file_path
stringlengths 11
79
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100
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Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean
|
midpoint_neg_self
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nV : Type u_1\nV' : Type ?u.135249\nP : Type ?u.135252\nP' : Type ?u.135255\ninst✝⁷ : Ring R\ninst✝⁶ : Invertible 2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx✝ y z : P\nx : V\n⊢ midpoint R (-x) x = 0",
"tactic": "simpa using midpoint_self_neg R (-x)"
}
] |
[
225,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Lp.lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : snorm' (fun x => ∑ i in Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\n⊢ (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : snorm' (fun x => ∑ i in Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i\n⊢ (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p",
"tactic": "have hp_pos : 0 < p := zero_lt_one.trans_le hp1"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : snorm' (fun x => ∑ i in Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\n⊢ (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : snorm' (fun x => ∑ i in Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\n⊢ (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p",
"tactic": "rw [← one_div_one_div p, @ENNReal.le_rpow_one_div_iff _ _ (1 / p) (by simp [hp_pos]),\n one_div_one_div p]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\n⊢ (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : snorm' (fun x => ∑ i in Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\n⊢ (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i",
"tactic": "simp_rw [snorm'] at hn"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhp_pos : 0 < p\nh_nnnorm_nonneg :\n (fun a => ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p) = fun a =>\n (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p\nhn :\n (∫⁻ (a : α), (fun x => ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) x - f i x‖‖₊ ^ p) a ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\n⊢ (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\nh_nnnorm_nonneg :\n (fun a => ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p) = fun a =>\n (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p\n⊢ (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i",
"tactic": "change\n (∫⁻ a, (fun x => ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) x - f i x‖‖₊ ^ p) a ∂μ) ^ (1 / p) ≤\n ∑' i, B i at hn"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhp_pos : 0 < p\nh_nnnorm_nonneg :\n (fun a => ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p) = fun a =>\n (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p\nhn :\n (∫⁻ (a : α), (fun x => ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) x - f i x‖‖₊ ^ p) a ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\n⊢ (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i",
"tactic": "rwa [h_nnnorm_nonneg] at hn"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : snorm' (fun x => ∑ i in Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\n⊢ 0 < 1 / p",
"tactic": "simp [hp_pos]"
},
{
"state_after": "case h\nα : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\na : α\n⊢ ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p =\n (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\n⊢ (fun a => ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p) = fun a =>\n (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p",
"tactic": "ext1 a"
},
{
"state_after": "case h.e_a\nα : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\na : α\n⊢ ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ = ∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊",
"state_before": "case h\nα : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\na : α\n⊢ ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p =\n (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p",
"tactic": "congr"
},
{
"state_after": "case h.e_a\nα : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\na : α\n⊢ ENNReal.ofReal ‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ =\n ∑ x in Finset.range (n + 1), ENNReal.ofReal ‖f (x + 1) a - f x a‖",
"state_before": "case h.e_a\nα : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\na : α\n⊢ ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ = ∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊",
"tactic": "simp_rw [← ofReal_norm_eq_coe_nnnorm]"
},
{
"state_after": "case h.e_a\nα : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\na : α\n⊢ ENNReal.ofReal ‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ =\n ENNReal.ofReal (∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖)\n\ncase h.e_a\nα : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\na : α\n⊢ ∀ (i : ℕ), i ∈ Finset.range (n + 1) → 0 ≤ ‖f (i + 1) a - f i a‖",
"state_before": "case h.e_a\nα : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\na : α\n⊢ ENNReal.ofReal ‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ =\n ∑ x in Finset.range (n + 1), ENNReal.ofReal ‖f (x + 1) a - f x a‖",
"tactic": "rw [← ENNReal.ofReal_sum_of_nonneg]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\na : α\n⊢ 0 ≤ ∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖",
"state_before": "case h.e_a\nα : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\na : α\n⊢ ENNReal.ofReal ‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ =\n ENNReal.ofReal (∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖)",
"tactic": "rw [Real.norm_of_nonneg _]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\na : α\n⊢ 0 ≤ ∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖",
"tactic": "exact Finset.sum_nonneg fun x _ => norm_nonneg _"
},
{
"state_after": "no goals",
"state_before": "case h.e_a\nα : Type u_1\nE : Type u_2\nF : Type ?u.8156900\nG : Type ?u.8156903\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhn : (∫⁻ (a : α), ↑‖∑ i in Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i\nhp_pos : 0 < p\na : α\n⊢ ∀ (i : ℕ), i ∈ Finset.range (n + 1) → 0 ≤ ‖f (i + 1) a - f i a‖",
"tactic": "exact fun x _ => norm_nonneg _"
}
] |
[
1362,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1340,
9
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
tsum_even_add_odd
|
[] |
[
791,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
788,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.le_rpow_add
|
[
{
"state_after": "case inl\nx✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\nH : 0 = x\n⊢ x ^ y * x ^ z ≤ x ^ (y + z)\n\ncase inr\nx✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\npos : 0 < x\n⊢ x ^ y * x ^ z ≤ x ^ (y + z)",
"state_before": "x✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\n⊢ x ^ y * x ^ z ≤ x ^ (y + z)",
"tactic": "rcases le_iff_eq_or_lt.1 hx with (H | pos)"
},
{
"state_after": "case pos\nx✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\nH : 0 = x\nh : y + z = 0\n⊢ x ^ y * x ^ z ≤ x ^ (y + z)\n\ncase neg\nx✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\nH : 0 = x\nh : ¬y + z = 0\n⊢ x ^ y * x ^ z ≤ x ^ (y + z)",
"state_before": "case inl\nx✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\nH : 0 = x\n⊢ x ^ y * x ^ z ≤ x ^ (y + z)",
"tactic": "by_cases h : y + z = 0"
},
{
"state_after": "case pos\nx✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\nH : 0 = x\nh : y + z = 0\n⊢ 0 ^ y * 0 ^ z ≤ 1",
"state_before": "case pos\nx✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\nH : 0 = x\nh : y + z = 0\n⊢ x ^ y * x ^ z ≤ x ^ (y + z)",
"tactic": "simp only [H.symm, h, rpow_zero]"
},
{
"state_after": "no goals",
"state_before": "case pos\nx✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\nH : 0 = x\nh : y + z = 0\n⊢ 0 ^ y * 0 ^ z ≤ 1",
"tactic": "calc\n (0 : ℝ) ^ y * 0 ^ z ≤ 1 * 1 :=\n mul_le_mul (zero_rpow_le_one y) (zero_rpow_le_one z) (zero_rpow_nonneg z) zero_le_one\n _ = 1 := by simp"
},
{
"state_after": "no goals",
"state_before": "x✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\nH : 0 = x\nh : y + z = 0\n⊢ 1 * 1 = 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case neg\nx✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\nH : 0 = x\nh : ¬y + z = 0\n⊢ x ^ y * x ^ z ≤ x ^ (y + z)",
"tactic": "simp [rpow_add', ← H, h]"
},
{
"state_after": "no goals",
"state_before": "case inr\nx✝ y✝ z✝ x : ℝ\nhx : 0 ≤ x\ny z : ℝ\npos : 0 < x\n⊢ x ^ y * x ^ z ≤ x ^ (y + z)",
"tactic": "simp [rpow_add pos]"
}
] |
[
208,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Order/RelClasses.lean
|
Set.not_unbounded_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nr✝ : α → α → Prop\ns✝ : β → β → Prop\nr : α → α → Prop\ns : Set α\n⊢ ¬Unbounded r s ↔ Bounded r s",
"tactic": "rw [not_iff_comm, not_bounded_iff]"
}
] |
[
543,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
542,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.of_mul_of
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝ : Mul α\nx y : α\n⊢ of x * of y = of (x * y)",
"tactic": "rw [mul_def, lift.of, lift.of]"
}
] |
[
418,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
417,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
|
UniformOnFun.hasBasis_uniformity
|
[] |
[
675,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
672,
11
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.coe_smul
|
[] |
[
594,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
593,
1
] |
Mathlib/Topology/Basic.lean
|
continuous_iff_continuousAt
|
[] |
[
1678,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1676,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.Subset.refl
|
[] |
[
393,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.smul_toEquiv
|
[] |
[
473,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
1
] |
Std/Data/Int/DivMod.lean
|
Int.dvd_of_mod_eq_zero
|
[] |
[
673,
52
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
672,
1
] |
Mathlib/RingTheory/TensorProduct.lean
|
Algebra.TensorProduct.congr_symm_apply
|
[] |
[
911,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
909,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.PushoutCocone.inl_colimit_cocone
|
[] |
[
1161,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1160,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.sqrtd_im
|
[] |
[
108,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.max
|
[] |
[
443,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
441,
11
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.type_pUnit
|
[] |
[
263,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Analysis/Fourier/FourierTransform.lean
|
Real.continuous_fourierChar
|
[] |
[
249,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.snoc_cast_add
|
[] |
[
489,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
487,
1
] |
Mathlib/Logic/Basic.lean
|
xor_false
|
[
{
"state_after": "case h.a\nx✝ : Prop\n⊢ Xor' False x✝ ↔ id x✝",
"state_before": "⊢ Xor' False = id",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.a\nx✝ : Prop\n⊢ Xor' False x✝ ↔ id x✝",
"tactic": "simp [Xor']"
}
] |
[
287,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
9
] |
Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean
|
CategoryTheory.Limits.Multicofork.toSigmaCofork_π
|
[] |
[
630,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
629,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.eval_apply
|
[] |
[
36,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
abs_nsmul
|
[
{
"state_after": "case inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na✝ b : α\nn : ℕ\na : α\nhneg : a ≤ 0\n⊢ abs (n • a) = n • abs a\n\ncase inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na✝ b : α\nn : ℕ\na : α\nhpos : 0 ≤ a\n⊢ abs (n • a) = n • abs a",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na✝ b : α\nn : ℕ\na : α\n⊢ abs (n • a) = n • abs a",
"tactic": "cases' le_total a 0 with hneg hpos"
},
{
"state_after": "case inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na✝ b : α\nn : ℕ\na : α\nhneg : a ≤ 0\n⊢ 0 ≤ n • -a",
"state_before": "case inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na✝ b : α\nn : ℕ\na : α\nhneg : a ≤ 0\n⊢ abs (n • a) = n • abs a",
"tactic": "rw [abs_of_nonpos hneg, ← abs_neg, ← neg_nsmul, abs_of_nonneg]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na✝ b : α\nn : ℕ\na : α\nhneg : a ≤ 0\n⊢ 0 ≤ n • -a",
"tactic": "exact nsmul_nonneg (neg_nonneg.mpr hneg) n"
},
{
"state_after": "case inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na✝ b : α\nn : ℕ\na : α\nhpos : 0 ≤ a\n⊢ 0 ≤ n • a",
"state_before": "case inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na✝ b : α\nn : ℕ\na : α\nhpos : 0 ≤ a\n⊢ abs (n • a) = n • abs a",
"tactic": "rw [abs_of_nonneg hpos, abs_of_nonneg]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na✝ b : α\nn : ℕ\na : α\nhpos : 0 ≤ a\n⊢ 0 ≤ n • a",
"tactic": "exact nsmul_nonneg hpos n"
}
] |
[
457,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.sin_two_mul
|
[
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ sin (2 * x) = 2 * sin x * cos x",
"tactic": "rw [two_mul, sin_add, two_mul, add_mul, mul_comm]"
}
] |
[
1038,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1037,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigO.const_mul_right'
|
[] |
[
1528,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1526,
1
] |
Mathlib/Algebra/Associated.lean
|
Associates.irreducible_mk
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\n⊢ (¬IsUnit a ∧ ∀ (a_1 b : Associates α), Associates.mk a = a_1 * b → IsUnit a_1 ∨ IsUnit b) ↔\n ¬IsUnit a ∧ ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b",
"state_before": "α : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\n⊢ Irreducible (Associates.mk a) ↔ Irreducible a",
"tactic": "simp only [irreducible_iff, isUnit_mk]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\n⊢ (∀ (a_1 b : Associates α), Associates.mk a = a_1 * b → IsUnit a_1 ∨ IsUnit b) ↔\n ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b",
"state_before": "α : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\n⊢ (¬IsUnit a ∧ ∀ (a_1 b : Associates α), Associates.mk a = a_1 * b → IsUnit a_1 ∨ IsUnit b) ↔\n ¬IsUnit a ∧ ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b",
"tactic": "apply and_congr Iff.rfl"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\n⊢ (∀ (a_1 b : Associates α), Associates.mk a = a_1 * b → IsUnit a_1 ∨ IsUnit b) →\n ∀ (a_2 b : α), a = a_2 * b → IsUnit a_2 ∨ IsUnit b\n\ncase mpr\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\n⊢ (∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b) →\n ∀ (a_2 b : Associates α), Associates.mk a = a_2 * b → IsUnit a_2 ∨ IsUnit b",
"state_before": "α : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\n⊢ (∀ (a_1 b : Associates α), Associates.mk a = a_1 * b → IsUnit a_1 ∨ IsUnit b) ↔\n ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\nx y : α\nh : ∀ (a b : Associates α), Associates.mk (x * y) = a * b → IsUnit a ∨ IsUnit b\n⊢ IsUnit x ∨ IsUnit y",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\n⊢ (∀ (a_1 b : Associates α), Associates.mk a = a_1 * b → IsUnit a_1 ∨ IsUnit b) →\n ∀ (a_2 b : α), a = a_2 * b → IsUnit a_2 ∨ IsUnit b",
"tactic": "rintro h x y rfl"
},
{
"state_after": "no goals",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\nx y : α\nh : ∀ (a b : Associates α), Associates.mk (x * y) = a * b → IsUnit a ∨ IsUnit b\n⊢ IsUnit x ∨ IsUnit y",
"tactic": "simpa [isUnit_mk] using h (Associates.mk x) (Associates.mk y) rfl"
},
{
"state_after": "case mpr\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\nh : ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b\nx y : Associates α\n⊢ Associates.mk a = x * y → IsUnit x ∨ IsUnit y",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\n⊢ (∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b) →\n ∀ (a_2 b : Associates α), Associates.mk a = a_2 * b → IsUnit a_2 ∨ IsUnit b",
"tactic": "intro h x y"
},
{
"state_after": "case mpr\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\nh : ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b\nx✝ y✝ : Associates α\nx y : α\na_eq : Associates.mk a = Quotient.mk (Associated.setoid α) x * Quotient.mk (Associated.setoid α) y\n⊢ IsUnit (Quotient.mk (Associated.setoid α) x) ∨ IsUnit (Quotient.mk (Associated.setoid α) y)",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\nh : ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b\nx y : Associates α\n⊢ Associates.mk a = x * y → IsUnit x ∨ IsUnit y",
"tactic": "refine' Quotient.inductionOn₂ x y fun x y a_eq => _"
},
{
"state_after": "case mpr.intro\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\nh : ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b\nx✝ y✝ : Associates α\nx y : α\na_eq✝ : Associates.mk a = Quotient.mk (Associated.setoid α) x * Quotient.mk (Associated.setoid α) y\nu : αˣ\na_eq : x * y * ↑u = a\n⊢ IsUnit (Quotient.mk (Associated.setoid α) x) ∨ IsUnit (Quotient.mk (Associated.setoid α) y)",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\nh : ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b\nx✝ y✝ : Associates α\nx y : α\na_eq : Associates.mk a = Quotient.mk (Associated.setoid α) x * Quotient.mk (Associated.setoid α) y\n⊢ IsUnit (Quotient.mk (Associated.setoid α) x) ∨ IsUnit (Quotient.mk (Associated.setoid α) y)",
"tactic": "rcases Quotient.exact a_eq.symm with ⟨u, a_eq⟩"
},
{
"state_after": "case mpr.intro\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\nh : ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b\nx✝ y✝ : Associates α\nx y : α\na_eq✝ : Associates.mk a = Quotient.mk (Associated.setoid α) x * Quotient.mk (Associated.setoid α) y\nu : αˣ\na_eq : x * (y * ↑u) = a\n⊢ IsUnit (Quotient.mk (Associated.setoid α) x) ∨ IsUnit (Quotient.mk (Associated.setoid α) y)",
"state_before": "case mpr.intro\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\nh : ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b\nx✝ y✝ : Associates α\nx y : α\na_eq✝ : Associates.mk a = Quotient.mk (Associated.setoid α) x * Quotient.mk (Associated.setoid α) y\nu : αˣ\na_eq : x * y * ↑u = a\n⊢ IsUnit (Quotient.mk (Associated.setoid α) x) ∨ IsUnit (Quotient.mk (Associated.setoid α) y)",
"tactic": "rw [mul_assoc] at a_eq"
},
{
"state_after": "case mpr.intro\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\nh : ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b\nx✝ y✝ : Associates α\nx y : α\na_eq✝ : Associates.mk a = Quotient.mk (Associated.setoid α) x * Quotient.mk (Associated.setoid α) y\nu : αˣ\na_eq : x * (y * ↑u) = a\n⊢ IsUnit (Associates.mk x) ∨ IsUnit (Associates.mk y)",
"state_before": "case mpr.intro\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\nh : ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b\nx✝ y✝ : Associates α\nx y : α\na_eq✝ : Associates.mk a = Quotient.mk (Associated.setoid α) x * Quotient.mk (Associated.setoid α) y\nu : αˣ\na_eq : x * (y * ↑u) = a\n⊢ IsUnit (Quotient.mk (Associated.setoid α) x) ∨ IsUnit (Quotient.mk (Associated.setoid α) y)",
"tactic": "show IsUnit (Associates.mk x) ∨ IsUnit (Associates.mk y)"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nα : Type u_1\nβ : Type ?u.320045\nγ : Type ?u.320048\nδ : Type ?u.320051\ninst✝ : CommMonoidWithZero α\na : α\nh : ∀ (a_1 b : α), a = a_1 * b → IsUnit a_1 ∨ IsUnit b\nx✝ y✝ : Associates α\nx y : α\na_eq✝ : Associates.mk a = Quotient.mk (Associated.setoid α) x * Quotient.mk (Associated.setoid α) y\nu : αˣ\na_eq : x * (y * ↑u) = a\n⊢ IsUnit (Associates.mk x) ∨ IsUnit (Associates.mk y)",
"tactic": "simpa [isUnit_mk] using h _ _ a_eq.symm"
}
] |
[
1041,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1030,
1
] |
Mathlib/Computability/Ackermann.lean
|
one_lt_ack_succ_right
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ 1 < ack 0 (n + 1)",
"tactic": "simp"
},
{
"state_after": "m n : ℕ\n⊢ 1 < ack m (ack (m + 1) n)",
"state_before": "m n : ℕ\n⊢ 1 < ack (m + 1) (n + 1)",
"tactic": "rw [ack_succ_succ]"
},
{
"state_after": "case intro\nm n h✝ : ℕ\nh : ack (m + 1) n = succ h✝\n⊢ 1 < ack m (ack (m + 1) n)",
"state_before": "m n : ℕ\n⊢ 1 < ack m (ack (m + 1) n)",
"tactic": "cases' exists_eq_succ_of_ne_zero (ack_pos (m + 1) n).ne' with h h"
},
{
"state_after": "case intro\nm n h✝ : ℕ\nh : ack (m + 1) n = succ h✝\n⊢ 1 < ack m (succ h✝)",
"state_before": "case intro\nm n h✝ : ℕ\nh : ack (m + 1) n = succ h✝\n⊢ 1 < ack m (ack (m + 1) n)",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case intro\nm n h✝ : ℕ\nh : ack (m + 1) n = succ h✝\n⊢ 1 < ack m (succ h✝)",
"tactic": "apply one_lt_ack_succ_right"
}
] |
[
135,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Data/Matrix/Basis.lean
|
Matrix.StdBasisMatrix.mul_of_ne
|
[
{
"state_after": "case a.h\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\n⊢ (stdBasisMatrix i j c ⬝ stdBasisMatrix k l d) a b = OfNat.ofNat 0 a b",
"state_before": "l✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\n⊢ stdBasisMatrix i j c ⬝ stdBasisMatrix k l d = 0",
"tactic": "ext (a b)"
},
{
"state_after": "case a.h\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\n⊢ (∑ j_1 : n, (if i = a ∧ j = j_1 then c else 0) * if k = j_1 ∧ l = b then d else 0) = OfNat.ofNat 0 a b",
"state_before": "case a.h\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\n⊢ (stdBasisMatrix i j c ⬝ stdBasisMatrix k l d) a b = OfNat.ofNat 0 a b",
"tactic": "simp only [mul_apply, boole_mul, stdBasisMatrix]"
},
{
"state_after": "case pos\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\nh₁ : i = a\n⊢ (∑ j_1 : n, (if i = a ∧ j = j_1 then c else 0) * if k = j_1 ∧ l = b then d else 0) = OfNat.ofNat 0 a b\n\ncase neg\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\nh₁ : ¬i = a\n⊢ (∑ j_1 : n, (if i = a ∧ j = j_1 then c else 0) * if k = j_1 ∧ l = b then d else 0) = OfNat.ofNat 0 a b",
"state_before": "case a.h\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\n⊢ (∑ j_1 : n, (if i = a ∧ j = j_1 then c else 0) * if k = j_1 ∧ l = b then d else 0) = OfNat.ofNat 0 a b",
"tactic": "by_cases h₁ : i = a"
},
{
"state_after": "case pos\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\nh₁ : i = a\n⊢ (∑ x : n, if (k = x ∧ l = b) ∧ j = x then c * d else 0) = 0",
"state_before": "case pos\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\nh₁ : i = a\n⊢ (∑ j_1 : n, (if i = a ∧ j = j_1 then c else 0) * if k = j_1 ∧ l = b then d else 0) = OfNat.ofNat 0 a b",
