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/Topology/Sets/Compacts.lean
|
TopologicalSpace.NonemptyCompacts.coe_toCompacts
|
[] |
[
257,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
Ideal.IsHomogeneous.toIdeal_homogeneousCore_eq_self
|
[
{
"state_after": "ι : Type u_1\nσ : Type u_2\nR : Type ?u.105177\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\n⊢ I ≤ homogeneousCore' 𝒜 I",
"state_before": "ι : Type u_1\nσ : Type u_2\nR : Type ?u.105177\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\n⊢ HomogeneousIdeal.toIdeal (homogeneousCore 𝒜 I) = I",
"tactic": "apply le_antisymm (I.homogeneousCore'_le 𝒜) _"
},
{
"state_after": "ι : Type u_1\nσ : Type u_2\nR : Type ?u.105177\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\nx : A\nhx : x ∈ I\n⊢ x ∈ homogeneousCore' 𝒜 I",
"state_before": "ι : Type u_1\nσ : Type u_2\nR : Type ?u.105177\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\n⊢ I ≤ homogeneousCore' 𝒜 I",
"tactic": "intro x hx"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nσ : Type u_2\nR : Type ?u.105177\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\nx : A\nhx : x ∈ I\n⊢ x ∈ homogeneousCore' 𝒜 I",
"tactic": "classical\nrw [← DirectSum.sum_support_decompose 𝒜 x]\nexact Ideal.sum_mem _ fun j _ => Ideal.subset_span ⟨⟨_, homogeneous_coe _⟩, h _ hx, rfl⟩"
},
{
"state_after": "ι : Type u_1\nσ : Type u_2\nR : Type ?u.105177\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\nx : A\nhx : x ∈ I\n⊢ ∑ i in Dfinsupp.support (↑(decompose 𝒜) x), ↑(↑(↑(decompose 𝒜) x) i) ∈ homogeneousCore' 𝒜 I",
"state_before": "ι : Type u_1\nσ : Type u_2\nR : Type ?u.105177\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\nx : A\nhx : x ∈ I\n⊢ x ∈ homogeneousCore' 𝒜 I",
"tactic": "rw [← DirectSum.sum_support_decompose 𝒜 x]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nσ : Type u_2\nR : Type ?u.105177\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\nx : A\nhx : x ∈ I\n⊢ ∑ i in Dfinsupp.support (↑(decompose 𝒜) x), ↑(↑(↑(decompose 𝒜) x) i) ∈ homogeneousCore' 𝒜 I",
"tactic": "exact Ideal.sum_mem _ fun j _ => Ideal.subset_span ⟨⟨_, homogeneous_coe _⟩, h _ hx, rfl⟩"
}
] |
[
212,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/NumberTheory/Divisors.lean
|
Nat.properDivisors_eq_singleton_one_iff_prime
|
[
{
"state_after": "case refine_1\nn : ℕ\n⊢ properDivisors n = {1} → Prime n\n\ncase refine_2\nn : ℕ\n⊢ Prime n → properDivisors n = {1}",
"state_before": "n : ℕ\n⊢ properDivisors n = {1} ↔ Prime n",
"tactic": "refine ⟨?_, ?_⟩"
},
{
"state_after": "case refine_1\nn : ℕ\nh : properDivisors n = {1}\n⊢ Prime n",
"state_before": "case refine_1\nn : ℕ\n⊢ properDivisors n = {1} → Prime n",
"tactic": "intro h"
},
{
"state_after": "case refine_1.refine'_1\nn : ℕ\nh : properDivisors n = {1}\n⊢ 2 ≤ n\n\ncase refine_1.refine'_2\nn : ℕ\nh : properDivisors n = {1}\nm : ℕ\nhdvd : m ∣ n\n⊢ m = 1 ∨ m = n",
"state_before": "case refine_1\nn : ℕ\nh : properDivisors n = {1}\n⊢ Prime n",
"tactic": "refine' Nat.prime_def_lt''.mpr ⟨_, fun m hdvd => _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine_1.refine'_1\nn : ℕ\nh : properDivisors n = {1}\n⊢ 2 ≤ n",
"tactic": "match n with\n| 0 => contradiction\n| 1 => contradiction\n| Nat.succ (Nat.succ n) => simp [succ_le_succ]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nh : properDivisors 0 = {1}\n⊢ 2 ≤ 0",
"tactic": "contradiction"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nh : properDivisors 1 = {1}\n⊢ 2 ≤ 1",
"tactic": "contradiction"
},
{
"state_after": "no goals",
"state_before": "n✝ n : ℕ\nh : properDivisors (succ (succ n)) = {1}\n⊢ 2 ≤ succ (succ n)",
"tactic": "simp [succ_le_succ]"
},
{
"state_after": "case refine_1.refine'_2\nn : ℕ\nh : properDivisors n = {1}\nm : ℕ\nhdvd : m ∣ n\n⊢ m ∣ n ∧ m < n ∨ m = n",
"state_before": "case refine_1.refine'_2\nn : ℕ\nh : properDivisors n = {1}\nm : ℕ\nhdvd : m ∣ n\n⊢ m = 1 ∨ m = n",
"tactic": "rw [← mem_singleton, ← h, mem_properDivisors]"
},
{
"state_after": "case refine_1.refine'_2.refine_2\nn : ℕ\nh : properDivisors n = {1}\nm : ℕ\nhdvd : m ∣ n\nthis : m ≤ n\n⊢ m ∣ n ∧ m < n ∨ m = n\n\ncase refine_1.refine'_2.refine_1\nn : ℕ\nh : properDivisors n = {1}\nm : ℕ\nhdvd : m ∣ n\n⊢ 0 < n",
"state_before": "case refine_1.refine'_2\nn : ℕ\nh : properDivisors n = {1}\nm : ℕ\nhdvd : m ∣ n\n⊢ m ∣ n ∧ m < n ∨ m = n",
"tactic": "have := Nat.le_of_dvd ?_ hdvd"
},
{
"state_after": "case refine_1.refine'_2.refine_2\nn : ℕ\nh : properDivisors n = {1}\nm : ℕ\nhdvd : m ∣ n\nthis : m ≤ n\n⊢ m < n ∨ m = n",
"state_before": "case refine_1.refine'_2.refine_2\nn : ℕ\nh : properDivisors n = {1}\nm : ℕ\nhdvd : m ∣ n\nthis : m ≤ n\n⊢ m ∣ n ∧ m < n ∨ m = n",
"tactic": "simp [hdvd, this]"
},
{
"state_after": "no goals",
"state_before": "case refine_1.refine'_2.refine_2\nn : ℕ\nh : properDivisors n = {1}\nm : ℕ\nhdvd : m ∣ n\nthis : m ≤ n\n⊢ m < n ∨ m = n",
"tactic": "exact (le_iff_eq_or_lt.mp this).symm"
},
{
"state_after": "case refine_1.refine'_2.refine_1\nn : ℕ\nh : properDivisors n = {1}\nm : ℕ\nhdvd : m ∣ n\nthis : n ≤ 0\n⊢ False",
"state_before": "case refine_1.refine'_2.refine_1\nn : ℕ\nh : properDivisors n = {1}\nm : ℕ\nhdvd : m ∣ n\n⊢ 0 < n",
"tactic": "by_contra'"
},
{
"state_after": "no goals",
"state_before": "case refine_1.refine'_2.refine_1\nn : ℕ\nh : properDivisors n = {1}\nm : ℕ\nhdvd : m ∣ n\nthis : n ≤ 0\n⊢ False",
"tactic": "simp [nonpos_iff_eq_zero.mp this, this] at h"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nn : ℕ\n⊢ Prime n → properDivisors n = {1}",
"tactic": "exact fun h => Prime.properDivisors h"
}
] |
[
407,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.erase_single_ne
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.165853\nγ : Type ?u.165856\nι : Type ?u.165859\nM : Type u_2\nM' : Type ?u.165865\nN : Type ?u.165868\nP : Type ?u.165871\nG : Type ?u.165874\nH : Type ?u.165877\nR : Type ?u.165880\nS : Type ?u.165883\ninst✝ : Zero M\na a' : α\nb : M\nh : a ≠ a'\ns : α\n⊢ ↑(erase a (single a' b)) s = ↑(single a' b) s",
"state_before": "α : Type u_1\nβ : Type ?u.165853\nγ : Type ?u.165856\nι : Type ?u.165859\nM : Type u_2\nM' : Type ?u.165865\nN : Type ?u.165868\nP : Type ?u.165871\nG : Type ?u.165874\nH : Type ?u.165877\nR : Type ?u.165880\nS : Type ?u.165883\ninst✝ : Zero M\na a' : α\nb : M\nh : a ≠ a'\n⊢ erase a (single a' b) = single a' b",
"tactic": "ext s"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.165853\nγ : Type ?u.165856\nι : Type ?u.165859\nM : Type u_2\nM' : Type ?u.165865\nN : Type ?u.165868\nP : Type ?u.165871\nG : Type ?u.165874\nH : Type ?u.165877\nR : Type ?u.165880\nS : Type ?u.165883\ninst✝ : Zero M\na a' : α\nb : M\nh : a ≠ a'\ns : α\nhs : s = a\n⊢ ↑(erase a (single a' b)) s = ↑(single a' b) s\n\ncase neg\nα : Type u_1\nβ : Type ?u.165853\nγ : Type ?u.165856\nι : Type ?u.165859\nM : Type u_2\nM' : Type ?u.165865\nN : Type ?u.165868\nP : Type ?u.165871\nG : Type ?u.165874\nH : Type ?u.165877\nR : Type ?u.165880\nS : Type ?u.165883\ninst✝ : Zero M\na a' : α\nb : M\nh : a ≠ a'\ns : α\nhs : ¬s = a\n⊢ ↑(erase a (single a' b)) s = ↑(single a' b) s",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.165853\nγ : Type ?u.165856\nι : Type ?u.165859\nM : Type u_2\nM' : Type ?u.165865\nN : Type ?u.165868\nP : Type ?u.165871\nG : Type ?u.165874\nH : Type ?u.165877\nR : Type ?u.165880\nS : Type ?u.165883\ninst✝ : Zero M\na a' : α\nb : M\nh : a ≠ a'\ns : α\n⊢ ↑(erase a (single a' b)) s = ↑(single a' b) s",
"tactic": "by_cases hs : s = a"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.165853\nγ : Type ?u.165856\nι : Type ?u.165859\nM : Type u_2\nM' : Type ?u.165865\nN : Type ?u.165868\nP : Type ?u.165871\nG : Type ?u.165874\nH : Type ?u.165877\nR : Type ?u.165880\nS : Type ?u.165883\ninst✝ : Zero M\na a' : α\nb : M\nh : a ≠ a'\ns : α\nhs : s = a\n⊢ ↑(erase a (single a' b)) s = ↑(single a' b) s",
"tactic": "rw [hs, erase_same, single_eq_of_ne h.symm]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.165853\nγ : Type ?u.165856\nι : Type ?u.165859\nM : Type u_2\nM' : Type ?u.165865\nN : Type ?u.165868\nP : Type ?u.165871\nG : Type ?u.165874\nH : Type ?u.165877\nR : Type ?u.165880\nS : Type ?u.165883\ninst✝ : Zero M\na a' : α\nb : M\nh : a ≠ a'\ns : α\nhs : ¬s = a\n⊢ ↑(erase a (single a' b)) s = ↑(single a' b) s",
"tactic": "rw [erase_ne hs]"
}
] |
[
664,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
661,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.map_add
|
[] |
[
1862,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1861,
1
] |
Mathlib/Topology/UniformSpace/Completion.lean
|
UniformSpace.Completion.induction_on₃
|
[] |
[
516,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
510,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.diff_subset_iff
|
[] |
[
1925,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1924,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
csSup_image2_eq_csSup_csSup
|
[
{
"state_after": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nι : Sort ?u.111757\ninst✝² : ConditionallyCompleteLattice α\ninst✝¹ : ConditionallyCompleteLattice β\ninst✝ : ConditionallyCompleteLattice γ\nf : α → β → γ\ns : Set α\nt : Set β\nl u : α → β → γ\nl₁ u₁ : β → γ → α\nl₂ u₂ : α → γ → β\nh₁ : ∀ (b : β), GaloisConnection (swap l b) (u₁ b)\nh₂ : ∀ (a : α), GaloisConnection (l a) (u₂ a)\nhs₀ : Set.Nonempty s\nhs₁ : BddAbove s\nht₀ : Set.Nonempty t\nht₁ : BddAbove t\nc : γ\n⊢ sSup (image2 l s t) ≤ c ↔ l (sSup s) (sSup t) ≤ c",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nι : Sort ?u.111757\ninst✝² : ConditionallyCompleteLattice α\ninst✝¹ : ConditionallyCompleteLattice β\ninst✝ : ConditionallyCompleteLattice γ\nf : α → β → γ\ns : Set α\nt : Set β\nl u : α → β → γ\nl₁ u₁ : β → γ → α\nl₂ u₂ : α → γ → β\nh₁ : ∀ (b : β), GaloisConnection (swap l b) (u₁ b)\nh₂ : ∀ (a : α), GaloisConnection (l a) (u₂ a)\nhs₀ : Set.Nonempty s\nhs₁ : BddAbove s\nht₀ : Set.Nonempty t\nht₁ : BddAbove t\n⊢ sSup (image2 l s t) = l (sSup s) (sSup t)",
