file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.add_one_le_succ
|
[
{
"state_after": "α β : Type u\nc : Cardinal\nthis : Set.Nonempty {c' | c < c'}\n⊢ c + 1 ≤ succ c",
"state_before": "α β : Type u\nc : Cardinal\n⊢ c + 1 ≤ succ c",
"tactic": "have : Set.Nonempty { c' | c < c' } := exists_gt c"
},
{
"state_after": "α β : Type u\nc : Cardinal\nthis : Set.Nonempty {c' | c < c'}\n⊢ ∀ (b : Cardinal), c < b → c + 1 ≤ b",
"state_before": "α β : Type u\nc : Cardinal\nthis : Set.Nonempty {c' | c < c'}\n⊢ c + 1 ≤ succ c",
"tactic": "simp_rw [succ_def, le_csInf_iff'' this, mem_setOf]"
},
{
"state_after": "α β : Type u\nc : Cardinal\nthis : Set.Nonempty {c' | c < c'}\nb : Cardinal\nhlt : c < b\n⊢ c + 1 ≤ b",
"state_before": "α β : Type u\nc : Cardinal\nthis : Set.Nonempty {c' | c < c'}\n⊢ ∀ (b : Cardinal), c < b → c + 1 ≤ b",
"tactic": "intro b hlt"
},
{
"state_after": "case mk.mk\nα β✝ : Type u\nc b : Cardinal\nβ γ : Type u\nthis : Set.Nonempty {c' | Quot.mk Setoid.r γ < c'}\nhlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β\n⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β",
"state_before": "α β : Type u\nc : Cardinal\nthis : Set.Nonempty {c' | c < c'}\nb : Cardinal\nhlt : c < b\n⊢ c + 1 ≤ b",
"tactic": "rcases b, c with ⟨⟨β⟩, ⟨γ⟩⟩"
},
{
"state_after": "case mk.mk.intro\nα β✝ : Type u\nc b : Cardinal\nβ γ : Type u\nthis : Set.Nonempty {c' | Quot.mk Setoid.r γ < c'}\nhlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β\nf : γ ↪ β\n⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β",
"state_before": "case mk.mk\nα β✝ : Type u\nc b : Cardinal\nβ γ : Type u\nthis : Set.Nonempty {c' | Quot.mk Setoid.r γ < c'}\nhlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β\n⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β",
"tactic": "cases' le_of_lt hlt with f"
},
{
"state_after": "case mk.mk.intro\nα β✝ : Type u\nc b : Cardinal\nβ γ : Type u\nthis✝ : Set.Nonempty {c' | Quot.mk Setoid.r γ < c'}\nhlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β\nf : γ ↪ β\nthis : ¬Surjective ↑f\n⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β",
"state_before": "case mk.mk.intro\nα β✝ : Type u\nc b : Cardinal\nβ γ : Type u\nthis : Set.Nonempty {c' | Quot.mk Setoid.r γ < c'}\nhlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β\nf : γ ↪ β\n⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β",
"tactic": "have : ¬Surjective f := fun hn => (not_le_of_lt hlt) (mk_le_of_surjective hn)"
},
{
"state_after": "case mk.mk.intro\nα β✝ : Type u\nc b : Cardinal\nβ γ : Type u\nthis✝ : Set.Nonempty {c' | Quot.mk Setoid.r γ < c'}\nhlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β\nf : γ ↪ β\nthis : ∃ x, ¬∃ a, ↑f a = x\n⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β",
"state_before": "case mk.mk.intro\nα β✝ : Type u\nc b : Cardinal\nβ γ : Type u\nthis✝ : Set.Nonempty {c' | Quot.mk Setoid.r γ < c'}\nhlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β\nf : γ ↪ β\nthis : ¬Surjective ↑f\n⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β",
"tactic": "simp only [Surjective, not_forall] at this"
},
{
"state_after": "case mk.mk.intro.intro\nα β✝ : Type u\nc b✝ : Cardinal\nβ γ : Type u\nthis : Set.Nonempty {c' | Quot.mk Setoid.r γ < c'}\nhlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β\nf : γ ↪ β\nb : β\nhb : ¬∃ a, ↑f a = b\n⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β",
"state_before": "case mk.mk.intro\nα β✝ : Type u\nc b : Cardinal\nβ γ : Type u\nthis✝ : Set.Nonempty {c' | Quot.mk Setoid.r γ < c'}\nhlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β\nf : γ ↪ β\nthis : ∃ x, ¬∃ a, ↑f a = x\n⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β",
"tactic": "rcases this with ⟨b, hb⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro\nα β✝ : Type u\nc b✝ : Cardinal\nβ γ : Type u\nthis : Set.Nonempty {c' | Quot.mk Setoid.r γ < c'}\nhlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β\nf : γ ↪ β\nb : β\nhb : ¬∃ a, ↑f a = b\n⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β",
"tactic": "calc\n (#γ) + 1 = (#Option γ) := mk_option.symm\n _ ≤ (#β) := (f.optionElim b hb).cardinal_le"
}
] |
[
828,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
816,
1
] |
Mathlib/Order/Atoms.lean
|
IsAtom.lt_iff
|
[] |
[
89,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/MeasureTheory/Integral/MeanInequalities.lean
|
ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr
|
[