"tactic": "simp only [h₁, true_and, mul_ite, ite_mul, zero_mul, mul_zero, ← ite_and, zero_apply]"
},
{
"state_after": "case pos\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\nh₁ : i = a\nx : n\nx✝ : x ∈ Finset.univ\n⊢ (if (k = x ∧ l = b) ∧ j = x then c * d else 0) = 0",
"state_before": "case pos\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\nh₁ : i = a\n⊢ (∑ x : n, if (k = x ∧ l = b) ∧ j = x then c * d else 0) = 0",
"tactic": "refine Finset.sum_eq_zero (fun x _ => ?_)"
},
{
"state_after": "case pos.hnc\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\nh₁ : i = a\nx : n\nx✝ : x ∈ Finset.univ\n⊢ ¬((k = x ∧ l = b) ∧ j = x)",
"state_before": "case pos\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\nh₁ : i = a\nx : n\nx✝ : x ∈ Finset.univ\n⊢ (if (k = x ∧ l = b) ∧ j = x then c * d else 0) = 0",
"tactic": "apply if_neg"
},
{
"state_after": "case pos.hnc.intro.intro\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh✝ : j ≠ k\nd : α\na : n\nh₁ : i = a\nx✝ : k ∈ Finset.univ\nh : j = k\n⊢ False",
"state_before": "case pos.hnc\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\nh₁ : i = a\nx : n\nx✝ : x ∈ Finset.univ\n⊢ ¬((k = x ∧ l = b) ∧ j = x)",
"tactic": "rintro ⟨⟨rfl, rfl⟩, h⟩"
},
{
"state_after": "no goals",
"state_before": "case pos.hnc.intro.intro\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh✝ : j ≠ k\nd : α\na : n\nh₁ : i = a\nx✝ : k ∈ Finset.univ\nh : j = k\n⊢ False",
"tactic": "contradiction"
},
{
"state_after": "no goals",
"state_before": "case neg\nl✝ : Type ?u.47153\nm : Type ?u.47156\nn : Type u_1\nR : Type ?u.47162\nα : Type u_2\ninst✝⁴ : DecidableEq l✝\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nk l : n\nh : j ≠ k\nd : α\na b : n\nh₁ : ¬i = a\n⊢ (∑ j_1 : n, (if i = a ∧ j = j_1 then c else 0) * if k = j_1 ∧ l = b then d else 0) = OfNat.ofNat 0 a b",
"tactic": "simp only [h₁, false_and, ite_false, mul_ite, zero_mul, mul_zero, ite_self,\n Finset.sum_const_zero, zero_apply]"
}
] |
[
217,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Data/Rat/Order.lean
|
Rat.lt_one_iff_num_lt_denom
|
[
{
"state_after": "no goals",
"state_before": "a b c q : ℚ\n⊢ q < 1 ↔ q.num < ↑q.den",
"tactic": "simp [Rat.lt_def]"
}
] |
[
297,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.lookupAll_cons_eq
|
[] |
[
266,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.EqOn.inter_preimage_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.7921\nι : Sort ?u.7924\nπ : α → Type ?u.7929\ns s₁ s₂ : Set α\nt✝ t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nheq : EqOn f₁ f₂ s\nt : Set β\nx : α\nhx : x ∈ s\n⊢ x ∈ f₁ ⁻¹' t ↔ x ∈ f₂ ⁻¹' t",
"tactic": "rw [mem_preimage, mem_preimage, heq hx]"
}
] |
[
218,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Data/Multiset/Sort.lean
|
Multiset.sort_singleton
|
[] |
[
68,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
aemeasurable_const
|
[] |
[
713,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
712,
1
] |
Mathlib/GroupTheory/Torsion.lean
|
IsTorsion.exponentExists
|
[] |
[
147,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Topology/UnitInterval.lean
|
unitInterval.coe_symm_eq
|
[] |
[
126,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Valid'.node
|
[] |
[
1070,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1067,
1
] |
Mathlib/Data/Polynomial/Module.lean
|
PolynomialModule.map_smul
|
[
{
"state_after": "case h0\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf : M →ₗ[R] M'\np : R[X]\nq : PolynomialModule R M\n⊢ ↑(map R' f) (p • 0) = Polynomial.map (algebraMap R R') p • ↑(map R' f) 0\n\ncase hadd\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf : M →ₗ[R] M'\np : R[X]\nq : PolynomialModule R M\n⊢ ∀ (f_1 g : PolynomialModule R M),\n ↑(map R' f) (p • f_1) = Polynomial.map (algebraMap R R') p • ↑(map R' f) f_1 →\n ↑(map R' f) (p • g) = Polynomial.map (algebraMap R R') p • ↑(map R' f) g →\n ↑(map R' f) (p • (f_1 + g)) = Polynomial.map (algebraMap R R') p • ↑(map R' f) (f_1 + g)\n\ncase hsingle\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf : M →ₗ[R] M'\np : R[X]\nq : PolynomialModule R M\n⊢ ∀ (a : ℕ) (b : M),\n ↑(map R' f) (p • ↑(single R a) b) = Polynomial.map (algebraMap R R') p • ↑(map R' f) (↑(single R a) b)",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf : M →ₗ[R] M'\np : R[X]\nq : PolynomialModule R M\n⊢ ↑(map R' f) (p • q) = Polynomial.map (algebraMap R R') p • ↑(map R' f) q",
"tactic": "apply induction_linear q"
},
{
"state_after": "case hsingle\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf : M →ₗ[R] M'\np : R[X]\nq : PolynomialModule R M\ni : ℕ\nm : M\n⊢ ↑(map R' f) (p • ↑(single R i) m) = Polynomial.map (algebraMap R R') p • ↑(map R' f) (↑(single R i) m)",
"state_before": "case hsingle\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf : M →ₗ[R] M'\np : R[X]\nq : PolynomialModule R M\n⊢ ∀ (a : ℕ) (b : M),\n ↑(map R' f) (p • ↑(single R a) b) = Polynomial.map (algebraMap R R') p • ↑(map R' f) (↑(single R a) b)",
"tactic": "intro i m"
},
{
"state_after": "case hsingle.h_add\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf : M →ₗ[R] M'\nq : PolynomialModule R M\ni : ℕ\nm : M\np✝ q✝ : R[X]\ne₁ : ↑(map R' f) (p✝ • ↑(single R i) m) = Polynomial.map (algebraMap R R') p✝ • ↑(map R' f) (↑(single R i) m)\ne₂ : ↑(map R' f) (q✝ • ↑(single R i) m) = Polynomial.map (algebraMap R R') q✝ • ↑(map R' f) (↑(single R i) m)\n⊢ ↑(map R' f) ((p✝ + q✝) • ↑(single R i) m) = Polynomial.map (algebraMap R R') (p✝ + q✝) • ↑(map R' f) (↑(single R i) m)\n\ncase hsingle.h_monomial\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf : M →ₗ[R] M'\nq : PolynomialModule R M\ni : ℕ\nm : M\nn✝ : ℕ\na✝ : R\n⊢ ↑(map R' f) (↑(monomial n✝) a✝ • ↑(single R i) m) =\n Polynomial.map (algebraMap R R') (↑(monomial n✝) a✝) • ↑(map R' f) (↑(single R i) m)",
"state_before": "case hsingle\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf : M →ₗ[R] M'\np : R[X]\nq : PolynomialModule R M\ni : ℕ\nm : M\n⊢ ↑(map R' f) (p • ↑(single R i) m) = Polynomial.map (algebraMap R R') p • ↑(map R' f) (↑(single R i) m)",
"tactic": "induction' p using Polynomial.induction_on' with _ _ e₁ e₂"
},
{
"state_after": "no goals",
"state_before": "case h0\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf : M →ₗ[R] M'\np : R[X]\nq : PolynomialModule R M\n⊢ ↑(map R' f) (p • 0) = Polynomial.map (algebraMap R R') p • ↑(map R' f) 0",
"tactic": "rw [smul_zero, map_zero, smul_zero]"
},
{
"state_after": "case hadd\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf✝ : M →ₗ[R] M'\np : R[X]\nq f g : PolynomialModule R M\ne₁ : ↑(map R' f✝) (p • f) = Polynomial.map (algebraMap R R') p • ↑(map R' f✝) f\ne₂ : ↑(map R' f✝) (p • g) = Polynomial.map (algebraMap R R') p • ↑(map R' f✝) g\n⊢ ↑(map R' f✝) (p • (f + g)) = Polynomial.map (algebraMap R R') p • ↑(map R' f✝) (f + g)",
"state_before": "case hadd\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf : M →ₗ[R] M'\np : R[X]\nq : PolynomialModule R M\n⊢ ∀ (f_1 g : PolynomialModule R M),\n ↑(map R' f) (p • f_1) = Polynomial.map (algebraMap R R') p • ↑(map R' f) f_1 →\n ↑(map R' f) (p • g) = Polynomial.map (algebraMap R R') p • ↑(map R' f) g →\n ↑(map R' f) (p • (f_1 + g)) = Polynomial.map (algebraMap R R') p • ↑(map R' f) (f_1 + g)",
"tactic": "intro f g e₁ e₂"
},
{
"state_after": "no goals",
"state_before": "case hadd\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf✝ : M →ₗ[R] M'\np : R[X]\nq f g : PolynomialModule R M\ne₁ : ↑(map R' f✝) (p • f) = Polynomial.map (algebraMap R R') p • ↑(map R' f✝) f\ne₂ : ↑(map R' f✝) (p • g) = Polynomial.map (algebraMap R R') p • ↑(map R' f✝) g\n⊢ ↑(map R' f✝) (p • (f + g)) = Polynomial.map (algebraMap R R') p • ↑(map R' f✝) (f + g)",
"tactic": "rw [smul_add, map_add, e₁, e₂, map_add, smul_add]"
},
{
"state_after": "no goals",
"state_before": "case hsingle.h_add\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf : M →ₗ[R] M'\nq : PolynomialModule R M\ni : ℕ\nm : M\np✝ q✝ : R[X]\ne₁ : ↑(map R' f) (p✝ • ↑(single R i) m) = Polynomial.map (algebraMap R R') p✝ • ↑(map R' f) (↑(single R i) m)\ne₂ : ↑(map R' f) (q✝ • ↑(single R i) m) = Polynomial.map (algebraMap R R') q✝ • ↑(map R' f) (↑(single R i) m)\n⊢ ↑(map R' f) ((p✝ + q✝) • ↑(single R i) m) = Polynomial.map (algebraMap R R') (p✝ + q✝) • ↑(map R' f) (↑(single R i) m)",
"tactic": "rw [add_smul, map_add, e₁, e₂, Polynomial.map_add, add_smul]"
},
{
"state_after": "no goals",
"state_before": "case hsingle.h_monomial\nR : Type u_1\nM : Type u_2\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nI : Ideal R\nS : Type ?u.432772\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R' M'\ninst✝² : Algebra R R'\ninst✝¹ : Module R M'\ninst✝ : IsScalarTower R R' M'\nf : M →ₗ[R] M'\nq : PolynomialModule R M\ni : ℕ\nm : M\nn✝ : ℕ\na✝ : R\n⊢ ↑(map R' f) (↑(monomial n✝) a✝ • ↑(single R i) m) =\n Polynomial.map (algebraMap R R') (↑(monomial n✝) a✝) • ↑(map R' f) (↑(single R i) m)",
"tactic": "rw [monomial_smul_single, map_single, Polynomial.map_monomial, map_single, monomial_smul_single,\n f.map_smul, algebraMap_smul]"
}
] |
[
264,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.swap_rightInverse
|
[] |
[
365,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
364,
1
] |
Mathlib/Init/Data/Sigma/Basic.lean
|
ex_of_psig
|
[] |
[
14,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
13,
1
] |
Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean
|
Complex.measurable_sin
|
[] |
[
98,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.edgeSet_mono
|
[] |
[
634,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
633,
1
] |
Mathlib/Analysis/Normed/Group/AddTorsor.lean
|
dist_vadd_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.11888\nV : Type u_2\nP : Type u_1\nW : Type ?u.11897\nQ : Type ?u.11900\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : NormedAddCommGroup W\ninst✝¹ : MetricSpace Q\ninst✝ : NormedAddTorsor W Q\nv : V\nx : P\n⊢ dist x (v +ᵥ x) = ‖v‖",
"tactic": "rw [dist_comm, dist_vadd_left]"
}
] |
[
128,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
dvd_mul
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\nk m n : α\n⊢ (∃ d₁ d₂, d₁ ∣ m ∧ d₂ ∣ n ∧ k = d₁ * d₂) → k ∣ m * n",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\nk m n : α\n⊢ k ∣ m * n ↔ ∃ d₁ d₂, d₁ ∣ m ∧ d₂ ∣ n ∧ k = d₁ * d₂",
"tactic": "refine' ⟨exists_dvd_and_dvd_of_dvd_mul, _⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\nm n d₁ d₂ : α\nhy : d₁ ∣ m\nhz : d₂ ∣ n\n⊢ d₁ * d₂ ∣ m * n",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\nk m n : α\n⊢ (∃ d₁ d₂, d₁ ∣ m ∧ d₂ ∣ n ∧ k = d₁ * d₂) → k ∣ m * n",
"tactic": "rintro ⟨d₁, d₂, hy, hz, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\nm n d₁ d₂ : α\nhy : d₁ ∣ m\nhz : d₂ ∣ n\n⊢ d₁ * d₂ ∣ m * n",
"tactic": "exact mul_dvd_mul hy hz"
}
] |
[
549,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
546,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.range_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.57313\ninst✝¹ : MeasurableSpace α\ninst✝ : DecidableEq γ\ng : β → γ\nf : α →ₛ β\n⊢ ↑(SimpleFunc.range (map g f)) = ↑(Finset.image g (SimpleFunc.range f))",
"tactic": "simp only [coe_range, coe_map, Finset.coe_image, range_comp]"
}
] |
[
323,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
