file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Topology/Basic.lean
|
IsOpen.inter_closure
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t✝ : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns t : Set α\nh : IsOpen s\n⊢ closure (s ∩ t)ᶜ ⊆ (s ∩ closure t)ᶜ",
"tactic": "simpa only [← interior_compl, compl_inter] using IsClosed.interior_union_left h.isClosed_compl"
}
] |
[
1390,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1388,
1
] |
Mathlib/Data/Real/ConjugateExponents.lean
|
Real.IsConjugateExponent.conjugate_eq
|
[] |
[
80,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Analysis/Seminorm.lean
|
balanced_ball_zero
|
[
{
"state_after": "R : Type ?u.1457964\nR' : Type ?u.1457967\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1457973\n𝕜₃ : Type ?u.1457976\n𝕝 : Type ?u.1457979\nE : Type u_2\nE₂ : Type ?u.1457985\nE₃ : Type ?u.1457988\nF : Type ?u.1457991\nG : Type ?u.1457994\nι : Type ?u.1457997\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nx : E\n⊢ Balanced 𝕜 (Seminorm.ball (normSeminorm 𝕜 E) 0 r)",
"state_before": "R : Type ?u.1457964\nR' : Type ?u.1457967\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1457973\n𝕜₃ : Type ?u.1457976\n𝕝 : Type ?u.1457979\nE : Type u_2\nE₂ : Type ?u.1457985\nE₃ : Type ?u.1457988\nF : Type ?u.1457991\nG : Type ?u.1457994\nι : Type ?u.1457997\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nx : E\n⊢ Balanced 𝕜 (Metric.ball 0 r)",
"tactic": "rw [← ball_normSeminorm 𝕜]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1457964\nR' : Type ?u.1457967\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1457973\n𝕜₃ : Type ?u.1457976\n𝕝 : Type ?u.1457979\nE : Type u_2\nE₂ : Type ?u.1457985\nE₃ : Type ?u.1457988\nF : Type ?u.1457991\nG : Type ?u.1457994\nι : Type ?u.1457997\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nx : E\n⊢ Balanced 𝕜 (Seminorm.ball (normSeminorm 𝕜 E) 0 r)",
"tactic": "exact (normSeminorm _ _).balanced_ball_zero r"
}
] |
[
1227,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1225,
1
] |
Mathlib/GroupTheory/Commutator.lean
|
Subgroup.commutator_mem_commutator
|
[] |
[
89,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
|
MvPolynomial.homogeneousComponent_eq_zero'
|
[
{
"state_after": "σ : Type u_1\nτ : Type ?u.355900\nR : Type u_2\nS : Type ?u.355906\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ∑ i in d.support, ↑d i ≠ n\n⊢ ∀ (x : σ →₀ ℕ), x ∈ filter (fun d => ∑ i in d.support, ↑d i = n) (support φ) → ↑(monomial x) (coeff x φ) = 0",
"state_before": "σ : Type u_1\nτ : Type ?u.355900\nR : Type u_2\nS : Type ?u.355906\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ∑ i in d.support, ↑d i ≠ n\n⊢ ↑(homogeneousComponent n) φ = 0",
"tactic": "rw [homogeneousComponent_apply, sum_eq_zero]"
},
{
"state_after": "σ : Type u_1\nτ : Type ?u.355900\nR : Type u_2\nS : Type ?u.355906\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ∑ i in d.support, ↑d i ≠ n\nd : σ →₀ ℕ\nhd : d ∈ filter (fun d => ∑ i in d.support, ↑d i = n) (support φ)\n⊢ ↑(monomial d) (coeff d φ) = 0",
"state_before": "σ : Type u_1\nτ : Type ?u.355900\nR : Type u_2\nS : Type ?u.355906\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ∑ i in d.support, ↑d i ≠ n\n⊢ ∀ (x : σ →₀ ℕ), x ∈ filter (fun d => ∑ i in d.support, ↑d i = n) (support φ) → ↑(monomial x) (coeff x φ) = 0",
"tactic": "intro d hd"
},
{
"state_after": "σ : Type u_1\nτ : Type ?u.355900\nR : Type u_2\nS : Type ?u.355906\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ∑ i in d.support, ↑d i ≠ n\nd : σ →₀ ℕ\nhd : d ∈ support φ ∧ ∑ i in d.support, ↑d i = n\n⊢ ↑(monomial d) (coeff d φ) = 0",
"state_before": "σ : Type u_1\nτ : Type ?u.355900\nR : Type u_2\nS : Type ?u.355906\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ∑ i in d.support, ↑d i ≠ n\nd : σ →₀ ℕ\nhd : d ∈ filter (fun d => ∑ i in d.support, ↑d i = n) (support φ)\n⊢ ↑(monomial d) (coeff d φ) = 0",
"tactic": "rw [mem_filter] at hd"
},
{
"state_after": "case h\nσ : Type u_1\nτ : Type ?u.355900\nR : Type u_2\nS : Type ?u.355906\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ∑ i in d.support, ↑d i ≠ n\nd : σ →₀ ℕ\nhd : d ∈ support φ ∧ ∑ i in d.support, ↑d i = n\n⊢ False",
"state_before": "σ : Type u_1\nτ : Type ?u.355900\nR : Type u_2\nS : Type ?u.355906\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ∑ i in d.support, ↑d i ≠ n\nd : σ →₀ ℕ\nhd : d ∈ support φ ∧ ∑ i in d.support, ↑d i = n\n⊢ ↑(monomial d) (coeff d φ) = 0",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case h\nσ : Type u_1\nτ : Type ?u.355900\nR : Type u_2\nS : Type ?u.355906\ninst✝ : CommSemiring R\nn : ℕ\nφ ψ : MvPolynomial σ R\nh : ∀ (d : σ →₀ ℕ), d ∈ support φ → ∑ i in d.support, ↑d i ≠ n\nd : σ →₀ ℕ\nhd : d ∈ support φ ∧ ∑ i in d.support, ↑d i = n\n⊢ False",
"tactic": "exact h _ hd.1 hd.2"
}
] |
[
297,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.coe_sSup
|
[] |
[
934,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
933,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.coe_iSup₂
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.56049\nγ : Type ?u.56052\nι : Sort u_2\nκ : ι → Sort u_3\ninst✝ : LE α\nS : Set (UpperSet α)\ns t : UpperSet α\na : α\nf : (i : ι) → κ i → UpperSet α\n⊢ ↑(⨆ (i : ι) (j : κ i), f i j) = ⋂ (i : ι) (j : κ i), ↑(f i j)",
"tactic": "simp_rw [coe_iSup]"
}
] |
[
562,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
561,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.Integrable.measure_ge_lt_top
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.989928\nδ : Type ?u.989931\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhf : Integrable f\nε : ℝ\nhε : 0 < ε\n⊢ ↑↑μ {x | ENNReal.ofReal ε ≤ ↑‖f x‖₊} < ⊤",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.989928\nδ : Type ?u.989931\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhf : Integrable f\nε : ℝ\nhε : 0 < ε\n⊢ ↑↑μ {x | ε ≤ ‖f x‖} < ⊤",
"tactic": "rw [show { x | ε ≤ ‖f x‖ } = { x | ENNReal.ofReal ε ≤ ‖f x‖₊ } by\n simp only [ENNReal.ofReal, Real.toNNReal_le_iff_le_coe, ENNReal.coe_le_coe, coe_nnnorm]]"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.989928\nδ : Type ?u.989931\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhf : Integrable f\nε : ℝ\nhε : 0 < ε\n⊢ ENNReal.ofReal ε ≠ 0\n\ncase refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.989928\nδ : Type ?u.989931\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhf : Integrable f\nε : ℝ\nhε : 0 < ε\n⊢ (ENNReal.ofReal ε)⁻¹ ^ ENNReal.toReal 1 * snorm f 1 μ ^ ENNReal.toReal 1 < ⊤",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.989928\nδ : Type ?u.989931\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhf : Integrable f\nε : ℝ\nhε : 0 < ε\n⊢ ↑↑μ {x | ENNReal.ofReal ε ≤ ↑‖f x‖₊} < ⊤",
"tactic": "refine' (meas_ge_le_mul_pow_snorm μ one_ne_zero ENNReal.one_ne_top hf.1 _).trans_lt _"
},
{
"state_after": "case refine'_2.a\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.989928\nδ : Type ?u.989931\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhf : Integrable f\nε : ℝ\nhε : 0 < ε\n⊢ (ENNReal.ofReal ε)⁻¹ ^ ENNReal.toReal 1 ≠ ⊤\n\ncase refine'_2.a\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.989928\nδ : Type ?u.989931\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhf : Integrable f\nε : ℝ\nhε : 0 < ε\n⊢ snorm f 1 μ ^ ENNReal.toReal 1 ≠ ⊤",
"state_before": "case refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.989928\nδ : Type ?u.989931\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhf : Integrable f\nε : ℝ\nhε : 0 < ε\n⊢ (ENNReal.ofReal ε)⁻¹ ^ ENNReal.toReal 1 * snorm f 1 μ ^ ENNReal.toReal 1 < ⊤",
"tactic": "apply ENNReal.mul_lt_top"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.a\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.989928\nδ : Type ?u.989931\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhf : Integrable f\nε : ℝ\nhε : 0 < ε\n⊢ snorm f 1 μ ^ ENNReal.toReal 1 ≠ ⊤",
"tactic": "simpa only [ENNReal.one_toReal, ENNReal.rpow_one] using\n (memℒp_one_iff_integrable.2 hf).snorm_ne_top"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.989928\nδ : Type ?u.989931\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhf : Integrable f\nε : ℝ\nhε : 0 < ε\n⊢ {x | ε ≤ ‖f x‖} = {x | ENNReal.ofReal ε ≤ ↑‖f x‖₊}",
"tactic": "simp only [ENNReal.ofReal, Real.toNNReal_le_iff_le_coe, ENNReal.coe_le_coe, coe_nnnorm]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.989928\nδ : Type ?u.989931\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhf : Integrable f\nε : ℝ\nhε : 0 < ε\n⊢ ENNReal.ofReal ε ≠ 0",