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "have hp0_ne : p ≠ 0 := (ne_of_lt hp0_lt).symm"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "have hp0 : 0 ≤ p := le_of_lt hp0_lt"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "have hq0_lt : 0 < q := lt_of_le_of_lt hp0 hpq"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "have hq0_ne : q ≠ 0 := (ne_of_lt hq0_lt).symm"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "have h_one_div_r : 1 / r = 1 / p - 1 / q := by rw [hpqr]; simp"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "have _ : r ≠ 0 := by\n have hr_inv_pos : 0 < 1 / r := by rwa [h_one_div_r, sub_pos, one_div_lt_one_div hq0_lt hp0_lt]\n rw [one_div, _root_.inv_pos] at hr_inv_pos\n exact (ne_of_lt hr_inv_pos).symm"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "let p2 := q / p"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "let q2 := p2.conjugateExponent"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "have hp2q2 : p2.IsConjugateExponent q2 :=\n Real.isConjugateExponent_conjugateExponent (by simp [_root_.lt_div_iff, hpq, hp0_lt])"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "calc\n (∫⁻ a : α, (f * g) a ^ p ∂μ) ^ (1 / p) = (∫⁻ a : α, f a ^ p * g a ^ p ∂μ) ^ (1 / p) := by\n simp_rw [Pi.mul_apply, ENNReal.mul_rpow_of_nonneg _ _ hp0]\n _ ≤ ((∫⁻ a, f a ^ (p * p2) ∂μ) ^ (1 / p2) *\n (∫⁻ a, g a ^ (p * q2) ∂μ) ^ (1 / q2)) ^ (1 / p) := by\n refine' ENNReal.rpow_le_rpow _ (by simp [hp0])\n simp_rw [ENNReal.rpow_mul]\n exact ENNReal.lintegral_mul_le_Lp_mul_Lq μ hp2q2 (hf.pow_const _) (hg.pow_const _)\n _ = (∫⁻ a : α, f a ^ q ∂μ) ^ (1 / q) * (∫⁻ a : α, g a ^ r ∂μ) ^ (1 / r) := by\n rw [@ENNReal.mul_rpow_of_nonneg _ _ (1 / p) (by simp [hp0]), ← ENNReal.rpow_mul, ←\n ENNReal.rpow_mul]\n have hpp2 : p * p2 = q := by\n symm\n rw [mul_comm, ← div_eq_iff hp0_ne]\n have hpq2 : p * q2 = r := by\n rw [← inv_inv r, ← one_div, ← one_div, h_one_div_r]\n field_simp [Real.conjugateExponent, hp0_ne, hq0_ne]\n simp_rw [div_mul_div_comm, mul_one, mul_comm p2, mul_comm q2, hpp2, hpq2]"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\n⊢ 1 / r = 1 / q + 1 / r - 1 / q",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\n⊢ 1 / r = 1 / p - 1 / q",
"tactic": "rw [hpqr]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\n⊢ 1 / r = 1 / q + 1 / r - 1 / q",
"tactic": "simp"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nhr_inv_pos : 0 < 1 / r\n⊢ r ≠ 0",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\n⊢ r ≠ 0",
"tactic": "have hr_inv_pos : 0 < 1 / r := by rwa [h_one_div_r, sub_pos, one_div_lt_one_div hq0_lt hp0_lt]"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nhr_inv_pos : 0 < r\n⊢ r ≠ 0",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nhr_inv_pos : 0 < 1 / r\n⊢ r ≠ 0",
"tactic": "rw [one_div, _root_.inv_pos] at hr_inv_pos"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nhr_inv_pos : 0 < r\n⊢ r ≠ 0",
"tactic": "exact (ne_of_lt hr_inv_pos).symm"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\n⊢ 0 < 1 / r",
"tactic": "rwa [h_one_div_r, sub_pos, one_div_lt_one_div hq0_lt hp0_lt]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\n⊢ 1 < p2",
"tactic": "simp [_root_.lt_div_iff, hpq, hp0_lt]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) = (∫⁻ (a : α), f a ^ p * g a ^ p ∂μ) ^ (1 / p)",
"tactic": "simp_rw [Pi.mul_apply, ENNReal.mul_rpow_of_nonneg _ _ hp0]"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ (∫⁻ (a : α), f a ^ p * g a ^ p ∂μ) ≤\n (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ (∫⁻ (a : α), f a ^ p * g a ^ p ∂μ) ^ (1 / p) ≤\n ((∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2)) ^ (1 / p)",
"tactic": "refine' ENNReal.rpow_le_rpow _ (by simp [hp0])"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ (∫⁻ (a : α), f a ^ p * g a ^ p ∂μ) ≤\n (∫⁻ (a : α), (f a ^ p) ^ (q / p) ∂μ) ^ (1 / (q / p)) *\n (∫⁻ (a : α), (g a ^ p) ^ Real.conjugateExponent (q / p) ∂μ) ^ (1 / Real.conjugateExponent (q / p))",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ (∫⁻ (a : α), f a ^ p * g a ^ p ∂μ) ≤\n (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2)",
"tactic": "simp_rw [ENNReal.rpow_mul]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ (∫⁻ (a : α), f a ^ p * g a ^ p ∂μ) ≤\n (∫⁻ (a : α), (f a ^ p) ^ (q / p) ∂μ) ^ (1 / (q / p)) *\n (∫⁻ (a : α), (g a ^ p) ^ Real.conjugateExponent (q / p) ∂μ) ^ (1 / Real.conjugateExponent (q / p))",
"tactic": "exact ENNReal.lintegral_mul_le_Lp_mul_Lq μ hp2q2 (hf.pow_const _) (hg.pow_const _)"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ 0 ≤ 1 / p",
"tactic": "simp [hp0]"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2 * (1 / p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2 * (1 / p)) =\n (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ ((∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2)) ^ (1 / p) =\n (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "rw [@ENNReal.mul_rpow_of_nonneg _ _ (1 / p) (by simp [hp0]), ← ENNReal.rpow_mul, ←\n ENNReal.rpow_mul]"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\nhpp2 : p * p2 = q\n⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2 * (1 / p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2 * (1 / p)) =\n (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2 * (1 / p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2 * (1 / p)) =\n (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "have hpp2 : p * p2 = q := by\n symm\n rw [mul_comm, ← div_eq_iff hp0_ne]"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\nhpp2 : p * p2 = q\nhpq2 : p * q2 = r\n⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2 * (1 / p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2 * (1 / p)) =\n (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\nhpp2 : p * p2 = q\n⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2 * (1 / p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2 * (1 / p)) =\n (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "have hpq2 : p * q2 = r := by\n rw [← inv_inv r, ← one_div, ← one_div, h_one_div_r]\n field_simp [Real.conjugateExponent, hp0_ne, hq0_ne]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\nhpp2 : p * p2 = q\nhpq2 : p * q2 = r\n⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2 * (1 / p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2 * (1 / p)) =\n (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)",
"tactic": "simp_rw [div_mul_div_comm, mul_one, mul_comm p2, mul_comm q2, hpp2, hpq2]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ 0 ≤ 1 / p",
"tactic": "simp [hp0]"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ q = p * p2",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ p * p2 = q",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n⊢ q = p * p2",
"tactic": "rw [mul_comm, ← div_eq_iff hp0_ne]"
},
{
"state_after": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\nhpp2 : p * p2 = q\n⊢ p * q2 = 1 / (1 / p - 1 / q)",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\nhpp2 : p * p2 = q\n⊢ p * q2 = r",
"tactic": "rw [← inv_inv r, ← one_div, ← one_div, h_one_div_r]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1456644\ninst✝¹ : MeasurableSpace α✝\nμ✝ : MeasureTheory.Measure α✝\nα : Type u_1\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : MeasureTheory.Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx✝ : r ≠ 0\np2 : ℝ := q / p\nq2 : ℝ := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\nhpp2 : p * p2 = q\n⊢ p * q2 = 1 / (1 / p - 1 / q)",
"tactic": "field_simp [Real.conjugateExponent, hp0_ne, hq0_ne]"
}
] |
[
254,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.pow_of_coprime
|
[
{
"state_after": "case pos\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh : IsPrimitiveRoot ζ k\ni : ℕ\nhi : Nat.coprime i k\nh0 : k = 0\n⊢ IsPrimitiveRoot (ζ ^ i) k\n\ncase neg\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh : IsPrimitiveRoot ζ k\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"state_before": "M : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh : IsPrimitiveRoot ζ k\ni : ℕ\nhi : Nat.coprime i k\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"tactic": "by_cases h0 : k = 0"
},
{
"state_after": "case neg.intro\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot (↑ζ) k\n⊢ IsPrimitiveRoot (↑ζ ^ i) k",
"state_before": "case neg\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh : IsPrimitiveRoot ζ k\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"tactic": "rcases h.isUnit (Nat.pos_of_ne_zero h0) with ⟨ζ, rfl⟩"
},
{
"state_after": "case neg.intro\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot (↑ζ) k\n⊢ IsPrimitiveRoot (↑(ζ ^ i)) k",
"state_before": "case neg.intro\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot (↑ζ) k\n⊢ IsPrimitiveRoot (↑ζ ^ i) k",
"tactic": "rw [← Units.val_pow_eq_pow_val]"
},
{
"state_after": "case neg.intro\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot ζ k\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"state_before": "case neg.intro\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot (↑ζ) k\n⊢ IsPrimitiveRoot (↑(ζ ^ i)) k",
"tactic": "rw [coe_units_iff] at h ⊢"
},
{
"state_after": "case neg.intro\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot ζ k\n⊢ ∀ (l : ℕ), (ζ ^ i) ^ l = 1 → k ∣ l",
"state_before": "case neg.intro\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot ζ k\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"tactic": "refine'\n { pow_eq_one := by rw [← pow_mul', pow_mul, h.pow_eq_one, one_pow]\n dvd_of_pow_eq_one := _ }"
},
{
"state_after": "case neg.intro\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l✝ : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot ζ k\nl : ℕ\nhl : (ζ ^ i) ^ l = 1\n⊢ k ∣ l",
"state_before": "case neg.intro\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot ζ k\n⊢ ∀ (l : ℕ), (ζ ^ i) ^ l = 1 → k ∣ l",
"tactic": "intro l hl"
},
{
"state_after": "case neg.intro.a\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l✝ : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot ζ k\nl : ℕ\nhl : (ζ ^ i) ^ l = 1\n⊢ ζ ^ l = 1",
"state_before": "case neg.intro\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l✝ : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot ζ k\nl : ℕ\nhl : (ζ ^ i) ^ l = 1\n⊢ k ∣ l",
"tactic": "apply h.dvd_of_pow_eq_one"
},