EuclideanGeometry.dist_div_tan_angle_of_angle_eq_pi_div_two
|
[
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₁ ≠ p₂ ∨ p₃ = p₂\n⊢ dist p₁ p₂ / Real.tan (∠ p₂ p₃ p₁) = dist p₃ p₂",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ∠ p₁ p₂ p₃ = π / 2\nh0 : p₁ ≠ p₂ ∨ p₃ = p₂\n⊢ dist p₁ p₂ / Real.tan (∠ p₂ p₃ p₁) = dist p₃ p₂",
"tactic": "rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←\n inner_neg_left, neg_vsub_eq_vsub_rev] at h"
},
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₂ -ᵥ p₃ = 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ dist p₁ p₂ / Real.tan (∠ p₂ p₃ p₁) = dist p₃ p₂",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₁ ≠ p₂ ∨ p₃ = p₂\n⊢ dist p₁ p₂ / Real.tan (∠ p₂ p₃ p₁) = dist p₃ p₂",
"tactic": "rw [eq_comm, ← @vsub_ne_zero V, ← @vsub_eq_zero_iff_eq V, or_comm] at h0"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₂ -ᵥ p₃ = 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ dist p₁ p₂ / Real.tan (∠ p₂ p₃ p₁) = dist p₃ p₂",
"tactic": "rw [angle, dist_eq_norm_vsub V p₁ p₂, dist_eq_norm_vsub' V p₃ p₂, ← vsub_add_vsub_cancel p₁ p₂ p₃,\n add_comm, norm_div_tan_angle_add_of_inner_eq_zero h h0]"
}
] |
[
522,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
LinearIndependent.independent_span_singleton
|
[
{
"state_after": "ι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\n⊢ Disjoint ((fun i => span R {v i}) i) (⨆ (j : ι) (_ : j ≠ i), (fun i => span R {v i}) j)",
"state_before": "ι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\n⊢ CompleteLattice.Independent fun i => span R {v i}",
"tactic": "refine' CompleteLattice.independent_def.mp fun i => _"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\n⊢ ((fun i => span R {v i}) i ⊓ ⨆ (j : ι) (_ : j ≠ i), (fun i => span R {v i}) j) ≤ ⊥",
"state_before": "ι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\n⊢ Disjoint ((fun i => span R {v i}) i) (⨆ (j : ι) (_ : j ≠ i), (fun i => span R {v i}) j)",
"tactic": "rw [disjoint_iff_inf_le]"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nm : M\nhm : m ∈ (fun i => span R {v i}) i ⊓ ⨆ (j : ι) (_ : j ≠ i), (fun i => span R {v i}) j\n⊢ m ∈ ⊥",
"state_before": "ι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\n⊢ ((fun i => span R {v i}) i ⊓ ⨆ (j : ι) (_ : j ≠ i), (fun i => span R {v i}) j) ≤ ⊥",
"tactic": "intro m hm"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nm : M\nhm : (∃ a, a • v i = m) ∧ m ∈ span R (range fun i_1 => v ↑i_1)\n⊢ m ∈ ⊥",
"state_before": "ι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nm : M\nhm : m ∈ (fun i => span R {v i}) i ⊓ ⨆ (j : ι) (_ : j ≠ i), (fun i => span R {v i}) j\n⊢ m ∈ ⊥",
"tactic": "simp only [mem_inf, mem_span_singleton, iSup_subtype', ← span_range_eq_iSup] at hm"
},
{
"state_after": "case intro.intro\nι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nr : R\nhm : r • v i ∈ span R (range fun i_1 => v ↑i_1)\n⊢ r • v i ∈ ⊥",
"state_before": "ι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nm : M\nhm : (∃ a, a • v i = m) ∧ m ∈ span R (range fun i_1 => v ↑i_1)\n⊢ m ∈ ⊥",
"tactic": "obtain ⟨⟨r, rfl⟩, hm⟩ := hm"
},
{
"state_after": "case intro.intro\nι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nr : R\nhm : r • v i ∈ span R (range fun i_1 => v ↑i_1)\n⊢ r = 0",
"state_before": "case intro.intro\nι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nr : R\nhm : r • v i ∈ span R (range fun i_1 => v ↑i_1)\n⊢ r • v i ∈ ⊥",
"tactic": "suffices r = 0 by simp [this]"
},
{
"state_after": "case intro.intro.a\nι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nr : R\nhm : r • v i ∈ span R (range fun i_1 => v ↑i_1)\n⊢ r • v i ∈ span R (v '' (univ \\ {i}))",
"state_before": "case intro.intro\nι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nr : R\nhm : r • v i ∈ span R (range fun i_1 => v ↑i_1)\n⊢ r = 0",
"tactic": "apply linearIndependent_iff_not_smul_mem_span.mp hv i"
},
{
"state_after": "case intro.intro.a\nι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nr : R\nhm : r • v i ∈ span R (range fun i_1 => v ↑i_1)\n⊢ v '' (univ \\ {i}) = range fun j => v ↑j",
"state_before": "case intro.intro.a\nι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nr : R\nhm : r • v i ∈ span R (range fun i_1 => v ↑i_1)\n⊢ r • v i ∈ span R (v '' (univ \\ {i}))",
"tactic": "suffices v '' (univ \\ {i}) = range fun j : { j // j ≠ i } => v j by rwa [this]"
},
{
"state_after": "case intro.intro.a.h\nι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nr : R\nhm : r • v i ∈ span R (range fun i_1 => v ↑i_1)\nx✝ : M\n⊢ x✝ ∈ v '' (univ \\ {i}) ↔ x✝ ∈ range fun j => v ↑j",
"state_before": "case intro.intro.a\nι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nr : R\nhm : r • v i ∈ span R (range fun i_1 => v ↑i_1)\n⊢ v '' (univ \\ {i}) = range fun j => v ↑j",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.a.h\nι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nr : R\nhm : r • v i ∈ span R (range fun i_1 => v ↑i_1)\nx✝ : M\n⊢ x✝ ∈ v '' (univ \\ {i}) ↔ x✝ ∈ range fun j => v ↑j",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nr : R\nhm : r • v i ∈ span R (range fun i_1 => v ↑i_1)\nthis : r = 0\n⊢ r • v i ∈ ⊥",
"tactic": "simp [this]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.484575\nR : Type u_1\nK : Type ?u.484581\nM : Type u_2\nM' : Type ?u.484587\nM'' : Type ?u.484590\nV : Type u\nV' : Type ?u.484595\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\nr : R\nhm : r • v i ∈ span R (range fun i_1 => v ↑i_1)\nthis : v '' (univ \\ {i}) = range fun j => v ↑j\n⊢ r • v i ∈ span R (v '' (univ \\ {i}))",
"tactic": "rwa [this]"
}
] |
[
867,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
855,
1
] |
Mathlib/CategoryTheory/Skeletal.lean
|
CategoryTheory.skeleton_skeletal
|
[
{
"state_after": "case intro\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nX Y : Skeleton C\nh : X ≅ Y\n⊢ X = Y",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\n⊢ Skeletal (Skeleton C)",
"tactic": "rintro X Y ⟨h⟩"
},
{
"state_after": "case intro\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nX Y : Skeleton C\nh : X ≅ Y\nthis : Quotient.out X ≈ Quotient.out Y\n⊢ X = Y",
"state_before": "case intro\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nX Y : Skeleton C\nh : X ≅ Y\n⊢ X = Y",
"tactic": "have : X.out ≈ Y.out := ⟨(fromSkeleton C).mapIso h⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nX Y : Skeleton C\nh : X ≅ Y\nthis : Quotient.out X ≈ Quotient.out Y\n⊢ X = Y",
"tactic": "simpa using Quotient.sound this"
}
] |
[
117,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Combinatorics/Quiver/Path.lean
|
Quiver.Path.comp_assoc
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\ninst✝ : Quiver V\na✝¹ b✝¹ c✝ d a b c x✝ : V\np : Path a b\nq : Path b c\nb✝ : V\nr : Path c b✝\na✝ : b✝ ⟶ x✝\n⊢ comp (comp p q) (cons r a✝) = comp p (comp q (cons r a✝))",
"tactic": "rw [comp_cons, comp_cons, comp_cons, comp_assoc p q r]"
}
] |
[
118,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
IsLocallyConstant.eq_const
|
[] |
[
172,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal_transpose
|
[
{
"state_after": "case a.h\nl : Type ?u.131023\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.131035\nq : Type ?u.131038\nm' : o → Type ?u.131043\nn' : o → Type ?u.131048\np' : o → Type ?u.131053\nR : Type ?u.131056\nS : Type ?u.131059\nα : Type u_3\nβ : Type ?u.131065\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : o → Matrix m n α\ni✝ : n × o\nx✝ : m × o\n⊢ (blockDiagonal M)ᵀ i✝ x✝ = blockDiagonal (fun k => (M k)ᵀ) i✝ x✝",
"state_before": "l : Type ?u.131023\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.131035\nq : Type ?u.131038\nm' : o → Type ?u.131043\nn' : o → Type ?u.131048\np' : o → Type ?u.131053\nR : Type ?u.131056\nS : Type ?u.131059\nα : Type u_3\nβ : Type ?u.131065\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : o → Matrix m n α\n⊢ (blockDiagonal M)ᵀ = blockDiagonal fun k => (M k)ᵀ",
"tactic": "ext"
},
{
"state_after": "case a.h\nl : Type ?u.131023\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.131035\nq : Type ?u.131038\nm' : o → Type ?u.131043\nn' : o → Type ?u.131048\np' : o → Type ?u.131053\nR : Type ?u.131056\nS : Type ?u.131059\nα : Type u_3\nβ : Type ?u.131065\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : o → Matrix m n α\ni✝ : n × o\nx✝ : m × o\n⊢ (if i✝.snd = x✝.snd then M i✝.snd x✝.fst i✝.fst else 0) = if i✝.snd = x✝.snd then M x✝.snd x✝.fst i✝.fst else 0",
"state_before": "case a.h\nl : Type ?u.131023\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.131035\nq : Type ?u.131038\nm' : o → Type ?u.131043\nn' : o → Type ?u.131048\np' : o → Type ?u.131053\nR : Type ?u.131056\nS : Type ?u.131059\nα : Type u_3\nβ : Type ?u.131065\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : o → Matrix m n α\ni✝ : n × o\nx✝ : m × o\n⊢ (blockDiagonal M)ᵀ i✝ x✝ = blockDiagonal (fun k => (M k)ᵀ) i✝ x✝",
"tactic": "simp only [transpose_apply, blockDiagonal_apply, eq_comm]"
},
{
"state_after": "case a.h.inl\nl : Type ?u.131023\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.131035\nq : Type ?u.131038\nm' : o → Type ?u.131043\nn' : o → Type ?u.131048\np' : o → Type ?u.131053\nR : Type ?u.131056\nS : Type ?u.131059\nα : Type u_3\nβ : Type ?u.131065\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : o → Matrix m n α\ni✝ : n × o\nx✝ : m × o\nh : i✝.snd = x✝.snd\n⊢ M i✝.snd x✝.fst i✝.fst = M x✝.snd x✝.fst i✝.fst\n\ncase a.h.inr\nl : Type ?u.131023\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.131035\nq : Type ?u.131038\nm' : o → Type ?u.131043\nn' : o → Type ?u.131048\np' : o → Type ?u.131053\nR : Type ?u.131056\nS : Type ?u.131059\nα : Type u_3\nβ : Type ?u.131065\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : o → Matrix m n α\ni✝ : n × o\nx✝ : m × o\nh : ¬i✝.snd = x✝.snd\n⊢ 0 = 0",
"state_before": "case a.h\nl : Type ?u.131023\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.131035\nq : Type ?u.131038\nm' : o → Type ?u.131043\nn' : o → Type ?u.131048\np' : o → Type ?u.131053\nR : Type ?u.131056\nS : Type ?u.131059\nα : Type u_3\nβ : Type ?u.131065\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : o → Matrix m n α\ni✝ : n × o\nx✝ : m × o\n⊢ (if i✝.snd = x✝.snd then M i✝.snd x✝.fst i✝.fst else 0) = if i✝.snd = x✝.snd then M x✝.snd x✝.fst i✝.fst else 0",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case a.h.inl\nl : Type ?u.131023\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.131035\nq : Type ?u.131038\nm' : o → Type ?u.131043\nn' : o → Type ?u.131048\np' : o → Type ?u.131053\nR : Type ?u.131056\nS : Type ?u.131059\nα : Type u_3\nβ : Type ?u.131065\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : o → Matrix m n α\ni✝ : n × o\nx✝ : m × o\nh : i✝.snd = x✝.snd\n⊢ M i✝.snd x✝.fst i✝.fst = M x✝.snd x✝.fst i✝.fst",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case a.h.inr\nl : Type ?u.131023\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.131035\nq : Type ?u.131038\nm' : o → Type ?u.131043\nn' : o → Type ?u.131048\np' : o → Type ?u.131053\nR : Type ?u.131056\nS : Type ?u.131059\nα : Type u_3\nβ : Type ?u.131065\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\nM : o → Matrix m n α\ni✝ : n × o\nx✝ : m × o\nh : ¬i✝.snd = x✝.snd\n⊢ 0 = 0",