"tactic": "simpa only [Ne.def, ENNReal.ofReal_eq_zero, not_le] using hε"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.a\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.989928\nδ : Type ?u.989931\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhf : Integrable f\nε : ℝ\nhε : 0 < ε\n⊢ (ENNReal.ofReal ε)⁻¹ ^ ENNReal.toReal 1 ≠ ⊤",
"tactic": "simpa only [ENNReal.one_toReal, ENNReal.rpow_one, Ne.def, ENNReal.inv_eq_top,\n ENNReal.ofReal_eq_zero, not_le] using hε"
}
] |
[
811,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.mk_add_moveLeft_inr
|
[] |
[
1533,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1530,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean
|
Real.integrable_of_summable_norm_Icc
|
[
{
"state_after": "case refine'_1\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\n⊢ ℤ → Compacts ℝ\n\ncase refine'_2\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\nn : ℤ\n⊢ ‖ContinuousMap.restrict (↑(?refine'_1 n)) f‖ * ENNReal.toReal (↑↑volume ↑(?refine'_1 n)) ≤\n ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\n\ncase refine'_3\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\n⊢ (⋃ (i : ℤ), ↑(?refine'_1 i)) = univ",
"state_before": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\n⊢ Integrable ↑f",
"tactic": "refine'\n @integrable_of_summable_norm_restrict ℝ ℤ E _ volume _ _ _ _ _ _ _ _\n (summable_of_nonneg_of_le\n (fun n : ℤ => mul_nonneg (norm_nonneg\n (f.restrict (⟨Icc (n : ℝ) ((n : ℝ) + 1), isCompact_Icc⟩ : Compacts ℝ)))\n ENNReal.toReal_nonneg)\n (fun n => _) hf) _"
},
{
"state_after": "case refine'_1\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\nn : ℤ\n⊢ Compacts ℝ",
"state_before": "case refine'_1\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\n⊢ ℤ → Compacts ℝ",
"tactic": "intro n"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\nn : ℤ\n⊢ Compacts ℝ",
"tactic": "exact ⟨Icc (n : ℝ) ((n : ℝ) + 1), isCompact_Icc⟩"
},
{
"state_after": "case refine'_2\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\nn : ℤ\n⊢ ∀ (x : ↑(Icc (↑n) (↑n + 1))),\n ‖↑(ContinuousMap.restrict (Icc (↑n) (↑n + 1)) f) x‖ ≤\n ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖",
"state_before": "case refine'_2\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\nn : ℤ\n⊢ ‖ContinuousMap.restrict (↑{ carrier := Icc (↑n) (↑n + 1), isCompact' := (_ : IsCompact (Icc (↑n) (↑n + 1))) }) f‖ *\n ENNReal.toReal (↑↑volume ↑{ carrier := Icc (↑n) (↑n + 1), isCompact' := (_ : IsCompact (Icc (↑n) (↑n + 1))) }) ≤\n ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖",
"tactic": "simp only [Compacts.coe_mk, Real.volume_Icc, add_sub_cancel', ENNReal.toReal_ofReal zero_le_one,\n mul_one, norm_le _ (norm_nonneg _)]"
},
{
"state_after": "case refine'_2\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\nn : ℤ\nx : ↑(Icc (↑n) (↑n + 1))\n⊢ ‖↑(ContinuousMap.restrict (Icc (↑n) (↑n + 1)) f) x‖ ≤\n ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖",
"state_before": "case refine'_2\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\nn : ℤ\n⊢ ∀ (x : ↑(Icc (↑n) (↑n + 1))),\n ‖↑(ContinuousMap.restrict (Icc (↑n) (↑n + 1)) f) x‖ ≤\n ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖",
"tactic": "intro x"
},
{
"state_after": "case refine'_2\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\nn : ℤ\nx : ↑(Icc (↑n) (↑n + 1))\nthis :\n ‖↑(ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n)))\n { val := ↑x - ↑n, property := (_ : 0 ≤ ↑x - ↑n ∧ ↑x - ↑n ≤ 1) }‖ ≤\n ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\n⊢ ‖↑(ContinuousMap.restrict (Icc (↑n) (↑n + 1)) f) x‖ ≤\n ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖",
"state_before": "case refine'_2\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\nn : ℤ\nx : ↑(Icc (↑n) (↑n + 1))\n⊢ ‖↑(ContinuousMap.restrict (Icc (↑n) (↑n + 1)) f) x‖ ≤\n ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖",
"tactic": "have := ((f.comp <| ContinuousMap.addRight n).restrict (Icc 0 1)).norm_coe_le_norm\n ⟨x - n, ⟨sub_nonneg.mpr x.2.1, sub_le_iff_le_add'.mpr x.2.2⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\nn : ℤ\nx : ↑(Icc (↑n) (↑n + 1))\nthis :\n ‖↑(ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n)))\n { val := ↑x - ↑n, property := (_ : 0 ≤ ↑x - ↑n ∧ ↑x - ↑n ≤ 1) }‖ ≤\n ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\n⊢ ‖↑(ContinuousMap.restrict (Icc (↑n) (↑n + 1)) f) x‖ ≤\n ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖",
"tactic": "simpa only [ContinuousMap.restrict_apply, comp_apply, coe_addRight, Subtype.coe_mk,\n sub_add_cancel] using this"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight ↑n))‖\n⊢ (⋃ (i : ℤ), ↑{ carrier := Icc (↑i) (↑i + 1), isCompact' := (_ : IsCompact (Icc (↑i) (↑i + 1))) }) = univ",
"tactic": "exact iUnion_Icc_int_cast ℝ"
}
] |
[
78,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Algebra/Ring/Regular.lean
|
isRightRegular_of_non_zero_divisor
|
[
{
"state_after": "α : Type u_1\ninst✝ : NonUnitalNonAssocRing α\nk : α\nh : ∀ (x : α), x * k = 0 → x = 0\nx y : α\nh' : x * k = y * k\n⊢ (x - y) * k = 0",
"state_before": "α : Type u_1\ninst✝ : NonUnitalNonAssocRing α\nk : α\nh : ∀ (x : α), x * k = 0 → x = 0\n⊢ IsRightRegular k",
"tactic": "refine' fun x y (h' : x * k = y * k) => sub_eq_zero.mp (h _ _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : NonUnitalNonAssocRing α\nk : α\nh : ∀ (x : α), x * k = 0 → x = 0\nx y : α\nh' : x * k = y * k\n⊢ (x - y) * k = 0",
"tactic": "rw [sub_mul, sub_eq_zero, h']"
}
] |
[
34,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
31,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean
|
map_finsupp_prod
|
[] |
[
223,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.prod_range_univ_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.55932\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nm₁ : α → γ\n⊢ ∀ (x : γ × β), x ∈ range m₁ ×ˢ univ ↔ x ∈ range fun p => (m₁ p.fst, p.snd)",
"tactic": "simp [range]"
}
] |
[
295,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.cof_succ
|
[
{
"state_after": "case a\nα : Type ?u.43015\nr : α → α → Prop\no : Ordinal\n⊢ cof (succ o) ≤ 1\n\ncase a\nα : Type ?u.43015\nr : α → α → Prop\no : Ordinal\n⊢ 1 ≤ cof (succ o)",
"state_before": "α : Type ?u.43015\nr : α → α → Prop\no : Ordinal\n⊢ cof (succ o) = 1",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ cof (succ (type r)) ≤ 1",
"state_before": "case a\nα : Type ?u.43015\nr : α → α → Prop\no : Ordinal\n⊢ cof (succ o) ≤ 1",
"tactic": "refine' inductionOn o fun α r _ => _"
},
{
"state_after": "case a\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ cof (type (Sum.Lex r EmptyRelation)) ≤ 1",
"state_before": "case a\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ cof (succ (type r)) ≤ 1",
"tactic": "change cof (type _) ≤ _"
},
{
"state_after": "case a\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ cof (type (Sum.Lex r EmptyRelation)) ≤ (#?m.43531)\n\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ Type u_1\n\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ (#?m.43531) = 1",
"state_before": "case a\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ cof (type (Sum.Lex r EmptyRelation)) ≤ 1",
"tactic": "rw [← (_ : (#_) = 1)]"
},
{
"state_after": "case a.h\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ Unbounded (Sum.Lex r EmptyRelation) ?a.S✝\n\ncase a.S\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ Set (α ⊕ PUnit)\n\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ (#↑?a.S✝) = 1",
"state_before": "case a\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ cof (type (Sum.Lex r EmptyRelation)) ≤ (#?m.43531)\n\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ Type u_1\n\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ (#?m.43531) = 1",
"tactic": "apply cof_type_le"
},
{
"state_after": "case a.h\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\na : α ⊕ PUnit\n⊢ ¬Sum.Lex r EmptyRelation (Sum.inr PUnit.unit) a",
"state_before": "case a.h\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ Unbounded (Sum.Lex r EmptyRelation) ?a.S✝",
"tactic": "refine' fun a => ⟨Sum.inr PUnit.unit, Set.mem_singleton _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case a.h\nα✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\na : α ⊕ PUnit\n⊢ ¬Sum.Lex r EmptyRelation (Sum.inr PUnit.unit) a",
"tactic": "rcases a with (a | ⟨⟨⟨⟩⟩⟩) <;> simp [EmptyRelation]"
