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/Algebra/GeomSum.lean
|
Commute.mul_geom_sum₂
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Ring α\nx y : α\nh : Commute x y\nn : ℕ\n⊢ (x - y) * ∑ i in range n, x ^ i * y ^ (n - 1 - i) = x ^ n - y ^ n",
"tactic": "rw [← neg_sub (y ^ n), ← h.mul_neg_geom_sum₂, ← neg_mul, neg_sub]"
}
] |
[
188,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Topology/Support.lean
|
HasCompactMulSupport.comp₂_left
|
[
{
"state_after": "X : Type ?u.16388\nα : Type u_1\nα' : Type ?u.16394\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nM : Type ?u.16406\nE : Type ?u.16409\nR : Type ?u.16412\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace α'\ninst✝² : One β\ninst✝¹ : One γ\ninst✝ : One δ\ng : β → γ\nf : α → β\nf₂ : α → γ\nm : β → γ → δ\nx : α\nhf : f =ᶠ[coclosedCompact α] 1\nhf₂ : f₂ =ᶠ[coclosedCompact α] 1\nhm : m 1 1 = 1\n⊢ (fun x => m (f x) (f₂ x)) =ᶠ[coclosedCompact α] 1",
"state_before": "X : Type ?u.16388\nα : Type u_1\nα' : Type ?u.16394\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nM : Type ?u.16406\nE : Type ?u.16409\nR : Type ?u.16412\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace α'\ninst✝² : One β\ninst✝¹ : One γ\ninst✝ : One δ\ng : β → γ\nf : α → β\nf₂ : α → γ\nm : β → γ → δ\nx : α\nhf : HasCompactMulSupport f\nhf₂ : HasCompactMulSupport f₂\nhm : m 1 1 = 1\n⊢ HasCompactMulSupport fun x => m (f x) (f₂ x)",
"tactic": "rw [hasCompactMulSupport_iff_eventuallyEq] at hf hf₂⊢"
},
{
"state_after": "no goals",
"state_before": "X : Type ?u.16388\nα : Type u_1\nα' : Type ?u.16394\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nM : Type ?u.16406\nE : Type ?u.16409\nR : Type ?u.16412\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace α'\ninst✝² : One β\ninst✝¹ : One γ\ninst✝ : One δ\ng : β → γ\nf : α → β\nf₂ : α → γ\nm : β → γ → δ\nx : α\nhf : f =ᶠ[coclosedCompact α] 1\nhf₂ : f₂ =ᶠ[coclosedCompact α] 1\nhm : m 1 1 = 1\n⊢ (fun x => m (f x) (f₂ x)) =ᶠ[coclosedCompact α] 1",
"tactic": "filter_upwards [hf, hf₂]using fun x hx hx₂ => by simp_rw [hx, hx₂, Pi.one_apply, hm]"
},
{
"state_after": "no goals",
"state_before": "X : Type ?u.16388\nα : Type u_1\nα' : Type ?u.16394\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nM : Type ?u.16406\nE : Type ?u.16409\nR : Type ?u.16412\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace α'\ninst✝² : One β\ninst✝¹ : One γ\ninst✝ : One δ\ng : β → γ\nf : α → β\nf₂ : α → γ\nm : β → γ → δ\nx✝ : α\nhf : f =ᶠ[coclosedCompact α] 1\nhf₂ : f₂ =ᶠ[coclosedCompact α] 1\nhm : m 1 1 = 1\nx : α\nhx : f x = OfNat.ofNat 1 x\nhx₂ : f₂ x = OfNat.ofNat 1 x\n⊢ m (f x) (f₂ x) = OfNat.ofNat 1 x",
"tactic": "simp_rw [hx, hx₂, Pi.one_apply, hm]"
}
] |
[
231,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/RingTheory/Polynomial/Bernstein.lean
|
bernsteinPolynomial.map
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\nS : Type u_2\ninst✝ : CommRing S\nf : R →+* S\nn ν : ℕ\n⊢ Polynomial.map f (bernsteinPolynomial R n ν) = bernsteinPolynomial S n ν",
"tactic": "simp [bernsteinPolynomial]"
}
] |
[
77,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean
|
MulChar.equivToUnitHom_symm_coe
|
[] |
[
259,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Order/Compare.lean
|
LT.lt.cmp_eq_gt
|
[] |
[
274,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
IsDedekindDomainInv.isDedekindDomain
|
[] |
[
352,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
351,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Function.Involutive.toPerm_involutive
|
[] |
[
1717,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1716,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
contDiffWithinAt_ext_coord_change
|
[] |
[
1260,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1256,
1
] |
Mathlib/Data/Nat/Order/Lemmas.lean
|
Nat.le_of_lt_add_of_dvd
|
[
{
"state_after": "case intro.intro\nm n k a b : ℕ\nh : n * a < n * b + n\n⊢ n * a ≤ n * b",
"state_before": "a b m n k : ℕ\nh : a < b + n\n⊢ n ∣ a → n ∣ b → a ≤ b",
"tactic": "rintro ⟨a, rfl⟩ ⟨b, rfl⟩"
},
{
"state_after": "case intro.intro\nm n k a b : ℕ\nh : n * a < n * (b + 1)\n⊢ n * a ≤ n * b",
"state_before": "case intro.intro\nm n k a b : ℕ\nh : n * a < n * b + n\n⊢ n * a ≤ n * b",
"tactic": "rw [← mul_add_one n] at h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nm n k a b : ℕ\nh : n * a < n * (b + 1)\n⊢ n * a ≤ n * b",
"tactic": "exact mul_le_mul_left' (lt_succ_iff.1 <| lt_of_mul_lt_mul_left h bot_le) _"
}
] |
[
222,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Data/Holor.lean
|
Holor.cprankMax_sum
|
[
{
"state_after": "no goals",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\n⊢ (∀ (x : β), x ∈ ∅ → CPRankMax n (f x)) → CPRankMax (Finset.card ∅ * n) (∑ x in ∅, f x)",
"tactic": "simp [CPRankMax.zero]"
},
{
"state_after": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx : β\ns : Finset β\nh_x_notin_s : ¬x ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x_1 : β), x_1 ∈ insert x s → CPRankMax n (f x_1)\n⊢ CPRankMax (Finset.card (insert x s) * n) (∑ x in insert x s, f x)",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\n⊢ ∀ ⦃a : β⦄ {s : Finset β},\n ¬a ∈ s →\n ((∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)) →\n (∀ (x : β), x ∈ insert a s → CPRankMax n (f x)) →\n CPRankMax (Finset.card (insert a s) * n) (∑ x in insert a s, f x)",
"tactic": "intro x s(h_x_notin_s : x ∉ s)ih h_cprank"
},
{
"state_after": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx : β\ns : Finset β\nh_x_notin_s : ¬x ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x_1 : β), x_1 ∈ insert x s → CPRankMax n (f x_1)\n⊢ CPRankMax ((Finset.card s + 1) * n) (f x + ∑ x in s, f x)",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx : β\ns : Finset β\nh_x_notin_s : ¬x ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x_1 : β), x_1 ∈ insert x s → CPRankMax n (f x_1)\n⊢ CPRankMax (Finset.card (insert x s) * n) (∑ x in insert x s, f x)",
"tactic": "simp only [Finset.sum_insert h_x_notin_s, Finset.card_insert_of_not_mem h_x_notin_s]"
},
{
"state_after": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx : β\ns : Finset β\nh_x_notin_s : ¬x ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x_1 : β), x_1 ∈ insert x s → CPRankMax n (f x_1)\n⊢ CPRankMax (Finset.card s * n + 1 * n) (f x + ∑ x in s, f x)",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx : β\ns : Finset β\nh_x_notin_s : ¬x ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x_1 : β), x_1 ∈ insert x s → CPRankMax n (f x_1)\n⊢ CPRankMax ((Finset.card s + 1) * n) (f x + ∑ x in s, f x)",
"tactic": "rw [Nat.right_distrib]"
},
{
"state_after": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx : β\ns : Finset β\nh_x_notin_s : ¬x ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x_1 : β), x_1 ∈ insert x s → CPRankMax n (f x_1)\n⊢ CPRankMax (n + Finset.card s * n) (f x + ∑ x in s, f x)",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx : β\ns : Finset β\nh_x_notin_s : ¬x ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x_1 : β), x_1 ∈ insert x s → CPRankMax n (f x_1)\n⊢ CPRankMax (Finset.card s * n + 1 * n) (f x + ∑ x in s, f x)",
"tactic": "simp only [Nat.one_mul, Nat.add_comm]"
},
{
"state_after": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx : β\ns : Finset β\nh_x_notin_s : ¬x ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x_1 : β), x_1 ∈ insert x s → CPRankMax n (f x_1)\nih' : CPRankMax (Finset.card s * n) (∑ x in s, f x)\n⊢ CPRankMax (n + Finset.card s * n) (f x + ∑ x in s, f x)",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx : β\ns : Finset β\nh_x_notin_s : ¬x ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x_1 : β), x_1 ∈ insert x s → CPRankMax n (f x_1)\n⊢ CPRankMax (n + Finset.card s * n) (f x + ∑ x in s, f x)",
"tactic": "have ih' : CPRankMax (Finset.card s * n) (∑ x in s, f x) := by\n apply ih\n intro (x : β)(h_x_in_s : x ∈ s)\n simp only [h_cprank, Finset.mem_insert_of_mem, h_x_in_s]"
},
{
"state_after": "no goals",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx : β\ns : Finset β\nh_x_notin_s : ¬x ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x_1 : β), x_1 ∈ insert x s → CPRankMax n (f x_1)\nih' : CPRankMax (Finset.card s * n) (∑ x in s, f x)\n⊢ CPRankMax (n + Finset.card s * n) (f x + ∑ x in s, f x)",
"tactic": "exact cprankMax_add (h_cprank x (Finset.mem_insert_self x s)) ih'"
},
{
"state_after": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx : β\ns : Finset β\nh_x_notin_s : ¬x ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x_1 : β), x_1 ∈ insert x s → CPRankMax n (f x_1)\n⊢ ∀ (x : β), x ∈ s → CPRankMax n (f x)",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx : β\ns : Finset β\nh_x_notin_s : ¬x ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x_1 : β), x_1 ∈ insert x s → CPRankMax n (f x_1)\n⊢ CPRankMax (Finset.card s * n) (∑ x in s, f x)",
"tactic": "apply ih"
},
{
"state_after": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx✝ : β\ns : Finset β\nh_x_notin_s : ¬x✝ ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x : β), x ∈ insert x✝ s → CPRankMax n (f x)\nx : β\nh_x_in_s : x ∈ s\n⊢ CPRankMax n (f x)",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx : β\ns : Finset β\nh_x_notin_s : ¬x ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x_1 : β), x_1 ∈ insert x s → CPRankMax n (f x_1)\n⊢ ∀ (x : β), x ∈ s → CPRankMax n (f x)",
"tactic": "intro (x : β)(h_x_in_s : x ∈ s)"
},
{
"state_after": "no goals",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝ : Ring α\nβ : Type u_1\nn : ℕ\ns✝ : Finset β\nf : β → Holor α ds\nthis : DecidableEq β := Classical.decEq β\nx✝ : β\ns : Finset β\nh_x_notin_s : ¬x✝ ∈ s\nih : (∀ (x : β), x ∈ s → CPRankMax n (f x)) → CPRankMax (Finset.card s * n) (∑ x in s, f x)\nh_cprank : ∀ (x : β), x ∈ insert x✝ s → CPRankMax n (f x)\nx : β\nh_x_in_s : x ∈ s\n⊢ CPRankMax n (f x)",
"tactic": "simp only [h_cprank, Finset.mem_insert_of_mem, h_x_in_s]"
}
] |
[
377,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
364,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.FinStronglyMeasurable.const_smul
|
[
{
"state_after": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.268899\nι : Type ?u.268902\ninst✝⁶ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝⁵ : TopologicalSpace β\n𝕜 : Type u_1\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : AddMonoid β\ninst✝² : Monoid 𝕜\ninst✝¹ : DistribMulAction 𝕜 β\ninst✝ : ContinuousSMul 𝕜 β\nhf : FinStronglyMeasurable f μ\nc : 𝕜\nn : ℕ\n⊢ ↑↑μ (support ↑((fun n => c • FinStronglyMeasurable.approx hf n) n)) < ⊤",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.268899\nι : Type ?u.268902\ninst✝⁶ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝⁵ : TopologicalSpace β\n𝕜 : Type u_1\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : AddMonoid β\ninst✝² : Monoid 𝕜\ninst✝¹ : DistribMulAction 𝕜 β\ninst✝ : ContinuousSMul 𝕜 β\nhf : FinStronglyMeasurable f μ\nc : 𝕜\n⊢ FinStronglyMeasurable (c • f) μ",
"tactic": "refine' ⟨fun n => c • hf.approx n, fun n => _, fun x => (hf.tendsto_approx x).const_smul c⟩"
},
{
"state_after": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.268899\nι : Type ?u.268902\ninst✝⁶ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝⁵ : TopologicalSpace β\n𝕜 : Type u_1\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : AddMonoid β\ninst✝² : Monoid 𝕜\ninst✝¹ : DistribMulAction 𝕜 β\ninst✝ : ContinuousSMul 𝕜 β\nhf : FinStronglyMeasurable f μ\nc : 𝕜\nn : ℕ\n⊢ ↑↑μ (support (c • ↑(FinStronglyMeasurable.approx hf n))) < ⊤",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.268899\nι : Type ?u.268902\ninst✝⁶ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝⁵ : TopologicalSpace β\n𝕜 : Type u_1\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : AddMonoid β\ninst✝² : Monoid 𝕜\ninst✝¹ : DistribMulAction 𝕜 β\ninst✝ : ContinuousSMul 𝕜 β\nhf : FinStronglyMeasurable f μ\nc : 𝕜\nn : ℕ\n⊢ ↑↑μ (support ↑((fun n => c • FinStronglyMeasurable.approx hf n) n)) < ⊤",
"tactic": "rw [SimpleFunc.coe_smul]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type ?u.268899\nι : Type ?u.268902\ninst✝⁶ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝⁵ : TopologicalSpace β\n𝕜 : Type u_1\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : AddMonoid β\ninst✝² : Monoid 𝕜\ninst✝¹ : DistribMulAction 𝕜 β\ninst✝ : ContinuousSMul 𝕜 β\nhf : FinStronglyMeasurable f μ\nc : 𝕜\nn : ℕ\n⊢ ↑↑μ (support (c • ↑(FinStronglyMeasurable.approx hf n))) < ⊤",
"tactic": "refine' (measure_mono (support_smul_subset_right c _)).trans_lt (hf.fin_support_approx n)"
}
] |
[
1101,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1096,
11
] |
Mathlib/Analysis/Convex/Caratheodory.lean
|
convexHull_eq_union
|
[
{
"state_after": "case h₁\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\n⊢ ↑(convexHull 𝕜).toOrderHom s ⊆\n ⋃ (t : Finset E) (_ : ↑t ⊆ s) (_ : AffineIndependent 𝕜 Subtype.val), ↑(convexHull 𝕜).toOrderHom ↑t\n\ncase h₂\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\n⊢ (⋃ (t : Finset E) (_ : ↑t ⊆ s) (_ : AffineIndependent 𝕜 Subtype.val), ↑(convexHull 𝕜).toOrderHom ↑t) ⊆\n ↑(convexHull 𝕜).toOrderHom s",
"state_before": "𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\n⊢ ↑(convexHull 𝕜).toOrderHom s =\n ⋃ (t : Finset E) (_ : ↑t ⊆ s) (_ : AffineIndependent 𝕜 Subtype.val), ↑(convexHull 𝕜).toOrderHom ↑t",
