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/GroupTheory/Subsemigroup/Operations.lean
|
MulHom.coe_srange
|
[] |
[
760,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
759,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
iteratedFDeriv_succ_apply_right
|
[
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nm : Fin (n + 1) → E\n⊢ ↑(iteratedFDerivWithin 𝕜 (n + 1) f univ x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f univ y) univ x) (init m)) (m (last n))",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nm : Fin (n + 1) → E\n⊢ ↑(iteratedFDeriv 𝕜 (n + 1) f x) m = ↑(↑(iteratedFDeriv 𝕜 n (fun y => fderiv 𝕜 f y) x) (init m)) (m (last n))",
"tactic": "rw [← iteratedFDerivWithin_univ, ← iteratedFDerivWithin_univ, ← fderivWithin_univ]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nm : Fin (n + 1) → E\n⊢ ↑(iteratedFDerivWithin 𝕜 (n + 1) f univ x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f univ y) univ x) (init m)) (m (last n))",
"tactic": "exact iteratedFDerivWithin_succ_apply_right uniqueDiffOn_univ (mem_univ _) _"
}
] |
[
1609,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1605,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.coe_trans
|
[] |
[
804,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
803,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
MultilinearMap.curryRight_apply
|
[] |
[
1386,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1384,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
convex_univ
|
[] |
[
94,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Data/Set/Function.lean
|
Function.invFunOn_eq
|
[] |
[
1223,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1222,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.union_subset_iff
|
[] |
[
830,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
829,
1
] |
Mathlib/MeasureTheory/Function/Egorov.lean
|
MeasureTheory.Egorov.iUnionNotConvergentSeq_subset
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\ninst✝³ : MetricSpace β\nμ : Measure α\nn : ℕ\ni j : ι\ns : Set α\nε : ℝ\nf : ι → α → β\ng : α → β\ninst✝² : SemilatticeSup ι\ninst✝¹ : Nonempty ι\ninst✝ : Countable ι\nhε : 0 < ε\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : ↑↑μ s ≠ ⊤\nhfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))\n⊢ (s ∩ ⋃ (i : ℕ), notConvergentSeq (fun n => f n) g i (notConvergentSeqLtIndex (_ : 0 < ε / 2) hf hg hsm hs hfg i)) ⊆ s",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\ninst✝³ : MetricSpace β\nμ : Measure α\nn : ℕ\ni j : ι\ns : Set α\nε : ℝ\nf : ι → α → β\ng : α → β\ninst✝² : SemilatticeSup ι\ninst✝¹ : Nonempty ι\ninst✝ : Countable ι\nhε : 0 < ε\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : ↑↑μ s ≠ ⊤\nhfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))\n⊢ iUnionNotConvergentSeq hε hf hg hsm hs hfg ⊆ s",
"tactic": "rw [iUnionNotConvergentSeq, ← Set.inter_iUnion]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\ninst✝³ : MetricSpace β\nμ : Measure α\nn : ℕ\ni j : ι\ns : Set α\nε : ℝ\nf : ι → α → β\ng : α → β\ninst✝² : SemilatticeSup ι\ninst✝¹ : Nonempty ι\ninst✝ : Countable ι\nhε : 0 < ε\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : ↑↑μ s ≠ ⊤\nhfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))\n⊢ (s ∩ ⋃ (i : ℕ), notConvergentSeq (fun n => f n) g i (notConvergentSeqLtIndex (_ : 0 < ε / 2) hf hg hsm hs hfg i)) ⊆ s",
"tactic": "exact Set.inter_subset_left _ _"
}
] |
[
169,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.X_dvd_iff
|
[
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\n⊢ (∀ (m : ℕ), m < 1 → ↑(coeff R m) φ = 0) ↔ ↑(coeff R 0) φ = 0",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\n⊢ X ∣ φ ↔ ↑(constantCoeff R) φ = 0",
"tactic": "rw [← pow_one (X : PowerSeries R), X_pow_dvd_iff, ← coeff_zero_eq_constantCoeff_apply]"
},
{
"state_after": "case mp\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : ∀ (m : ℕ), m < 1 → ↑(coeff R m) φ = 0\n⊢ ↑(coeff R 0) φ = 0\n\ncase mpr\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : ↑(coeff R 0) φ = 0\n⊢ ∀ (m : ℕ), m < 1 → ↑(coeff R m) φ = 0",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\n⊢ (∀ (m : ℕ), m < 1 → ↑(coeff R m) φ = 0) ↔ ↑(coeff R 0) φ = 0",
"tactic": "constructor <;> intro h"
},
{
"state_after": "no goals",
"state_before": "case mp\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : ∀ (m : ℕ), m < 1 → ↑(coeff R m) φ = 0\n⊢ ↑(coeff R 0) φ = 0",
"tactic": "exact h 0 zero_lt_one"
},
{
"state_after": "case mpr\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : ↑(coeff R 0) φ = 0\nm : ℕ\nhm : m < 1\n⊢ ↑(coeff R m) φ = 0",
"state_before": "case mpr\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : ↑(coeff R 0) φ = 0\n⊢ ∀ (m : ℕ), m < 1 → ↑(coeff R m) φ = 0",
"tactic": "intro m hm"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : ↑(coeff R 0) φ = 0\nm : ℕ\nhm : m < 1\n⊢ ↑(coeff R m) φ = 0",
"tactic": "rwa [Nat.eq_zero_of_le_zero (Nat.le_of_succ_le_succ hm)]"
}
] |
[
1724,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1719,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
LinearMap.eqOn_span'
|
[] |
[
959,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
957,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
countable_cover_nhdsWithin_of_sigma_compact
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.164016\nπ : ι → Type ?u.164021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t : Set α\ninst✝ : SigmaCompactSpace α\nf : α → Set α\ns : Set α\nhs : IsClosed s\nhf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x\n⊢ ∃ t x, Set.Countable t ∧ s ⊆ ⋃ (x : α) (_ : x ∈ t), f x",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.164016\nπ : ι → Type ?u.164021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t : Set α\ninst✝ : SigmaCompactSpace α\nf : α → Set α\ns : Set α\nhs : IsClosed s\nhf : ∀ (x : α), x ∈ s → f x ∈ 𝓝[s] x\n⊢ ∃ t x, Set.Countable t ∧ s ⊆ ⋃ (x : α) (_ : x ∈ t), f x",
"tactic": "simp only [nhdsWithin, mem_inf_principal] at hf"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.164016\nπ : ι → Type ?u.164021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t✝ : Set α\ninst✝ : SigmaCompactSpace α\nf : α → Set α\ns : Set α\nhs : IsClosed s\nhf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x\nt : ℕ → Finset α\nht : ∀ (n : ℕ) (x : α), x ∈ t n → x ∈ compactCovering α n ∩ s\nhsub : ∀ (n : ℕ), compactCovering α n ∩ s ⊆ ⋃ (x : α) (_ : x ∈ t n), {x_1 | x_1 ∈ s → x_1 ∈ f x}\n⊢ ∃ t x, Set.Countable t ∧ s ⊆ ⋃ (x : α) (_ : x ∈ t), f x",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.164016\nπ : ι → Type ?u.164021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t : Set α\ninst✝ : SigmaCompactSpace α\nf : α → Set α\ns : Set α\nhs : IsClosed s\nhf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x\n⊢ ∃ t x, Set.Countable t ∧ s ⊆ ⋃ (x : α) (_ : x ∈ t), f x",
"tactic": "choose t ht hsub using fun n =>\n ((isCompact_compactCovering α n).inter_right hs).elim_nhds_subcover _ fun x hx => hf x hx.right"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.164016\nπ : ι → Type ?u.164021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t✝ : Set α\ninst✝ : SigmaCompactSpace α\nf : α → Set α\ns : Set α\nhs : IsClosed s\nhf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x\nt : ℕ → Finset α\nht : ∀ (n : ℕ) (x : α), x ∈ t n → x ∈ compactCovering α n ∩ s\nhsub : ∀ (n : ℕ), compactCovering α n ∩ s ⊆ ⋃ (x : α) (_ : x ∈ t n), {x_1 | x_1 ∈ s → x_1 ∈ f x}\nx : α\nhx : x ∈ s\n⊢ ∃ i j, x ∈ f i",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.164016\nπ : ι → Type ?u.164021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t✝ : Set α\ninst✝ : SigmaCompactSpace α\nf : α → Set α\ns : Set α\nhs : IsClosed s\nhf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x\nt : ℕ → Finset α\nht : ∀ (n : ℕ) (x : α), x ∈ t n → x ∈ compactCovering α n ∩ s\nhsub : ∀ (n : ℕ), compactCovering α n ∩ s ⊆ ⋃ (x : α) (_ : x ∈ t n), {x_1 | x_1 ∈ s → x_1 ∈ f x}\n⊢ ∃ t x, Set.Countable t ∧ s ⊆ ⋃ (x : α) (_ : x ∈ t), f x",
"tactic": "refine'\n ⟨⋃ n, (t n : Set α), iUnion_subset fun n x hx => (ht n x hx).2,\n countable_iUnion fun n => (t n).countable_toSet, fun x hx => mem_iUnion₂.2 _⟩"
},
{
"state_after": "case intro\nα : Type u\nβ : Type v\nι : Type ?u.164016\nπ : ι → Type ?u.164021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t✝ : Set α\ninst✝ : SigmaCompactSpace α\nf : α → Set α\ns : Set α\nhs : IsClosed s\nhf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x\nt : ℕ → Finset α\nht : ∀ (n : ℕ) (x : α), x ∈ t n → x ∈ compactCovering α n ∩ s\nhsub : ∀ (n : ℕ), compactCovering α n ∩ s ⊆ ⋃ (x : α) (_ : x ∈ t n), {x_1 | x_1 ∈ s → x_1 ∈ f x}\nx : α\nhx : x ∈ s\nn : ℕ\nhn : x ∈ compactCovering α n\n⊢ ∃ i j, x ∈ f i",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.164016\nπ : ι → Type ?u.164021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t✝ : Set α\ninst✝ : SigmaCompactSpace α\nf : α → Set α\ns : Set α\nhs : IsClosed s\nhf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x\nt : ℕ → Finset α\nht : ∀ (n : ℕ) (x : α), x ∈ t n → x ∈ compactCovering α n ∩ s\nhsub : ∀ (n : ℕ), compactCovering α n ∩ s ⊆ ⋃ (x : α) (_ : x ∈ t n), {x_1 | x_1 ∈ s → x_1 ∈ f x}\nx : α\nhx : x ∈ s\n⊢ ∃ i j, x ∈ f i",
"tactic": "rcases exists_mem_compactCovering x with ⟨n, hn⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.164016\nπ : ι → Type ?u.164021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t✝ : Set α\ninst✝ : SigmaCompactSpace α\nf : α → Set α\ns : Set α\nhs : IsClosed s\nhf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x\nt : ℕ → Finset α\nht : ∀ (n : ℕ) (x : α), x ∈ t n → x ∈ compactCovering α n ∩ s\nhsub : ∀ (n : ℕ), compactCovering α n ∩ s ⊆ ⋃ (x : α) (_ : x ∈ t n), {x_1 | x_1 ∈ s → x_1 ∈ f x}\nx : α\nhx : x ∈ s\nn : ℕ\nhn : x ∈ compactCovering α n\ny : α\nhyt : y ∈ t n\nhyf : x ∈ s → x ∈ f y\n⊢ ∃ i j, x ∈ f i",
"state_before": "case intro\nα : Type u\nβ : Type v\nι : Type ?u.164016\nπ : ι → Type ?u.164021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t✝ : Set α\ninst✝ : SigmaCompactSpace α\nf : α → Set α\ns : Set α\nhs : IsClosed s\nhf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x\nt : ℕ → Finset α\nht : ∀ (n : ℕ) (x : α), x ∈ t n → x ∈ compactCovering α n ∩ s\nhsub : ∀ (n : ℕ), compactCovering α n ∩ s ⊆ ⋃ (x : α) (_ : x ∈ t n), {x_1 | x_1 ∈ s → x_1 ∈ f x}\nx : α\nhx : x ∈ s\nn : ℕ\nhn : x ∈ compactCovering α n\n⊢ ∃ i j, x ∈ f i",