"tactic": "refine' eq_of_forall_ge_iff fun c => _"
},
{
"state_after": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nι : Sort ?u.111757\ninst✝² : ConditionallyCompleteLattice α\ninst✝¹ : ConditionallyCompleteLattice β\ninst✝ : ConditionallyCompleteLattice γ\nf : α → β → γ\ns : Set α\nt : Set β\nl u : α → β → γ\nl₁ u₁ : β → γ → α\nl₂ u₂ : α → γ → β\nh₁ : ∀ (b : β), GaloisConnection (swap l b) (u₁ b)\nh₂ : ∀ (a : α), GaloisConnection (l a) (u₂ a)\nhs₀ : Set.Nonempty s\nhs₁ : BddAbove s\nht₀ : Set.Nonempty t\nht₁ : BddAbove t\nc : γ\n⊢ (∀ (i₂ : β), i₂ ∈ t → ∀ (i₁ : α), i₁ ∈ s → l i₁ i₂ ≤ c) ↔ ∀ (b : β), b ∈ t → b ≤ u₂ (sSup s) c",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nι : Sort ?u.111757\ninst✝² : ConditionallyCompleteLattice α\ninst✝¹ : ConditionallyCompleteLattice β\ninst✝ : ConditionallyCompleteLattice γ\nf : α → β → γ\ns : Set α\nt : Set β\nl u : α → β → γ\nl₁ u₁ : β → γ → α\nl₂ u₂ : α → γ → β\nh₁ : ∀ (b : β), GaloisConnection (swap l b) (u₁ b)\nh₂ : ∀ (a : α), GaloisConnection (l a) (u₂ a)\nhs₀ : Set.Nonempty s\nhs₁ : BddAbove s\nht₀ : Set.Nonempty t\nht₁ : BddAbove t\nc : γ\n⊢ sSup (image2 l s t) ≤ c ↔ l (sSup s) (sSup t) ≤ c",
"tactic": "rw [csSup_le_iff (hs₁.image2 (fun _ => (h₁ _).monotone_l) (fun _ => (h₂ _).monotone_l) ht₁)\n (hs₀.image2 ht₀),\n forall_image2_iff, forall₂_swap, (h₂ _).le_iff_le, csSup_le_iff ht₁ ht₀]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nι : Sort ?u.111757\ninst✝² : ConditionallyCompleteLattice α\ninst✝¹ : ConditionallyCompleteLattice β\ninst✝ : ConditionallyCompleteLattice γ\nf : α → β → γ\ns : Set α\nt : Set β\nl u : α → β → γ\nl₁ u₁ : β → γ → α\nl₂ u₂ : α → γ → β\nh₁ : ∀ (b : β), GaloisConnection (swap l b) (u₁ b)\nh₂ : ∀ (a : α), GaloisConnection (l a) (u₂ a)\nhs₀ : Set.Nonempty s\nhs₁ : BddAbove s\nht₀ : Set.Nonempty t\nht₁ : BddAbove t\nc : γ\n⊢ (∀ (i₂ : β), i₂ ∈ t → ∀ (i₁ : α), i₁ ∈ s → l i₁ i₂ ≤ c) ↔ ∀ (b : β), b ∈ t → b ≤ u₂ (sSup s) c",
"tactic": "simp_rw [← (h₂ _).le_iff_le, (h₁ _).le_iff_le, csSup_le_iff hs₁ hs₀]"
}
] |
[
1388,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1381,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.omega_pos
|
[] |
[
182,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.dvd_sub
|
[
{
"state_after": "no goals",
"state_before": "k m n : Nat\nH : n ≤ m\nh₁ : k ∣ m\nh₂ : k ∣ n\n⊢ k ∣ m - n + n",
"tactic": "rwa [Nat.sub_add_cancel H]"
}
] |
[
680,
63
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
679,
1
] |
Mathlib/Data/Finite/Set.lean
|
Finite.of_injective_finite_range
|
[] |
[
37,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
1
] |
Mathlib/Topology/Constructions.lean
|
continuous_uncurry_of_discreteTopology_left
|
[] |
[
681,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
677,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter_inter
|
[] |
[
552,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
551,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.image_eq_target_inter_inv_preimage
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.32478\nδ : Type ?u.32481\ne : LocalEquiv α β\ne' : LocalEquiv β γ\ns : Set α\nh : s ⊆ e.source\n⊢ ↑e '' s = e.target ∩ ↑(LocalEquiv.symm e) ⁻¹' s",
"tactic": "rw [← e.image_source_inter_eq', inter_eq_self_of_subset_right h]"
}
] |
[
496,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
494,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.finite_const_le_of_tsum_ne_top
|
[
{
"state_after": "α : Type ?u.301039\nβ : Type ?u.301042\nγ : Type ?u.301045\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε✝ ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf g : α → ℝ≥0∞\nι : Type u_1\na : ι → ℝ≥0∞\ntsum_ne_top : (∑' (i : ι), a i) ≠ ⊤\nε : ℝ≥0∞\nε_ne_zero : ε ≠ 0\nh : ¬Set.Finite {i | ε ≤ a i}\n⊢ False",
"state_before": "α : Type ?u.301039\nβ : Type ?u.301042\nγ : Type ?u.301045\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε✝ ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf g : α → ℝ≥0∞\nι : Type u_1\na : ι → ℝ≥0∞\ntsum_ne_top : (∑' (i : ι), a i) ≠ ⊤\nε : ℝ≥0∞\nε_ne_zero : ε ≠ 0\n⊢ Set.Finite {i | ε ≤ a i}",
"tactic": "by_contra h"
},
{
"state_after": "α : Type ?u.301039\nβ : Type ?u.301042\nγ : Type ?u.301045\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε✝ ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf g : α → ℝ≥0∞\nι : Type u_1\na : ι → ℝ≥0∞\ntsum_ne_top : (∑' (i : ι), a i) ≠ ⊤\nε : ℝ≥0∞\nε_ne_zero : ε ≠ 0\nh : ¬Set.Finite {i | ε ≤ a i}\nthis : Infinite ↑{i | ε ≤ a i}\n⊢ False",
"state_before": "α : Type ?u.301039\nβ : Type ?u.301042\nγ : Type ?u.301045\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε✝ ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf g : α → ℝ≥0∞\nι : Type u_1\na : ι → ℝ≥0∞\ntsum_ne_top : (∑' (i : ι), a i) ≠ ⊤\nε : ℝ≥0∞\nε_ne_zero : ε ≠ 0\nh : ¬Set.Finite {i | ε ≤ a i}\n⊢ False",
"tactic": "have := Infinite.to_subtype h"
},
{
"state_after": "α : Type ?u.301039\nβ : Type ?u.301042\nγ : Type ?u.301045\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε✝ ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf g : α → ℝ≥0∞\nι : Type u_1\na : ι → ℝ≥0∞\ntsum_ne_top : (∑' (i : ι), a i) ≠ ⊤\nε : ℝ≥0∞\nε_ne_zero : ε ≠ 0\nh : ¬Set.Finite {i | ε ≤ a i}\nthis : Infinite ↑{i | ε ≤ a i}\n⊢ ⊤ ≤ ∑' (i : ι), a i",
"state_before": "α : Type ?u.301039\nβ : Type ?u.301042\nγ : Type ?u.301045\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε✝ ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf g : α → ℝ≥0∞\nι : Type u_1\na : ι → ℝ≥0∞\ntsum_ne_top : (∑' (i : ι), a i) ≠ ⊤\nε : ℝ≥0∞\nε_ne_zero : ε ≠ 0\nh : ¬Set.Finite {i | ε ≤ a i}\nthis : Infinite ↑{i | ε ≤ a i}\n⊢ False",
"tactic": "refine tsum_ne_top (top_unique ?_)"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.301039\nβ : Type ?u.301042\nγ : Type ?u.301045\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε✝ ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf g : α → ℝ≥0∞\nι : Type u_1\na : ι → ℝ≥0∞\ntsum_ne_top : (∑' (i : ι), a i) ≠ ⊤\nε : ℝ≥0∞\nε_ne_zero : ε ≠ 0\nh : ¬Set.Finite {i | ε ≤ a i}\nthis : Infinite ↑{i | ε ≤ a i}\n⊢ ⊤ ≤ ∑' (i : ι), a i",
"tactic": "calc ⊤ = ∑' _ : { i | ε ≤ a i }, ε := (tsum_const_eq_top_of_ne_zero ε_ne_zero).symm\n_ ≤ ∑' i, a i := tsum_le_tsum_of_inj (↑) Subtype.val_injective (fun _ _ => zero_le _)\n (fun i => i.2) ENNReal.summable ENNReal.summable"
}
] |
[
1047,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1040,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sSup_empty
|
[] |
[
485,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean
|
CategoryTheory.GrothendieckTopology.sheafifyCompIso_inv_eq_sheafifyLift
|
[
{
"state_after": "case a\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝⁴ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\n⊢ toSheafify J (P ⋙ F) ≫ (sheafifyCompIso J F P).inv = whiskerRight (toSheafify J P) F",
"state_before": "C : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝⁴ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\n⊢ (sheafifyCompIso J F P).inv =\n sheafifyLift J (whiskerRight (toSheafify J P) F) (_ : Presheaf.IsSheaf J (sheafify J P ⋙ F))",
"tactic": "apply J.sheafifyLift_unique"
},
{
"state_after": "case a\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝⁴ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\n⊢ toSheafify J (P ⋙ F) = whiskerRight (toSheafify J P) F ≫ (sheafifyCompIso J F P).hom",
"state_before": "case a\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝⁴ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\n⊢ toSheafify J (P ⋙ F) ≫ (sheafifyCompIso J F P).inv = whiskerRight (toSheafify J P) F",
"tactic": "rw [Iso.comp_inv_eq]"
},
{
"state_after": "no goals",
"state_before": "case a\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝⁴ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : Cᵒᵖ ⥤ D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\n⊢ toSheafify J (P ⋙ F) = whiskerRight (toSheafify J P) F ≫ (sheafifyCompIso J F P).hom",
"tactic": "simp"
}
] |
[
158,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/NumberTheory/Bernoulli.lean
|
bernoulli_eq_bernoulli'_of_ne_one
|
[
{
"state_after": "case pos\nA : Type ?u.635032\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhn : n ≠ 1\nh0 : n = 0\n⊢ bernoulli n = bernoulli' n\n\ncase neg\nA : Type ?u.635032\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhn : n ≠ 1\nh0 : ¬n = 0\n⊢ bernoulli n = bernoulli' n",
"state_before": "A : Type ?u.635032\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhn : n ≠ 1\n⊢ bernoulli n = bernoulli' n",
"tactic": "by_cases h0 : n = 0"
},
{
"state_after": "case neg\nA : Type ?u.635032\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhn : n ≠ 1\nh0 : ¬n = 0\n⊢ (-1) ^ (n % 2) * bernoulli' n = bernoulli' n",
"state_before": "case neg\nA : Type ?u.635032\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhn : n ≠ 1\nh0 : ¬n = 0\n⊢ bernoulli n = bernoulli' n",
"tactic": "rw [bernoulli, neg_one_pow_eq_pow_mod_two]"
},
{
"state_after": "case neg.inl\nA : Type ?u.635032\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhn : n ≠ 1\nh0 : ¬n = 0\nh : n % 2 = 0\n⊢ (-1) ^ (n % 2) * bernoulli' n = bernoulli' n\n\ncase neg.inr\nA : Type ?u.635032\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhn : n ≠ 1\nh0 : ¬n = 0\nh : n % 2 = 1\n⊢ (-1) ^ (n % 2) * bernoulli' n = bernoulli' n",