"tactic": "apply Set.Subset.antisymm"
},
{
"state_after": "case h₁\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\n⊢ x ∈ ⋃ (t : Finset E) (_ : ↑t ⊆ s) (_ : AffineIndependent 𝕜 Subtype.val), ↑(convexHull 𝕜).toOrderHom ↑t",
"state_before": "case h₁\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\n⊢ ↑(convexHull 𝕜).toOrderHom s ⊆\n ⋃ (t : Finset E) (_ : ↑t ⊆ s) (_ : AffineIndependent 𝕜 Subtype.val), ↑(convexHull 𝕜).toOrderHom ↑t",
"tactic": "intro x hx"
},
{
"state_after": "case h₁\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\n⊢ ∃ i, ↑i ⊆ s ∧ AffineIndependent 𝕜 Subtype.val ∧ x ∈ ↑(convexHull 𝕜).toOrderHom ↑i",
"state_before": "case h₁\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\n⊢ x ∈ ⋃ (t : Finset E) (_ : ↑t ⊆ s) (_ : AffineIndependent 𝕜 Subtype.val), ↑(convexHull 𝕜).toOrderHom ↑t",
"tactic": "simp only [exists_prop, Set.mem_iUnion]"
},
{
"state_after": "no goals",
"state_before": "case h₁\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\n⊢ ∃ i, ↑i ⊆ s ∧ AffineIndependent 𝕜 Subtype.val ∧ x ∈ ↑(convexHull 𝕜).toOrderHom ↑i",
"tactic": "exact ⟨Caratheodory.minCardFinsetOfMemConvexHull hx,\n Caratheodory.minCardFinsetOfMemConvexHull_subseteq hx,\n Caratheodory.affineIndependent_minCardFinsetOfMemConvexHull hx,\n Caratheodory.mem_minCardFinsetOfMemConvexHull hx⟩"
},
{
"state_after": "case h₂.convert_5.convert_5.convert_5\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\ni✝² : Finset E\ni✝¹ : ↑i✝² ⊆ s\ni✝ : AffineIndependent 𝕜 Subtype.val\n⊢ ↑(convexHull 𝕜).toOrderHom ↑i✝² ⊆ ↑(convexHull 𝕜).toOrderHom s",
"state_before": "case h₂\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\n⊢ (⋃ (t : Finset E) (_ : ↑t ⊆ s) (_ : AffineIndependent 𝕜 Subtype.val), ↑(convexHull 𝕜).toOrderHom ↑t) ⊆\n ↑(convexHull 𝕜).toOrderHom s",
"tactic": "iterate 3 convert Set.iUnion_subset _; intro"
},
{
"state_after": "no goals",
"state_before": "case h₂.convert_5.convert_5.convert_5\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\ni✝² : Finset E\ni✝¹ : ↑i✝² ⊆ s\ni✝ : AffineIndependent 𝕜 Subtype.val\n⊢ ↑(convexHull 𝕜).toOrderHom ↑i✝² ⊆ ↑(convexHull 𝕜).toOrderHom s",
"tactic": "exact convexHull_mono ‹_›"
},
{
"state_after": "case h₂.convert_5.convert_5.convert_5\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\ni✝¹ : Finset E\ni✝ : ↑i✝¹ ⊆ s\n⊢ AffineIndependent 𝕜 Subtype.val → ↑(convexHull 𝕜).toOrderHom ↑i✝¹ ⊆ ↑(convexHull 𝕜).toOrderHom s",
"state_before": "case h₂.convert_5.convert_5\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\ni✝¹ : Finset E\ni✝ : ↑i✝¹ ⊆ s\n⊢ (⋃ (_ : AffineIndependent 𝕜 Subtype.val), ↑(convexHull 𝕜).toOrderHom ↑i✝¹) ⊆ ↑(convexHull 𝕜).toOrderHom s",
"tactic": "convert Set.iUnion_subset _"
},
{
"state_after": "case h₂.convert_5.convert_5.convert_5\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\ni✝² : Finset E\ni✝¹ : ↑i✝² ⊆ s\ni✝ : AffineIndependent 𝕜 Subtype.val\n⊢ ↑(convexHull 𝕜).toOrderHom ↑i✝² ⊆ ↑(convexHull 𝕜).toOrderHom s",
"state_before": "case h₂.convert_5.convert_5.convert_5\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\ni✝¹ : Finset E\ni✝ : ↑i✝¹ ⊆ s\n⊢ AffineIndependent 𝕜 Subtype.val → ↑(convexHull 𝕜).toOrderHom ↑i✝¹ ⊆ ↑(convexHull 𝕜).toOrderHom s",
"tactic": "intro"
}
] |
[
166,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.getLeft_swap
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.23182\nδ : Type ?u.23185\nx : α ⊕ β\n⊢ getLeft (swap x) = getRight x",
"tactic": "cases x <;> rfl"
}
] |
[
377,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.mem_uIcc_of_ge
|
[] |
[
929,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
928,
1
] |
Mathlib/NumberTheory/PythagoreanTriples.lean
|
PythagoreanTriple.mul_iff
|
[
{
"state_after": "x y z : ℤ\nh : PythagoreanTriple x y z\nk : ℤ\nhk : k ≠ 0\n⊢ PythagoreanTriple (k * x) (k * y) (k * z) → PythagoreanTriple x y z",
"state_before": "x y z : ℤ\nh : PythagoreanTriple x y z\nk : ℤ\nhk : k ≠ 0\n⊢ PythagoreanTriple (k * x) (k * y) (k * z) ↔ PythagoreanTriple x y z",
"tactic": "refine' ⟨_, fun h => h.mul k⟩"
},
{
"state_after": "x y z : ℤ\nh : PythagoreanTriple x y z\nk : ℤ\nhk : k ≠ 0\n⊢ k * x * (k * x) + k * y * (k * y) = k * z * (k * z) → x * x + y * y = z * z",
"state_before": "x y z : ℤ\nh : PythagoreanTriple x y z\nk : ℤ\nhk : k ≠ 0\n⊢ PythagoreanTriple (k * x) (k * y) (k * z) → PythagoreanTriple x y z",
"tactic": "simp only [PythagoreanTriple]"
},
{
"state_after": "x y z : ℤ\nh✝ : PythagoreanTriple x y z\nk : ℤ\nhk : k ≠ 0\nh : k * x * (k * x) + k * y * (k * y) = k * z * (k * z)\n⊢ x * x + y * y = z * z",
"state_before": "x y z : ℤ\nh : PythagoreanTriple x y z\nk : ℤ\nhk : k ≠ 0\n⊢ k * x * (k * x) + k * y * (k * y) = k * z * (k * z) → x * x + y * y = z * z",
"tactic": "intro h"
},
{
"state_after": "x y z : ℤ\nh✝ : PythagoreanTriple x y z\nk : ℤ\nhk : k ≠ 0\nh : k * x * (k * x) + k * y * (k * y) = k * z * (k * z)\n⊢ (x * x + y * y) * (k * k) = z * z * (k * k)",
"state_before": "x y z : ℤ\nh✝ : PythagoreanTriple x y z\nk : ℤ\nhk : k ≠ 0\nh : k * x * (k * x) + k * y * (k * y) = k * z * (k * z)\n⊢ x * x + y * y = z * z",
"tactic": "rw [← mul_left_inj' (mul_ne_zero hk hk)]"
},
{
"state_after": "no goals",
"state_before": "x y z : ℤ\nh✝ : PythagoreanTriple x y z\nk : ℤ\nhk : k ≠ 0\nh : k * x * (k * x) + k * y * (k * y) = k * z * (k * z)\n⊢ (x * x + y * y) * (k * k) = z * z * (k * k)",
"tactic": "convert h using 1 <;> ring"
}
] |
[
94,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.ncard_pos
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.6628\ns t : Set α\na b x y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\n⊢ 0 < ncard s ↔ Set.Nonempty s",
"tactic": "rw [pos_iff_ne_zero, Ne.def, ncard_eq_zero hs, nonempty_iff_ne_empty]"
}
] |
[
113,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
Subtype.edist_eq
|
[] |
[
412,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
412,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iSup₂_le_iSup
|
[] |
[
833,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
832,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.injective
|
[] |
[
278,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
11
] |
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
[
{
"state_after": "case intro.mk\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nH : b * ↑(algebraMap R B) ↑(a₀, y).snd = ↑(algebraMap R B) (a₀, y).fst\n⊢ ∃ a n, ¬x ∣ a ∧ selfZpow x B n * ↑(algebraMap R B) a = b",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\n⊢ ∃ a n, ¬x ∣ a ∧ selfZpow x B n * ↑(algebraMap R B) a = b",
"tactic": "obtain ⟨⟨a₀, y⟩, H⟩ := surj (Submonoid.powers x) b"
},
{
"state_after": "case intro.mk.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nH : b * ↑(algebraMap R B) ↑(a₀, y).snd = ↑(algebraMap R B) (a₀, y).fst\nd : ℕ\nhy : x ^ d = ↑y\n⊢ ∃ a n, ¬x ∣ a ∧ selfZpow x B n * ↑(algebraMap R B) a = b",
"state_before": "case intro.mk\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nH : b * ↑(algebraMap R B) ↑(a₀, y).snd = ↑(algebraMap R B) (a₀, y).fst\n⊢ ∃ a n, ¬x ∣ a ∧ selfZpow x B n * ↑(algebraMap R B) a = b",
"tactic": "obtain ⟨d, hy⟩ := (Submonoid.mem_powers_iff y.1 x).mp y.2"
},
{
"state_after": "case intro.mk.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nH : b * ↑(algebraMap R B) ↑(a₀, y).snd = ↑(algebraMap R B) (a₀, y).fst\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\n⊢ ∃ a n, ¬x ∣ a ∧ selfZpow x B n * ↑(algebraMap R B) a = b",
"state_before": "case intro.mk.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nH : b * ↑(algebraMap R B) ↑(a₀, y).snd = ↑(algebraMap R B) (a₀, y).fst\nd : ℕ\nhy : x ^ d = ↑y\n⊢ ∃ a n, ¬x ∣ a ∧ selfZpow x B n * ↑(algebraMap R B) a = b",
"tactic": "have ha₀ : a₀ ≠ 0 := by\n haveI :=\n @isDomain_of_le_nonZeroDivisors B _ R _ _ _ (Submonoid.powers x) _\n (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)\n simp only [map_zero, ← hy, map_pow] at H\n apply ((injective_iff_map_eq_zero' (algebraMap R B)).mp _ a₀).mpr.mt\n rw [← H]\n apply mul_ne_zero hb (pow_ne_zero _ _)\n exact\n IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors B\n (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)\n (mem_nonZeroDivisors_iff_ne_zero.mpr hx.ne_zero)\n exact IsLocalization.injective B (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)"
},
{
"state_after": "case intro.mk.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * ↑(algebraMap R B) (x ^ d) = ↑(algebraMap R B) a₀\n⊢ ∃ a n, ¬x ∣ a ∧ selfZpow x B n * ↑(algebraMap R B) a = b",
"state_before": "case intro.mk.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nH : b * ↑(algebraMap R B) ↑(a₀, y).snd = ↑(algebraMap R B) (a₀, y).fst\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\n⊢ ∃ a n, ¬x ∣ a ∧ selfZpow x B n * ↑(algebraMap R B) a = b",
"tactic": "simp only [← hy] at H"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * ↑(algebraMap R B) (x ^ d) = ↑(algebraMap R B) a₀\n⊢ ∃ a n, ¬x ∣ a ∧ selfZpow x B n * ↑(algebraMap R B) a = b",
"tactic": "classical\nobtain ⟨m, a, hyp1, hyp2⟩ := max_power_factor ha₀ hx\nrefine' ⟨a, m - d, _⟩\nrw [← mk'_one (M := Submonoid.powers x) B, selfZpow_pow_sub, selfZpow_coe_nat, selfZpow_coe_nat,\n ← map_pow _ _ d, mul_comm _ b, H, hyp2, map_mul, map_pow _ _ m]\nexact ⟨hyp1, congr_arg _ (IsLocalization.mk'_one _ _)⟩"
},
{
"state_after": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nH : b * ↑(algebraMap R B) ↑(a₀, y).snd = ↑(algebraMap R B) (a₀, y).fst\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\n⊢ a₀ ≠ 0",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nH : b * ↑(algebraMap R B) ↑(a₀, y).snd = ↑(algebraMap R B) (a₀, y).fst\nd : ℕ\nhy : x ^ d = ↑y\n⊢ a₀ ≠ 0",
"tactic": "haveI :=\n @isDomain_of_le_nonZeroDivisors B _ R _ _ _ (Submonoid.powers x) _\n (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)"
},
{
"state_after": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ a₀ ≠ 0",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nH : b * ↑(algebraMap R B) ↑(a₀, y).snd = ↑(algebraMap R B) (a₀, y).fst\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\n⊢ a₀ ≠ 0",
"tactic": "simp only [map_zero, ← hy, map_pow] at H"
},
{
"state_after": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ ¬↑(algebraMap R B) a₀ = 0\n\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ Function.Injective ↑(algebraMap R B)",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ a₀ ≠ 0",
"tactic": "apply ((injective_iff_map_eq_zero' (algebraMap R B)).mp _ a₀).mpr.mt"
},
{
"state_after": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ ¬b * ↑(algebraMap R B) x ^ d = 0\n\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ Function.Injective ↑(algebraMap R B)",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ ¬↑(algebraMap R B) a₀ = 0\n\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ Function.Injective ↑(algebraMap R B)",
"tactic": "rw [← H]"
},
{
"state_after": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ ↑(algebraMap R B) x ≠ 0\n\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ Function.Injective ↑(algebraMap R B)",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ ¬b * ↑(algebraMap R B) x ^ d = 0\n\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ Function.Injective ↑(algebraMap R B)",
"tactic": "apply mul_ne_zero hb (pow_ne_zero _ _)"
},
{
"state_after": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ Function.Injective ↑(algebraMap R B)",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ ↑(algebraMap R B) x ≠ 0\n\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ Function.Injective ↑(algebraMap R B)",
"tactic": "exact\n IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors B\n (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)\n (mem_nonZeroDivisors_iff_ne_zero.mpr hx.ne_zero)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nthis : IsDomain B\nH : b * ↑(algebraMap R B) x ^ d = ↑(algebraMap R B) a₀\n⊢ Function.Injective ↑(algebraMap R B)",
"tactic": "exact IsLocalization.injective B (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * ↑(algebraMap R B) (x ^ d) = ↑(algebraMap R B) a₀\nm : ℕ\na : R\nhyp1 : ¬x ∣ a\nhyp2 : a₀ = x ^ m * a\n⊢ ∃ a n, ¬x ∣ a ∧ selfZpow x B n * ↑(algebraMap R B) a = b",
"state_before": "case intro.mk.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * ↑(algebraMap R B) (x ^ d) = ↑(algebraMap R B) a₀\n⊢ ∃ a n, ¬x ∣ a ∧ selfZpow x B n * ↑(algebraMap R B) a = b",
"tactic": "obtain ⟨m, a, hyp1, hyp2⟩ := max_power_factor ha₀ hx"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * ↑(algebraMap R B) (x ^ d) = ↑(algebraMap R B) a₀\nm : ℕ\na : R\nhyp1 : ¬x ∣ a\nhyp2 : a₀ = x ^ m * a\n⊢ ¬x ∣ a ∧ selfZpow x B (↑m - ↑d) * ↑(algebraMap R B) a = b",
"state_before": "case intro.mk.intro.intro.intro.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * ↑(algebraMap R B) (x ^ d) = ↑(algebraMap R B) a₀\nm : ℕ\na : R\nhyp1 : ¬x ∣ a\nhyp2 : a₀ = x ^ m * a\n⊢ ∃ a n, ¬x ∣ a ∧ selfZpow x B n * ↑(algebraMap R B) a = b",
"tactic": "refine' ⟨a, m - d, _⟩"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * ↑(algebraMap R B) (x ^ d) = ↑(algebraMap R B) a₀\nm : ℕ\na : R\nhyp1 : ¬x ∣ a\nhyp2 : a₀ = x ^ m * a\n⊢ ¬x ∣ a ∧ ↑(algebraMap R B) x ^ m * mk' B a 1 = ↑(algebraMap R B) x ^ m * ↑(algebraMap R B) a",
"state_before": "case intro.mk.intro.intro.intro.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * ↑(algebraMap R B) (x ^ d) = ↑(algebraMap R B) a₀\nm : ℕ\na : R\nhyp1 : ¬x ∣ a\nhyp2 : a₀ = x ^ m * a\n⊢ ¬x ∣ a ∧ selfZpow x B (↑m - ↑d) * ↑(algebraMap R B) a = b",
