = (c • ↑↑(Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p))) x\nhx2 : ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) x = ↑↑(c • f) x\nhx3 : ↑↑(Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x = ↑↑f x\nhx4 : ↑↑(c • f) x = (c • ↑↑f) x\n⊢ ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) x = ↑↑(c • Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.322244\nE : Type ?u.322247\nF : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\nc : 𝕜\nf : { x // x ∈ Lp F p }\ns : Set α\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s,\n ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) x = ↑↑(c • f) x →\n ↑↑(Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x = ↑↑f x →\n ↑↑(c • f) x = (c • ↑↑f) x →\n ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) x = ↑↑(c • Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x",
"tactic": "refine'\n (Lp.coeFn_smul c (Memℒp.toLp f ((Lp.memℒp f).restrict s))).mono fun x hx1 hx2 hx3 hx4 => _"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.322244\nE : Type ?u.322247\nF : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\nc : 𝕜\nf : { x // x ∈ Lp F p }\ns : Set α\nx : α\nhx1 : ↑↑(c • Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x = (c • ↑↑(Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p))) x\nhx2 : ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) x = ↑↑(c • f) x\nhx3 : ↑↑(Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x = ↑↑f x\nhx4 : ↑↑(c • f) x = (c • ↑↑f) x\n⊢ ↑↑(Memℒp.toLp ↑↑(c • f) (_ : Memℒp (↑↑(c • f)) p)) x = ↑↑(c • Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)) x",
"tactic": "rw [hx2, hx1, Pi.smul_apply, hx3, hx4, Pi.smul_apply]"
}
] |
[
896,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
888,
1
] |
Mathlib/Topology/MetricSpace/Gluing.lean
|
Metric.Sum.one_le_dist_inl_inr
|
[] |
[
223,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/CategoryTheory/Limits/Types.lean
|
CategoryTheory.Limits.Types.Limit.lift_π_apply'
|
[] |
[
215,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Data/List/Prime.lean
|
mem_list_primes_of_dvd_prod
|
[
{
"state_after": "case intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : Unique Mˣ\np : M\nhp : Prime p\nL : List M\nhL : ∀ (q : M), q ∈ L → Prime q\nhpL : p ∣ prod L\nx : M\nhx1 : x ∈ L\nhx2 : p ∣ x\n⊢ p ∈ L",
"state_before": "M : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : Unique Mˣ\np : M\nhp : Prime p\nL : List M\nhL : ∀ (q : M), q ∈ L → Prime q\nhpL : p ∣ prod L\n⊢ p ∈ L",
"tactic": "obtain ⟨x, hx1, hx2⟩ := hp.dvd_prod_iff.mp hpL"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : Unique Mˣ\np : M\nhp : Prime p\nL : List M\nhL : ∀ (q : M), q ∈ L → Prime q\nhpL : p ∣ prod L\nx : M\nhx1 : x ∈ L\nhx2 : p ∣ x\n⊢ p ∈ L",
"tactic": "rwa [(prime_dvd_prime_iff_eq hp (hL x hx1)).mp hx2]"
}
] |
[
58,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.logb_ne_zero_of_pos_of_ne_one_of_base_lt_one
|
[] |
[
339,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.one_mul
|
[
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n✝ : σ →₀ ℕ\nφ ψ : MvPowerSeries σ R\nn : σ →₀ ℕ\n⊢ ↑(coeff R n) (1 * φ) = ↑(coeff R n) φ",
"tactic": "simpa using coeff_add_monomial_mul 0 n φ 1"
}
] |
[
270,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
11
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.QuasiMeasurePreserving.preimage_iterate_ae_eq
|
[
{
"state_after": "case zero\nα : Type u_1\nβ : Type ?u.411803\nγ : Type ?u.411806\nδ : Type ?u.411809\nι : Type ?u.411812\nR : Type ?u.411815\nR' : Type ?u.411818\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\ns : Set α\nf : α → α\nhf : QuasiMeasurePreserving f\nhs : f ⁻¹' s =ᵐ[μ] s\n⊢ f^[Nat.zero] ⁻¹' s =ᵐ[μ] s\n\ncase succ\nα : Type u_1\nβ : Type ?u.411803\nγ : Type ?u.411806\nδ : Type ?u.411809\nι : Type ?u.411812\nR : Type ?u.411815\nR' : Type ?u.411818\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\ns : Set α\nf : α → α\nhf : QuasiMeasurePreserving f\nhs : f ⁻¹' s =ᵐ[μ] s\nk : ℕ\nih : f^[k] ⁻¹' s =ᵐ[μ] s\n⊢ f^[Nat.succ k] ⁻¹' s =ᵐ[μ] s",
