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/RingTheory/Ideal/Operations.lean
|
Ideal.le_comap_map
|
[] |
[
1431,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1430,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
norm_div_le
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.79045\n𝕜 : Type ?u.79048\nα : Type ?u.79051\nι : Type ?u.79054\nκ : Type ?u.79057\nE : Type u_1\nF : Type ?u.79063\nG : Type ?u.79066\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na✝ a₁ a₂ b✝ b₁ b₂ : E\nr r₁ r₂ : ℝ\na b : E\n⊢ ‖a / b‖ ≤ ‖a‖ + ‖b‖",
"tactic": "simpa [dist_eq_norm_div] using dist_triangle a 1 b"
}
] |
[
541,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
one_div_neg
|
[] |
[
80,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Order/Filter/Extr.lean
|
isMaxOn_univ_iff
|
[] |
[
147,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.ae_bdd_liminf_atTop_rpow_of_snorm_bdd
|
[
{
"state_after": "case pos\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ENNReal.toReal p = 0\n⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n => ↑‖f n x‖₊ ^ ENNReal.toReal p) atTop < ⊤\n\ncase neg\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ¬ENNReal.toReal p = 0\n⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n => ↑‖f n x‖₊ ^ ENNReal.toReal p) atTop < ⊤",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\n⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n => ↑‖f n x‖₊ ^ ENNReal.toReal p) atTop < ⊤",
"tactic": "by_cases hp0 : p.toReal = 0"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ¬ENNReal.toReal p = 0\nhp : p ≠ 0\n⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n => ↑‖f n x‖₊ ^ ENNReal.toReal p) atTop < ⊤",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ¬ENNReal.toReal p = 0\n⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n => ↑‖f n x‖₊ ^ ENNReal.toReal p) atTop < ⊤",
"tactic": "have hp : p ≠ 0 := fun h => by simp [h] at hp0"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ¬ENNReal.toReal p = 0\nhp : p ≠ 0\nhp' : p ≠ ⊤\n⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n => ↑‖f n x‖₊ ^ ENNReal.toReal p) atTop < ⊤",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ¬ENNReal.toReal p = 0\nhp : p ≠ 0\n⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n => ↑‖f n x‖₊ ^ ENNReal.toReal p) atTop < ⊤",
"tactic": "have hp' : p ≠ ∞ := fun h => by simp [h] at hp0"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ¬ENNReal.toReal p = 0\nhp : p ≠ 0\nhp' : p ≠ ⊤\n⊢ liminf (fun n => ∫⁻ (a : α), ↑‖f n a‖₊ ^ ENNReal.toReal p ∂μ) atTop ≤ ↑R ^ ENNReal.toReal p",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ¬ENNReal.toReal p = 0\nhp : p ≠ 0\nhp' : p ≠ ⊤\n⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n => ↑‖f n x‖₊ ^ ENNReal.toReal p) atTop < ⊤",
"tactic": "refine'\n ae_lt_top (measurable_liminf fun n => (hfmeas n).nnnorm.coe_nnreal_ennreal.pow_const p.toReal)\n (lt_of_le_of_lt\n (lintegral_liminf_le fun n => (hfmeas n).nnnorm.coe_nnreal_ennreal.pow_const p.toReal)\n (lt_of_le_of_lt _\n (ENNReal.rpow_lt_top_of_nonneg ENNReal.toReal_nonneg ENNReal.coe_ne_top :\n (R : ℝ≥0∞) ^ p.toReal < ∞))).ne"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhp0 : ¬ENNReal.toReal p = 0\nhp : p ≠ 0\nhp' : p ≠ ⊤\nhbdd : ∀ (n : ℕ), (∫⁻ (x : α), ↑‖f n x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) ≤ ↑R\n⊢ liminf (fun n => ∫⁻ (a : α), ↑‖f n a‖₊ ^ ENNReal.toReal p ∂μ) atTop ≤ ↑R ^ ENNReal.toReal p",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ¬ENNReal.toReal p = 0\nhp : p ≠ 0\nhp' : p ≠ ⊤\n⊢ liminf (fun n => ∫⁻ (a : α), ↑‖f n a‖₊ ^ ENNReal.toReal p ∂μ) atTop ≤ ↑R ^ ENNReal.toReal p",
"tactic": "simp_rw [snorm_eq_lintegral_rpow_nnnorm hp hp'] at hbdd"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhp0 : ¬ENNReal.toReal p = 0\nhp : p ≠ 0\nhp' : p ≠ ⊤\nhbdd : ∀ (n : ℕ), (∫⁻ (x : α), ↑‖f n x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) ≤ ↑R\n⊢ sSup {a | ∃ a_1, ∀ (b : ℕ), b ≥ a_1 → a ≤ ∫⁻ (a : α), ↑‖f b a‖₊ ^ ENNReal.toReal p ∂μ} ≤ ↑R ^ ENNReal.toReal p",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhp0 : ¬ENNReal.toReal p = 0\nhp : p ≠ 0\nhp' : p ≠ ⊤\nhbdd : ∀ (n : ℕ), (∫⁻ (x : α), ↑‖f n x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) ≤ ↑R\n⊢ liminf (fun n => ∫⁻ (a : α), ↑‖f n a‖₊ ^ ENNReal.toReal p ∂μ) atTop ≤ ↑R ^ ENNReal.toReal p",
"tactic": "simp_rw [liminf_eq, eventually_atTop]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhp0 : ¬ENNReal.toReal p = 0\nhp : p ≠ 0\nhp' : p ≠ ⊤\nhbdd : ∀ (n : ℕ), (∫⁻ (x : α), ↑‖f n x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) ≤ ↑R\n⊢ sSup {a | ∃ a_1, ∀ (b : ℕ), b ≥ a_1 → a ≤ ∫⁻ (a : α), ↑‖f b a‖₊ ^ ENNReal.toReal p ∂μ} ≤ ↑R ^ ENNReal.toReal p",
"tactic": "exact\n sSup_le fun b ⟨a, ha⟩ =>\n (ha a le_rfl).trans ((ENNReal.rpow_one_div_le_iff (ENNReal.toReal_pos hp hp')).1 (hbdd _))"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ENNReal.toReal p = 0\n⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n => 1) atTop < ⊤",
"state_before": "case pos\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ENNReal.toReal p = 0\n⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n => ↑‖f n x‖₊ ^ ENNReal.toReal p) atTop < ⊤",
"tactic": "simp only [hp0, ENNReal.rpow_zero]"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ENNReal.toReal p = 0\nx : α\n⊢ liminf (fun n => 1) atTop < ⊤",
"state_before": "case pos\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ENNReal.toReal p = 0\n⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n => 1) atTop < ⊤",
"tactic": "refine' eventually_of_forall fun x => _"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ENNReal.toReal p = 0\nx : α\n⊢ 1 < ⊤",
"state_before": "case pos\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ENNReal.toReal p = 0\nx : α\n⊢ liminf (fun n => 1) atTop < ⊤",
"tactic": "rw [liminf_const (1 : ℝ≥0∞)]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ENNReal.toReal p = 0\nx : α\n⊢ 1 < ⊤",
"tactic": "exact ENNReal.one_lt_top"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ¬ENNReal.toReal p = 0\nh : p = 0\n⊢ False",
"tactic": "simp [h] at hp0"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.6222998\nG : Type ?u.6223001\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), snorm (f n) p μ ≤ ↑R\nhp0 : ¬ENNReal.toReal p = 0\nhp : p ≠ 0\nh : p = ⊤\n⊢ False",
"tactic": "simp [h] at hp0"
}
] |
[
1645,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1624,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.span_union
|
[] |
[
133,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Data/Prod/Basic.lean
|
Function.Surjective.Prod_map
|
[] |
[
322,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/Data/Set/Sups.lean
|
Set.infs_subset
|
[] |
[
241,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.mem_graph_iff
|
[
{
"state_after": "case mk\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.477455\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf : E →ₗ.[R] F\nfst✝ : E\nsnd✝ : F\n⊢ (fst✝, snd✝) ∈ graph f ↔ ∃ y, ↑y = (fst✝, snd✝).fst ∧ ↑f y = (fst✝, snd✝).snd",
"state_before": "R : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.477455\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf : E →ₗ.[R] F\nx : E × F\n⊢ x ∈ graph f ↔ ∃ y, ↑y = x.fst ∧ ↑f y = x.snd",
"tactic": "cases x"
},
{
"state_after": "no goals",
"state_before": "case mk\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.477455\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf : E →ₗ.[R] F\nfst✝ : E\nsnd✝ : F\n⊢ (fst✝, snd✝) ∈ graph f ↔ ∃ y, ↑y = (fst✝, snd✝).fst ∧ ↑f y = (fst✝, snd✝).snd",
"tactic": "simp_rw [mem_graph_iff', Prod.mk.inj_iff]"
}
] |
[
733,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
730,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
IsCompact.isComplete
|
[] |
[
594,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
593,
11
] |
Mathlib/Data/Finsupp/Order.lean
|
Finsupp.add_sub_single_one
|
[] |
[
262,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
AddMonoidAlgebra.single_zero
|
[] |
[
1193,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1193,
1
] |
Mathlib/Data/List/Pairwise.lean
|
List.pairwise_iff_get
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na : α\nl : List α\n⊢ ∀ (i j : Fin (length [])), i < j → R (get [] i) (get [] j)",
"state_before": "α : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na : α\nl : List α\n⊢ Pairwise R [] ↔ ∀ (i j : Fin (length [])), i < j → R (get [] i) (get [] j)",
"tactic": "simp only [Pairwise.nil, true_iff_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na : α\nl : List α\n⊢ ∀ (i j : Fin (length [])), i < j → R (get [] i) (get [] j)",
"tactic": "exact fun i j _h => (Nat.not_lt_zero j).elim j.2"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\n⊢ ((∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)) ↔\n ∀ (i j : Fin (length (a :: l))), i < j → R (get (a :: l) i) (get (a :: l) j)",
"state_before": "α : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\n⊢ Pairwise R (a :: l) ↔ ∀ (i j : Fin (length (a :: l))), i < j → R (get (a :: l) i) (get (a :: l) j)",
"tactic": "rw [pairwise_cons, pairwise_iff_get]"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\ni j : Fin (length (a :: l))\nhij : i < j\n⊢ R (get (a :: l) i) (get (a :: l) j)\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : ∀ (i j : Fin (length (a :: l))), i < j → R (get (a :: l) i) (get (a :: l) j)\na' : α\nm : a' ∈ l\n⊢ R a a'\n\ncase refine'_3\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : ∀ (i j : Fin (length (a :: l))), i < j → R (get (a :: l) i) (get (a :: l) j)\ni j : Fin (length l)\nhij : i < j\n⊢ R (get l i) (get l j)",
"state_before": "α : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\n⊢ ((∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)) ↔\n ∀ (i j : Fin (length (a :: l))), i < j → R (get (a :: l) i) (get (a :: l) j)",
"tactic": "refine'\n ⟨fun H i j hij => _, fun H =>\n ⟨fun a' m => _, fun i j hij => _⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : ∀ (i j : Fin (length (a :: l))), i < j → R (get (a :: l) i) (get (a :: l) j)\ni j : Fin (length l)\nhij : i < j\n⊢ R (get l i) (get l j)",
"tactic": ". simpa using H i.succ j.succ (show i.1.succ < j.1.succ from Nat.succ_lt_succ hij)"
},
{
"state_after": "case refine'_1.mk\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\ni : Fin (length (a :: l))\nj : ℕ\nhj : j < length (a :: l)\nhij : i < { val := j, isLt := hj }\n⊢ R (get (a :: l) i) (get (a :: l) { val := j, isLt := hj })",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\ni j : Fin (length (a :: l))\nhij : i < j\n⊢ R (get (a :: l) i) (get (a :: l) j)",
"tactic": "cases' j with j hj"
},
{
"state_after": "case refine'_1.mk.zero\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\ni : Fin (length (a :: l))\nhj : zero < length (a :: l)\nhij : i < { val := zero, isLt := hj }\n⊢ R (get (a :: l) i) (get (a :: l) { val := zero, isLt := hj })\n\ncase refine'_1.mk.succ\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\ni : Fin (length (a :: l))\nj : ℕ\nhj : succ j < length (a :: l)\nhij : i < { val := succ j, isLt := hj }\n⊢ R (get (a :: l) i) (get (a :: l) { val := succ j, isLt := hj })",
"state_before": "case refine'_1.mk\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\ni : Fin (length (a :: l))\nj : ℕ\nhj : j < length (a :: l)\nhij : i < { val := j, isLt := hj }\n⊢ R (get (a :: l) i) (get (a :: l) { val := j, isLt := hj })",
"tactic": "cases' j with j"
},
{
"state_after": "case refine'_1.mk.succ.mk\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\nj : ℕ\nhj : succ j < length (a :: l)\ni : ℕ\nhi : i < length (a :: l)\nhij : { val := i, isLt := hi } < { val := succ j, isLt := hj }\n⊢ R (get (a :: l) { val := i, isLt := hi }) (get (a :: l) { val := succ j, isLt := hj })",
"state_before": "case refine'_1.mk.succ\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\ni : Fin (length (a :: l))\nj : ℕ\nhj : succ j < length (a :: l)\nhij : i < { val := succ j, isLt := hj }\n⊢ R (get (a :: l) i) (get (a :: l) { val := succ j, isLt := hj })",
"tactic": "cases' i with i hi"
},
{
"state_after": "case refine'_1.mk.succ.mk.zero\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\nj : ℕ\nhj : succ j < length (a :: l)\nhi : zero < length (a :: l)\nhij : { val := zero, isLt := hi } < { val := succ j, isLt := hj }\n⊢ R (get (a :: l) { val := zero, isLt := hi }) (get (a :: l) { val := succ j, isLt := hj })\n\ncase refine'_1.mk.succ.mk.succ\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\nj : ℕ\nhj : succ j < length (a :: l)\ni : ℕ\nhi : succ i < length (a :: l)\nhij : { val := succ i, isLt := hi } < { val := succ j, isLt := hj }\n⊢ R (get (a :: l) { val := succ i, isLt := hi }) (get (a :: l) { val := succ j, isLt := hj })",
"state_before": "case refine'_1.mk.succ.mk\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\nj : ℕ\nhj : succ j < length (a :: l)\ni : ℕ\nhi : i < length (a :: l)\nhij : { val := i, isLt := hi } < { val := succ j, isLt := hj }\n⊢ R (get (a :: l) { val := i, isLt := hi }) (get (a :: l) { val := succ j, isLt := hj })",
"tactic": "cases' i with i"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.mk.zero\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\ni : Fin (length (a :: l))\nhj : zero < length (a :: l)\nhij : i < { val := zero, isLt := hj }\n⊢ R (get (a :: l) i) (get (a :: l) { val := zero, isLt := hj })",
"tactic": "exact (Nat.not_lt_zero _).elim hij"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.mk.succ.mk.zero\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\nj : ℕ\nhj : succ j < length (a :: l)\nhi : zero < length (a :: l)\nhij : { val := zero, isLt := hi } < { val := succ j, isLt := hj }\n⊢ R (get (a :: l) { val := zero, isLt := hi }) (get (a :: l) { val := succ j, isLt := hj })",
"tactic": "exact H.1 _ (get_mem l _ _)"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.mk.succ.mk.succ\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : (∀ (a' : α), a' ∈ l → R a a') ∧ ∀ (i j : Fin (length l)), i < j → R (get l i) (get l j)\nj : ℕ\nhj : succ j < length (a :: l)\ni : ℕ\nhi : succ i < length (a :: l)\nhij : { val := succ i, isLt := hi } < { val := succ j, isLt := hj }\n⊢ R (get (a :: l) { val := succ i, isLt := hi }) (get (a :: l) { val := succ j, isLt := hj })",