"tactic": "rw [← mk'_one (M := Submonoid.powers x) B, selfZpow_pow_sub, selfZpow_coe_nat, selfZpow_coe_nat,\n ← map_pow _ _ d, mul_comm _ b, H, hyp2, map_mul, map_pow _ _ m]"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro.intro.intro.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nx : R\nB : Type u_1\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : IsLocalization.Away x B\ninst✝² : IsDomain R\ninst✝¹ : NormalizationMonoid R\ninst✝ : UniqueFactorizationMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : { x_1 // x_1 ∈ Submonoid.powers x }\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * ↑(algebraMap R B) (x ^ d) = ↑(algebraMap R B) a₀\nm : ℕ\na : R\nhyp1 : ¬x ∣ a\nhyp2 : a₀ = x ^ m * a\n⊢ ¬x ∣ a ∧ ↑(algebraMap R B) x ^ m * mk' B a 1 = ↑(algebraMap R B) x ^ m * ↑(algebraMap R B) a",
"tactic": "exact ⟨hyp1, congr_arg _ (IsLocalization.mk'_one _ _)⟩"
}
] |
[
334,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.nsmul_lt_aleph0_iff
|
[
{
"state_after": "no goals",
"state_before": "case zero\nα β : Type u\na : Cardinal\n⊢ Nat.zero • a < ℵ₀ ↔ Nat.zero = 0 ∨ a < ℵ₀",
"tactic": "simpa using nat_lt_aleph0 0"
},
{
"state_after": "case succ\nα β : Type u\na : Cardinal\nn : ℕ\n⊢ Nat.succ n • a < ℵ₀ ↔ a < ℵ₀",
"state_before": "case succ\nα β : Type u\na : Cardinal\nn : ℕ\n⊢ Nat.succ n • a < ℵ₀ ↔ Nat.succ n = 0 ∨ a < ℵ₀",
"tactic": "simp only [Nat.succ_ne_zero, false_or_iff]"
},
{
"state_after": "case succ.zero\nα β : Type u\na : Cardinal\n⊢ Nat.succ Nat.zero • a < ℵ₀ ↔ a < ℵ₀\n\ncase succ.succ\nα β : Type u\na : Cardinal\nn : ℕ\nih : Nat.succ n • a < ℵ₀ ↔ a < ℵ₀\n⊢ Nat.succ (Nat.succ n) • a < ℵ₀ ↔ a < ℵ₀",
"state_before": "case succ\nα β : Type u\na : Cardinal\nn : ℕ\n⊢ Nat.succ n • a < ℵ₀ ↔ a < ℵ₀",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case succ.succ\nα β : Type u\na : Cardinal\nn : ℕ\nih : Nat.succ n • a < ℵ₀ ↔ a < ℵ₀\n⊢ Nat.succ (Nat.succ n) • a < ℵ₀ ↔ a < ℵ₀",
"tactic": "rw [succ_nsmul, add_lt_aleph0_iff, ih, and_self_iff]"
},
{
"state_after": "no goals",
"state_before": "case succ.zero\nα β : Type u\na : Cardinal\n⊢ Nat.succ Nat.zero • a < ℵ₀ ↔ a < ℵ₀",
"tactic": "simp"
}
] |
[
1546,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1539,
1
] |
Mathlib/RingTheory/Filtration.lean
|
Ideal.Filtration.iInf_N
|
[] |
[
182,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.mem_map
|
[] |
[
267,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.finrank_eq_four
|
[
{
"state_after": "no goals",
"state_before": "S : Type ?u.259837\nT : Type ?u.259840\nR : Type u_1\ninst✝¹ : CommRing R\nc₁ c₂ r x y z : R\na b c : ℍ[R,c₁,c₂]\ninst✝ : StrongRankCondition R\n⊢ FiniteDimensional.finrank R ℍ[R,c₁,c₂] = 4",
"tactic": "rw [FiniteDimensional.finrank, rank_eq_four, Cardinal.toNat_ofNat]"
}
] |
[
575,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
mul_ne_zero
|
[] |
[
93,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.not_infinite_neg
|
[] |
[
604,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
604,
1
] |
Mathlib/Order/Iterate.lean
|
Function.Commute.iterate_pos_lt_of_map_lt'
|
[] |
[
189,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Topology/Constructions.lean
|
nhds_subtype
|
[] |
[
229,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/RingTheory/WittVector/Basic.lean
|
WittVector.map_coeff
|
[] |
[
294,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_subtype
|
[
{
"state_after": "ι : Type ?u.383791\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\nF : Fintype { x // x ∈ s }\nh : ∀ (x : α), x ∈ s ↔ (fun x => x ∈ s) x\n⊢ ∏ a in s, f a = ∏ a : { x // x ∈ s }, f ↑a",
"state_before": "ι : Type ?u.383791\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\np : α → Prop\nF : Fintype (Subtype p)\ns : Finset α\nh : ∀ (x : α), x ∈ s ↔ p x\nf : α → β\nthis : (fun x => x ∈ s) = p\n⊢ ∏ a in s, f a = ∏ a : Subtype p, f ↑a",
"tactic": "subst p"
},
{
"state_after": "ι : Type ?u.383791\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\nF : Fintype { x // x ∈ s }\nh : ∀ (x : α), x ∈ s ↔ (fun x => x ∈ s) x\n⊢ ∏ i : { x // x ∈ s }, f ↑i = ∏ a : { x // x ∈ s }, f ↑a",
"state_before": "ι : Type ?u.383791\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\nF : Fintype { x // x ∈ s }\nh : ∀ (x : α), x ∈ s ↔ (fun x => x ∈ s) x\n⊢ ∏ a in s, f a = ∏ a : { x // x ∈ s }, f ↑a",
"tactic": "rw [← prod_coe_sort]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.383791\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\nF : Fintype { x // x ∈ s }\nh : ∀ (x : α), x ∈ s ↔ (fun x => x ∈ s) x\n⊢ ∏ i : { x // x ∈ s }, f ↑i = ∏ a : { x // x ∈ s }, f ↑a",
"tactic": "congr!"
}
] |
[
927,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
922,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
continuousOn_iff_isClosed
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.315580\nδ : Type ?u.315583\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns : Set α\nthis : ∀ (t : Set β), IsClosed (restrict s f ⁻¹' t) ↔ ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s\n⊢ ContinuousOn f s ↔ ∀ (t : Set β), IsClosed t → ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.315580\nδ : Type ?u.315583\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns : Set α\n⊢ ContinuousOn f s ↔ ∀ (t : Set β), IsClosed t → ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s",
"tactic": "have : ∀ t, IsClosed (s.restrict f ⁻¹' t) ↔ ∃ u : Set α, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s := by\n intro t\n rw [isClosed_induced_iff, Set.restrict_eq, Set.preimage_comp]\n simp only [Subtype.preimage_coe_eq_preimage_coe_iff, eq_comm]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.315580\nδ : Type ?u.315583\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns : Set α\nthis : ∀ (t : Set β), IsClosed (restrict s f ⁻¹' t) ↔ ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s\n⊢ (∀ (s_1 : Set β), IsClosed s_1 → IsClosed (restrict s f ⁻¹' s_1)) ↔\n ∀ (t : Set β), IsClosed t → ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.315580\nδ : Type ?u.315583\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns : Set α\nthis : ∀ (t : Set β), IsClosed (restrict s f ⁻¹' t) ↔ ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s\n⊢ ContinuousOn f s ↔ ∀ (t : Set β), IsClosed t → ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s",
"tactic": "rw [continuousOn_iff_continuous_restrict, continuous_iff_isClosed]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.315580\nδ : Type ?u.315583\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns : Set α\nthis : ∀ (t : Set β), IsClosed (restrict s f ⁻¹' t) ↔ ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s\n⊢ (∀ (s_1 : Set β), IsClosed s_1 → IsClosed (restrict s f ⁻¹' s_1)) ↔\n ∀ (t : Set β), IsClosed t → ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s",
"tactic": "simp only [this]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.315580\nδ : Type ?u.315583\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns : Set α\nt : Set β\n⊢ IsClosed (restrict s f ⁻¹' t) ↔ ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.315580\nδ : Type ?u.315583\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns : Set α\n⊢ ∀ (t : Set β), IsClosed (restrict s f ⁻¹' t) ↔ ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s",
"tactic": "intro t"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.315580\nδ : Type ?u.315583\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns : Set α\nt : Set β\n⊢ (∃ t_1, IsClosed t_1 ∧ Subtype.val ⁻¹' t_1 = Subtype.val ⁻¹' (f ⁻¹' t)) ↔ ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.315580\nδ : Type ?u.315583\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns : Set α\nt : Set β\n⊢ IsClosed (restrict s f ⁻¹' t) ↔ ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s",
"tactic": "rw [isClosed_induced_iff, Set.restrict_eq, Set.preimage_comp]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.315580\nδ : Type ?u.315583\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns : Set α\nt : Set β\n⊢ (∃ t_1, IsClosed t_1 ∧ Subtype.val ⁻¹' t_1 = Subtype.val ⁻¹' (f ⁻¹' t)) ↔ ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s",
"tactic": "simp only [Subtype.preimage_coe_eq_preimage_coe_iff, eq_comm]"
}
] |
[
651,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
645,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.ListBlank.tail_map
|
[
{
"state_after": "Γ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nl : ListBlank Γ\n⊢ tail (map f (cons (head l) (tail l))) = map f (tail l)",
"state_before": "Γ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nl : ListBlank Γ\n⊢ tail (map f l) = map f (tail l)",
"tactic": "conv => lhs; rw [← ListBlank.cons_head_tail l]"
},
{
"state_after": "no goals",
"state_before": "Γ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nl : ListBlank Γ\n⊢ tail (map f (cons (head l) (tail l))) = map f (tail l)",
"tactic": "exact Quotient.inductionOn' l fun a ↦ rfl"
}
] |
[
408,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
|
BilinForm.mul_toMatrix'
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.834155\nM✝ : Type ?u.834158\ninst✝¹⁸ : Semiring R\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module R M✝\nR₁ : Type ?u.834194\nM₁ : Type ?u.834197\ninst✝¹⁵ : Ring R₁\ninst✝¹⁴ : AddCommGroup M₁\ninst✝¹³ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type ?u.834809\ninst✝¹² : CommSemiring R₂\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : Module R₂ M₂\nR₃ : Type ?u.834996\nM₃ : Type ?u.834999\ninst✝⁹ : CommRing R₃\ninst✝⁸ : AddCommGroup M₃\ninst✝⁷ : Module R₃ M₃\nV : Type ?u.835587\nK : Type ?u.835590\ninst✝⁶ : Field K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nB✝ : BilinForm R M✝\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_2\no : Type ?u.836807\ninst✝³ : Fintype n\ninst✝² : Fintype o\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq o\nB : BilinForm R₂ (n → R₂)\nM : Matrix n n R₂\n⊢ M ⬝ ↑toMatrix' B = ↑toMatrix' (compLeft B (↑Matrix.toLin' Mᵀ))",
"tactic": "simp only [toMatrix'_compLeft, transpose_transpose, toMatrix'_toLin']"
}
] |
[
265,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
cauchySeq_const
|
[] |
[
194,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.coe_le
|
[] |
[
534,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_nonempty
|
[] |
[
520,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
519,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
lowerClosure_prod
|
[
{
"state_after": "case a.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.195438\nι : Sort ?u.195441\nκ : ι → Sort ?u.195446\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set α\nt : Set β\nx✝ : α × β\n⊢ x✝ ∈ ↑(lowerClosure (s ×ˢ t)) ↔ x✝ ∈ ↑(lowerClosure s ×ˢ lowerClosure t)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.195438\nι : Sort ?u.195441\nκ : ι → Sort ?u.195446\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set α\nt : Set β\n⊢ lowerClosure (s ×ˢ t) = lowerClosure s ×ˢ lowerClosure t",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.195438\nι : Sort ?u.195441\nκ : ι → Sort ?u.195446\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set α\nt : Set β\nx✝ : α × β\n⊢ x✝ ∈ ↑(lowerClosure (s ×ˢ t)) ↔ x✝ ∈ ↑(lowerClosure s ×ˢ lowerClosure t)",
"tactic": "simp [Prod.le_def, @and_and_and_comm _ (_ ∈ t)]"
}
] |
[
1759,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1756,
1
] |
Mathlib/MeasureTheory/Measure/Sub.lean
|
MeasureTheory.Measure.sub_eq_zero_of_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nh : μ ≤ ν\n⊢ μ ≤ 0 + ν",
"tactic": "rwa [zero_add]"
}
] |
[
50,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Algebra/DirectSum/Module.lean
|
DirectSum.coe_toModule_eq_coe_toAddMonoid
|
[] |
[
114,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/CategoryTheory/Subterminal.lean
|
CategoryTheory.isSubterminal_of_isIso_diag
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\nA : C\ninst✝¹ : HasBinaryProduct A A\ninst✝ : IsIso (diag A)\nZ : C\nf g : Z ⟶ A\nthis : prod.fst = prod.snd\n⊢ f = g",
"state_before": "C : Type u₁\ninst✝² : Category C\nA : C\ninst✝¹ : HasBinaryProduct A A\ninst✝ : IsIso (diag A)\nZ : C\nf g : Z ⟶ A\n⊢ f = g",
"tactic": "have : (Limits.prod.fst : A ⨯ A ⟶ _) = Limits.prod.snd := by simp [← cancel_epi (diag A)]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nA : C\ninst✝¹ : HasBinaryProduct A A\ninst✝ : IsIso (diag A)\nZ : C\nf g : Z ⟶ A\nthis : prod.fst = prod.snd\n⊢ f = g",
"tactic": "rw [← prod.lift_fst f g, this, prod.lift_snd]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nA : C\ninst✝¹ : HasBinaryProduct A A\ninst✝ : IsIso (diag A)\nZ : C\nf g : Z ⟶ A\n⊢ prod.fst = prod.snd",
"tactic": "simp [← cancel_epi (diag A)]"
}
] |
[
112,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.ae_lt_top
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\n⊢ ↑↑μ {a | f a = ⊤} = 0",
"state_before": "α : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\n⊢ ∀ᵐ (x : α) ∂μ, f x < ⊤",
"tactic": "simp_rw [ae_iff, ENNReal.not_lt_top]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nh : ¬↑↑μ {a | f a = ⊤} = 0\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\n⊢ ↑↑μ {a | f a = ⊤} = 0",