"state_before": "α : Type u_1\nβ : Type ?u.411803\nγ : Type ?u.411806\nδ : Type ?u.411809\nι : Type ?u.411812\nR : Type ?u.411815\nR' : Type ?u.411818\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\ns : Set α\nf : α → α\nhf : QuasiMeasurePreserving f\nk : ℕ\nhs : f ⁻¹' s =ᵐ[μ] s\n⊢ f^[k] ⁻¹' s =ᵐ[μ] s",
"tactic": "induction' k with k ih"
},
{
"state_after": "case succ\nα : Type u_1\nβ : Type ?u.411803\nγ : Type ?u.411806\nδ : Type ?u.411809\nι : Type ?u.411812\nR : Type ?u.411815\nR' : Type ?u.411818\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\ns : Set α\nf : α → α\nhf : QuasiMeasurePreserving f\nhs : f ⁻¹' s =ᵐ[μ] s\nk : ℕ\nih : f^[k] ⁻¹' s =ᵐ[μ] s\n⊢ f ⁻¹' (f^[k] ⁻¹' s) =ᵐ[μ] s",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.411803\nγ : Type ?u.411806\nδ : Type ?u.411809\nι : Type ?u.411812\nR : Type ?u.411815\nR' : Type ?u.411818\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\ns : Set α\nf : α → α\nhf : QuasiMeasurePreserving f\nhs : f ⁻¹' s =ᵐ[μ] s\nk : ℕ\nih : f^[k] ⁻¹' s =ᵐ[μ] s\n⊢ f^[Nat.succ k] ⁻¹' s =ᵐ[μ] s",
"tactic": "rw [iterate_succ, preimage_comp]"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.411803\nγ : Type ?u.411806\nδ : Type ?u.411809\nι : Type ?u.411812\nR : Type ?u.411815\nR' : Type ?u.411818\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\ns : Set α\nf : α → α\nhf : QuasiMeasurePreserving f\nhs : f ⁻¹' s =ᵐ[μ] s\nk : ℕ\nih : f^[k] ⁻¹' s =ᵐ[μ] s\n⊢ f ⁻¹' (f^[k] ⁻¹' s) =ᵐ[μ] s",
"tactic": "exact EventuallyEq.trans (hf.preimage_ae_eq ih) hs"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\nβ : Type ?u.411803\nγ : Type ?u.411806\nδ : Type ?u.411809\nι : Type ?u.411812\nR : Type ?u.411815\nR' : Type ?u.411818\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\ns : Set α\nf : α → α\nhf : QuasiMeasurePreserving f\nhs : f ⁻¹' s =ᵐ[μ] s\n⊢ f^[Nat.zero] ⁻¹' s =ᵐ[μ] s",
"tactic": "rfl"
}
] |
[
2529,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2525,
1
] |
Mathlib/Topology/MetricSpace/IsometricSMul.lean
|
edist_div_right
|
[
{
"state_after": "no goals",
"state_before": "M : Type u\nG : Type v\nX : Type w\ninst✝⁶ : PseudoEMetricSpace X\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G X\ninst✝³ : IsometricSMul G X\ninst✝² : DivInvMonoid M\ninst✝¹ : PseudoEMetricSpace M\ninst✝ : IsometricSMul Mᵐᵒᵖ M\na b c : M\n⊢ edist (a / c) (b / c) = edist a b",
"tactic": "simp only [div_eq_mul_inv, edist_mul_right]"
}
] |
[
124,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
Subspace.dualAnnihilator_inf_eq
|
[
{
"state_after": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\n⊢ dualAnnihilator (W ⊓ W') ≤ dualAnnihilator W ⊔ dualAnnihilator W'",
"state_before": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\n⊢ dualAnnihilator (W ⊓ W') = dualAnnihilator W ⊔ dualAnnihilator W'",
"tactic": "refine' le_antisymm _ (sup_dualAnnihilator_le_inf W W')"
},
{
"state_after": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\n⊢ dualAnnihilator (W ⊓ W') ≤ dualAnnihilator W ⊔ dualAnnihilator W'",
"state_before": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\n⊢ dualAnnihilator (W ⊓ W') ≤ dualAnnihilator W ⊔ dualAnnihilator W'",
"tactic": "let F : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := (Submodule.mkQ W).prod (Submodule.mkQ W')"
},
{
"state_after": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\n⊢ dualAnnihilator (W ⊓ W') ≤ dualAnnihilator W ⊔ dualAnnihilator W'",
"state_before": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\n⊢ dualAnnihilator (W ⊓ W') ≤ dualAnnihilator W ⊔ dualAnnihilator W'",
"tactic": "have : LinearMap.ker F = W ⊓ W' := by simp only [LinearMap.ker_prod, ker_mkQ]"
},
{
"state_after": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\n⊢ LinearMap.range (LinearMap.dualMap F) ≤ dualAnnihilator W ⊔ dualAnnihilator W'",
"state_before": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\n⊢ dualAnnihilator (W ⊓ W') ≤ dualAnnihilator W ⊔ dualAnnihilator W'",
"tactic": "rw [← this, ← LinearMap.range_dualMap_eq_dualAnnihilator_ker]"
},
{
"state_after": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\nφ : Dual K V₁\n⊢ φ ∈ LinearMap.range (LinearMap.dualMap F) → φ ∈ dualAnnihilator W ⊔ dualAnnihilator W'",