"tactic": "rfl"
}
] |
[
388,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
382,
1
] |
Mathlib/Data/Fintype/Lattice.lean
|
Finite.exists_min
|
[
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : Finite α\ninst✝¹ : Nonempty α\ninst✝ : LinearOrder β\nf : α → β\nval✝ : Fintype α\n⊢ ∃ x₀, ∀ (x : α), f x₀ ≤ f x",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : Finite α\ninst✝¹ : Nonempty α\ninst✝ : LinearOrder β\nf : α → β\n⊢ ∃ x₀, ∀ (x : α), f x₀ ≤ f x",
"tactic": "cases nonempty_fintype α"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : Finite α\ninst✝¹ : Nonempty α\ninst✝ : LinearOrder β\nf : α → β\nval✝ : Fintype α\n⊢ ∃ x₀, ∀ (x : α), f x₀ ≤ f x",
"tactic": "simpa using exists_min_image univ f univ_nonempty"
}
] |
[
71,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean
|
balancedCore_mem_nhds_zero
|
[
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\n⊢ balancedCore 𝕜 U ∈ 𝓝 0",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\n⊢ balancedCore 𝕜 U ∈ 𝓝 0",
"tactic": "obtain ⟨r, V, hr, hV, hrVU⟩ : ∃ (r : ℝ)(V : Set E),\n 0 < r ∧ V ∈ 𝓝 (0 : E) ∧ ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U := by\n have h : Filter.Tendsto (fun x : 𝕜 × E => x.fst • x.snd) (𝓝 (0, 0)) (𝓝 0) :=\n continuous_smul.tendsto' (0, 0) _ (smul_zero _)\n simpa only [← Prod.exists', ← Prod.forall', ← and_imp, ← and_assoc, exists_prop] using\n h.basis_left (NormedAddCommGroup.nhds_zero_basis_norm_lt.prod_nhds (𝓝 _).basis_sets) U hU"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : 0 < ‖y‖\nhyr : ‖y‖ < r\n⊢ balancedCore 𝕜 U ∈ 𝓝 0",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\n⊢ balancedCore 𝕜 U ∈ 𝓝 0",
"tactic": "rcases NormedField.exists_norm_lt 𝕜 hr with ⟨y, hy₀, hyr⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\n⊢ balancedCore 𝕜 U ∈ 𝓝 0",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : 0 < ‖y‖\nhyr : ‖y‖ < r\n⊢ balancedCore 𝕜 U ∈ 𝓝 0",
"tactic": "rw [norm_pos_iff] at hy₀"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\nthis : y • V ∈ 𝓝 0\n⊢ balancedCore 𝕜 U ∈ 𝓝 0",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\n⊢ balancedCore 𝕜 U ∈ 𝓝 0",
"tactic": "have : y • V ∈ 𝓝 (0 : E) := (set_smul_mem_nhds_zero_iff hy₀).mpr hV"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\nthis : y • V ∈ 𝓝 0\na : 𝕜\nha : ‖a‖ ≤ 1\n⊢ a • y • V ⊆ U",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\nthis : y • V ∈ 𝓝 0\n⊢ balancedCore 𝕜 U ∈ 𝓝 0",
"tactic": "refine' Filter.mem_of_superset this (subset_balancedCore (mem_of_mem_nhds hU) fun a ha => _)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\nthis : y • V ∈ 𝓝 0\na : 𝕜\nha : ‖a‖ ≤ 1\n⊢ (a * y) • V ⊆ U",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\nthis : y • V ∈ 𝓝 0\na : 𝕜\nha : ‖a‖ ≤ 1\n⊢ a • y • V ⊆ U",
"tactic": "rw [smul_smul]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\nthis : y • V ∈ 𝓝 0\na : 𝕜\nha : ‖a‖ ≤ 1\nz : E\nhz : z ∈ V\n⊢ (fun x => (a * y) • x) z ∈ U",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\nthis : y • V ∈ 𝓝 0\na : 𝕜\nha : ‖a‖ ≤ 1\n⊢ (a * y) • V ⊆ U",
"tactic": "rintro _ ⟨z, hz, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\nthis : y • V ∈ 𝓝 0\na : 𝕜\nha : ‖a‖ ≤ 1\nz : E\nhz : z ∈ V\n⊢ ‖a * y‖ < r",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\nthis : y • V ∈ 𝓝 0\na : 𝕜\nha : ‖a‖ ≤ 1\nz : E\nhz : z ∈ V\n⊢ (fun x => (a * y) • x) z ∈ U",
"tactic": "refine' hrVU _ _ _ hz"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\nthis : y • V ∈ 𝓝 0\na : 𝕜\nha : ‖a‖ ≤ 1\nz : E\nhz : z ∈ V\n⊢ ‖a‖ * ‖y‖ < 1 * r",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\nthis : y • V ∈ 𝓝 0\na : 𝕜\nha : ‖a‖ ≤ 1\nz : E\nhz : z ∈ V\n⊢ ‖a * y‖ < r",
"tactic": "rw [norm_mul, ← one_mul r]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nr : ℝ\nV : Set E\nhr : 0 < r\nhV : V ∈ 𝓝 0\nhrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U\ny : 𝕜\nhy₀ : y ≠ 0\nhyr : ‖y‖ < r\nthis : y • V ∈ 𝓝 0\na : 𝕜\nha : ‖a‖ ≤ 1\nz : E\nhz : z ∈ V\n⊢ ‖a‖ * ‖y‖ < 1 * r",
"tactic": "exact mul_lt_mul' ha hyr (norm_nonneg y) one_pos"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst • x.snd) (𝓝 (0, 0)) (𝓝 0)\n⊢ ∃ r V, 0 < r ∧ V ∈ 𝓝 0 ∧ ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\n⊢ ∃ r V, 0 < r ∧ V ∈ 𝓝 0 ∧ ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U",
"tactic": "have h : Filter.Tendsto (fun x : 𝕜 × E => x.fst • x.snd) (𝓝 (0, 0)) (𝓝 0) :=\n continuous_smul.tendsto' (0, 0) _ (smul_zero _)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.92160\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst • x.snd) (𝓝 (0, 0)) (𝓝 0)\n⊢ ∃ r V, 0 < r ∧ V ∈ 𝓝 0 ∧ ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U",
"tactic": "simpa only [← Prod.exists', ← Prod.forall', ← and_imp, ← and_assoc, exists_prop] using\n h.basis_left (NormedAddCommGroup.nhds_zero_basis_norm_lt.prod_nhds (𝓝 _).basis_sets) U hU"
}
] |
[
262,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/RingTheory/Derivation/Basic.lean
|
Derivation.coe_sub_linearMap
|
[] |
[
451,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
450,
1
] |
Mathlib/Topology/Constructions.lean
|
continuous_isRight
|
[] |
[
880,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
879,
1
] |
Mathlib/Analysis/Normed/Order/Lattice.lean
|
norm_sup_le_add
|
[
{
"state_after": "α : Type u_1\ninst✝ : NormedLatticeAddCommGroup α\nx y : α\nh : ‖x ⊔ y - 0 ⊔ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖\n⊢ ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖",
"state_before": "α : Type u_1\ninst✝ : NormedLatticeAddCommGroup α\nx y : α\n⊢ ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖",
"tactic": "have h : ‖x ⊔ y - 0 ⊔ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖ := norm_sup_sub_sup_le_add_norm x y 0 0"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : NormedLatticeAddCommGroup α\nx y : α\nh : ‖x ⊔ y - 0 ⊔ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖\n⊢ ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖",
"tactic": "simpa only [sup_idem, sub_zero] using h"
}
] |
[
156,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
orderOf_one
|
[
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\n⊢ orderOf 1 = 1",
"tactic": "rw [orderOf, ← minimalPeriod_id (x := (1:G)), ← one_mul_eq_id]"
}
] |
[
221,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/Order/Ideal.lean
|
Order.Ideal.principal_bot
|
[] |
[
323,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Structure.funMap_apply₁
|
[] |
[
430,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.ofReal_pow
|
[] |
[
253,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.mem_rootSet_of_ne
|
[] |
[
950,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
948,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.dvd_iff_div_factorization_eq_tsub
|
[
{
"state_after": "d n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\n⊢ factorization (n / d) = factorization n - factorization d → d ∣ n",
"state_before": "d n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\n⊢ d ∣ n ↔ factorization (n / d) = factorization n - factorization d",
"tactic": "refine' ⟨factorization_div, _⟩"
},
{
"state_after": "case inl\nd : ℕ\nhd : d ≠ 0\nhdn : d ≤ d\n⊢ factorization (d / d) = factorization d - factorization d → d ∣ d\n\ncase inr\nd n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\nhd_lt_n : d < n\n⊢ factorization (n / d) = factorization n - factorization d → d ∣ n",
"state_before": "d n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\n⊢ factorization (n / d) = factorization n - factorization d → d ∣ n",
"tactic": "rcases eq_or_lt_of_le hdn with (rfl | hd_lt_n)"
},
{
"state_after": "case inr\nd n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\nhd_lt_n : d < n\nh1 : n / d ≠ 0\n⊢ factorization (n / d) = factorization n - factorization d → d ∣ n",
"state_before": "case inr\nd n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\nhd_lt_n : d < n\n⊢ factorization (n / d) = factorization n - factorization d → d ∣ n",
"tactic": "have h1 : n / d ≠ 0 := fun H => Nat.lt_asymm hd_lt_n ((Nat.div_eq_zero_iff hd.bot_lt).mp H)"
},
{
"state_after": "case inr\nd n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\nhd_lt_n : d < n\nh1 : n / d ≠ 0\nh : factorization (n / d) = factorization n - factorization d\n⊢ d ∣ n",
"state_before": "case inr\nd n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\nhd_lt_n : d < n\nh1 : n / d ≠ 0\n⊢ factorization (n / d) = factorization n - factorization d → d ∣ n",
"tactic": "intro h"
},
{
"state_after": "case inr\nd n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\nhd_lt_n : d < n\nh1 : n / d ≠ 0\nh : factorization (n / d) = factorization n - factorization d\n⊢ n ≤ n / d * d",
"state_before": "case inr\nd n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\nhd_lt_n : d < n\nh1 : n / d ≠ 0\nh : factorization (n / d) = factorization n - factorization d\n⊢ d ∣ n",
"tactic": "rw [dvd_iff_le_div_mul n d]"
},
{
"state_after": "case inr\nd n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\nhd_lt_n : d < n\nh1 : n / d ≠ 0\nh : factorization (n / d) = factorization n - factorization d\nh2 : ¬n ≤ n / d * d\n⊢ False",
"state_before": "case inr\nd n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\nhd_lt_n : d < n\nh1 : n / d ≠ 0\nh : factorization (n / d) = factorization n - factorization d\n⊢ n ≤ n / d * d",
"tactic": "by_contra h2"
},
{
"state_after": "case inr.intro\nd n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\nhd_lt_n : d < n\nh1 : n / d ≠ 0\nh : factorization (n / d) = factorization n - factorization d\nh2 : ¬n ≤ n / d * d\np : ℕ\nhp : ↑(factorization (n / d * d)) p < ↑(factorization n) p\n⊢ False",
"state_before": "case inr\nd n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\nhd_lt_n : d < n\nh1 : n / d ≠ 0\nh : factorization (n / d) = factorization n - factorization d\nh2 : ¬n ≤ n / d * d\n⊢ False",
"tactic": "cases' exists_factorization_lt_of_lt (mul_ne_zero h1 hd) (not_le.mp h2) with p hp"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nd n : ℕ\nhd : d ≠ 0\nhdn : d ≤ n\nhd_lt_n : d < n\nh1 : n / d ≠ 0\nh : factorization (n / d) = factorization n - factorization d\nh2 : ¬n ≤ n / d * d\np : ℕ\nhp : ↑(factorization (n / d * d)) p < ↑(factorization n) p\n⊢ False",
"tactic": "rwa [factorization_mul h1 hd, add_apply, ← lt_tsub_iff_right, h, tsub_apply,\n lt_self_iff_false] at hp"
},
{
"state_after": "no goals",
"state_before": "case inl\nd : ℕ\nhd : d ≠ 0\nhdn : d ≤ d\n⊢ factorization (d / d) = factorization d - factorization d → d ∣ d",
"tactic": "simp"
}
] |
[
591,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
581,
1
] |
Std/Logic.lean
|
Exists.imp'
|
[] |
[
365,
29
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
363,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.map_apply
|
[] |
[
309,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
308,
1
] |
Std/Data/BinomialHeap.lean
|
Std.BinomialHeapImp.Heap.realSize_tail?
|
[
{
"state_after": "α : Type u_1\nle : α → α → Bool\ns' s : Heap α\n⊢ Option.map (fun x => x.snd) (deleteMin le s) = some s' → realSize s = realSize s' + 1",
"state_before": "α : Type u_1\nle : α → α → Bool\ns' s : Heap α\n⊢ tail? le s = some s' → realSize s = realSize s' + 1",
"tactic": "simp only [Heap.tail?]"