"tactic": "exact H.2 _ _ (Nat.lt_of_succ_lt_succ hij)"
},
{
"state_after": "case refine'_2.intro.refl\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : ∀ (i j : Fin (length (a :: l))), i < j → R (get (a :: l) i) (get (a :: l) j)\nn : Fin (length l)\nm : get l n ∈ l\n⊢ R a (get l n)",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : ∀ (i j : Fin (length (a :: l))), i < j → R (get (a :: l) i) (get (a :: l) j)\na' : α\nm : a' ∈ l\n⊢ R a a'",
"tactic": "rcases get_of_mem m with ⟨n, h, rfl⟩"
},
{
"state_after": "case refine'_2.intro.refl\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : ∀ (i j : Fin (length (a :: l))), i < j → R (get (a :: l) i) (get (a :: l) j)\nn : Fin (length l)\nm : get l n ∈ l\nthis :\n R (get (a :: l) { val := 0, isLt := (_ : 0 < length (a :: l)) })\n (get (a :: l) { val := ↑(Fin.succ n), isLt := (_ : succ ↑n < succ (length l)) })\n⊢ R a (get l n)",
"state_before": "case refine'_2.intro.refl\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : ∀ (i j : Fin (length (a :: l))), i < j → R (get (a :: l) i) (get (a :: l) j)\nn : Fin (length l)\nm : get l n ∈ l\n⊢ R a (get l n)",
"tactic": "have := H ⟨0, show 0 < (a::l).length from Nat.succ_pos _⟩ ⟨n.succ, Nat.succ_lt_succ n.2⟩\n (Nat.succ_pos n)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.refl\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : ∀ (i j : Fin (length (a :: l))), i < j → R (get (a :: l) i) (get (a :: l) j)\nn : Fin (length l)\nm : get l n ∈ l\nthis :\n R (get (a :: l) { val := 0, isLt := (_ : 0 < length (a :: l)) })\n (get (a :: l) { val := ↑(Fin.succ n), isLt := (_ : succ ↑n < succ (length l)) })\n⊢ R a (get l n)",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.44959\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\na : α\nl : List α\nH : ∀ (i j : Fin (length (a :: l))), i < j → R (get (a :: l) i) (get (a :: l) j)\ni j : Fin (length l)\nhij : i < j\n⊢ R (get l i) (get l j)",
"tactic": "simpa using H i.succ j.succ (show i.1.succ < j.1.succ from Nat.succ_lt_succ hij)"
}
] |
[
332,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
1
] |
Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean
|
GeneralizedContinuedFraction.convergents'Aux_stable_step_of_terminated
|
[
{
"state_after": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn m : ℕ\ninst✝ : DivisionRing K\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s n = none\n⊢ convergents'Aux s (n + 1) = convergents'Aux s n",
"state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn m : ℕ\ninst✝ : DivisionRing K\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.TerminatedAt s n\n⊢ convergents'Aux s (n + 1) = convergents'Aux s n",
"tactic": "change s.get? n = none at terminated_at_n"
},
{
"state_after": "case zero\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n = none\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s Nat.zero = none\n⊢ convergents'Aux s (Nat.zero + 1) = convergents'Aux s Nat.zero\n\ncase succ\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn m : ℕ\ninst✝ : DivisionRing K\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s n = none\n⊢ convergents'Aux s (n + 1) = convergents'Aux s n",
"tactic": "induction' n with n IH generalizing s"
},
{
"state_after": "case succ\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"state_before": "case zero\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n = none\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s Nat.zero = none\n⊢ convergents'Aux s (Nat.zero + 1) = convergents'Aux s Nat.zero\n\ncase succ\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"tactic": "case zero => simp only [convergents'Aux, terminated_at_n, Stream'.Seq.head]"
},
{
"state_after": "no goals",
"state_before": "case succ\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"tactic": "case succ =>\n cases' s_head_eq : s.head with gp_head\n case none => simp only [convergents'Aux, s_head_eq]\n case some =>\n have : s.tail.TerminatedAt n := by\n simp only [Stream'.Seq.TerminatedAt, s.get?_tail, terminated_at_n]\n have := IH this\n rw [convergents'Aux] at this\n simp [this, Nat.add_eq, add_zero, convergents'Aux, s_head_eq]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n = none\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s Nat.zero = none\n⊢ convergents'Aux s (Nat.zero + 1) = convergents'Aux s Nat.zero",
"tactic": "simp only [convergents'Aux, terminated_at_n, Stream'.Seq.head]"
},
{
"state_after": "case none\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ns_head_eq : Stream'.Seq.head s = none\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n\ncase some\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"tactic": "cases' s_head_eq : s.head with gp_head"
},
{
"state_after": "case some\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"state_before": "case none\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ns_head_eq : Stream'.Seq.head s = none\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n\ncase some\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"tactic": "case none => simp only [convergents'Aux, s_head_eq]"
},
{
"state_after": "no goals",
"state_before": "case some\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"tactic": "case some =>\n have : s.tail.TerminatedAt n := by\n simp only [Stream'.Seq.TerminatedAt, s.get?_tail, terminated_at_n]\n have := IH this\n rw [convergents'Aux] at this\n simp [this, Nat.add_eq, add_zero, convergents'Aux, s_head_eq]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ns_head_eq : Stream'.Seq.head s = none\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"tactic": "simp only [convergents'Aux, s_head_eq]"
},
{
"state_after": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis : Stream'.Seq.TerminatedAt (Stream'.Seq.tail s) n\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"tactic": "have : s.tail.TerminatedAt n := by\n simp only [Stream'.Seq.TerminatedAt, s.get?_tail, terminated_at_n]"
},
{
"state_after": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis✝ : Stream'.Seq.TerminatedAt (Stream'.Seq.tail s) n\nthis : convergents'Aux (Stream'.Seq.tail s) (n + 1) = convergents'Aux (Stream'.Seq.tail s) n\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis : Stream'.Seq.TerminatedAt (Stream'.Seq.tail s) n\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"tactic": "have := IH this"
},
{
"state_after": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis✝ : Stream'.Seq.TerminatedAt (Stream'.Seq.tail s) n\nthis :\n (match Stream'.Seq.head (Stream'.Seq.tail s) with\n | none => 0\n | some gp => gp.a / (gp.b + convergents'Aux (Stream'.Seq.tail (Stream'.Seq.tail s)) n)) =\n convergents'Aux (Stream'.Seq.tail s) n\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis✝ : Stream'.Seq.TerminatedAt (Stream'.Seq.tail s) n\nthis : convergents'Aux (Stream'.Seq.tail s) (n + 1) = convergents'Aux (Stream'.Seq.tail s) n\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"tactic": "rw [convergents'Aux] at this"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis✝ : Stream'.Seq.TerminatedAt (Stream'.Seq.tail s) n\nthis :\n (match Stream'.Seq.head (Stream'.Seq.tail s) with\n | none => 0\n | some gp => gp.a / (gp.b + convergents'Aux (Stream'.Seq.tail (Stream'.Seq.tail s)) n)) =\n convergents'Aux (Stream'.Seq.tail s) n\n⊢ convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)",
"tactic": "simp [this, Nat.add_eq, add_zero, convergents'Aux, s_head_eq]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn✝ m : ℕ\ninst✝ : DivisionRing K\ns✝ : Stream'.Seq (Pair K)\nterminated_at_n✝ : Stream'.Seq.get? s✝ n✝ = none\nn : ℕ\nIH : ∀ {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none → convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n⊢ Stream'.Seq.TerminatedAt (Stream'.Seq.tail s) n",
"tactic": "simp only [Stream'.Seq.TerminatedAt, s.get?_tail, terminated_at_n]"
}
] |
[
61,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
edist_toMul
|
[] |
[
1202,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1201,
1
] |
Mathlib/CategoryTheory/Endomorphism.lean
|
CategoryTheory.End.one_def
|
[] |
[
64,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Data/Set/Sigma.lean
|
Set.mem_sigma_iff
|
[] |
[
63,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Topology/ExtremallyDisconnected.lean
|
CompactT2.Projective.extremallyDisconnected
|
[
{
"state_after": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\n⊢ IsOpen (closure U)",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\n⊢ ExtremallyDisconnected X",
"tactic": "refine' { open_closure := fun U hU => _ }"
},
{
"state_after": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\n⊢ IsOpen (closure U)",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\n⊢ IsOpen (closure U)",
"tactic": "let Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}"
},
{
"state_after": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\n⊢ IsOpen (closure U)",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\n⊢ IsOpen (closure U)",
"tactic": "let Z₂ : Set (X × Bool) := closure U ×ˢ {false}"
},
{
"state_after": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\n⊢ IsOpen (closure U)",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\n⊢ IsOpen (closure U)",
"tactic": "let Z : Set (X × Bool) := Z₁ ∪ Z₂"
},
{
"state_after": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\n⊢ IsOpen (closure U)",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\n⊢ IsOpen (closure U)",
"tactic": "have hZ₁₂ : Disjoint Z₁ Z₂ := disjoint_left.2 fun x hx₁ hx₂ => by cases hx₁.2.symm.trans hx₂.2"
},
{
"state_after": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\n⊢ IsOpen (closure U)",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\n⊢ IsOpen (closure U)",
"tactic": "have hZ₁ : IsClosed Z₁ := hU.isClosed_compl.prod (T1Space.t1 _)"
},
{
"state_after": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\n⊢ IsOpen (closure U)",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\n⊢ IsOpen (closure U)",
"tactic": "have hZ₂ : IsClosed Z₂ := isClosed_closure.prod (T1Space.t1 false)"
},
{
"state_after": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\n⊢ IsOpen (closure U)",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\n⊢ IsOpen (closure U)",
"tactic": "have hZ : IsClosed Z := hZ₁.union hZ₂"
},
{
"state_after": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\n⊢ IsOpen (closure U)",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\n⊢ IsOpen (closure U)",
"tactic": "let f : Z → X := Prod.fst ∘ Subtype.val"
},
{
"state_after": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\n⊢ IsOpen (closure U)",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\n⊢ IsOpen (closure U)",
"tactic": "have f_cont : Continuous f := continuous_fst.comp continuous_subtype_val"
},
{
"state_after": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\n⊢ IsOpen (closure U)",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\n⊢ IsOpen (closure U)",
"tactic": "haveI : CompactSpace Z := isCompact_iff_compactSpace.mp hZ.isCompact"
},
{
"state_after": "case intro.intro\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\n⊢ IsOpen (closure U)",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\n⊢ IsOpen (closure U)",
"tactic": "obtain ⟨g, hg, g_sec⟩ := h continuous_id f_cont f_sur"
},
{
"state_after": "case intro.intro\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\n⊢ IsOpen (closure U)",
"state_before": "case intro.intro\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\n⊢ IsOpen (closure U)",
"tactic": "let φ := Subtype.val ∘ g"
},
{
"state_after": "case intro.intro\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\n⊢ IsOpen (closure U)",
"state_before": "case intro.intro\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\n⊢ IsOpen (closure U)",
"tactic": "have hφ : Continuous φ := continuous_subtype_val.comp hg"
},
{
"state_after": "case intro.intro\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\n⊢ IsOpen (closure U)",
"state_before": "case intro.intro\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\n⊢ IsOpen (closure U)",
"tactic": "have hφ₁ : ∀ x, (φ x).1 = x := congr_fun g_sec"
},
{
"state_after": "case intro.intro.refine'_1\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\n⊢ U ⊆ φ ⁻¹' Z₂\n\ncase intro.intro.refine'_2\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nx : X\nhx : x ∈ φ ⁻¹' Z₂\n⊢ x ∈ closure U",
"state_before": "case intro.intro\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\n⊢ closure U = φ ⁻¹' Z₂",
"tactic": "refine' (closure_minimal _ <| hZ₂.preimage hφ).antisymm fun x hx => _"
},
{
"state_after": "no goals",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nx : X × Bool\nhx₁ : x ∈ Z₁\nhx₂ : x ∈ Z₂\n⊢ False",
"tactic": "cases hx₁.2.symm.trans hx₂.2"
},
{
"state_after": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nx : X\n⊢ ∃ a, f a = x",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\n⊢ Surjective f",
"tactic": "intro x"
},
{
"state_after": "case pos\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nx : X\nhx : x ∈ U\n⊢ ∃ a, f a = x\n\ncase neg\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nx : X\nhx : ¬x ∈ U\n⊢ ∃ a, f a = x",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nx : X\n⊢ ∃ a, f a = x",
"tactic": "by_cases hx : x ∈ U"
},
{
"state_after": "no goals",
"state_before": "case pos\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nx : X\nhx : x ∈ U\n⊢ ∃ a, f a = x",
"tactic": "exact ⟨⟨(x, false), Or.inr ⟨subset_closure hx, Set.mem_singleton _⟩⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case neg\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nx : X\nhx : ¬x ∈ U\n⊢ ∃ a, f a = x",
"tactic": "exact ⟨⟨(x, true), Or.inl ⟨hx, Set.mem_singleton _⟩⟩, rfl⟩"
},
{