"state_before": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\n⊢ LinearMap.range (LinearMap.dualMap F) ≤ dualAnnihilator W ⊔ dualAnnihilator W'",
"tactic": "intro φ"
},
{
"state_after": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\nφ : Dual K V₁\n⊢ (∃ y, ↑(LinearMap.dualMap F) y = φ) → φ ∈ dualAnnihilator W ⊔ dualAnnihilator W'",
"state_before": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\nφ : Dual K V₁\n⊢ φ ∈ LinearMap.range (LinearMap.dualMap F) → φ ∈ dualAnnihilator W ⊔ dualAnnihilator W'",
"tactic": "rw [LinearMap.mem_range]"
},
{
"state_after": "case intro\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\nx : Dual K ((V₁ ⧸ W) × V₁ ⧸ W')\n⊢ ↑(LinearMap.dualMap F) x ∈ dualAnnihilator W ⊔ dualAnnihilator W'",
"state_before": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\nφ : Dual K V₁\n⊢ (∃ y, ↑(LinearMap.dualMap F) y = φ) → φ ∈ dualAnnihilator W ⊔ dualAnnihilator W'",
"tactic": "rintro ⟨x, rfl⟩"
},
{
"state_after": "case intro\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\nx : Dual K ((V₁ ⧸ W) × V₁ ⧸ W')\n⊢ ∃ y, y ∈ dualAnnihilator W ∧ ∃ z, z ∈ dualAnnihilator W' ∧ y + z = ↑(LinearMap.dualMap F) x",
"state_before": "case intro\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\nx : Dual K ((V₁ ⧸ W) × V₁ ⧸ W')\n⊢ ↑(LinearMap.dualMap F) x ∈ dualAnnihilator W ⊔ dualAnnihilator W'",
"tactic": "rw [Submodule.mem_sup]"
},
{
"state_after": "case intro.intro.mk\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\na : Dual K (V₁ ⧸ W)\nb : Dual K (V₁ ⧸ W')\n⊢ ∃ y,\n y ∈ dualAnnihilator W ∧\n ∃ z,\n z ∈ dualAnnihilator W' ∧ y + z = ↑(LinearMap.dualMap F) (↑(dualProdDualEquivDual K (V₁ ⧸ W) (V₁ ⧸ W')) (a, b))",
"state_before": "case intro\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\nx : Dual K ((V₁ ⧸ W) × V₁ ⧸ W')\n⊢ ∃ y, y ∈ dualAnnihilator W ∧ ∃ z, z ∈ dualAnnihilator W' ∧ y + z = ↑(LinearMap.dualMap F) x",
"tactic": "obtain ⟨⟨a, b⟩, rfl⟩ := (dualProdDualEquivDual K (V₁ ⧸ W) (V₁ ⧸ W')).surjective x"
},
{
"state_after": "case intro.intro.mk.intro\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\nb : Dual K (V₁ ⧸ W')\na' : { x // x ∈ dualAnnihilator W }\n⊢ ∃ y,\n y ∈ dualAnnihilator W ∧\n ∃ z,\n z ∈ dualAnnihilator W' ∧\n y + z =\n ↑(LinearMap.dualMap F)\n (↑(dualProdDualEquivDual K (V₁ ⧸ W) (V₁ ⧸ W'))\n (↑(LinearEquiv.symm (dualQuotEquivDualAnnihilator W)) a', b))",
"state_before": "case intro.intro.mk\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\na : Dual K (V₁ ⧸ W)\nb : Dual K (V₁ ⧸ W')\n⊢ ∃ y,\n y ∈ dualAnnihilator W ∧\n ∃ z,\n z ∈ dualAnnihilator W' ∧ y + z = ↑(LinearMap.dualMap F) (↑(dualProdDualEquivDual K (V₁ ⧸ W) (V₁ ⧸ W')) (a, b))",
"tactic": "obtain ⟨a', rfl⟩ := (dualQuotEquivDualAnnihilator W).symm.surjective a"
},
{
"state_after": "case intro.intro.mk.intro.intro\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\na' : { x // x ∈ dualAnnihilator W }\nb' : { x // x ∈ dualAnnihilator W' }\n⊢ ∃ y,\n y ∈ dualAnnihilator W ∧\n ∃ z,\n z ∈ dualAnnihilator W' ∧\n y + z =\n ↑(LinearMap.dualMap F)\n (↑(dualProdDualEquivDual K (V₁ ⧸ W) (V₁ ⧸ W'))\n (↑(LinearEquiv.symm (dualQuotEquivDualAnnihilator W)) a',\n ↑(LinearEquiv.symm (dualQuotEquivDualAnnihilator W')) b'))",
"state_before": "case intro.intro.mk.intro\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\nb : Dual K (V₁ ⧸ W')\na' : { x // x ∈ dualAnnihilator W }\n⊢ ∃ y,\n y ∈ dualAnnihilator W ∧\n ∃ z,\n z ∈ dualAnnihilator W' ∧\n y + z =\n ↑(LinearMap.dualMap F)\n (↑(dualProdDualEquivDual K (V₁ ⧸ W) (V₁ ⧸ W'))\n (↑(LinearEquiv.symm (dualQuotEquivDualAnnihilator W)) a', b))",
"tactic": "obtain ⟨b', rfl⟩ := (dualQuotEquivDualAnnihilator W').symm.surjective b"
},
{
"state_after": "case intro.intro.mk.intro.intro\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\na' : { x // x ∈ dualAnnihilator W }\nb' : { x // x ∈ dualAnnihilator W' }\n⊢ ↑a' + ↑b' =\n ↑(LinearMap.dualMap F)\n (↑(dualProdDualEquivDual K (V₁ ⧸ W) (V₁ ⧸ W'))\n (↑(LinearEquiv.symm (dualQuotEquivDualAnnihilator W)) a',\n ↑(LinearEquiv.symm (dualQuotEquivDualAnnihilator W')) b'))",