"state_before": "case neg\nA : Type ?u.635032\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhn : n ≠ 1\nh0 : ¬n = 0\n⊢ (-1) ^ (n % 2) * bernoulli' n = bernoulli' n",
"tactic": "cases' mod_two_eq_zero_or_one n with h h"
},
{
"state_after": "no goals",
"state_before": "case pos\nA : Type ?u.635032\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhn : n ≠ 1\nh0 : n = 0\n⊢ bernoulli n = bernoulli' n",
"tactic": "simp [h0]"
},
{
"state_after": "no goals",
"state_before": "case neg.inl\nA : Type ?u.635032\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhn : n ≠ 1\nh0 : ¬n = 0\nh : n % 2 = 0\n⊢ (-1) ^ (n % 2) * bernoulli' n = bernoulli' n",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\nA : Type ?u.635032\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhn : n ≠ 1\nh0 : ¬n = 0\nh : n % 2 = 1\n⊢ (-1) ^ (n % 2) * bernoulli' n = bernoulli' n",
"tactic": "simp [bernoulli'_odd_eq_zero (odd_iff.mpr h) (one_lt_iff_ne_zero_and_ne_one.mpr ⟨h0, hn⟩)]"
}
] |
[
225,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
FormalMultilinearSeries.summable_nnnorm_mul_pow
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.220483\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np✝ : FormalMultilinearSeries 𝕜 E F\nr✝ : ℝ≥0\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nh : ↑r < radius p\n⊢ Summable fun a => ↑(‖p a‖₊ * r ^ a)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.220483\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np✝ : FormalMultilinearSeries 𝕜 E F\nr✝ : ℝ≥0\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nh : ↑r < radius p\n⊢ Summable fun n => ‖p n‖₊ * r ^ n",
"tactic": "rw [← NNReal.summable_coe]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.220483\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np✝ : FormalMultilinearSeries 𝕜 E F\nr✝ : ℝ≥0\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nh : ↑r < radius p\n⊢ Summable fun a => ‖p a‖ * ↑r ^ a",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.220483\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np✝ : FormalMultilinearSeries 𝕜 E F\nr✝ : ℝ≥0\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nh : ↑r < radius p\n⊢ Summable fun a => ↑(‖p a‖₊ * r ^ a)",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.220483\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np✝ : FormalMultilinearSeries 𝕜 E F\nr✝ : ℝ≥0\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nh : ↑r < radius p\n⊢ Summable fun a => ‖p a‖ * ↑r ^ a",
"tactic": "exact p.summable_norm_mul_pow h"
}
] |
[
303,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.integral_integral_sub'
|
[] |
[
418,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
415,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.cast_strictMono
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.55640\nι : Type ?u.55643\nα : Type ?u.55646\nβ : Type ?u.55649\nK : Type u_1\ninst✝ : LinearOrderedField K\nm n : ℚ\n⊢ m < n → ↑m < ↑n",
"tactic": "simpa only [sub_pos, cast_sub] using @cast_pos_of_pos K _ (n - m)"
}
] |
[
306,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/Data/Set/Pointwise/BigOperators.lean
|
Set.image_fintype_prod_pi
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\nβ : Type ?u.90981\nF : Type ?u.90984\ninst✝³ : CommMonoid α\ninst✝² : CommMonoid β\ninst✝¹ : MonoidHomClass F α β\ninst✝ : Fintype ι\nS : ι → Set α\n⊢ (fun f => ∏ i : ι, f i) '' pi univ S = ∏ i : ι, S i",
"tactic": "simpa only [Finset.coe_univ] using image_finset_prod_pi Finset.univ S"
}
] |
[
191,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Iio_filter_lt
|
[
{
"state_after": "case a\nι : Type ?u.150465\nα✝ : Type ?u.150468\ninst✝³ : LinearOrder α✝\ninst✝² : LocallyFiniteOrder α✝\na✝¹ b✝ : α✝\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrderBot α\na b a✝ : α\n⊢ a✝ ∈ filter (fun x => x < b) (Iio a) ↔ a✝ ∈ Iio (min a b)",
"state_before": "ι : Type ?u.150465\nα✝ : Type ?u.150468\ninst✝³ : LinearOrder α✝\ninst✝² : LocallyFiniteOrder α✝\na✝ b✝ : α✝\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrderBot α\na b : α\n⊢ filter (fun x => x < b) (Iio a) = Iio (min a b)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Type ?u.150465\nα✝ : Type ?u.150468\ninst✝³ : LinearOrder α✝\ninst✝² : LocallyFiniteOrder α✝\na✝¹ b✝ : α✝\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrderBot α\na b a✝ : α\n⊢ a✝ ∈ filter (fun x => x < b) (Iio a) ↔ a✝ ∈ Iio (min a b)",
"tactic": "simp [and_assoc]"
}
] |
[
811,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
808,
1
] |
Mathlib/Topology/DiscreteQuotient.lean
|
DiscreteQuotient.leComap_id
|
[] |
[
292,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.HasLimit.isoOfNatIso_inv_π
|
[] |
[
358,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/MeasureTheory/Group/Prod.lean
|
MeasureTheory.ae_measure_preimage_mul_right_lt_top
|
[
{
"state_after": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\n⊢ ∀ (s_1 : Set G),\n MeasurableSet s_1 →\n ↑↑μ s_1 < ⊤ → ↑↑(Measure.inv ν) s_1 < ⊤ → ∀ᵐ (x : G) ∂Measure.restrict μ s_1, ↑↑ν ((fun y => y * x) ⁻¹' s) < ⊤",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\n⊢ ∀ᵐ (x : G) ∂μ, ↑↑ν ((fun y => y * x) ⁻¹' s) < ⊤",
"tactic": "refine' ae_of_forall_measure_lt_top_ae_restrict' ν.inv _ _"
},
{
"state_after": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑(Measure.inv ν) A < ⊤\n⊢ ∀ᵐ (x : G) ∂Measure.restrict μ A, ↑↑ν ((fun y => y * x) ⁻¹' s) < ⊤",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\n⊢ ∀ (s_1 : Set G),\n MeasurableSet s_1 →\n ↑↑μ s_1 < ⊤ → ↑↑(Measure.inv ν) s_1 < ⊤ → ∀ᵐ (x : G) ∂Measure.restrict μ s_1, ↑↑ν ((fun y => y * x) ⁻¹' s) < ⊤",
"tactic": "intro A hA _ h3A"
},
{
"state_after": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑ν A⁻¹ < ⊤\n⊢ ∀ᵐ (x : G) ∂Measure.restrict μ A, ↑↑ν ((fun y => y * x) ⁻¹' s) < ⊤",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑(Measure.inv ν) A < ⊤\n⊢ ∀ᵐ (x : G) ∂Measure.restrict μ A, ↑↑ν ((fun y => y * x) ⁻¹' s) < ⊤",
"tactic": "simp only [ν.inv_apply] at h3A"
},
{
"state_after": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑ν A⁻¹ < ⊤\n⊢ (∫⁻ (x : G) in A, ↑↑ν ((fun y => y * x) ⁻¹' s) ∂μ) ≠ ⊤",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑ν A⁻¹ < ⊤\n⊢ ∀ᵐ (x : G) ∂Measure.restrict μ A, ↑↑ν ((fun y => y * x) ⁻¹' s) < ⊤",
"tactic": "apply ae_lt_top (measurable_measure_mul_right ν sm)"
},
{
"state_after": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑ν A⁻¹ < ⊤\nh1 : (↑↑μ s * ∫⁻ (y : G), indicator A⁻¹ 1 y ∂ν) = ∫⁻ (x : G), ↑↑ν ((fun z => z * x) ⁻¹' s) * indicator A⁻¹ 1 x⁻¹ ∂μ\n⊢ (∫⁻ (x : G) in A, ↑↑ν ((fun y => y * x) ⁻¹' s) ∂μ) ≠ ⊤",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑ν A⁻¹ < ⊤\n⊢ (∫⁻ (x : G) in A, ↑↑ν ((fun y => y * x) ⁻¹' s) ∂μ) ≠ ⊤",
"tactic": "have h1 := measure_mul_lintegral_eq μ ν sm (A⁻¹.indicator 1) (measurable_one.indicator hA.inv)"
},
{
"state_after": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑ν A⁻¹ < ⊤\nh1 : (↑↑μ s * ∫⁻ (a : G) in A⁻¹, OfNat.ofNat 1 a ∂ν) = ∫⁻ (x : G), ↑↑ν ((fun z => z * x) ⁻¹' s) * indicator A⁻¹ 1 x⁻¹ ∂μ\n⊢ (∫⁻ (x : G) in A, ↑↑ν ((fun y => y * x) ⁻¹' s) ∂μ) ≠ ⊤",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑ν A⁻¹ < ⊤\nh1 : (↑↑μ s * ∫⁻ (y : G), indicator A⁻¹ 1 y ∂ν) = ∫⁻ (x : G), ↑↑ν ((fun z => z * x) ⁻¹' s) * indicator A⁻¹ 1 x⁻¹ ∂μ\n⊢ (∫⁻ (x : G) in A, ↑↑ν ((fun y => y * x) ⁻¹' s) ∂μ) ≠ ⊤",
"tactic": "rw [lintegral_indicator _ hA.inv] at h1"
},
{
"state_after": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑ν A⁻¹ < ⊤\nh1 : ↑↑μ s * ↑↑ν A⁻¹ = ∫⁻ (x : G), indicator A (fun a => ↑↑ν ((fun y => y * a) ⁻¹' s)) x ∂μ\n⊢ (∫⁻ (x : G) in A, ↑↑ν ((fun y => y * x) ⁻¹' s) ∂μ) ≠ ⊤",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑ν A⁻¹ < ⊤\nh1 : (↑↑μ s * ∫⁻ (a : G) in A⁻¹, OfNat.ofNat 1 a ∂ν) = ∫⁻ (x : G), ↑↑ν ((fun z => z * x) ⁻¹' s) * indicator A⁻¹ 1 x⁻¹ ∂μ\n⊢ (∫⁻ (x : G) in A, ↑↑ν ((fun y => y * x) ⁻¹' s) ∂μ) ≠ ⊤",
"tactic": "simp_rw [Pi.one_apply, set_lintegral_one, ← image_inv, indicator_image inv_injective, image_inv, ←\n indicator_mul_right _ fun x => ν ((fun y => y * x) ⁻¹' s), Function.comp, Pi.one_apply,\n mul_one] at h1"
},
{
"state_after": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑ν A⁻¹ < ⊤\nh1 : ↑↑μ s * ↑↑ν A⁻¹ = ∫⁻ (x : G), indicator A (fun a => ↑↑ν ((fun y => y * a) ⁻¹' s)) x ∂μ\n⊢ ↑↑μ s * ↑↑ν A⁻¹ ≠ ⊤",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑ν A⁻¹ < ⊤\nh1 : ↑↑μ s * ↑↑ν A⁻¹ = ∫⁻ (x : G), indicator A (fun a => ↑↑ν ((fun y => y * a) ⁻¹' s)) x ∂μ\n⊢ (∫⁻ (x : G) in A, ↑↑ν ((fun y => y * x) ⁻¹' s) ∂μ) ≠ ⊤",
"tactic": "rw [← lintegral_indicator _ hA, ← h1]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nsm : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nA : Set G\nhA : MeasurableSet A\na✝ : ↑↑μ A < ⊤\nh3A : ↑↑ν A⁻¹ < ⊤\nh1 : ↑↑μ s * ↑↑ν A⁻¹ = ∫⁻ (x : G), indicator A (fun a => ↑↑ν ((fun y => y * a) ⁻¹' s)) x ∂μ\n⊢ ↑↑μ s * ↑↑ν A⁻¹ ≠ ⊤",
"tactic": "exact ENNReal.mul_ne_top hμs h3A.ne"
}
] |
[
291,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/GroupTheory/GroupAction/Quotient.lean
|
Subgroup.quotientCenterEmbedding_apply
|
[] |
[
403,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.vsub_subset_vsub
|
[] |
[
665,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
664,
1
] |
Mathlib/CategoryTheory/Limits/FunctorCategory.lean
|
CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_inv_colimit_map
|
[
{