{
"state_after": "case neg.intro.a\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l✝ : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot ζ k\nl : ℕ\nhl : (ζ ^ i) ^ l = 1\n⊢ ζ ^ (↑i * ↑l * Nat.gcdA i k) * (ζ ^ (↑k * Nat.gcdB i k)) ^ l = 1",
"state_before": "case neg.intro.a\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l✝ : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot ζ k\nl : ℕ\nhl : (ζ ^ i) ^ l = 1\n⊢ ζ ^ l = 1",
"tactic": "rw [← pow_one ζ, ← zpow_ofNat ζ, ← hi.gcd_eq_one, Nat.gcd_eq_gcd_ab, zpow_add, mul_pow,\n ← zpow_ofNat, ← zpow_mul, mul_right_comm]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.a\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l✝ : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot ζ k\nl : ℕ\nhl : (ζ ^ i) ^ l = 1\n⊢ ζ ^ (↑i * ↑l * Nat.gcdA i k) * (ζ ^ (↑k * Nat.gcdB i k)) ^ l = 1",
"tactic": "simp only [zpow_mul, hl, h.pow_eq_one, one_zpow, one_pow, one_mul, zpow_ofNat]"
},
{
"state_after": "case pos\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nl : ℕ\nζ : M\nf : F\ni : ℕ\nh : IsPrimitiveRoot ζ 0\nhi : Nat.coprime i 0\n⊢ IsPrimitiveRoot (ζ ^ i) 0",
"state_before": "case pos\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh : IsPrimitiveRoot ζ k\ni : ℕ\nhi : Nat.coprime i k\nh0 : k = 0\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"tactic": "subst k"
},
{
"state_after": "no goals",
"state_before": "case pos\nM : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nl : ℕ\nζ : M\nf : F\ni : ℕ\nh : IsPrimitiveRoot ζ 0\nhi : Nat.coprime i 0\n⊢ IsPrimitiveRoot (ζ ^ i) 0",
"tactic": "simp_all only [pow_one, Nat.coprime_zero_right]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.1908173\nG : Type ?u.1908176\nR : Type ?u.1908179\nS : Type ?u.1908182\nF : Type ?u.1908185\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nf : F\ni : ℕ\nhi : Nat.coprime i k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot ζ k\n⊢ (ζ ^ i) ^ k = 1",
"tactic": "rw [← pow_mul', pow_mul, h.pow_eq_one, one_pow]"
}
] |
[
431,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
418,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.adj_getVert_succ
|
[
{
"state_after": "no goals",
"state_before": "case nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v u✝ : V\ni : ℕ\nhi : i < length nil\n⊢ Adj G (getVert nil i) (getVert nil (i + 1))",
"tactic": "cases hi"
},
{
"state_after": "case cons.zero\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v u✝ v✝ w✝ : V\nhxy : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : ∀ {i : ℕ}, i < length p✝ → Adj G (getVert p✝ i) (getVert p✝ (i + 1))\nhi : Nat.zero < length (cons hxy p✝)\n⊢ Adj G (getVert (cons hxy p✝) Nat.zero) (getVert (cons hxy p✝) (Nat.zero + 1))\n\ncase cons.succ\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v u✝ v✝ w✝ : V\nhxy : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : ∀ {i : ℕ}, i < length p✝ → Adj G (getVert p✝ i) (getVert p✝ (i + 1))\nn✝ : ℕ\nhi : Nat.succ n✝ < length (cons hxy p✝)\n⊢ Adj G (getVert (cons hxy p✝) (Nat.succ n✝)) (getVert (cons hxy p✝) (Nat.succ n✝ + 1))",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v u✝ v✝ w✝ : V\nhxy : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : ∀ {i : ℕ}, i < length p✝ → Adj G (getVert p✝ i) (getVert p✝ (i + 1))\ni : ℕ\nhi : i < length (cons hxy p✝)\n⊢ Adj G (getVert (cons hxy p✝) i) (getVert (cons hxy p✝) (i + 1))",
"tactic": "cases i"
},
{
"state_after": "no goals",
"state_before": "case cons.zero\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v u✝ v✝ w✝ : V\nhxy : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : ∀ {i : ℕ}, i < length p✝ → Adj G (getVert p✝ i) (getVert p✝ (i + 1))\nhi : Nat.zero < length (cons hxy p✝)\n⊢ Adj G (getVert (cons hxy p✝) Nat.zero) (getVert (cons hxy p✝) (Nat.zero + 1))",
"tactic": "simp [getVert, hxy]"
},
{
"state_after": "no goals",
"state_before": "case cons.succ\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v u✝ v✝ w✝ : V\nhxy : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : ∀ {i : ℕ}, i < length p✝ → Adj G (getVert p✝ i) (getVert p✝ (i + 1))\nn✝ : ℕ\nhi : Nat.succ n✝ < length (cons hxy p✝)\n⊢ Adj G (getVert (cons hxy p✝) (Nat.succ n✝)) (getVert (cons hxy p✝) (Nat.succ n✝ + 1))",
"tactic": "exact ih (Nat.succ_lt_succ_iff.1 hi)"
}
] |
[
242,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
Real.volume_interval
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.242227\ninst✝ : Fintype ι\na b : ℝ\n⊢ ↑↑volume (uIcc a b) = ofReal (abs (b - a))",
"tactic": "rw [← Icc_min_max, volume_Icc, max_sub_min_eq_abs]"
}
] |
[
144,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
Isometry.antilipschitz
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1422\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nf : α → β\nx✝ y✝ z : α\ns : Set α\nh : Isometry f\nx y : α\n⊢ edist x y ≤ ↑1 * edist (f x) (f y)",
"tactic": "simp only [h x y, ENNReal.coe_one, one_mul, le_refl]"