},
{
"state_after": "α✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ ↑1 = 1",
"state_before": "α✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ (#↑{Sum.inr PUnit.unit}) = 1",
"tactic": "rw [Cardinal.mk_fintype, Set.card_singleton]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.43015\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\n⊢ ↑1 = 1",
"tactic": "simp"
},
{
"state_after": "case a\nα : Type ?u.43015\nr : α → α → Prop\no : Ordinal\n⊢ 0 < cof (succ o)",
"state_before": "case a\nα : Type ?u.43015\nr : α → α → Prop\no : Ordinal\n⊢ 1 ≤ cof (succ o)",
"tactic": "rw [← Cardinal.succ_zero, succ_le_iff]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type ?u.43015\nr : α → α → Prop\no : Ordinal\n⊢ 0 < cof (succ o)",
"tactic": "simpa [lt_iff_le_and_ne, Cardinal.zero_le] using fun h =>\n succ_ne_zero o (cof_eq_zero.1 (Eq.symm h))"
}
] |
[
513,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
501,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.deleteEdges_le
|
[
{
"state_after": "ι : Sort ?u.157022\n𝕜 : Type ?u.157025\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ns : Set (Sym2 V)\nv✝ : V\n⊢ ∀ ⦃w : V⦄, Adj (deleteEdges G s) v✝ w → Adj G v✝ w",
"state_before": "ι : Sort ?u.157022\n𝕜 : Type ?u.157025\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ns : Set (Sym2 V)\n⊢ deleteEdges G s ≤ G",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.157022\n𝕜 : Type ?u.157025\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ns : Set (Sym2 V)\nv✝ : V\n⊢ ∀ ⦃w : V⦄, Adj (deleteEdges G s) v✝ w → Adj G v✝ w",
"tactic": "simp (config := { contextual := true })"
}
] |
[
1152,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1150,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
mem_skewAdjointMatricesSubmodule
|
[
{
"state_after": "R : Type u_1\nR₁ : Type ?u.2628894\nR₂ : Type ?u.2628897\nM : Type ?u.2628900\nM₁ : Type ?u.2628903\nM₂ : Type ?u.2628906\nM₁' : Type ?u.2628909\nM₂' : Type ?u.2628912\nn : Type u_2\nm : Type ?u.2628918\nn' : Type ?u.2628921\nm' : Type ?u.2628924\nι : Type ?u.2628927\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\n⊢ Matrix.IsAdjointPair (-J) J A₁ A₁ ↔ Matrix.IsSkewAdjoint J A₁",
"state_before": "R : Type u_1\nR₁ : Type ?u.2628894\nR₂ : Type ?u.2628897\nM : Type ?u.2628900\nM₁ : Type ?u.2628903\nM₂ : Type ?u.2628906\nM₁' : Type ?u.2628909\nM₂' : Type ?u.2628912\nn : Type u_2\nm : Type ?u.2628918\nn' : Type ?u.2628921\nm' : Type ?u.2628924\nι : Type ?u.2628927\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\n⊢ A₁ ∈ skewAdjointMatricesSubmodule J ↔ Matrix.IsSkewAdjoint J A₁",
"tactic": "erw [mem_pairSelfAdjointMatricesSubmodule]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.2628894\nR₂ : Type ?u.2628897\nM : Type ?u.2628900\nM₁ : Type ?u.2628903\nM₂ : Type ?u.2628906\nM₁' : Type ?u.2628909\nM₂' : Type ?u.2628912\nn : Type u_2\nm : Type ?u.2628918\nn' : Type ?u.2628921\nm' : Type ?u.2628924\nι : Type ?u.2628927\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\n⊢ Matrix.IsAdjointPair (-J) J A₁ A₁ ↔ Matrix.IsSkewAdjoint J A₁",
"tactic": "simp [Matrix.IsSkewAdjoint, Matrix.IsAdjointPair]"
}
] |
[
664,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
661,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.dotProduct_mulVec
|
[
{
"state_after": "l : Type ?u.840566\nm : Type u_2\nn : Type u_1\no : Type ?u.840575\nm' : o → Type ?u.840580\nn' : o → Type ?u.840585\nR : Type u_3\nS : Type ?u.840591\nα : Type v\nβ : Type w\nγ : Type ?u.840598\ninst✝³ : NonUnitalNonAssocSemiring α\ninst✝² : Fintype n\ninst✝¹ : Fintype m\ninst✝ : NonUnitalSemiring R\nv : m → R\nA : Matrix m n R\nw : n → R\n⊢ ∑ x : m, ∑ x_1 : n, v x * (A x x_1 * w x_1) = ∑ x : n, ∑ x_1 : m, v x_1 * (A x_1 x * w x)",
"state_before": "l : Type ?u.840566\nm : Type u_2\nn : Type u_1\no : Type ?u.840575\nm' : o → Type ?u.840580\nn' : o → Type ?u.840585\nR : Type u_3\nS : Type ?u.840591\nα : Type v\nβ : Type w\nγ : Type ?u.840598\ninst✝³ : NonUnitalNonAssocSemiring α\ninst✝² : Fintype n\ninst✝¹ : Fintype m\ninst✝ : NonUnitalSemiring R\nv : m → R\nA : Matrix m n R\nw : n → R\n⊢ v ⬝ᵥ mulVec A w = vecMul v A ⬝ᵥ w",
"tactic": "simp only [dotProduct, vecMul, mulVec, Finset.mul_sum, Finset.sum_mul, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.840566\nm : Type u_2\nn : Type u_1\no : Type ?u.840575\nm' : o → Type ?u.840580\nn' : o → Type ?u.840585\nR : Type u_3\nS : Type ?u.840591\nα : Type v\nβ : Type w\nγ : Type ?u.840598\ninst✝³ : NonUnitalNonAssocSemiring α\ninst✝² : Fintype n\ninst✝¹ : Fintype m\ninst✝ : NonUnitalSemiring R\nv : m → R\nA : Matrix m n R\nw : n → R\n⊢ ∑ x : m, ∑ x_1 : n, v x * (A x x_1 * w x_1) = ∑ x : n, ∑ x_1 : m, v x_1 * (A x_1 x * w x)",
"tactic": "exact Finset.sum_comm"
}
] |
[
1715,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1712,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
IntervalIntegrable.comp_sub_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.7559135\n𝕜 : Type ?u.7559138\nE : Type u_1\nF : Type ?u.7559144\nA : Type ?u.7559147\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntervalIntegrable f volume a b\nc : ℝ\n⊢ IntervalIntegrable (fun x => f (x - c)) volume (a + c) (b + c)",
"tactic": "simpa only [sub_neg_eq_add] using IntervalIntegrable.comp_add_right hf (-c)"
}
] |
[
338,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Algebra/Order/Pointwise.lean
|
csSup_mul
|
[] |
[
150,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean
|
SimpleGraph.IsSRGWith.top
|
[
{
"state_after": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nv w : V\nh : Adj ⊤ v w\n⊢ v ≠ w",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nv w : V\nh : Adj ⊤ v w\n⊢ Fintype.card ↑(commonNeighbors ⊤ v w) = Fintype.card V - 2",
"tactic": "rw [card_commonNeighbors_top]"
},
{
"state_after": "no goals",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nv w : V\nh : Adj ⊤ v w\n⊢ v ≠ w",
"tactic": "exact h"
}
] |
[
80,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.PullbackCone.equalizer_ext
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nW✝ X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nt : PullbackCone f g\nW : C\nk l : W ⟶ t.pt\nh₀ : k ≫ fst t = l ≫ fst t\nh₁ : k ≫ snd t = l ≫ snd t\n⊢ k ≫ t.π.app WalkingCospan.left ≫ (cospan f g).map inl = l ≫ t.π.app WalkingCospan.left ≫ (cospan f g).map inl",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nW✝ X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nt : PullbackCone f g\nW : C\nk l : W ⟶ t.pt\nh₀ : k ≫ fst t = l ≫ fst t\nh₁ : k ≫ snd t = l ≫ snd t\n⊢ k ≫ t.π.app none = l ≫ t.π.app none",
"tactic": "rw [← t.w inl]"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nW✝ X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nt : PullbackCone f g\nW : C\nk l : W ⟶ t.pt\nh₀ : k ≫ fst t = l ≫ fst t\nh₁ : k ≫ snd t = l ≫ snd t\n⊢ k ≫ fst t ≫ f = l ≫ fst t ≫ f",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nW✝ X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nt : PullbackCone f g\nW : C\nk l : W ⟶ t.pt\nh₀ : k ≫ fst t = l ≫ fst t\nh₁ : k ≫ snd t = l ≫ snd t\n⊢ k ≫ t.π.app WalkingCospan.left ≫ (cospan f g).map inl = l ≫ t.π.app WalkingCospan.left ≫ (cospan f g).map inl",
"tactic": "dsimp [h₀]"
}
] |
[
620,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
616,
1
] |
Mathlib/Algebra/Group/Defs.lean
|
mul_inv_self
|
[] |
[
1101,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1100,
1
] |
Mathlib/RingTheory/DedekindDomain/Dvr.lean
|
IsLocalization.isDedekindDomain
|
[
{
"state_after": "R : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\n⊢ IsDedekindDomain Aₘ",
"state_before": "R : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\n⊢ IsDedekindDomain Aₘ",
"tactic": "have h : ∀ y : M, IsUnit (algebraMap A (FractionRing A) y) := by\n rintro ⟨y, hy⟩\n exact IsUnit.mk0 _ (mt IsFractionRing.to_map_eq_zero_iff.mp (nonZeroDivisors.ne_zero (hM hy)))"
},
{
"state_after": "R : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\n⊢ IsDedekindDomain Aₘ",
"state_before": "R : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\n⊢ IsDedekindDomain Aₘ",
"tactic": "letI : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (IsLocalization.lift h)"
},
{
"state_after": "R : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis : IsScalarTower A Aₘ (FractionRing A)\n⊢ IsDedekindDomain Aₘ",
"state_before": "R : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\n⊢ IsDedekindDomain Aₘ",