"tactic": "by_contra h"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nh : ¬↑↑μ {a | f a = ⊤} = 0\n⊢ ⊤ ≤ ∫⁻ (x : α), f x ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nh : ¬↑↑μ {a | f a = ⊤} = 0\n⊢ False",
"tactic": "apply h2f.lt_top.not_le"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nh : ¬↑↑μ {a | f a = ⊤} = 0\nthis : indicator (f ⁻¹' {⊤}) ⊤ ≤ f\n⊢ ⊤ ≤ ∫⁻ (x : α), f x ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nh : ¬↑↑μ {a | f a = ⊤} = 0\n⊢ ⊤ ≤ ∫⁻ (x : α), f x ∂μ",
"tactic": "have : (f ⁻¹' {∞}).indicator ⊤ ≤ f := by\n intro x\n by_cases hx : x ∈ f ⁻¹' {∞} <;> [simpa [indicator_of_mem hx]; simp [indicator_of_not_mem hx]]"
},
{
"state_after": "case h.e'_3\nα : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nh : ¬↑↑μ {a | f a = ⊤} = 0\nthis : indicator (f ⁻¹' {⊤}) ⊤ ≤ f\n⊢ ⊤ = ∫⁻ (a : α), indicator (f ⁻¹' {⊤}) ⊤ a ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nh : ¬↑↑μ {a | f a = ⊤} = 0\nthis : indicator (f ⁻¹' {⊤}) ⊤ ≤ f\n⊢ ⊤ ≤ ∫⁻ (x : α), f x ∂μ",
"tactic": "convert lintegral_mono this"
},
{
"state_after": "case h.e'_3\nα : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nh : ¬↑↑μ {a | f a = ⊤} = 0\nthis : indicator (f ⁻¹' {⊤}) ⊤ ≤ f\n⊢ ⊤ = ∫⁻ (a : α) in f ⁻¹' {⊤}, ⊤ a ∂μ",
"state_before": "case h.e'_3\nα : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nh : ¬↑↑μ {a | f a = ⊤} = 0\nthis : indicator (f ⁻¹' {⊤}) ⊤ ≤ f\n⊢ ⊤ = ∫⁻ (a : α), indicator (f ⁻¹' {⊤}) ⊤ a ∂μ",
"tactic": "rw [lintegral_indicator _ (hf (measurableSet_singleton ∞))]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nα : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nh : ¬↑↑μ {a | f a = ⊤} = 0\nthis : indicator (f ⁻¹' {⊤}) ⊤ ≤ f\n⊢ ⊤ = ∫⁻ (a : α) in f ⁻¹' {⊤}, ⊤ a ∂μ",
"tactic": "simp [ENNReal.top_mul', preimage, h]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nh : ¬↑↑μ {a | f a = ⊤} = 0\nx : α\n⊢ indicator (f ⁻¹' {⊤}) ⊤ x ≤ f x",
"state_before": "α : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nh : ¬↑↑μ {a | f a = ⊤} = 0\n⊢ indicator (f ⁻¹' {⊤}) ⊤ ≤ f",
"tactic": "intro x"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1736279\nγ : Type ?u.1736282\nδ : Type ?u.1736285\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nh2f : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nh : ¬↑↑μ {a | f a = ⊤} = 0\nx : α\n⊢ indicator (f ⁻¹' {⊤}) ⊤ x ≤ f x",
"tactic": "by_cases hx : x ∈ f ⁻¹' {∞} <;> [simpa [indicator_of_mem hx]; simp [indicator_of_not_mem hx]]"
}
] |
[
1505,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1495,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.exists_infinite_primes
|
[] |
[
484,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
475,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean
|
AlgebraicGeometry.StructureSheaf.const_mul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nf₁ f₂ g₁ g₂ : R\nU : Opens ↑(PrimeSpectrum.Top R)\nhu₁ : ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ U → g₁ ∈ Ideal.primeCompl x.asIdeal\nhu₂ : ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ U → g₂ ∈ Ideal.primeCompl x.asIdeal\nx : { x // x ∈ U.op.unop }\n⊢ ↑(const R (f₁ * f₂) (g₁ * g₂) U (_ : ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ U → g₁ * g₂ ∈ Ideal.primeCompl x.asIdeal))\n x =\n ↑(const R f₁ g₁ U hu₁ * const R f₂ g₂ U hu₂) x",
"tactic": "convert IsLocalization.mk'_mul _ f₁ f₂ ⟨g₁, hu₁ x x.2⟩ ⟨g₂, hu₂ x x.2⟩"
}
] |
[
382,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/RingTheory/PrincipalIdealDomain.lean
|
PrincipalIdealRing.factors_spec
|
[
{
"state_after": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na : R\nh : a ≠ 0\n⊢ (∀ (b : R),\n (b ∈\n if h : a = 0 then ∅\n else choose (_ : ∃ f, (∀ (b : R), b ∈ f → Irreducible b) ∧ Associated (Multiset.prod f) a)) →\n Irreducible b) ∧\n Associated\n (Multiset.prod\n (if h : a = 0 then ∅\n else choose (_ : ∃ f, (∀ (b : R), b ∈ f → Irreducible b) ∧ Associated (Multiset.prod f) a)))\n a",
"state_before": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na : R\nh : a ≠ 0\n⊢ (∀ (b : R), b ∈ factors a → Irreducible b) ∧ Associated (Multiset.prod (factors a)) a",
"tactic": "unfold factors"
},
{
"state_after": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na : R\nh : a ≠ 0\n⊢ (∀ (b : R),\n b ∈ choose (_ : ∃ f, (∀ (b : R), b ∈ f → Irreducible b) ∧ Associated (Multiset.prod f) a) → Irreducible b) ∧\n Associated (Multiset.prod (choose (_ : ∃ f, (∀ (b : R), b ∈ f → Irreducible b) ∧ Associated (Multiset.prod f) a))) a",
"state_before": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na : R\nh : a ≠ 0\n⊢ (∀ (b : R),\n (b ∈\n if h : a = 0 then ∅\n else choose (_ : ∃ f, (∀ (b : R), b ∈ f → Irreducible b) ∧ Associated (Multiset.prod f) a)) →\n Irreducible b) ∧\n Associated\n (Multiset.prod\n (if h : a = 0 then ∅\n else choose (_ : ∃ f, (∀ (b : R), b ∈ f → Irreducible b) ∧ Associated (Multiset.prod f) a)))\n a",
"tactic": "rw [dif_neg h]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na : R\nh : a ≠ 0\n⊢ (∀ (b : R),\n b ∈ choose (_ : ∃ f, (∀ (b : R), b ∈ f → Irreducible b) ∧ Associated (Multiset.prod f) a) → Irreducible b) ∧\n Associated (Multiset.prod (choose (_ : ∃ f, (∀ (b : R), b ∈ f → Irreducible b) ∧ Associated (Multiset.prod f) a))) a",
"tactic": "exact Classical.choose_spec (WfDvdMonoid.exists_factors a h)"
}
] |
[
286,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Std/Data/Array/Init/Lemmas.lean
|
Array.get_push_eq
|
[
{
"state_after": "α : Type u_1\na : Array α\nx : α\n⊢ List.get (a.data ++ [x])\n { val := size a, isLt := (_ : (fun xs i => i < size xs) { data := a.data ++ [x] } (size a)) } =\n x",
"state_before": "α : Type u_1\na : Array α\nx : α\n⊢ (push a x)[size a] = x",
"tactic": "simp only [push, getElem_eq_data_get, List.concat_eq_append]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na : Array α\nx : α\n⊢ List.get (a.data ++ [x])\n { val := size a, isLt := (_ : (fun xs i => i < size xs) { data := a.data ++ [x] } (size a)) } =\n x",
"tactic": "rw [List.get_append_right] <;> simp [getElem_eq_data_get]"
}
] |
[
133,
60
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
131,
9
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.neg_of
|
[] |
[
338,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.eq_top_iff_forall_le
|
[] |
[
394,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
392,
1
] |
Mathlib/Data/Multiset/Bind.lean
|
Multiset.bind_assoc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.34299\na : α\ns✝ t : Multiset α\nf✝ g✝ : α → Multiset β\ns : Multiset α\nf : α → Multiset β\ng : β → Multiset γ\n⊢ bind (bind 0 f) g = bind 0 fun a => bind (f a) g",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.34299\na : α\ns✝ t : Multiset α\nf✝ g✝ : α → Multiset β\ns : Multiset α\nf : α → Multiset β\ng : β → Multiset γ\n⊢ ∀ ⦃a : α⦄ {s : Multiset α},\n (bind (bind s f) g = bind s fun a => bind (f a) g) →\n bind (bind (a ::ₘ s) f) g = bind (a ::ₘ s) fun a => bind (f a) g",
"tactic": "simp (config := { contextual := true })"
}
] |
[
174,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.forall_coe_one_iff_one
|
[] |
[
641,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
640,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
differentiableAt_add_const_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.207343\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.207438\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nc : F\nh : DifferentiableAt 𝕜 (fun y => f y + c) x\n⊢ DifferentiableAt 𝕜 f x",
"tactic": "simpa using h.add_const (-c)"
}
] |
[
216,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.map_bot
|
[] |
[
357,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.mul_smul'
|
[
{
"state_after": "case h\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn : ℕ\n⊢ ↑((f * g) • h) n = ↑(f • g • h) n",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\n⊢ (f * g) • h = f • g • h",
"tactic": "ext n"
},
{
"state_after": "case h\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn : ℕ\n⊢ ∑ x in Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst,\n ↑f x.snd.fst • ↑g x.snd.snd • ↑h x.fst.snd =\n ∑ x in Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.snd,\n ↑f x.fst.fst • ↑g x.snd.fst • ↑h x.snd.snd",
"state_before": "case h\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn : ℕ\n⊢ ↑((f * g) • h) n = ↑(f • g • h) n",
"tactic": "simp only [mul_apply, smul_apply, sum_smul, mul_smul, smul_sum, Finset.sum_sigma']"
},
{
"state_after": "case h.hi\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn : ℕ\n⊢ ∀ (a : (_ : ℕ × ℕ) × ℕ × ℕ) (ha : a ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst),\n ?h.i a ha ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.snd\n\ncase h.h\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn : ℕ\n⊢ ∀ (a : (_ : ℕ × ℕ) × ℕ × ℕ) (ha : a ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst),\n ↑f a.snd.fst • ↑g a.snd.snd • ↑h a.fst.snd =\n ↑f (?h.i a ha).fst.fst • ↑g (?h.i a ha).snd.fst • ↑h (?h.i a ha).snd.snd\n\ncase h.i_inj\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn : ℕ\n⊢ ∀ (a₁ a₂ : (_ : ℕ × ℕ) × ℕ × ℕ) (ha₁ : a₁ ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst)\n (ha₂ : a₂ ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst),\n ?h.i a₁ ha₁ = ?h.i a₂ ha₂ → a₁ = a₂\n\ncase h.i_surj\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn : ℕ\n⊢ ∀ (b : (_ : ℕ × ℕ) × ℕ × ℕ),\n (b ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.snd) → ∃ a ha, b = ?h.i a ha\n\ncase h.i\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn : ℕ\n⊢ (a : (_ : ℕ × ℕ) × ℕ × ℕ) →\n (a ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ",
"state_before": "case h\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn : ℕ\n⊢ ∑ x in Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst,\n ↑f x.snd.fst • ↑g x.snd.snd • ↑h x.fst.snd =\n ∑ x in Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.snd,\n ↑f x.fst.fst • ↑g x.snd.fst • ↑h x.snd.snd",
"tactic": "apply Finset.sum_bij"
},
{
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"tactic": "rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ _H"
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{
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"tactic": "exact ⟨(k, l * j), (l, j)⟩"
},
{
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"tactic": "rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H"
},
{
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"tactic": "simp only [Finset.mem_sigma, mem_divisorsAntidiagonal] at H⊢"
},
{
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"tactic": "rcases H with ⟨⟨rfl, n0⟩, rfl, i0⟩"
},
{
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"tactic": "refine' ⟨⟨(mul_assoc _ _ _).symm, n0⟩, trivial, _⟩"
},
{
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"tactic": "rw [mul_ne_zero_iff] at *"
},
{
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"tactic": "exact ⟨i0.2, n0.2⟩"
},
{
"state_after": "case h.h.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn i j k l : ℕ\n_H : { fst := (i, j), snd := (k, l) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst\n⊢ ↑f { fst := (i, j), snd := (k, l) }.snd.fst •\n ↑g { fst := (i, j), snd := (k, l) }.snd.snd • ↑h { fst := (i, j), snd := (k, l) }.fst.snd =\n ↑f\n (Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n { fst := (i, j), snd := (k, l) }\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n _H).fst.fst •\n ↑g\n (Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n { fst := (i, j), snd := (k, l) }\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n _H).snd.fst •\n ↑h\n (Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n { fst := (i, j), snd := (k, l) }\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n _H).snd.snd",
"state_before": "case h.h\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn : ℕ\n⊢ ∀ (a : (_ : ℕ × ℕ) × ℕ × ℕ) (ha : a ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst),\n ↑f a.snd.fst • ↑g a.snd.snd • ↑h a.fst.snd =\n ↑f\n (Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ) a\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n ha).fst.fst •\n ↑g\n (Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n a\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n ha).snd.fst •\n ↑h\n (Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n a\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n ha).snd.snd",
"tactic": "rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ _H"
},
{
"state_after": "no goals",
"state_before": "case h.h.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn i j k l : ℕ\n_H : { fst := (i, j), snd := (k, l) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst\n⊢ ↑f { fst := (i, j), snd := (k, l) }.snd.fst •\n ↑g { fst := (i, j), snd := (k, l) }.snd.snd • ↑h { fst := (i, j), snd := (k, l) }.fst.snd =\n ↑f\n (Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n { fst := (i, j), snd := (k, l) }\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n _H).fst.fst •\n ↑g\n (Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n { fst := (i, j), snd := (k, l) }\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n _H).snd.fst •\n ↑h\n (Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n { fst := (i, j), snd := (k, l) }\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n _H).snd.snd",
"tactic": "simp only [mul_assoc]"
},
{