"state_after": "case w\nC : Type u\ninst✝⁴ : Category C\nD : Type u'\ninst✝³ : Category D\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : HasColimitsOfShape J C\nF : J ⥤ K ⥤ C\ni j : K\nf : i ⟶ j\nj✝ : J\n⊢ colimit.ι (F ⋙ (evaluation K C).obj i) j✝ ≫ (colimitObjIsoColimitCompEvaluation F i).inv ≫ (colimit F).map f =\n colimit.ι (F ⋙ (evaluation K C).obj i) j✝ ≫\n colimMap (whiskerLeft F ((evaluation K C).map f)) ≫ (colimitObjIsoColimitCompEvaluation F j).inv",
"state_before": "C : Type u\ninst✝⁴ : Category C\nD : Type u'\ninst✝³ : Category D\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : HasColimitsOfShape J C\nF : J ⥤ K ⥤ C\ni j : K\nf : i ⟶ j\n⊢ (colimitObjIsoColimitCompEvaluation F i).inv ≫ (colimit F).map f =\n colimMap (whiskerLeft F ((evaluation K C).map f)) ≫ (colimitObjIsoColimitCompEvaluation F j).inv",
"tactic": "ext"
},
{
"state_after": "case w\nC : Type u\ninst✝⁴ : Category C\nD : Type u'\ninst✝³ : Category D\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : HasColimitsOfShape J C\nF : J ⥤ K ⥤ C\ni j : K\nf : i ⟶ j\nj✝ : J\n⊢ colimit.ι (F ⋙ (evaluation K C).obj i) j✝ ≫ (colimitObjIsoColimitCompEvaluation F i).inv ≫ (colimit F).map f =\n colimit.ι (F ⋙ (evaluation K C).obj i) j✝ ≫\n colimMap (whiskerLeft F ((evaluation K C).map f)) ≫ (colimitObjIsoColimitCompEvaluation F j).inv",
"state_before": "case w\nC : Type u\ninst✝⁴ : Category C\nD : Type u'\ninst✝³ : Category D\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : HasColimitsOfShape J C\nF : J ⥤ K ⥤ C\ni j : K\nf : i ⟶ j\nj✝ : J\n⊢ colimit.ι (F ⋙ (evaluation K C).obj i) j✝ ≫ (colimitObjIsoColimitCompEvaluation F i).inv ≫ (colimit F).map f =\n colimit.ι (F ⋙ (evaluation K C).obj i) j✝ ≫\n colimMap (whiskerLeft F ((evaluation K C).map f)) ≫ (colimitObjIsoColimitCompEvaluation F j).inv",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u\ninst✝⁴ : Category C\nD : Type u'\ninst✝³ : Category D\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : HasColimitsOfShape J C\nF : J ⥤ K ⥤ C\ni j : K\nf : i ⟶ j\nj✝ : J\n⊢ colimit.ι (F ⋙ (evaluation K C).obj i) j✝ ≫ (colimitObjIsoColimitCompEvaluation F i).inv ≫ (colimit F).map f =\n colimit.ι (F ⋙ (evaluation K C).obj i) j✝ ≫\n colimMap (whiskerLeft F ((evaluation K C).map f)) ≫ (colimitObjIsoColimitCompEvaluation F j).inv",
"tactic": "simp"
}
] |
[
306,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/RingTheory/WittVector/Defs.lean
|
WittVector.one_coeff_zero
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\n⊢ ↑(aeval (Function.uncurry fun i => ![].coeff)) (wittOne p 0) = 1",
"tactic": "simp only [wittOne_zero_eq_one, AlgHom.map_one]"
}
] |
[
342,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Mathlib/Data/List/Forall2.lean
|
List.Forall₂.mp
|
[] |
[
50,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/Data/Int/GCD.lean
|
Int.gcd_one_left
|
[] |
[
285,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.coeff_eq_zero_of_natDegree_lt
|
[
{
"state_after": "case h\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : natDegree p < n\n⊢ degree p < ↑n",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : natDegree p < n\n⊢ coeff p n = 0",
"tactic": "apply coeff_eq_zero_of_degree_lt"
},
{
"state_after": "case pos\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : natDegree p < n\nhp : p = 0\n⊢ degree p < ↑n\n\ncase neg\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : natDegree p < n\nhp : ¬p = 0\n⊢ degree p < ↑n",
"state_before": "case h\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : natDegree p < n\n⊢ degree p < ↑n",
"tactic": "by_cases hp : p = 0"
},
{
"state_after": "case pos\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nh : natDegree 0 < n\n⊢ degree 0 < ↑n",
"state_before": "case pos\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : natDegree p < n\nhp : p = 0\n⊢ degree p < ↑n",
"tactic": "subst hp"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nh : natDegree 0 < n\n⊢ degree 0 < ↑n",
"tactic": "exact WithBot.bot_lt_coe n"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nh : natDegree p < n\nhp : ¬p = 0\n⊢ degree p < ↑n",
"tactic": "rwa [degree_eq_natDegree hp, Nat.cast_withBot, Nat.cast_withBot, WithBot.coe_lt_coe]"
}
] |
[
355,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
348,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.updateRow_conjTranspose
|
[
{
"state_after": "l : Type ?u.1227385\nm : Type u_2\nn : Type u_1\no : Type ?u.1227394\nm' : o → Type ?u.1227399\nn' : o → Type ?u.1227404\nR : Type ?u.1227407\nS : Type ?u.1227410\nα : Type v\nβ : Type w\nγ : Type ?u.1227417\nM : Matrix m n α\ni : m\nj : n\nb : n → α\nc : m → α\ninst✝¹ : DecidableEq n\ninst✝ : Star α\n⊢ (updateColumn (map M star) j (star c))ᵀ = (updateColumn (map M star) j (star ∘ c))ᵀ",
"state_before": "l : Type ?u.1227385\nm : Type u_2\nn : Type u_1\no : Type ?u.1227394\nm' : o → Type ?u.1227399\nn' : o → Type ?u.1227404\nR : Type ?u.1227407\nS : Type ?u.1227410\nα : Type v\nβ : Type w\nγ : Type ?u.1227417\nM : Matrix m n α\ni : m\nj : n\nb : n → α\nc : m → α\ninst✝¹ : DecidableEq n\ninst✝ : Star α\n⊢ updateRow Mᴴ j (star c) = (updateColumn M j c)ᴴ",
"tactic": "rw [conjTranspose, conjTranspose, transpose_map, transpose_map, updateRow_transpose,\n map_updateColumn]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.1227385\nm : Type u_2\nn : Type u_1\no : Type ?u.1227394\nm' : o → Type ?u.1227399\nn' : o → Type ?u.1227404\nR : Type ?u.1227407\nS : Type ?u.1227410\nα : Type v\nβ : Type w\nγ : Type ?u.1227417\nM : Matrix m n α\ni : m\nj : n\nb : n → α\nc : m → α\ninst✝¹ : DecidableEq n\ninst✝ : Star α\n⊢ (updateColumn (map M star) j (star c))ᵀ = (updateColumn (map M star) j (star ∘ c))ᵀ",
"tactic": "rfl"
}
] |
[
2831,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2827,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
|
MeasureTheory.Measure.add_haar_smul_of_nonneg
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.1940240\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nr : ℝ\nhr : 0 ≤ r\ns : Set E\n⊢ ↑↑μ (r • s) = ENNReal.ofReal (r ^ finrank ℝ E) * ↑↑μ s",
"tactic": "rw [add_haar_smul, abs_pow, abs_of_nonneg hr]"
}
] |
[
366,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
364,
1
] |
Mathlib/Algebra/Order/Hom/Monoid.lean
|
OrderMonoidWithZeroHom.comp_apply
|
[] |
[
688,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
687,
1
] |
Mathlib/Data/Part.lean
|
Part.mul_mem_mul
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.56876\nγ : Type ?u.56879\ninst✝ : Mul α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ get (a * b)\n (_ : ∃ h, ((fun b_1 => (fun y => map y ((fun x => b) ())) (get ((fun x x_1 => x * x_1) <$> a) b_1)) h).Dom) =\n get a (_ : a.Dom) * get b (_ : b.Dom)",
"state_before": "α : Type u_1\nβ : Type ?u.56876\nγ : Type ?u.56879\ninst✝ : Mul α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ get (a * b)\n (_ : ∃ h, ((fun b_1 => (fun y => map y ((fun x => b) ())) (get ((fun x x_1 => x * x_1) <$> a) b_1)) h).Dom) =\n ma * mb",
"tactic": "simp [← ha.2, ← hb.2]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.56876\nγ : Type ?u.56879\ninst✝ : Mul α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ get (a * b)\n (_ : ∃ h, ((fun b_1 => (fun y => map y ((fun x => b) ())) (get ((fun x x_1 => x * x_1) <$> a) b_1)) h).Dom) =\n get a (_ : a.Dom) * get b (_ : b.Dom)",
"tactic": "rfl"
}
] |
[
719,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
718,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.span_singleton_le_span_singleton
|
[] |
[
508,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.toNat_add_toNat_neg_eq_natAbs
|
[] |
[
1406,
29
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1403,
9
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.mapQ_pow
|
[
{
"state_after": "case zero\nR : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.292638\nM₂ : Type ?u.292641\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nf : M →ₗ[R] M\nh : p ≤ comap f p\nk : ℕ\nh'✝ : optParam (p ≤ comap (f ^ k) p) (_ : p ≤ comap (f ^ k) p)\nh' : optParam (p ≤ comap (f ^ Nat.zero) p) (_ : p ≤ comap (f ^ Nat.zero) p)\n⊢ mapQ p p (f ^ Nat.zero) h' = mapQ p p f h ^ Nat.zero\n\ncase succ\nR : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.292638\nM₂ : Type ?u.292641\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nf : M →ₗ[R] M\nh : p ≤ comap f p\nk✝ : ℕ\nh'✝ : optParam (p ≤ comap (f ^ k✝) p) (_ : p ≤ comap (f ^ k✝) p)\nk : ℕ\nih : ∀ (h' : optParam (p ≤ comap (f ^ k) p) (_ : p ≤ comap (f ^ k) p)), mapQ p p (f ^ k) h' = mapQ p p f h ^ k\nh' : optParam (p ≤ comap (f ^ Nat.succ k) p) (_ : p ≤ comap (f ^ Nat.succ k) p)\n⊢ mapQ p p (f ^ Nat.succ k) h' = mapQ p p f h ^ Nat.succ k",
"state_before": "R : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.292638\nM₂ : Type ?u.292641\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nf : M →ₗ[R] M\nh : p ≤ comap f p\nk : ℕ\nh' : optParam (p ≤ comap (f ^ k) p) (_ : p ≤ comap (f ^ k) p)\n⊢ mapQ p p (f ^ k) h' = mapQ p p f h ^ k",
"tactic": "induction' k with k ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.292638\nM₂ : Type ?u.292641\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nf : M →ₗ[R] M\nh : p ≤ comap f p\nk : ℕ\nh'✝ : optParam (p ≤ comap (f ^ k) p) (_ : p ≤ comap (f ^ k) p)\nh' : optParam (p ≤ comap (f ^ Nat.zero) p) (_ : p ≤ comap (f ^ Nat.zero) p)\n⊢ mapQ p p (f ^ Nat.zero) h' = mapQ p p f h ^ Nat.zero",
"tactic": "simp [LinearMap.one_eq_id]"
},
{