"tactic": "haveI : IsScalarTower A Aₘ (FractionRing A) :=\n IsScalarTower.of_algebraMap_eq fun x => (IsLocalization.lift_eq h x).symm"
},
{
"state_after": "R : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\n⊢ IsDedekindDomain Aₘ",
"state_before": "R : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis : IsScalarTower A Aₘ (FractionRing A)\n⊢ IsDedekindDomain Aₘ",
"tactic": "haveI : IsFractionRing Aₘ (FractionRing A) :=\n IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M _ _"
},
{
"state_after": "case refine'_1\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\n⊢ IsNoetherianRing Aₘ\n\ncase refine'_2\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\n⊢ Ring.DimensionLEOne Aₘ\n\ncase refine'_3\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\n⊢ ∀ {x : FractionRing A}, IsIntegral Aₘ x → ∃ y, ↑(algebraMap Aₘ (FractionRing A)) y = x",
"state_before": "R : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\n⊢ IsDedekindDomain Aₘ",
"tactic": "refine' (isDedekindDomain_iff _ (FractionRing A)).mpr ⟨_, _, _⟩"
},
{
"state_after": "case mk\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\ny : A\nhy : y ∈ M\n⊢ IsUnit (↑(algebraMap A (FractionRing A)) ↑{ val := y, property := hy })",
"state_before": "R : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\n⊢ ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)",
"tactic": "rintro ⟨y, hy⟩"
},
{
"state_after": "no goals",
"state_before": "case mk\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\ny : A\nhy : y ∈ M\n⊢ IsUnit (↑(algebraMap A (FractionRing A)) ↑{ val := y, property := hy })",
"tactic": "exact IsUnit.mk0 _ (mt IsFractionRing.to_map_eq_zero_iff.mp (nonZeroDivisors.ne_zero (hM hy)))"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\n⊢ IsNoetherianRing Aₘ",
"tactic": "exact IsLocalization.isNoetherianRing M _ (by infer_instance)"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\n⊢ IsNoetherianRing A",
"tactic": "infer_instance"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\n⊢ Ring.DimensionLEOne Aₘ",
"tactic": "exact IsDedekindDomain.dimensionLEOne.localization Aₘ hM"
},
{
"state_after": "case refine'_3\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\nx : FractionRing A\nhx : IsIntegral Aₘ x\n⊢ ∃ y, ↑(algebraMap Aₘ (FractionRing A)) y = x",
"state_before": "case refine'_3\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\n⊢ ∀ {x : FractionRing A}, IsIntegral Aₘ x → ∃ y, ↑(algebraMap Aₘ (FractionRing A)) y = x",
"tactic": "intro x hx"
},
{
"state_after": "case refine'_3.intro.mk\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\nx : FractionRing A\nhx : IsIntegral Aₘ x\ny : A\ny_mem : y ∈ M\nhy : IsIntegral A ({ val := y, property := y_mem } • x)\n⊢ ∃ y, ↑(algebraMap Aₘ (FractionRing A)) y = x",
"state_before": "case refine'_3\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\nx : FractionRing A\nhx : IsIntegral Aₘ x\n⊢ ∃ y, ↑(algebraMap Aₘ (FractionRing A)) y = x",
"tactic": "obtain ⟨⟨y, y_mem⟩, hy⟩ := hx.exists_multiple_integral_of_isLocalization M _"
},
{
"state_after": "case refine'_3.intro.mk.intro\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\nx : FractionRing A\nhx : IsIntegral Aₘ x\ny : A\ny_mem : y ∈ M\nhy : IsIntegral A ({ val := y, property := y_mem } • x)\nz : A\nhz : ↑(algebraMap A (FractionRing A)) z = { val := y, property := y_mem } • x\n⊢ ∃ y, ↑(algebraMap Aₘ (FractionRing A)) y = x",
"state_before": "case refine'_3.intro.mk\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\nx : FractionRing A\nhx : IsIntegral Aₘ x\ny : A\ny_mem : y ∈ M\nhy : IsIntegral A ({ val := y, property := y_mem } • x)\n⊢ ∃ y, ↑(algebraMap Aₘ (FractionRing A)) y = x",
"tactic": "obtain ⟨z, hz⟩ := (isIntegrallyClosed_iff _).mp IsDedekindDomain.isIntegrallyClosed hy"
},
{
"state_after": "case refine'_3.intro.mk.intro\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\nx : FractionRing A\nhx : IsIntegral Aₘ x\ny : A\ny_mem : y ∈ M\nhy : IsIntegral A ({ val := y, property := y_mem } • x)\nz : A\nhz : ↑(algebraMap A (FractionRing A)) z = { val := y, property := y_mem } • x\n⊢ ↑(algebraMap A (FractionRing A)) z = ↑(algebraMap A (FractionRing A)) ↑{ val := y, property := y_mem } * x",
"state_before": "case refine'_3.intro.mk.intro\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\nx : FractionRing A\nhx : IsIntegral Aₘ x\ny : A\ny_mem : y ∈ M\nhy : IsIntegral A ({ val := y, property := y_mem } • x)\nz : A\nhz : ↑(algebraMap A (FractionRing A)) z = { val := y, property := y_mem } • x\n⊢ ∃ y, ↑(algebraMap Aₘ (FractionRing A)) y = x",
"tactic": "refine' ⟨IsLocalization.mk' Aₘ z ⟨y, y_mem⟩, (IsLocalization.lift_mk'_spec _ _ _ _).mpr _⟩"
},
{
"state_after": "case refine'_3.intro.mk.intro\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\nx : FractionRing A\nhx : IsIntegral Aₘ x\ny : A\ny_mem : y ∈ M\nhy : IsIntegral A ({ val := y, property := y_mem } • x)\nz : A\nhz : ↑(algebraMap A (FractionRing A)) z = { val := y, property := y_mem } • x\n⊢ { val := y, property := y_mem } • x = ↑{ val := y, property := y_mem } • x",
"state_before": "case refine'_3.intro.mk.intro\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\nx : FractionRing A\nhx : IsIntegral Aₘ x\ny : A\ny_mem : y ∈ M\nhy : IsIntegral A ({ val := y, property := y_mem } • x)\nz : A\nhz : ↑(algebraMap A (FractionRing A)) z = { val := y, property := y_mem } • x\n⊢ ↑(algebraMap A (FractionRing A)) z = ↑(algebraMap A (FractionRing A)) ↑{ val := y, property := y_mem } * x",
"tactic": "rw [hz, ← Algebra.smul_def]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3.intro.mk.intro\nR : Type ?u.6256\nA : Type u_1\nK : Type ?u.6262\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\ninst✝⁵ : Field K\ninst✝⁴ : IsDedekindDomain A\nM : Submonoid A\nhM : M ≤ A⁰\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization M Aₘ\nh : ∀ (y : { x // x ∈ M }), IsUnit (↑(algebraMap A (FractionRing A)) ↑y)\nthis✝¹ : Algebra Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)\nthis✝ : IsScalarTower A Aₘ (FractionRing A)\nthis : IsFractionRing Aₘ (FractionRing A)\nx : FractionRing A\nhx : IsIntegral Aₘ x\ny : A\ny_mem : y ∈ M\nhy : IsIntegral A ({ val := y, property := y_mem } • x)\nz : A\nhz : ↑(algebraMap A (FractionRing A)) z = { val := y, property := y_mem } • x\n⊢ { val := y, property := y_mem } • x = ↑{ val := y, property := y_mem } • x",
"tactic": "rfl"
}
] |
[
110,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalSubsemiring.mem_prod
|
[] |
[
784,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
782,
1
] |
Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean
|
CategoryTheory.Functor.map_sum
|
[] |
[
106,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.inter_prod
|
[
{
"state_after": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.12527\nδ : Type ?u.12530\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\n⊢ (x, y) ∈ (s₁ ∩ s₂) ×ˢ t ↔ (x, y) ∈ s₁ ×ˢ t ∩ s₂ ×ˢ t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.12527\nδ : Type ?u.12530\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\n⊢ (s₁ ∩ s₂) ×ˢ t = s₁ ×ˢ t ∩ s₂ ×ˢ t",
"tactic": "ext ⟨x, y⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.12527\nδ : Type ?u.12530\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\n⊢ (x, y) ∈ (s₁ ∩ s₂) ×ˢ t ↔ (x, y) ∈ s₁ ×ˢ t ∩ s₂ ×ˢ t",
"tactic": "simp only [← and_and_right, mem_inter_iff, mem_prod]"
}
] |
[
163,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
LowerSemicontinuousAt.lowerSemicontinuousWithinAt
|
[] |
[
146,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/CategoryTheory/MorphismProperty.lean
|
CategoryTheory.MorphismProperty.StableUnderComposition.universally
|
[
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type ?u.85650\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : MorphismProperty.universally P f\nhg : MorphismProperty.universally P g\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH : IsPullback f' i₁ i₂ (f ≫ g)\n⊢ P f'",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.85650\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\n⊢ StableUnderComposition (MorphismProperty.universally P)",
"tactic": "intro X Y Z f g hf hg X' Z' i₁ i₂ f' H"
},