}
] |
[
89,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Std/Data/Array/Init/Lemmas.lean
|
Array.foldr_push
|
[] |
[
87,
73
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
86,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.nonempty_sUnion
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.122213\nγ : Type ?u.122216\nι : Sort ?u.122219\nι' : Sort ?u.122222\nι₂ : Sort ?u.122225\nκ : ι → Sort ?u.122230\nκ₁ : ι → Sort ?u.122235\nκ₂ : ι → Sort ?u.122240\nκ' : ι' → Sort ?u.122245\nS : Set (Set α)\n⊢ Set.Nonempty (⋃₀ S) ↔ ∃ s, s ∈ S ∧ Set.Nonempty s",
"tactic": "simp [nonempty_iff_ne_empty]"
}
] |
[
1118,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1117,
1
] |
Mathlib/Data/Polynomial/Derivative.lean
|
Polynomial.derivative_X_sub_C_pow
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\nc : R\nm : ℕ\n⊢ ↑derivative ((X - ↑C c) ^ m) = ↑C ↑m * (X - ↑C c) ^ (m - 1)",
"tactic": "rw [derivative_pow, derivative_X_sub_C, mul_one]"
}
] |
[
665,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
663,
1
] |
Mathlib/Data/List/Count.lean
|
List.count_cons'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl✝ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\n⊢ count a (b :: l) = count a l + if a = b then 1 else 0",
"tactic": "conv =>\nsimp [count, countp_cons]\nlhs\nsimp only [eq_comm]"
}
] |
[
181,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
AlternatingMap.map_update_update
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_4\ninst✝¹⁵ : Semiring R\nM : Type u_3\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\nN : Type u_2\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : Module R N\nP : Type ?u.414052\ninst✝¹⁰ : AddCommMonoid P\ninst✝⁹ : Module R P\nM' : Type ?u.414082\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nN' : Type ?u.414470\ninst✝⁶ : AddCommGroup N'\ninst✝⁵ : Module R N'\nι : Type u_1\nι' : Type ?u.414861\nι'' : Type ?u.414864\nM₂ : Type ?u.414867\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₂\nM₃ : Type ?u.414897\ninst✝² : AddCommMonoid M₃\ninst✝¹ : Module R M₃\nf f' : AlternatingMap R M N ι\ng g₂ : AlternatingMap R M N' ι\ng' : AlternatingMap R M' N' ι\nv : ι → M\nv' : ι → M'\ninst✝ : DecidableEq ι\ni j : ι\nhij : i ≠ j\nm : M\n⊢ update (update v i m) j m i = update (update v i m) j m j",
"tactic": "rw [Function.update_same, Function.update_noteq hij, Function.update_same]"
}
] |
[
686,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
683,
1
] |
Mathlib/Analysis/Fourier/FourierTransform.lean
|
VectorFourier.fourierIntegral_comp_add_right
|
[
{
"state_after": "case h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\n⊢ fourierIntegral e μ L (f ∘ fun v => v + v₀) w = ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • fourierIntegral e μ L f w",
"state_before": "𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\n⊢ fourierIntegral e μ L (f ∘ fun v => v + v₀) = fun w =>\n ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • fourierIntegral e μ L f w",
"tactic": "ext1 w"
},
{
"state_after": "case h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\n⊢ (∫ (v : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L v) w))) • f (v + v₀) ∂μ) =\n ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • ∫ (v : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L v) w))) • f v ∂μ",
"state_before": "case h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\n⊢ fourierIntegral e μ L (f ∘ fun v => v + v₀) w = ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • fourierIntegral e μ L f w",
"tactic": "dsimp only [fourierIntegral, Function.comp_apply]"
},
{
"state_after": "case h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\n⊢ (∫ (v : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L (v + v₀ - v₀)) w))) • f (v + v₀) ∂μ) =\n ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • ∫ (v : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L v) w))) • f v ∂μ",
"state_before": "case h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\n⊢ (∫ (v : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L v) w))) • f (v + v₀) ∂μ) =\n ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • ∫ (v : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L v) w))) • f v ∂μ",
"tactic": "conv in L _ => rw [← add_sub_cancel v v₀]"
},
{
"state_after": "case h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\n⊢ (∫ (x : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L (x - v₀)) w))) • f x ∂μ) =\n ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • ∫ (v : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L v) w))) • f v ∂μ",
"state_before": "case h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\n⊢ (∫ (v : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L (v + v₀ - v₀)) w))) • f (v + v₀) ∂μ) =\n ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • ∫ (v : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L v) w))) • f v ∂μ",