"state_before": "case intro.intro.mk.intro.intro\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\na' : { x // x ∈ dualAnnihilator W }\nb' : { x // x ∈ dualAnnihilator W' }\n⊢ ∃ y,\n y ∈ dualAnnihilator W ∧\n ∃ z,\n z ∈ dualAnnihilator W' ∧\n y + z =\n ↑(LinearMap.dualMap F)\n (↑(dualProdDualEquivDual K (V₁ ⧸ W) (V₁ ⧸ W'))\n (↑(LinearEquiv.symm (dualQuotEquivDualAnnihilator W)) a',\n ↑(LinearEquiv.symm (dualQuotEquivDualAnnihilator W')) b'))",
"tactic": "use a', a'.property, b', b'.property"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.mk.intro.intro\nK : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\nthis : LinearMap.ker F = W ⊓ W'\na' : { x // x ∈ dualAnnihilator W }\nb' : { x // x ∈ dualAnnihilator W' }\n⊢ ↑a' + ↑b' =\n ↑(LinearMap.dualMap F)\n (↑(dualProdDualEquivDual K (V₁ ⧸ W) (V₁ ⧸ W'))\n (↑(LinearEquiv.symm (dualQuotEquivDualAnnihilator W)) a',\n ↑(LinearEquiv.symm (dualQuotEquivDualAnnihilator W')) b'))",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝⁴ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := LinearMap.prod (mkQ W) (mkQ W')\n⊢ LinearMap.ker F = W ⊓ W'",
"tactic": "simp only [LinearMap.ker_prod, ker_mkQ]"
}
] |
[
1430,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1415,
1
] |
Mathlib/Order/Bounds/OrderIso.lean
|
OrderIso.isLUB_image'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α ≃o β\ns : Set α\nx : α\n⊢ IsLUB (↑f '' s) (↑f x) ↔ IsLUB s x",
"tactic": "rw [isLUB_image, f.symm_apply_apply]"
}
] |
[
42,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
src/lean/Init/Data/Nat/Log2.lean
|
Nat.log2_le_self
|
[
{
"state_after": "n : Nat\n⊢ (if n ≥ 2 then log2 (n / 2) + 1 else 0) ≤ n",
"state_before": "n : Nat\n⊢ log2 n ≤ n",
"tactic": "unfold Nat.log2"
},
{
"state_after": "case inl\nn : Nat\nh✝ : n ≥ 2\n⊢ log2 (n / 2) + 1 ≤ n\n\ncase inr\nn : Nat\nh✝ : ¬n ≥ 2\n⊢ 0 ≤ n",
"state_before": "n : Nat\n⊢ (if n ≥ 2 then log2 (n / 2) + 1 else 0) ≤ n",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case inl\nn : Nat\nh✝ : n ≥ 2\n⊢ log2 (n / 2) + 1 ≤ n",
"tactic": "next h =>\nhave := log2_le_self (n / 2)\nexact Nat.lt_of_le_of_lt this (Nat.div_lt_self (Nat.le_of_lt h) (by decide))"
},
{
"state_after": "n : Nat\nh : n ≥ 2\nthis : log2 (n / 2) ≤ n / 2\n⊢ log2 (n / 2) + 1 ≤ n",
"state_before": "n : Nat\nh : n ≥ 2\n⊢ log2 (n / 2) + 1 ≤ n",
"tactic": "have := log2_le_self (n / 2)"
},
{
"state_after": "no goals",
"state_before": "n : Nat\nh : n ≥ 2\nthis : log2 (n / 2) ≤ n / 2\n⊢ log2 (n / 2) + 1 ≤ n",
"tactic": "exact Nat.lt_of_le_of_lt this (Nat.div_lt_self (Nat.le_of_lt h) (by decide))"
},
{
"state_after": "no goals",
"state_before": "n : Nat\nh : n ≥ 2\nthis : log2 (n / 2) ≤ n / 2\n⊢ 1 < 2",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "case inr\nn : Nat\nh✝ : ¬n ≥ 2\n⊢ 0 ≤ n",
"tactic": "apply Nat.zero_le"
},
{
"state_after": "no goals",
"state_before": "n : Nat\na✝ : ∀ (y : Nat), (invImage (fun a => sizeOf a) instWellFoundedRelation).1 y n → log2 y ≤ y\nh✝ : n ≥ 2\n⊢ (invImage (fun a => sizeOf a) instWellFoundedRelation).1 (n / 2) n",
"tactic": "exact Nat.log2_terminates _ ‹_›"
}
] |
[
37,
46
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
31,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.coeff_erase
|
[
{
"state_after": "case ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn i : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ coeff (erase n { toFinsupp := toFinsupp✝ }) i = if i = n then 0 else coeff { toFinsupp := toFinsupp✝ } i",
"state_before": "R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np✝ q p : R[X]\nn i : ℕ\n⊢ coeff (erase n p) i = if i = n then 0 else coeff p i",
"tactic": "rcases p with ⟨⟩"
},
{
"state_after": "case ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn i : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ↑(Finsupp.erase n toFinsupp✝) i = if i = n then 0 else ↑toFinsupp✝ i",
"state_before": "case ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn i : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ coeff (erase n { toFinsupp := toFinsupp✝ }) i = if i = n then 0 else coeff { toFinsupp := toFinsupp✝ } i",