"tactic": "rcases mem_iUnion₂.1 (hsub n ⟨hn, hx⟩) with ⟨y, hyt : y ∈ t n, hyf : x ∈ s → x ∈ f y⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.164016\nπ : ι → Type ?u.164021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t✝ : Set α\ninst✝ : SigmaCompactSpace α\nf : α → Set α\ns : Set α\nhs : IsClosed s\nhf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x\nt : ℕ → Finset α\nht : ∀ (n : ℕ) (x : α), x ∈ t n → x ∈ compactCovering α n ∩ s\nhsub : ∀ (n : ℕ), compactCovering α n ∩ s ⊆ ⋃ (x : α) (_ : x ∈ t n), {x_1 | x_1 ∈ s → x_1 ∈ f x}\nx : α\nhx : x ∈ s\nn : ℕ\nhn : x ∈ compactCovering α n\ny : α\nhyt : y ∈ t n\nhyf : x ∈ s → x ∈ f y\n⊢ ∃ i j, x ∈ f i",
"tactic": "exact ⟨y, mem_iUnion.2 ⟨n, hyt⟩, hyf hx⟩"
}
] |
[
1403,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1393,
1
] |
Mathlib/Topology/Order/Basic.lean
|
nhdsWithin_Ioc_eq_nhdsWithin_Iic
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderClosedTopology α\na✝ b✝ : α\ninst✝ : TopologicalSpace γ\na b : α\nh : a < b\n⊢ 𝓝[Ioc a b] b = 𝓝[Iic b] b",
"tactic": "simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual"
}
] |
[
599,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
598,
1
] |
Mathlib/GroupTheory/Solvable.lean
|
solvable_of_solvable_injective
|
[] |
[
155,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
TendstoUniformlyOnFilter.prod
|
[] |
[
306,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.mem_inf
|
[] |
[
718,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
717,
1
] |
Mathlib/Computability/Halting.lean
|
Nat.Partrec'.vec_iff
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\nf : Vector ℕ m → Vector ℕ n\nh : Vec f\n⊢ Computable f",
"tactic": "simpa only [ofFn_get] using vector_ofFn fun i => to_part (h i)"
}
] |
[
437,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
435,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
IsGLB.dual
|
[] |
[
167,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.comap_le_comap_iff_of_surjective
|
[] |
[
505,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
1
] |
Mathlib/CategoryTheory/StructuredArrow.lean
|
CategoryTheory.CostructuredArrow.eqToHom_left
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nT T' T'' : D\nY✝ Y' : C\nS : C ⥤ D\nX Y : CostructuredArrow S T\nh : X = Y\n⊢ X.left = Y.left",
"tactic": "rw [h]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nT T' T'' : D\nY Y' : C\nS : C ⥤ D\nX : CostructuredArrow S T\n⊢ (eqToHom (_ : X = X)).left = eqToHom (_ : X.left = X.left)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nT T' T'' : D\nY✝ Y' : C\nS : C ⥤ D\nX Y : CostructuredArrow S T\nh : X = Y\n⊢ (eqToHom h).left = eqToHom (_ : X.left = Y.left)",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nT T' T'' : D\nY Y' : C\nS : C ⥤ D\nX : CostructuredArrow S T\n⊢ (eqToHom (_ : X = X)).left = eqToHom (_ : X.left = X.left)",
"tactic": "simp only [eqToHom_refl, id_left]"
}
] |
[
331,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/Data/ENat/Basic.lean
|
ENat.one_le_iff_pos
|
[] |
[
197,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Data/Set/Sups.lean
|
Set.Nonempty.sups
|
[] |
[
120,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
11
] |
Mathlib/Analysis/Calculus/LocalExtr.lean
|
IsLocalMin.fderiv_eq_zero
|
[] |
[
199,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Order/Filter/Pi.lean
|
Filter.map_eval_pi
|
[
{
"state_after": "ι : Type u_2\nα : ι → Type u_1\nf✝ f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\nf : (i : ι) → Filter (α i)\ninst✝ : ∀ (i : ι), NeBot (f i)\ni : ι\ns : Set (α i)\nhs : s ∈ map (eval i) (pi f)\n⊢ s ∈ f i",
"state_before": "ι : Type u_2\nα : ι → Type u_1\nf✝ f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\nf : (i : ι) → Filter (α i)\ninst✝ : ∀ (i : ι), NeBot (f i)\ni : ι\n⊢ map (eval i) (pi f) = f i",
"tactic": "refine' le_antisymm (tendsto_eval_pi f i) fun s hs => _"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type u_2\nα : ι → Type u_1\nf✝ f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\nf : (i : ι) → Filter (α i)\ninst✝ : ∀ (i : ι), NeBot (f i)\ni : ι\ns : Set (α i)\nhs : s ∈ map (eval i) (pi f)\nI : Set ι\nhIf : Set.Finite I\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhI : Set.pi I t ⊆ eval i ⁻¹' s\n⊢ s ∈ f i",
"state_before": "ι : Type u_2\nα : ι → Type u_1\nf✝ f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\nf : (i : ι) → Filter (α i)\ninst✝ : ∀ (i : ι), NeBot (f i)\ni : ι\ns : Set (α i)\nhs : s ∈ map (eval i) (pi f)\n⊢ s ∈ f i",
"tactic": "rcases mem_pi.1 (mem_map.1 hs) with ⟨I, hIf, t, htf, hI⟩"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type u_2\nα : ι → Type u_1\nf✝ f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\nf : (i : ι) → Filter (α i)\ninst✝ : ∀ (i : ι), NeBot (f i)\ni : ι\ns : Set (α i)\nhs : s ∈ map (eval i) (pi f)\nI : Set ι\nhIf : Set.Finite I\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhI : eval i '' Set.pi I t ⊆ s\n⊢ s ∈ f i",
"state_before": "case intro.intro.intro.intro\nι : Type u_2\nα : ι → Type u_1\nf✝ f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\nf : (i : ι) → Filter (α i)\ninst✝ : ∀ (i : ι), NeBot (f i)\ni : ι\ns : Set (α i)\nhs : s ∈ map (eval i) (pi f)\nI : Set ι\nhIf : Set.Finite I\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhI : Set.pi I t ⊆ eval i ⁻¹' s\n⊢ s ∈ f i",
"tactic": "rw [← image_subset_iff] at hI"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type u_2\nα : ι → Type u_1\nf✝ f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\nf : (i : ι) → Filter (α i)\ninst✝ : ∀ (i : ι), NeBot (f i)\ni : ι\ns : Set (α i)\nhs : s ∈ map (eval i) (pi f)\nI : Set ι\nhIf : Set.Finite I\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhI : eval i '' Set.pi I t ⊆ s\n⊢ Set.Nonempty (Set.pi I t)",
"state_before": "case intro.intro.intro.intro\nι : Type u_2\nα : ι → Type u_1\nf✝ f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\nf : (i : ι) → Filter (α i)\ninst✝ : ∀ (i : ι), NeBot (f i)\ni : ι\ns : Set (α i)\nhs : s ∈ map (eval i) (pi f)\nI : Set ι\nhIf : Set.Finite I\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhI : eval i '' Set.pi I t ⊆ s\n⊢ s ∈ f i",
"tactic": "refine' mem_of_superset (htf i) ((subset_eval_image_pi _ _).trans hI)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nι : Type u_2\nα : ι → Type u_1\nf✝ f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\nf : (i : ι) → Filter (α i)\ninst✝ : ∀ (i : ι), NeBot (f i)\ni : ι\ns : Set (α i)\nhs : s ∈ map (eval i) (pi f)\nI : Set ι\nhIf : Set.Finite I\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhI : eval i '' Set.pi I t ⊆ s\n⊢ Set.Nonempty (Set.pi I t)",
"tactic": "exact nonempty_of_mem (pi_mem_pi hIf fun i _ => htf i)"
}
] |
[
174,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.repr_apply_eq
|
[
{
"state_after": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\nf_i : M →ₗ[R] R :=\n {\n toAddHom :=\n { toFun := fun x => f x i,\n map_add' := (_ : ∀ (x x_1 : M), (fun x => f x i) (x + x_1) = (fun x => f x i) x + (fun x => f x i) x_1) },\n map_smul' := (_ : ∀ (x : R) (x_1 : M), f (x • x_1) i = x * f x_1 i) }\n⊢ ↑(↑b.repr x) i = f x i",
"state_before": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\n⊢ ↑(↑b.repr x) i = f x i",
"tactic": "let f_i : M →ₗ[R] R :=\n { toFun := fun x => f x i\n map_add' := fun _ _ => by dsimp only []; rw [hadd, Pi.add_apply]\n map_smul' := fun _ _ => by simp [hsmul, Pi.smul_apply] }"
},
{
"state_after": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\nf_i : M →ₗ[R] R :=\n {\n toAddHom :=\n { toFun := fun x => f x i,\n map_add' := (_ : ∀ (x x_1 : M), (fun x => f x i) (x + x_1) = (fun x => f x i) x + (fun x => f x i) x_1) },\n map_smul' := (_ : ∀ (x : R) (x_1 : M), f (x • x_1) i = x * f x_1 i) }\nthis : LinearMap.comp (Finsupp.lapply i) ↑b.repr = f_i\n⊢ ↑(↑b.repr x) i = f x i",
"state_before": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\nf_i : M →ₗ[R] R :=\n {\n toAddHom :=\n { toFun := fun x => f x i,\n map_add' := (_ : ∀ (x x_1 : M), (fun x => f x i) (x + x_1) = (fun x => f x i) x + (fun x => f x i) x_1) },\n map_smul' := (_ : ∀ (x : R) (x_1 : M), f (x • x_1) i = x * f x_1 i) }\n⊢ ↑(↑b.repr x) i = f x i",
"tactic": "have : Finsupp.lapply i ∘ₗ ↑b.repr = f_i := by\n refine' b.ext fun j => _\n show b.repr (b j) i = f (b j) i\n rw [b.repr_self, f_eq]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\nf_i : M →ₗ[R] R :=\n {\n toAddHom :=\n { toFun := fun x => f x i,\n map_add' := (_ : ∀ (x x_1 : M), (fun x => f x i) (x + x_1) = (fun x => f x i) x + (fun x => f x i) x_1) },\n map_smul' := (_ : ∀ (x : R) (x_1 : M), f (x • x_1) i = x * f x_1 i) }\nthis : LinearMap.comp (Finsupp.lapply i) ↑b.repr = f_i\n⊢ ↑(↑b.repr x) i = f x i",
"tactic": "calc\n b.repr x i = f_i x := by\n { rw [← this]\n rfl }\n _ = f x i := rfl"
},
{
"state_after": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝² : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\nx✝¹ x✝ : M\n⊢ f (x✝¹ + x✝) i = f x✝¹ i + f x✝ i",
"state_before": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝² : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\nx✝¹ x✝ : M\n⊢ (fun x => f x i) (x✝¹ + x✝) = (fun x => f x i) x✝¹ + (fun x => f x i) x✝",
"tactic": "dsimp only []"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝² : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\nx✝¹ x✝ : M\n⊢ f (x✝¹ + x✝) i = f x✝¹ i + f x✝ i",
"tactic": "rw [hadd, Pi.add_apply]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝² : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\nx✝¹ : R\nx✝ : M\n⊢ AddHom.toFun\n { toFun := fun x => f x i,\n map_add' := (_ : ∀ (x x_1 : M), (fun x => f x i) (x + x_1) = (fun x => f x i) x + (fun x => f x i) x_1) }\n (x✝¹ • x✝) =\n ↑(RingHom.id R) x✝¹ •\n AddHom.toFun\n { toFun := fun x => f x i,\n map_add' := (_ : ∀ (x x_1 : M), (fun x => f x i) (x + x_1) = (fun x => f x i) x + (fun x => f x i) x_1) }\n x✝",
"tactic": "simp [hsmul, Pi.smul_apply]"
},
{