"state_after": "case succ\nR : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.292638\nM₂ : Type ?u.292641\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nf : M →ₗ[R] M\nh : p ≤ comap f p\nk✝ : ℕ\nh'✝ : optParam (p ≤ comap (f ^ k✝) p) (_ : p ≤ comap (f ^ k✝) p)\nk : ℕ\nih : ∀ (h' : optParam (p ≤ comap (f ^ k) p) (_ : p ≤ comap (f ^ k) p)), mapQ p p (f ^ k) h' = mapQ p p f h ^ k\nh' : optParam (p ≤ comap (f ^ Nat.succ k) p) (_ : p ≤ comap (f ^ Nat.succ k) p)\n⊢ mapQ p p (comp (f ^ k) f) (_ : p ≤ comap (comp (f ^ k) f) p) = comp (mapQ p p f h ^ k) (mapQ p p f h)",
"state_before": "case succ\nR : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.292638\nM₂ : Type ?u.292641\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nf : M →ₗ[R] M\nh : p ≤ comap f p\nk✝ : ℕ\nh'✝ : optParam (p ≤ comap (f ^ k✝) p) (_ : p ≤ comap (f ^ k✝) p)\nk : ℕ\nih : ∀ (h' : optParam (p ≤ comap (f ^ k) p) (_ : p ≤ comap (f ^ k) p)), mapQ p p (f ^ k) h' = mapQ p p f h ^ k\nh' : optParam (p ≤ comap (f ^ Nat.succ k) p) (_ : p ≤ comap (f ^ Nat.succ k) p)\n⊢ mapQ p p (f ^ Nat.succ k) h' = mapQ p p f h ^ Nat.succ k",
"tactic": "simp only [LinearMap.iterate_succ]"
},
{
"state_after": "case h.e'_3.h.e'_20\nR : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.292638\nM₂ : Type ?u.292641\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nf : M →ₗ[R] M\nh : p ≤ comap f p\nk✝ : ℕ\nh'✝ : optParam (p ≤ comap (f ^ k✝) p) (_ : p ≤ comap (f ^ k✝) p)\nk : ℕ\nih : ∀ (h' : optParam (p ≤ comap (f ^ k) p) (_ : p ≤ comap (f ^ k) p)), mapQ p p (f ^ k) h' = mapQ p p f h ^ k\nh' : optParam (p ≤ comap (f ^ Nat.succ k) p) (_ : p ≤ comap (f ^ Nat.succ k) p)\n⊢ mapQ p p f h ^ k = mapQ p p (f ^ k) (_ : p ≤ comap (f ^ k) p)",
"state_before": "case succ\nR : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.292638\nM₂ : Type ?u.292641\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nf : M →ₗ[R] M\nh : p ≤ comap f p\nk✝ : ℕ\nh'✝ : optParam (p ≤ comap (f ^ k✝) p) (_ : p ≤ comap (f ^ k✝) p)\nk : ℕ\nih : ∀ (h' : optParam (p ≤ comap (f ^ k) p) (_ : p ≤ comap (f ^ k) p)), mapQ p p (f ^ k) h' = mapQ p p f h ^ k\nh' : optParam (p ≤ comap (f ^ Nat.succ k) p) (_ : p ≤ comap (f ^ Nat.succ k) p)\n⊢ mapQ p p (comp (f ^ k) f) (_ : p ≤ comap (comp (f ^ k) f) p) = comp (mapQ p p f h ^ k) (mapQ p p f h)",
"tactic": "convert mapQ_comp p p p f (f ^ k) h (p.le_comap_pow_of_le_comap h k)\n (h.trans (comap_mono <| p.le_comap_pow_of_le_comap h k))"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_20\nR : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.292638\nM₂ : Type ?u.292641\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nf : M →ₗ[R] M\nh : p ≤ comap f p\nk✝ : ℕ\nh'✝ : optParam (p ≤ comap (f ^ k✝) p) (_ : p ≤ comap (f ^ k✝) p)\nk : ℕ\nih : ∀ (h' : optParam (p ≤ comap (f ^ k) p) (_ : p ≤ comap (f ^ k) p)), mapQ p p (f ^ k) h' = mapQ p p f h ^ k\nh' : optParam (p ≤ comap (f ^ Nat.succ k) p) (_ : p ≤ comap (f ^ Nat.succ k) p)\n⊢ mapQ p p f h ^ k = mapQ p p (f ^ k) (_ : p ≤ comap (f ^ k) p)",
"tactic": "exact (ih _).symm"
}
] |
[
458,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.hasPullbacks_of_hasLimit_cospan
|
[] |
[
2674,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2672,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
continuousMultilinearCurryRightEquiv_symm_apply
|
[] |
[
1574,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1571,
1
] |
Std/Data/List/Lemmas.lean
|
List.replaceF_cons
|
[] |
[
1307,
34
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1304,
1
] |
Mathlib/Data/PFunctor/Univariate/M.lean
|
PFunctor.Approx.cofixA_eq_zero
|
[] |
[
52,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometry.injective
|
[] |
[
164,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
11
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithBot.map_one
|
[] |
[
533,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
532,
11
] |
Mathlib/SetTheory/Ordinal/Exponential.lean
|
Ordinal.opow_isLimit
|
[] |
[
128,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.IsPartition.existsUnique
|
[
{
"state_after": "case intro.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nh✝ : IsPartition π\nhx✝ : x ∈ I\nJ : Box ι\nh : J ∈ π\nhx : x ∈ J\n⊢ ∃! J x_1, x ∈ J",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nh : IsPartition π\nhx : x ∈ I\n⊢ ∃! J x_1, x ∈ J",
"tactic": "rcases h x hx with ⟨J, h, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nh✝ : IsPartition π\nhx✝ : x ∈ I\nJ : Box ι\nh : J ∈ π\nhx : x ∈ J\n⊢ ∃! J x_1, x ∈ J",
"tactic": "exact ExistsUnique.intro₂ J h hx fun J' h' hx' => π.eq_of_mem_of_mem h' h hx' hx"
}
] |
[
748,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
745,
11
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.coe_pow
|
[] |
[
607,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
606,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
Basis.op_nnnorm_le
|
[
{
"state_after": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑(equivFunL v)\n⊢ ‖↑u e‖₊ ≤ Fintype.card ι • ‖φ‖₊ * M * ‖e‖₊",
"state_before": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M\ne : E\n⊢ ‖↑u e‖₊ ≤ Fintype.card ι • ‖↑(equivFunL v)‖₊ * M * ‖e‖₊",
"tactic": "set φ := v.equivFunL.toContinuousLinearMap"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑(equivFunL v)\n⊢ ‖↑u e‖₊ ≤ Fintype.card ι • ‖φ‖₊ * M * ‖e‖₊",
"tactic": "calc\n ‖u e‖₊ = ‖u (∑ i, v.equivFun e i • v i)‖₊ := by rw [v.sum_equivFun]\n _ = ‖∑ i, v.equivFun e i • (u <| v i)‖₊ := by simp [u.map_sum, LinearMap.map_smul]\n _ ≤ ∑ i, ‖v.equivFun e i • (u <| v i)‖₊ := (nnnorm_sum_le _ _)\n _ = ∑ i, ‖v.equivFun e i‖₊ * ‖u (v i)‖₊ := by simp only [nnnorm_smul]\n _ ≤ ∑ i, ‖v.equivFun e i‖₊ * M := by gcongr; apply hu\n _ = (∑ i, ‖v.equivFun e i‖₊) * M := Finset.sum_mul.symm\n _ ≤ Fintype.card ι • (‖φ‖₊ * ‖e‖₊) * M := by\n gcongr\n calc\n (∑ i, ‖v.equivFun e i‖₊) ≤ Fintype.card ι • ‖φ e‖₊ := Pi.sum_nnnorm_apply_le_nnnorm _\n _ ≤ Fintype.card ι • (‖φ‖₊ * ‖e‖₊) := nsmul_le_nsmul_of_le_right (φ.le_op_nnnorm e) _\n _ = Fintype.card ι • ‖φ‖₊ * M * ‖e‖₊ := by simp only [smul_mul_assoc, mul_right_comm]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑(equivFunL v)\n⊢ ‖↑u e‖₊ = ‖↑u (∑ i : ι, ↑(equivFun v) e i • ↑v i)‖₊",
"tactic": "rw [v.sum_equivFun]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑(equivFunL v)\n⊢ ‖↑u (∑ i : ι, ↑(equivFun v) e i • ↑v i)‖₊ = ‖∑ i : ι, ↑(equivFun v) e i • ↑u (↑v i)‖₊",
"tactic": "simp [u.map_sum, LinearMap.map_smul]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑(equivFunL v)\n⊢ ∑ i : ι, ‖↑(equivFun v) e i • ↑u (↑v i)‖₊ = ∑ i : ι, ‖↑(equivFun v) e i‖₊ * ‖↑u (↑v i)‖₊",
"tactic": "simp only [nnnorm_smul]"
},
{
"state_after": "case h.bc\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑(equivFunL v)\ni✝ : ι\na✝ : i✝ ∈ Finset.univ\n⊢ ‖↑u (↑v i✝)‖₊ ≤ M",
"state_before": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑(equivFunL v)\n⊢ ∑ i : ι, ‖↑(equivFun v) e i‖₊ * ‖↑u (↑v i)‖₊ ≤ ∑ i : ι, ‖↑(equivFun v) e i‖₊ * M",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h.bc\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑(equivFunL v)\ni✝ : ι\na✝ : i✝ ∈ Finset.univ\n⊢ ‖↑u (↑v i✝)‖₊ ≤ M",
"tactic": "apply hu"
},
{
"state_after": "case bc\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑(equivFunL v)\n⊢ ∑ i : ι, ‖↑(equivFun v) e i‖₊ ≤ Fintype.card ι • (‖φ‖₊ * ‖e‖₊)",
"state_before": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑(equivFunL v)\n⊢ (∑ i : ι, ‖↑(equivFun v) e i‖₊) * M ≤ Fintype.card ι • (‖φ‖₊ * ‖e‖₊) * M",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case bc\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑(equivFunL v)\n⊢ ∑ i : ι, ‖↑(equivFun v) e i‖₊ ≤ Fintype.card ι • (‖φ‖₊ * ‖e‖₊)",
"tactic": "calc\n (∑ i, ‖v.equivFun e i‖₊) ≤ Fintype.card ι • ‖φ e‖₊ := Pi.sum_nnnorm_apply_le_nnnorm _\n _ ≤ Fintype.card ι • (‖φ‖₊ * ‖e‖₊) := nsmul_le_nsmul_of_le_right (φ.le_op_nnnorm e) _"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ≥0\nhu : ∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M\ne : E\nφ : E →L[𝕜] ι → 𝕜 := ↑(equivFunL v)\n⊢ Fintype.card ι • (‖φ‖₊ * ‖e‖₊) * M = Fintype.card ι • ‖φ‖₊ * M * ‖e‖₊",
"tactic": "simp only [smul_mul_assoc, mul_right_comm]"
}
] |
[
290,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/Data/PFunctor/Univariate/Basic.lean
|
PFunctor.map_eq
|
[] |
[
65,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
11
] |
Mathlib/Data/Erased.lean
|
Erased.pure_def
|
[] |
[
146,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/CategoryTheory/Monoidal/OfHasFiniteProducts.lean
|
CategoryTheory.monoidalOfHasFiniteProducts.leftUnitor_hom
|
[] |
[
104,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Topology/LocallyFinite.lean
|
LocallyFinite.continuousOn_iUnion
|
[] |
[
109,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.iInf_option_toFinset
|
[] |
[
1943,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1942,
1
] |
Mathlib/Data/Polynomial/Coeff.lean
|
Polynomial.mul_X_pow_injective
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nP Q : R[X]\nhPQ : (fun P => X ^ n * P) P = (fun P => X ^ n * P) Q\n⊢ P = Q",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\n⊢ Function.Injective fun P => X ^ n * P",
"tactic": "intro P Q hPQ"
},
{
"state_after": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nP Q : R[X]\nhPQ : X ^ n * P = X ^ n * Q\n⊢ P = Q",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nP Q : R[X]\nhPQ : (fun P => X ^ n * P) P = (fun P => X ^ n * P) Q\n⊢ P = Q",
"tactic": "simp only at hPQ"
},
{
"state_after": "case a\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nP Q : R[X]\nhPQ : X ^ n * P = X ^ n * Q\ni : ℕ\n⊢ coeff P i = coeff Q i",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nP Q : R[X]\nhPQ : X ^ n * P = X ^ n * Q\n⊢ P = Q",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nP Q : R[X]\nhPQ : X ^ n * P = X ^ n * Q\ni : ℕ\n⊢ coeff P i = coeff Q i",