"tactic": "simp only [erase_def, coeff]"
},
{
"state_after": "no goals",
"state_before": "case ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn i : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ↑(Finsupp.erase n toFinsupp✝) i = if i = n then 0 else ↑toFinsupp✝ i",
"tactic": "exact ite_congr rfl (fun _ => rfl) (fun _ => rfl)"
}
] |
[
1052,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1047,
1
] |
Mathlib/SetTheory/Cardinal/Continuum.lean
|
Cardinal.continuum_add_aleph0
|
[] |
[
132,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.yn_ge_n
|
[] |
[
429,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
1
] |
Mathlib/SetTheory/Ordinal/Principal.lean
|
Ordinal.nfp_le_of_principal
|
[] |
[
89,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
comp_comp_symm_mem_uniformity_sets
|
[
{
"state_after": "case intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.53210\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nw : Set (α × α)\nw_in : w ∈ 𝓤 α\nleft✝ : SymmetricRel w\nw_sub : w ○ w ⊆ s\n⊢ ∃ t, t ∈ 𝓤 α ∧ SymmetricRel t ∧ t ○ t ○ t ⊆ s",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.53210\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\n⊢ ∃ t, t ∈ 𝓤 α ∧ SymmetricRel t ∧ t ○ t ○ t ⊆ s",
"tactic": "rcases comp_symm_mem_uniformity_sets hs with ⟨w, w_in, _, w_sub⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.53210\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nw : Set (α × α)\nw_in : w ∈ 𝓤 α\nleft✝ : SymmetricRel w\nw_sub : w ○ w ⊆ s\nt : Set (α × α)\nt_in : t ∈ 𝓤 α\nt_symm : SymmetricRel t\nt_sub : t ○ t ⊆ w\n⊢ ∃ t, t ∈ 𝓤 α ∧ SymmetricRel t ∧ t ○ t ○ t ⊆ s",
"state_before": "case intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.53210\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nw : Set (α × α)\nw_in : w ∈ 𝓤 α\nleft✝ : SymmetricRel w\nw_sub : w ○ w ⊆ s\n⊢ ∃ t, t ∈ 𝓤 α ∧ SymmetricRel t ∧ t ○ t ○ t ⊆ s",
"tactic": "rcases comp_symm_mem_uniformity_sets w_in with ⟨t, t_in, t_symm, t_sub⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.53210\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nw : Set (α × α)\nw_in : w ∈ 𝓤 α\nleft✝ : SymmetricRel w\nw_sub : w ○ w ⊆ s\nt : Set (α × α)\nt_in : t ∈ 𝓤 α\nt_symm : SymmetricRel t\nt_sub : t ○ t ⊆ w\n⊢ t ○ t ○ t ⊆ s",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.53210\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nw : Set (α × α)\nw_in : w ∈ 𝓤 α\nleft✝ : SymmetricRel w\nw_sub : w ○ w ⊆ s\nt : Set (α × α)\nt_in : t ∈ 𝓤 α\nt_symm : SymmetricRel t\nt_sub : t ○ t ⊆ w\n⊢ ∃ t, t ∈ 𝓤 α ∧ SymmetricRel t ∧ t ○ t ○ t ⊆ s",
"tactic": "use t, t_in, t_symm"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.53210\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nw : Set (α × α)\nw_in : w ∈ 𝓤 α\nleft✝ : SymmetricRel w\nw_sub : w ○ w ⊆ s\nt : Set (α × α)\nt_in : t ∈ 𝓤 α\nt_symm : SymmetricRel t\nt_sub : t ○ t ⊆ w\nthis : t ⊆ t ○ t\n⊢ t ○ t ○ t ⊆ s",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.53210\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nw : Set (α × α)\nw_in : w ∈ 𝓤 α\nleft✝ : SymmetricRel w\nw_sub : w ○ w ⊆ s\nt : Set (α × α)\nt_in : t ∈ 𝓤 α\nt_symm : SymmetricRel t\nt_sub : t ○ t ⊆ w\n⊢ t ○ t ○ t ⊆ s",
"tactic": "have : t ⊆ t ○ t := subset_comp_self_of_mem_uniformity t_in"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.53210\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nw : Set (α × α)\nw_in : w ∈ 𝓤 α\nleft✝ : SymmetricRel w\nw_sub : w ○ w ⊆ s\nt : Set (α × α)\nt_in : t ∈ 𝓤 α\nt_symm : SymmetricRel t\nt_sub : t ○ t ⊆ w\nthis : t ⊆ t ○ t\n⊢ t ○ t ○ t ⊆ s",
"tactic": "calc\n t ○ t ○ t ⊆ w ○ t := compRel_mono t_sub Subset.rfl\n _ ⊆ w ○ (t ○ t) := compRel_mono Subset.rfl this\n _ ⊆ w ○ w := compRel_mono Subset.rfl t_sub\n _ ⊆ s := w_sub"
}
] |
[
604,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
593,
1
] |
Mathlib/GroupTheory/GroupAction/Group.lean
|
MulAction.injective₀
|
[] |
[
229,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
11
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.valid'_nil
|
[] |
[