"state_after": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\nf_i : M →ₗ[R] R :=\n {\n toAddHom :=\n { toFun := fun x => f x i,\n map_add' := (_ : ∀ (x x_1 : M), (fun x => f x i) (x + x_1) = (fun x => f x i) x + (fun x => f x i) x_1) },\n map_smul' := (_ : ∀ (x : R) (x_1 : M), f (x • x_1) i = x * f x_1 i) }\nj : ι\n⊢ ↑(LinearMap.comp (Finsupp.lapply i) ↑b.repr) (↑b j) = ↑f_i (↑b j)",
"state_before": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\nf_i : M →ₗ[R] R :=\n {\n toAddHom :=\n { toFun := fun x => f x i,\n map_add' := (_ : ∀ (x x_1 : M), (fun x => f x i) (x + x_1) = (fun x => f x i) x + (fun x => f x i) x_1) },\n map_smul' := (_ : ∀ (x : R) (x_1 : M), f (x • x_1) i = x * f x_1 i) }\n⊢ LinearMap.comp (Finsupp.lapply i) ↑b.repr = f_i",
"tactic": "refine' b.ext fun j => _"
},
{
"state_after": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\nf_i : M →ₗ[R] R :=\n {\n toAddHom :=\n { toFun := fun x => f x i,\n map_add' := (_ : ∀ (x x_1 : M), (fun x => f x i) (x + x_1) = (fun x => f x i) x + (fun x => f x i) x_1) },\n map_smul' := (_ : ∀ (x : R) (x_1 : M), f (x • x_1) i = x * f x_1 i) }\nj : ι\n⊢ ↑(↑b.repr (↑b j)) i = f (↑b j) i",
"state_before": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\nf_i : M →ₗ[R] R :=\n {\n toAddHom :=\n { toFun := fun x => f x i,\n map_add' := (_ : ∀ (x x_1 : M), (fun x => f x i) (x + x_1) = (fun x => f x i) x + (fun x => f x i) x_1) },\n map_smul' := (_ : ∀ (x : R) (x_1 : M), f (x • x_1) i = x * f x_1 i) }\nj : ι\n⊢ ↑(LinearMap.comp (Finsupp.lapply i) ↑b.repr) (↑b j) = ↑f_i (↑b j)",
"tactic": "show b.repr (b j) i = f (b j) i"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.220743\nR : Type u_2\nR₂ : Type ?u.220749\nK : Type ?u.220752\nM : Type u_3\nM' : Type ?u.220758\nM'' : Type ?u.220761\nV : Type u\nV' : Type ?u.220766\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nR₁ : Type ?u.220878\ninst✝⁴ : Semiring R₁\nσ : R →+* R₁\nσ' : R₁ →+* R\ninst✝³ : RingHomInvPair σ σ'\ninst✝² : RingHomInvPair σ' σ\nM₁ : Type ?u.221082\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nf : M → ι → R\nhadd : ∀ (x y : M), f (x + y) = f x + f y\nhsmul : ∀ (c : R) (x : M), f (c • x) = c • f x\nf_eq : ∀ (i : ι), f (↑b i) = ↑(Finsupp.single i 1)\nx : M\ni : ι\nf_i : M →ₗ[R] R :=\n {\n toAddHom :=\n { toFun := fun x => f x i,\n map_add' := (_ : ∀ (x x_1 : M), (fun x => f x i) (x + x_1) = (fun x => f x i) x + (fun x => f x i) x_1) },\n map_smul' := (_ : ∀ (x : R) (x_1 : M), f (x • x_1) i = x * f x_1 i) }\nj : ι\n⊢ ↑(↑b.repr (↑b j)) i = f (↑b j) i",
"tactic": "rw [b.repr_self, f_eq]"
}
] |
[
330,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.countp_congr
|
[
{
"state_after": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\ns : l₁ ~ l₂\np p' : α → Bool\nhp : ∀ (x : α), x ∈ l₁ → p x = p' x\n⊢ countp p l₁ = countp p' l₁",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\ns : l₁ ~ l₂\np p' : α → Bool\nhp : ∀ (x : α), x ∈ l₁ → p x = p' x\n⊢ countp p l₁ = countp p' l₂",
"tactic": "rw [← s.countp_eq p']"
},
{
"state_after": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\np p' : α → Bool\nhp : ∀ (x : α), x ∈ l₁ → p x = p' x\n⊢ countp p l₁ = countp p' l₁",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\ns : l₁ ~ l₂\np p' : α → Bool\nhp : ∀ (x : α), x ∈ l₁ → p x = p' x\n⊢ countp p l₁ = countp p' l₁",
"tactic": "clear s"
},
{
"state_after": "case nil\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\np p' : α → Bool\nhp✝ : ∀ (x : α), x ∈ l₁ → p x = p' x\nhp : ∀ (x : α), x ∈ [] → p x = p' x\n⊢ countp p [] = countp p' []\n\ncase cons\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\np p' : α → Bool\nhp✝ : ∀ (x : α), x ∈ l₁ → p x = p' x\ny : α\ns : List α\nhs : (∀ (x : α), x ∈ s → p x = p' x) → countp p s = countp p' s\nhp : ∀ (x : α), x ∈ y :: s → p x = p' x\n⊢ countp p (y :: s) = countp p' (y :: s)",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\np p' : α → Bool\nhp : ∀ (x : α), x ∈ l₁ → p x = p' x\n⊢ countp p l₁ = countp p' l₁",
"tactic": "induction' l₁ with y s hs"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\np p' : α → Bool\nhp✝ : ∀ (x : α), x ∈ l₁ → p x = p' x\nhp : ∀ (x : α), x ∈ [] → p x = p' x\n⊢ countp p [] = countp p' []",
"tactic": "rfl"
},
{
"state_after": "case cons\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\np p' : α → Bool\nhp✝ : ∀ (x : α), x ∈ l₁ → p x = p' x\ny : α\ns : List α\nhs : (∀ (x : α), x ∈ s → p x = p' x) → countp p s = countp p' s\nhp : p y = p' y ∧ ∀ (a : α), a ∈ s → p a = p' a\n⊢ countp p (y :: s) = countp p' (y :: s)",
"state_before": "case cons\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\np p' : α → Bool\nhp✝ : ∀ (x : α), x ∈ l₁ → p x = p' x\ny : α\ns : List α\nhs : (∀ (x : α), x ∈ s → p x = p' x) → countp p s = countp p' s\nhp : ∀ (x : α), x ∈ y :: s → p x = p' x\n⊢ countp p (y :: s) = countp p' (y :: s)",
"tactic": "simp only [mem_cons, forall_eq_or_imp] at hp"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\np p' : α → Bool\nhp✝ : ∀ (x : α), x ∈ l₁ → p x = p' x\ny : α\ns : List α\nhs : (∀ (x : α), x ∈ s → p x = p' x) → countp p s = countp p' s\nhp : p y = p' y ∧ ∀ (a : α), a ∈ s → p a = p' a\n⊢ countp p (y :: s) = countp p' (y :: s)",
"tactic": "simp only [countp_cons, hs hp.2, hp.1]"
}
] |
[
489,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.approx_apply
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\n⊢ ↑(Finset.sup (Finset.range n) fun k => restrict (const α (i k)) {a | i k ≤ f a}) a =\n Finset.sup (Finset.range n) fun k => if i k ≤ f a then i k else 0",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\n⊢ ↑(approx i f n) a = Finset.sup (Finset.range n) fun k => if i k ≤ f a then i k else 0",
"tactic": "dsimp only [approx]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\n⊢ (Finset.sup (Finset.range n) fun c => ↑(restrict (const α (i c)) {a | i c ≤ f a}) a) =\n Finset.sup (Finset.range n) fun k => if i k ≤ f a then i k else 0",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\n⊢ ↑(Finset.sup (Finset.range n) fun k => restrict (const α (i k)) {a | i k ≤ f a}) a =\n Finset.sup (Finset.range n) fun k => if i k ≤ f a then i k else 0",
"tactic": "rw [finset_sup_apply]"
},
{
"state_after": "case e_f\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\n⊢ (fun c => ↑(restrict (const α (i c)) {a | i c ≤ f a}) a) = fun k => if i k ≤ f a then i k else 0",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\n⊢ (Finset.sup (Finset.range n) fun c => ↑(restrict (const α (i c)) {a | i c ≤ f a}) a) =\n Finset.sup (Finset.range n) fun k => if i k ≤ f a then i k else 0",
"tactic": "congr"
},
{
"state_after": "case e_f.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\nk : ℕ\n⊢ ↑(restrict (const α (i k)) {a | i k ≤ f a}) a = if i k ≤ f a then i k else 0",
"state_before": "case e_f\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\n⊢ (fun c => ↑(restrict (const α (i c)) {a | i c ≤ f a}) a) = fun k => if i k ≤ f a then i k else 0",
"tactic": "funext k"
},
{
"state_after": "case e_f.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\nk : ℕ\n⊢ indicator {a | i k ≤ f a} (↑(const α (i k))) a = if i k ≤ f a then i k else 0\n\ncase e_f.h.hs\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\nk : ℕ\n⊢ MeasurableSet {a | i k ≤ f a}",
"state_before": "case e_f.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\nk : ℕ\n⊢ ↑(restrict (const α (i k)) {a | i k ≤ f a}) a = if i k ≤ f a then i k else 0",
"tactic": "rw [restrict_apply]"
},
{
"state_after": "case e_f.h.hs\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\nk : ℕ\n⊢ MeasurableSet {a | i k ≤ f a}",
"state_before": "case e_f.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\nk : ℕ\n⊢ indicator {a | i k ≤ f a} (↑(const α (i k))) a = if i k ≤ f a then i k else 0\n\ncase e_f.h.hs\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\nk : ℕ\n⊢ MeasurableSet {a | i k ≤ f a}",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case e_f.h.hs\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.821186\nδ : Type ?u.821189\ninst✝⁷ : MeasurableSpace α\nK : Type ?u.821195\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\nk : ℕ\n⊢ MeasurableSet {a | i k ≤ f a}",
"tactic": "exact hf measurableSet_Ici"
}
] |
[
842,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
833,
1
] |
Mathlib/Deprecated/Group.lean
|
RingHom.to_isMonoidHom
|
[] |
[
376,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
bddAbove_Ico
|
[] |
[
664,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
663,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.restrict_apply₀
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.278939\nγ : Type ?u.278942\nδ : Type ?u.278945\nι : Type ?u.278948\nR : Type ?u.278951\nR' : Type ?u.278954\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s'✝ t✝ : Set α\nht : NullMeasurableSet t✝\ns' : Set α\nhs' : MeasurableSet s'\nt : Set α\n⊢ ↑↑μ (s ∩ t) = ↑↑μ (s ∩ t ∩ s') + ↑↑μ ((s ∩ t) \\ s')",
"state_before": "α : Type u_1\nβ : Type ?u.278939\nγ : Type ?u.278942\nδ : Type ?u.278945\nι : Type ?u.278948\nR : Type ?u.278951\nR' : Type ?u.278954\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s'✝ t✝ : Set α\nht : NullMeasurableSet t✝\ns' : Set α\nhs' : MeasurableSet s'\nt : Set α\n⊢ ↑(↑(OuterMeasure.restrict s) ↑μ) t =\n ↑(↑(OuterMeasure.restrict s) ↑μ) (t ∩ s') + ↑(↑(OuterMeasure.restrict s) ↑μ) (t \\ s')",
"tactic": "suffices μ (s ∩ t) = μ (s ∩ t ∩ s') + μ ((s ∩ t) \\ s') by\n simpa [← Set.inter_assoc, Set.inter_comm _ s, ← inter_diff_assoc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.278939\nγ : Type ?u.278942\nδ : Type ?u.278945\nι : Type ?u.278948\nR : Type ?u.278951\nR' : Type ?u.278954\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s'✝ t✝ : Set α\nht : NullMeasurableSet t✝\ns' : Set α\nhs' : MeasurableSet s'\nt : Set α\n⊢ ↑↑μ (s ∩ t) = ↑↑μ (s ∩ t ∩ s') + ↑↑μ ((s ∩ t) \\ s')",
"tactic": "exact le_toOuterMeasure_caratheodory _ _ hs' _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.278939\nγ : Type ?u.278942\nδ : Type ?u.278945\nι : Type ?u.278948\nR : Type ?u.278951\nR' : Type ?u.278954\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s'✝ t✝ : Set α\nht : NullMeasurableSet t✝\ns' : Set α\nhs' : MeasurableSet s'\nt : Set α\nthis : ↑↑μ (s ∩ t) = ↑↑μ (s ∩ t ∩ s') + ↑↑μ ((s ∩ t) \\ s')\n⊢ ↑(↑(OuterMeasure.restrict s) ↑μ) t =\n ↑(↑(OuterMeasure.restrict s) ↑μ) (t ∩ s') + ↑(↑(OuterMeasure.restrict s) ↑μ) (t \\ s')",