"tactic": "rw [integral_add_right_eq_self fun v : V => e[-L (v - v₀) w] • f v]"
},
{
"state_after": "case h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\n⊢ (∫ (x : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L (x - v₀)) w))) • f x ∂μ) =\n ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • ∫ (v : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L v) w))) • f v ∂μ",
"state_before": "case h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\n⊢ (∫ (x : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L (x - v₀)) w))) • f x ∂μ) =\n ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • ∫ (v : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L v) w))) • f v ∂μ",
"tactic": "dsimp only"
},
{
"state_after": "case h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\n⊢ (∫ (x : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L (x - v₀)) w))) • f x ∂μ) =\n ∫ (a : V), ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L a) w))) • f a ∂μ",
"state_before": "case h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\n⊢ (∫ (x : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L (x - v₀)) w))) • f x ∂μ) =\n ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • ∫ (v : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L v) w))) • f v ∂μ",
"tactic": "rw [← integral_smul]"
},
{
"state_after": "case h.e_f.h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\nv : V\n⊢ ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L (v - v₀)) w))) • f v =\n ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L v) w))) • f v",
"state_before": "case h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\n⊢ (∫ (x : V), ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L (x - v₀)) w))) • f x ∂μ) =\n ∫ (a : V), ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L a) w))) • f a ∂μ",
"tactic": "congr 1 with v"
},
{
"state_after": "no goals",
"state_before": "case h.e_f.h\n𝕜 : Type u_2\ninst✝¹⁰ : CommRing 𝕜\nV : Type u_1\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module 𝕜 V\ninst✝⁷ : MeasurableSpace V\nW : Type u_3\ninst✝⁶ : AddCommGroup W\ninst✝⁵ : Module 𝕜 W\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableAdd V\ne : Multiplicative 𝕜 →* { x // x ∈ 𝕊 }\nμ : MeasureTheory.Measure V\ninst✝ : Measure.IsAddRightInvariant μ\nL : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜\nf : V → E\nv₀ : V\nw : W\nv : V\n⊢ ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L (v - v₀)) w))) • f v =\n ↑(↑e (↑Multiplicative.ofAdd (↑(↑L v₀) w))) • ↑(↑e (↑Multiplicative.ofAdd (-↑(↑L v) w))) • f v",
"tactic": "rw [← smul_assoc, smul_eq_mul, ← Submonoid.coe_mul, ← e.map_mul, ← ofAdd_add, ←\n LinearMap.neg_apply, ← sub_eq_add_neg, ← LinearMap.sub_apply, LinearMap.map_sub, neg_sub]"
}
] |
[
117,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Order/Basic.lean
|
LE.le.le_or_le
|
[] |
[
294,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.comap_comp
|
[] |
[
264,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/LinearAlgebra/Pi.lean
|
Submodule.le_comap_single_pi
|
[
{
"state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝³ : Semiring R\nφ : ι → Type u_1\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\nI : Set ι\np✝ q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (φ i)\ni : ι\nx : φ i\nhx : x ∈ p i\n⊢ x ∈ comap (single i) (pi Set.univ p)",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝³ : Semiring R\nφ : ι → Type u_1\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\nI : Set ι\np✝ q : (i : ι) → Submodule R (φ i)\nx : (i : ι) → φ i\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (φ i)\ni : ι\n⊢ p i ≤ comap (single i) (pi Set.univ p)",
"tactic": "intro x hx"
},
{
"state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝³ : Semiring R\nφ : ι → Type u_1\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\nI : Set ι\np✝ q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (φ i)\ni : ι\nx : φ i\nhx : x ∈ p i\n⊢ ∀ (i_1 : ι), i_1 ∈ Set.univ → ↑(single i) x i_1 ∈ p i_1",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝³ : Semiring R\nφ : ι → Type u_1\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\nI : Set ι\np✝ q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (φ i)\ni : ι\nx : φ i\nhx : x ∈ p i\n⊢ x ∈ comap (single i) (pi Set.univ p)",
"tactic": "rw [Submodule.mem_comap, Submodule.mem_pi]"
},
{
"state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝³ : Semiring R\nφ : ι → Type u_1\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\nI : Set ι\np✝ q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (φ i)\ni : ι\nx : φ i\nhx : x ∈ p i\nj : ι\n⊢ ↑(single i) x j ∈ p j",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝³ : Semiring R\nφ : ι → Type u_1\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\nI : Set ι\np✝ q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (φ i)\ni : ι\nx : φ i\nhx : x ∈ p i\n⊢ ∀ (i_1 : ι), i_1 ∈ Set.univ → ↑(single i) x i_1 ∈ p i_1",