1060,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1059,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.Eventually.exists
|
[] |
[
1300,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1298,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_sum_le
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type u_1\ns : Finset ι\nf : ι → R[X]\n⊢ degree (∑ i in ∅, f i) ≤ sup ∅ fun b => degree (f b)",
"tactic": "simp only [sum_empty, sup_empty, degree_zero, le_refl]"
},
{
"state_after": "R : Type u\nS : Type v\na✝ b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type u_1\ns✝ : Finset ι\nf : ι → R[X]\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih : degree (∑ i in s, f i) ≤ sup s fun b => degree (f b)\n⊢ degree (f a + ∑ x in s, f x) ≤ max (degree (f a)) (degree (∑ i in s, f i))",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type u_1\ns✝ : Finset ι\nf : ι → R[X]\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih : degree (∑ i in s, f i) ≤ sup s fun b => degree (f b)\n⊢ degree (∑ i in insert a s, f i) ≤ max (degree (f a)) (degree (∑ i in s, f i))",
"tactic": "rw [sum_insert has]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type u_1\ns✝ : Finset ι\nf : ι → R[X]\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih : degree (∑ i in s, f i) ≤ sup s fun b => degree (f b)\n⊢ degree (f a + ∑ x in s, f x) ≤ max (degree (f a)) (degree (∑ i in s, f i))",
"tactic": "exact degree_add_le _ _"
},
{
"state_after": "R : Type u\nS : Type v\na✝ b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type u_1\ns✝ : Finset ι\nf : ι → R[X]\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih : degree (∑ i in s, f i) ≤ sup s fun b => degree (f b)\n⊢ max (degree (f a)) (degree (∑ i in s, f i)) ≤ max (degree (f a)) (sup s fun b => degree (f b))",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type u_1\ns✝ : Finset ι\nf : ι → R[X]\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih : degree (∑ i in s, f i) ≤ sup s fun b => degree (f b)\n⊢ max (degree (f a)) (degree (∑ i in s, f i)) ≤ sup (insert a s) fun b => degree (f b)",
"tactic": "rw [sup_insert, sup_eq_max]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type u_1\ns✝ : Finset ι\nf : ι → R[X]\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih : degree (∑ i in s, f i) ≤ sup s fun b => degree (f b)\n⊢ max (degree (f a)) (degree (∑ i in s, f i)) ≤ max (degree (f a)) (sup s fun b => degree (f b))",
"tactic": "exact max_le_max le_rfl ih"
}
] |
[
762,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
755,
1
] |
Mathlib/Data/Nat/Bits.lean
|
Nat.bodd_bit0
|
[] |
[
47,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
BoundedLatticeHom.coe_comp_sup_hom
|
[] |
[
1322,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1320,
1
] |
Mathlib/GroupTheory/Index.lean
|
Subgroup.relindex_inf_ne_zero
|
[
{
"state_after": "G : Type u_1\ninst✝ : Group G\nH K L : Subgroup G\nhK : relindex K L ≠ 0\nhH : relindex H (K ⊓ L) ≠ 0\n⊢ relindex (H ⊓ K) L ≠ 0",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K L : Subgroup G\nhH : relindex H L ≠ 0\nhK : relindex K L ≠ 0\n⊢ relindex (H ⊓ K) L ≠ 0",
"tactic": "replace hH : H.relindex (K ⊓ L) ≠ 0 := mt (relindex_eq_zero_of_le_right inf_le_right) hH"
},
{
"state_after": "G : Type u_1\ninst✝ : Group G\nH K L : Subgroup G\nhK : relindex (K ⊓ L) L ≠ 0\nhH : relindex (H ⊓ (K ⊓ L)) (K ⊓ L) ≠ 0\n⊢ relindex (H ⊓ K ⊓ L) L ≠ 0",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K L : Subgroup G\nhK : relindex K L ≠ 0\nhH : relindex H (K ⊓ L) ≠ 0\n⊢ relindex (H ⊓ K) L ≠ 0",
"tactic": "rw [← inf_relindex_right] at hH hK⊢"
},
{
"state_after": "G : Type u_1\ninst✝ : Group G\nH K L : Subgroup G\nhK : relindex (K ⊓ L) L ≠ 0\nhH : relindex (H ⊓ (K ⊓ L)) (K ⊓ L) ≠ 0\n⊢ relindex (H ⊓ (K ⊓ L)) L ≠ 0",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K L : Subgroup G\nhK : relindex (K ⊓ L) L ≠ 0\nhH : relindex (H ⊓ (K ⊓ L)) (K ⊓ L) ≠ 0\n⊢ relindex (H ⊓ K ⊓ L) L ≠ 0",
"tactic": "rw [inf_assoc]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K L : Subgroup G\nhK : relindex (K ⊓ L) L ≠ 0\nhH : relindex (H ⊓ (K ⊓ L)) (K ⊓ L) ≠ 0\n⊢ relindex (H ⊓ (K ⊓ L)) L ≠ 0",