"tactic": "simpa [← Set.inter_assoc, Set.inter_comm _ s, ← inter_diff_assoc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.278939\nγ : Type ?u.278942\nδ : Type ?u.278945\nι : Type ?u.278948\nR : Type ?u.278951\nR' : Type ?u.278954\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nht : NullMeasurableSet t\n⊢ ↑(↑(OuterMeasure.restrict s) ↑μ) t = ↑↑μ (t ∩ s)",
"tactic": "simp only [OuterMeasure.restrict_apply]"
}
] |
[
1519,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1514,
1
] |
Mathlib/Order/Filter/Partial.lean
|
Filter.tendsto_iff_rtendsto
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl₁ : Filter α\nl₂ : Filter β\nf : α → β\n⊢ Tendsto f l₁ l₂ ↔ RTendsto (Function.graph f) l₁ l₂",
"tactic": "simp [tendsto_def, Function.graph, rtendsto_def, Rel.core, Set.preimage]"
}
] |
[
200,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Algebra/ContinuedFractions/Translations.lean
|
GeneralizedContinuedFraction.num_eq_conts_a
|
[] |
[
97,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
Right.mul_le_one
|
[] |
[
823,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
820,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
sub_pow_char
|
[] |
[
287,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/MeasureTheory/Function/Floor.lean
|
Int.measurable_floor
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.120\nR : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : LinearOrderedRing R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : OpensMeasurableSpace R\nx : R\n⊢ MeasurableSet (floor ⁻¹' {⌊x⌋})",
"tactic": "simpa only [Int.preimage_floor_singleton] using measurableSet_Ico"
}
] |
[
30,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
28,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.IsImage.symm
|
[] |
[
371,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
370,
11
] |
Mathlib/Data/Nat/GCD/Basic.lean
|
Nat.coprime_one_right_iff
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ coprime n 1 ↔ True",
"tactic": "simp [coprime]"
}
] |
[
212,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
WfDvdMonoid.induction_on_irreducible
|
[] |
[
109,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
tsub_tsub_cancel_of_le
|
[] |
[
299,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff
|
[
{
"state_after": "θ : ℝ\nk : ℤ\n⊢ -π < θ - 2 * ↑k * π ∧ θ - 2 * ↑k * π ≤ π ↔ (2 * ↑k - 1) * π < θ ∧ θ ≤ (2 * ↑k + 1) * π",
"state_before": "θ : ℝ\nk : ℤ\n⊢ toReal ↑θ = θ - 2 * ↑k * π ↔ θ ∈ Set.Ioc ((2 * ↑k - 1) * π) ((2 * ↑k + 1) * π)",
"tactic": "rw [← sub_zero (θ : Angle), ← zsmul_zero k, ← coe_two_pi, ← coe_zsmul, ← coe_sub, zsmul_eq_mul, ←\n mul_assoc, mul_comm (k : ℝ), toReal_coe_eq_self_iff, Set.mem_Ioc]"
},
{
"state_after": "no goals",
"state_before": "θ : ℝ\nk : ℤ\n⊢ -π < θ - 2 * ↑k * π ∧ θ - 2 * ↑k * π ≤ π ↔ (2 * ↑k - 1) * π < θ ∧ θ ≤ (2 * ↑k + 1) * π",
"tactic": "exact ⟨fun h => ⟨by linarith, by linarith⟩, fun h => ⟨by linarith, by linarith⟩⟩"
},
{
"state_after": "no goals",
"state_before": "θ : ℝ\nk : ℤ\nh : -π < θ - 2 * ↑k * π ∧ θ - 2 * ↑k * π ≤ π\n⊢ (2 * ↑k - 1) * π < θ",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "θ : ℝ\nk : ℤ\nh : -π < θ - 2 * ↑k * π ∧ θ - 2 * ↑k * π ≤ π\n⊢ θ ≤ (2 * ↑k + 1) * π",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "θ : ℝ\nk : ℤ\nh : (2 * ↑k - 1) * π < θ ∧ θ ≤ (2 * ↑k + 1) * π\n⊢ -π < θ - 2 * ↑k * π",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "θ : ℝ\nk : ℤ\nh : (2 * ↑k - 1) * π < θ ∧ θ ≤ (2 * ↑k + 1) * π\n⊢ θ - 2 * ↑k * π ≤ π",
"tactic": "linarith"
}
] |
[
685,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
681,
1
] |
Mathlib/Topology/Order.lean
|
gc_nhds
|
[
{
"state_after": "α : Type u\nβ : Type v\na : α\nf : Filter α\nt : TopologicalSpace α\n⊢ nhdsAdjoint a f ≤ t ↔ ∀ (s : Set α), a ∈ s → IsOpen s → s ∈ f",
"state_before": "α : Type u\nβ : Type v\na : α\nf : Filter α\nt : TopologicalSpace α\n⊢ nhdsAdjoint a f ≤ t ↔ f ≤ (fun t => 𝓝 a) t",
"tactic": "rw [le_nhds_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\na : α\nf : Filter α\nt : TopologicalSpace α\n⊢ nhdsAdjoint a f ≤ t ↔ ∀ (s : Set α), a ∈ s → IsOpen s → s ∈ f",
"tactic": "exact ⟨fun H s hs has => H _ has hs, fun H s has hs => H _ hs has⟩"
}
] |
[
589,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
587,
1
] |
Mathlib/Data/PFunctor/Multivariate/Basic.lean
|
MvPFunctor.liftP_iff
|
[
{
"state_after": "case mp\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\n⊢ LiftP p x → ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : Fin2 n) (j : B P a i), p (f i j)\n\ncase mpr\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\n⊢ (∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : Fin2 n) (j : B P a i), p (f i j)) → LiftP p x",
"state_before": "n m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\n⊢ LiftP p x ↔ ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : Fin2 n) (j : B P a i), p (f i j)",
"tactic": "constructor"
},
{
"state_after": "case mpr.intro.intro.intro\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\na : P.A\nf : B P a ⟹ α\nxeq : x = { fst := a, snd := f }\npf : ∀ (i : Fin2 n) (j : B P a i), p (f i j)\n⊢ LiftP p x",
"state_before": "case mpr\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\n⊢ (∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : Fin2 n) (j : B P a i), p (f i j)) → LiftP p x",
"tactic": "rintro ⟨a, f, xeq, pf⟩"
},
{
"state_after": "case mpr.intro.intro.intro\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\na : P.A\nf : B P a ⟹ α\nxeq : x = { fst := a, snd := f }\npf : ∀ (i : Fin2 n) (j : B P a i), p (f i j)\n⊢ (fun i => Subtype.val) <$$> { fst := a, snd := fun i j => { val := f i j, property := (_ : p (f i j)) } } = x",
"state_before": "case mpr.intro.intro.intro\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\na : P.A\nf : B P a ⟹ α\nxeq : x = { fst := a, snd := f }\npf : ∀ (i : Fin2 n) (j : B P a i), p (f i j)\n⊢ LiftP p x",
"tactic": "use ⟨a, fun i j => ⟨f i j, pf i j⟩⟩"
},
{
"state_after": "case mpr.intro.intro.intro\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\na : P.A\nf : B P a ⟹ α\nxeq : x = { fst := a, snd := f }\npf : ∀ (i : Fin2 n) (j : B P a i), p (f i j)\n⊢ (fun i => Subtype.val) <$$> { fst := a, snd := fun i j => { val := f i j, property := (_ : p (f i j)) } } =\n { fst := a, snd := f }",
"state_before": "case mpr.intro.intro.intro\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\na : P.A\nf : B P a ⟹ α\nxeq : x = { fst := a, snd := f }\npf : ∀ (i : Fin2 n) (j : B P a i), p (f i j)\n⊢ (fun i => Subtype.val) <$$> { fst := a, snd := fun i j => { val := f i j, property := (_ : p (f i j)) } } = x",
"tactic": "rw [xeq]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\na : P.A\nf : B P a ⟹ α\nxeq : x = { fst := a, snd := f }\npf : ∀ (i : Fin2 n) (j : B P a i), p (f i j)\n⊢ (fun i => Subtype.val) <$$> { fst := a, snd := fun i j => { val := f i j, property := (_ : p (f i j)) } } =\n { fst := a, snd := f }",
"tactic": "rfl"
},
{
"state_after": "case mp.intro\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\ny : Obj P fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\n⊢ ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : Fin2 n) (j : B P a i), p (f i j)",
"state_before": "case mp\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\n⊢ LiftP p x → ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : Fin2 n) (j : B P a i), p (f i j)",
"tactic": "rintro ⟨y, hy⟩"
},
{
"state_after": "case mp.intro.mk\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\ny : Obj P fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : P.A\nf : B P a ⟹ fun i => Subtype p\nh : y = { fst := a, snd := f }\n⊢ ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : Fin2 n) (j : B P a i), p (f i j)",
"state_before": "case mp.intro\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\ny : Obj P fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\n⊢ ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : Fin2 n) (j : B P a i), p (f i j)",
"tactic": "cases' h : y with a f"
},
{
"state_after": "case mp.intro.mk\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\ny : Obj P fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : P.A\nf : B P a ⟹ fun i => Subtype p\nh : y = { fst := a, snd := f }\n⊢ x = { fst := a, snd := fun i j => ↑(f i j) }",
"state_before": "case mp.intro.mk\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\ny : Obj P fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : P.A\nf : B P a ⟹ fun i => Subtype p\nh : y = { fst := a, snd := f }\n⊢ ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : Fin2 n) (j : B P a i), p (f i j)",
"tactic": "refine' ⟨a, fun i j => (f i j).val, _, fun i j => (f i j).property⟩"
},
{
"state_after": "case mp.intro.mk\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\ny : Obj P fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : P.A\nf : B P a ⟹ fun i => Subtype p\nh : y = { fst := a, snd := f }\n⊢ { fst := a, snd := (fun i => Subtype.val) ⊚ f } = { fst := a, snd := fun i j => ↑(f i j) }",
"state_before": "case mp.intro.mk\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\ny : Obj P fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : P.A\nf : B P a ⟹ fun i => Subtype p\nh : y = { fst := a, snd := f }\n⊢ x = { fst := a, snd := fun i j => ↑(f i j) }",
"tactic": "rw [← hy, h, map_eq]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.mk\nn m : ℕ\nP : MvPFunctor n\nQ : Fin2 n → MvPFunctor m\nα✝ β : TypeVec m\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\nx : Obj P α\ny : Obj P fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : P.A\nf : B P a ⟹ fun i => Subtype p\nh : y = { fst := a, snd := f }\n⊢ { fst := a, snd := (fun i => Subtype.val) ⊚ f } = { fst := a, snd := fun i j => ↑(f i j) }",
"tactic": "rfl"
}
] |
[
169,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/CategoryTheory/Adjunction/Limits.lean
|
CategoryTheory.Adjunction.hasColimit_comp_equivalence
|
[] |
[
142,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Analysis/Convex/Side.lean
|
AffineSubspace.SSameSide.trans_sOppSide
|
[] |
[
574,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
572,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
Padic.exi_rat_seq_conv
|
[