"state_after": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis✝ : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nthis : closure U = φ ⁻¹' Z₂\n⊢ IsClosed (g ⁻¹' (Subtype.val ⁻¹' Z₁))\n\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis✝ : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nthis : closure U = φ ⁻¹' Z₂\n⊢ (Z₁ ∪ Z₂) ∩ (Z₁ ∩ Z₂) = ∅",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis✝ : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nthis : closure U = φ ⁻¹' Z₂\n⊢ IsOpen (closure U)",
"tactic": "rw [this, Set.preimage_comp, ← isClosed_compl_iff, ← preimage_compl, ←\n preimage_subtype_coe_eq_compl Subset.rfl]"
},
{
"state_after": "no goals",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis✝ : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nthis : closure U = φ ⁻¹' Z₂\n⊢ IsClosed (g ⁻¹' (Subtype.val ⁻¹' Z₁))",
"tactic": "exact hZ₁.preimage hφ"
},
{
"state_after": "no goals",
"state_before": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis✝ : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nthis : closure U = φ ⁻¹' Z₂\n⊢ (Z₁ ∪ Z₂) ∩ (Z₁ ∩ Z₂) = ∅",
"tactic": "rw [hZ₁₂.inter_eq, inter_empty]"
},
{
"state_after": "case intro.intro.refine'_1\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nx : X\nhx : x ∈ U\n⊢ x ∈ φ ⁻¹' Z₂",
"state_before": "case intro.intro.refine'_1\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\n⊢ U ⊆ φ ⁻¹' Z₂",
"tactic": "intro x hx"
},
{
"state_after": "case intro.intro.refine'_1\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis✝ : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nx : X\nhx : x ∈ U\nthis : φ x ∈ Z₁ ∪ Z₂\n⊢ x ∈ φ ⁻¹' Z₂",
"state_before": "case intro.intro.refine'_1\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nx : X\nhx : x ∈ U\n⊢ x ∈ φ ⁻¹' Z₂",
"tactic": "have : φ x ∈ Z₁ ∪ Z₂ := (g x).2"
},
{
"state_after": "case intro.intro.refine'_1.inl\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ✝ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nx : X\nhx : x ∈ U\nhφ : φ x ∈ Z₁\n⊢ x ∈ φ ⁻¹' Z₂\n\ncase intro.intro.refine'_1.inr\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ✝ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nx : X\nhx : x ∈ U\nhφ : φ x ∈ Z₂\n⊢ x ∈ φ ⁻¹' Z₂",
"state_before": "case intro.intro.refine'_1\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis✝ : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nx : X\nhx : x ∈ U\nthis : φ x ∈ Z₁ ∪ Z₂\n⊢ x ∈ φ ⁻¹' Z₂",
"tactic": "cases' this with hφ hφ"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1.inl\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ✝ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nx : X\nhx : x ∈ U\nhφ : φ x ∈ Z₁\n⊢ x ∈ φ ⁻¹' Z₂",
"tactic": "exact ((hφ₁ x ▸ hφ.1) hx).elim"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1.inr\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ✝ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nx : X\nhx : x ∈ U\nhφ : φ x ∈ Z₂\n⊢ x ∈ φ ⁻¹' Z₂",
"tactic": "exact hφ"
},
{
"state_after": "case intro.intro.refine'_2\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nx : X\nhx : x ∈ φ ⁻¹' Z₂\n⊢ (φ x).fst ∈ closure U",
"state_before": "case intro.intro.refine'_2\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nx : X\nhx : x ∈ φ ⁻¹' Z₂\n⊢ x ∈ closure U",
"tactic": "rw [← hφ₁ x]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_2\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsClosed Z₂\nhZ : IsClosed Z\nf : ↑Z → X := Prod.fst ∘ Subtype.val\nf_cont : Continuous f\nf_sur : Surjective f\nthis : CompactSpace ↑Z\ng : X → ↑Z\nhg : Continuous g\ng_sec : f ∘ g = id\nφ : X → X × Bool := Subtype.val ∘ g\nhφ : Continuous φ\nhφ₁ : ∀ (x : X), (φ x).fst = x\nx : X\nhx : x ∈ φ ⁻¹' Z₂\n⊢ (φ x).fst ∈ closure U",
"tactic": "exact hx.1"
}
] |
[
121,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
11
] |
Mathlib/Algebra/Hom/NonUnitalAlg.lean
|
NonUnitalAlgHom.snd_prod
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\nB : Type w\nC : Type w₁\ninst✝⁶ : Monoid R\ninst✝⁵ : NonUnitalNonAssocSemiring A\ninst✝⁴ : DistribMulAction R A\ninst✝³ : NonUnitalNonAssocSemiring B\ninst✝² : DistribMulAction R B\ninst✝¹ : NonUnitalNonAssocSemiring C\ninst✝ : DistribMulAction R C\nf : A →ₙₐ[R] B\ng : A →ₙₐ[R] C\n⊢ comp (snd R B C) (prod f g) = g",
"tactic": "rfl"
}
] |
[
357,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
Submodule.smul_closure_subset
|
[] |
[
145,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.mk'_sec
|
[] |
[
249,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/Algebra/Algebra/Operations.lean
|
Submodule.comap_unop_one
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ comap (↑(LinearEquiv.symm (opLinearEquiv R))) 1 = 1",
"tactic": "rw [← map_equiv_eq_comap_symm, map_op_one]"
}
] |
[
156,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
|
Real.iter_deriv_exp
|
[
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nn : ℕ\n⊢ (deriv^[n + 1]) exp = exp",
"tactic": "rw [iterate_succ_apply, deriv_exp, iter_deriv_exp n]"
}
] |
[
210,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.divides_sq_eq_zero
|
[
{
"state_after": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\nh : x * x = d * y * y\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\n⊢ False",
"state_before": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\nh : x * x = d * y * y\ng : ℕ := Nat.gcd x y\ngpos : g > 0\n⊢ False",
"tactic": "let ⟨m, n, co, (hx : x = m * g), (hy : y = n * g)⟩ := Nat.exists_coprime gpos"
},
{
"state_after": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\n⊢ False",
"state_before": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\nh : x * x = d * y * y\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\n⊢ False",
"tactic": "rw [hx, hy] at h"
},
{
"state_after": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\nthis : m * m = d * (n * n)\n⊢ False",
"state_before": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\n⊢ False",
"tactic": "have : m * m = d * (n * n) := by\n refine mul_left_cancel₀ (mul_pos gpos gpos).ne' ?_\n calc\n g * g * (m * m)\n _ = m * g * (m * g) := by ring\n _ = d * (n * g) * (n * g) := h\n _ = g * g * (d * (n * n)) := by ring"
},
{
"state_after": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\nthis : m * m = d * (n * n)\nco2 : Nat.coprime (m * m) (n * n)\n⊢ False",
"state_before": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\nthis : m * m = d * (n * n)\n⊢ False",
"tactic": "have co2 :=\n let co1 := co.mul_right co\n co1.mul co1"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\nthis : m * m = d * (n * n)\nco2 : Nat.coprime (m * m) (n * n)\n⊢ False",
"tactic": "exact\n Nonsquare.ns d m\n (Nat.dvd_antisymm (by rw [this]; apply dvd_mul_right) <|\n co2.dvd_of_dvd_mul_right <| by simp [this])"
},
{
"state_after": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\n⊢ g * g * (m * m) = g * g * (d * (n * n))",
"state_before": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\n⊢ m * m = d * (n * n)",
"tactic": "refine mul_left_cancel₀ (mul_pos gpos gpos).ne' ?_"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\n⊢ g * g * (m * m) = g * g * (d * (n * n))",
"tactic": "calc\n g * g * (m * m)\n _ = m * g * (m * g) := by ring\n _ = d * (n * g) * (n * g) := h\n _ = g * g * (d * (n * n)) := by ring"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\n⊢ g * g * (m * m) = m * g * (m * g)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\n⊢ d * (n * g) * (n * g) = g * g * (d * (n * n))",
"tactic": "ring"
},
{
"state_after": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\nthis : m * m = d * (n * n)\nco2 : Nat.coprime (m * m) (n * n)\n⊢ d ∣ d * (n * n)",
"state_before": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\nthis : m * m = d * (n * n)\nco2 : Nat.coprime (m * m) (n * n)\n⊢ d ∣ m * m",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\nthis : m * m = d * (n * n)\nco2 : Nat.coprime (m * m) (n * n)\n⊢ d ∣ d * (n * n)",
"tactic": "apply dvd_mul_right"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := Nat.gcd x y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : Nat.coprime m n\nhx : x = m * g\nhy : y = n * g\nthis : m * m = d * (n * n)\nco2 : Nat.coprime (m * m) (n * n)\n⊢ m * m ∣ d * (n * n)",
"tactic": "simp [this]"
}
] |
[
912,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
891,
1
] |
Mathlib/Analysis/Normed/Group/HomCompletion.lean
|
NormedAddCommGroup.norm_toCompl
|
[] |
[
150,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Topology/Separation.lean
|
lim_eq
|
[] |
[
1031,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1030,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean
|
DifferentiableAt.norm
|
[] |
[
239,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Algebra/Module/Injective.lean
|
Module.Baer.extensionOfMax_le
|
[
{
"state_after": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nx : { x // x ∈ (extensionOfMax i f).toLinearPMap.domain }\nx' : { x // x ∈ (extensionOfMaxAdjoin i f h y).toLinearPMap.domain }\nEQ : ↑x = ↑x'\n⊢ ↑(extensionOfMaxAdjoin i f h y).toLinearPMap x' = ↑(extensionOfMax i f).toLinearPMap x",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nx : { x // x ∈ (extensionOfMax i f).toLinearPMap.domain }\nx' : { x // x ∈ (extensionOfMaxAdjoin i f h y).toLinearPMap.domain }\nEQ : ↑x = ↑x'\n⊢ ↑(extensionOfMax i f).toLinearPMap x = ↑(extensionOfMaxAdjoin i f h y).toLinearPMap x'",
"tactic": "symm"
},
{
"state_after": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nx : { x // x ∈ (extensionOfMax i f).toLinearPMap.domain }\nx' : { x // x ∈ (extensionOfMaxAdjoin i f h y).toLinearPMap.domain }\nEQ : ↑x = ↑x'\n⊢ ExtensionOfMaxAdjoin.extensionToFun i f h x' = ↑(extensionOfMax i f).toLinearPMap x",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nx : { x // x ∈ (extensionOfMax i f).toLinearPMap.domain }\nx' : { x // x ∈ (extensionOfMaxAdjoin i f h y).toLinearPMap.domain }\nEQ : ↑x = ↑x'\n⊢ ↑(extensionOfMaxAdjoin i f h y).toLinearPMap x' = ↑(extensionOfMax i f).toLinearPMap x",
"tactic": "change ExtensionOfMaxAdjoin.extensionToFun i f h _ = _"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nx : { x // x ∈ (extensionOfMax i f).toLinearPMap.domain }\nx' : { x // x ∈ (extensionOfMaxAdjoin i f h y).toLinearPMap.domain }\nEQ : ↑x = ↑x'\n⊢ ExtensionOfMaxAdjoin.extensionToFun i f h x' = ↑(extensionOfMax i f).toLinearPMap x",
"tactic": "rw [ExtensionOfMaxAdjoin.extensionToFun_wd i f h x' x 0 (by simp [EQ]), map_zero,\n add_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nx : { x // x ∈ (extensionOfMax i f).toLinearPMap.domain }\nx' : { x // x ∈ (extensionOfMaxAdjoin i f h y).toLinearPMap.domain }\nEQ : ↑x = ↑x'\n⊢ ↑x' = ↑x + 0 • y",
"tactic": "simp [EQ]"
}
] |
[
438,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/Geometry/Euclidean/Sphere/Basic.lean
|
EuclideanGeometry.Cospherical.affineIndependent_of_mem_of_ne
|
[
{
"state_after": "case refine'_1\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nhs : Cospherical s\np₁ p₂ p₃ : P\nh₁ : p₁ ∈ s\nh₂ : p₂ ∈ s\nh₃ : p₃ ∈ s\nh₁₂ : p₁ ≠ p₂\nh₁₃ : p₁ ≠ p₃\nh₂₃ : p₂ ≠ p₃\n⊢ Set.range ![p₁, p₂, p₃] ⊆ s\n\ncase refine'_2\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nhs : Cospherical s\np₁ p₂ p₃ : P\nh₁ : p₁ ∈ s\nh₂ : p₂ ∈ s\nh₃ : p₃ ∈ s\nh₁₂ : p₁ ≠ p₂\nh₁₃ : p₁ ≠ p₃\nh₂₃ : p₂ ≠ p₃\n⊢ Function.Injective ![p₁, p₂, p₃]",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nhs : Cospherical s\np₁ p₂ p₃ : P\nh₁ : p₁ ∈ s\nh₂ : p₂ ∈ s\nh₃ : p₃ ∈ s\nh₁₂ : p₁ ≠ p₂\nh₁₃ : p₁ ≠ p₃\nh₂₃ : p₂ ≠ p₃\n⊢ AffineIndependent ℝ ![p₁, p₂, p₃]",
"tactic": "refine' hs.affineIndependent _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nhs : Cospherical s\np₁ p₂ p₃ : P\nh₁ : p₁ ∈ s\nh₂ : p₂ ∈ s\nh₃ : p₃ ∈ s\nh₁₂ : p₁ ≠ p₂\nh₁₃ : p₁ ≠ p₃\nh₂₃ : p₂ ≠ p₃\n⊢ Set.range ![p₁, p₂, p₃] ⊆ s",
"tactic": "simp [h₁, h₂, h₃, Set.insert_subset]"
},
{
"state_after": "case refine'_2\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nhs : Cospherical s\np₁ p₂ p₃ : P\nh₁ : p₁ ∈ s\nh₂ : p₂ ∈ s\nh₃ : p₃ ∈ s\nh₁₂ : p₁ ≠ p₂\nh₁₃ : p₁ ≠ p₃\nh₂₃ : p₂ ≠ p₃\n⊢ ¬p₁ ∈ Set.range ![p₂, p₃] ∧ ¬p₂ ∈ Set.range ![p₃] ∧ Function.Injective ![p₃]",
"state_before": "case refine'_2\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nhs : Cospherical s\np₁ p₂ p₃ : P\nh₁ : p₁ ∈ s\nh₂ : p₂ ∈ s\nh₃ : p₃ ∈ s\nh₁₂ : p₁ ≠ p₂\nh₁₃ : p₁ ≠ p₃\nh₂₃ : p₂ ≠ p₃\n⊢ Function.Injective ![p₁, p₂, p₃]",
"tactic": "erw [Fin.cons_injective_iff, Fin.cons_injective_iff]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nhs : Cospherical s\np₁ p₂ p₃ : P\nh₁ : p₁ ∈ s\nh₂ : p₂ ∈ s\nh₃ : p₃ ∈ s\nh₁₂ : p₁ ≠ p₂\nh₁₃ : p₁ ≠ p₃\nh₂₃ : p₂ ≠ p₃\n⊢ ¬p₁ ∈ Set.range ![p₂, p₃] ∧ ¬p₂ ∈ Set.range ![p₃] ∧ Function.Injective ![p₃]",
"tactic": "simp [h₁₂, h₁₃, h₂₃, Function.Injective]"
}
] |
[
286,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
|
SimpleGraph.left_nonuniformWitnesses_subset
|
[
{
"state_after": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬IsUniform G ε s t\n⊢ (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧ ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))),\n Exists.choose\n (_ :\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤\n ↑(card\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))))) ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤\n ↑(abs\n (edgeDensity G\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))))\n t' -\n edgeDensity G s t)))).fst ⊆\n s",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬IsUniform G ε s t\n⊢ (nonuniformWitnesses G ε s t).fst ⊆ s",
"tactic": "rw [nonuniformWitnesses, dif_pos h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬IsUniform G ε s t\n⊢ (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧ ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))),\n Exists.choose\n (_ :\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤\n ↑(card\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))))) ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤\n ↑(abs\n (edgeDensity G\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))))\n t' -\n edgeDensity G s t)))).fst ⊆\n s",