"tactic": "rw [← coeff_X_pow_mul P n i, hPQ, coeff_X_pow_mul Q n i]"
}
] |
[
303,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.ae_eq_of_ae_le_of_lintegral_le
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹\n⊢ f =ᵐ[μ] g",
"state_before": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\n⊢ f =ᵐ[μ] g",
"tactic": "have : ∀ n : ℕ, ∀ᵐ x ∂μ, g x < f x + (n : ℝ≥0∞)⁻¹ := by\n intro n\n simp only [ae_iff, not_lt]\n have : (∫⁻ x, f x ∂μ) + (↑n)⁻¹ * μ { x : α | f x + (n : ℝ≥0∞)⁻¹ ≤ g x } ≤ ∫⁻ x, f x ∂μ :=\n (lintegral_add_mul_meas_add_le_le_lintegral hfg hg n⁻¹).trans hgf\n rw [(ENNReal.cancel_of_ne hf).add_le_iff_nonpos_right, nonpos_iff_eq_zero, mul_eq_zero] at this\n exact this.resolve_left (ENNReal.inv_ne_zero.2 (ENNReal.nat_ne_top _))"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹\nx : α\nhlt : ∀ (i : ℕ), g x < f x + (↑i)⁻¹\nhle : f x ≤ g x\n⊢ g x ≤ f x",
"state_before": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹\n⊢ f =ᵐ[μ] g",
"tactic": "refine' hfg.mp ((ae_all_iff.2 this).mono fun x hlt hle => hle.antisymm _)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nthis✝ : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹\nx : α\nhlt : ∀ (i : ℕ), g x < f x + (↑i)⁻¹\nhle : f x ≤ g x\nthis : Tendsto (fun n => f x + (↑n)⁻¹) atTop (𝓝 (f x))\n⊢ g x ≤ f x\n\ncase this\nα : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹\nx : α\nhlt : ∀ (i : ℕ), g x < f x + (↑i)⁻¹\nhle : f x ≤ g x\n⊢ Tendsto (fun n => f x + (↑n)⁻¹) atTop (𝓝 (f x))",
"state_before": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹\nx : α\nhlt : ∀ (i : ℕ), g x < f x + (↑i)⁻¹\nhle : f x ≤ g x\n⊢ g x ≤ f x",
"tactic": "suffices : Tendsto (fun n : ℕ => f x + (n : ℝ≥0∞)⁻¹) atTop (𝓝 (f x))"
},
{
"state_after": "case this\nα : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹\nx : α\nhlt : ∀ (i : ℕ), g x < f x + (↑i)⁻¹\nhle : f x ≤ g x\n⊢ Tendsto (fun n => f x + (↑n)⁻¹) atTop (𝓝 (f x))",
"state_before": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nthis✝ : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹\nx : α\nhlt : ∀ (i : ℕ), g x < f x + (↑i)⁻¹\nhle : f x ≤ g x\nthis : Tendsto (fun n => f x + (↑n)⁻¹) atTop (𝓝 (f x))\n⊢ g x ≤ f x\n\ncase this\nα : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹\nx : α\nhlt : ∀ (i : ℕ), g x < f x + (↑i)⁻¹\nhle : f x ≤ g x\n⊢ Tendsto (fun n => f x + (↑n)⁻¹) atTop (𝓝 (f x))",
"tactic": "exact ge_of_tendsto' this fun i => (hlt i).le"
},
{
"state_after": "no goals",
"state_before": "case this\nα : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹\nx : α\nhlt : ∀ (i : ℕ), g x < f x + (↑i)⁻¹\nhle : f x ≤ g x\n⊢ Tendsto (fun n => f x + (↑n)⁻¹) atTop (𝓝 (f x))",
"tactic": "simpa only [inv_top, add_zero] using\n tendsto_const_nhds.add (ENNReal.tendsto_inv_iff.2 ENNReal.tendsto_nat_nhds_top)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nn : ℕ\n⊢ ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹",
"state_before": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\n⊢ ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹",
"tactic": "intro n"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nn : ℕ\n⊢ ↑↑μ {a | f a + (↑n)⁻¹ ≤ g a} = 0",
"state_before": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nn : ℕ\n⊢ ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹",
"tactic": "simp only [ae_iff, not_lt]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nn : ℕ\nthis : (∫⁻ (x : α), f x ∂μ) + (↑n)⁻¹ * ↑↑μ {x | f x + (↑n)⁻¹ ≤ g x} ≤ ∫⁻ (x : α), f x ∂μ\n⊢ ↑↑μ {a | f a + (↑n)⁻¹ ≤ g a} = 0",
"state_before": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nn : ℕ\n⊢ ↑↑μ {a | f a + (↑n)⁻¹ ≤ g a} = 0",
"tactic": "have : (∫⁻ x, f x ∂μ) + (↑n)⁻¹ * μ { x : α | f x + (n : ℝ≥0∞)⁻¹ ≤ g x } ≤ ∫⁻ x, f x ∂μ :=\n (lintegral_add_mul_meas_add_le_le_lintegral hfg hg n⁻¹).trans hgf"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nn : ℕ\nthis : (↑n)⁻¹ = 0 ∨ ↑↑μ {x | f x + (↑n)⁻¹ ≤ g x} = 0\n⊢ ↑↑μ {a | f a + (↑n)⁻¹ ≤ g a} = 0",
"state_before": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nn : ℕ\nthis : (∫⁻ (x : α), f x ∂μ) + (↑n)⁻¹ * ↑↑μ {x | f x + (↑n)⁻¹ ≤ g x} ≤ ∫⁻ (x : α), f x ∂μ\n⊢ ↑↑μ {a | f a + (↑n)⁻¹ ≤ g a} = 0",
"tactic": "rw [(ENNReal.cancel_of_ne hf).add_le_iff_nonpos_right, nonpos_iff_eq_zero, mul_eq_zero] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.967242\nγ : Type ?u.967245\nδ : Type ?u.967248\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhfg : f ≤ᵐ[μ] g\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhg : AEMeasurable g\nhgf : (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ\nn : ℕ\nthis : (↑n)⁻¹ = 0 ∨ ↑↑μ {x | f x + (↑n)⁻¹ ≤ g x} = 0\n⊢ ↑↑μ {a | f a + (↑n)⁻¹ ≤ g a} = 0",
"tactic": "exact this.resolve_left (ENNReal.inv_ne_zero.2 (ENNReal.nat_ne_top _))"
}
] |
[
868,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
855,
1
] |
Mathlib/Data/Polynomial/UnitTrinomial.lean
|
Polynomial.trinomial_support
|
[] |
[
125,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Tactic/CancelDenoms.lean
|
CancelDenoms.derive_trans
|
[] |
[
189,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.lift_comp_castAddHom
|
[] |
[
1222,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1221,
1
] |
Mathlib/Data/Finset/Slice.lean
|
Finset.pairwiseDisjoint_slice
|
[] |
[
165,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean
|
Polynomial.degree_annIdealGenerator_le_of_mem
|
[] |
[
154,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
IsIntegral.pow_iff
|
[] |
[
673,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
672,
1
] |
src/lean/Init/Data/Nat/Linear.lean
|
Nat.Linear.denote_monomialToExpr
|
[
{
"state_after": "ctx : Context\nk : Nat\nv : Var\n⊢ Expr.denote ctx (bif v == fixedVar then Expr.num k else bif k == 1 then Expr.var v else Expr.mulL k (Expr.var v)) =\n k * Var.denote ctx v",
"state_before": "ctx : Context\nk : Nat\nv : Var\n⊢ Expr.denote ctx (monomialToExpr k v) = k * Var.denote ctx v",
"tactic": "simp [monomialToExpr]"
},
{
"state_after": "case inl\nctx : Context\nk : Nat\nv : Var\nh : (v == fixedVar) = true\n⊢ k = k * Var.denote ctx v\n\ncase inr\nctx : Context\nk : Nat\nv : Var\nh : ¬(v == fixedVar) = true\n⊢ Expr.denote ctx (bif k == 1 then Expr.var v else Expr.mulL k (Expr.var v)) = k * Var.denote ctx v",
"state_before": "ctx : Context\nk : Nat\nv : Var\n⊢ Expr.denote ctx (bif v == fixedVar then Expr.num k else bif k == 1 then Expr.var v else Expr.mulL k (Expr.var v)) =\n k * Var.denote ctx v",
"tactic": "by_cases h : v == fixedVar <;> simp [h, Expr.denote]"
},
{
"state_after": "no goals",
"state_before": "case inl\nctx : Context\nk : Nat\nv : Var\nh : (v == fixedVar) = true\n⊢ k = k * Var.denote ctx v",
"tactic": "simp [eq_of_beq h, Var.denote]"
},
{
"state_after": "case inr.inl\nctx : Context\nk : Nat\nv : Var\nh✝ : ¬(v == fixedVar) = true\nh : (k == 1) = true\n⊢ Var.denote ctx v = k * Var.denote ctx v",
"state_before": "case inr\nctx : Context\nk : Nat\nv : Var\nh : ¬(v == fixedVar) = true\n⊢ Expr.denote ctx (bif k == 1 then Expr.var v else Expr.mulL k (Expr.var v)) = k * Var.denote ctx v",
"tactic": "by_cases h : k == 1 <;> simp [h, Expr.denote]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nctx : Context\nk : Nat\nv : Var\nh✝ : ¬(v == fixedVar) = true\nh : (k == 1) = true\n⊢ Var.denote ctx v = k * Var.denote ctx v",
"tactic": "simp [eq_of_beq h]"
}
] |
[
699,
70
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
695,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
|
MeasureTheory.Measure.prod_eq
|
[] |
[
508,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
1
] |
Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
|
GromovHausdorff.candidatesB_nonempty
|
[] |
[
238,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
9
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_X_add_C
|
[
{
"state_after": "R : Type u\nS : Type v\na✝ b c d : R\nn m : ℕ\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\na : R\nthis : degree (↑C a) < degree X\n⊢ degree (X + ↑C a) = 1",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn m : ℕ\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\na : R\n⊢ degree (X + ↑C a) = 1",
"tactic": "have : degree (C a) < degree (X : R[X]) :=\n calc\n degree (C a) ≤ 0 := degree_C_le\n _ < 1 := (WithBot.some_lt_some.mpr zero_lt_one)\n _ = degree X := degree_X.symm"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn m : ℕ\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\na : R\nthis : degree (↑C a) < degree X\n⊢ degree (X + ↑C a) = 1",
"tactic": "rw [degree_add_eq_left_of_degree_lt this, degree_X]"
}
] |
[
1364,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1358,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.not_isUnit_X_sub_C
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.2110936\nB' : Type ?u.2110939\na b : R\nn : ℕ\ninst✝³ : CommSemiring A'\ninst✝² : Semiring B'\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nr : R\nx✝ : IsUnit (X - ↑C r)\ng : R[X]\n_hfg : (X - ↑C r) * g = 1\nhgf : g * (X - ↑C r) = 1\n⊢ 0 = 1",
"tactic": "erw [← eval_mul_X_sub_C, hgf, eval_one]"
}
] |
[
507,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
inv_pos_le_iff_one_le_mul
|
[
{
"state_after": "ι : Type ?u.39032\nα : Type u_1\nβ : Type ?u.39038\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 0 < a\n⊢ 1 / a ≤ b ↔ 1 ≤ b * a",
"state_before": "ι : Type ?u.39032\nα : Type u_1\nβ : Type ?u.39038\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 0 < a\n⊢ a⁻¹ ≤ b ↔ 1 ≤ b * a",