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type ?u.85650\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : MorphismProperty.universally P f\nhg : MorphismProperty.universally P g\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH : IsPullback f' i₁ i₂ (f ≫ g)\nthis : pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g) ≫ pullback.fst = f'\n⊢ P f'",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.85650\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : MorphismProperty.universally P f\nhg : MorphismProperty.universally P g\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH : IsPullback f' i₁ i₂ (f ≫ g)\n⊢ P f'",
"tactic": "have := pullback.lift_fst _ _ (H.w.trans (Category.assoc _ _ _).symm)"
},
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type ?u.85650\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : MorphismProperty.universally P f\nhg : MorphismProperty.universally P g\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH✝ : IsPullback f' i₁ i₂ (f ≫ g)\nH : IsPullback (pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g) ≫ pullback.fst) i₁ i₂ (f ≫ g)\nthis : pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g) ≫ pullback.fst = f'\n⊢ P (pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g) ≫ pullback.fst)",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.85650\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : MorphismProperty.universally P f\nhg : MorphismProperty.universally P g\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH : IsPullback f' i₁ i₂ (f ≫ g)\nthis : pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g) ≫ pullback.fst = f'\n⊢ P f'",
"tactic": "rw [← this] at H⊢"
},
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type ?u.85650\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : MorphismProperty.universally P f\nhg : MorphismProperty.universally P g\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH✝ : IsPullback f' i₁ i₂ (f ≫ g)\nH : IsPullback (pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g) ≫ pullback.fst) i₁ i₂ (f ≫ g)\nthis : pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g) ≫ pullback.fst = f'\n⊢ P (pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g))",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.85650\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : MorphismProperty.universally P f\nhg : MorphismProperty.universally P g\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH✝ : IsPullback f' i₁ i₂ (f ≫ g)\nH : IsPullback (pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g) ≫ pullback.fst) i₁ i₂ (f ≫ g)\nthis : pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g) ≫ pullback.fst = f'\n⊢ P (pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g) ≫ pullback.fst)",
"tactic": "apply hP _ _ _ (hg _ _ _ <| IsPullback.of_hasPullback _ _)"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.85650\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : MorphismProperty.universally P f\nhg : MorphismProperty.universally P g\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH✝ : IsPullback f' i₁ i₂ (f ≫ g)\nH : IsPullback (pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g) ≫ pullback.fst) i₁ i₂ (f ≫ g)\nthis : pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g) ≫ pullback.fst = f'\n⊢ P (pullback.lift f' (i₁ ≫ f) (_ : f' ≫ i₂ = (i₁ ≫ f) ≫ g))",
"tactic": "exact hf _ _ _ (H.of_right (pullback.lift_snd _ _ _) (IsPullback.of_hasPullback i₂ g))"
}
] |
[
593,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
587,
1
] |
Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
|
GeneralizedContinuedFraction.IntFractPair.stream_zero
|
[] |
[
64,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Std/Data/List/Lemmas.lean
|
List.filterMap_cons_none
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → Option β\na : α\nl : List α\nh : f a = none\n⊢ filterMap f (a :: l) = filterMap f l",
"tactic": "simp only [filterMap, h]"
}
] |
[
1157,
72
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1156,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.sup_inf_sup
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.182498\nα : Type u_3\nβ : Type ?u.182504\nγ : Type ?u.182507\nι : Type u_1\nκ : Type u_2\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns✝ : Finset ι\nt✝ : Finset κ\nf✝ : ι → α\ng✝ : κ → α\na : α\ns : Finset ι\nt : Finset κ\nf : ι → α\ng : κ → α\n⊢ sup s f ⊓ sup t g = sup (s ×ˢ t) fun i => f i.fst ⊓ g i.snd",
"tactic": "simp_rw [Finset.sup_inf_distrib_right, Finset.sup_inf_distrib_left, sup_product_left]"
}
] |
[
537,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoMod_eq_fract_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\np : α\nhp : 0 < p\nb : α\n⊢ toIcoMod hp 0 b = Int.fract (b / p) * p",
"tactic": "simp [toIcoMod_eq_add_fract_mul]"
}
] |
[
1026,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1025,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
measurable_coe_ennreal_ereal
|
[] |
[
2012,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2011,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
reflection_mem_subspace_orthogonal_precomplement_eq_neg
|
[] |
[
977,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
975,
1
] |
Mathlib/Analysis/Normed/Ring/Seminorm.lean
|
RingSeminorm.ext
|
[] |
[
109,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.sin_ofReal_re
|
[] |
[
947,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
946,
1
] |
Mathlib/RingTheory/Localization/AtPrime.lean
|
IsLocalization.AtPrime.comap_maximalIdeal
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁶ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra R S\nP : Type ?u.70150\ninst✝³ : CommSemiring P\nA : Type ?u.70156\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\nI : Ideal R\nhI : Ideal.IsPrime I\ninst✝ : IsLocalization.AtPrime S I\nh : optParam (LocalRing S) (_ : LocalRing S)\nx : R\n⊢ x ∈ Ideal.comap (algebraMap R S) (LocalRing.maximalIdeal S) ↔ x ∈ I",
"tactic": "simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x"
}
] |
[
157,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
Subalgebra.eq_bot_of_rank_le_one
|
[
{
"state_after": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\n⊢ S = ⊥",
"state_before": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n⊢ S = ⊥",
"tactic": "nontriviality E"
},
{
"state_after": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\n⊢ S = ⊥",
"state_before": "K : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\n⊢ S = ⊥",
"tactic": "obtain ⟨m, _, he⟩ := Cardinal.exists_nat_eq_of_le_nat (h.trans_eq Nat.cast_one.symm)"
},
{
"state_after": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\nthis : FiniteDimensional F { x // x ∈ S }\n⊢ S = ⊥",
"state_before": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\n⊢ S = ⊥",
"tactic": "haveI : FiniteDimensional F S := finiteDimensional_of_rank_eq_nat he"
},
{
"state_after": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\nthis : FiniteDimensional F { x // x ∈ S }\n⊢ ¬↑toSubmodule ⊥ < ↑toSubmodule S",
"state_before": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\nthis : FiniteDimensional F { x // x ∈ S }\n⊢ S = ⊥",
"tactic": "rw [← not_bot_lt_iff, ← Subalgebra.toSubmodule.lt_iff_lt]"
},
{
"state_after": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\nthis✝ : FiniteDimensional F { x // x ∈ S }\nthis : FiniteDimensional F { x // x ∈ ↑toSubmodule S }\n⊢ ¬↑toSubmodule ⊥ < ↑toSubmodule S",
"state_before": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\nthis : FiniteDimensional F { x // x ∈ S }\n⊢ ¬↑toSubmodule ⊥ < ↑toSubmodule S",
"tactic": "haveI : FiniteDimensional F (Subalgebra.toSubmodule S) :=\n S.toSubmoduleEquiv.symm.finiteDimensional"
},
{
"state_after": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\nthis✝ : FiniteDimensional F { x // x ∈ S }\nthis : FiniteDimensional F { x // x ∈ ↑toSubmodule S }\nhl : ↑toSubmodule ⊥ < ↑toSubmodule S\n⊢ ↑(finrank F { x // x ∈ ↑toSubmodule S }) ≤ ↑(finrank F { x // x ∈ ↑toSubmodule ⊥ })",
"state_before": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\nthis✝ : FiniteDimensional F { x // x ∈ S }\nthis : FiniteDimensional F { x // x ∈ ↑toSubmodule S }\n⊢ ¬↑toSubmodule ⊥ < ↑toSubmodule S",
"tactic": "refine fun hl => (Submodule.finrank_lt_finrank_of_lt hl).not_le (natCast_le.1 ?_)"
},
{
"state_after": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\nthis✝ : FiniteDimensional F { x // x ∈ S }\nthis : FiniteDimensional F { x // x ∈ ↑toSubmodule S }\nhl : ↑toSubmodule ⊥ < ↑toSubmodule S\n⊢ Module.rank F { x // x ∈ S } ≤ Module.rank F { x // x ∈ ⊥ }",