"tactic": "rintro j -"
},
{
"state_after": "case pos\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝³ : Semiring R\nφ : ι → Type u_1\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\nI : Set ι\np✝ q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (φ i)\ni : ι\nx : φ i\nhx : x ∈ p i\nj : ι\nh : j = i\n⊢ ↑(single i) x j ∈ p j\n\ncase neg\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝³ : Semiring R\nφ : ι → Type u_1\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\nI : Set ι\np✝ q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (φ i)\ni : ι\nx : φ i\nhx : x ∈ p i\nj : ι\nh : ¬j = i\n⊢ ↑(single i) x j ∈ p j",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝³ : Semiring R\nφ : ι → Type u_1\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\nI : Set ι\np✝ q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (φ i)\ni : ι\nx : φ i\nhx : x ∈ p i\nj : ι\n⊢ ↑(single i) x j ∈ p j",
"tactic": "by_cases h : j = i"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝³ : Semiring R\nφ : ι → Type u_1\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\nI : Set ι\np✝ q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (φ i)\ni : ι\nx : φ i\nhx : x ∈ p i\nj : ι\nh : j = i\n⊢ ↑(single i) x j ∈ p j",
"tactic": "rwa [h, LinearMap.coe_single, Pi.single_eq_same]"
},
{
"state_after": "case neg\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝³ : Semiring R\nφ : ι → Type u_1\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\nI : Set ι\np✝ q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (φ i)\ni : ι\nx : φ i\nhx : x ∈ p i\nj : ι\nh : ¬j = i\n⊢ 0 ∈ p j",
"state_before": "case neg\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝³ : Semiring R\nφ : ι → Type u_1\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\nI : Set ι\np✝ q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (φ i)\ni : ι\nx : φ i\nhx : x ∈ p i\nj : ι\nh : ¬j = i\n⊢ ↑(single i) x j ∈ p j",
"tactic": "rw [LinearMap.coe_single, Pi.single_eq_of_ne h]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝³ : Semiring R\nφ : ι → Type u_1\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\nI : Set ι\np✝ q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (φ i)\ni : ι\nx : φ i\nhx : x ∈ p i\nj : ι\nh : ¬j = i\n⊢ 0 ∈ p j",
"tactic": "exact (p j).zero_mem"
}
] |
[
331,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
HasDerivWithinAt.sin
|
[] |
[
838,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
836,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.exists_congr_left
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Sort u\nβ✝ : Sort v\nγ : Sort w\nα : Sort u_1\nβ : Sort u_2\nf : α ≃ β\np : α → Prop\nx✝ : ∃ a, p a\na : α\nh : p a\n⊢ p (↑f.symm (↑f a))",
"tactic": "simpa using h"
}
] |
[
921,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
919,
11
] |
Mathlib/Order/UpperLower/Basic.lean
|
IsLowerSet.inter
|
[] |
[
118,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.coeIdeal_span_singleton
|
[
{
"state_after": "case a\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : R\ny : P\n⊢ y ∈ ↑(Ideal.span {x}) ↔ y ∈ spanSingleton S (↑(algebraMap R P) x)",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : R\n⊢ ↑(Ideal.span {x}) = spanSingleton S (↑(algebraMap R P) x)",
"tactic": "ext y"
},
{
"state_after": "case a\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : R\ny : P\n⊢ (∃ x', x' ∈ Ideal.span {x} ∧ ↑(algebraMap R P) x' = y) ↔ ∃ z, z • ↑(algebraMap R P) x = y",
"state_before": "case a\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : R\ny : P\n⊢ y ∈ ↑(Ideal.span {x}) ↔ y ∈ spanSingleton S (↑(algebraMap R P) x)",
"tactic": "refine' (mem_coeIdeal S).trans (Iff.trans _ (mem_spanSingleton S).symm)"
},
{
"state_after": "case a.mp\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : R\ny : P\n⊢ (∃ x', x' ∈ Ideal.span {x} ∧ ↑(algebraMap R P) x' = y) → ∃ z, z • ↑(algebraMap R P) x = y\n\ncase a.mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : R\ny : P\n⊢ (∃ z, z • ↑(algebraMap R P) x = y) → ∃ x', x' ∈ Ideal.span {x} ∧ ↑(algebraMap R P) x' = y",
"state_before": "case a\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : R\ny : P\n⊢ (∃ x', x' ∈ Ideal.span {x} ∧ ↑(algebraMap R P) x' = y) ↔ ∃ z, z • ↑(algebraMap R P) x = y",
"tactic": "constructor"
},
{
"state_after": "case a.mp.intro.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx y' : R\nhy' : y' ∈ Ideal.span {x}\n⊢ ∃ z, z • ↑(algebraMap R P) x = ↑(algebraMap R P) y'",
"state_before": "case a.mp\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : R\ny : P\n⊢ (∃ x', x' ∈ Ideal.span {x} ∧ ↑(algebraMap R P) x' = y) → ∃ z, z • ↑(algebraMap R P) x = y",
"tactic": "rintro ⟨y', hy', rfl⟩"
},
{
"state_after": "case a.mp.intro.intro.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx x' : R\nhy' : x' • x ∈ Ideal.span {x}\n⊢ ∃ z, z • ↑(algebraMap R P) x = ↑(algebraMap R P) (x' • x)",