"tactic": "rw [inv_eq_one_div]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.39032\nα : Type u_1\nβ : Type ?u.39038\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 0 < a\n⊢ 1 / a ≤ b ↔ 1 ≤ b * a",
"tactic": "exact div_le_iff ha"
}
] |
[
212,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.eq_zero_or_opow_omega_le_of_mul_eq_right
|
[
{
"state_after": "case inl\na b : Ordinal\nhab : a * b = b\nha : a = 0\n⊢ b = 0 ∨ a ^ ω ≤ b\n\ncase inr\na b : Ordinal\nhab : a * b = b\nha : 0 < a\n⊢ b = 0 ∨ a ^ ω ≤ b",
"state_before": "a b : Ordinal\nhab : a * b = b\n⊢ b = 0 ∨ a ^ ω ≤ b",
"tactic": "cases' eq_zero_or_pos a with ha ha"
},
{
"state_after": "case inr\na b : Ordinal\nhab : a * b = b\nha : 0 < a\n⊢ ¬b = 0 → a ^ ω ≤ b",
"state_before": "case inr\na b : Ordinal\nhab : a * b = b\nha : 0 < a\n⊢ b = 0 ∨ a ^ ω ≤ b",
"tactic": "rw [or_iff_not_imp_left]"
},
{
"state_after": "case inr\na b : Ordinal\nhab : a * b = b\nha : 0 < a\nhb : ¬b = 0\n⊢ a ^ ω ≤ b",
"state_before": "case inr\na b : Ordinal\nhab : a * b = b\nha : 0 < a\n⊢ ¬b = 0 → a ^ ω ≤ b",
"tactic": "intro hb"
},
{
"state_after": "case inr\na b : Ordinal\nhab : a * b = b\nha : 0 < a\nhb : ¬b = 0\n⊢ nfp (fun x => a * x) 1 ≤ b",
"state_before": "case inr\na b : Ordinal\nhab : a * b = b\nha : 0 < a\nhb : ¬b = 0\n⊢ a ^ ω ≤ b",
"tactic": "rw [← nfp_mul_one ha]"
},
{
"state_after": "case inr\na b : Ordinal\nhab : a * b = b\nha : 0 < a\nhb : 1 ≤ b\n⊢ nfp (fun x => a * x) 1 ≤ b",
"state_before": "case inr\na b : Ordinal\nhab : a * b = b\nha : 0 < a\nhb : ¬b = 0\n⊢ nfp (fun x => a * x) 1 ≤ b",
"tactic": "rw [← Ne.def, ← one_le_iff_ne_zero] at hb"
},
{
"state_after": "no goals",
"state_before": "case inr\na b : Ordinal\nhab : a * b = b\nha : 0 < a\nhb : 1 ≤ b\n⊢ nfp (fun x => a * x) 1 ≤ b",
"tactic": "exact nfp_le_fp (mul_isNormal ha).monotone hb (le_of_eq hab)"
},
{
"state_after": "case inl\na b : Ordinal\nhab : a * b = b\nha : a = 0\n⊢ b = 0 ∨ 0 ≤ b",
"state_before": "case inl\na b : Ordinal\nhab : a * b = b\nha : a = 0\n⊢ b = 0 ∨ a ^ ω ≤ b",
"tactic": "rw [ha, zero_opow omega_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "case inl\na b : Ordinal\nhab : a * b = b\nha : a = 0\n⊢ b = 0 ∨ 0 ≤ b",
"tactic": "exact Or.inr (Ordinal.zero_le b)"
}
] |
[
675,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
666,
1
] |
Mathlib/Algebra/Order/Interval.lean
|
NonemptyInterval.length_zero
|
[] |
[
652,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
651,
1
] |
Mathlib/GroupTheory/Solvable.lean
|
IsSimpleGroup.comm_iff_isSolvable
|
[
{
"state_after": "case zero\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nhn : derivedSeries G Nat.zero = ⊥\n⊢ ∀ (a b : G), a * b = b * a\n\ncase succ\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nn✝ : ℕ\nhn : derivedSeries G (Nat.succ n✝) = ⊥\n⊢ ∀ (a b : G), a * b = b * a",
"state_before": "G : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nn : ℕ\nhn : derivedSeries G n = ⊥\n⊢ ∀ (a b : G), a * b = b * a",
"tactic": "cases n"
},
{
"state_after": "case zero\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nhn : derivedSeries G Nat.zero = ⊥\na b : G\n⊢ a * b = b * a",
"state_before": "case zero\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nhn : derivedSeries G Nat.zero = ⊥\n⊢ ∀ (a b : G), a * b = b * a",
"tactic": "intro a b"
},
{
"state_after": "case zero.refine'_2\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nhn : derivedSeries G Nat.zero = ⊥\na b : G\n⊢ b * a ∈ derivedSeries G Nat.zero",
"state_before": "case zero.refine'_2\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nhn : derivedSeries G Nat.zero = ⊥\na b : G\n⊢ b * a ∈ ⊥",
"tactic": "rw [← hn]"
},
{
"state_after": "no goals",
"state_before": "case zero.refine'_2\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nhn : derivedSeries G Nat.zero = ⊥\na b : G\n⊢ b * a ∈ derivedSeries G Nat.zero",
"tactic": "exact mem_top _"
},
{
"state_after": "case succ\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nn✝ : ℕ\nhn : _root_.commutator G = ⊥\n⊢ ∀ (a b : G), a * b = b * a",
"state_before": "case succ\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nn✝ : ℕ\nhn : derivedSeries G (Nat.succ n✝) = ⊥\n⊢ ∀ (a b : G), a * b = b * a",
"tactic": "rw [IsSimpleGroup.derivedSeries_succ] at hn"
},
{
"state_after": "case succ\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nn✝ : ℕ\nhn : _root_.commutator G = ⊥\na b : G\n⊢ a * b = b * a",
"state_before": "case succ\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nn✝ : ℕ\nhn : _root_.commutator G = ⊥\n⊢ ∀ (a b : G), a * b = b * a",
"tactic": "intro a b"
},
{
"state_after": "case succ\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nn✝ : ℕ\nhn : _root_.commutator G = ⊥\na b : G\n⊢ a * b * a⁻¹ * b⁻¹ ∈ closure (commutatorSet G)",
"state_before": "case succ\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nn✝ : ℕ\nhn : _root_.commutator G = ⊥\na b : G\n⊢ a * b = b * a",
"tactic": "rw [← mul_inv_eq_one, mul_inv_rev, ← mul_assoc, ← mem_bot, ← hn, commutator_eq_closure]"
},
{
"state_after": "no goals",
"state_before": "case succ\nG : Type u_1\nG' : Type ?u.84178\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nn✝ : ℕ\nhn : _root_.commutator G = ⊥\na b : G\n⊢ a * b * a⁻¹ * b⁻¹ ∈ closure (commutatorSet G)",
"tactic": "exact subset_closure ⟨a, b, rfl⟩"
}
] |
[
202,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
src/lean/Init/Core.lean
|
Eq.substr
|
[] |
[
551,
10
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
550,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
himp_le_himp_himp_himp
|
[
{
"state_after": "ι : Type ?u.30631\nα : Type u_1\nβ : Type ?u.30637\ninst✝ : GeneralizedHeytingAlgebra α\na b c d : α\n⊢ c ⊓ (b ⊓ a) ≤ c",
"state_before": "ι : Type ?u.30631\nα : Type u_1\nβ : Type ?u.30637\ninst✝ : GeneralizedHeytingAlgebra α\na b c d : α\n⊢ b ⇨ c ≤ (a ⇨ b) ⇨ a ⇨ c",
"tactic": "rw [le_himp_iff, le_himp_iff, inf_assoc, himp_inf_self, ← inf_assoc, himp_inf_self, inf_assoc]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.30631\nα : Type u_1\nβ : Type ?u.30637\ninst✝ : GeneralizedHeytingAlgebra α\na b c d : α\n⊢ c ⊓ (b ⊓ a) ≤ c",
"tactic": "exact inf_le_left"
}
] |
[
389,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.isUnit_constantCoeff
|
[] |
[
527,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
525,
1
] |
Mathlib/CategoryTheory/Adjunction/Limits.lean
|
CategoryTheory.Adjunction.hasColimit_of_comp_equivalence
|
[] |
[
149,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.div_lt_of_lt_mul
|
[] |
[
404,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
11
] |
Mathlib/Topology/MetricSpace/PiNat.lean
|
PiNat.shortestPrefixDiff_pos
|
[
{
"state_after": "case intro\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\ny : (n : ℕ) → E n\nhy : y ∈ s\n⊢ 0 < shortestPrefixDiff x s",
"state_before": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\n⊢ 0 < shortestPrefixDiff x s",
"tactic": "rcases hne with ⟨y, hy⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\ny : (n : ℕ) → E n\nhy : y ∈ s\n⊢ 0 < shortestPrefixDiff x s",
"tactic": "exact (zero_le _).trans_lt (firstDiff_lt_shortestPrefixDiff hs hx hy)"
}
] |
[
528,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
525,
1
] |
Mathlib/RepresentationTheory/Basic.lean
|
Representation.asAlgebraHom_single_one
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_2\nG : Type u_3\nV : Type u_1\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\ng : G\n⊢ ↑(asAlgebraHom ρ) (Finsupp.single g 1) = ↑ρ g",
"tactic": "simp"
}
] |
[
97,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
iteratedFDeriv_add_apply'
|
[] |
[
1252,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1249,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.mem_closedBall_self
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.804396\nR' : Type ?u.804399\n𝕜 : Type u_2\n𝕜₂ : Type ?u.804405\n𝕜₃ : Type ?u.804408\n𝕝 : Type ?u.804411\nE : Type u_1\nE₂ : Type ?u.804417\nE₃ : Type ?u.804420\nF : Type ?u.804423\nG : Type ?u.804426\nι : Type ?u.804429\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np : Seminorm 𝕜 E\nx y : E\nr : ℝ\nhr : 0 ≤ r\n⊢ x ∈ closedBall p x r",
"tactic": "simp [hr]"
}
] |
[
659,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
659,
1
] |
Mathlib/Algebra/MonoidAlgebra/Degree.lean
|
AddMonoidAlgebra.le_inf_support_pow
|
[
{
"state_after": "case refine'_1\nR : Type u_3\nA : Type u_2\nT : Type u_1\nB : Type ?u.55973\nι : Type ?u.55976\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegt0 : 0 ≤ degt 0\ndegtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)\nn : ℕ\nf : AddMonoidAlgebra R A\n⊢ ↑OrderDual.ofDual (degt 0) ≤ ↑OrderDual.ofDual 0\n\ncase refine'_2\nR : Type u_3\nA : Type u_2\nT : Type u_1\nB : Type ?u.55973\nι : Type ?u.55976\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegt0 : 0 ≤ degt 0\ndegtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)\nn : ℕ\nf : AddMonoidAlgebra R A\na b : A\n⊢ ↑OrderDual.ofDual (degt (a + b)) ≤ ↑OrderDual.ofDual (degt a + degt b)",
"state_before": "R : Type u_3\nA : Type u_2\nT : Type u_1\nB : Type ?u.55973\nι : Type ?u.55976\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegt0 : 0 ≤ degt 0\ndegtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)\nn : ℕ\nf : AddMonoidAlgebra R A\n⊢ n • Finset.inf f.support degt ≤ Finset.inf (f ^ n).support degt",
"tactic": "refine' OrderDual.ofDual_le_ofDual.mpr <| sup_support_pow_le (OrderDual.ofDual_le_ofDual.mp _)\n (fun a b => OrderDual.ofDual_le_ofDual.mp _) n f"
},
{