"state_before": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\nthis✝ : FiniteDimensional F { x // x ∈ S }\nthis : FiniteDimensional F { x // x ∈ ↑toSubmodule S }\nhl : ↑toSubmodule ⊥ < ↑toSubmodule S\n⊢ ↑(finrank F { x // x ∈ ↑toSubmodule S }) ≤ ↑(finrank F { x // x ∈ ↑toSubmodule ⊥ })",
"tactic": "iterate 2 rw [Subalgebra.finrank_toSubmodule, finrank_eq_rank]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\nthis✝ : FiniteDimensional F { x // x ∈ S }\nthis : FiniteDimensional F { x // x ∈ ↑toSubmodule S }\nhl : ↑toSubmodule ⊥ < ↑toSubmodule S\n⊢ Module.rank F { x // x ∈ S } ≤ Module.rank F { x // x ∈ ⊥ }",
"tactic": "exact h.trans_eq Subalgebra.rank_bot.symm"
},
{
"state_after": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\nthis✝ : FiniteDimensional F { x // x ∈ S }\nthis : FiniteDimensional F { x // x ∈ ↑toSubmodule S }\nhl : ↑toSubmodule ⊥ < ↑toSubmodule S\n⊢ Module.rank F { x // x ∈ S } ≤ Module.rank F { x // x ∈ ⊥ }",
"state_before": "case intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Ring E\ninst✝ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x ∈ S } ≤ 1\n✝ : Nontrivial E\nm : ℕ\nleft✝ : m ≤ 1\nhe : Module.rank F { x // x ∈ S } = ↑m\nthis✝ : FiniteDimensional F { x // x ∈ S }\nthis : FiniteDimensional F { x // x ∈ ↑toSubmodule S }\nhl : ↑toSubmodule ⊥ < ↑toSubmodule S\n⊢ Module.rank F { x // x ∈ S } ≤ ↑(finrank F { x // x ∈ ↑toSubmodule ⊥ })",
"tactic": "rw [Subalgebra.finrank_toSubmodule, finrank_eq_rank]"
}
] |
[
1397,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1385,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.CompactOpens.coe_sdiff
|
[] |
[
564,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
563,
1
] |
Mathlib/Tactic/NormNum/Basic.lean
|
Mathlib.Meta.NormNum.isInt_pow
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Ring α\nn✝¹ : ℤ\nn✝ : ℕ\n⊢ ↑n✝¹ ^ ↑n✝ = ↑(Int.pow n✝¹ n✝)",
"tactic": "simp"
}
] |
[
432,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/Data/Polynomial/Mirror.lean
|
Polynomial.mirror_C
|
[] |
[
60,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.mul_bind
|
[] |
[
1071,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1070,
1
] |
Mathlib/CategoryTheory/Limits/IsLimit.lean
|
CategoryTheory.Limits.IsColimit.existsUnique
|
[] |
[
618,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
616,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
HasCompactSupport.iteratedFDeriv
|
[
{
"state_after": "case zero\n𝕜 : 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\nhf : HasCompactSupport f\n⊢ HasCompactSupport (_root_.iteratedFDeriv 𝕜 Nat.zero f)\n\ncase succ\n𝕜 : 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\nhf : HasCompactSupport f\nn : ℕ\nIH : HasCompactSupport (_root_.iteratedFDeriv 𝕜 n f)\n⊢ HasCompactSupport (_root_.iteratedFDeriv 𝕜 (Nat.succ n) f)",
"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\nhf : HasCompactSupport f\nn : ℕ\n⊢ HasCompactSupport (_root_.iteratedFDeriv 𝕜 n f)",
"tactic": "induction' n with n IH"
},
{
"state_after": "case zero\n𝕜 : 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\nhf : HasCompactSupport f\n⊢ HasCompactSupport (↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) ∘ f)",
"state_before": "case zero\n𝕜 : 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\nhf : HasCompactSupport f\n⊢ HasCompactSupport (_root_.iteratedFDeriv 𝕜 Nat.zero f)",
"tactic": "rw [iteratedFDeriv_zero_eq_comp]"
},
{
"state_after": "case zero\n𝕜 : 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\nhf : HasCompactSupport f\n⊢ ↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) 0 = 0",
"state_before": "case zero\n𝕜 : 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\nhf : HasCompactSupport f\n⊢ HasCompactSupport (↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) ∘ f)",
"tactic": "apply hf.comp_left"
},
{
"state_after": "no goals",
"state_before": "case zero\n𝕜 : 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\nhf : HasCompactSupport f\n⊢ ↑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) 0 = 0",
"tactic": "exact LinearIsometryEquiv.map_zero _"
},
{
"state_after": "case succ\n𝕜 : 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\nhf : HasCompactSupport f\nn : ℕ\nIH : HasCompactSupport (_root_.iteratedFDeriv 𝕜 n f)\n⊢ HasCompactSupport (↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ fderiv 𝕜 (_root_.iteratedFDeriv 𝕜 n f))",
"state_before": "case succ\n𝕜 : 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\nhf : HasCompactSupport f\nn : ℕ\nIH : HasCompactSupport (_root_.iteratedFDeriv 𝕜 n f)\n⊢ HasCompactSupport (_root_.iteratedFDeriv 𝕜 (Nat.succ n) f)",
"tactic": "rw [iteratedFDeriv_succ_eq_comp_left]"
},
{
"state_after": "case succ\n𝕜 : 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\nhf : HasCompactSupport f\nn : ℕ\nIH : HasCompactSupport (_root_.iteratedFDeriv 𝕜 n f)\n⊢ ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) 0 = 0",
"state_before": "case succ\n𝕜 : 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\nhf : HasCompactSupport f\nn : ℕ\nIH : HasCompactSupport (_root_.iteratedFDeriv 𝕜 n f)\n⊢ HasCompactSupport (↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ fderiv 𝕜 (_root_.iteratedFDeriv 𝕜 n f))",
"tactic": "apply (IH.fderiv 𝕜).comp_left"
},
{
"state_after": "no goals",
"state_before": "case succ\n𝕜 : 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\nhf : HasCompactSupport f\nn : ℕ\nIH : HasCompactSupport (_root_.iteratedFDeriv 𝕜 n f)\n⊢ ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) 0 = 0",
"tactic": "exact LinearIsometryEquiv.map_zero _"
}
] |
[
1564,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1556,
1
] |
Mathlib/CategoryTheory/Opposites.lean
|
CategoryTheory.NatTrans.removeOp_id
|
[] |
[
356,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.eq_zero_or_pos
|
[] |
[
290,
31
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
288,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
lowerClosure_min
|
[] |
[
1304,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1303,
1
] |
Mathlib/Tactic/NormNum/Basic.lean
|
Mathlib.Meta.NormNum.isInt_lt_true
|
[] |
[
700,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
698,
1
] |
Mathlib/RingTheory/TensorProduct.lean
|
TensorProduct.AlgebraTensorModule.smul_eq_lsmul_rTensor
|
[] |
[
82,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Analysis/NormedSpace/Banach.lean
|
ContinuousLinearMap.NonlinearRightInverse.right_inv
|
[] |
[
49,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Analysis/Convex/Side.lean
|
AffineSubspace.not_wOppSide_bot
|
[] |
[
234,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.comap_id
|
[] |
[
1639,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1638,
1
] |
Mathlib/Algebra/Order/Hom/Monoid.lean
|
OrderMonoidWithZeroHom.comp_id
|
[] |
[
708,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
708,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
IsLeast.isGreatest_image2_of_isGreatest
|
[] |
[
1542,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1539,
1
] |
Mathlib/RingTheory/Subsemiring/Pointwise.lean
|
Subsemiring.pointwise_smul_le_iff₀
|
[] |
[
169,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Std/Logic.lean
|
exists_false
|
[] |
[
430,
69
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
430,
9
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.Prime.eq_two_or_odd
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Prime p\nh : p % 2 = 0\n⊢ ¬2 = 1",
"tactic": "decide"
}
] |
[
497,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
LinearMap.dualMap_apply'
|
[] |
[
206,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisConnection.isGLB_u_image
|
[] |
[
139,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
mul_lt_of_le_one_of_lt_of_nonneg
|
[] |
[
831,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
829,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
ContDiffWithinAt.cosh
|
[] |
[
1117,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1115,
1
] |
Mathlib/RingTheory/WittVector/StructurePolynomial.lean
|
wittStructureInt_vars
|
[
{
"state_after": "p : ℕ\nR : Type ?u.2119290\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℤ\nn : ℕ\nthis : Function.Injective ↑(Int.castRingHom ℚ)\n⊢ vars (wittStructureInt p Φ n) ⊆ Finset.univ ×ˢ Finset.range (n + 1)",
"state_before": "p : ℕ\nR : Type ?u.2119290\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℤ\nn : ℕ\n⊢ vars (wittStructureInt p Φ n) ⊆ Finset.univ ×ˢ Finset.range (n + 1)",
"tactic": "have : Function.Injective (Int.castRingHom ℚ) := Int.cast_injective"
},
{