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\n⊢ ↑padicNormE (↑f i - ↑(limSeq f i)) < ε",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\n⊢ ∃ N, ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε",
"tactic": "refine' (exists_nat_gt (1 / ε)).imp fun N hN i hi ↦ _"
},
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ ↑padicNormE (↑f i - ↑(limSeq f i)) < ε",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\n⊢ ↑padicNormE (↑f i - ↑(limSeq f i)) < ε",
"tactic": "have h := Classical.choose_spec (rat_dense' (f i) (div_nat_pos i))"
},
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ 1 ≤ (↑i + 1) * ε",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ ↑padicNormE (↑f i - ↑(limSeq f i)) < ε",
"tactic": "refine' lt_of_lt_of_le h ((div_le_iff' <| by exact_mod_cast succ_pos _).mpr _)"
},
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ 1 ≤ ↑i * ε + 1 * ε",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ 1 ≤ (↑i + 1) * ε",
"tactic": "rw [right_distrib]"
},
{
"state_after": "case hbc\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ 1 ≤ ↑i * ε\n\ncase ha\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ 0 ≤ 1 * ε",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ 1 ≤ ↑i * ε + 1 * ε",
"tactic": "apply le_add_of_le_of_nonneg"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ 0 < ↑i + 1",
"tactic": "exact_mod_cast succ_pos _"
},
{
"state_after": "no goals",
"state_before": "case hbc\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ 1 ≤ ↑i * ε",
"tactic": "exact (div_le_iff hε).mp (le_trans (le_of_lt hN) (by exact_mod_cast hi))"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ ↑N ≤ ↑i",
"tactic": "exact_mod_cast hi"
},
{
"state_after": "case ha.a\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ 0 < 1 * ε",
"state_before": "case ha\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ 0 ≤ 1 * ε",
"tactic": "apply le_of_lt"
},
{
"state_after": "no goals",
"state_before": "case ha.a\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : 0 < ε\nN : ℕ\nhN : 1 / ε < ↑N\ni : ℕ\nhi : i ≥ N\nh : ↑padicNormE (↑f i - ↑(Classical.choose (_ : ∃ r, ↑padicNormE (↑f i - ↑r) < 1 / (↑i + 1)))) < 1 / (↑i + 1)\n⊢ 0 < 1 * ε",
"tactic": "simpa"
}
] |
[
700,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
691,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.sum_add_sum
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.347212\nγ : Type ?u.347215\nδ : Type ?u.347218\nι : Type ?u.347221\nR : Type ?u.347224\nR' : Type ?u.347227\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν✝ ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nμ ν : ℕ → Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(sum μ + sum ν) s = ↑↑(sum fun n => μ n + ν n) s",
"state_before": "α : Type u_1\nβ : Type ?u.347212\nγ : Type ?u.347215\nδ : Type ?u.347218\nι : Type ?u.347221\nR : Type ?u.347224\nR' : Type ?u.347227\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν✝ ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nμ ν : ℕ → Measure α\n⊢ sum μ + sum ν = sum fun n => μ n + ν n",
"tactic": "ext1 s hs"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.347212\nγ : Type ?u.347215\nδ : Type ?u.347218\nι : Type ?u.347221\nR : Type ?u.347224\nR' : Type ?u.347227\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν✝ ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nμ ν : ℕ → Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(sum μ + sum ν) s = ↑↑(sum fun n => μ n + ν n) s",
"tactic": "simp only [add_apply, sum_apply _ hs, Pi.add_apply, coe_add,\n tsum_add ENNReal.summable ENNReal.summable]"
}
] |
[
2129,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2126,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.limsup_eq_sInf_sSup
|
[
{
"state_after": "case refine'_1\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\n⊢ limsup a F ≤ sInf ((fun I => sSup (a '' I)) '' F.sets)\n\ncase refine'_2\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\n⊢ sInf ((fun I => sSup (a '' I)) '' F.sets) ≤ limsup a F",
"state_before": "α : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\n⊢ limsup a F = sInf ((fun I => sSup (a '' I)) '' F.sets)",
"tactic": "refine' le_antisymm _ _"
},
{
"state_after": "case refine'_1\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\n⊢ sInf {a_1 | ∀ᶠ (n : ι) in F, a n ≤ a_1} ≤ sInf ((fun I => sSup (a '' I)) '' F.sets)",
"state_before": "case refine'_1\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\n⊢ limsup a F ≤ sInf ((fun I => sSup (a '' I)) '' F.sets)",
"tactic": "rw [limsup_eq]"
},
{
"state_after": "case refine'_1\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\nx : R\nhx : x ∈ (fun I => sSup (a '' I)) '' F.sets\n⊢ x ∈ {a_1 | ∀ᶠ (n : ι) in F, a n ≤ a_1}",
"state_before": "case refine'_1\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\n⊢ sInf {a_1 | ∀ᶠ (n : ι) in F, a n ≤ a_1} ≤ sInf ((fun I => sSup (a '' I)) '' F.sets)",
"tactic": "refine' sInf_le_sInf fun x hx => _"
},
{
"state_after": "case refine'_1.intro.intro\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\nx : R\nhx : x ∈ (fun I => sSup (a '' I)) '' F.sets\nI : Set ι\nI_mem_F : I ∈ F.sets\nhI : sSup (a '' I) = x\n⊢ x ∈ {a_1 | ∀ᶠ (n : ι) in F, a n ≤ a_1}",
"state_before": "case refine'_1\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\nx : R\nhx : x ∈ (fun I => sSup (a '' I)) '' F.sets\n⊢ x ∈ {a_1 | ∀ᶠ (n : ι) in F, a n ≤ a_1}",
"tactic": "rcases(mem_image _ F.sets x).mp hx with ⟨I, ⟨I_mem_F, hI⟩⟩"
},
{
"state_after": "case h\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\nx : R\nhx : x ∈ (fun I => sSup (a '' I)) '' F.sets\nI : Set ι\nI_mem_F : I ∈ F.sets\nhI : sSup (a '' I) = x\ni : ι\nhi : i ∈ I\n⊢ a i ≤ x",
"state_before": "case refine'_1.intro.intro\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\nx : R\nhx : x ∈ (fun I => sSup (a '' I)) '' F.sets\nI : Set ι\nI_mem_F : I ∈ F.sets\nhI : sSup (a '' I) = x\n⊢ x ∈ {a_1 | ∀ᶠ (n : ι) in F, a n ≤ a_1}",
"tactic": "filter_upwards [I_mem_F]with i hi"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\nx : R\nhx : x ∈ (fun I => sSup (a '' I)) '' F.sets\nI : Set ι\nI_mem_F : I ∈ F.sets\nhI : sSup (a '' I) = x\ni : ι\nhi : i ∈ I\n⊢ a i ≤ x",
"tactic": "exact hI ▸ le_sSup (mem_image_of_mem _ hi)"
},
{
"state_after": "case refine'_2\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\nb : R\nhb : b ∈ {a_1 | ∀ᶠ (n : R) in map a F, n ≤ a_1}\n⊢ ∀ (b_1 : R), b_1 ∈ a '' (a ⁻¹' {x | (fun n => n ≤ b) x}) → b_1 ≤ b",
"state_before": "case refine'_2\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\n⊢ sInf ((fun I => sSup (a '' I)) '' F.sets) ≤ limsup a F",
"tactic": "refine'\n le_sInf_iff.mpr fun b hb =>\n sInf_le_of_le (mem_image_of_mem _ <| Filter.mem_sets.mpr hb) <| sSup_le _"
},
{
"state_after": "case refine'_2.intro.intro\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\nb : R\nhb : b ∈ {a_1 | ∀ᶠ (n : R) in map a F, n ≤ a_1}\nw✝ : ι\nh : w✝ ∈ a ⁻¹' {x | (fun n => n ≤ b) x}\n⊢ a w✝ ≤ b",
"state_before": "case refine'_2\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\nb : R\nhb : b ∈ {a_1 | ∀ᶠ (n : R) in map a F, n ≤ a_1}\n⊢ ∀ (b_1 : R), b_1 ∈ a '' (a ⁻¹' {x | (fun n => n ≤ b) x}) → b_1 ≤ b",
"tactic": "rintro _ ⟨_, h, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\nα : Type ?u.120102\nβ : Type ?u.120105\nγ : Type ?u.120108\nι✝ : Type ?u.120111\ninst✝¹ : CompleteLattice α\nι : Type u_1\nR : Type u_2\nF : Filter ι\ninst✝ : CompleteLattice R\na : ι → R\nb : R\nhb : b ∈ {a_1 | ∀ᶠ (n : R) in map a F, n ≤ a_1}\nw✝ : ι\nh : w✝ ∈ a ⁻¹' {x | (fun n => n ≤ b) x}\n⊢ a w✝ ≤ b",
"tactic": "exact h"
}
] |
[
793,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
781,
1
] |
Mathlib/LinearAlgebra/SModEq.lean
|
SModEq.mono
|
[] |
[
63,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Data/Int/Order/Basic.lean
|
Int.le_induction
|
[
{
"state_after": "case refine_1\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn : ℤ\n⊢ m ≤ m → P m\n\ncase refine_2\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn : ℤ\n⊢ ∀ (k : ℤ), m ≤ k → (m ≤ k → P k) → m ≤ k + 1 → P (k + 1)\n\ncase refine_3\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn : ℤ\n⊢ ∀ (k : ℤ), k ≤ m → (m ≤ k → P k) → m ≤ k - 1 → P (k - 1)",
"state_before": "P : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn : ℤ\n⊢ m ≤ n → P n",
"tactic": "refine Int.inductionOn' n m ?_ ?_ ?_"
},
{
"state_after": "case refine_1\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn : ℤ\na✝ : m ≤ m\n⊢ P m",
"state_before": "case refine_1\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn : ℤ\n⊢ m ≤ m → P m",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn : ℤ\na✝ : m ≤ m\n⊢ P m",
"tactic": "exact h0"
},
{
"state_after": "case refine_2\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn k : ℤ\nhle : m ≤ k\nhi : m ≤ k → P k\na✝ : m ≤ k + 1\n⊢ P (k + 1)",
"state_before": "case refine_2\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn : ℤ\n⊢ ∀ (k : ℤ), m ≤ k → (m ≤ k → P k) → m ≤ k + 1 → P (k + 1)",
"tactic": "intro k hle hi _"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn k : ℤ\nhle : m ≤ k\nhi : m ≤ k → P k\na✝ : m ≤ k + 1\n⊢ P (k + 1)",
"tactic": "exact h1 k hle (hi hle)"
},
{
"state_after": "case refine_3\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn k : ℤ\nhle : k ≤ m\na✝ : m ≤ k → P k\nhle' : m ≤ k - 1\n⊢ P (k - 1)",
"state_before": "case refine_3\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn : ℤ\n⊢ ∀ (k : ℤ), k ≤ m → (m ≤ k → P k) → m ≤ k - 1 → P (k - 1)",
"tactic": "intro k hle _ hle'"
},
{
"state_after": "case refine_3.h\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn k : ℤ\nhle : k ≤ m\na✝ : m ≤ k → P k\nhle' : m ≤ k - 1\n⊢ False",
"state_before": "case refine_3\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn k : ℤ\nhle : k ≤ m\na✝ : m ≤ k → P k\nhle' : m ≤ k - 1\n⊢ P (k - 1)",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case refine_3.h\nP : ℤ → Prop\nm : ℤ\nh0 : P m\nh1 : ∀ (n : ℤ), m ≤ n → P n → P (n + 1)\nn k : ℤ\nhle : k ≤ m\na✝ : m ≤ k → P k\nhle' : m ≤ k - 1\n⊢ False",
"tactic": "exact lt_irrefl k (le_sub_one_iff.mp (hle.trans hle'))"
}
] |
[
175,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
11
] |
Mathlib/Data/Set/Finite.lean
|
Set.iInf_iSup_of_monotone
|
[] |
[
1513,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1510,