},
{
"state_after": "α : Type u_1\nle : α → α → Bool\ns' s : Heap α\neq : Option.map (fun x => x.snd) (deleteMin le s) = some s'\n⊢ realSize s = realSize s' + 1",
"state_before": "α : Type u_1\nle : α → α → Bool\ns' s : Heap α\n⊢ Option.map (fun x => x.snd) (deleteMin le s) = some s' → realSize s = realSize s' + 1",
"tactic": "intro eq"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nle : α → α → Bool\ns' s : Heap α\neq : Option.map (fun x => x.snd) (deleteMin le s) = some s'\n⊢ realSize s = realSize s' + 1",
"tactic": "match eq₂ : s.deleteMin le, eq with\n| some (a, tl), rfl => exact realSize_deleteMin eq₂"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nle : α → α → Bool\ns' s : Heap α\neq : Option.map (fun x => x.snd) (deleteMin le s) = some s'\na : α\ntl : Heap α\neq₂ : deleteMin le s = some (a, tl)\n⊢ realSize s = realSize ((fun x => x.snd) (a, tl)) + 1",
"tactic": "exact realSize_deleteMin eq₂"
}
] |
[
252,
54
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
248,
1
] |
Mathlib/Algebra/Order/Pi.lean
|
Function.const_le_one
|
[] |
[
151,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.union_add_inter
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ s ∪ t + s ∩ t ≤ s + t\n\ncase a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ s + t ≤ s ∪ t + s ∩ t",
"state_before": "α : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ s ∪ t + s ∩ t = s + t",
"tactic": "apply _root_.le_antisymm"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ s + s ∩ t ∪ (t + s ∩ t) ≤ s + t",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ s ∪ t + s ∩ t ≤ s + t",
"tactic": "rw [union_add_distrib]"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ t + s ∩ t ≤ s + t",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ s + s ∩ t ∪ (t + s ∩ t) ≤ s + t",
"tactic": "refine' union_le (add_le_add_left (inter_le_right _ _) _) _"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ s ∩ t + t ≤ s + t",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ t + s ∩ t ≤ s + t",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ s ∩ t + t ≤ s + t",
"tactic": "exact add_le_add_right (inter_le_left _ _) _"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ t + s ≤ (s ∪ t + s) ∩ (s ∪ t + t)",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ s + t ≤ s ∪ t + s ∩ t",
"tactic": "rw [add_comm, add_inter_distrib]"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ t + s ≤ s ∪ t + t",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ t + s ≤ (s ∪ t + s) ∩ (s ∪ t + t)",
"tactic": "refine' le_inter (add_le_add_right (le_union_right _ _) _) _"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ s + t ≤ s ∪ t + t",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ t + s ≤ s ∪ t + t",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.190902\nγ : Type ?u.190905\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\n⊢ s + t ≤ s ∪ t + t",
"tactic": "exact add_le_add_right (le_union_left _ _) _"
}
] |
[
1892,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1883,
1
] |
Mathlib/Data/Nat/Dist.lean
|
Nat.dist_pos_of_ne
|
[
{
"state_after": "i j : ℕ\nhne : i ≠ j\nh : i < j\n⊢ 0 < j - i",
"state_before": "i j : ℕ\nhne : i ≠ j\nh : i < j\n⊢ 0 < dist i j",
"tactic": "rw [dist_eq_sub_of_le (le_of_lt h)]"
},
{
"state_after": "no goals",
"state_before": "i j : ℕ\nhne : i ≠ j\nh : i < j\n⊢ 0 < j - i",
"tactic": "apply tsub_pos_of_lt h"
},
{
"state_after": "no goals",
"state_before": "i j : ℕ\nhne : i ≠ j\nh : i = j\n⊢ 0 < dist i j",
"tactic": "contradiction"
},
{
"state_after": "i j : ℕ\nhne : i ≠ j\nh : i > j\n⊢ 0 < i - j",
"state_before": "i j : ℕ\nhne : i ≠ j\nh : i > j\n⊢ 0 < dist i j",
"tactic": "rw [dist_eq_sub_of_le_right (le_of_lt h)]"
},
{
"state_after": "no goals",
"state_before": "i j : ℕ\nhne : i ≠ j\nh : i > j\n⊢ 0 < i - j",
"tactic": "apply tsub_pos_of_lt h"
}
] |
[
124,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Algebra/GeomSum.lean
|
geom_sum_Ico_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Ring α\nx : α\nm n : ℕ\nhmn : m ≤ n\n⊢ (∑ i in Ico m n, x ^ i) * (x - 1) = x ^ n - x ^ m",
"tactic": "rw [sum_Ico_eq_sub _ hmn, sub_mul, geom_sum_mul, geom_sum_mul, sub_sub_sub_cancel_right]"
}
] |
[
335,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
SubmonoidClass.coe_multiset_prod
|
[] |
[
59,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_atTop_diagonal
|
[
{
"state_after": "ι : Type ?u.282150\nι' : Type ?u.282153\nα : Type u_1\nβ : Type ?u.282159\nγ : Type ?u.282162\ninst✝ : SemilatticeSup α\n⊢ Tendsto (fun a => (a, a)) atTop (atTop ×ˢ atTop)",
"state_before": "ι : Type ?u.282150\nι' : Type ?u.282153\nα : Type u_1\nβ : Type ?u.282159\nγ : Type ?u.282162\ninst✝ : SemilatticeSup α\n⊢ Tendsto (fun a => (a, a)) atTop atTop",
"tactic": "rw [← prod_atTop_atTop_eq]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.282150\nι' : Type ?u.282153\nα : Type u_1\nβ : Type ?u.282159\nγ : Type ?u.282162\ninst✝ : SemilatticeSup α\n⊢ Tendsto (fun a => (a, a)) atTop (atTop ×ˢ atTop)",
"tactic": "exact tendsto_id.prod_mk tendsto_id"
}
] |
[
1437,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1435,
1
] |
Mathlib/Order/Bounded.lean
|
Set.bounded_gt_Ico
|
[] |
[
254,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Std/Data/Array/Lemmas.lean
|
Array.size_ofFn_go
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nn : Nat\nf : Fin n → α\ni : Nat\nacc : Array α\n⊢ size (ofFn.go f i acc) = size acc + (n - i)",
"tactic": "if hin : i < n then\n unfold ofFn.go\n have : 1 + (n - (i + 1)) = n - i :=\n Nat.sub_sub .. ▸ Nat.add_sub_cancel' (Nat.le_sub_of_add_le (Nat.add_comm .. ▸ hin))\n rw [dif_pos hin, size_ofFn_go f (i+1), size_push, Nat.add_assoc, this]\nelse\n have : n - i = 0 := Nat.sub_eq_zero_of_le (Nat.le_of_not_lt hin)\n unfold ofFn.go\n simp [hin, this]"
},
{
"state_after": "α : Type u_1\nn : Nat\nf : Fin n → α\ni : Nat\nacc : Array α\nhin : i < n\n⊢ size (if h : i < n then ofFn.go f (i + 1) (push acc (f { val := i, isLt := h })) else acc) = size acc + (n - i)",
"state_before": "α : Type u_1\nn : Nat\nf : Fin n → α\ni : Nat\nacc : Array α\nhin : i < n\n⊢ size (ofFn.go f i acc) = size acc + (n - i)",
"tactic": "unfold ofFn.go"
},
{
"state_after": "α : Type u_1\nn : Nat\nf : Fin n → α\ni : Nat\nacc : Array α\nhin : i < n\nthis : 1 + (n - (i + 1)) = n - i\n⊢ size (if h : i < n then ofFn.go f (i + 1) (push acc (f { val := i, isLt := h })) else acc) = size acc + (n - i)",
"state_before": "α : Type u_1\nn : Nat\nf : Fin n → α\ni : Nat\nacc : Array α\nhin : i < n\n⊢ size (if h : i < n then ofFn.go f (i + 1) (push acc (f { val := i, isLt := h })) else acc) = size acc + (n - i)",
"tactic": "have : 1 + (n - (i + 1)) = n - i :=\n Nat.sub_sub .. ▸ Nat.add_sub_cancel' (Nat.le_sub_of_add_le (Nat.add_comm .. ▸ hin))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nn : Nat\nf : Fin n → α\ni : Nat\nacc : Array α\nhin : i < n\nthis : 1 + (n - (i + 1)) = n - i\n⊢ size (if h : i < n then ofFn.go f (i + 1) (push acc (f { val := i, isLt := h })) else acc) = size acc + (n - i)",
"tactic": "rw [dif_pos hin, size_ofFn_go f (i+1), size_push, Nat.add_assoc, this]"
},
{
"state_after": "α : Type u_1\nn : Nat\nf : Fin n → α\ni : Nat\nacc : Array α\nhin : ¬i < n\nthis : n - i = 0\n⊢ size (ofFn.go f i acc) = size acc + (n - i)",
"state_before": "α : Type u_1\nn : Nat\nf : Fin n → α\ni : Nat\nacc : Array α\nhin : ¬i < n\n⊢ size (ofFn.go f i acc) = size acc + (n - i)",
"tactic": "have : n - i = 0 := Nat.sub_eq_zero_of_le (Nat.le_of_not_lt hin)"
},
{
"state_after": "α : Type u_1\nn : Nat\nf : Fin n → α\ni : Nat\nacc : Array α\nhin : ¬i < n\nthis : n - i = 0\n⊢ size (if h : i < n then ofFn.go f (i + 1) (push acc (f { val := i, isLt := h })) else acc) = size acc + (n - i)",
"state_before": "α : Type u_1\nn : Nat\nf : Fin n → α\ni : Nat\nacc : Array α\nhin : ¬i < n\nthis : n - i = 0\n⊢ size (ofFn.go f i acc) = size acc + (n - i)",
"tactic": "unfold ofFn.go"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nn : Nat\nf : Fin n → α\ni : Nat\nacc : Array α\nhin : ¬i < n\nthis : n - i = 0\n⊢ size (if h : i < n then ofFn.go f (i + 1) (push acc (f { val := i, isLt := h })) else acc) = size acc + (n - i)",
"tactic": "simp [hin, this]"
}
] |
[
278,
26
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
267,
9
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.one_apply
|
[] |
[
536,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/Data/Polynomial/Laurent.lean
|
LaurentPolynomial.commute_T
|
[
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\nf : R[T;T⁻¹]\nm : ℤ\na : R\n⊢ AddMonoidAlgebra.single (n + (0 + m)) (1 * (a * 1)) = AddMonoidAlgebra.single (0 + m + n) (a * 1 * 1)",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\nf : R[T;T⁻¹]\nm : ℤ\na : R\n⊢ T n * (↑C a * T m) = ↑C a * T m * T n",
"tactic": "rw [T, T, ← single_eq_C, single_mul_single, single_mul_single, single_mul_single]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nn : ℤ\nf : R[T;T⁻¹]\nm : ℤ\na : R\n⊢ AddMonoidAlgebra.single (n + (0 + m)) (1 * (a * 1)) = AddMonoidAlgebra.single (0 + m + n) (a * 1 * 1)",
"tactic": "simp [add_comm]"
}
] |
[
323,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.destruct_cons
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\ns : WSeq α\n⊢ Computation.destruct (destruct (cons a s)) = Sum.inl (some (a, s))",
"tactic": "simp [destruct, cons, Computation.rmap]"
}
] |
[
634,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
633,
1
] |
src/lean/Init/Data/Nat/SOM.lean
|
Nat.SOM.Mon.append_denote
|
[
{
"state_after": "no goals",
"state_before": "ctx : Context\nm₁ m₂ : Mon\n⊢ denote ctx (m₁ ++ m₂) = denote ctx m₁ * denote ctx m₂",
"tactic": "match m₁ with\n| [] => simp! [Nat.one_mul]\n| v :: m₁ => simp! [append_denote ctx m₁ m₂, Nat.mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "ctx : Context\nm₁ m₂ : Mon\n⊢ denote ctx ([] ++ m₂) = denote ctx [] * denote ctx m₂",
"tactic": "simp! [Nat.one_mul]"
},
{
"state_after": "no goals",
"state_before": "ctx : Context\nm₁✝ m₂ : Mon\nv : Var\nm₁ : List Var\n⊢ denote ctx (v :: m₁ ++ m₂) = denote ctx (v :: m₁) * denote ctx m₂",
"tactic": "simp! [append_denote ctx m₁ m₂, Nat.mul_assoc]"
}
] |
[
107,
62
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
104,
1
] |
Mathlib/Data/Polynomial/Div.lean
|
Polynomial.modByMonic_X_sub_C_eq_C_eval
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\n⊢ p %ₘ (X - ↑C a) = ↑C (eval a p)",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n⊢ p %ₘ (X - ↑C a) = ↑C (eval a p)",
"tactic": "nontriviality R"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\nh : eval a (p %ₘ (X - ↑C a)) = eval a p\n⊢ p %ₘ (X - ↑C a) = ↑C (eval a p)",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\n⊢ p %ₘ (X - ↑C a) = ↑C (eval a p)",
"tactic": "have h : (p %ₘ (X - C a)).eval a = p.eval a := by\n rw [modByMonic_eq_sub_mul_div _ (monic_X_sub_C a), eval_sub, eval_mul, eval_sub, eval_X,\n eval_C, sub_self, MulZeroClass.zero_mul, sub_zero]"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\nh : eval a (p %ₘ (X - ↑C a)) = eval a p\nthis : degree (p %ₘ (X - ↑C a)) < 1\n⊢ p %ₘ (X - ↑C a) = ↑C (eval a p)",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\nh : eval a (p %ₘ (X - ↑C a)) = eval a p\n⊢ p %ₘ (X - ↑C a) = ↑C (eval a p)",
"tactic": "have : degree (p %ₘ (X - C a)) < 1 :=\n degree_X_sub_C a ▸ degree_modByMonic_lt p (monic_X_sub_C a)"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\nh : coeff (p %ₘ (X - ↑C a)) 0 = eval a p\nthis✝ : degree (p %ₘ (X - ↑C a)) < 1\nthis : degree (p %ₘ (X - ↑C a)) ≤ 0\n⊢ p %ₘ (X - ↑C a) = ↑C (eval a p)",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\nh : eval a (p %ₘ (X - ↑C a)) = eval a p\nthis✝ : degree (p %ₘ (X - ↑C a)) < 1\nthis : degree (p %ₘ (X - ↑C a)) ≤ 0\n⊢ p %ₘ (X - ↑C a) = ↑C (eval a p)",
"tactic": "rw [eq_C_of_degree_le_zero this, eval_C] at h"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\nh : coeff (p %ₘ (X - ↑C a)) 0 = eval a p\nthis✝ : degree (p %ₘ (X - ↑C a)) < 1\nthis : degree (p %ₘ (X - ↑C a)) ≤ 0\n⊢ p %ₘ (X - ↑C a) = ↑C (eval a p)",
"tactic": "rw [eq_C_of_degree_le_zero this, h]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\n⊢ eval a (p %ₘ (X - ↑C a)) = eval a p",