"state_after": "case h.i_inj.mk.mk.mk.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn a b c d i j k l : ℕ\nH₁ : { fst := (a, b), snd := (c, d) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst\nH₂ : { fst := (i, j), snd := (k, l) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst\n⊢ Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n { fst := (a, b), snd := (c, d) }\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n H₁ =\n Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n { fst := (i, j), snd := (k, l) }\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n H₂ →\n { fst := (a, b), snd := (c, d) } = { fst := (i, j), snd := (k, l) }",
"state_before": "case h.i_inj\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn : ℕ\n⊢ ∀ (a₁ a₂ : (_ : ℕ × ℕ) × ℕ × ℕ) (ha₁ : a₁ ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst)\n (ha₂ : a₂ ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst),\n Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ) a₁\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n ha₁ =\n Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ) a₂\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n ha₂ →\n a₁ = a₂",
"tactic": "rintro ⟨⟨a, b⟩, ⟨c, d⟩⟩ ⟨⟨i, j⟩, ⟨k, l⟩⟩ H₁ H₂"
},
{
"state_after": "case h.i_inj.mk.mk.mk.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn a b c d i j k l : ℕ\nH₁ : (a * b = n ∧ n ≠ 0) ∧ c * d = a ∧ a ≠ 0\nH₂ : (i * j = n ∧ n ≠ 0) ∧ k * l = i ∧ i ≠ 0\n⊢ { fst := (c, d * b), snd := (d, b) } = { fst := (k, l * j), snd := (l, j) } →\n { fst := (a, b), snd := (c, d) } = { fst := (i, j), snd := (k, l) }",
"state_before": "case h.i_inj.mk.mk.mk.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn a b c d i j k l : ℕ\nH₁ : { fst := (a, b), snd := (c, d) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst\nH₂ : { fst := (i, j), snd := (k, l) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst\n⊢ Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n { fst := (a, b), snd := (c, d) }\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n H₁ =\n Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n { fst := (i, j), snd := (k, l) }\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n H₂ →\n { fst := (a, b), snd := (c, d) } = { fst := (i, j), snd := (k, l) }",
"tactic": "simp only [Finset.mem_sigma, mem_divisorsAntidiagonal, and_imp, Prod.mk.inj_iff, add_comm,\n heq_iff_eq] at H₁ H₂⊢"
},
{
"state_after": "case h.i_inj.mk.mk.mk.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn a b c d i j k l : ℕ\nH₁ : (a * b = n ∧ n ≠ 0) ∧ c * d = a ∧ a ≠ 0\nH₂ : (i * j = n ∧ n ≠ 0) ∧ k * l = i ∧ i ≠ 0\n⊢ c = k → d * b = l * j → d = l → b = j → (a = i ∧ b = j) ∧ c = k ∧ d = l",
"state_before": "case h.i_inj.mk.mk.mk.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn a b c d i j k l : ℕ\nH₁ : (a * b = n ∧ n ≠ 0) ∧ c * d = a ∧ a ≠ 0\nH₂ : (i * j = n ∧ n ≠ 0) ∧ k * l = i ∧ i ≠ 0\n⊢ { fst := (c, d * b), snd := (d, b) } = { fst := (k, l * j), snd := (l, j) } →\n { fst := (a, b), snd := (c, d) } = { fst := (i, j), snd := (k, l) }",
"tactic": "simp only [Sigma.mk.inj_iff, Prod.mk.injEq, heq_eq_eq, and_imp]"
},
{
"state_after": "case h.i_inj.mk.mk.mk.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh✝ : ArithmeticFunction M\nn a b c d i k : ℕ\nH₁ : (a * b = n ∧ n ≠ 0) ∧ c * d = a ∧ a ≠ 0\nh : c = k\nH₂ : (i * b = n ∧ n ≠ 0) ∧ k * d = i ∧ i ≠ 0\nh2 : d * b = d * b\n⊢ (a = i ∧ b = b) ∧ c = k ∧ d = d",
"state_before": "case h.i_inj.mk.mk.mk.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn a b c d i j k l : ℕ\nH₁ : (a * b = n ∧ n ≠ 0) ∧ c * d = a ∧ a ≠ 0\nH₂ : (i * j = n ∧ n ≠ 0) ∧ k * l = i ∧ i ≠ 0\n⊢ c = k → d * b = l * j → d = l → b = j → (a = i ∧ b = j) ∧ c = k ∧ d = l",
"tactic": "rintro h h2 rfl rfl"
},
{
"state_after": "case h.i_inj.mk.mk.mk.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn a b c d i : ℕ\nH₁ : (a * b = n ∧ n ≠ 0) ∧ c * d = a ∧ a ≠ 0\nh2 : d * b = d * b\nH₂ : (i * b = n ∧ n ≠ 0) ∧ c * d = i ∧ i ≠ 0\n⊢ (a = i ∧ b = b) ∧ c = c ∧ d = d",
"state_before": "case h.i_inj.mk.mk.mk.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh✝ : ArithmeticFunction M\nn a b c d i k : ℕ\nH₁ : (a * b = n ∧ n ≠ 0) ∧ c * d = a ∧ a ≠ 0\nh : c = k\nH₂ : (i * b = n ∧ n ≠ 0) ∧ k * d = i ∧ i ≠ 0\nh2 : d * b = d * b\n⊢ (a = i ∧ b = b) ∧ c = k ∧ d = d",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "case h.i_inj.mk.mk.mk.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn a b c d i : ℕ\nH₁ : (a * b = n ∧ n ≠ 0) ∧ c * d = a ∧ a ≠ 0\nh2 : d * b = d * b\nH₂ : (i * b = n ∧ n ≠ 0) ∧ c * d = i ∧ i ≠ 0\n⊢ (a = i ∧ b = b) ∧ c = c ∧ d = d",
"tactic": "exact ⟨⟨Eq.trans H₁.2.1.symm H₂.2.1, rfl⟩, rfl, rfl⟩"
},
{
"state_after": "case h.i_surj.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn i j k l : ℕ\nH : { fst := (i, j), snd := (k, l) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.snd\n⊢ ∃ a ha,\n { fst := (i, j), snd := (k, l) } =\n Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ) a\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n ha",
"state_before": "case h.i_surj\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn : ℕ\n⊢ ∀ (b : (_ : ℕ × ℕ) × ℕ × ℕ),\n (b ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.snd) →\n ∃ a ha,\n b =\n Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ) a\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n ha",
"tactic": "rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H"
},
{
"state_after": "case h.i_surj.mk.mk.mk.refine'_1\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn i j k l : ℕ\nH : { fst := (i, j), snd := (k, l) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.snd\n⊢ { fst := (i * k, l), snd := (i, k) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst\n\ncase h.i_surj.mk.mk.mk.refine'_2\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn i j k l : ℕ\nH : { fst := (i, j), snd := (k, l) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.snd\n⊢ { fst := (i, j), snd := (k, l) } =\n Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n { fst := (i * k, l), snd := (i, k) }\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n ?h.i_surj.mk.mk.mk.refine'_1",
"state_before": "case h.i_surj.mk.mk.mk\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn i j k l : ℕ\nH : { fst := (i, j), snd := (k, l) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.snd\n⊢ ∃ a ha,\n { fst := (i, j), snd := (k, l) } =\n Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ) a\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n ha",
"tactic": "refine' ⟨⟨(i * k, l), (i, k)⟩, _, _⟩"
},
{
"state_after": "case h.i_surj.mk.mk.mk.refine'_1\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn i j k l : ℕ\nH : (i * j = n ∧ n ≠ 0) ∧ k * l = j ∧ j ≠ 0\n⊢ (i * k * l = n ∧ n ≠ 0) ∧ True ∧ i * k ≠ 0",
"state_before": "case h.i_surj.mk.mk.mk.refine'_1\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn i j k l : ℕ\nH : { fst := (i, j), snd := (k, l) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.snd\n⊢ { fst := (i * k, l), snd := (i, k) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst",
"tactic": "simp only [Finset.mem_sigma, mem_divisorsAntidiagonal] at H⊢"
},
{
"state_after": "case h.i_surj.mk.mk.mk.refine'_1.intro.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\ni k l : ℕ\nn0 : i * (k * l) ≠ 0\nj0 : k * l ≠ 0\n⊢ (i * k * l = i * (k * l) ∧ i * (k * l) ≠ 0) ∧ True ∧ i * k ≠ 0",
"state_before": "case h.i_surj.mk.mk.mk.refine'_1\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn i j k l : ℕ\nH : (i * j = n ∧ n ≠ 0) ∧ k * l = j ∧ j ≠ 0\n⊢ (i * k * l = n ∧ n ≠ 0) ∧ True ∧ i * k ≠ 0",
"tactic": "rcases H with ⟨⟨rfl, n0⟩, rfl, j0⟩"
},
{
"state_after": "case h.i_surj.mk.mk.mk.refine'_1.intro.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\ni k l : ℕ\nn0 : i * (k * l) ≠ 0\nj0 : k * l ≠ 0\n⊢ i * k ≠ 0",
"state_before": "case h.i_surj.mk.mk.mk.refine'_1.intro.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\ni k l : ℕ\nn0 : i * (k * l) ≠ 0\nj0 : k * l ≠ 0\n⊢ (i * k * l = i * (k * l) ∧ i * (k * l) ≠ 0) ∧ True ∧ i * k ≠ 0",
"tactic": "refine' ⟨⟨mul_assoc _ _ _, n0⟩, trivial, _⟩"
},
{
"state_after": "case h.i_surj.mk.mk.mk.refine'_1.intro.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\ni k l : ℕ\nn0 : i ≠ 0 ∧ k * l ≠ 0\nj0 : k ≠ 0 ∧ l ≠ 0\n⊢ i ≠ 0 ∧ k ≠ 0",
"state_before": "case h.i_surj.mk.mk.mk.refine'_1.intro.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\ni k l : ℕ\nn0 : i * (k * l) ≠ 0\nj0 : k * l ≠ 0\n⊢ i * k ≠ 0",
"tactic": "rw [mul_ne_zero_iff] at *"
},
{
"state_after": "no goals",
"state_before": "case h.i_surj.mk.mk.mk.refine'_1.intro.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\ni k l : ℕ\nn0 : i ≠ 0 ∧ k * l ≠ 0\nj0 : k ≠ 0 ∧ l ≠ 0\n⊢ i ≠ 0 ∧ k ≠ 0",
"tactic": "exact ⟨n0.1, j0.1⟩"
},
{
"state_after": "case h.i_surj.mk.mk.mk.refine'_2\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn i j k l : ℕ\nH : (i * j = n ∧ ¬n = 0) ∧ k * l = j ∧ ¬j = 0\n⊢ { fst := (i, j), snd := (k, l) } = { fst := (i, k * l), snd := (k, l) }",
"state_before": "case h.i_surj.mk.mk.mk.refine'_2\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn i j k l : ℕ\nH : { fst := (i, j), snd := (k, l) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.snd\n⊢ { fst := (i, j), snd := (k, l) } =\n Sigma.casesOn (motive := fun x =>\n (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) → (_ : ℕ × ℕ) × ℕ × ℕ)\n { fst := (i * k, l), snd := (i, k) }\n (fun fst snd =>\n Prod.casesOn (motive := fun x =>\n ({ fst := x, snd := snd } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n fst fun i j =>\n Prod.casesOn (motive := fun x =>\n ({ fst := (i, j), snd := x } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst) →\n (_ : ℕ × ℕ) × ℕ × ℕ)\n snd fun k l _H => { fst := (k, l * j), snd := (l, j) })\n (_ :\n { fst := (i * k, l), snd := (i, k) } ∈\n Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.fst)",
"tactic": "simp only [true_and_iff, mem_divisorsAntidiagonal, and_true_iff, Prod.mk.inj_iff,\n eq_self_iff_true, Ne.def, mem_sigma, heq_iff_eq] at H⊢"
},
{
"state_after": "no goals",
"state_before": "case h.i_surj.mk.mk.mk.refine'_2\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ArithmeticFunction R\nh : ArithmeticFunction M\nn i j k l : ℕ\nH : (i * j = n ∧ ¬n = 0) ∧ k * l = j ∧ ¬j = 0\n⊢ { fst := (i, j), snd := (k, l) } = { fst := (i, k * l), snd := (k, l) }",
"tactic": "rw [H.2.1]"
}
] |
[
334,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Algebra/GradedMonoid.lean
|
List.dProd_cons
|
[] |
[
403,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
NNReal.rpow_one
|
[] |
[
72,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Topology/Bornology/Basic.lean
|
Bornology.isBounded_ofBounded_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.6781\nα : Type u_1\nβ : Type ?u.6787\ns : Set α\nB : Set (Set α)\nempty_mem : ∅ ∈ B\nsubset_mem : ∀ (s₁ : Set α), s₁ ∈ B → ∀ (s₂ : Set α), s₂ ⊆ s₁ → s₂ ∈ B\nunion_mem : ∀ (s₁ : Set α), s₁ ∈ B → ∀ (s₂ : Set α), s₂ ∈ B → s₁ ∪ s₂ ∈ B\nsUnion_univ : ∀ (x : α), {x} ∈ B\n⊢ IsBounded s ↔ s ∈ B",
"tactic": "rw [@isBounded_def _ (ofBounded B empty_mem subset_mem union_mem sUnion_univ), ← Filter.mem_sets,\n ofBounded_cobounded_sets, Set.mem_setOf_eq, compl_compl]"
}
] |
[
256,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.Monic.irreducible_iff_natDegree
|
[
{
"state_after": "case pos\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nhp : Monic p\nhp1 : p = 1\n⊢ Irreducible p ↔ p ≠ 1 ∧ ∀ (f g : R[X]), Monic f → Monic g → f * g = p → natDegree f = 0 ∨ natDegree g = 0\n\ncase neg\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nhp : Monic p\nhp1 : ¬p = 1\n⊢ Irreducible p ↔ p ≠ 1 ∧ ∀ (f g : R[X]), Monic f → Monic g → f * g = p → natDegree f = 0 ∨ natDegree g = 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nhp : Monic p\n⊢ Irreducible p ↔ p ≠ 1 ∧ ∀ (f g : R[X]), Monic f → Monic g → f * g = p → natDegree f = 0 ∨ natDegree g = 0",
"tactic": "by_cases hp1 : p = 1"
},
{
"state_after": "case neg\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nhp : Monic p\nhp1 : ¬p = 1\n⊢ (∀ (f g : R[X]), Monic f → Monic g → f * g = p → f = 1 ∨ g = 1) ↔\n ∀ (f g : R[X]), Monic f → Monic g → f * g = p → natDegree f = 0 ∨ natDegree g = 0",
"state_before": "case neg\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nhp : Monic p\nhp1 : ¬p = 1\n⊢ Irreducible p ↔ p ≠ 1 ∧ ∀ (f g : R[X]), Monic f → Monic g → f * g = p → natDegree f = 0 ∨ natDegree g = 0",
"tactic": "rw [irreducible_of_monic hp hp1, and_iff_right hp1]"
},
{
"state_after": "case neg\nR : Type u\nS : Type v\nT : Type w\na✝ b✝ : R\nn : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nhp : Monic p\nhp1 : ¬p = 1\na b : R[X]\nha : Monic a\nhb : Monic b\n⊢ a * b = p → a = 1 ∨ b = 1 ↔ a * b = p → natDegree a = 0 ∨ natDegree b = 0",
"state_before": "case neg\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nhp : Monic p\nhp1 : ¬p = 1\n⊢ (∀ (f g : R[X]), Monic f → Monic g → f * g = p → f = 1 ∨ g = 1) ↔\n ∀ (f g : R[X]), Monic f → Monic g → f * g = p → natDegree f = 0 ∨ natDegree g = 0",
"tactic": "refine' forall₄_congr fun a b ha hb => _"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\nT : Type w\na✝ b✝ : R\nn : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nhp : Monic p\nhp1 : ¬p = 1\na b : R[X]\nha : Monic a\nhb : Monic b\n⊢ a * b = p → a = 1 ∨ b = 1 ↔ a * b = p → natDegree a = 0 ∨ natDegree b = 0",