"state_after": "case refine'_2\nR : Type u_3\nA : Type u_2\nT : Type u_1\nB : Type ?u.55973\nι : Type ?u.55976\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegt0 : 0 ≤ degt 0\ndegtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)\nn : ℕ\nf : AddMonoidAlgebra R A\na b : A\n⊢ ↑OrderDual.ofDual (degt (a + b)) ≤ ↑OrderDual.ofDual (degt a + degt b)",
"state_before": "case refine'_1\nR : Type u_3\nA : Type u_2\nT : Type u_1\nB : Type ?u.55973\nι : Type ?u.55976\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegt0 : 0 ≤ degt 0\ndegtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)\nn : ℕ\nf : AddMonoidAlgebra R A\n⊢ ↑OrderDual.ofDual (degt 0) ≤ ↑OrderDual.ofDual 0\n\ncase refine'_2\nR : Type u_3\nA : Type u_2\nT : Type u_1\nB : Type ?u.55973\nι : Type ?u.55976\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegt0 : 0 ≤ degt 0\ndegtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)\nn : ℕ\nf : AddMonoidAlgebra R A\na b : A\n⊢ ↑OrderDual.ofDual (degt (a + b)) ≤ ↑OrderDual.ofDual (degt a + degt b)",
"tactic": "exact degt0"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u_3\nA : Type u_2\nT : Type u_1\nB : Type ?u.55973\nι : Type ?u.55976\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegt0 : 0 ≤ degt 0\ndegtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)\nn : ℕ\nf : AddMonoidAlgebra R A\na b : A\n⊢ ↑OrderDual.ofDual (degt (a + b)) ≤ ↑OrderDual.ofDual (degt a + degt b)",
"tactic": "exact degtm _ _"
}
] |
[
130,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Data/Polynomial/Coeff.lean
|
Polynomial.coeff_X_mul_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\n⊢ coeff (X * p) 0 = 0",
"tactic": "simp"
}
] |
[
152,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.lsub_eq_blsub
|
[] |
[
1776,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1774,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.const_smul
|
[] |
[
363,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.isIso_limit_cone_parallelPair_of_eq
|
[] |
[
862,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
860,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.iSup_option_toFinset
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.432384\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.432393\nι : Type ?u.432396\nκ : Type ?u.432399\ninst✝ : CompleteLattice β\no : Option α\nf : α → β\n⊢ (⨆ (x : α) (_ : x ∈ Option.toFinset o), f x) = ⨆ (x : α) (_ : x ∈ o), f x",
"tactic": "simp"
}
] |
[
1939,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1938,
1
] |
Mathlib/Data/Polynomial/Reverse.lean
|
Polynomial.reverse_trailingCoeff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf✝ f : R[X]\n⊢ trailingCoeff (reverse f) = leadingCoeff f",
"tactic": "rw [trailingCoeff, reverse_natTrailingDegree, coeff_zero_reverse]"
}
] |
[
313,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
exp_add
|
[] |
[
631,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
629,
1
] |
Mathlib/GroupTheory/GroupAction/ConjAct.lean
|
ConjAct.units_smul_def
|
[] |
[
165,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.mem_iInf
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_2\nG' : Type ?u.188426\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.188435\ninst✝ : AddGroup A\nH K : Subgroup G\nι : Sort u_1\nS : ι → Subgroup G\nx : G\n⊢ (x ∈ ⨅ (i : ι), S i) ↔ ∀ (i : ι), x ∈ S i",
"tactic": "simp only [iInf, mem_sInf, Set.forall_range_iff]"
}
] |
[
1001,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1000,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
map_extChartAt_nhdsWithin_eq_image
|
[] |
[
1168,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1165,
1
] |
Mathlib/Data/List/Basic.lean
|
List.enum_cons'
|
[] |
[
3928,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3926,
1
] |
Mathlib/Data/List/Basic.lean
|
List.map₂Right_cons_cons
|
[] |
[
4153,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4151,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.target_subset_preimage_source
|
[] |
[
546,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
545,
1
] |
Mathlib/RingTheory/Localization/Integral.lean
|
IsLocalization.integerNormalization_map_to_map
|
[
{
"state_after": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.42950\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\nb : { x // x ∈ M }\nhb : ∀ (i : ℕ), ↑(algebraMap R S) (coeff (integerNormalization M p) i) = ↑b • coeff p i\ni : ℕ\n⊢ ↑(algebraMap R S) (coeff (integerNormalization M p) i) = ↑b • coeff p i",
"state_before": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.42950\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\nb : { x // x ∈ M }\nhb : ∀ (i : ℕ), ↑(algebraMap R S) (coeff (integerNormalization M p) i) = ↑b • coeff p i\ni : ℕ\n⊢ coeff (Polynomial.map (algebraMap R S) (integerNormalization M p)) i = coeff (↑b • p) i",
"tactic": "rw [coeff_map, coeff_smul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.42950\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\nb : { x // x ∈ M }\nhb : ∀ (i : ℕ), ↑(algebraMap R S) (coeff (integerNormalization M p) i) = ↑b • coeff p i\ni : ℕ\n⊢ ↑(algebraMap R S) (coeff (integerNormalization M p) i) = ↑b • coeff p i",
"tactic": "exact hb i"
}
] |
[
106,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.coe_le_iff
|
[] |
[
715,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
715,
1
] |
Mathlib/Order/Filter/Partial.lean
|
Filter.rtendsto'_def
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nr : Rel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ l₁ ≤\n { sets := Rel.image (fun s t => Rel.preimage r s ⊆ t) l₂.sets,\n univ_sets := (_ : ∃ x, x ∈ l₂.sets ∧ (fun s t => Rel.preimage r s ⊆ t) x Set.univ),\n sets_of_superset :=\n (_ :\n ∀ {x y : Set α},\n x ∈ Rel.image (fun s t => Rel.preimage r s ⊆ t) l₂.sets →\n x ⊆ y → y ∈ Rel.image (fun s t => Rel.preimage r s ⊆ t) l₂.sets),\n inter_sets :=\n (_ :\n ∀ {x y : Set α},\n x ∈ Rel.image (fun s t => Rel.preimage r s ⊆ t) l₂.sets →\n y ∈ Rel.image (fun s t => Rel.preimage r s ⊆ t) l₂.sets →\n x ∩ y ∈ Rel.image (fun s t => Rel.preimage r s ⊆ t) l₂.sets) } ↔\n ∀ (s : Set β), s ∈ l₂ → Rel.preimage r s ∈ l₁",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nr : Rel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ RTendsto' r l₁ l₂ ↔ ∀ (s : Set β), s ∈ l₂ → Rel.preimage r s ∈ l₁",
"tactic": "unfold RTendsto' rcomap'"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nr : Rel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ (x : Set α) (x_1 : Set β), x_1 ∈ l₂ → Rel.preimage r x_1 ⊆ x → x ∈ l₁) ↔\n ∀ (s : Set β), s ∈ l₂ → Rel.preimage r s ∈ l₁",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nr : Rel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ l₁ ≤\n { sets := Rel.image (fun s t => Rel.preimage r s ⊆ t) l₂.sets,\n univ_sets := (_ : ∃ x, x ∈ l₂.sets ∧ (fun s t => Rel.preimage r s ⊆ t) x Set.univ),\n sets_of_superset :=\n (_ :\n ∀ {x y : Set α},\n x ∈ Rel.image (fun s t => Rel.preimage r s ⊆ t) l₂.sets →\n x ⊆ y → y ∈ Rel.image (fun s t => Rel.preimage r s ⊆ t) l₂.sets),\n inter_sets :=\n (_ :\n ∀ {x y : Set α},\n x ∈ Rel.image (fun s t => Rel.preimage r s ⊆ t) l₂.sets →\n y ∈ Rel.image (fun s t => Rel.preimage r s ⊆ t) l₂.sets →\n x ∩ y ∈ Rel.image (fun s t => Rel.preimage r s ⊆ t) l₂.sets) } ↔\n ∀ (s : Set β), s ∈ l₂ → Rel.preimage r s ∈ l₁",
"tactic": "simp [le_def, Rel.mem_image]"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nr : Rel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ (x : Set α) (x_1 : Set β), x_1 ∈ l₂ → Rel.preimage r x_1 ⊆ x → x ∈ l₁) →\n ∀ (s : Set β), s ∈ l₂ → Rel.preimage r s ∈ l₁\n\ncase mpr\nα : Type u\nβ : Type v\nγ : Type w\nr : Rel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ (s : Set β), s ∈ l₂ → Rel.preimage r s ∈ l₁) →\n ∀ (x : Set α) (x_1 : Set β), x_1 ∈ l₂ → Rel.preimage r x_1 ⊆ x → x ∈ l₁",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nr : Rel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ (x : Set α) (x_1 : Set β), x_1 ∈ l₂ → Rel.preimage r x_1 ⊆ x → x ∈ l₁) ↔\n ∀ (s : Set β), s ∈ l₂ → Rel.preimage r s ∈ l₁",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nr : Rel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ (x : Set α) (x_1 : Set β), x_1 ∈ l₂ → Rel.preimage r x_1 ⊆ x → x ∈ l₁) →\n ∀ (s : Set β), s ∈ l₂ → Rel.preimage r s ∈ l₁",
"tactic": "exact fun h s hs => h _ _ hs Set.Subset.rfl"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u\nβ : Type v\nγ : Type w\nr : Rel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ (s : Set β), s ∈ l₂ → Rel.preimage r s ∈ l₁) →\n ∀ (x : Set α) (x_1 : Set β), x_1 ∈ l₂ → Rel.preimage r x_1 ⊆ x → x ∈ l₁",
"tactic": "exact fun h s t ht => mem_of_superset (h t ht)"
}
] |
[
195,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
|
isLittleO_log_rpow_nhds_zero
|
[
{
"state_after": "no goals",
"state_before": "r : ℝ\nhr : r < 0\nx : ℝ\nhx : x ∈ Set.Icc 0 1\n⊢ x ∈ {x | (fun x => (fun x => -abs (log x) ^ 1) x = log x) x}",
"tactic": "simp [abs_of_nonpos (log_nonpos hx.1 hx.2)]"
}
] |
[
325,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MonoidHom.map_mul_inv
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nM : Type ?u.226815\nN : Type ?u.226818\nP : Type ?u.226821\nG : Type ?u.226824\nH : Type ?u.226827\nF : Type ?u.226830\ninst✝³ : Group G\ninst✝² : CommGroup H\ninst✝¹ : Group α\ninst✝ : DivisionMonoid β\nf : α →* β\ng h : α\n⊢ ↑f (g * h⁻¹) = ↑f g * (↑f h)⁻¹",
"tactic": "simp"
}
] |
[
1559,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1558,
11
] |
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