1
] |
Mathlib/Algebra/Lie/Normalizer.lean
|
LieSubmodule.gc_top_lie_normalizer
|
[] |
[
95,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/LinearAlgebra/Matrix/Symmetric.lean
|
Matrix.IsSymm.apply
|
[] |
[
55,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/FieldTheory/Subfield.lean
|
Subfield.coe_mul
|
[] |
[
382,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.x_sub_y_dvd_pow_lem
|
[
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\ny2 y1 y0 yn1 yn0 xn1 xn0 ay a2 : ℤ\n⊢ (a2 * yn1 - yn0) * ay + y2 - (a2 * xn1 - xn0) = y2 - a2 * y1 + y0 + a2 * (yn1 * ay + y1 - xn1) - (yn0 * ay + y0 - xn0)",
"tactic": "ring"
}
] |
[
570,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
567,
1
] |
Mathlib/Data/Prod/TProd.lean
|
List.TProd.elim_mk
|
[
{
"state_after": "case pos\nι : Type u\nα : ι → Type v\ni✝ j✝ : ι\nl : List ι\nf✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\ni : ι\nis : List ι\nf : (i : ι) → α i\nj : ι\nhj : j ∈ i :: is\nhji : j = i\n⊢ TProd.elim (TProd.mk (i :: is) f) hj = f j\n\ncase neg\nι : Type u\nα : ι → Type v\ni✝ j✝ : ι\nl : List ι\nf✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\ni : ι\nis : List ι\nf : (i : ι) → α i\nj : ι\nhj : j ∈ i :: is\nhji : ¬j = i\n⊢ TProd.elim (TProd.mk (i :: is) f) hj = f j",
"state_before": "ι : Type u\nα : ι → Type v\ni✝ j✝ : ι\nl : List ι\nf✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\ni : ι\nis : List ι\nf : (i : ι) → α i\nj : ι\nhj : j ∈ i :: is\n⊢ TProd.elim (TProd.mk (i :: is) f) hj = f j",
"tactic": "by_cases hji : j = i"
},
{
"state_after": "case pos\nι : Type u\nα : ι → Type v\ni j✝ : ι\nl : List ι\nf✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\nis : List ι\nf : (i : ι) → α i\nj : ι\nhj : j ∈ j :: is\n⊢ TProd.elim (TProd.mk (j :: is) f) hj = f j",
"state_before": "case pos\nι : Type u\nα : ι → Type v\ni✝ j✝ : ι\nl : List ι\nf✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\ni : ι\nis : List ι\nf : (i : ι) → α i\nj : ι\nhj : j ∈ i :: is\nhji : j = i\n⊢ TProd.elim (TProd.mk (i :: is) f) hj = f j",
"tactic": "subst hji"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type u\nα : ι → Type v\ni j✝ : ι\nl : List ι\nf✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\nis : List ι\nf : (i : ι) → α i\nj : ι\nhj : j ∈ j :: is\n⊢ TProd.elim (TProd.mk (j :: is) f) hj = f j",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type u\nα : ι → Type v\ni✝ j✝ : ι\nl : List ι\nf✝ : (i : ι) → α i\ninst✝ : DecidableEq ι\ni : ι\nis : List ι\nf : (i : ι) → α i\nj : ι\nhj : j ∈ i :: is\nhji : ¬j = i\n⊢ TProd.elim (TProd.mk (i :: is) f) hj = f j",
"tactic": "rw [TProd.elim_of_ne _ hji, snd_mk, elim_mk is]"
}
] |
[
115,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
differentiableWithinAt_const_sub_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.545914\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.546009\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\n⊢ DifferentiableWithinAt 𝕜 (fun y => c - f y) s x ↔ DifferentiableWithinAt 𝕜 f s x",
"tactic": "simp [sub_eq_add_neg]"
}
] |
[
626,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
624,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.hasFiniteIntegral_zero_measure
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.721423\nδ : Type ?u.721426\nm✝ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nm : MeasurableSpace α\nf : α → β\n⊢ HasFiniteIntegral f",
"tactic": "simp only [HasFiniteIntegral, lintegral_zero_measure, WithTop.zero_lt_top]"
}
] |
[
226,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/Data/Finset/Prod.lean
|
Finset.map_swap_product
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.17685\ns✝ s' : Finset α\nt✝ t' : Finset β\na : α\nb : β\ns : Finset α\nt : Finset β\n⊢ (fun a => ↑{ toFun := Prod.swap, inj' := (_ : Function.Injective Prod.swap) } a) '' ↑t ×ˢ ↑s = ↑s ×ˢ ↑t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.17685\ns✝ s' : Finset α\nt✝ t' : Finset β\na : α\nb : β\ns : Finset α\nt : Finset β\n⊢ ↑(map { toFun := Prod.swap, inj' := (_ : Function.Injective Prod.swap) } (t ×ˢ s)) = ↑(s ×ˢ t)",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.17685\ns✝ s' : Finset α\nt✝ t' : Finset β\na : α\nb : β\ns : Finset α\nt : Finset β\n⊢ (fun a => ↑{ toFun := Prod.swap, inj' := (_ : Function.Injective Prod.swap) } a) '' ↑t ×ˢ ↑s = ↑s ×ˢ ↑t",
"tactic": "exact Set.image_swap_prod _ _"
}
] |
[
109,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Data/LazyList/Basic.lean
|
LazyList.mem_cons
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nx y : α\nys : Thunk (LazyList α)\n⊢ x ∈ cons y ys ↔ x = y ∨ x ∈ Thunk.get ys",
"tactic": "simp [Membership.mem, LazyList.Mem]"
}
] |
[
258,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.smul_apply_eq_smul_apply_inv_smul
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝³ : Group G\nH K : Subgroup G\nS T✝ : Set G\nF : Type u_2\ninst✝² : Group F\ninst✝¹ : MulAction F G\ninst✝ : QuotientAction F H\nf : F\nT : ↑(leftTransversals ↑H)\nq : G ⧸ H\n⊢ ↑(↑(toEquiv (_ : ↑(f • T) ∈ leftTransversals ↑H)) q) = f • ↑(↑(toEquiv (_ : ↑T ∈ leftTransversals ↑H)) (f⁻¹ • q))",
"tactic": "rw [smul_toEquiv, smul_inv_smul]"
}
] |
[
480,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.prod_eq_bot
|
[
{
"state_after": "R✝ : Type u\nι : Type ?u.301378\ninst✝² : CommSemiring R✝\nI J K L : Ideal R✝\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ns : Multiset (Ideal R)\n⊢ (∃ r x, r = 0) ↔ ∃ I, I ∈ s ∧ I = 0",
"state_before": "R✝ : Type u\nι : Type ?u.301378\ninst✝² : CommSemiring R✝\nI J K L : Ideal R✝\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ns : Multiset (Ideal R)\n⊢ Multiset.prod s = ⊥ ↔ ∃ I, I ∈ s ∧ I = ⊥",
"tactic": "rw [bot_eq_zero, prod_zero_iff_exists_zero]"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type u\nι : Type ?u.301378\ninst✝² : CommSemiring R✝\nI J K L : Ideal R✝\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ns : Multiset (Ideal R)\n⊢ (∃ r x, r = 0) ↔ ∃ I, I ∈ s ∧ I = 0",
"tactic": "simp"
}
] |
[
823,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
820,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.iSup_comap_le
|
[] |
[
1531,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1529,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
abs_eq_zero
|
[] |
[
184,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.vanishingIdeal_iUnion
|
[] |
[
354,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.eq_singleton_iff_unique_mem
|
[] |
[
1363,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1362,
1
] |
Mathlib/GroupTheory/Finiteness.lean
|
AddGroup.fg_def
|
[] |
[
301,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Algebra/Module/LocalizedModule.lean
|
IsLocalizedModule.smul_injective
|
[] |
[
910,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
909,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.coe_zpow
|
[] |
[
1342,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1341,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.of_apply
|
[] |
[
124,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.EqOnSource.eqOn
|
[] |
[
962,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
961,
1
] |
Mathlib/Algebra/FreeMonoid/Basic.lean
|
FreeMonoid.lift_ofList
|
[] |
[
243,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Topology/Sheaves/Stalks.lean
|
TopCat.Presheaf.isIso_iff_stalkFunctor_map_iso
|
[] |
[
637,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
633,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.eq_symm_apply
|
[] |
[
168,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.mk'_toSubmonoid
|
[] |
[
318,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
1
] |
Mathlib/FieldTheory/IntermediateField.lean
|
IntermediateField.coe_zero
|
[] |
[
268,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
11
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.coe_iInf
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\nι : Sort u_1\nS : ι → IntermediateField F E\n⊢ ↑(iInf S) = ⋂ (i : ι), ↑(S i)",
"tactic": "simp [iInf]"
}
] |
[
164,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
EuclideanGeometry.collinear_of_angle_eq_zero
|
[] |
[
447,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
445,
1
] |
Mathlib/Analysis/Normed/Order/Lattice.lean
|
continuous_pos
|
[] |
[
209,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
measure_Ico_lt_top
|
[] |
[
4594,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4593,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Cardinal.sum_lt_lift_of_isRegular
|
[] |
[
1115,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1113,
1
] |
Std/Logic.lean
|
Decidable.imp_iff_not_or
|
[] |
[
556,
39
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
555,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.drop.aux_none
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nn : ℕ\n⊢ Computation.bind (Computation.pure none) (aux n) = Computation.pure none",
"tactic": "rw [ret_bind, drop.aux_none n]"
}
] |
[
821,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
817,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.castSucc_pred_eq_pred_castSucc
|
[
{
"state_after": "case mk\nn m val✝ : ℕ\nisLt✝ : val✝ < n + 1\nha : { val := val✝, isLt := isLt✝ } ≠ 0\nha' : optParam (↑castSucc { val := val✝, isLt := isLt✝ } ≠ 0) (_ : ↑castSucc { val := val✝, isLt := isLt✝ } ≠ 0)\n⊢ ↑castSucc (pred { val := val✝, isLt := isLt✝ } ha) = pred (↑castSucc { val := val✝, isLt := isLt✝ }) ha'",
"state_before": "n m : ℕ\na : Fin (n + 1)\nha : a ≠ 0\nha' : optParam (↑castSucc a ≠ 0) (_ : ↑castSucc a ≠ 0)\n⊢ ↑castSucc (pred a ha) = pred (↑castSucc a) ha'",
"tactic": "cases a"
},
{
"state_after": "no goals",
"state_before": "case mk\nn m val✝ : ℕ\nisLt✝ : val✝ < n + 1\nha : { val := val✝, isLt := isLt✝ } ≠ 0\nha' : optParam (↑castSucc { val := val✝, isLt := isLt✝ } ≠ 0) (_ : ↑castSucc { val := val✝, isLt := isLt✝ } ≠ 0)\n⊢ ↑castSucc (pred { val := val✝, isLt := isLt✝ } ha) = pred (↑castSucc { val := val✝, isLt := isLt✝ }) ha'",
"tactic": "rfl"
}
] |
[
2408,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2405,
1
] |
Mathlib/Order/Cover.lean
|
Prod.mk_wcovby_mk_iff
|
[