"tactic": "rw [ha.natDegree_eq_zero_iff_eq_one, hb.natDegree_eq_zero_iff_eq_one]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nhp : Monic p\nhp1 : p = 1\n⊢ Irreducible p ↔ p ≠ 1 ∧ ∀ (f g : R[X]), Monic f → Monic g → f * g = p → natDegree f = 0 ∨ natDegree g = 0",
"tactic": "simp [hp1]"
}
] |
[
296,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
LinearMap.subtype_compMultilinearMap_codRestrict
|
[] |
[
803,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.revOrderIso_symm_apply
|
[] |
[
516,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
contDiff_of_differentiable_iteratedFDeriv
|
[] |
[
1666,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1663,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
isOpen_setOf_linearIndependent
|
[] |
[
262,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/MeasureTheory/Constructions/Pi.lean
|
MeasureTheory.volume_preserving_piEquivPiSubtypeProd
|
[] |
[
753,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
750,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.coe_countp
|
[] |
[
2199,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2198,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.Tape.mk'_left_right₀
|
[
{
"state_after": "case mk\nΓ : Type u_1\ninst✝ : Inhabited Γ\nhead✝ : Γ\nleft✝ right✝ : ListBlank Γ\n⊢ mk' { head := head✝, left := left✝, right := right✝ }.left\n (right₀ { head := head✝, left := left✝, right := right✝ }) =\n { head := head✝, left := left✝, right := right✝ }",
"state_before": "Γ : Type u_1\ninst✝ : Inhabited Γ\nT : Tape Γ\n⊢ mk' T.left (right₀ T) = T",
"tactic": "cases T"
},
{
"state_after": "no goals",
"state_before": "case mk\nΓ : Type u_1\ninst✝ : Inhabited Γ\nhead✝ : Γ\nleft✝ right✝ : ListBlank Γ\n⊢ mk' { head := head✝, left := left✝, right := right✝ }.left\n (right₀ { head := head✝, left := left✝, right := right✝ }) =\n { head := head✝, left := left✝, right := right✝ }",
"tactic": "simp only [Tape.right₀, Tape.mk', ListBlank.head_cons, ListBlank.tail_cons, eq_self_iff_true,\n and_self_iff]"
}
] |
[
576,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
573,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval_monomial
|
[] |
[
367,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/Order/JordanHolder.lean
|
CompositionSeries.Equivalent.trans
|
[] |
[
642,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
639,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_mod_two
|
[
{
"state_after": "case ofNat\na✝ : ℕ\n⊢ ↑(Int.natAbs (Int.ofNat a✝)) = ↑(Int.ofNat a✝)\n\ncase negSucc\na✝ : ℕ\n⊢ ↑(Int.natAbs (Int.negSucc a✝)) = ↑(Int.negSucc a✝)",
"state_before": "a : ℤ\n⊢ ↑(Int.natAbs a) = ↑a",
"tactic": "cases a"
},
{
"state_after": "no goals",
"state_before": "case ofNat\na✝ : ℕ\n⊢ ↑(Int.natAbs (Int.ofNat a✝)) = ↑(Int.ofNat a✝)",
"tactic": "simp only [Int.natAbs_ofNat, Int.cast_ofNat, Int.ofNat_eq_coe]"
},
{
"state_after": "no goals",
"state_before": "case negSucc\na✝ : ℕ\n⊢ ↑(Int.natAbs (Int.negSucc a✝)) = ↑(Int.negSucc a✝)",
"tactic": "simp only [neg_eq_self_mod_two, Nat.cast_succ, Int.natAbs, Int.cast_negSucc]"
}
] |
[
835,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
832,
1
] |
Mathlib/Data/List/Basic.lean
|
List.nthLe_length_sub_one
|
[
{
"state_after": "ι : Type ?u.89110\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nh : length [] - 1 < length []\n⊢ False",
"state_before": "ι : Type ?u.89110\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nh : length l - 1 < length l\n⊢ l ≠ []",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.89110\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nh : length [] - 1 < length []\n⊢ False",
"tactic": "exact Nat.lt_irrefl 0 h"
}
] |
[
1363,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1361,
1
] |
Mathlib/Algebra/Module/GradedModule.lean
|
DirectSum.Gmodule.of_smul_of
|
[] |
[
109,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.comapDomain'_apply
|
[] |
[
1366,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1364,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.eq_of_direction_eq_of_nonempty_of_le
|
[] |
[
661,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/Data/Multiset/Dedup.lean
|
Multiset.count_dedup
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.3980\ninst✝ : DecidableEq α\nm : Multiset α\na : α\nx✝ : List α\n⊢ List.count a (List.dedup x✝) = if a ∈ x✝ then 1 else 0",
"state_before": "α : Type u_1\nβ : Type ?u.3980\ninst✝ : DecidableEq α\nm : Multiset α\na : α\nx✝ : List α\n⊢ count a (dedup (Quot.mk Setoid.r x✝)) = if a ∈ Quot.mk Setoid.r x✝ then 1 else 0",
"tactic": "simp only [quot_mk_to_coe'', coe_dedup, mem_coe, List.mem_dedup, coe_nodup, coe_count]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3980\ninst✝ : DecidableEq α\nm : Multiset α\na : α\nx✝ : List α\n⊢ List.count a (List.dedup x✝) = if a ∈ x✝ then 1 else 0",
"tactic": "apply List.count_dedup _ _"
}
] |
[
93,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
MeasurableSet.subtype_image
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.60270\nγ : Type ?u.60273\nδ : Type ?u.60276\nδ' : Type ?u.60279\nι : Sort uι\ns✝ t u✝ : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ns : Set α\nhs : MeasurableSet s\nu : Set α\nhu : MeasurableSet u\n⊢ MeasurableSet (Subtype.val '' (Subtype.val ⁻¹' u))",
"state_before": "α : Type u_1\nβ : Type ?u.60270\nγ : Type ?u.60273\nδ : Type ?u.60276\nδ' : Type ?u.60279\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ns : Set α\nt : Set ↑s\nhs : MeasurableSet s\n⊢ MeasurableSet t → MeasurableSet (Subtype.val '' t)",
"tactic": "rintro ⟨u, hu, rfl⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.60270\nγ : Type ?u.60273\nδ : Type ?u.60276\nδ' : Type ?u.60279\nι : Sort uι\ns✝ t u✝ : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ns : Set α\nhs : MeasurableSet s\nu : Set α\nhu : MeasurableSet u\n⊢ MeasurableSet (u ∩ s)",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.60270\nγ : Type ?u.60273\nδ : Type ?u.60276\nδ' : Type ?u.60279\nι : Sort uι\ns✝ t u✝ : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ns : Set α\nhs : MeasurableSet s\nu : Set α\nhu : MeasurableSet u\n⊢ MeasurableSet (Subtype.val '' (Subtype.val ⁻¹' u))",
"tactic": "rw [Subtype.image_preimage_coe]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.60270\nγ : Type ?u.60273\nδ : Type ?u.60276\nδ' : Type ?u.60279\nι : Sort uι\ns✝ t u✝ : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ns : Set α\nhs : MeasurableSet s\nu : Set α\nhu : MeasurableSet u\n⊢ MeasurableSet (u ∩ s)",
"tactic": "exact hu.inter hs"
}
] |
[
558,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
554,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
derivWithin_sinh
|
[] |
[
901,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
899,
1
] |
Mathlib/Data/Sym/Basic.lean
|
Sym.append_inj_left
|
[] |
[
505,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
1
] |
Mathlib/Computability/Ackermann.lean
|
ack_one
|
[
{
"state_after": "case zero\n\n⊢ ack 1 zero = zero + 2\n\ncase succ\nn : ℕ\nIH : ack 1 n = n + 2\n⊢ ack 1 (succ n) = succ n + 2",
"state_before": "n : ℕ\n⊢ ack 1 n = n + 2",
"tactic": "induction' n with n IH"
},
{
"state_after": "no goals",
"state_before": "case zero\n\n⊢ ack 1 zero = zero + 2",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case succ\nn : ℕ\nIH : ack 1 n = n + 2\n⊢ ack 1 (succ n) = succ n + 2",
"tactic": "simp [IH]"
}
] |
[
87,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Order/Cover.lean
|
Covby.unique_right
|
[] |
[
451,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
450,
1
] |
Mathlib/Data/Real/CauSeqCompletion.lean
|
CauSeq.lim_eq_zero_iff
|
[
{
"state_after": "α : Type u_1\ninst✝³ : LinearOrderedField α\nβ : Type u_2\ninst✝² : Ring β\nabv : β → α\ninst✝¹ : IsAbsoluteValue abv\ninst✝ : IsComplete β abv\nf : CauSeq β abv\nh : lim f = 0\nhf : f ≈ const abv (lim f)\n⊢ LimZero f",
"state_before": "α : Type u_1\ninst✝³ : LinearOrderedField α\nβ : Type u_2\ninst✝² : Ring β\nabv : β → α\ninst✝¹ : IsAbsoluteValue abv\ninst✝ : IsComplete β abv\nf : CauSeq β abv\nh : lim f = 0\n⊢ LimZero f",
"tactic": "have hf := equiv_lim f"
},
{
"state_after": "α : Type u_1\ninst✝³ : LinearOrderedField α\nβ : Type u_2\ninst✝² : Ring β\nabv : β → α\ninst✝¹ : IsAbsoluteValue abv\ninst✝ : IsComplete β abv\nf : CauSeq β abv\nh : lim f = 0\nhf : f ≈ const abv 0\n⊢ LimZero f",
"state_before": "α : Type u_1\ninst✝³ : LinearOrderedField α\nβ : Type u_2\ninst✝² : Ring β\nabv : β → α\ninst✝¹ : IsAbsoluteValue abv\ninst✝ : IsComplete β abv\nf : CauSeq β abv\nh : lim f = 0\nhf : f ≈ const abv (lim f)\n⊢ LimZero f",
"tactic": "rw [h] at hf"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : LinearOrderedField α\nβ : Type u_2\ninst✝² : Ring β\nabv : β → α\ninst✝¹ : IsAbsoluteValue abv\ninst✝ : IsComplete β abv\nf : CauSeq β abv\nh : lim f = 0\nhf : f ≈ const abv 0\n⊢ LimZero f",
"tactic": "exact (limZero_congr hf).mpr (const_limZero.mpr rfl)"
},
{
"state_after": "α : Type u_1\ninst✝³ : LinearOrderedField α\nβ : Type u_2\ninst✝² : Ring β\nabv : β → α\ninst✝¹ : IsAbsoluteValue abv\ninst✝ : IsComplete β abv\nf : CauSeq β abv\nh : LimZero f\nh₁ : f = f - const abv 0\n⊢ lim f = 0",
"state_before": "α : Type u_1\ninst✝³ : LinearOrderedField α\nβ : Type u_2\ninst✝² : Ring β\nabv : β → α\ninst✝¹ : IsAbsoluteValue abv\ninst✝ : IsComplete β abv\nf : CauSeq β abv\nh : LimZero f\n⊢ lim f = 0",
"tactic": "have h₁ : f = f - const abv 0 := ext fun n => by simp [sub_apply, const_apply]"
},
{
"state_after": "α : Type u_1\ninst✝³ : LinearOrderedField α\nβ : Type u_2\ninst✝² : Ring β\nabv : β → α\ninst✝¹ : IsAbsoluteValue abv\ninst✝ : IsComplete β abv\nf : CauSeq β abv\nh : LimZero (f - const abv 0)\nh₁ : f = f - const abv 0\n⊢ lim f = 0",
"state_before": "α : Type u_1\ninst✝³ : LinearOrderedField α\nβ : Type u_2\ninst✝² : Ring β\nabv : β → α\ninst✝¹ : IsAbsoluteValue abv\ninst✝ : IsComplete β abv\nf : CauSeq β abv\nh : LimZero f\nh₁ : f = f - const abv 0\n⊢ lim f = 0",
"tactic": "rw [h₁] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : LinearOrderedField α\nβ : Type u_2\ninst✝² : Ring β\nabv : β → α\ninst✝¹ : IsAbsoluteValue abv\ninst✝ : IsComplete β abv\nf : CauSeq β abv\nh : LimZero (f - const abv 0)\nh₁ : f = f - const abv 0\n⊢ lim f = 0",
"tactic": "exact lim_eq_of_equiv_const h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : LinearOrderedField α\nβ : Type u_2\ninst✝² : Ring β\nabv : β → α\ninst✝¹ : IsAbsoluteValue abv\ninst✝ : IsComplete β abv\nf : CauSeq β abv\nh : LimZero f\nn : ℕ\n⊢ ↑f n = ↑(f - const abv 0) n",
"tactic": "simp [sub_apply, const_apply]"
}
] |
[
408,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.isUnit_iff_bijective
|
[
{
"state_after": "no goals",
"state_before": "f✝ g f : CircleDeg1Lift\nh : Bijective ↑f\nx y : ℝ\nhxy : x ≤ y\n⊢ ↑f (↑(Equiv.ofBijective (↑f) h).symm x) ≤ ↑f (↑(Equiv.ofBijective (↑f) h).symm y)",
"tactic": "simp only [Equiv.ofBijective_apply_symm_apply f h, hxy]"
},
{
"state_after": "no goals",
"state_before": "f✝ g f : CircleDeg1Lift\nh : Bijective ↑f\nx : ℝ\n⊢ ↑f\n (↑{ toFun := ↑(Equiv.ofBijective (↑f) h).symm,\n monotone' :=\n (_ : ∀ (x y : ℝ), x ≤ y → ↑(Equiv.ofBijective (↑f) h).symm x ≤ ↑(Equiv.ofBijective (↑f) h).symm y) }\n (x + 1)) =\n ↑f\n (↑{ toFun := ↑(Equiv.ofBijective (↑f) h).symm,\n monotone' :=\n (_ : ∀ (x y : ℝ), x ≤ y → ↑(Equiv.ofBijective (↑f) h).symm x ≤ ↑(Equiv.ofBijective (↑f) h).symm y) }\n x +\n 1)",
"tactic": "simp only [Equiv.ofBijective_apply_symm_apply f, f.map_add_one]"
}
] |
[
262,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/Algebra/Category/ModuleCat/Biproducts.lean
|
ModuleCat.binaryProductLimitCone_cone_π_app_left
|
[] |
[
65,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval₂_at_apply
|
[
{
"state_after": "R : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\nr : R\n⊢ ∑ n in support p, ↑f (coeff p n) * ↑f r ^ n = ∑ x in support p, ↑f (coeff p x * r ^ x)",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\nr : R\n⊢ eval₂ f (↑f r) p = ↑f (eval r p)",
"tactic": "rw [eval₂_eq_sum, eval_eq_sum, sum, sum, f.map_sum]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\nr : R\n⊢ ∑ n in support p, ↑f (coeff p n) * ↑f r ^ n = ∑ x in support p, ↑f (coeff p x * r ^ x)",
"tactic": "simp only [f.map_mul, f.map_pow]"
}
] |
[
335,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/LinearAlgebra/Pi.lean
|
Submodule.biInf_comap_proj
|
[
{
"state_after": "case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝² : Semiring R\nφ : ι → Type u_1\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx✝ x : (i : ι) → φ i\n⊢ (x ∈ ⨅ (i : ι) (_ : i ∈ I), comap (proj i) (p i)) ↔ x ∈ pi I p",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝² : Semiring R\nφ : ι → Type u_1\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx : (i : ι) → φ i\n⊢ (⨅ (i : ι) (_ : i ∈ I), comap (proj i) (p i)) = pi I p",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝² : Semiring R\nφ : ι → Type u_1\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx✝ x : (i : ι) → φ i\n⊢ (x ∈ ⨅ (i : ι) (_ : i ∈ I), comap (proj i) (p i)) ↔ x ∈ pi I p",
"tactic": "simp"
}
] |
[
303,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.coe_mul
|
[] |
[
194,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Order/BoundedOrder.lean
|
min_top_right
|
[] |
[
851,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
850,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.oreDiv_mul_char