"state_after": "p : ℕ\nR : Type ?u.2119290\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℤ\nn : ℕ\nthis : Function.Injective ↑(Int.castRingHom ℚ)\n⊢ vars (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) n) ⊆ Finset.univ ×ˢ Finset.range (n + 1)",
"state_before": "p : ℕ\nR : Type ?u.2119290\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℤ\nn : ℕ\nthis : Function.Injective ↑(Int.castRingHom ℚ)\n⊢ vars (wittStructureInt p Φ n) ⊆ Finset.univ ×ˢ Finset.range (n + 1)",
"tactic": "rw [← vars_map_of_injective _ this, map_wittStructureInt]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type ?u.2119290\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℤ\nn : ℕ\nthis : Function.Injective ↑(Int.castRingHom ℚ)\n⊢ vars (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) n) ⊆ Finset.univ ×ˢ Finset.range (n + 1)",
"tactic": "apply wittStructureRat_vars"
}
] |
[
414,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.le_inter_iff
|
[] |
[
1819,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1818,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.lt_or_equiv_or_gf
|
[
{
"state_after": "x y : PGame\n⊢ x < y ∨ (x ≈ y) ∨ y < x ∨ x ‖ y",
"state_before": "x y : PGame\n⊢ x < y ∨ (x ≈ y) ∨ y ⧏ x",
"tactic": "rw [lf_iff_lt_or_fuzzy, Fuzzy.swap_iff]"
},
{
"state_after": "no goals",
"state_before": "x y : PGame\n⊢ x < y ∨ (x ≈ y) ∨ y < x ∨ x ‖ y",
"tactic": "exact lt_or_equiv_or_gt_or_fuzzy x y"
}
] |
[
1004,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1002,
1
] |
Mathlib/Algebra/EuclideanDomain/Basic.lean
|
EuclideanDomain.mod_one
|
[] |
[
75,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
IsometryEquiv.toEquiv_inj
|
[] |
[
316,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
9
] |
Mathlib/Analysis/Calculus/FDeriv/Linear.lean
|
IsBoundedLinearMap.differentiable
|
[] |
[
146,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.factors_lemma
|
[
{
"state_after": "case h\nk : ℕ\n⊢ 1 < k + 2",
"state_before": "k : ℕ\n⊢ k + 2 ≠ 1",
"tactic": "apply Nat.ne_of_gt"
},
{
"state_after": "case h.a\nk : ℕ\n⊢ 0 < k + 1",
"state_before": "case h\nk : ℕ\n⊢ 1 < k + 2",
"tactic": "apply Nat.succ_lt_succ"
},
{
"state_after": "no goals",
"state_before": "case h.a\nk : ℕ\n⊢ 0 < k + 1",
"tactic": "apply Nat.zero_lt_succ"
}
] |
[
541,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
536,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffAt.snd''
|
[] |
[
840,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
838,
1
] |
Mathlib/Data/TwoPointing.lean
|
TwoPointing.sum_snd
|
[] |
[
140,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Order/Circular.lean
|
btw_refl_left_right
|
[] |
[
295,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.NullMeasurableSet.mono
|
[] |
[
2669,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2668,
1
] |
Mathlib/Algebra/Parity.lean
|
isSquare_one
|
[] |
[
73,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Algebra/Homology/Homotopy.lean
|
Homotopy.mkInductiveAux₃
|
[
{
"state_after": "ι : Type ?u.344609\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nP Q : ChainComplex V ℕ\ne : P ⟶ Q\nzero : X P 0 ⟶ X Q 1\ncomm_zero : Hom.f e 0 = zero ≫ d Q 1 0\none : X P 1 ⟶ X Q 2\ncomm_one : Hom.f e 1 = d P 1 0 ≫ zero + one ≫ d Q 2 1\nsucc :\n (n : ℕ) →\n (p :\n (f : X P n ⟶ X Q (n + 1)) ×'\n (f' : X P (n + 1) ⟶ X Q (n + 2)) ×' Hom.f e (n + 1) = d P (n + 1) n ≫ f + f' ≫ d Q (n + 2) (n + 1)) →\n (f'' : X P (n + 2) ⟶ X Q (n + 3)) ×' Hom.f e (n + 2) = d P (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ d Q (n + 3) (n + 2)\ni : ℕ\n⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q (_ : i + 1 = i + 1)).hom =\n (xNextIso P (_ : i + 1 = i + 1)).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1)).fst",
"state_before": "ι : Type ?u.344609\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh✝ k : D ⟶ E\ni✝ : ι\nP Q : ChainComplex V ℕ\ne : P ⟶ Q\nzero : X P 0 ⟶ X Q 1\ncomm_zero : Hom.f e 0 = zero ≫ d Q 1 0\none : X P 1 ⟶ X Q 2\ncomm_one : Hom.f e 1 = d P 1 0 ≫ zero + one ≫ d Q 2 1\nsucc :\n (n : ℕ) →\n (p :\n (f : X P n ⟶ X Q (n + 1)) ×'\n (f' : X P (n + 1) ⟶ X Q (n + 2)) ×' Hom.f e (n + 1) = d P (n + 1) n ≫ f + f' ≫ d Q (n + 2) (n + 1)) →\n (f'' : X P (n + 2) ⟶ X Q (n + 3)) ×' Hom.f e (n + 2) = d P (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ d Q (n + 3) (n + 2)\ni j : ℕ\nh : i + 1 = j\n⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q h).hom =\n (xNextIso P h).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ j).fst",
"tactic": "subst j"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.344609\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nP Q : ChainComplex V ℕ\ne : P ⟶ Q\nzero : X P 0 ⟶ X Q 1\ncomm_zero : Hom.f e 0 = zero ≫ d Q 1 0\none : X P 1 ⟶ X Q 2\ncomm_one : Hom.f e 1 = d P 1 0 ≫ zero + one ≫ d Q 2 1\nsucc :\n (n : ℕ) →\n (p :\n (f : X P n ⟶ X Q (n + 1)) ×'\n (f' : X P (n + 1) ⟶ X Q (n + 2)) ×' Hom.f e (n + 1) = d P (n + 1) n ≫ f + f' ≫ d Q (n + 2) (n + 1)) →\n (f'' : X P (n + 2) ⟶ X Q (n + 3)) ×' Hom.f e (n + 2) = d P (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ d Q (n + 3) (n + 2)\ni : ℕ\n⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q (_ : i + 1 = i + 1)).hom =\n (xNextIso P (_ : i + 1 = i + 1)).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1)).fst",
"tactic": "rcases i with (_ | _ | i) <;> simp [mkInductiveAux₂]"
}
] |
[
535,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
531,
1
] |
Mathlib/CategoryTheory/EqToHom.lean
|
CategoryTheory.Functor.congr_hom
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\n⊢ F.map f = eqToHom (_ : F.obj X = F.obj X) ≫ F.map f ≫ eqToHom (_ : F.obj Y = F.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G : C ⥤ D\nh : F = G\nX Y : C\nf : X ⟶ Y\n⊢ F.map f = eqToHom (_ : F.obj X = G.obj X) ≫ G.map f ≫ eqToHom (_ : G.obj Y = F.obj Y)",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\n⊢ F.map f = eqToHom (_ : F.obj X = F.obj X) ≫ F.map f ≫ eqToHom (_ : F.obj Y = F.obj Y)",
"tactic": "simp"
}
] |
[
211,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Topology/Sets/Opens.lean
|
TopologicalSpace.Opens.coe_finset_inf
|
[] |
[
217,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Data/Set/NAry.lean
|
Set.mem_image2
|
[] |
[
44,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.ConnectedComponent.connectedComponentMk_eq_of_adj
|
[] |
[
2016,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2014,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.sign_neg
|
[
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ sign (-θ) = -sign θ",
"tactic": "simp_rw [sign, sin_neg, Left.sign_neg]"
}
] |
[
867,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
866,
1
] |
Mathlib/Data/Multiset/Fold.lean
|
Multiset.fold_cons_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4822\nop : α → α → α\nhc : IsCommutative α op\nha : IsAssociative α op\nb a : α\ns : Multiset α\n⊢ fold op b (a ::ₘ s) = op (fold op b s) a",
"tactic": "simp [hc.comm]"
}
] |
[
68,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
lipschitzWith_iff_dist_le_mul
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\n⊢ (∀ (x y : α), ↑(nndist (f x) (f y)) ≤ ↑K * ↑(nndist x y)) ↔ ∀ (x y : α), ↑(nndist (f x) (f y)) ≤ ↑K * ↑(nndist x y)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\n⊢ LipschitzWith K f ↔ ∀ (x y : α), dist (f x) (f y) ≤ ↑K * dist x y",
"tactic": "simp only [LipschitzWith, edist_nndist, dist_nndist]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\n⊢ (∀ (x y : α), ↑(nndist (f x) (f y)) ≤ ↑K * ↑(nndist x y)) ↔ ∀ (x y : α), ↑(nndist (f x) (f y)) ≤ ↑K * ↑(nndist x y)",
"tactic": "norm_cast"
}
] |
[
62,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Order/Atoms.lean
|
isCoatom_dual_iff_isAtom
|
[] |
[
125,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Computability/Encoding.lean
|
Computability.inclusionBoolΓ'_injective
|
[] |
[
95,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Submodule.annihilator_mono
|
[] |
[
96,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.arccos_of_le_neg_one
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x ≤ -1\n⊢ arccos x = π",
"tactic": "rw [arccos, arcsin_of_le_neg_one hx, sub_neg_eq_add, add_halves']"
}
] |
[
416,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
415,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.sub_le_sub
|
[] |
[
1037,
42
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1036,
11
] |
Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