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na a₁ a₂ : α\nb b₁ b₂ : β\nx y : α × β\nh : (a₁, b₁) ⩿ (a₂, b₂)\n⊢ a₁ ⩿ a₂ ∧ b₁ = b₂ ∨ b₁ ⩿ b₂ ∧ a₁ = a₂\n\ncase refine'_2\nα : Type u_2\nβ : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na a₁ a₂ : α\nb b₁ b₂ : β\nx y : α × β\n⊢ a₁ ⩿ a₂ ∧ b₁ = b₂ ∨ b₁ ⩿ b₂ ∧ a₁ = a₂ → (a₁, b₁) ⩿ (a₂, b₂)",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na a₁ a₂ : α\nb b₁ b₂ : β\nx y : α × β\n⊢ (a₁, b₁) ⩿ (a₂, b₂) ↔ a₁ ⩿ a₂ ∧ b₁ = b₂ ∨ b₁ ⩿ b₂ ∧ a₁ = a₂",
"tactic": "refine' ⟨fun h => _, _⟩"
},
{
"state_after": "case refine'_1.inl\nα : Type u_2\nβ : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na a₁ : α\nb b₁ b₂ : β\nx y : α × β\nh : (a₁, b₁) ⩿ (a₁, b₂)\n⊢ a₁ ⩿ a₁ ∧ b₁ = b₂ ∨ b₁ ⩿ b₂ ∧ a₁ = a₁\n\ncase refine'_1.inr\nα : Type u_2\nβ : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na a₁ a₂ : α\nb b₁ : β\nx y : α × β\nh : (a₁, b₁) ⩿ (a₂, b₁)\n⊢ a₁ ⩿ a₂ ∧ b₁ = b₁ ∨ b₁ ⩿ b₁ ∧ a₁ = a₂",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na a₁ a₂ : α\nb b₁ b₂ : β\nx y : α × β\nh : (a₁, b₁) ⩿ (a₂, b₂)\n⊢ a₁ ⩿ a₂ ∧ b₁ = b₂ ∨ b₁ ⩿ b₂ ∧ a₁ = a₂",
"tactic": "obtain rfl | rfl : a₁ = a₂ ∨ b₁ = b₂ := fst_eq_or_snd_eq_of_wcovby h"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inl\nα : Type u_2\nβ : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na a₁ : α\nb b₁ b₂ : β\nx y : α × β\nh : (a₁, b₁) ⩿ (a₁, b₂)\n⊢ a₁ ⩿ a₁ ∧ b₁ = b₂ ∨ b₁ ⩿ b₂ ∧ a₁ = a₁",
"tactic": "exact Or.inr ⟨mk_wcovby_mk_iff_right.1 h, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inr\nα : Type u_2\nβ : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na a₁ a₂ : α\nb b₁ : β\nx y : α × β\nh : (a₁, b₁) ⩿ (a₂, b₁)\n⊢ a₁ ⩿ a₂ ∧ b₁ = b₁ ∨ b₁ ⩿ b₁ ∧ a₁ = a₂",
"tactic": "exact Or.inl ⟨mk_wcovby_mk_iff_left.1 h, rfl⟩"
},
{
"state_after": "case refine'_2.inl.intro\nα : Type u_2\nβ : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na a₁ a₂ : α\nb b₁ : β\nx y : α × β\nh : a₁ ⩿ a₂\n⊢ (a₁, b₁) ⩿ (a₂, b₁)\n\ncase refine'_2.inr.intro\nα : Type u_2\nβ : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na a₁ : α\nb b₁ b₂ : β\nx y : α × β\nh : b₁ ⩿ b₂\n⊢ (a₁, b₁) ⩿ (a₁, b₂)",
"state_before": "case refine'_2\nα : Type u_2\nβ : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na a₁ a₂ : α\nb b₁ b₂ : β\nx y : α × β\n⊢ a₁ ⩿ a₂ ∧ b₁ = b₂ ∨ b₁ ⩿ b₂ ∧ a₁ = a₂ → (a₁, b₁) ⩿ (a₂, b₂)",
"tactic": "rintro (⟨h, rfl⟩ | ⟨h, rfl⟩)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.inl.intro\nα : Type u_2\nβ : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na a₁ a₂ : α\nb b₁ : β\nx y : α × β\nh : a₁ ⩿ a₂\n⊢ (a₁, b₁) ⩿ (a₂, b₁)",
"tactic": "exact mk_wcovby_mk_iff_left.2 h"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.inr.intro\nα : Type u_2\nβ : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na a₁ : α\nb b₁ b₂ : β\nx y : α × β\nh : b₁ ⩿ b₂\n⊢ (a₁, b₁) ⩿ (a₁, b₂)",
"tactic": "exact mk_wcovby_mk_iff_right.2 h"
}
] |
[
543,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
536,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.fst_image_prod_subset
|
[] |
[
353,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.tan_nat_mul_pi
|
[] |
[
1319,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1318,
1
] |
Mathlib/Logic/Basic.lean
|
eq_true_eq_id
|
[
{
"state_after": "case h\nx✝ : Prop\n⊢ (True = x✝) = id x✝",
"state_before": "⊢ Eq True = id",
"tactic": "funext _"
},
{
"state_after": "no goals",
"state_before": "case h\nx✝ : Prop\n⊢ (True = x✝) = id x✝",
"tactic": "simp only [true_iff, id.def, eq_iff_iff]"
}
] |
[
170,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
9
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
zpowersEquivZpowers_apply
|
[
{
"state_after": "G : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝² : Group G\ninst✝¹ : AddGroup A\ninst✝ : Finite G\nh : orderOf x = orderOf y\nn : ℕ\n⊢ ↑(Fin.cast h).toEquiv { val := n % orderOf x, isLt := (_ : n % orderOf x < orderOf x) } =\n { val := n % orderOf y, isLt := (_ : n % orderOf y < orderOf y) }",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝² : Group G\ninst✝¹ : AddGroup A\ninst✝ : Finite G\nh : orderOf x = orderOf y\nn : ℕ\n⊢ ↑(zpowersEquivZpowers h) { val := x ^ n, property := (_ : ∃ y, (fun x x_1 => x ^ x_1) x y = x ^ n) } =\n { val := y ^ n, property := (_ : ∃ y_1, (fun x x_1 => x ^ x_1) y y_1 = y ^ n) }",
"tactic": "rw [zpowersEquivZpowers, Equiv.trans_apply, Equiv.trans_apply, finEquivZpowers_symm_apply, ←\n Equiv.eq_symm_apply, finEquivZpowers_symm_apply]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝² : Group G\ninst✝¹ : AddGroup A\ninst✝ : Finite G\nh : orderOf x = orderOf y\nn : ℕ\n⊢ ↑(Fin.cast h).toEquiv { val := n % orderOf x, isLt := (_ : n % orderOf x < orderOf x) } =\n { val := n % orderOf y, isLt := (_ : n % orderOf y < orderOf y) }",
"tactic": "simp [h]"
}
] |
[
884,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
880,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
orderOf_eq_iff
|
[
{
"state_after": "G : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\n⊢ (if h : 1 ∈ periodicPts fun x_1 => x * x_1 then Nat.find h else 0) = n ↔\n IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\n⊢ orderOf x = n ↔ x ^ n = 1 ∧ ∀ (m : ℕ), m < n → 0 < m → x ^ m ≠ 1",
"tactic": "simp_rw [Ne, ← isPeriodicPt_mul_iff_pow_eq_one, orderOf, minimalPeriod]"
},
{
"state_after": "case inl\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : 1 ∈ periodicPts fun x_1 => x * x_1\n⊢ Nat.find h1 = n ↔\n IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1\n\ncase inr\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : ¬1 ∈ periodicPts fun x_1 => x * x_1\n⊢ 0 = n ↔ IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\n⊢ (if h : 1 ∈ periodicPts fun x_1 => x * x_1 then Nat.find h else 0) = n ↔\n IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1",
"tactic": "split_ifs with h1"
},
{
"state_after": "no goals",
"state_before": "case inl\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : 1 ∈ periodicPts fun x_1 => x * x_1\n⊢ Nat.find h1 = n ↔\n IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1",
"tactic": "classical\nrw [find_eq_iff]\nsimp only [h, true_and]\npush_neg\nrfl"
},
{
"state_after": "case inl\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : 1 ∈ periodicPts fun x_1 => x * x_1\n⊢ ((n > 0 ∧ IsPeriodicPt (fun x_1 => x * x_1) n 1) ∧\n ∀ (n_1 : ℕ), n_1 < n → ¬(n_1 > 0 ∧ IsPeriodicPt (fun x_1 => x * x_1) n_1 1)) ↔\n IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1",
"state_before": "case inl\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : 1 ∈ periodicPts fun x_1 => x * x_1\n⊢ Nat.find h1 = n ↔\n IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1",
"tactic": "rw [find_eq_iff]"
},
{
"state_after": "case inl\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : 1 ∈ periodicPts fun x_1 => x * x_1\n⊢ (IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧\n ∀ (n_1 : ℕ), n_1 < n → ¬(n_1 > 0 ∧ IsPeriodicPt (fun x_1 => x * x_1) n_1 1)) ↔\n IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1",
"state_before": "case inl\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : 1 ∈ periodicPts fun x_1 => x * x_1\n⊢ ((n > 0 ∧ IsPeriodicPt (fun x_1 => x * x_1) n 1) ∧\n ∀ (n_1 : ℕ), n_1 < n → ¬(n_1 > 0 ∧ IsPeriodicPt (fun x_1 => x * x_1) n_1 1)) ↔\n IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1",
"tactic": "simp only [h, true_and]"
},
{
"state_after": "case inl\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : 1 ∈ periodicPts fun x_1 => x * x_1\n⊢ (IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (n_1 : ℕ), n_1 < n → n_1 > 0 → ¬IsPeriodicPt (fun x_1 => x * x_1) n_1 1) ↔\n IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1",
"state_before": "case inl\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : 1 ∈ periodicPts fun x_1 => x * x_1\n⊢ (IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧\n ∀ (n_1 : ℕ), n_1 < n → ¬(n_1 > 0 ∧ IsPeriodicPt (fun x_1 => x * x_1) n_1 1)) ↔\n IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1",
"tactic": "push_neg"
},
{
"state_after": "no goals",
"state_before": "case inl\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : 1 ∈ periodicPts fun x_1 => x * x_1\n⊢ (IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (n_1 : ℕ), n_1 < n → n_1 > 0 → ¬IsPeriodicPt (fun x_1 => x * x_1) n_1 1) ↔\n IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1",
"tactic": "rfl"
},
{
"state_after": "case inr\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : ¬1 ∈ periodicPts fun x_1 => x * x_1\n⊢ ¬(IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1)",
"state_before": "case inr\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : ¬1 ∈ periodicPts fun x_1 => x * x_1\n⊢ 0 = n ↔ IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1",
"tactic": "rw [iff_false_left h.ne]"
},
{
"state_after": "case inr.intro\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : ¬1 ∈ periodicPts fun x_1 => x * x_1\nh' : IsPeriodicPt (fun x_1 => x * x_1) n 1\n⊢ False",
"state_before": "case inr\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : ¬1 ∈ periodicPts fun x_1 => x * x_1\n⊢ ¬(IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ (m : ℕ), m < n → 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1)",
"tactic": "rintro ⟨h', -⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nG : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\nh : 0 < n\nh1 : ¬1 ∈ periodicPts fun x_1 => x * x_1\nh' : IsPeriodicPt (fun x_1 => x * x_1) n 1\n⊢ False",
"tactic": "exact h1 ⟨n, h, h'⟩"
}
] |
[
195,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.ae_restrict_iff'₀
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.576111\nγ : Type ?u.576114\nδ : Type ?u.576117\nι : Type ?u.576120\nR : Type ?u.576123\nR' : Type ?u.576126\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\np : α → Prop\nhs : NullMeasurableSet s\nh : ∀ᵐ (x : α) ∂μ, x ∈ s → p x\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, p x",
"state_before": "α : Type u_1\nβ : Type ?u.576111\nγ : Type ?u.576114\nδ : Type ?u.576117\nι : Type ?u.576120\nR : Type ?u.576123\nR' : Type ?u.576126\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\np : α → Prop\nhs : NullMeasurableSet s\n⊢ (∀ᵐ (x : α) ∂Measure.restrict μ s, p x) ↔ ∀ᵐ (x : α) ∂μ, x ∈ s → p x",