|
[] |
[
264,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
lp.memℓp_of_tendsto
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nF : ι → { x // x ∈ lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nC : ℝ\nleft✝ : C > 0\nhCF' : ∀ (x : { x // x ∈ lp E p }), x ∈ Set.range F → ‖x‖ ≤ C\n⊢ Memℓp f p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nF : ι → { x // x ∈ lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\n⊢ Memℓp f p",
"tactic": "obtain ⟨C, _, hCF'⟩ := hF.exists_pos_norm_le"
},
{
"state_after": "case intro.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nF : ι → { x // x ∈ lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nC : ℝ\nleft✝ : C > 0\nhCF' : ∀ (x : { x // x ∈ lp E p }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\n⊢ Memℓp f p",
"state_before": "case intro.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nF : ι → { x // x ∈ lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nC : ℝ\nleft✝ : C > 0\nhCF' : ∀ (x : { x // x ∈ lp E p }), x ∈ Set.range F → ‖x‖ ≤ C\n⊢ Memℓp f p",
"tactic": "have hCF : ∀ k, ‖F k‖ ≤ C := fun k => hCF' _ ⟨k, rfl⟩"
},
{
"state_after": "case intro.intro.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\nf : (a : α) → E a\nC : ℝ\nleft✝ : C > 0\n_i : Fact (1 ≤ ⊤)\nF : ι → { x // x ∈ lp E ⊤ }\nhF : Metric.Bounded (Set.range F)\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nhCF' : ∀ (x : { x // x ∈ lp E ⊤ }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\n⊢ Memℓp f ⊤\n\ncase intro.intro.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nF : ι → { x // x ∈ lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nC : ℝ\nleft✝ : C > 0\nhCF' : ∀ (x : { x // x ∈ lp E p }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\nhp : p < ⊤\n⊢ Memℓp f p",
"state_before": "case intro.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nF : ι → { x // x ∈ lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nC : ℝ\nleft✝ : C > 0\nhCF' : ∀ (x : { x // x ∈ lp E p }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\n⊢ Memℓp f p",
"tactic": "rcases eq_top_or_lt_top p with (rfl | hp)"
},
{
"state_after": "case intro.intro.inl.hf\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\nf : (a : α) → E a\nC : ℝ\nleft✝ : C > 0\n_i : Fact (1 ≤ ⊤)\nF : ι → { x // x ∈ lp E ⊤ }\nhF : Metric.Bounded (Set.range F)\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nhCF' : ∀ (x : { x // x ∈ lp E ⊤ }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\n⊢ BddAbove (Set.range fun i => ‖f i‖)",
"state_before": "case intro.intro.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\nf : (a : α) → E a\nC : ℝ\nleft✝ : C > 0\n_i : Fact (1 ≤ ⊤)\nF : ι → { x // x ∈ lp E ⊤ }\nhF : Metric.Bounded (Set.range F)\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nhCF' : ∀ (x : { x // x ∈ lp E ⊤ }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\n⊢ Memℓp f ⊤",
"tactic": "apply memℓp_infty"
},
{
"state_after": "case intro.intro.inl.hf\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\nf : (a : α) → E a\nC : ℝ\nleft✝ : C > 0\n_i : Fact (1 ≤ ⊤)\nF : ι → { x // x ∈ lp E ⊤ }\nhF : Metric.Bounded (Set.range F)\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nhCF' : ∀ (x : { x // x ∈ lp E ⊤ }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\n⊢ C ∈ upperBounds (Set.range fun i => ‖f i‖)",
"state_before": "case intro.intro.inl.hf\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\nf : (a : α) → E a\nC : ℝ\nleft✝ : C > 0\n_i : Fact (1 ≤ ⊤)\nF : ι → { x // x ∈ lp E ⊤ }\nhF : Metric.Bounded (Set.range F)\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nhCF' : ∀ (x : { x // x ∈ lp E ⊤ }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\n⊢ BddAbove (Set.range fun i => ‖f i‖)",
"tactic": "use C"
},
{
"state_after": "case intro.intro.inl.hf.intro\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\nf : (a : α) → E a\nC : ℝ\nleft✝ : C > 0\n_i : Fact (1 ≤ ⊤)\nF : ι → { x // x ∈ lp E ⊤ }\nhF : Metric.Bounded (Set.range F)\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nhCF' : ∀ (x : { x // x ∈ lp E ⊤ }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\na : α\n⊢ (fun i => ‖f i‖) a ≤ C",
"state_before": "case intro.intro.inl.hf\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\nf : (a : α) → E a\nC : ℝ\nleft✝ : C > 0\n_i : Fact (1 ≤ ⊤)\nF : ι → { x // x ∈ lp E ⊤ }\nhF : Metric.Bounded (Set.range F)\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nhCF' : ∀ (x : { x // x ∈ lp E ⊤ }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\n⊢ C ∈ upperBounds (Set.range fun i => ‖f i‖)",
"tactic": "rintro _ ⟨a, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inl.hf.intro\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\nf : (a : α) → E a\nC : ℝ\nleft✝ : C > 0\n_i : Fact (1 ≤ ⊤)\nF : ι → { x // x ∈ lp E ⊤ }\nhF : Metric.Bounded (Set.range F)\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nhCF' : ∀ (x : { x // x ∈ lp E ⊤ }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\na : α\n⊢ (fun i => ‖f i‖) a ≤ C",
"tactic": "refine' norm_apply_le_of_tendsto (eventually_of_forall hCF) hf a"
},
{
"state_after": "case intro.intro.inr.hf\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nF : ι → { x // x ∈ lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nC : ℝ\nleft✝ : C > 0\nhCF' : ∀ (x : { x // x ∈ lp E p }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\nhp : p < ⊤\n⊢ ∀ (s : Finset α), ∑ i in s, ‖f i‖ ^ ENNReal.toReal p ≤ ?intro.intro.inr.C\n\ncase intro.intro.inr.C\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nF : ι → { x // x ∈ lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nC : ℝ\nleft✝ : C > 0\nhCF' : ∀ (x : { x // x ∈ lp E p }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\nhp : p < ⊤\n⊢ ℝ",
"state_before": "case intro.intro.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nF : ι → { x // x ∈ lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nC : ℝ\nleft✝ : C > 0\nhCF' : ∀ (x : { x // x ∈ lp E p }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\nhp : p < ⊤\n⊢ Memℓp f p",
"tactic": "apply memℓp_gen'"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inr.hf\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nF : ι → { x // x ∈ lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nC : ℝ\nleft✝ : C > 0\nhCF' : ∀ (x : { x // x ∈ lp E p }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\nhp : p < ⊤\n⊢ ∀ (s : Finset α), ∑ i in s, ‖f i‖ ^ ENNReal.toReal p ≤ ?intro.intro.inr.C\n\ncase intro.intro.inr.C\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_3\nl : Filter ι\ninst✝ : NeBot l\n_i : Fact (1 ≤ p)\nF : ι → { x // x ∈ lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : α) → E a\nhf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)\nC : ℝ\nleft✝ : C > 0\nhCF' : ∀ (x : { x // x ∈ lp E p }), x ∈ Set.range F → ‖x‖ ≤ C\nhCF : ∀ (k : ι), ‖F k‖ ≤ C\nhp : p < ⊤\n⊢ ℝ",
"tactic": "exact sum_rpow_le_of_tendsto hp.ne (eventually_of_forall hCF) hf"
}
] |
[
1185,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1175,
1
] |
Mathlib/Init/Algebra/Order.lean
|
compare_eq_iff_eq
|
[
{
"state_after": "α : Type u\ninst✝ : LinearOrder α\na b : α\n⊢ (if a < b then Ordering.lt else if a = b then Ordering.eq else Ordering.gt) = Ordering.eq ↔ a = b",
"state_before": "α : Type u\ninst✝ : LinearOrder α\na b : α\n⊢ compare a b = Ordering.eq ↔ a = b",
"tactic": "rw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq]"
},
{
"state_after": "case inl\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝ : a < b\n⊢ False ↔ a = b\n\ncase inr.inl\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝¹ : ¬a < b\nh✝ : a = b\n⊢ True ↔ a = b\n\ncase inr.inr\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝¹ : ¬a < b\nh✝ : ¬a = b\n⊢ False ↔ a = b",
"state_before": "α : Type u\ninst✝ : LinearOrder α\na b : α\n⊢ (if a < b then Ordering.lt else if a = b then Ordering.eq else Ordering.gt) = Ordering.eq ↔ a = b",
"tactic": "split_ifs <;> simp only []"
},
{
"state_after": "case inr.inl\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝¹ : ¬a < b\nh✝ : a = b\n⊢ True ↔ a = b\n\ncase inr.inr\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝¹ : ¬a < b\nh✝ : ¬a = b\n⊢ False ↔ a = b",
"state_before": "case inl\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝ : a < b\n⊢ False ↔ a = b\n\ncase inr.inl\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝¹ : ¬a < b\nh✝ : a = b\n⊢ True ↔ a = b\n\ncase inr.inr\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝¹ : ¬a < b\nh✝ : ¬a = b\n⊢ False ↔ a = b",
"tactic": "case _ h => exact false_iff_iff.2 <| ne_iff_lt_or_gt.2 <| .inl h"
},
{
"state_after": "case inr.inr\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝¹ : ¬a < b\nh✝ : ¬a = b\n⊢ False ↔ a = b",
"state_before": "case inr.inl\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝¹ : ¬a < b\nh✝ : a = b\n⊢ True ↔ a = b\n\ncase inr.inr\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝¹ : ¬a < b\nh✝ : ¬a = b\n⊢ False ↔ a = b",
"tactic": "case _ _ h => exact true_iff_iff.2 h"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝¹ : ¬a < b\nh✝ : ¬a = b\n⊢ False ↔ a = b",
"tactic": "case _ _ h => exact false_iff_iff.2 h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : LinearOrder α\na b : α\nh : a < b\n⊢ False ↔ a = b",
"tactic": "exact false_iff_iff.2 <| ne_iff_lt_or_gt.2 <| .inl h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : LinearOrder α\na b : α\nh✝ : ¬a < b\nh : a = b\n⊢ True ↔ a = b",
"tactic": "exact true_iff_iff.2 h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : LinearOrder α\na b : α\nh✝ : ¬a < b\nh : ¬a = b\n⊢ False ↔ a = b",
"tactic": "exact false_iff_iff.2 h"
}
] |
[
440,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
435,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioi_subset_Ioi
|
[] |
[
602,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
602,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
eq_zero_of_sameRay_self_neg
|
[
{
"state_after": "R : Type u_1\ninst✝⁵ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.156166\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nx y : M\ninst✝ : NoZeroSMulDivisors R M\nh : SameRay R x (-x)\n✝ : Nontrivial M\n⊢ x = 0",
"state_before": "R : Type u_1\ninst✝⁵ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.156166\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nx y : M\ninst✝ : NoZeroSMulDivisors R M\nh : SameRay R x (-x)\n⊢ x = 0",
"tactic": "nontriviality M"
},
{
"state_after": "R : Type u_1\ninst✝⁵ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.156166\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nx y : M\ninst✝ : NoZeroSMulDivisors R M\nh : SameRay R x (-x)\n✝ : Nontrivial M\nthis : Nontrivial R\n⊢ x = 0",
"state_before": "R : Type u_1\ninst✝⁵ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.156166\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nx y : M\ninst✝ : NoZeroSMulDivisors R M\nh : SameRay R x (-x)\n✝ : Nontrivial M\n⊢ x = 0",
"tactic": "haveI : Nontrivial R := Module.nontrivial R M"
},
{
"state_after": "R : Type u_1\ninst✝⁵ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.156166\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nx y : M\ninst✝ : NoZeroSMulDivisors R M\nh : SameRay R x (-x)\n✝ : Nontrivial M\nthis : Nontrivial R\n⊢ SameRay R x (-1 • x)",
"state_before": "R : Type u_1\ninst✝⁵ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.156166\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nx y : M\ninst✝ : NoZeroSMulDivisors R M\nh : SameRay R x (-x)\n✝ : Nontrivial M\nthis : Nontrivial R\n⊢ x = 0",
"tactic": "refine' eq_zero_of_sameRay_neg_smul_right (neg_lt_zero.2 (zero_lt_one' R)) _"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁵ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.156166\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nx y : M\ninst✝ : NoZeroSMulDivisors R M\nh : SameRay R x (-x)\n✝ : Nontrivial M\nthis : Nontrivial R\n⊢ SameRay R x (-1 • x)",
"tactic": "rwa [neg_one_smul]"
}
] |
[
423,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.measure_eq_left_of_subset_of_measure_add_eq
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.136128\nγ : Type ?u.136131\nδ : Type ?u.136134\nι : Type ?u.136137\nR : Type ?u.136140\nR' : Type ?u.136143\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s t : Set α\nh : ↑↑(μ + ν) t ≠ ⊤\nh' : s ⊆ t\nh'' : ↑↑(μ + ν) s = ↑↑(μ + ν) t\n⊢ ↑↑μ t ≤ ↑↑μ s",
"state_before": "α : Type u_1\nβ : Type ?u.136128\nγ : Type ?u.136131\nδ : Type ?u.136134\nι : Type ?u.136137\nR : Type ?u.136140\nR' : Type ?u.136143\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s t : Set α\nh : ↑↑(μ + ν) t ≠ ⊤\nh' : s ⊆ t\nh'' : ↑↑(μ + ν) s = ↑↑(μ + ν) t\n⊢ ↑↑μ s = ↑↑μ t",
"tactic": "refine' le_antisymm (measure_mono h') _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.136128\nγ : Type ?u.136131\nδ : Type ?u.136134\nι : Type ?u.136137\nR : Type ?u.136140\nR' : Type ?u.136143\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s t : Set α\nh : ↑↑(μ + ν) t ≠ ⊤\nh' : s ⊆ t\nh'' : ↑↑(μ + ν) s = ↑↑(μ + ν) t\nthis : ↑↑μ t + ↑↑ν t ≤ ↑↑μ s + ↑↑ν t\n⊢ ↑↑μ t ≤ ↑↑μ s",
"state_before": "α : Type u_1\nβ : Type ?u.136128\nγ : Type ?u.136131\nδ : Type ?u.136134\nι : Type ?u.136137\nR : Type ?u.136140\nR' : Type ?u.136143\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s t : Set α\nh : ↑↑(μ + ν) t ≠ ⊤\nh' : s ⊆ t\nh'' : ↑↑(μ + ν) s = ↑↑(μ + ν) t\n⊢ ↑↑μ t ≤ ↑↑μ s",