"tactic": "exact (not_isUniform_iff.1 h).choose_spec.1"
}
] |
[
131,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiff_neg
|
[] |
[
1263,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1262,
1
] |
Mathlib/Data/Pi/Algebra.lean
|
Function.injective_pi_map
|
[] |
[
418,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
mul_lt_one_of_nonneg_of_lt_one_right
|
[] |
[
333,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisCoinsertion.isLUB_of_l_image
|
[] |
[
850,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
848,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.PartiallyWellOrderedOn.mono
|
[] |
[
248,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.ext
|
[
{
"state_after": "case h.h.a\nM : Type u_1\nN : Type ?u.7026\nP : Type ?u.7029\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nc✝ c d : Con M\nH : ∀ (x y : M), ↑c x y ↔ ↑d x y\nx✝¹ x✝ : M\n⊢ r x✝¹ x✝ ↔ r x✝¹ x✝",
"state_before": "M : Type u_1\nN : Type ?u.7026\nP : Type ?u.7029\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nc✝ c d : Con M\nH : ∀ (x y : M), ↑c x y ↔ ↑d x y\n⊢ r = r",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h.a\nM : Type u_1\nN : Type ?u.7026\nP : Type ?u.7029\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nc✝ c d : Con M\nH : ∀ (x y : M), ↑c x y ↔ ↑d x y\nx✝¹ x✝ : M\n⊢ r x✝¹ x✝ ↔ r x✝¹ x✝",
"tactic": "apply H"
}
] |
[
195,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.mul_assoc
|
[
{
"state_after": "no goals",
"state_before": "a b c : Int\n⊢ a * b * c = a * (b * c)",
"tactic": "cases a <;> cases b <;> cases c <;> simp [Nat.mul_assoc]"
}
] |
[
398,
59
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
397,
11
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
norm_expSeries_summable_of_mem_ball'
|
[
{
"state_after": "𝕂 : Type u_2\n𝔸 : Type u_1\n𝔹 : Type ?u.75852\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : NormedAlgebra 𝕂 𝔹\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ Summable (norm ∘ fun n => (↑n !)⁻¹ • x ^ n)",
"state_before": "𝕂 : Type u_2\n𝔸 : Type u_1\n𝔹 : Type ?u.75852\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : NormedAlgebra 𝕂 𝔹\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ Summable fun n => ‖(↑n !)⁻¹ • x ^ n‖",
"tactic": "change Summable (norm ∘ _)"
},
{
"state_after": "𝕂 : Type u_2\n𝔸 : Type u_1\n𝔹 : Type ?u.75852\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : NormedAlgebra 𝕂 𝔹\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ Summable (norm ∘ fun n => ↑(expSeries 𝕂 𝔸 n) fun x_1 => x)",
"state_before": "𝕂 : Type u_2\n𝔸 : Type u_1\n𝔹 : Type ?u.75852\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : NormedAlgebra 𝕂 𝔹\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ Summable (norm ∘ fun n => (↑n !)⁻¹ • x ^ n)",
"tactic": "rw [← expSeries_apply_eq']"
},
{
"state_after": "no goals",
"state_before": "𝕂 : Type u_2\n𝔸 : Type u_1\n𝔹 : Type ?u.75852\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : NormedAlgebra 𝕂 𝔹\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ Summable (norm ∘ fun n => ↑(expSeries 𝕂 𝔸 n) fun x_1 => x)",
"tactic": "exact norm_expSeries_summable_of_mem_ball x hx"
}
] |
[
209,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/Algebra/Parity.lean
|
even_iff_two_dvd
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.63447\nα : Type u_1\nβ : Type ?u.63453\nR : Type ?u.63456\ninst✝¹ : Semiring α\ninst✝ : Semiring β\nm n a : α\n⊢ Even a ↔ 2 ∣ a",
"tactic": "simp [Even, Dvd.dvd, two_mul]"
}
] |
[
250,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.iUnion_ball_nat_succ
|
[] |
[
473,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/Algebra/GCDMonoid/Finset.lean
|
Finset.gcd_mul_left
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.34528\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns s₁ s₂ : Finset β\nf : β → α\na : α\n⊢ (gcd ∅ fun x => a * f x) = ↑normalize a * gcd ∅ f\n\ncase refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.34528\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns s₁ s₂ : Finset β\nf : β → α\na : α\n⊢ ∀ ⦃a_1 : β⦄ {s : Finset β},\n ¬a_1 ∈ s →\n (gcd s fun x => a * f x) = ↑normalize a * gcd s f →\n (gcd (insert a_1 s) fun x => a * f x) = ↑normalize a * gcd (insert a_1 s) f",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.34528\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns s₁ s₂ : Finset β\nf : β → α\na : α\n⊢ (gcd s fun x => a * f x) = ↑normalize a * gcd s f",
"tactic": "refine' s.induction_on _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.34528\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns s₁ s₂ : Finset β\nf : β → α\na : α\n⊢ (gcd ∅ fun x => a * f x) = ↑normalize a * gcd ∅ f",
"tactic": "simp"
},
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.34528\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns s₁ s₂ : Finset β\nf : β → α\na : α\nb : β\nt : Finset β\na✝ : ¬b ∈ t\nh : (gcd t fun x => a * f x) = ↑normalize a * gcd t f\n⊢ (gcd (insert b t) fun x => a * f x) = ↑normalize a * gcd (insert b t) f",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.34528\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns s₁ s₂ : Finset β\nf : β → α\na : α\n⊢ ∀ ⦃a_1 : β⦄ {s : Finset β},\n ¬a_1 ∈ s →\n (gcd s fun x => a * f x) = ↑normalize a * gcd s f →\n (gcd (insert a_1 s) fun x => a * f x) = ↑normalize a * gcd (insert a_1 s) f",
"tactic": "intro b t _ h"
},
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.34528\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns s₁ s₂ : Finset β\nf : β → α\na : α\nb : β\nt : Finset β\na✝ : ¬b ∈ t\nh : (gcd t fun x => a * f x) = ↑normalize a * gcd t f\n⊢ GCDMonoid.gcd (a * f b) (↑normalize a * gcd t f) = GCDMonoid.gcd (a * f b) (a * gcd t f)",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.34528\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns s₁ s₂ : Finset β\nf : β → α\na : α\nb : β\nt : Finset β\na✝ : ¬b ∈ t\nh : (gcd t fun x => a * f x) = ↑normalize a * gcd t f\n⊢ (gcd (insert b t) fun x => a * f x) = ↑normalize a * gcd (insert b t) f",
"tactic": "rw [gcd_insert, gcd_insert, h, ← gcd_mul_left]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.34528\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns s₁ s₂ : Finset β\nf : β → α\na : α\nb : β\nt : Finset β\na✝ : ¬b ∈ t\nh : (gcd t fun x => a * f x) = ↑normalize a * gcd t f\n⊢ GCDMonoid.gcd (a * f b) (↑normalize a * gcd t f) = GCDMonoid.gcd (a * f b) (a * gcd t f)",
"tactic": "apply ((normalize_associated a).mul_right _).gcd_eq_right"
}
] |
[
252,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
8
] |
Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
|
BilinForm.toMatrix_symm
|
[] |
[
327,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
326,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.lTensorHomToHomLTensor_apply
|
[] |
[
846,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
844,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
NNReal.rpow_sub
|
[] |
[
100,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.neighborSet_inf
|
[] |
[
503,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/GroupTheory/PresentedGroup.lean
|
PresentedGroup.to_group_eq_one_of_mem_closure
|
[] |
[
73,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sup_biSup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.103547\nβ₂ : Type ?u.103550\nγ : Type ?u.103553\nι : Sort u_1\nι' : Sort ?u.103559\nκ : ι → Sort ?u.103564\nκ' : ι' → Sort ?u.103569\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na✝ b : α\np : ι → Prop\nf : (i : ι) → p i → α\na : α\nh : ∃ i, p i\n⊢ (a ⊔ ⨆ (i : ι) (h : p i), f i h) = ⨆ (i : ι) (h : p i), a ⊔ f i h",
"tactic": "simpa only [sup_comm] using @biSup_sup α _ _ p _ _ h"
}
] |
[
1275,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1273,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
AeStronglyMeasurable.stronglyMeasurableAtFilter_of_mem
|
[] |
[
68,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Icc_subset_Icc_union_Ioc
|
[] |
[
1580,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1579,
1
] |
Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean
|
Geometry.SimplicialComplex.face_subset_face_iff
|
[] |
[
185,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Data/List/Range.lean
|
List.pairwise_lt_range'
|
[] |
[
47,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Std/Data/List/Basic.lean
|
List.replicateTR_loop_eq
|
[
{
"state_after": "α✝ : Type u_1\na : α✝\nacc : List α✝\nn : Nat\n⊢ replicate n a ++ a :: acc = replicate n a ++ ([a] ++ acc)",
"state_before": "α✝ : Type u_1\na : α✝\nacc : List α✝\nn : Nat\n⊢ replicateTR.loop a (n + 1) acc = replicate (n + 1) a ++ acc",
"tactic": "rw [← replicateTR_loop_replicate_eq _ 1 n, replicate, replicate,\nreplicateTR.loop, replicateTR_loop_eq n, replicateTR_loop_eq n, append_assoc]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\na : α✝\nacc : List α✝\nn : Nat\n⊢ replicate n a ++ a :: acc = replicate n a ++ ([a] ++ acc)",
"tactic": "rfl"
}
] |
[
211,
87
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
208,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.integrableOn_iUnion_of_summable_norm_restrict
|
[
{
"state_after": "α : Type u_1\nβ : Type u_3\nE : Type u_2\nF : Type ?u.308351\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : Countable β\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : C(α, E)\ns : β → Compacts α\nhf : Summable fun i => ‖ContinuousMap.restrict (↑(s i)) f‖ * ENNReal.toReal (↑↑μ ↑(s i))\ni : β\n⊢ (∫ (a : α) in ↑(s i), ‖↑f a‖ ∂μ) ≤ ‖ContinuousMap.restrict (↑(s i)) f‖ * ENNReal.toReal (↑↑μ ↑(s i))",
"state_before": "α : Type u_1\nβ : Type u_3\nE : Type u_2\nF : Type ?u.308351\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : Countable β\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : C(α, E)\ns : β → Compacts α\nhf : Summable fun i => ‖ContinuousMap.restrict (↑(s i)) f‖ * ENNReal.toReal (↑↑μ ↑(s i))\n⊢ IntegrableOn (↑f) (⋃ (i : β), ↑(s i))",
"tactic": "refine'\n integrableOn_iUnion_of_summable_integral_norm (fun i => (s i).isCompact.isClosed.measurableSet)\n (fun i => (map_continuous f).continuousOn.integrableOn_compact (s i).isCompact)\n (summable_of_nonneg_of_le (fun ι => integral_nonneg fun x => norm_nonneg _) (fun i => _) hf)"
},
{
"state_after": "α : Type u_1\nβ : Type u_3\nE : Type u_2\nF : Type ?u.308351\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : Countable β\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : C(α, E)\ns : β → Compacts α\nhf : Summable fun i => ‖ContinuousMap.restrict (↑(s i)) f‖ * ENNReal.toReal (↑↑μ ↑(s i))\ni : β\n⊢ ‖∫ (x : α) in ↑(s i), ‖↑f x‖ ∂μ‖ ≤ ‖ContinuousMap.restrict (↑(s i)) f‖ * ENNReal.toReal (↑↑μ ↑(s i))",
"state_before": "α : Type u_1\nβ : Type u_3\nE : Type u_2\nF : Type ?u.308351\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : Countable β\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : C(α, E)\ns : β → Compacts α\nhf : Summable fun i => ‖ContinuousMap.restrict (↑(s i)) f‖ * ENNReal.toReal (↑↑μ ↑(s i))\ni : β\n⊢ (∫ (a : α) in ↑(s i), ‖↑f a‖ ∂μ) ≤ ‖ContinuousMap.restrict (↑(s i)) f‖ * ENNReal.toReal (↑↑μ ↑(s i))",
"tactic": "rw [← (Real.norm_of_nonneg (integral_nonneg fun a => norm_nonneg _) : ‖_‖ = ∫ x in s i, ‖f x‖ ∂μ)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nE : Type u_2\nF : Type ?u.308351\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : Countable β\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : C(α, E)\ns : β → Compacts α\nhf : Summable fun i => ‖ContinuousMap.restrict (↑(s i)) f‖ * ENNReal.toReal (↑↑μ ↑(s i))\ni : β\n⊢ ‖∫ (x : α) in ↑(s i), ‖↑f x‖ ∂μ‖ ≤ ‖ContinuousMap.restrict (↑(s i)) f‖ * ENNReal.toReal (↑↑μ ↑(s i))",
"tactic": "exact\n norm_set_integral_le_of_norm_le_const' (s i).isCompact.measure_lt_top\n (s i).isCompact.isClosed.measurableSet fun x hx =>\n (norm_norm (f x)).symm ▸ (f.restrict (s i : Set α)).norm_coe_le_norm ⟨x, hx⟩"
}
] |
[
818,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
807,
1
] |
Mathlib/Algebra/ModEq.lean
|
AddCommGroup.modEq_refl
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddCommGroup α\np a✝ a₁ a₂ b b₁ b₂ c : α\nn : ℕ\nz : ℤ\na : α\n⊢ a - a = 0 • p",
"tactic": "simp"
}
] |
[
57,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.coe_ringHom_injective
|
[] |
[
203,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
Int.natAbs_pow
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.267830\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\nn : ℤ\nk : ℕ\n⊢ natAbs (n ^ k) = natAbs n ^ k",
"tactic": "induction' k with k ih <;> [rfl; rw [pow_succ', Int.natAbs_mul, pow_succ', ih]]"
}
] |
[
551,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
550,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
Memℓp.of_exponent_ge
|
[
{
"state_after": "case inl.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ 0\n⊢ Memℓp f 0\n\ncase inr.inl.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\n⊢ Memℓp f ⊤\n\ncase inr.inr.inl.intro\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\nf : (i : α) → E i\nhp : 0 < ENNReal.toReal p\nhfq : Memℓp f 0\nhpq : 0 ≤ p\n⊢ Memℓp f p\n\ncase inr.inr.inr.inl.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f ⊤\nhpq : ⊤ ≤ ⊤\n⊢ Memℓp f ⊤\n\ncase inr.inr.inr.inr.inl.intro\nα : Type u_1\nE : α → Type u_2\np q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nq : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhq : 0 < ENNReal.toReal q\nhpq : q ≤ ⊤\n⊢ Memℓp f ⊤\n\ncase inr.inr.inr.inr.inr.intro.intro\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\n⊢ Memℓp f p",
"state_before": "α : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\n⊢ Memℓp f p",
"tactic": "rcases ENNReal.trichotomy₂ hpq with\n (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ | ⟨rfl, hp⟩ | ⟨rfl, rfl⟩ | ⟨hq, rfl⟩ | ⟨hq, _, hpq'⟩)"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ 0\n⊢ Memℓp f 0",
"tactic": "exact hfq"
},
{
"state_after": "case inr.inl.intro.hf\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\n⊢ BddAbove (Set.range fun i => ‖f i‖)",