"tactic": "rw [modByMonic_eq_sub_mul_div _ (monic_X_sub_C a), eval_sub, eval_mul, eval_sub, eval_X,\n eval_C, sub_self, MulZeroClass.zero_mul, sub_zero]"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\nh : eval a (p %ₘ (X - ↑C a)) = eval a p\n⊢ degree (p %ₘ (X - ↑C a)) < 1 → degree (p %ₘ (X - ↑C a)) ≤ 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\nh : eval a (p %ₘ (X - ↑C a)) = eval a p\nthis : degree (p %ₘ (X - ↑C a)) < 1\n⊢ degree (p %ₘ (X - ↑C a)) ≤ 0",
"tactic": "revert this"
},
{
"state_after": "case none\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\nh : eval a (p %ₘ (X - ↑C a)) = eval a p\n⊢ none < 1 → none ≤ 0\n\ncase some\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\nh : eval a (p %ₘ (X - ↑C a)) = eval a p\nval✝ : ℕ\n⊢ some val✝ < 1 → some val✝ ≤ 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\nh : eval a (p %ₘ (X - ↑C a)) = eval a p\n⊢ degree (p %ₘ (X - ↑C a)) < 1 → degree (p %ₘ (X - ↑C a)) ≤ 0",
"tactic": "cases degree (p %ₘ (X - C a))"
},
{
"state_after": "no goals",
"state_before": "case none\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\nh : eval a (p %ₘ (X - ↑C a)) = eval a p\n⊢ none < 1 → none ≤ 0",
"tactic": "exact fun _ => bot_le"
},
{
"state_after": "no goals",
"state_before": "case some\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\na : R\n✝ : Nontrivial R\nh : eval a (p %ₘ (X - ↑C a)) = eval a p\nval✝ : ℕ\n⊢ some val✝ < 1 → some val✝ ≤ 0",
"tactic": "exact fun h => WithBot.some_le_some.2 (Nat.le_of_lt_succ (WithBot.some_lt_some.1 h))"
}
] |
[
437,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
HasDerivAt.hasDerivWithinAt
|
[] |
[
381,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
380,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingHom.comp_id
|
[] |
[
324,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/Data/List/Perm.lean
|
List.perm_nil
|
[] |
[
169,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/FieldTheory/IntermediateField.lean
|
IntermediateField.copy_eq
|
[] |
[
142,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/MeasureTheory/Lattice.lean
|
AEMeasurable.const_inf
|
[] |
[
188,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Algebra/BigOperators/Option.lean
|
Finset.prod_insertNone
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nf : Option α → M\ns : Finset α\n⊢ ∏ x in ↑insertNone s, f x = f none * ∏ x in s, f (some x)",
"tactic": "simp [insertNone]"
}
] |
[
31,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
30,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
AlternatingMap.congr_arg
|
[] |
[
134,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/CategoryTheory/Functor/Basic.lean
|
CategoryTheory.Functor.map_dite
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nE : Type u₃\ninst✝¹ : Category E\nF : C ⥤ D\nX Y : C\nP : Prop\ninst✝ : Decidable P\nf : P → (X ⟶ Y)\ng : ¬P → (X ⟶ Y)\n⊢ F.map (if h : P then f h else g h) = if h : P then F.map (f h) else F.map (g h)",
"tactic": "aesop_cat"
}
] |
[
143,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.not_isRoot_C
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na✝ b : R\nm n : ℕ\ninst✝ : Semiring R\np q r✝ : R[X]\nx r a : R\nhr : r ≠ 0\n⊢ ¬IsRoot (↑C r) a",
"tactic": "simpa using hr"
}
] |
[
516,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/Topology/Hom/Open.lean
|
ContinuousOpenMap.coe_comp
|
[] |
[
133,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/CategoryTheory/MorphismProperty.lean
|
CategoryTheory.MorphismProperty.isomorphisms.infer_property
|
[] |
[
416,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
415,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.div_eq_one
|
[] |
[
292,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
11
] |
Mathlib/Analysis/NormedSpace/Dual.lean
|
NormedSpace.inclusionInDoubleDual_norm_le
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type ?u.27970\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\n⊢ ‖ContinuousLinearMap.id 𝕜 (Dual 𝕜 E)‖ ≤ 1",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type ?u.27970\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\n⊢ ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1",
"tactic": "rw [inclusionInDoubleDual_norm_eq]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type ?u.27970\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\n⊢ ‖ContinuousLinearMap.id 𝕜 (Dual 𝕜 E)‖ ≤ 1",
"tactic": "exact ContinuousLinearMap.norm_id_le"
}
] |
[
107,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Order/Directed.lean
|
ScottContinuous.monotone
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\n⊢ f a ≤ f b",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\n⊢ Monotone f",
"tactic": "intro a b hab"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\ne1 : IsLUB (f '' {a, b}) (f b)\n⊢ f a ∈ f '' {a, b}",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\ne1 : IsLUB (f '' {a, b}) (f b)\n⊢ f a ≤ f b",
"tactic": "apply e1.1"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\ne1 : IsLUB (f '' {a, b}) (f b)\n⊢ f a ∈ {f a, f b}",
"state_before": "case a\nα : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\ne1 : IsLUB (f '' {a, b}) (f b)\n⊢ f a ∈ f '' {a, b}",
"tactic": "rw [Set.image_pair]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\ne1 : IsLUB (f '' {a, b}) (f b)\n⊢ f a ∈ {f a, f b}",
"tactic": "exact Set.mem_insert _ _"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\n⊢ Set.Nonempty {a, b}\n\ncase a\nα : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\n⊢ DirectedOn (fun x x_1 => x ≤ x_1) {a, b}\n\ncase a\nα : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\n⊢ IsLUB {a, b} b",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\n⊢ IsLUB (f '' {a, b}) (f b)",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\n⊢ Set.Nonempty {a, b}",
"tactic": "exact Set.insert_nonempty _ _"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\n⊢ DirectedOn (fun x x_1 => x ≤ x_1) {a, b}",
"tactic": "exact directedOn_pair le_refl hab"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\n⊢ IsLeast (Set.Ici b) b",
"state_before": "case a\nα : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\n⊢ IsLUB {a, b} b",
"tactic": "rw [IsLUB, upperBounds_insert, upperBounds_singleton,\n Set.inter_eq_self_of_subset_right (Set.Ici_subset_Ici.mpr hab)]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nι : Sort w\nr r' s : α → α → Prop\ninst✝¹ : Preorder α\na✝ : α\ninst✝ : Preorder β\nf : α → β\nh : ScottContinuous f\na b : α\nhab : a ≤ b\n⊢ IsLeast (Set.Ici b) b",
"tactic": "exact isLeast_Ici"
}
] |
[
367,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
11
] |
Std/Data/List/Lemmas.lean
|
List.eraseP_cons_of_neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\na : α\nl : List α\np : α → Bool\nh : ¬p a = true\n⊢ eraseP p (a :: l) = a :: eraseP p l",
"tactic": "simp [eraseP_cons, h]"
}
] |
[
935,
68
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
934,
9
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.dvd_ord_proj_of_dvd
|
[] |
[
528,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
div_le_inv_mul_iff
|
[
{
"state_after": "α : Type u\ninst✝³ : Group α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\n⊢ a * a ≤ b * b ↔ a ≤ b",
"state_before": "α : Type u\ninst✝³ : Group α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\n⊢ a / b ≤ a⁻¹ * b ↔ a ≤ b",
"tactic": "rw [div_eq_mul_inv, mul_inv_le_inv_mul_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝³ : Group α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\n⊢ a * a ≤ b * b ↔ a ≤ b",
"tactic": "exact\n ⟨fun h => not_lt.mp fun k => not_lt.mpr h (mul_lt_mul_of_lt_of_lt k k), fun h =>\n mul_le_mul' h h⟩"
}
] |
[
1065,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1060,
1
] |
Mathlib/Data/List/Basic.lean
|
List.mem_enumFrom
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\n⊢ (i, x) ∈ enumFrom j [] → j ≤ i ∧ i < j + length [] ∧ x ∈ []",
"tactic": "simp [enumFrom]"
},
{
"state_after": "ι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\n⊢ i = j ∧ x = y ∨ (i, x) ∈ enumFrom (j + 1) ys → j ≤ i ∧ i < j + (length ys + 1) ∧ (x = y ∨ x ∈ ys)",
"state_before": "ι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\n⊢ (i, x) ∈ enumFrom j (y :: ys) → j ≤ i ∧ i < j + length (y :: ys) ∧ x ∈ y :: ys",
"tactic": "suffices\n i = j ∧ x = y ∨ (i, x) ∈ enumFrom (j + 1) ys →\n j ≤ i ∧ i < j + (length ys + 1) ∧ (x = y ∨ x ∈ ys)\n by simpa [enumFrom, mem_enumFrom ys]"
},
{
"state_after": "case inl\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : i = j ∧ x = y\n⊢ j ≤ i ∧ i < j + (length ys + 1) ∧ (x = y ∨ x ∈ ys)\n\ncase inr\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : (i, x) ∈ enumFrom (j + 1) ys\n⊢ j ≤ i ∧ i < j + (length ys + 1) ∧ (x = y ∨ x ∈ ys)",
"state_before": "ι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\n⊢ i = j ∧ x = y ∨ (i, x) ∈ enumFrom (j + 1) ys → j ≤ i ∧ i < j + (length ys + 1) ∧ (x = y ∨ x ∈ ys)",
"tactic": "rintro (h | h)"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nthis : i = j ∧ x = y ∨ (i, x) ∈ enumFrom (j + 1) ys → j ≤ i ∧ i < j + (length ys + 1) ∧ (x = y ∨ x ∈ ys)\n⊢ (i, x) ∈ enumFrom j (y :: ys) → j ≤ i ∧ i < j + length (y :: ys) ∧ x ∈ y :: ys",
"tactic": "simpa [enumFrom, mem_enumFrom ys]"
},
{
"state_after": "case inl\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : i = j ∧ x = y\n⊢ i < i + (length ys + 1)",
"state_before": "case inl\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : i = j ∧ x = y\n⊢ j ≤ i ∧ i < j + (length ys + 1) ∧ (x = y ∨ x ∈ ys)",
"tactic": "refine' ⟨le_of_eq h.1.symm, h.1 ▸ _, Or.inl h.2⟩"
},
{
"state_after": "case inl.h\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : i = j ∧ x = y\n⊢ 0 < length ys + 1",
"state_before": "case inl\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : i = j ∧ x = y\n⊢ i < i + (length ys + 1)",
"tactic": "apply Nat.lt_add_of_pos_right"
},
{
"state_after": "no goals",
"state_before": "case inl.h\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : i = j ∧ x = y\n⊢ 0 < length ys + 1",
"tactic": "simp"
},
{
"state_after": "case inr\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : (i, x) ∈ enumFrom (j + 1) ys\nhji : j + 1 ≤ i\nhijlen : i < j + 1 + length ys\nhmem : x ∈ ys\n⊢ j ≤ i ∧ i < j + (length ys + 1) ∧ (x = y ∨ x ∈ ys)",
"state_before": "case inr\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : (i, x) ∈ enumFrom (j + 1) ys\n⊢ j ≤ i ∧ i < j + (length ys + 1) ∧ (x = y ∨ x ∈ ys)",
"tactic": "have ⟨hji, hijlen, hmem⟩ := mem_enumFrom _ h"
},
{
"state_after": "case inr.refine'_1\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : (i, x) ∈ enumFrom (j + 1) ys\nhji : j + 1 ≤ i\nhijlen : i < j + 1 + length ys\nhmem : x ∈ ys\n⊢ j ≤ i\n\ncase inr.refine'_2\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : (i, x) ∈ enumFrom (j + 1) ys\nhji : j + 1 ≤ i\nhijlen : i < j + 1 + length ys\nhmem : x ∈ ys\n⊢ i < j + (length ys + 1)\n\ncase inr.refine'_3\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : (i, x) ∈ enumFrom (j + 1) ys\nhji : j + 1 ≤ i\nhijlen : i < j + 1 + length ys\nhmem : x ∈ ys\n⊢ x = y ∨ x ∈ ys",
"state_before": "case inr\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : (i, x) ∈ enumFrom (j + 1) ys\nhji : j + 1 ≤ i\nhijlen : i < j + 1 + length ys\nhmem : x ∈ ys\n⊢ j ≤ i ∧ i < j + (length ys + 1) ∧ (x = y ∨ x ∈ ys)",
"tactic": "refine' ⟨_, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_1\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : (i, x) ∈ enumFrom (j + 1) ys\nhji : j + 1 ≤ i\nhijlen : i < j + 1 + length ys\nhmem : x ∈ ys\n⊢ j ≤ i",
"tactic": "exact le_trans (Nat.le_succ _) hji"
},
{
"state_after": "case h.e'_4\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : (i, x) ∈ enumFrom (j + 1) ys\nhji : j + 1 ≤ i\nhijlen : i < j + 1 + length ys\nhmem : x ∈ ys\n⊢ j + (length ys + 1) = j + 1 + length ys",
"state_before": "case inr.refine'_2\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : (i, x) ∈ enumFrom (j + 1) ys\nhji : j + 1 ≤ i\nhijlen : i < j + 1 + length ys\nhmem : x ∈ ys\n⊢ i < j + (length ys + 1)",