"tactic": "have : μ t + ν t ≤ μ s + ν t :=\n calc\n μ t + ν t = μ s + ν s := h''.symm\n _ ≤ μ s + ν t := add_le_add le_rfl (measure_mono h')"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.136128\nγ : Type ?u.136131\nδ : Type ?u.136134\nι : Type ?u.136137\nR : Type ?u.136140\nR' : Type ?u.136143\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s t : Set α\nh : ↑↑(μ + ν) t ≠ ⊤\nh' : s ⊆ t\nh'' : ↑↑(μ + ν) s = ↑↑(μ + ν) t\nthis : ↑↑μ t + ↑↑ν t ≤ ↑↑μ s + ↑↑ν t\n⊢ ↑↑ν t ≠ ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.136128\nγ : Type ?u.136131\nδ : Type ?u.136134\nι : Type ?u.136137\nR : Type ?u.136140\nR' : Type ?u.136143\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s t : Set α\nh : ↑↑(μ + ν) t ≠ ⊤\nh' : s ⊆ t\nh'' : ↑↑(μ + ν) s = ↑↑(μ + ν) t\nthis : ↑↑μ t + ↑↑ν t ≤ ↑↑μ s + ↑↑ν t\n⊢ ↑↑μ t ≤ ↑↑μ s",
"tactic": "apply ENNReal.le_of_add_le_add_right _ this"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.136128\nγ : Type ?u.136131\nδ : Type ?u.136134\nι : Type ?u.136137\nR : Type ?u.136140\nR' : Type ?u.136143\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s t : Set α\nh' : s ⊆ t\nh'' : ↑↑(μ + ν) s = ↑↑(μ + ν) t\nthis : ↑↑μ t + ↑↑ν t ≤ ↑↑μ s + ↑↑ν t\nh : ¬↑↑μ t = ⊤ ∧ ¬↑↑ν t = ⊤\n⊢ ↑↑ν t ≠ ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.136128\nγ : Type ?u.136131\nδ : Type ?u.136134\nι : Type ?u.136137\nR : Type ?u.136140\nR' : Type ?u.136143\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s t : Set α\nh : ↑↑(μ + ν) t ≠ ⊤\nh' : s ⊆ t\nh'' : ↑↑(μ + ν) s = ↑↑(μ + ν) t\nthis : ↑↑μ t + ↑↑ν t ≤ ↑↑μ s + ↑↑ν t\n⊢ ↑↑ν t ≠ ⊤",
"tactic": "simp only [not_or, ENNReal.add_eq_top, Pi.add_apply, Ne.def, coe_add] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.136128\nγ : Type ?u.136131\nδ : Type ?u.136134\nι : Type ?u.136137\nR : Type ?u.136140\nR' : Type ?u.136143\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s t : Set α\nh' : s ⊆ t\nh'' : ↑↑(μ + ν) s = ↑↑(μ + ν) t\nthis : ↑↑μ t + ↑↑ν t ≤ ↑↑μ s + ↑↑ν t\nh : ¬↑↑μ t = ⊤ ∧ ¬↑↑ν t = ⊤\n⊢ ↑↑ν t ≠ ⊤",
"tactic": "exact h.2"
}
] |
[
925,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
916,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iUnion
|
[] |
[
1193,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1190,
1
] |
Mathlib/Order/WellFounded.lean
|
WellFounded.not_lt_min
|
[] |
[
74,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Algebra/Module/LinearMap.lean
|
image_smul_set
|
[] |
[
388,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_comp
|
[
{
"state_after": "ι : Type ?u.722181\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq γ\nf : γ → β\ng : α → γ\n⊢ ∀ (a : α), a ∈ s → (∃ a_1, a_1 ∈ s ∧ g a_1 = g a) ∧ a ∈ s",
"state_before": "ι : Type ?u.722181\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq γ\nf : γ → β\ng : α → γ\n⊢ ∀ (a : α) (ha : a ∈ s),\n (fun a _ha => { fst := g a, snd := a }) a ha ∈ Finset.sigma (image g s) fun b => filter (fun a => g a = b) s",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.722181\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq γ\nf : γ → β\ng : α → γ\n⊢ ∀ (a : α), a ∈ s → (∃ a_1, a_1 ∈ s ∧ g a_1 = g a) ∧ a ∈ s",
"tactic": "tauto"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.722181\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq γ\nf : γ → β\ng : α → γ\n⊢ ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s),\n (fun a _ha => { fst := g a, snd := a }) a₁ ha₁ = (fun a _ha => { fst := g a, snd := a }) a₂ ha₂ → a₁ = a₂",
"tactic": "simp"
},
{
"state_after": "case mk\nι : Type ?u.722181\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq γ\nf : γ → β\ng : α → γ\nb_fst : γ\nb_snd : α\nH : { fst := b_fst, snd := b_snd } ∈ Finset.sigma (image g s) fun b => filter (fun a => g a = b) s\n⊢ ∃ a ha, { fst := b_fst, snd := b_snd } = (fun a _ha => { fst := g a, snd := a }) a ha",
"state_before": "ι : Type ?u.722181\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq γ\nf : γ → β\ng : α → γ\n⊢ ∀ (b : (_ : γ) × α),\n (b ∈ Finset.sigma (image g s) fun b => filter (fun a => g a = b) s) →\n ∃ a ha, b = (fun a _ha => { fst := g a, snd := a }) a ha",
"tactic": "rintro ⟨b_fst, b_snd⟩ H"
},
{
"state_after": "case mk\nι : Type ?u.722181\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq γ\nf : γ → β\ng : α → γ\nb_fst : γ\nb_snd : α\nH : (∃ a, a ∈ s ∧ g a = b_fst) ∧ b_snd ∈ s ∧ g b_snd = b_fst\n⊢ ∃ a ha, { fst := b_fst, snd := b_snd } = (fun a _ha => { fst := g a, snd := a }) a ha",
"state_before": "case mk\nι : Type ?u.722181\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq γ\nf : γ → β\ng : α → γ\nb_fst : γ\nb_snd : α\nH : { fst := b_fst, snd := b_snd } ∈ Finset.sigma (image g s) fun b => filter (fun a => g a = b) s\n⊢ ∃ a ha, { fst := b_fst, snd := b_snd } = (fun a _ha => { fst := g a, snd := a }) a ha",
"tactic": "simp only [mem_image, exists_prop, mem_filter, mem_sigma, decide_eq_true_eq] at H"
},
{
"state_after": "no goals",
"state_before": "case mk\nι : Type ?u.722181\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq γ\nf : γ → β\ng : α → γ\nb_fst : γ\nb_snd : α\nH : (∃ a, a ∈ s ∧ g a = b_fst) ∧ b_snd ∈ s ∧ g b_snd = b_fst\n⊢ ∃ a ha, { fst := b_fst, snd := b_snd } = (fun a _ha => { fst := g a, snd := a }) a ha",
"tactic": "tauto"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.722181\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq γ\nf : γ → β\ng : α → γ\nb : γ\n_hb : b ∈ image g s\n⊢ ∀ (x : α), x ∈ filter (fun a => g a = b) s → f (g x) = f b",
"tactic": "simp (config := { contextual := true })"
}
] |
[
1532,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1518,
1
] |
Std/Data/List/Lemmas.lean
|
List.mem_bind_of_mem
|
[] |
[
200,
47
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
199,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.map_finsupp_sum
|
[] |
[
277,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
11
] |
Mathlib/Algebra/Lie/Classical.lean
|
LieAlgebra.Orthogonal.pd_inv
|
[
{
"state_after": "n : Type ?u.121810\np : Type ?u.121813\nq : Type ?u.121816\nl : Type u_1\nR : Type u₂\ninst✝⁸ : DecidableEq n\ninst✝⁷ : DecidableEq p\ninst✝⁶ : DecidableEq q\ninst✝⁵ : DecidableEq l\ninst✝⁴ : CommRing R\ninst✝³ : Fintype p\ninst✝² : Fintype q\ninst✝¹ : Fintype l\ninst✝ : Invertible 2\n⊢ fromBlocks (1 ⬝ (⅟2 • 1ᵀ) + (-1) ⬝ (⅟2 • (-1)ᵀ)) (1 ⬝ (⅟2 • 1ᵀ) + (-1) ⬝ (⅟2 • 1ᵀ)) (1 ⬝ (⅟2 • 1ᵀ) + 1 ⬝ (⅟2 • (-1)ᵀ))\n (1 ⬝ (⅟2 • 1ᵀ) + 1 ⬝ (⅟2 • 1ᵀ)) =\n 1",
"state_before": "n : Type ?u.121810\np : Type ?u.121813\nq : Type ?u.121816\nl : Type u_1\nR : Type u₂\ninst✝⁸ : DecidableEq n\ninst✝⁷ : DecidableEq p\ninst✝⁶ : DecidableEq q\ninst✝⁵ : DecidableEq l\ninst✝⁴ : CommRing R\ninst✝³ : Fintype p\ninst✝² : Fintype q\ninst✝¹ : Fintype l\ninst✝ : Invertible 2\n⊢ PD l R * ⅟2 • (PD l R)ᵀ = 1",
"tactic": "rw [PD, Matrix.fromBlocks_transpose, Matrix.fromBlocks_smul, Matrix.mul_eq_mul,\n Matrix.fromBlocks_multiply]"
},
{
"state_after": "no goals",
"state_before": "n : Type ?u.121810\np : Type ?u.121813\nq : Type ?u.121816\nl : Type u_1\nR : Type u₂\ninst✝⁸ : DecidableEq n\ninst✝⁷ : DecidableEq p\ninst✝⁶ : DecidableEq q\ninst✝⁵ : DecidableEq l\ninst✝⁴ : CommRing R\ninst✝³ : Fintype p\ninst✝² : Fintype q\ninst✝¹ : Fintype l\ninst✝ : Invertible 2\n⊢ fromBlocks (1 ⬝ (⅟2 • 1ᵀ) + (-1) ⬝ (⅟2 • (-1)ᵀ)) (1 ⬝ (⅟2 • 1ᵀ) + (-1) ⬝ (⅟2 • 1ᵀ)) (1 ⬝ (⅟2 • 1ᵀ) + 1 ⬝ (⅟2 • (-1)ᵀ))\n (1 ⬝ (⅟2 • 1ᵀ) + 1 ⬝ (⅟2 • 1ᵀ)) =\n 1",
"tactic": "simp"
}
] |
[
285,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/Analysis/ODE/Gronwall.lean
|
dist_le_of_trajectories_ODE_of_mem_set
|
[
{
"state_after": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\nf_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (f t)) (v t (f t)) ≤ 0\n⊢ ∀ (t : ℝ), t ∈ Icc a b → dist (f t) (g t) ≤ δ * exp (K * (t - a))",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\n⊢ ∀ (t : ℝ), t ∈ Icc a b → dist (f t) (g t) ≤ δ * exp (K * (t - a))",
"tactic": "have f_bound : ∀ t ∈ Ico a b, dist (v t (f t)) (v t (f t)) ≤ 0 := by intros; rw [dist_self]"
},
{
"state_after": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\nf_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (f t)) (v t (f t)) ≤ 0\ng_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (g t)) (v t (g t)) ≤ 0\n⊢ ∀ (t : ℝ), t ∈ Icc a b → dist (f t) (g t) ≤ δ * exp (K * (t - a))",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\nf_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (f t)) (v t (f t)) ≤ 0\n⊢ ∀ (t : ℝ), t ∈ Icc a b → dist (f t) (g t) ≤ δ * exp (K * (t - a))",
"tactic": "have g_bound : ∀ t ∈ Ico a b, dist (v t (g t)) (v t (g t)) ≤ 0 := by intros; rw [dist_self]"
},
{
"state_after": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\nf_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (f t)) (v t (f t)) ≤ 0\ng_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (g t)) (v t (g t)) ≤ 0\nt : ℝ\nht : t ∈ Icc a b\n⊢ dist (f t) (g t) ≤ δ * exp (K * (t - a))",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\nf_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (f t)) (v t (f t)) ≤ 0\ng_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (g t)) (v t (g t)) ≤ 0\n⊢ ∀ (t : ℝ), t ∈ Icc a b → dist (f t) (g t) ≤ δ * exp (K * (t - a))",
"tactic": "intro t ht"
},
{
"state_after": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\nf_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (f t)) (v t (f t)) ≤ 0\ng_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (g t)) (v t (g t)) ≤ 0\nt : ℝ\nht : t ∈ Icc a b\nthis : dist (f t) (g t) ≤ gronwallBound δ K (0 + 0) (t - a)\n⊢ dist (f t) (g t) ≤ δ * exp (K * (t - a))",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\nf_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (f t)) (v t (f t)) ≤ 0\ng_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (g t)) (v t (g t)) ≤ 0\nt : ℝ\nht : t ∈ Icc a b\n⊢ dist (f t) (g t) ≤ δ * exp (K * (t - a))",
"tactic": "have :=\n dist_le_of_approx_trajectories_ODE_of_mem_set hv hf hf' f_bound hfs hg hg' g_bound hgs ha t ht"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\nf_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (f t)) (v t (f t)) ≤ 0\ng_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (g t)) (v t (g t)) ≤ 0\nt : ℝ\nht : t ∈ Icc a b\nthis : dist (f t) (g t) ≤ gronwallBound δ K (0 + 0) (t - a)\n⊢ dist (f t) (g t) ≤ δ * exp (K * (t - a))",
"tactic": "rwa [zero_add, gronwallBound_ε0] at this"
},
{
"state_after": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\nt✝ : ℝ\na✝ : t✝ ∈ Ico a b\n⊢ dist (v t✝ (f t✝)) (v t✝ (f t✝)) ≤ 0",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\n⊢ ∀ (t : ℝ), t ∈ Ico a b → dist (v t (f t)) (v t (f t)) ≤ 0",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\nt✝ : ℝ\na✝ : t✝ ∈ Ico a b\n⊢ dist (v t✝ (f t✝)) (v t✝ (f t✝)) ≤ 0",
"tactic": "rw [dist_self]"
},
{
"state_after": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\nf_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (f t)) (v t (f t)) ≤ 0\nt✝ : ℝ\na✝ : t✝ ∈ Ico a b\n⊢ dist (v t✝ (g t✝)) (v t✝ (g t✝)) ≤ 0",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\nf_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (f t)) (v t (f t)) ≤ 0\n⊢ ∀ (t : ℝ), t ∈ Ico a b → dist (v t (g t)) (v t (g t)) ≤ 0",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.46221\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ\nhv : ∀ (t : ℝ) (x : E), x ∈ s t → ∀ (y : E), y ∈ s t → dist (v t x) (v t y) ≤ K * dist x y\nf g : ℝ → E\na b δ : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : ∀ (t : ℝ), t ∈ Ico a b → f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ (t : ℝ), t ∈ Ico a b → HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : ∀ (t : ℝ), t ∈ Ico a b → g t ∈ s t\nha : dist (f a) (g a) ≤ δ\nf_bound : ∀ (t : ℝ), t ∈ Ico a b → dist (v t (f t)) (v t (f t)) ≤ 0\nt✝ : ℝ\na✝ : t✝ ∈ Ico a b\n⊢ dist (v t✝ (g t✝)) (v t✝ (g t✝)) ≤ 0",
"tactic": "rw [dist_self]"
}
] |
[
214,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.card_support_eq_one'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.135476\nγ : Type ?u.135479\nι : Type ?u.135482\nM : Type u_2\nM' : Type ?u.135488\nN : Type ?u.135491\nP : Type ?u.135494\nG : Type ?u.135497\nH : Type ?u.135500\nR : Type ?u.135503\nS : Type ?u.135506\ninst✝ : Zero M\na a' : α\nb : M\nf : α →₀ M\n⊢ card f.support = 1 ↔ ∃ a b x, f = single a b",
"tactic": "simp only [card_eq_one, support_eq_singleton']"
}
] |
[
490,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_map
|
[
{
"state_after": "ι : Type ?u.273965\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\ne : α ↪ γ\nf : γ → β\n⊢ Multiset.prod (Multiset.map ((fun x => f x) ∘ ↑e) s.val) = ∏ x in s, f (↑e x)",
"state_before": "ι : Type ?u.273965\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\ne : α ↪ γ\nf : γ → β\n⊢ ∏ x in map e s, f x = ∏ x in s, f (↑e x)",
"tactic": "rw [Finset.prod, Finset.map_val, Multiset.map_map]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.273965\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\ne : α ↪ γ\nf : γ → β\n⊢ Multiset.prod (Multiset.map ((fun x => f x) ∘ ↑e) s.val) = ∏ x in s, f (↑e x)",
"tactic": "rfl"
}
] |
[
374,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
CochainComplex.mk'_X_1
|
[] |
[
1043,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1042,
1
] |
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