"state_before": "case inr.inl.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\n⊢ Memℓp f ⊤",
"tactic": "apply memℓp_infty"
},
{
"state_after": "case inr.inl.intro.hf.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\nC : ℝ\nhC : C ∈ upperBounds ((fun i => ‖f i‖) '' {i | f i ≠ 0})\n⊢ BddAbove (Set.range fun i => ‖f i‖)",
"state_before": "case inr.inl.intro.hf\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\n⊢ BddAbove (Set.range fun i => ‖f i‖)",
"tactic": "obtain ⟨C, hC⟩ := (hfq.finite_dsupport.image fun i => ‖f i‖).bddAbove"
},
{
"state_after": "case inr.inl.intro.hf.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\nC : ℝ\nhC : C ∈ upperBounds ((fun i => ‖f i‖) '' {i | f i ≠ 0})\n⊢ max 0 C ∈ upperBounds (Set.range fun i => ‖f i‖)",
"state_before": "case inr.inl.intro.hf.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\nC : ℝ\nhC : C ∈ upperBounds ((fun i => ‖f i‖) '' {i | f i ≠ 0})\n⊢ BddAbove (Set.range fun i => ‖f i‖)",
"tactic": "use max 0 C"
},
{
"state_after": "case inr.inl.intro.hf.intro.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\nC : ℝ\nhC : C ∈ upperBounds ((fun i => ‖f i‖) '' {i | f i ≠ 0})\ni : α\n⊢ (fun i => ‖f i‖) i ≤ max 0 C",
"state_before": "case inr.inl.intro.hf.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\nC : ℝ\nhC : C ∈ upperBounds ((fun i => ‖f i‖) '' {i | f i ≠ 0})\n⊢ max 0 C ∈ upperBounds (Set.range fun i => ‖f i‖)",
"tactic": "rintro x ⟨i, rfl⟩"
},
{
"state_after": "case pos\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\nC : ℝ\nhC : C ∈ upperBounds ((fun i => ‖f i‖) '' {i | f i ≠ 0})\ni : α\nhi : f i = 0\n⊢ (fun i => ‖f i‖) i ≤ max 0 C\n\ncase neg\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\nC : ℝ\nhC : C ∈ upperBounds ((fun i => ‖f i‖) '' {i | f i ≠ 0})\ni : α\nhi : ¬f i = 0\n⊢ (fun i => ‖f i‖) i ≤ max 0 C",
"state_before": "case inr.inl.intro.hf.intro.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\nC : ℝ\nhC : C ∈ upperBounds ((fun i => ‖f i‖) '' {i | f i ≠ 0})\ni : α\n⊢ (fun i => ‖f i‖) i ≤ max 0 C",
"tactic": "by_cases hi : f i = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\nC : ℝ\nhC : C ∈ upperBounds ((fun i => ‖f i‖) '' {i | f i ≠ 0})\ni : α\nhi : f i = 0\n⊢ (fun i => ‖f i‖) i ≤ max 0 C",
"tactic": "simp [hi]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f 0\nhpq : 0 ≤ ⊤\nC : ℝ\nhC : C ∈ upperBounds ((fun i => ‖f i‖) '' {i | f i ≠ 0})\ni : α\nhi : ¬f i = 0\n⊢ (fun i => ‖f i‖) i ≤ max 0 C",
"tactic": "exact (hC ⟨i, hi, rfl⟩).trans (le_max_right _ _)"
},
{
"state_after": "case inr.inr.inl.intro.hf\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\nf : (i : α) → E i\nhp : 0 < ENNReal.toReal p\nhfq : Memℓp f 0\nhpq : 0 ≤ p\n⊢ Summable fun i => ‖f i‖ ^ ENNReal.toReal p",
"state_before": "case inr.inr.inl.intro\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\nf : (i : α) → E i\nhp : 0 < ENNReal.toReal p\nhfq : Memℓp f 0\nhpq : 0 ≤ p\n⊢ Memℓp f p",
"tactic": "apply memℓp_gen"
},
{
"state_after": "case inr.inr.inl.intro.hf\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\nf : (i : α) → E i\nhp : 0 < ENNReal.toReal p\nhfq : Memℓp f 0\nhpq : 0 ≤ p\nthis : ∀ (i : α), ¬i ∈ Set.Finite.toFinset (_ : Set.Finite {i | f i ≠ 0}) → ‖f i‖ ^ ENNReal.toReal p = 0\n⊢ Summable fun i => ‖f i‖ ^ ENNReal.toReal p",
"state_before": "case inr.inr.inl.intro.hf\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\nf : (i : α) → E i\nhp : 0 < ENNReal.toReal p\nhfq : Memℓp f 0\nhpq : 0 ≤ p\n⊢ Summable fun i => ‖f i‖ ^ ENNReal.toReal p",
"tactic": "have : ∀ (i) (_ : i ∉ hfq.finite_dsupport.toFinset), ‖f i‖ ^ p.toReal = 0 := by\n intro i hi\n have : f i = 0 := by simpa using hi\n simp [this, Real.zero_rpow hp.ne']"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inl.intro.hf\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\nf : (i : α) → E i\nhp : 0 < ENNReal.toReal p\nhfq : Memℓp f 0\nhpq : 0 ≤ p\nthis : ∀ (i : α), ¬i ∈ Set.Finite.toFinset (_ : Set.Finite {i | f i ≠ 0}) → ‖f i‖ ^ ENNReal.toReal p = 0\n⊢ Summable fun i => ‖f i‖ ^ ENNReal.toReal p",
"tactic": "exact summable_of_ne_finset_zero this"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\nf : (i : α) → E i\nhp : 0 < ENNReal.toReal p\nhfq : Memℓp f 0\nhpq : 0 ≤ p\ni : α\nhi : ¬i ∈ Set.Finite.toFinset (_ : Set.Finite {i | f i ≠ 0})\n⊢ ‖f i‖ ^ ENNReal.toReal p = 0",
"state_before": "α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\nf : (i : α) → E i\nhp : 0 < ENNReal.toReal p\nhfq : Memℓp f 0\nhpq : 0 ≤ p\n⊢ ∀ (i : α), ¬i ∈ Set.Finite.toFinset (_ : Set.Finite {i | f i ≠ 0}) → ‖f i‖ ^ ENNReal.toReal p = 0",
"tactic": "intro i hi"
},
{
"state_after": "α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\nf : (i : α) → E i\nhp : 0 < ENNReal.toReal p\nhfq : Memℓp f 0\nhpq : 0 ≤ p\ni : α\nhi : ¬i ∈ Set.Finite.toFinset (_ : Set.Finite {i | f i ≠ 0})\nthis : f i = 0\n⊢ ‖f i‖ ^ ENNReal.toReal p = 0",
"state_before": "α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\nf : (i : α) → E i\nhp : 0 < ENNReal.toReal p\nhfq : Memℓp f 0\nhpq : 0 ≤ p\ni : α\nhi : ¬i ∈ Set.Finite.toFinset (_ : Set.Finite {i | f i ≠ 0})\n⊢ ‖f i‖ ^ ENNReal.toReal p = 0",
"tactic": "have : f i = 0 := by simpa using hi"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\nf : (i : α) → E i\nhp : 0 < ENNReal.toReal p\nhfq : Memℓp f 0\nhpq : 0 ≤ p\ni : α\nhi : ¬i ∈ Set.Finite.toFinset (_ : Set.Finite {i | f i ≠ 0})\nthis : f i = 0\n⊢ ‖f i‖ ^ ENNReal.toReal p = 0",
"tactic": "simp [this, Real.zero_rpow hp.ne']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\nf : (i : α) → E i\nhp : 0 < ENNReal.toReal p\nhfq : Memℓp f 0\nhpq : 0 ≤ p\ni : α\nhi : ¬i ∈ Set.Finite.toFinset (_ : Set.Finite {i | f i ≠ 0})\n⊢ f i = 0",
"tactic": "simpa using hi"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inl.intro\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhfq : Memℓp f ⊤\nhpq : ⊤ ≤ ⊤\n⊢ Memℓp f ⊤",
"tactic": "exact hfq"
},
{
"state_after": "case inr.inr.inr.inr.inl.intro.hf\nα : Type u_1\nE : α → Type u_2\np q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nq : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhq : 0 < ENNReal.toReal q\nhpq : q ≤ ⊤\n⊢ BddAbove (Set.range fun i => ‖f i‖)",
"state_before": "case inr.inr.inr.inr.inl.intro\nα : Type u_1\nE : α → Type u_2\np q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nq : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhq : 0 < ENNReal.toReal q\nhpq : q ≤ ⊤\n⊢ Memℓp f ⊤",
"tactic": "apply memℓp_infty"
},
{
"state_after": "case inr.inr.inr.inr.inl.intro.hf.intro\nα : Type u_1\nE : α → Type u_2\np q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nq : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhq : 0 < ENNReal.toReal q\nhpq : q ≤ ⊤\nA : ℝ\nhA : A ∈ upperBounds (Set.range fun i => ‖f i‖ ^ ENNReal.toReal q)\n⊢ BddAbove (Set.range fun i => ‖f i‖)",
"state_before": "case inr.inr.inr.inr.inl.intro.hf\nα : Type u_1\nE : α → Type u_2\np q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nq : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhq : 0 < ENNReal.toReal q\nhpq : q ≤ ⊤\n⊢ BddAbove (Set.range fun i => ‖f i‖)",
"tactic": "obtain ⟨A, hA⟩ := (hfq.summable hq).tendsto_cofinite_zero.bddAbove_range_of_cofinite"
},
{
"state_after": "case inr.inr.inr.inr.inl.intro.hf.intro\nα : Type u_1\nE : α → Type u_2\np q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nq : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhq : 0 < ENNReal.toReal q\nhpq : q ≤ ⊤\nA : ℝ\nhA : A ∈ upperBounds (Set.range fun i => ‖f i‖ ^ ENNReal.toReal q)\n⊢ A ^ (ENNReal.toReal q)⁻¹ ∈ upperBounds (Set.range fun i => ‖f i‖)",
"state_before": "case inr.inr.inr.inr.inl.intro.hf.intro\nα : Type u_1\nE : α → Type u_2\np q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nq : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhq : 0 < ENNReal.toReal q\nhpq : q ≤ ⊤\nA : ℝ\nhA : A ∈ upperBounds (Set.range fun i => ‖f i‖ ^ ENNReal.toReal q)\n⊢ BddAbove (Set.range fun i => ‖f i‖)",
"tactic": "use A ^ q.toReal⁻¹"
},
{
"state_after": "case inr.inr.inr.inr.inl.intro.hf.intro.intro\nα : Type u_1\nE : α → Type u_2\np q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nq : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhq : 0 < ENNReal.toReal q\nhpq : q ≤ ⊤\nA : ℝ\nhA : A ∈ upperBounds (Set.range fun i => ‖f i‖ ^ ENNReal.toReal q)\ni : α\n⊢ (fun i => ‖f i‖) i ≤ A ^ (ENNReal.toReal q)⁻¹",
"state_before": "case inr.inr.inr.inr.inl.intro.hf.intro\nα : Type u_1\nE : α → Type u_2\np q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nq : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhq : 0 < ENNReal.toReal q\nhpq : q ≤ ⊤\nA : ℝ\nhA : A ∈ upperBounds (Set.range fun i => ‖f i‖ ^ ENNReal.toReal q)\n⊢ A ^ (ENNReal.toReal q)⁻¹ ∈ upperBounds (Set.range fun i => ‖f i‖)",
"tactic": "rintro x ⟨i, rfl⟩"
},
{
"state_after": "case inr.inr.inr.inr.inl.intro.hf.intro.intro\nα : Type u_1\nE : α → Type u_2\np q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nq : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhq : 0 < ENNReal.toReal q\nhpq : q ≤ ⊤\nA : ℝ\nhA : A ∈ upperBounds (Set.range fun i => ‖f i‖ ^ ENNReal.toReal q)\ni : α\nthis : 0 ≤ ‖f i‖ ^ ENNReal.toReal q\n⊢ (fun i => ‖f i‖) i ≤ A ^ (ENNReal.toReal q)⁻¹",
"state_before": "case inr.inr.inr.inr.inl.intro.hf.intro.intro\nα : Type u_1\nE : α → Type u_2\np q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nq : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhq : 0 < ENNReal.toReal q\nhpq : q ≤ ⊤\nA : ℝ\nhA : A ∈ upperBounds (Set.range fun i => ‖f i‖ ^ ENNReal.toReal q)\ni : α\n⊢ (fun i => ‖f i‖) i ≤ A ^ (ENNReal.toReal q)⁻¹",
"tactic": "have : 0 ≤ ‖f i‖ ^ q.toReal := Real.rpow_nonneg_of_nonneg (norm_nonneg _) _"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inr.inl.intro.hf.intro.intro\nα : Type u_1\nE : α → Type u_2\np q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nq : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhq : 0 < ENNReal.toReal q\nhpq : q ≤ ⊤\nA : ℝ\nhA : A ∈ upperBounds (Set.range fun i => ‖f i‖ ^ ENNReal.toReal q)\ni : α\nthis : 0 ≤ ‖f i‖ ^ ENNReal.toReal q\n⊢ (fun i => ‖f i‖) i ≤ A ^ (ENNReal.toReal q)⁻¹",
"tactic": "simpa [← Real.rpow_mul, mul_inv_cancel hq.ne'] using\n Real.rpow_le_rpow this (hA ⟨i, rfl⟩) (inv_nonneg.mpr hq.le)"
},
{
"state_after": "case inr.inr.inr.inr.inr.intro.intro.hf\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\n⊢ Summable fun i => ‖f i‖ ^ ENNReal.toReal p",
"state_before": "case inr.inr.inr.inr.inr.intro.intro\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\n⊢ Memℓp f p",
"tactic": "apply memℓp_gen"
},
{
"state_after": "case inr.inr.inr.inr.inr.intro.intro.hf\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\n⊢ Summable fun i => ‖f i‖ ^ ENNReal.toReal p",
"state_before": "case inr.inr.inr.inr.inr.intro.intro.hf\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\n⊢ Summable fun i => ‖f i‖ ^ ENNReal.toReal p",
"tactic": "have hf' := hfq.summable hq"
},
{
"state_after": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_1\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\n⊢ Set.Finite {i | 1 ≤ ‖f i‖}\n\ncase inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\n⊢ {x | (fun i => ‖‖f i‖ ^ ENNReal.toReal p‖ ≤ ‖f i‖ ^ ENNReal.toReal q) x}ᶜ ⊆ {i | 1 ≤ ‖f i‖}",
"state_before": "case inr.inr.inr.inr.inr.intro.intro.hf\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\n⊢ Summable fun i => ‖f i‖ ^ ENNReal.toReal p",
"tactic": "refine' summable_of_norm_bounded_eventually _ hf' (@Set.Finite.subset _ { i | 1 ≤ ‖f i‖ } _ _ _)"
},
{
"state_after": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_1\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\nH : Set.Finite {x | 1 ≤ ‖f x‖ ^ ENNReal.toReal q}\n⊢ Set.Finite {i | 1 ≤ ‖f i‖}",
"state_before": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_1\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\n⊢ Set.Finite {i | 1 ≤ ‖f i‖}",
"tactic": "have H : { x : α | 1 ≤ ‖f x‖ ^ q.toReal }.Finite := by\n simpa using eventually_lt_of_tendsto_lt (by norm_num) hf'.tendsto_cofinite_zero"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_1\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\nH : Set.Finite {x | 1 ≤ ‖f x‖ ^ ENNReal.toReal q}\n⊢ Set.Finite {i | 1 ≤ ‖f i‖}",
"tactic": "exact H.subset fun i hi => Real.one_le_rpow hi hq.le"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\n⊢ Set.Finite {x | 1 ≤ ‖f x‖ ^ ENNReal.toReal q}",
"tactic": "simpa using eventually_lt_of_tendsto_lt (by norm_num) hf'.tendsto_cofinite_zero"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\n⊢ 0 < 1",
"tactic": "norm_num"
},
{
"state_after": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\n⊢ ∀ (i : α), ¬abs (‖f i‖ ^ ENNReal.toReal p) ≤ ‖f i‖ ^ ENNReal.toReal q → 1 ≤ ‖f i‖",
"state_before": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\n⊢ {x | (fun i => ‖‖f i‖ ^ ENNReal.toReal p‖ ≤ ‖f i‖ ^ ENNReal.toReal q) x}ᶜ ⊆ {i | 1 ≤ ‖f i‖}",
"tactic": "show ∀ i, ¬|‖f i‖ ^ p.toReal| ≤ ‖f i‖ ^ q.toReal → 1 ≤ ‖f i‖"
},
{
"state_after": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\ni : α\nhi : ¬abs (‖f i‖ ^ ENNReal.toReal p) ≤ ‖f i‖ ^ ENNReal.toReal q\n⊢ 1 ≤ ‖f i‖",
"state_before": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\n⊢ ∀ (i : α), ¬abs (‖f i‖ ^ ENNReal.toReal p) ≤ ‖f i‖ ^ ENNReal.toReal q → 1 ≤ ‖f i‖",
"tactic": "intro i hi"
},
{
"state_after": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\ni : α\nhi : ¬abs (‖f i‖ ^ ENNReal.toReal p) ≤ ‖f i‖ ^ ENNReal.toReal q\nthis : 0 ≤ ‖f i‖ ^ ENNReal.toReal p\n⊢ 1 ≤ ‖f i‖",
"state_before": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\ni : α\nhi : ¬abs (‖f i‖ ^ ENNReal.toReal p) ≤ ‖f i‖ ^ ENNReal.toReal q\n⊢ 1 ≤ ‖f i‖",
"tactic": "have : 0 ≤ ‖f i‖ ^ p.toReal := Real.rpow_nonneg_of_nonneg (norm_nonneg _) p.toReal"
},
{
"state_after": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\ni : α\nthis : 0 ≤ ‖f i‖ ^ ENNReal.toReal p\nhi : ¬‖f i‖ ^ ENNReal.toReal p ≤ ‖f i‖ ^ ENNReal.toReal q\n⊢ 1 ≤ ‖f i‖",