"tactic": "exact relindex_ne_zero_trans hH hK"
}
] |
[
425,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.eqOnSource_refl
|
[] |
[
829,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
828,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
RingHom.map_multiset_sum
|
[] |
[
259,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
11
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.filter_single_neg
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ns : Finset ι\nx✝ : (i : ↑↑s) → β ↑i\ni✝ : ι\np : ι → Prop\ninst✝ : DecidablePred p\ni : ι\nx : β i\nh : ¬p i\n⊢ filter p (single i x) = 0",
"tactic": "rw [filter_single, if_neg h]"
}
] |
[
716,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
715,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
DifferentiableOn.snd
|
[] |
[
312,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
310,
11
] |
Mathlib/Algebra/Support.lean
|
Function.support_one
|
[] |
[
310,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.single_mul_single
|
[
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.365398\ninst✝¹ : Semiring k\ninst✝ : Mul G\na₁ a₂ : G\nb₁ b₂ : k\n⊢ (sum (single a₂ b₂) fun a₂ b₂ => single (a₁ * a₂) (0 * b₂)) = 0",
"tactic": "simp only [zero_mul, single_zero, sum_zero]"
},
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.365398\ninst✝¹ : Semiring k\ninst✝ : Mul G\na₁ a₂ : G\nb₁ b₂ : k\n⊢ single (a₁ * a₂) (b₁ * 0) = 0",
"tactic": "rw [mul_zero, single_zero]"
}
] |
[
461,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.tendsto_cos_neg_pi_div_two
|
[
{
"state_after": "case h1\n\n⊢ Tendsto cos (𝓝[Ioi (-(π / 2))] (-(π / 2))) (𝓝 0)\n\ncase h2\n\n⊢ ∀ᶠ (x : ℝ) in 𝓝[Ioi (-(π / 2))] (-(π / 2)), cos x ∈ Ioi 0",
"state_before": "⊢ Tendsto cos (𝓝[Ioi (-(π / 2))] (-(π / 2))) (𝓝[Ioi 0] 0)",
"tactic": "apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within"
},
{
"state_after": "case h.e'_5.h.e'_3\n\n⊢ 0 = cos (-(π / 2))",
"state_before": "case h1\n\n⊢ Tendsto cos (𝓝[Ioi (-(π / 2))] (-(π / 2))) (𝓝 0)",
"tactic": "convert continuous_cos.continuousWithinAt.tendsto"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.e'_3\n\n⊢ 0 = cos (-(π / 2))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h2\n\n⊢ ∀ᶠ (x : ℝ) in 𝓝[Ioi (-(π / 2))] (-(π / 2)), cos x ∈ Ioi 0",
"tactic": "filter_upwards [Ioo_mem_nhdsWithin_Ioi\n (left_mem_Ico.mpr (neg_lt_self pi_div_two_pos))]with x hx using cos_pos_of_mem_Ioo hx"
}
] |
[
1087,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1082,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.smul_apply
|
[] |
[
182,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean
|
Polynomial.IsWeaklyEisensteinAt.exists_mem_adjoin_mul_eq_pow_natDegree
|
[
{
"state_after": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) +\n ∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i =\n 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : ↑(aeval x) f = 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)",
"tactic": "rw [aeval_def, Polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,\n sum_insert not_mem_range_self, sum_range, (hmo.map (algebraMap R S)).coeff_natDegree,\n one_mul] at hx"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) +\n ∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i =\n 0\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)",
"tactic": "replace hx := eq_neg_of_add_eq_zero_left hx"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i\nthis : ∀ (n : ℕ), n < natDegree f → p ∣ coeff f n\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)",
"tactic": "have : ∀ n < f.natDegree, p ∣ f.coeff n := by\n intro n hn\n refine' mem_span_singleton.1 (by simpa using hf.mem hn)"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i\nφ : ℕ → R\nhφ : ∀ (n : ℕ), n < natDegree f → coeff f n = p * φ n\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i\nthis : ∀ (n : ℕ), n < natDegree f → p ∣ coeff f n\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)",