"tactic": "refine' ⟨fun h => ae_imp_of_ae_restrict h, fun h => _⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.576111\nγ : Type ?u.576114\nδ : Type ?u.576117\nι : Type ?u.576120\nR : Type ?u.576123\nR' : Type ?u.576126\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\np : α → Prop\nhs : NullMeasurableSet s\nh : ∀ᵐ (x : α) ∂μ, x ∈ s → p x\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, p x",
"tactic": "filter_upwards [ae_restrict_mem₀ hs, ae_restrict_of_ae h]with x hx h'x using h'x hx"
}
] |
[
2840,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2837,
1
] |
Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean
|
Asymptotics.IsEquivalent.add_isLittleO
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v w : α → β\nl : Filter α\nhuv : u ~[l] v\nhwv : w =o[l] v\n⊢ u + w ~[l] v",
"tactic": "simpa only [IsEquivalent, add_sub_right_comm] using huv.add hwv"
}
] |
[
169,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean
|
MulChar.inv_apply'
|
[] |
[
373,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
1
] |
Std/Data/List/Lemmas.lean
|
List.filter_nil
|
[] |
[
1106,
68
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1106,
9
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.single_mem_supported
|
[] |
[
215,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Data/Nat/Choose/Basic.lean
|
Nat.choose_mul_factorial_mul_factorial
|
[
{
"state_after": "no goals",
"state_before": "x✝ : ℕ\nhk : x✝ ≤ 0\n⊢ choose 0 x✝ * x✝! * (0 - x✝)! = 0!",
"tactic": "simp [Nat.eq_zero_of_le_zero hk]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nx✝ : 0 ≤ n + 1\n⊢ choose (n + 1) 0 * 0! * (n + 1 - 0)! = (n + 1)!",
"tactic": "simp"
},
{
"state_after": "case inl\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\n⊢ choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!\n\ncase inr\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k = n + 1\n⊢ choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!",
"state_before": "n k : ℕ\nhk : succ k ≤ n + 1\n⊢ choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!",
"tactic": "cases' lt_or_eq_of_le hk with hk₁ hk₁"
},
{
"state_after": "case inl\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\n⊢ choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!",
"state_before": "case inl\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\n⊢ choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!",
"tactic": "have h : choose n k * k.succ ! * (n - k)! = (k + 1) * n ! := by\n rw [← choose_mul_factorial_mul_factorial (le_of_succ_le_succ hk)];\n simp [factorial_succ, mul_comm, mul_left_comm, mul_assoc]"
},
{
"state_after": "case inl\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh₁ : (n - k)! = (n - k) * (n - succ k)!\n⊢ choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!",
"state_before": "case inl\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\n⊢ choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!",
"tactic": "have h₁ : (n - k)! = (n - k) * (n - k.succ)! := by\n rw [← succ_sub_succ, succ_sub (le_of_lt_succ hk₁), factorial_succ]"
},
{
"state_after": "case inl\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh₁ : (n - k)! = (n - k) * (n - succ k)!\nh₂ : choose n (succ k) * (succ k)! * ((n - k) * (n - succ k)!) = (n - k) * n !\n⊢ choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!",
"state_before": "case inl\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh₁ : (n - k)! = (n - k) * (n - succ k)!\n⊢ choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!",
"tactic": "have h₂ : choose n (succ k) * k.succ ! * ((n - k) * (n - k.succ)!) = (n - k) * n ! := by\n rw [← choose_mul_factorial_mul_factorial (le_of_lt_succ hk₁)];\n simp [factorial_succ, mul_comm, mul_left_comm, mul_assoc]"
},
{
"state_after": "case inl\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh₁ : (n - k)! = (n - k) * (n - succ k)!\nh₂ : choose n (succ k) * (succ k)! * ((n - k) * (n - succ k)!) = (n - k) * n !\nh₃ : k * n ! ≤ n * n !\n⊢ choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!",
"state_before": "case inl\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh₁ : (n - k)! = (n - k) * (n - succ k)!\nh₂ : choose n (succ k) * (succ k)! * ((n - k) * (n - succ k)!) = (n - k) * n !\n⊢ choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!",
"tactic": "have h₃ : k * n ! ≤ n * n ! := Nat.mul_le_mul_right _ (le_of_succ_le_succ hk)"
},
{
"state_after": "no goals",
"state_before": "case inl\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh₁ : (n - k)! = (n - k) * (n - succ k)!\nh₂ : choose n (succ k) * (succ k)! * ((n - k) * (n - succ k)!) = (n - k) * n !\nh₃ : k * n ! ≤ n * n !\n⊢ choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!",
"tactic": "rw [choose_succ_succ, add_mul, add_mul, succ_sub_succ, h, h₁, h₂, add_mul, tsub_mul,\n factorial_succ, ← add_tsub_assoc_of_le h₃, add_assoc, ← add_mul, add_tsub_cancel_left,\n add_comm]"
},
{
"state_after": "n k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\n⊢ choose n k * (succ k)! * (n - k)! = (k + 1) * (choose n k * k ! * (n - k)!)",
"state_before": "n k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\n⊢ choose n k * (succ k)! * (n - k)! = (k + 1) * n !",
"tactic": "rw [← choose_mul_factorial_mul_factorial (le_of_succ_le_succ hk)]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\n⊢ choose n k * (succ k)! * (n - k)! = (k + 1) * (choose n k * k ! * (n - k)!)",
"tactic": "simp [factorial_succ, mul_comm, mul_left_comm, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\n⊢ (n - k)! = (n - k) * (n - succ k)!",
"tactic": "rw [← succ_sub_succ, succ_sub (le_of_lt_succ hk₁), factorial_succ]"
},
{
"state_after": "n k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh₁ : (n - k)! = (n - k) * (n - succ k)!\n⊢ choose n (succ k) * (succ k)! * ((n - k) * (n - succ k)!) = (n - k) * (choose n (succ k) * (succ k)! * (n - succ k)!)",
"state_before": "n k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh₁ : (n - k)! = (n - k) * (n - succ k)!\n⊢ choose n (succ k) * (succ k)! * ((n - k) * (n - succ k)!) = (n - k) * n !",
"tactic": "rw [← choose_mul_factorial_mul_factorial (le_of_lt_succ hk₁)]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh₁ : (n - k)! = (n - k) * (n - succ k)!\n⊢ choose n (succ k) * (succ k)! * ((n - k) * (n - succ k)!) = (n - k) * (choose n (succ k) * (succ k)! * (n - succ k)!)",
"tactic": "simp [factorial_succ, mul_comm, mul_left_comm, mul_assoc]"
},
{
"state_after": "case inr\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k = n + 1\n⊢ choose (n + 1) (n + 1) * (n + 1)! * (n + 1 - (n + 1))! = (n + 1)!",
"state_before": "case inr\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k = n + 1\n⊢ choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!",
"tactic": "rw [hk₁]"
},
{
"state_after": "no goals",
"state_before": "case inr\nn k : ℕ\nhk : succ k ≤ n + 1\nhk₁ : succ k = n + 1\n⊢ choose (n + 1) (n + 1) * (n + 1)! * (n + 1 - (n + 1))! = (n + 1)!",
"tactic": "simp [hk₁, mul_comm, choose, tsub_self]"
}
] |
[
146,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
rel_sup_add
|
[
{
"state_after": "case h.e'_1.h.e'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.378311\nδ : Type ?u.378314\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Countable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\nR : α → α → Prop\nm_iSup : ∀ (s : ℕ → β), R (m (⨆ (i : ℕ), s i)) (∑' (i : ℕ), m (s i))\ns₁ s₂ : β\n⊢ s₁ ⊔ s₂ = ⨆ (b : Bool), bif b then s₁ else s₂\n\ncase h.e'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.378311\nδ : Type ?u.378314\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Countable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\nR : α → α → Prop\nm_iSup : ∀ (s : ℕ → β), R (m (⨆ (i : ℕ), s i)) (∑' (i : ℕ), m (s i))\ns₁ s₂ : β\n⊢ m s₁ + m s₂ = ∑' (b : Bool), m (bif b then s₁ else s₂)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.378311\nδ : Type ?u.378314\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Countable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\nR : α → α → Prop\nm_iSup : ∀ (s : ℕ → β), R (m (⨆ (i : ℕ), s i)) (∑' (i : ℕ), m (s i))\ns₁ s₂ : β\n⊢ R (m (s₁ ⊔ s₂)) (m s₁ + m s₂)",
"tactic": "convert rel_iSup_tsum m m0 R m_iSup fun b => cond b s₁ s₂"
},
{
"state_after": "no goals",
"state_before": "case h.e'_1.h.e'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.378311\nδ : Type ?u.378314\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Countable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\nR : α → α → Prop\nm_iSup : ∀ (s : ℕ → β), R (m (⨆ (i : ℕ), s i)) (∑' (i : ℕ), m (s i))\ns₁ s₂ : β\n⊢ s₁ ⊔ s₂ = ⨆ (b : Bool), bif b then s₁ else s₂",
"tactic": "simp only [iSup_bool_eq, cond]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.378311\nδ : Type ?u.378314\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Countable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\nR : α → α → Prop\nm_iSup : ∀ (s : ℕ → β), R (m (⨆ (i : ℕ), s i)) (∑' (i : ℕ), m (s i))\ns₁ s₂ : β\n⊢ m s₁ + m s₂ = ∑' (b : Bool), m (bif b then s₁ else s₂)",
"tactic": "rw [tsum_fintype, Fintype.sum_bool, cond, cond]"
}
] |
[
765,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
760,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
IsLocallyConstant.isClosed_fiber
|
[] |
[
79,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/CategoryTheory/Preadditive/Basic.lean
|
CategoryTheory.Preadditive.zsmul_comp
|
[] |
[
178,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/Data/Nat/Bits.lean
|
Nat.pos_of_bit0_pos
|
[
{
"state_after": "case zero\nn : ℕ\nh : 0 < bit0 zero\n⊢ 0 < zero\n\ncase succ\nn n✝ : ℕ\nh : 0 < bit0 (succ n✝)\n⊢ 0 < succ n✝",
"state_before": "n✝ n : ℕ\nh : 0 < bit0 n\n⊢ 0 < n",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\nn : ℕ\nh : 0 < bit0 zero\n⊢ 0 < zero",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case succ\nn n✝ : ℕ\nh : 0 < bit0 (succ n✝)\n⊢ 0 < succ n✝",
"tactic": "apply succ_pos"
}
] |
[
119,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Iic_subset_Iio
|
[] |
[
432,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
431,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Iio_subset_Iic
|
[] |
[
619,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
618,
1
] |
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