"state_before": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\ni : α\nhi : ¬abs (‖f i‖ ^ ENNReal.toReal p) ≤ ‖f i‖ ^ ENNReal.toReal q\nthis : 0 ≤ ‖f i‖ ^ ENNReal.toReal p\n⊢ 1 ≤ ‖f i‖",
"tactic": "simp only [abs_of_nonneg, this] at hi"
},
{
"state_after": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\ni : α\nthis : 0 ≤ ‖f i‖ ^ ENNReal.toReal p\nhi : ¬1 ≤ ‖f i‖\n⊢ ‖f i‖ ^ ENNReal.toReal p ≤ ‖f i‖ ^ ENNReal.toReal q",
"state_before": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\ni : α\nthis : 0 ≤ ‖f i‖ ^ ENNReal.toReal p\nhi : ¬‖f i‖ ^ ENNReal.toReal p ≤ ‖f i‖ ^ ENNReal.toReal q\n⊢ 1 ≤ ‖f i‖",
"tactic": "contrapose! hi"
},
{
"state_after": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\ni : α\nthis : 0 ≤ ‖f i‖ ^ ENNReal.toReal p\nhi : ‖f i‖ < 1\n⊢ ‖f i‖ ^ ENNReal.toReal p ≤ ‖f i‖ ^ ENNReal.toReal q",
"state_before": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\ni : α\nthis : 0 ≤ ‖f i‖ ^ ENNReal.toReal p\nhi : ¬1 ≤ ‖f i‖\n⊢ ‖f i‖ ^ ENNReal.toReal p ≤ ‖f i‖ ^ ENNReal.toReal q",
"tactic": "rw [not_le] at hi"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\nα : Type u_1\nE : α → Type u_2\np✝ q✝ : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nf : (i : α) → E i\nhfq : Memℓp f q\nhpq : q ≤ p\nhq : 0 < ENNReal.toReal q\nleft✝ : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q ≤ ENNReal.toReal p\nhf' : Summable fun i => ‖f i‖ ^ ENNReal.toReal q\ni : α\nthis : 0 ≤ ‖f i‖ ^ ENNReal.toReal p\nhi : ‖f i‖ < 1\n⊢ ‖f i‖ ^ ENNReal.toReal p ≤ ‖f i‖ ^ ENNReal.toReal q",
"tactic": "exact Real.rpow_le_rpow_of_exponent_ge' (norm_nonneg _) hi.le hq.le hpq'"
}
] |
[
217,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.some_single_some
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.325058\nγ : Type ?u.325061\nι : Type ?u.325064\nM : Type u_1\nM' : Type ?u.325070\nN : Type ?u.325073\nP : Type ?u.325076\nG : Type ?u.325079\nH : Type ?u.325082\nR : Type ?u.325085\nS : Type ?u.325088\ninst✝ : Zero M\na : α\nm : M\n⊢ some (single (Option.some a) m) = single a m",
"tactic": "classical\n ext b\n simp [single_apply]"
},
{
"state_after": "case h\nα : Type u_2\nβ : Type ?u.325058\nγ : Type ?u.325061\nι : Type ?u.325064\nM : Type u_1\nM' : Type ?u.325070\nN : Type ?u.325073\nP : Type ?u.325076\nG : Type ?u.325079\nH : Type ?u.325082\nR : Type ?u.325085\nS : Type ?u.325088\ninst✝ : Zero M\na : α\nm : M\nb : α\n⊢ ↑(some (single (Option.some a) m)) b = ↑(single a m) b",
"state_before": "α : Type u_2\nβ : Type ?u.325058\nγ : Type ?u.325061\nι : Type ?u.325064\nM : Type u_1\nM' : Type ?u.325070\nN : Type ?u.325073\nP : Type ?u.325076\nG : Type ?u.325079\nH : Type ?u.325082\nR : Type ?u.325085\nS : Type ?u.325088\ninst✝ : Zero M\na : α\nm : M\n⊢ some (single (Option.some a) m) = single a m",
"tactic": "ext b"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\nβ : Type ?u.325058\nγ : Type ?u.325061\nι : Type ?u.325064\nM : Type u_1\nM' : Type ?u.325070\nN : Type ?u.325073\nP : Type ?u.325076\nG : Type ?u.325079\nH : Type ?u.325082\nR : Type ?u.325085\nS : Type ?u.325088\ninst✝ : Zero M\na : α\nm : M\nb : α\n⊢ ↑(some (single (Option.some a) m)) b = ↑(single a m) b",
"tactic": "simp [single_apply]"
}
] |
[
839,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
835,
1
] |
Mathlib/Algebra/Group/Pi.lean
|
Pi.mulSingle_apply_commute
|
[
{
"state_after": "case inl\nι : Type ?u.74703\nα : Type ?u.74706\nI : Type u\nf : I → Type v\nx✝ y : (i : I) → f i\ni✝ j : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\nx : (i : I) → f i\ni : I\n⊢ Commute (mulSingle i (x i)) (mulSingle i (x i))\n\ncase inr\nι : Type ?u.74703\nα : Type ?u.74706\nI : Type u\nf : I → Type v\nx✝ y : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\nx : (i : I) → f i\ni j : I\nhij : i ≠ j\n⊢ Commute (mulSingle i (x i)) (mulSingle j (x j))",
"state_before": "ι : Type ?u.74703\nα : Type ?u.74706\nI : Type u\nf : I → Type v\nx✝ y : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\nx : (i : I) → f i\ni j : I\n⊢ Commute (mulSingle i (x i)) (mulSingle j (x j))",
"tactic": "obtain rfl | hij := Decidable.eq_or_ne i j"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type ?u.74703\nα : Type ?u.74706\nI : Type u\nf : I → Type v\nx✝ y : (i : I) → f i\ni✝ j : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\nx : (i : I) → f i\ni : I\n⊢ Commute (mulSingle i (x i)) (mulSingle i (x i))",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Type ?u.74703\nα : Type ?u.74706\nI : Type u\nf : I → Type v\nx✝ y : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\nx : (i : I) → f i\ni j : I\nhij : i ≠ j\n⊢ Commute (mulSingle i (x i)) (mulSingle j (x j))",
"tactic": "exact Pi.mulSingle_commute hij _ _"
}
] |
[
562,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
558,
1
] |
Mathlib/Order/BoundedOrder.lean
|
bot_lt_top
|
[] |
[
903,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
902,
1
] |
Mathlib/Order/Bounded.lean
|
Set.unbounded_inter_ge
|
[] |
[
445,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/Topology/Basic.lean
|
pure_le_nhds
|
[] |
[
1050,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1050,
1
] |
Std/Data/List/Lemmas.lean
|
List.suffix_refl
|
[] |
[
1580,
56
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1580,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measure_eq_measure_of_between_null_diff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.27550\nγ : Type ?u.27553\nδ : Type ?u.27556\nι : Type ?u.27559\nR : Type ?u.27562\nR' : Type ?u.27565\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁✝ s₂✝ t s₁ s₂ s₃ : Set α\nh12 : s₁ ⊆ s₂\nh23 : s₂ ⊆ s₃\nh_nulldiff : ↑↑μ (s₃ \\ s₁) = 0\nle12 : ↑↑μ s₁ ≤ ↑↑μ s₂\n⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃",
"state_before": "α : Type u_1\nβ : Type ?u.27550\nγ : Type ?u.27553\nδ : Type ?u.27556\nι : Type ?u.27559\nR : Type ?u.27562\nR' : Type ?u.27565\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁✝ s₂✝ t s₁ s₂ s₃ : Set α\nh12 : s₁ ⊆ s₂\nh23 : s₂ ⊆ s₃\nh_nulldiff : ↑↑μ (s₃ \\ s₁) = 0\n⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃",
"tactic": "have le12 : μ s₁ ≤ μ s₂ := measure_mono h12"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.27550\nγ : Type ?u.27553\nδ : Type ?u.27556\nι : Type ?u.27559\nR : Type ?u.27562\nR' : Type ?u.27565\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁✝ s₂✝ t s₁ s₂ s₃ : Set α\nh12 : s₁ ⊆ s₂\nh23 : s₂ ⊆ s₃\nh_nulldiff : ↑↑μ (s₃ \\ s₁) = 0\nle12 : ↑↑μ s₁ ≤ ↑↑μ s₂\nle23 : ↑↑μ s₂ ≤ ↑↑μ s₃\n⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃",
"state_before": "α : Type u_1\nβ : Type ?u.27550\nγ : Type ?u.27553\nδ : Type ?u.27556\nι : Type ?u.27559\nR : Type ?u.27562\nR' : Type ?u.27565\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁✝ s₂✝ t s₁ s₂ s₃ : Set α\nh12 : s₁ ⊆ s₂\nh23 : s₂ ⊆ s₃\nh_nulldiff : ↑↑μ (s₃ \\ s₁) = 0\nle12 : ↑↑μ s₁ ≤ ↑↑μ s₂\n⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃",
"tactic": "have le23 : μ s₂ ≤ μ s₃ := measure_mono h23"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.27550\nγ : Type ?u.27553\nδ : Type ?u.27556\nι : Type ?u.27559\nR : Type ?u.27562\nR' : Type ?u.27565\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁✝ s₂✝ t s₁ s₂ s₃ : Set α\nh12 : s₁ ⊆ s₂\nh23 : s₂ ⊆ s₃\nh_nulldiff : ↑↑μ (s₃ \\ s₁) = 0\nle12 : ↑↑μ s₁ ≤ ↑↑μ s₂\nle23 : ↑↑μ s₂ ≤ ↑↑μ s₃\nkey : ↑↑μ s₃ ≤ ↑↑μ s₁\n⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃",
"state_before": "α : Type u_1\nβ : Type ?u.27550\nγ : Type ?u.27553\nδ : Type ?u.27556\nι : Type ?u.27559\nR : Type ?u.27562\nR' : Type ?u.27565\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁✝ s₂✝ t s₁ s₂ s₃ : Set α\nh12 : s₁ ⊆ s₂\nh23 : s₂ ⊆ s₃\nh_nulldiff : ↑↑μ (s₃ \\ s₁) = 0\nle12 : ↑↑μ s₁ ≤ ↑↑μ s₂\nle23 : ↑↑μ s₂ ≤ ↑↑μ s₃\n⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃",
"tactic": "have key : μ s₃ ≤ μ s₁ :=\n calc\n μ s₃ = μ (s₃ \\ s₁ ∪ s₁) := by rw [diff_union_of_subset (h12.trans h23)]\n _ ≤ μ (s₃ \\ s₁) + μ s₁ := (measure_union_le _ _)\n _ = μ s₁ := by simp only [h_nulldiff, zero_add]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.27550\nγ : Type ?u.27553\nδ : Type ?u.27556\nι : Type ?u.27559\nR : Type ?u.27562\nR' : Type ?u.27565\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁✝ s₂✝ t s₁ s₂ s₃ : Set α\nh12 : s₁ ⊆ s₂\nh23 : s₂ ⊆ s₃\nh_nulldiff : ↑↑μ (s₃ \\ s₁) = 0\nle12 : ↑↑μ s₁ ≤ ↑↑μ s₂\nle23 : ↑↑μ s₂ ≤ ↑↑μ s₃\nkey : ↑↑μ s₃ ≤ ↑↑μ s₁\n⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃",
"tactic": "exact ⟨le12.antisymm (le23.trans key), le23.antisymm (key.trans le12)⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.27550\nγ : Type ?u.27553\nδ : Type ?u.27556\nι : Type ?u.27559\nR : Type ?u.27562\nR' : Type ?u.27565\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁✝ s₂✝ t s₁ s₂ s₃ : Set α\nh12 : s₁ ⊆ s₂\nh23 : s₂ ⊆ s₃\nh_nulldiff : ↑↑μ (s₃ \\ s₁) = 0\nle12 : ↑↑μ s₁ ≤ ↑↑μ s₂\nle23 : ↑↑μ s₂ ≤ ↑↑μ s₃\n⊢ ↑↑μ s₃ = ↑↑μ (s₃ \\ s₁ ∪ s₁)",
"tactic": "rw [diff_union_of_subset (h12.trans h23)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.27550\nγ : Type ?u.27553\nδ : Type ?u.27556\nι : Type ?u.27559\nR : Type ?u.27562\nR' : Type ?u.27565\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁✝ s₂✝ t s₁ s₂ s₃ : Set α\nh12 : s₁ ⊆ s₂\nh23 : s₂ ⊆ s₃\nh_nulldiff : ↑↑μ (s₃ \\ s₁) = 0\nle12 : ↑↑μ s₁ ≤ ↑↑μ s₂\nle23 : ↑↑μ s₂ ≤ ↑↑μ s₃\n⊢ ↑↑μ (s₃ \\ s₁) + ↑↑μ s₁ = ↑↑μ s₁",
"tactic": "simp only [h_nulldiff, zero_add]"
}
] |
[
278,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.mul
|
[] |
[
285,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
11
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.rpow_pos_of_nonneg
|
[
{
"state_after": "case pos\np : ℝ\nx : ℝ≥0∞\nhx_pos : 0 < x\nhp_nonneg : 0 ≤ p\nhp_zero : p = 0\n⊢ 0 < x ^ p\n\ncase neg\np : ℝ\nx : ℝ≥0∞\nhx_pos : 0 < x\nhp_nonneg : 0 ≤ p\nhp_zero : ¬p = 0\n⊢ 0 < x ^ p",
"state_before": "p : ℝ\nx : ℝ≥0∞\nhx_pos : 0 < x\nhp_nonneg : 0 ≤ p\n⊢ 0 < x ^ p",
"tactic": "by_cases hp_zero : p = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\np : ℝ\nx : ℝ≥0∞\nhx_pos : 0 < x\nhp_nonneg : 0 ≤ p\nhp_zero : p = 0\n⊢ 0 < x ^ p",
"tactic": "simp [hp_zero, zero_lt_one]"
},
{
"state_after": "case neg\np : ℝ\nx : ℝ≥0∞\nhx_pos : 0 < x\nhp_nonneg : 0 ≤ p\nhp_zero : p ≠ 0\n⊢ 0 < x ^ p",
"state_before": "case neg\np : ℝ\nx : ℝ≥0∞\nhx_pos : 0 < x\nhp_nonneg : 0 ≤ p\nhp_zero : ¬p = 0\n⊢ 0 < x ^ p",
"tactic": "rw [← Ne.def] at hp_zero"
},
{
"state_after": "case neg\np : ℝ\nx : ℝ≥0∞\nhx_pos : 0 < x\nhp_nonneg : 0 ≤ p\nhp_zero : p ≠ 0\nhp_pos : 0 < p\n⊢ 0 < x ^ p",
"state_before": "case neg\np : ℝ\nx : ℝ≥0∞\nhx_pos : 0 < x\nhp_nonneg : 0 ≤ p\nhp_zero : p ≠ 0\n⊢ 0 < x ^ p",
"tactic": "have hp_pos := lt_of_le_of_ne hp_nonneg hp_zero.symm"
},
{
"state_after": "case neg\np : ℝ\nx : ℝ≥0∞\nhx_pos : 0 < x\nhp_nonneg : 0 ≤ p\nhp_zero : p ≠ 0\nhp_pos : 0 < p\n⊢ 0 ^ p < x ^ p",
"state_before": "case neg\np : ℝ\nx : ℝ≥0∞\nhx_pos : 0 < x\nhp_nonneg : 0 ≤ p\nhp_zero : p ≠ 0\nhp_pos : 0 < p\n⊢ 0 < x ^ p",
"tactic": "rw [← zero_rpow_of_pos hp_pos]"
},
{
"state_after": "no goals",
"state_before": "case neg\np : ℝ\nx : ℝ≥0∞\nhx_pos : 0 < x\nhp_nonneg : 0 ≤ p\nhp_zero : p ≠ 0\nhp_pos : 0 < p\n⊢ 0 ^ p < x ^ p",
"tactic": "exact rpow_lt_rpow hx_pos hp_pos"
}
] |
[
655,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
649,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Antitone.map_iInf_of_continuousAt'
|
[] |
[
2757,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2755,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.HasBasis.limsup_eq_iInf_iSup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type ?u.109072\nι : Type u_2\ninst✝ : CompleteLattice α\np : ι → Prop\ns : ι → Set β\nf : Filter β\nu : β → α\nh : HasBasis f p s\n⊢ (⨅ (i : ι) (_ : p i), sSup (u '' s i)) = ⨅ (i : ι) (_ : p i), ⨆ (a : β) (_ : a ∈ s i), u a",
"tactic": "simp only [sSup_image, id]"
}
] |
[
709,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
707,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
div_div_div_cancel_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.74626\nβ : Type ?u.74629\nG : Type u_1\ninst✝ : CommGroup G\na✝ b✝ c✝ d a b c : G\n⊢ c / a / (c / b) = b / a",
"tactic": "rw [← inv_div b c, div_inv_eq_mul, mul_comm, div_mul_div_cancel']"
}
] |
[
1013,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1012,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.degree_le'
|
[] |
[
815,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
813,
1
] |
Mathlib/Algebra/Star/Basic.lean
|
MulOpposite.op_star
|
[] |
[
602,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
601,
1
] |
Mathlib/Analysis/Normed/Group/Pointwise.lean
|
closedBall_div_singleton
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ closedBall x δ / {y} = closedBall (x / y) δ",
"tactic": "simp [div_eq_mul_inv]"
}
] |
[
176,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
sub_one_div_inv_le_two
|
[
{
"state_after": "ι : Type ?u.187185\nα : Type u_1\nβ : Type ?u.187191\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\na2 : 2 ≤ a\n⊢ 2⁻¹ ≤ 1 - 1 / a",
"state_before": "ι : Type ?u.187185\nα : Type u_1\nβ : Type ?u.187191\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\na2 : 2 ≤ a\n⊢ (1 - 1 / a)⁻¹ ≤ 2",
"tactic": "refine' (inv_le_inv_of_le (inv_pos.2 <| zero_lt_two' α) _).trans_eq (inv_inv (2 : α))"
},
{
"state_after": "ι : Type ?u.187185\nα : Type u_1\nβ : Type ?u.187191\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\na2 : 2 ≤ a\n⊢ 1 / a ≤ 1 - 2⁻¹",
"state_before": "ι : Type ?u.187185\nα : Type u_1\nβ : Type ?u.187191\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\na2 : 2 ≤ a\n⊢ 2⁻¹ ≤ 1 - 1 / a",
"tactic": "rw [le_sub_iff_add_le, add_comm, ←le_sub_iff_add_le]"
},
{
"state_after": "case h.e'_4\nι : Type ?u.187185\nα : Type u_1\nβ : Type ?u.187191\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\na2 : 2 ≤ a\n⊢ 1 - 2⁻¹ = 2⁻¹",
"state_before": "ι : Type ?u.187185\nα : Type u_1\nβ : Type ?u.187191\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\na2 : 2 ≤ a\n⊢ 1 / a ≤ 1 - 2⁻¹",
"tactic": "convert (one_div a).le.trans (inv_le_inv_of_le zero_lt_two a2) using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nι : Type ?u.187185\nα : Type u_1\nβ : Type ?u.187191\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\na2 : 2 ≤ a\n⊢ 1 - 2⁻¹ = 2⁻¹",