"tactic": "convert hijlen using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : (i, x) ∈ enumFrom (j + 1) ys\nhji : j + 1 ≤ i\nhijlen : i < j + 1 + length ys\nhmem : x ∈ ys\n⊢ j + (length ys + 1) = j + 1 + length ys",
"tactic": "ac_rfl"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_3\nι : Type ?u.432423\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx : α\ni j : ℕ\ny : α\nys : List α\nh : (i, x) ∈ enumFrom (j + 1) ys\nhji : j + 1 ≤ i\nhijlen : i < j + 1 + length ys\nhmem : x ∈ ys\n⊢ x = y ∨ x ∈ ys",
"tactic": "simp [hmem]"
}
] |
[
3862,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3846,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_neg_I
|
[
{
"state_after": "no goals",
"state_before": "⊢ arg (-I) = -(π / 2)",
"tactic": "simp [arg, le_refl]"
}
] |
[
210,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Data/Polynomial/Degree/CardPowDegree.lean
|
Polynomial.cardPowDegree_isEuclidean
|
[
{
"state_after": "Fq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\n⊢ ↑cardPowDegree p < ↑cardPowDegree q ↔ degree p < degree q",
"state_before": "Fq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\n⊢ ↑cardPowDegree p < ↑cardPowDegree q ↔ EuclideanDomain.r p q",
"tactic": "show cardPowDegree p < cardPowDegree q ↔ degree p < degree q"
},
{
"state_after": "Fq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\n⊢ ((if p = 0 then 0 else ↑(Fintype.card Fq) ^ natDegree p) < if q = 0 then 0 else ↑(Fintype.card Fq) ^ natDegree q) ↔\n degree p < degree q",
"state_before": "Fq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\n⊢ ↑cardPowDegree p < ↑cardPowDegree q ↔ degree p < degree q",
"tactic": "simp only [cardPowDegree_apply]"
},
{
"state_after": "case inl.inl\nFq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : p = 0\nhq : q = 0\n⊢ 0 < 0 ↔ degree p < degree q\n\ncase inl.inr\nFq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : p = 0\nhq : ¬q = 0\n⊢ 0 < ↑(Fintype.card Fq) ^ natDegree q ↔ degree p < degree q\n\ncase inr.inl\nFq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : ¬p = 0\nhq : q = 0\n⊢ ↑(Fintype.card Fq) ^ natDegree p < 0 ↔ degree p < degree q\n\ncase inr.inr\nFq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : ¬p = 0\nhq : ¬q = 0\n⊢ ↑(Fintype.card Fq) ^ natDegree p < ↑(Fintype.card Fq) ^ natDegree q ↔ degree p < degree q",
"state_before": "Fq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\n⊢ ((if p = 0 then 0 else ↑(Fintype.card Fq) ^ natDegree p) < if q = 0 then 0 else ↑(Fintype.card Fq) ^ natDegree q) ↔\n degree p < degree q",
"tactic": "split_ifs with hp hq hq"
},
{
"state_after": "no goals",
"state_before": "case inl.inl\nFq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : p = 0\nhq : q = 0\n⊢ 0 < 0 ↔ degree p < degree q",
"tactic": "simp only [hp, hq, lt_self_iff_false]"
},
{
"state_after": "no goals",
"state_before": "case inl.inr\nFq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : p = 0\nhq : ¬q = 0\n⊢ 0 < ↑(Fintype.card Fq) ^ natDegree q ↔ degree p < degree q",
"tactic": "simp only [hp, hq, degree_zero, Ne.def, bot_lt_iff_ne_bot, degree_eq_bot, pow_pos,\n not_false_iff]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nFq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : ¬p = 0\nhq : q = 0\n⊢ ↑(Fintype.card Fq) ^ natDegree p < 0 ↔ degree p < degree q",
"tactic": "simp only [hp, hq, degree_zero, not_lt_bot, (pow_pos _).not_lt]"
},
{
"state_after": "case inr.inr\nFq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : ¬p = 0\nhq : ¬q = 0\n⊢ 1 < ↑(Fintype.card Fq)",
"state_before": "case inr.inr\nFq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : ¬p = 0\nhq : ¬q = 0\n⊢ ↑(Fintype.card Fq) ^ natDegree p < ↑(Fintype.card Fq) ^ natDegree q ↔ degree p < degree q",
"tactic": "rw [degree_eq_natDegree hp, degree_eq_natDegree hq, Nat.cast_withBot, Nat.cast_withBot,\n WithBot.coe_lt_coe, pow_lt_pow_iff]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nFq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : ¬p = 0\nhq : ¬q = 0\n⊢ 1 < ↑(Fintype.card Fq)",
"tactic": "exact_mod_cast @Fintype.one_lt_card Fq _ _"
}
] |
[
109,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Init/Logic.lean
|
if_t_t
|
[] |
[
354,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
VitaliFamily.ae_tendsto_average_norm_sub
|
[
{
"state_after": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\n⊢ Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)",
"state_before": "α : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)",
"tactic": "filter_upwards [v.ae_tendsto_lintegral_nnnorm_sub_div hf, v.ae_eventually_measure_pos] with x hx\n h'x"
},
{
"state_after": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis :\n Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (ENNReal.toReal 0))\n⊢ Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)",
"state_before": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\n⊢ Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)",
"tactic": "have := (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp hx"
},
{
"state_after": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\n⊢ Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)",
"state_before": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis :\n Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (ENNReal.toReal 0))\n⊢ Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)",
"tactic": "simp only [ENNReal.zero_toReal] at this"
},
{
"state_after": "α : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\n⊢ (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) =ᶠ[filterAt v x] fun a =>\n ⨍ (y : α) in a, ‖f y - f x‖ ∂μ",
"state_before": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\n⊢ Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0)",
"tactic": "apply Tendsto.congr' _ this"
},
{
"state_after": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\n⊢ (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) a = ⨍ (y : α) in a, ‖f y - f x‖ ∂μ",
"state_before": "α : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\n⊢ (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) =ᶠ[filterAt v x] fun a =>\n ⨍ (y : α) in a, ‖f y - f x‖ ∂μ",
"tactic": "filter_upwards [h'x, v.eventually_measure_lt_top x] with a _ h'a"
},
{
"state_after": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\n⊢ (ENNReal.toReal (↑↑μ a))⁻¹ * ENNReal.toReal (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) =\n (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (x_1 : α) in a, ‖f x_1 - f x‖ ∂μ",
"state_before": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\n⊢ (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) a = ⨍ (y : α) in a, ‖f y - f x‖ ∂μ",
"tactic": "simp only [Function.comp_apply, ENNReal.toReal_div, set_average_eq, div_eq_inv_mul]"
},
{
"state_after": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\nA : IntegrableOn (fun y => ↑‖f y - f x‖₊) a\n⊢ (ENNReal.toReal (↑↑μ a))⁻¹ * ENNReal.toReal (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) =\n (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (x_1 : α) in a, ‖f x_1 - f x‖ ∂μ",
"state_before": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\n⊢ (ENNReal.toReal (↑↑μ a))⁻¹ * ENNReal.toReal (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) =\n (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (x_1 : α) in a, ‖f x_1 - f x‖ ∂μ",
"tactic": "have A : IntegrableOn (fun y => (‖f y - f x‖₊ : ℝ)) a μ := by\n simp_rw [coe_nnnorm]\n exact (hf.integrableOn.sub (integrableOn_const.2 (Or.inr h'a))).norm"
},
{
"state_after": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\nA : IntegrableOn (fun y => ↑‖f y - f x‖₊) a\n⊢ ((ENNReal.toReal (↑↑μ a))⁻¹ * ∫ (a : α) in a, ↑‖f a - f x‖₊ ∂μ) =\n (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (x_1 : α) in a, ‖f x_1 - f x‖ ∂μ\n\ncase h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\nA : IntegrableOn (fun y => ↑‖f y - f x‖₊) a\n⊢ 0 ≤ ∫ (a : α) in a, ↑‖f a - f x‖₊ ∂μ",
"state_before": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\nA : IntegrableOn (fun y => ↑‖f y - f x‖₊) a\n⊢ (ENNReal.toReal (↑↑μ a))⁻¹ * ENNReal.toReal (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) =\n (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (x_1 : α) in a, ‖f x_1 - f x‖ ∂μ",
"tactic": "rw [lintegral_coe_eq_integral _ A, ENNReal.toReal_ofReal]"
},
{
"state_after": "α : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\n⊢ IntegrableOn (fun y => ‖f y - f x‖) a",
"state_before": "α : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\n⊢ IntegrableOn (fun y => ↑‖f y - f x‖₊) a",
"tactic": "simp_rw [coe_nnnorm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\n⊢ IntegrableOn (fun y => ‖f y - f x‖) a",
"tactic": "exact (hf.integrableOn.sub (integrableOn_const.2 (Or.inr h'a))).norm"
},
{
"state_after": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\nA : IntegrableOn (fun y => ↑‖f y - f x‖₊) a\n⊢ ((ENNReal.toReal (↑↑μ a))⁻¹ * ∫ (a : α) in a, ‖f a - f x‖ ∂μ) =\n (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (x_1 : α) in a, ‖f x_1 - f x‖ ∂μ",
"state_before": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\nA : IntegrableOn (fun y => ↑‖f y - f x‖₊) a\n⊢ ((ENNReal.toReal (↑↑μ a))⁻¹ * ∫ (a : α) in a, ↑‖f a - f x‖₊ ∂μ) =\n (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (x_1 : α) in a, ‖f x_1 - f x‖ ∂μ",
"tactic": "simp_rw [coe_nnnorm]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\nA : IntegrableOn (fun y => ↑‖f y - f x‖₊) a\n⊢ ((ENNReal.toReal (↑↑μ a))⁻¹ * ∫ (a : α) in a, ‖f a - f x‖ ∂μ) =\n (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (x_1 : α) in a, ‖f x_1 - f x‖ ∂μ",
"tactic": "rfl"
},
{
"state_after": "case h.hf\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\nA : IntegrableOn (fun y => ↑‖f y - f x‖₊) a\n⊢ 0 ≤ fun a => ↑‖f a - f x‖₊",
"state_before": "case h\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\nA : IntegrableOn (fun y => ↑‖f y - f x‖₊) a\n⊢ 0 ≤ ∫ (a : α) in a, ↑‖f a - f x‖₊ ∂μ",
"tactic": "apply integral_nonneg"
},
{
"state_after": "case h.hf\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx✝ : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x✝‖₊ ∂μ) / ↑↑μ a) (filterAt v x✝) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x✝, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x✝‖₊ ∂μ) / ↑↑μ a) (filterAt v x✝) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\nA : IntegrableOn (fun y => ↑‖f y - f x✝‖₊) a\nx : α\n⊢ OfNat.ofNat 0 x ≤ (fun a => ↑‖f a - f x✝‖₊) x",
"state_before": "case h.hf\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\nA : IntegrableOn (fun y => ↑‖f y - f x‖₊) a\n⊢ 0 ≤ fun a => ↑‖f a - f x‖₊",
"tactic": "intro x"
},
{
"state_after": "no goals",
"state_before": "case h.hf\nα : Type u_2\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nf : α → E\nhf : Integrable f\nx✝ : α\nhx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x✝‖₊ ∂μ) / ↑↑μ a) (filterAt v x✝) (𝓝 0)\nh'x : ∀ᶠ (a : Set α) in filterAt v x✝, 0 < ↑↑μ a\nthis : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x✝‖₊ ∂μ) / ↑↑μ a) (filterAt v x✝) (𝓝 0)\na : Set α\na✝ : 0 < ↑↑μ a\nh'a : ↑↑μ a < ⊤\nA : IntegrableOn (fun y => ↑‖f y - f x✝‖₊) a\nx : α\n⊢ OfNat.ofNat 0 x ≤ (fun a => ↑‖f a - f x✝‖₊) x",
"tactic": "exact NNReal.coe_nonneg _"
}
] |
[
914,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
897,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.add_sub_assoc
|
[
{
"state_after": "case intro\nm k : Nat\nh : k ≤ m\nn w✝ : Nat\nh✝ : k + w✝ = m\n⊢ n + m - k = n + (m - k)",
"state_before": "m k : Nat\nh : k ≤ m\nn : Nat\n⊢ n + m - k = n + (m - k)",
"tactic": "cases Nat.le.dest h"
},
{
"state_after": "case intro\nm k : Nat\nh : k ≤ m\nn l : Nat\nhl : k + l = m\n⊢ n + m - k = n + (m - k)",
"state_before": "case intro\nm k : Nat\nh : k ≤ m\nn w✝ : Nat\nh✝ : k + w✝ = m\n⊢ n + m - k = n + (m - k)",
"tactic": "rename_i l hl"
},
{
"state_after": "no goals",
"state_before": "case intro\nm k : Nat\nh : k ≤ m\nn l : Nat\nhl : k + l = m\n⊢ n + m - k = n + (m - k)",
"tactic": "rw [← hl, Nat.add_sub_cancel_left, Nat.add_comm k, ← Nat.add_assoc, Nat.add_sub_cancel]"
}
] |
[
605,
89
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
602,
11
] |
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