"tactic": "choose! φ hφ using this"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nφ : ℕ → R\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)),\n ↑(algebraMap R S) p * (↑(algebraMap R S) (φ ↑i) * x ^ ↑i)\nhφ : ∀ (n : ℕ), n < natDegree f → coeff f n = p * φ n\n⊢ ∃ y,\n y ∈ adjoin R {x} ∧\n ↑(algebraMap R S) p * y =\n ↑(algebraMap R S) p *\n (-1 * ∑ x_1 : Fin (natDegree (Polynomial.map (algebraMap R S) f)), ↑(algebraMap R S) (φ ↑x_1) * x ^ ↑x_1)",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nφ : ℕ → R\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)),\n ↑(algebraMap R S) p * (↑(algebraMap R S) (φ ↑i) * x ^ ↑i)\nhφ : ∀ (n : ℕ), n < natDegree f → coeff f n = p * φ n\n⊢ ∃ y, y ∈ adjoin R {x} ∧ ↑(algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)",
"tactic": "rw [hx, ← mul_sum, neg_eq_neg_one_mul, ← mul_assoc (-1 : S), mul_comm (-1 : S), mul_assoc]"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nφ : ℕ → R\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)),\n ↑(algebraMap R S) p * (↑(algebraMap R S) (φ ↑i) * x ^ ↑i)\nhφ : ∀ (n : ℕ), n < natDegree f → coeff f n = p * φ n\n⊢ -1 * ∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), ↑(algebraMap R S) (φ ↑i) * x ^ ↑i ∈ adjoin R {x}",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nφ : ℕ → R\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)),\n ↑(algebraMap R S) p * (↑(algebraMap R S) (φ ↑i) * x ^ ↑i)\nhφ : ∀ (n : ℕ), n < natDegree f → coeff f n = p * φ n\n⊢ ∃ y,\n y ∈ adjoin R {x} ∧\n ↑(algebraMap R S) p * y =\n ↑(algebraMap R S) p *\n (-1 * ∑ x_1 : Fin (natDegree (Polynomial.map (algebraMap R S) f)), ↑(algebraMap R S) (φ ↑x_1) * x ^ ↑x_1)",
"tactic": "refine'\n ⟨-1 * ∑ i : Fin (f.map (algebraMap R S)).natDegree, (algebraMap R S) (φ i.1) * x ^ i.1, _, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nφ : ℕ → R\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)),\n ↑(algebraMap R S) p * (↑(algebraMap R S) (φ ↑i) * x ^ ↑i)\nhφ : ∀ (n : ℕ), n < natDegree f → coeff f n = p * φ n\n⊢ -1 * ∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), ↑(algebraMap R S) (φ ↑i) * x ^ ↑i ∈ adjoin R {x}",
"tactic": "exact\n Subalgebra.mul_mem _ (Subalgebra.neg_mem _ (Subalgebra.one_mem _))\n (Subalgebra.sum_mem _ fun i _ =>\n Subalgebra.mul_mem _ (Subalgebra.algebraMap_mem _ _)\n (Subalgebra.pow_mem _ (subset_adjoin (Set.mem_singleton x)) _))"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i\nn : ℕ\nhn : n < natDegree f\n⊢ p ∣ coeff f n",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i\n⊢ ∀ (n : ℕ), n < natDegree f → p ∣ coeff f n",
"tactic": "intro n hn"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i\nn : ℕ\nhn : n < natDegree f\n⊢ p ∣ coeff f n",
"tactic": "refine' mem_span_singleton.1 (by simpa using hf.mem hn)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nhf✝ : IsWeaklyEisensteinAt f 𝓟\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : Monic f\nhf : IsWeaklyEisensteinAt f (Submodule.span R {p})\nhx :\n x ^ natDegree (Polynomial.map (algebraMap R S) f) =\n -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i\nn : ℕ\nhn : n < natDegree f\n⊢ coeff f n ∈ span {p}",
"tactic": "simpa using hf.mem hn"
}
] |
[
112,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.nat_cast_eq_nat_cast_iff'
|
[] |
[
471,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
470,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.eventually_principal
|
[] |
[
1232,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1231,
1
] |
Mathlib/Algebra/Group/TypeTags.lean
|
toMul_sub
|
[] |
[
347,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
1
] |
Mathlib/Data/Nat/Log.lean
|
Nat.clog_monotone
|
[] |
[
350,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
AntitoneOn.inv
|
[] |
[
1305,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1304,
1
] |
Mathlib/SetTheory/Ordinal/Exponential.lean
|
Ordinal.log_def
|
[
{
"state_after": "no goals",
"state_before": "b : Ordinal\nh : 1 < b\nx : Ordinal\n⊢ log b x = pred (sInf {o | x < b ^ o})",
"tactic": "simp only [log, dif_pos h]"
}
] |
[
267,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_erase_lt_of_mem
|
[] |
[
146,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
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