"tactic": "rw [sub_eq_iff_eq_add, ←two_mul, mul_inv_cancel two_ne_zero]"
}
] |
[
931,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
923,
1
] |
Mathlib/Data/List/Cycle.lean
|
List.prev_ne_cons_cons
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.86825\ninst✝ : DecidableEq α\nl : List α\nx y z : α\nh : x ∈ y :: z :: l\nhy : x ≠ y\nhz : x ≠ z\n⊢ x ∈ z :: l",
"tactic": "simpa [hy] using h"
},
{
"state_after": "case nil\nα : Type u_1\ninst✝ : DecidableEq α\nx y z : α\nhy : x ≠ y\nhz : x ≠ z\nh : x ∈ [y, z]\n⊢ prev [y, z] x h = prev [z] x (_ : x ∈ [z])\n\ncase cons\nα : Type u_1\ninst✝ : DecidableEq α\nx y z : α\nhy : x ≠ y\nhz : x ≠ z\nhead✝ : α\ntail✝ : List α\nh : x ∈ y :: z :: head✝ :: tail✝\n⊢ prev (y :: z :: head✝ :: tail✝) x h = prev (z :: head✝ :: tail✝) x (_ : x ∈ z :: head✝ :: tail✝)",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx y z : α\nh : x ∈ y :: z :: l\nhy : x ≠ y\nhz : x ≠ z\n⊢ prev (y :: z :: l) x h = prev (z :: l) x (_ : x ∈ z :: l)",
"tactic": "cases l"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\ninst✝ : DecidableEq α\nx y z : α\nhy : x ≠ y\nhz : x ≠ z\nh : x ∈ [y, z]\n⊢ prev [y, z] x h = prev [z] x (_ : x ∈ [z])",
"tactic": "simp [hy, hz] at h"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\ninst✝ : DecidableEq α\nx y z : α\nhy : x ≠ y\nhz : x ≠ z\nhead✝ : α\ntail✝ : List α\nh : x ∈ y :: z :: head✝ :: tail✝\n⊢ prev (y :: z :: head✝ :: tail✝) x h = prev (z :: head✝ :: tail✝) x (_ : x ∈ z :: head✝ :: tail✝)",
"tactic": "rw [prev, dif_neg hy, if_neg hz]"
}
] |
[
245,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/Data/FunLike/Basic.lean
|
FunLike.congr_fun
|
[] |
[
188,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
11
] |
Std/Classes/LawfulMonad.lean
|
SatisfiesM.seq
|
[
{
"state_after": "m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ SatisfiesM q (Seq.seq (Subtype.val <$> f) fun x_1 => Subtype.val <$> x)",
"state_before": "m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx : m α\nhf : SatisfiesM p₁ f\nhx : SatisfiesM p₂ x\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\n⊢ SatisfiesM q (Seq.seq f fun x_1 => x)",
"tactic": "match f, x, hf, hx with | _, _, ⟨f, rfl⟩, ⟨x, rfl⟩ => ?_"
},
{
"state_after": "m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Subtype.val <$>\n Seq.seq\n ((fun x x_1 =>\n match x with\n | { val := a, property := h₁ } =>\n match x_1 with\n | { val := b, property := h₂ } => { val := a b, property := (_ : q (a b)) }) <$>\n f)\n fun x_1 => x) =\n Seq.seq (Subtype.val <$> f) fun x_1 => Subtype.val <$> x",
"state_before": "m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ SatisfiesM q (Seq.seq (Subtype.val <$> f) fun x_1 => Subtype.val <$> x)",
"tactic": "refine ⟨(fun ⟨a, h₁⟩ ⟨b, h₂⟩ => ⟨a b, H h₁ h₂⟩) <$> f <*> x, ?_⟩"
},
{
"state_after": "m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Seq.seq (pure Subtype.val) fun x_1 =>\n Seq.seq\n (Seq.seq (pure fun x x_2 => { val := Subtype.val x x_2.val, property := (_ : q (Subtype.val x x_2.val)) })\n fun x => f)\n fun x_2 => x) =\n Seq.seq (Seq.seq (pure Subtype.val) fun x => f) fun x_1 => Seq.seq (pure Subtype.val) fun x_2 => x",
"state_before": "m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Subtype.val <$>\n Seq.seq\n ((fun x x_1 =>\n match x with\n | { val := a, property := h₁ } =>\n match x_1 with\n | { val := b, property := h₂ } => { val := a b, property := (_ : q (a b)) }) <$>\n f)\n fun x_1 => x) =\n Seq.seq (Subtype.val <$> f) fun x_1 => Subtype.val <$> x",
"tactic": "simp only [← pure_seq]"
},
{
"state_after": "m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Seq.seq\n (Seq.seq\n (pure\n (Function.comp Subtype.val ∘ fun x x_1 =>\n { val := Subtype.val x x_1.val, property := (_ : q (Subtype.val x x_1.val)) }))\n fun x => f)\n fun x_1 => x) =\n Seq.seq ((fun h => h Subtype.val) <$> Function.comp <$> Seq.seq (pure Subtype.val) fun x => f) fun x_1 => x",
"state_before": "m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Seq.seq (pure Subtype.val) fun x_1 =>\n Seq.seq\n (Seq.seq (pure fun x x_2 => { val := Subtype.val x x_2.val, property := (_ : q (Subtype.val x x_2.val)) })\n fun x => f)\n fun x_2 => x) =\n Seq.seq (Seq.seq (pure Subtype.val) fun x => f) fun x_1 => Seq.seq (pure Subtype.val) fun x_2 => x",
"tactic": "simp [SatisfiesM, seq_assoc]"
},
{
"state_after": "m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Seq.seq\n (Seq.seq\n (pure\n (Function.comp Subtype.val ∘ fun x x_1 =>\n { val := Subtype.val x x_1.val, property := (_ : q (Subtype.val x x_1.val)) }))\n fun x => f)\n fun x_1 => x) =\n Seq.seq\n (Seq.seq (pure fun h => h Subtype.val) fun x =>\n Seq.seq (pure Function.comp) fun x => Seq.seq (pure Subtype.val) fun x => f)\n fun x_1 => x",
"state_before": "m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Seq.seq\n (Seq.seq\n (pure\n (Function.comp Subtype.val ∘ fun x x_1 =>\n { val := Subtype.val x x_1.val, property := (_ : q (Subtype.val x x_1.val)) }))\n fun x => f)\n fun x_1 => x) =\n Seq.seq ((fun h => h Subtype.val) <$> Function.comp <$> Seq.seq (pure Subtype.val) fun x => f) fun x_1 => x",
"tactic": "simp only [← pure_seq]"
},
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nα α✝ : Type u_1\np₁ : (α → α✝) → Prop\nf✝ : m (α → α✝)\np₂ : α → Prop\nq : α✝ → Prop\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nx✝ : m α\nhf : SatisfiesM p₁ f✝\nhx : SatisfiesM p₂ x✝\nH : ∀ {f : α → α✝} {a : α}, p₁ f → p₂ a → q (f a)\nf : m { a // p₁ a }\nx : m { a // p₂ a }\n⊢ (Seq.seq\n (Seq.seq\n (pure\n (Function.comp Subtype.val ∘ fun x x_1 =>\n { val := Subtype.val x x_1.val, property := (_ : q (Subtype.val x x_1.val)) }))\n fun x => f)\n fun x_1 => x) =\n Seq.seq\n (Seq.seq (pure fun h => h Subtype.val) fun x =>\n Seq.seq (pure Function.comp) fun x => Seq.seq (pure Subtype.val) fun x => f)\n fun x_1 => x",
"tactic": "simp [seq_assoc, Function.comp]"
}
] |
[
130,
58
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
124,
11
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
Convex.lipschitzOnWith_of_nnnorm_derivWithin_le
|
[] |
[
663,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
660,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Restrict.lean
|
AffineMap.restrict.coe_apply
|
[] |
[
61,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Tactic/Ring/Basic.lean
|
Mathlib.Tactic.Ring.add_overlap_pf
|
[
{
"state_after": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na b x : R\ne : ℕ\n⊢ x ^ e * a + x ^ e * b = x ^ e * (a + b)",
"state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na b c x : R\ne : ℕ\npq_pf : a + b = c\n⊢ x ^ e * a + x ^ e * b = x ^ e * c",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na b x : R\ne : ℕ\n⊢ x ^ e * a + x ^ e * b = x ^ e * (a + b)",
"tactic": "simp [mul_add]"
}
] |
[
278,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Analysis/SpecialFunctions/CompareExp.lean
|
Complex.IsExpCmpFilter.isLittleO_im_pow_exp_re
|
[
{
"state_after": "no goals",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\n⊢ ∀ (x : ℂ), ((fun z => z.im ^ ↑n) ^ 2) x = (x.im ^ ↑n) ^ 2",
"tactic": "simp"
},
{
"state_after": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nx✝ : ℂ\n⊢ (x✝.im ^ n) ^ 2 = x✝.im ^ (2 * n)",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nx✝ : ℂ\n⊢ (x✝.im ^ ↑n) ^ 2 = x✝.im ^ (2 * ↑n)",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nx✝ : ℂ\n⊢ (x✝.im ^ n) ^ 2 = x✝.im ^ (2 * n)",
"tactic": "rw [pow_mul']"
},
{
"state_after": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nthis : (fun z => z.im ^ ↑(2 * n)) =O[l] fun z => Real.exp z.re\n⊢ (fun z => z.im ^ (2 * ↑n)) =O[l] fun z => Real.exp z.re",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\n⊢ (fun z => z.im ^ (2 * ↑n)) =O[l] fun z => Real.exp z.re",
"tactic": "have := hl.isBigO_im_pow_re (2 * n)"
},
{
"state_after": "no goals",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nthis : (fun z => z.im ^ ↑(2 * n)) =O[l] fun z => Real.exp z.re\n⊢ (fun z => z.im ^ (2 * ↑n)) =O[l] fun z => Real.exp z.re",
"tactic": "norm_cast at *"
},
{
"state_after": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nx✝ : ℂ\n⊢ Real.exp x✝.re = Real.exp x✝.re ^ 1",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nx✝ : ℂ\n⊢ Real.exp x✝.re = Real.exp x✝.re ^ 1",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nx✝ : ℂ\n⊢ Real.exp x✝.re = Real.exp x✝.re ^ 1",
"tactic": "rw [pow_one]"
},
{
"state_after": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nthis : ((fun x => x ^ 1) ∘ Real.exp ∘ re) =o[l] ((fun x => x ^ 2) ∘ Real.exp ∘ re)\n⊢ (fun z => Real.exp z.re ^ 1) =o[l] fun z => Real.exp z.re ^ 2",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\n⊢ (fun z => Real.exp z.re ^ 1) =o[l] fun z => Real.exp z.re ^ 2",
"tactic": "have := (isLittleO_pow_pow_atTop_of_lt one_lt_two).comp_tendsto <|\n Real.tendsto_exp_atTop.comp hl.tendsto_re"
},
{
"state_after": "no goals",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nthis : ((fun x => x ^ 1) ∘ Real.exp ∘ re) =o[l] ((fun x => x ^ 2) ∘ Real.exp ∘ re)\n⊢ (fun z => Real.exp z.re ^ 1) =o[l] fun z => Real.exp z.re ^ 2",
"tactic": "simpa only [pow_one, Real.rpow_one, Real.rpow_two]"
},
{
"state_after": "no goals",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\n⊢ ∀ (x : ℂ), Real.exp x.re ^ 2 = ((fun z => Real.exp z.re) ^ 2) x",
"tactic": "simp"
}
] |
[
126,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Data/Nat/Totient.lean
|
Nat.prime_iff_card_units
|
[
{
"state_after": "case inl\np : ℕ\ninst✝ : Fintype (ZMod p)ˣ\nhp : p = 0\n⊢ Prime p ↔ Fintype.card (ZMod p)ˣ = p - 1\n\ncase inr\np : ℕ\ninst✝ : Fintype (ZMod p)ˣ\nhp : NeZero p\n⊢ Prime p ↔ Fintype.card (ZMod p)ˣ = p - 1",
"state_before": "p : ℕ\ninst✝ : Fintype (ZMod p)ˣ\n⊢ Prime p ↔ Fintype.card (ZMod p)ˣ = p - 1",
"tactic": "cases' eq_zero_or_neZero p with hp hp"
},
{
"state_after": "no goals",
"state_before": "case inr\np : ℕ\ninst✝ : Fintype (ZMod p)ˣ\nhp : NeZero p\n⊢ Prime p ↔ Fintype.card (ZMod p)ˣ = p - 1",
"tactic": "rw [ZMod.card_units_eq_totient, Nat.totient_eq_iff_prime <| NeZero.pos p]"
},
{
"state_after": "case inl\ninst✝ : Fintype (ZMod 0)ˣ\n⊢ Prime 0 ↔ Fintype.card (ZMod 0)ˣ = 0 - 1",
"state_before": "case inl\np : ℕ\ninst✝ : Fintype (ZMod p)ˣ\nhp : p = 0\n⊢ Prime p ↔ Fintype.card (ZMod p)ˣ = p - 1",
"tactic": "subst hp"
},
{
"state_after": "case inl\ninst✝ : Fintype (ZMod 0)ˣ\n⊢ ¬Fintype.card ℤˣ = 0",
"state_before": "case inl\ninst✝ : Fintype (ZMod 0)ˣ\n⊢ Prime 0 ↔ Fintype.card (ZMod 0)ˣ = 0 - 1",
"tactic": "simp only [ZMod, not_prime_zero, false_iff_iff, zero_tsub]"
},
{
"state_after": "case inl\ninst✝ : Fintype (ZMod 0)ˣ\n⊢ Fintype.card ℤˣ ≠ 0",
"state_before": "case inl\ninst✝ : Fintype (ZMod 0)ˣ\n⊢ ¬Fintype.card ℤˣ = 0",
"tactic": "suffices Fintype.card ℤˣ ≠ 0 by convert this"
},
{
"state_after": "no goals",
"state_before": "case inl\ninst✝ : Fintype (ZMod 0)ˣ\n⊢ Fintype.card ℤˣ ≠ 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "inst✝ : Fintype (ZMod 0)ˣ\nthis : Fintype.card ℤˣ ≠ 0\n⊢ ¬Fintype.card ℤˣ = 0",
"tactic": "convert this"
}
] |
[
262,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_eq_nhds_of_re_neg_of_im_pos
|
[
{
"state_after": "x z : ℂ\nhx_re : x.re < 0\nhx_im : 0 < x.im\nh_forall_nhds : ∀ᶠ (y : ℂ) in 𝓝 x, y.re < 0 ∧ 0 < y.im\n⊢ arg =ᶠ[𝓝 x] fun x => arcsin ((-x).im / ↑abs x) + π\n\ncase h_forall_nhds\nx z : ℂ\nhx_re : x.re < 0\nhx_im : 0 < x.im\n⊢ ∀ᶠ (y : ℂ) in 𝓝 x, y.re < 0 ∧ 0 < y.im",
"state_before": "x z : ℂ\nhx_re : x.re < 0\nhx_im : 0 < x.im\n⊢ arg =ᶠ[𝓝 x] fun x => arcsin ((-x).im / ↑abs x) + π",
"tactic": "suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im"
},
{
"state_after": "case h_forall_nhds\nx z : ℂ\nhx_re : x.re < 0\nhx_im : 0 < x.im\n⊢ ∀ᶠ (y : ℂ) in 𝓝 x, y.re < 0 ∧ 0 < y.im",
"state_before": "x z : ℂ\nhx_re : x.re < 0\nhx_im : 0 < x.im\nh_forall_nhds : ∀ᶠ (y : ℂ) in 𝓝 x, y.re < 0 ∧ 0 < y.im\n⊢ arg =ᶠ[𝓝 x] fun x => arcsin ((-x).im / ↑abs x) + π\n\ncase h_forall_nhds\nx z : ℂ\nhx_re : x.re < 0\nhx_im : 0 < x.im\n⊢ ∀ᶠ (y : ℂ) in 𝓝 x, y.re < 0 ∧ 0 < y.im",
"tactic": "exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le"
},
{
"state_after": "case h_forall_nhds\nx z : ℂ\nhx_re : x.re < 0\nhx_im : 0 < x.im\n⊢ IsOpen fun y => y.re < 0 ∧ 0 < y.im",
"state_before": "case h_forall_nhds\nx z : ℂ\nhx_re : x.re < 0\nhx_im : 0 < x.im\n⊢ ∀ᶠ (y : ℂ) in 𝓝 x, y.re < 0 ∧ 0 < y.im",
"tactic": "refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im)"
},
{
"state_after": "no goals",
"state_before": "case h_forall_nhds\nx z : ℂ\nhx_re : x.re < 0\nhx_im : 0 < x.im\n⊢ IsOpen fun y => y.re < 0 ∧ 0 < y.im",
"tactic": "exact\n IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im)"
}
] |
[
521,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Real.dist_le_of_mem_Icc_01
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.197176\nι : Type ?u.197179\ninst✝ : PseudoMetricSpace α\nx y : ℝ\nhx : x ∈ Icc 0 1\nhy : y ∈ Icc 0 1\n⊢ dist x y ≤ 1",
"tactic": "simpa only [sub_zero] using Real.dist_le_of_mem_Icc hx hy"
}
] |
[
1386,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1385,
1
] |
Mathlib/Data/QPF/Multivariate/Constructions/Const.lean
|
MvQPF.Const.get_map
|
[] |
[
72,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
tendstoUniformly_iff_seq_tendstoUniformly
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nl : Filter ι\ninst✝ : IsCountablyGenerated l\n⊢ TendstoUniformlyOn F f l univ ↔ ∀ (u : ℕ → ι), Tendsto u atTop l → TendstoUniformlyOn (fun n => F (u n)) f atTop univ",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nl : Filter ι\ninst✝ : IsCountablyGenerated l\n⊢ TendstoUniformly F f l ↔ ∀ (u : ℕ → ι), Tendsto u atTop l → TendstoUniformly (fun n => F (u n)) f atTop",
"tactic": "simp_rw [← tendstoUniformlyOn_univ]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nl : Filter ι\ninst✝ : IsCountablyGenerated l\n⊢ TendstoUniformlyOn F f l univ ↔ ∀ (u : ℕ → ι), Tendsto u atTop l → TendstoUniformlyOn (fun n => F (u n)) f atTop univ",