|
Matrix.toBilin_toMatrix
|
[] |
[
349,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
347,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
MonoidHomClass.antilipschitz_of_bound
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type u_1\n𝕜 : Type ?u.369462\nα : Type ?u.369465\nι : Type ?u.369468\nκ : Type ?u.369471\nE : Type u_2\nF : Type u_3\nG : Type ?u.369480\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\nK : ℝ≥0\nh : ∀ (x : E), ‖x‖ ≤ ↑K * ‖↑f x‖\nx y : E\n⊢ dist x y ≤ ↑K * dist (↑f x) (↑f y)",
"tactic": "simpa only [dist_eq_norm_div, map_div] using h (x / y)"
}
] |
[
1010,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1007,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.deleteEdges_verts
|
[] |
[
1038,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1037,
1
] |
Mathlib/Logic/Equiv/Fin.lean
|
finCongr_symm
|
[] |
[
118,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
9
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.sin_pos_of_mem_Ioo
|
[] |
[
414,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
413,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.range_pair_subset
|
[
{
"state_after": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.78105\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nf : α → β\ng : α → γ\nthis : (fun x => (f x, g x)) = Prod.map f g ∘ fun x => (x, x)\n⊢ (range fun x => (f x, g x)) ⊆ range f ×ˢ range g",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.78105\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nf : α → β\ng : α → γ\n⊢ (range fun x => (f x, g x)) ⊆ range f ×ˢ range g",
"tactic": "have : (fun x => (f x, g x)) = Prod.map f g ∘ fun x => (x, x) := funext fun x => rfl"
},
{
"state_after": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.78105\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nf : α → β\ng : α → γ\nthis : (fun x => (f x, g x)) = Prod.map f g ∘ fun x => (x, x)\n⊢ range (Prod.map f g ∘ fun x => (x, x)) ⊆ range (Prod.map f g)",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.78105\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nf : α → β\ng : α → γ\nthis : (fun x => (f x, g x)) = Prod.map f g ∘ fun x => (x, x)\n⊢ (range fun x => (f x, g x)) ⊆ range f ×ˢ range g",
"tactic": "rw [this, ← range_prod_map]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.78105\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nf : α → β\ng : α → γ\nthis : (fun x => (f x, g x)) = Prod.map f g ∘ fun x => (x, x)\n⊢ range (Prod.map f g ∘ fun x => (x, x)) ⊆ range (Prod.map f g)",
"tactic": "apply range_comp_subset_range"
}
] |
[
307,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
303,
1
] |
Mathlib/Algebra/Order/Ring/Abs.lean
|
abs_cases
|
[
{
"state_after": "case pos\nα : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b c a : α\nh : 0 ≤ a\n⊢ abs a = a ∧ 0 ≤ a ∨ abs a = -a ∧ a < 0\n\ncase neg\nα : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b c a : α\nh : ¬0 ≤ a\n⊢ abs a = a ∧ 0 ≤ a ∨ abs a = -a ∧ a < 0",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b c a : α\n⊢ abs a = a ∧ 0 ≤ a ∨ abs a = -a ∧ a < 0",
"tactic": "by_cases h : 0 ≤ a"
},
{
"state_after": "case pos.h\nα : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b c a : α\nh : 0 ≤ a\n⊢ abs a = a ∧ 0 ≤ a",
"state_before": "case pos\nα : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b c a : α\nh : 0 ≤ a\n⊢ abs a = a ∧ 0 ≤ a ∨ abs a = -a ∧ a < 0",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case pos.h\nα : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b c a : α\nh : 0 ≤ a\n⊢ abs a = a ∧ 0 ≤ a",
"tactic": "exact ⟨abs_eq_self.mpr h, h⟩"
},
{
"state_after": "case neg.h\nα : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b c a : α\nh : ¬0 ≤ a\n⊢ abs a = -a ∧ a < 0",
"state_before": "case neg\nα : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b c a : α\nh : ¬0 ≤ a\n⊢ abs a = a ∧ 0 ≤ a ∨ abs a = -a ∧ a < 0",
"tactic": "right"
},
{
"state_after": "case neg.h\nα : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b c a : α\nh : a < 0\n⊢ abs a = -a ∧ a < 0",
"state_before": "case neg.h\nα : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b c a : α\nh : ¬0 ≤ a\n⊢ abs a = -a ∧ a < 0",
"tactic": "push_neg at h"
},
{
"state_after": "no goals",
"state_before": "case neg.h\nα : Type u_1\ninst✝ : LinearOrderedRing α\na✝ b c a : α\nh : a < 0\n⊢ abs a = -a ∧ a < 0",
"tactic": "exact ⟨abs_eq_neg_self.mpr (le_of_lt h), h⟩"
}
] |
[
77,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
div_nonpos_of_nonpos_of_nonneg
|
[
{
"state_after": "ι : Type ?u.10909\nα : Type u_1\nβ : Type ?u.10915\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : a ≤ 0\nhb : 0 ≤ b\n⊢ a * b⁻¹ ≤ 0",
"state_before": "ι : Type ?u.10909\nα : Type u_1\nβ : Type ?u.10915\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : a ≤ 0\nhb : 0 ≤ b\n⊢ a / b ≤ 0",
"tactic": "rw [div_eq_mul_inv]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.10909\nα : Type u_1\nβ : Type ?u.10915\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : a ≤ 0\nhb : 0 ≤ b\n⊢ a * b⁻¹ ≤ 0",
"tactic": "exact mul_nonpos_of_nonpos_of_nonneg ha (inv_nonneg.2 hb)"
}
] |
[
103,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.surjective
|
[] |
[
203,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
11
] |
Mathlib/Data/List/BigOperators/Basic.lean
|
List.tail_sum
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.161222\nα : Type ?u.161225\nM : Type ?u.161228\nN : Type ?u.161231\nP : Type ?u.161234\nM₀ : Type ?u.161237\nG : Type ?u.161240\nR : Type ?u.161243\nL : List ℕ\n⊢ sum (tail L) = sum L - headI L",
"tactic": "rw [← headI_add_tail_sum L, add_comm, @add_tsub_cancel_right]"
}
] |
[
613,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
612,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.ofReal_finsupp_sum
|
[] |
[
243,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.isFiniteMeasure_withDensity
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1762622\nγ : Type ?u.1762625\nδ : Type ?u.1762628\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : (∫⁻ (a : α), f a ∂μ) ≠ ⊤\n⊢ ↑↑(withDensity μ f) univ < ⊤",
"tactic": "rwa [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ, lt_top_iff_ne_top]"
}
] |
[
1605,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1602,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.ofSet_trans
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.72442\nδ : Type ?u.72445\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\ns : Set α\nhs : IsOpen s\n⊢ (LocalHomeomorph.trans (ofSet s hs) e).toLocalEquiv.source = (LocalHomeomorph.restr e s).toLocalEquiv.source",
"tactic": "simp [hs.interior_eq, inter_comm]"
}
] |
[
878,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
877,
1
] |
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.snorm_rpow_two_norm_lt_top
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.42419\nF : Type u_2\n𝕜 : Type ?u.42425\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp F 2 }\nh_two : ENNReal.ofReal 2 = 2\n⊢ snorm (fun x => ‖↑↑f x‖ ^ 2) 1 μ < ⊤",
"state_before": "α : Type u_1\nE : Type ?u.42419\nF : Type u_2\n𝕜 : Type ?u.42425\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp F 2 }\n⊢ snorm (fun x => ‖↑↑f x‖ ^ 2) 1 μ < ⊤",
"tactic": "have h_two : ENNReal.ofReal (2 : ℝ) = 2 := by simp [zero_le_one]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.42419\nF : Type u_2\n𝕜 : Type ?u.42425\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp F 2 }\nh_two : ENNReal.ofReal 2 = 2\n⊢ snorm (↑↑f) 2 μ ^ 2 < ⊤",
"state_before": "α : Type u_1\nE : Type ?u.42419\nF : Type u_2\n𝕜 : Type ?u.42425\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp F 2 }\nh_two : ENNReal.ofReal 2 = 2\n⊢ snorm (fun x => ‖↑↑f x‖ ^ 2) 1 μ < ⊤",
"tactic": "rw [snorm_norm_rpow f zero_lt_two, one_mul, h_two]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.42419\nF : Type u_2\n𝕜 : Type ?u.42425\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp F 2 }\nh_two : ENNReal.ofReal 2 = 2\n⊢ snorm (↑↑f) 2 μ ^ 2 < ⊤",
"tactic": "exact ENNReal.rpow_lt_top_of_nonneg zero_le_two (Lp.snorm_ne_top f)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.42419\nF : Type u_2\n𝕜 : Type ?u.42425\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp F 2 }\n⊢ ENNReal.ofReal 2 = 2",
"tactic": "simp [zero_le_one]"
}
] |
[
128,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
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