"tactic": "exact tendstoUniformlyOn_iff_seq_tendstoUniformlyOn"
}
] |
[
585,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
581,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.Finite_of_ncard_pos
|
[] |
[
125,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/LinearAlgebra/BilinearMap.lean
|
LinearMap.llcomp_apply'
|
[] |
[
322,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/Data/Nat/Choose/Basic.lean
|
Nat.multichoose_eq
|
[
{
"state_after": "no goals",
"state_before": "x✝ : ℕ\n⊢ multichoose x✝ 0 = choose (x✝ + 0 - 1) 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\n⊢ multichoose 0 (k + 1) = choose (0 + (k + 1) - 1) (k + 1)",
"tactic": "simp"
},
{
"state_after": "n k : ℕ\nthis : n + (k + 1) < n + 1 + (k + 1)\n⊢ multichoose (n + 1) (k + 1) = choose (n + 1 + (k + 1) - 1) (k + 1)",
"state_before": "n k : ℕ\n⊢ multichoose (n + 1) (k + 1) = choose (n + 1 + (k + 1) - 1) (k + 1)",
"tactic": "have : n + (k + 1) < (n + 1) + (k + 1) := add_lt_add_right (Nat.lt_succ_self _) _"
},
{
"state_after": "n k : ℕ\nthis✝ : n + (k + 1) < n + 1 + (k + 1)\nthis : n + 1 + k < n + 1 + (k + 1)\n⊢ multichoose (n + 1) (k + 1) = choose (n + 1 + (k + 1) - 1) (k + 1)",
"state_before": "n k : ℕ\nthis : n + (k + 1) < n + 1 + (k + 1)\n⊢ multichoose (n + 1) (k + 1) = choose (n + 1 + (k + 1) - 1) (k + 1)",
"tactic": "have : (n + 1) + k < (n + 1) + (k + 1) := add_lt_add_left (Nat.lt_succ_self _) _"
},
{
"state_after": "n k : ℕ\nthis✝ : n + (k + 1) < n + 1 + (k + 1)\nthis : n + 1 + k < n + 1 + (k + 1)\n⊢ multichoose (n + 1) k + multichoose n (k + 1) = choose (n + k) k + choose (n + k) (succ k)",
"state_before": "n k : ℕ\nthis✝ : n + (k + 1) < n + 1 + (k + 1)\nthis : n + 1 + k < n + 1 + (k + 1)\n⊢ multichoose (n + 1) (k + 1) = choose (n + 1 + (k + 1) - 1) (k + 1)",
"tactic": "erw [multichoose_succ_succ, add_comm, Nat.succ_add_sub_one, ← add_assoc, Nat.choose_succ_succ]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\nthis✝ : n + (k + 1) < n + 1 + (k + 1)\nthis : n + 1 + k < n + 1 + (k + 1)\n⊢ multichoose (n + 1) k + multichoose n (k + 1) = choose (n + k) k + choose (n + k) (succ k)",
"tactic": "simp [multichoose_eq n (k+1), multichoose_eq (n+1) k]"
}
] |
[
404,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
395,
1
] |
Mathlib/MeasureTheory/Decomposition/Jordan.lean
|
MeasureTheory.SignedMeasure.mutuallySingular_ennreal_iff
|
[
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\n⊢ s ⟂ᵥ μ → totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ\n\ncase mpr\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\n⊢ totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ → s ⟂ᵥ μ",
"state_before": "α : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\n⊢ s ⟂ᵥ μ ↔ totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro.intro\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0\n⊢ totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\n⊢ s ⟂ᵥ μ → totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ",
"tactic": "rintro ⟨u, hmeas, hu₁, hu₂⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i\nhi₃ : VectorMeasure.restrict s (iᶜ) ≤ VectorMeasure.restrict 0 (iᶜ)\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s (iᶜ) (_ : MeasurableSet (iᶜ)) hi₃\n⊢ totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ",
"state_before": "case mp.intro.intro.intro\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0\n⊢ totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ",
"tactic": "obtain ⟨i, hi₁, hi₂, hi₃, hpos, hneg⟩ := s.toJordanDecomposition_spec"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i\nhi₃ : VectorMeasure.restrict s (iᶜ) ≤ VectorMeasure.restrict 0 (iᶜ)\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s (iᶜ) (_ : MeasurableSet (iᶜ)) hi₃\n⊢ ↑↑(totalVariation s) u = 0\n\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i\nhi₃ : VectorMeasure.restrict s (iᶜ) ≤ VectorMeasure.restrict 0 (iᶜ)\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s (iᶜ) (_ : MeasurableSet (iᶜ)) hi₃\n⊢ ↑↑(VectorMeasure.ennrealToMeasure μ) (uᶜ) = 0",
"state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i\nhi₃ : VectorMeasure.restrict s (iᶜ) ≤ VectorMeasure.restrict 0 (iᶜ)\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s (iᶜ) (_ : MeasurableSet (iᶜ)) hi₃\n⊢ totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ",
"tactic": "refine' ⟨u, hmeas, _, _⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i\nhi₃ : VectorMeasure.restrict s (iᶜ) ≤ VectorMeasure.restrict 0 (iᶜ)\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s (iᶜ) (_ : MeasurableSet (iᶜ)) hi₃\n⊢ ↑{ val := ↑s (i ∩ u), property := (_ : 0 ≤ ↑s (i ∩ u)) } +\n ↑{ val := -↑s (iᶜ ∩ u), property := (_ : 0 ≤ -↑s (iᶜ ∩ u)) } =\n 0",
"state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i\nhi₃ : VectorMeasure.restrict s (iᶜ) ≤ VectorMeasure.restrict 0 (iᶜ)\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s (iᶜ) (_ : MeasurableSet (iᶜ)) hi₃\n⊢ ↑↑(totalVariation s) u = 0",
"tactic": "rw [totalVariation, Measure.add_apply, hpos, hneg, toMeasureOfZeroLE_apply _ _ _ hmeas,\n toMeasureOfLEZero_apply _ _ _ hmeas]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i\nhi₃ : VectorMeasure.restrict s (iᶜ) ≤ VectorMeasure.restrict 0 (iᶜ)\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s (iᶜ) (_ : MeasurableSet (iᶜ)) hi₃\n⊢ ↑{ val := ↑s (i ∩ u), property := (_ : 0 ≤ ↑s (i ∩ u)) } +\n ↑{ val := -↑s (iᶜ ∩ u), property := (_ : 0 ≤ -↑s (iᶜ ∩ u)) } =\n 0",
"tactic": "simp [hu₁ _ (Set.inter_subset_right _ _), ← NNReal.eq_iff]"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i\nhi₃ : VectorMeasure.restrict s (iᶜ) ≤ VectorMeasure.restrict 0 (iᶜ)\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s (iᶜ) (_ : MeasurableSet (iᶜ)) hi₃\n⊢ ↑μ (uᶜ) = 0",
"state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i\nhi₃ : VectorMeasure.restrict s (iᶜ) ≤ VectorMeasure.restrict 0 (iᶜ)\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s (iᶜ) (_ : MeasurableSet (iᶜ)) hi₃\n⊢ ↑↑(VectorMeasure.ennrealToMeasure μ) (uᶜ) = 0",
"tactic": "rw [VectorMeasure.ennrealToMeasure_apply hmeas.compl]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ (t : Set α), t ⊆ u → ↑s t = 0\nhu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑μ t = 0\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i\nhi₃ : VectorMeasure.restrict s (iᶜ) ≤ VectorMeasure.restrict 0 (iᶜ)\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi₁ hi₂\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s (iᶜ) (_ : MeasurableSet (iᶜ)) hi₃\n⊢ ↑μ (uᶜ) = 0",
"tactic": "exact hu₂ _ (Set.Subset.refl _)"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ↑↑(totalVariation s) u = 0\nhu₂ : ↑↑(VectorMeasure.ennrealToMeasure μ) (uᶜ) = 0\n⊢ s ⟂ᵥ μ",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\n⊢ totalVariation s ⟂ₘ VectorMeasure.ennrealToMeasure μ → s ⟂ᵥ μ",
"tactic": "rintro ⟨u, hmeas, hu₁, hu₂⟩"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ↑↑(totalVariation s) u = 0\nhu₂ : ↑↑(VectorMeasure.ennrealToMeasure μ) (uᶜ) = 0\nt : Set α\nhtv : t ⊆ uᶜ\nhmt : MeasurableSet t\n⊢ ↑μ t = 0",
"state_before": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ↑↑(totalVariation s) u = 0\nhu₂ : ↑↑(VectorMeasure.ennrealToMeasure μ) (uᶜ) = 0\n⊢ s ⟂ᵥ μ",
"tactic": "refine'\n VectorMeasure.MutuallySingular.mk u hmeas\n (fun t htu _ => null_of_totalVariation_zero _ (measure_mono_null htu hu₁)) fun t htv hmt =>\n _"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ↑↑(totalVariation s) u = 0\nhu₂ : ↑↑(VectorMeasure.ennrealToMeasure μ) (uᶜ) = 0\nt : Set α\nhtv : t ⊆ uᶜ\nhmt : MeasurableSet t\n⊢ ↑↑(VectorMeasure.ennrealToMeasure μ) t = 0",
"state_before": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ↑↑(totalVariation s) u = 0\nhu₂ : ↑↑(VectorMeasure.ennrealToMeasure μ) (uᶜ) = 0\nt : Set α\nhtv : t ⊆ uᶜ\nhmt : MeasurableSet t\n⊢ ↑μ t = 0",
"tactic": "rw [← VectorMeasure.ennrealToMeasure_apply hmt]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.98874\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ↑↑(totalVariation s) u = 0\nhu₂ : ↑↑(VectorMeasure.ennrealToMeasure μ) (uᶜ) = 0\nt : Set α\nhtv : t ⊆ uᶜ\nhmt : MeasurableSet t\n⊢ ↑↑(VectorMeasure.ennrealToMeasure μ) t = 0",
"tactic": "exact measure_mono_null htv hu₂"
}
] |
[
592,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/Analysis/Convex/Star.lean
|
starConvex_iff_openSegment_subset
|
[] |
[
198,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Algebra/Order/Field/Power.lean
|
Nat.zpow_pos_of_pos
|
[
{
"state_after": "case ha\nα : Type u_1\ninst✝ : LinearOrderedSemifield α\na✝ b c d e : α\nm n✝ : ℤ\na : ℕ\nh : 0 < a\nn : ℤ\n⊢ 0 < ↑a",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedSemifield α\na✝ b c d e : α\nm n✝ : ℤ\na : ℕ\nh : 0 < a\nn : ℤ\n⊢ 0 < ↑a ^ n",
"tactic": "apply zpow_pos_of_pos"
},
{
"state_after": "no goals",
"state_before": "case ha\nα : Type u_1\ninst✝ : LinearOrderedSemifield α\na✝ b c d e : α\nm n✝ : ℤ\na : ℕ\nh : 0 < a\nn : ℤ\n⊢ 0 < ↑a",
"tactic": "exact_mod_cast h"
}
] |
[
52,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
11
] |
Std/Data/Int/Lemmas.lean
|
Int.neg_ne_zero
|
[] |
[
92,
76
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
92,
11
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Icc_diff_both
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.89689\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na✝ b✝ c : α\ninst✝ : DecidableEq α\na b : α\n⊢ Icc a b \\ {a, b} = Ioo a b",
"tactic": "simp [← coe_inj]"
}
] |
[
561,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
561,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Trivialization.symm_apply_apply_mk
|
[] |
[
622,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
620,
1
] |
Mathlib/Topology/Maps.lean
|
QuotientMap.of_inverse
|
[] |
[
310,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
308,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.sinh_add
|
[
{
"state_after": "x y : ℂ\n⊢ (exp x - exp (-x)) * (exp y + exp (-y)) + (exp x + exp (-x)) * (exp y - exp (-y)) =\n 2 * (exp x * exp y - exp (-x) * exp (-y))",
"state_before": "x y : ℂ\n⊢ sinh (x + y) = sinh x * cosh y + cosh x * sinh y",
"tactic": "rw [← mul_right_inj' (two_ne_zero' ℂ), two_sinh, exp_add, neg_add, exp_add, eq_comm, mul_add, ←\n mul_assoc, two_sinh, mul_left_comm, two_sinh, ← mul_right_inj' (two_ne_zero' ℂ), mul_add,\n mul_left_comm, two_cosh, ← mul_assoc, two_cosh]"
},
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ (exp x - exp (-x)) * (exp y + exp (-y)) + (exp x + exp (-x)) * (exp y - exp (-y)) =\n 2 * (exp x * exp y - exp (-x) * exp (-y))",
"tactic": "exact sinh_add_aux"
}
] |
[
615,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
611,
1
] |
Mathlib/Data/MvPolynomial/Division.lean
|
MvPolynomial.divMonomial_add_modMonomial
|
[] |
[
142,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/InformationTheory/Hamming.lean
|
Hamming.toHamming_sub
|
[] |
[
383,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/Topology/ContinuousFunction/Ordered.lean
|
ContinuousMap.sup_apply
|
[] |
[
80,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Data/List/Basic.lean
|
List.zipLeft_eq_zipLeft'
|
[
{
"state_after": "ι : Type ?u.477219\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nas✝ : List α\nb : β\nbs✝ : List β\nas : List α\nbs : List β\n⊢ zipWithLeft Prod.mk as bs = (zipWithLeft' Prod.mk as bs).fst",
"state_before": "ι : Type ?u.477219\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nas✝ : List α\nb : β\nbs✝ : List β\nas : List α\nbs : List β\n⊢ zipLeft as bs = (zipLeft' as bs).fst",
"tactic": "rw [zipLeft, zipLeft']"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.477219\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nas✝ : List α\nb : β\nbs✝ : List β\nas : List α\nbs : List β\n⊢ zipWithLeft Prod.mk as bs = (zipWithLeft' Prod.mk as bs).fst",
"tactic": "cases as with\n| nil => rfl\n| cons _ atl =>\n cases bs with\n | nil => rfl\n | cons _ btl => rw [zipWithLeft, zipWithLeft', cons_inj]; exact @zipLeft_eq_zipLeft' atl btl"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.477219\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nas : List α\nb : β\nbs✝ bs : List β\n⊢ zipWithLeft Prod.mk [] bs = (zipWithLeft' Prod.mk [] bs).fst",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.477219\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nas : List α\nb : β\nbs✝ bs : List β\nhead✝ : α\natl : List α\n⊢ zipWithLeft Prod.mk (head✝ :: atl) bs = (zipWithLeft' Prod.mk (head✝ :: atl) bs).fst",
"tactic": "cases bs with\n| nil => rfl\n| cons _ btl => rw [zipWithLeft, zipWithLeft', cons_inj]; exact @zipLeft_eq_zipLeft' atl btl"
},
{
"state_after": "no goals",
"state_before": "case cons.nil\nι : Type ?u.477219\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nas : List α\nb : β\nbs : List β\nhead✝ : α\natl : List α\n⊢ zipWithLeft Prod.mk (head✝ :: atl) [] = (zipWithLeft' Prod.mk (head✝ :: atl) []).fst",
"tactic": "rfl"
},
{
"state_after": "case cons.cons\nι : Type ?u.477219\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nas : List α\nb : β\nbs : List β\nhead✝¹ : α\natl : List α\nhead✝ : β\nbtl : List β\n⊢ zipWithLeft Prod.mk atl btl = (zipWithLeft' Prod.mk atl btl).fst",
"state_before": "case cons.cons\nι : Type ?u.477219\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nas : List α\nb : β\nbs : List β\nhead✝¹ : α\natl : List α\nhead✝ : β\nbtl : List β\n⊢ zipWithLeft Prod.mk (head✝¹ :: atl) (head✝ :: btl) = (zipWithLeft' Prod.mk (head✝¹ :: atl) (head✝ :: btl)).fst",
"tactic": "rw [zipWithLeft, zipWithLeft', cons_inj]"
},
{
"state_after": "no goals",
"state_before": "case cons.cons\nι : Type ?u.477219\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nas : List α\nb : β\nbs : List β\nhead✝¹ : α\natl : List α\nhead✝ : β\nbtl : List β\n⊢ zipWithLeft Prod.mk atl btl = (zipWithLeft' Prod.mk atl btl).fst",
"tactic": "exact @zipLeft_eq_zipLeft' atl btl"
}
] |
[
4204,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4197,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.