"state_before": "case a.mp.intro.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx y' : R\nhy' : y' ∈ Ideal.span {x}\n⊢ ∃ z, z • ↑(algebraMap R P) x = ↑(algebraMap R P) y'",
"tactic": "obtain ⟨x', rfl⟩ := Submodule.mem_span_singleton.mp hy'"
},
{
"state_after": "case a.mp.intro.intro.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx x' : R\nhy' : x' • x ∈ Ideal.span {x}\n⊢ x' • ↑(algebraMap R P) x = ↑(algebraMap R P) (x' • x)",
"state_before": "case a.mp.intro.intro.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx x' : R\nhy' : x' • x ∈ Ideal.span {x}\n⊢ ∃ z, z • ↑(algebraMap R P) x = ↑(algebraMap R P) (x' • x)",
"tactic": "use x'"
},
{
"state_after": "no goals",
"state_before": "case a.mp.intro.intro.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx x' : R\nhy' : x' • x ∈ Ideal.span {x}\n⊢ x' • ↑(algebraMap R P) x = ↑(algebraMap R P) (x' • x)",
"tactic": "rw [smul_eq_mul, RingHom.map_mul, Algebra.smul_def]"
},
{
"state_after": "case a.mpr.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx y' : R\n⊢ ∃ x', x' ∈ Ideal.span {x} ∧ ↑(algebraMap R P) x' = y' • ↑(algebraMap R P) x",
"state_before": "case a.mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : R\ny : P\n⊢ (∃ z, z • ↑(algebraMap R P) x = y) → ∃ x', x' ∈ Ideal.span {x} ∧ ↑(algebraMap R P) x' = y",
"tactic": "rintro ⟨y', rfl⟩"
},
{
"state_after": "case a.mpr.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx y' : R\n⊢ ↑(algebraMap R P) (y' * x) = y' • ↑(algebraMap R P) x",
"state_before": "case a.mpr.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx y' : R\n⊢ ∃ x', x' ∈ Ideal.span {x} ∧ ↑(algebraMap R P) x' = y' • ↑(algebraMap R P) x",
"tactic": "refine' ⟨y' * x, Submodule.mem_span_singleton.mpr ⟨y', rfl⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "case a.mpr.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1339818\ninst✝³ : CommRing R₁\nK : Type ?u.1339824\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx y' : R\n⊢ ↑(algebraMap R P) (y' * x) = y' • ↑(algebraMap R P) x",
"tactic": "rw [RingHom.map_mul, Algebra.smul_def]"
}
] |
[
1392,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1381,
1
] |
Mathlib/LinearAlgebra/BilinearMap.lean
|
LinearMap.compr₂_apply
|
[] |
[
384,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/Data/Nat/Log.lean
|
Nat.log_eq_one_iff
|
[] |
[
170,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Algebra/Order/Hom/Monoid.lean
|
OrderMonoidHom.mul_apply
|
[] |
[
524,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/Algebra/Algebra/Tower.lean
|
IsScalarTower.coe_toAlgHom'
|
[] |
[
134,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/NumberTheory/Bernoulli.lean
|
bernoulliPowerSeries_mul_exp_sub_one
|
[
{
"state_after": "case h\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ↑(coeff A n) (bernoulliPowerSeries A * (exp A - 1)) = ↑(coeff A n) X",
"state_before": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ bernoulliPowerSeries A * (exp A - 1) = X",
"tactic": "ext n"
},
{
"state_after": "case h.zero\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ ↑(coeff A zero) (bernoulliPowerSeries A * (exp A - 1)) = ↑(coeff A zero) X\n\ncase h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ↑(coeff A (succ n)) (bernoulliPowerSeries A * (exp A - 1)) = ↑(coeff A (succ n)) X",
"state_before": "case h\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ↑(coeff A n) (bernoulliPowerSeries A * (exp A - 1)) = ↑(coeff A n) X",
"tactic": "cases' n with n n"
},
{
"state_after": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ↑(algebraMap ℚ A) (bernoulli (n + 1) / ((↑n + 1) * ↑(Nat.add n 0)!)) * (↑(algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +\n ↑(algebraMap ℚ A) (∑ x in antidiagonal n, bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(Nat.add x.snd 0)!)⁻¹) =\n if succ n = 1 then 1 else 0",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ↑(coeff A (succ n)) (bernoulliPowerSeries A * (exp A - 1)) = ↑(coeff A (succ n)) X",
"tactic": "simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,\n coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,\n sub_zero, RingHom.map_one, add_eq_zero_iff, if_false, _root_.inv_one, zero_add, one_ne_zero,\n MulZeroClass.mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]"
},
{
"state_after": "case h.succ.zero\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ ↑(algebraMap ℚ A) (bernoulli (zero + 1) / ((↑zero + 1) * ↑(Nat.add zero 0)!)) * (↑(algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +\n ↑(algebraMap ℚ A)\n (∑ x in antidiagonal zero, bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(Nat.add x.snd 0)!)⁻¹) =\n if succ zero = 1 then 1 else 0\n\ncase h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ↑(algebraMap ℚ A) (bernoulli (succ n + 1) / ((↑(succ n) + 1) * ↑(Nat.add (succ n) 0)!)) *\n (↑(algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +\n ↑(algebraMap ℚ A)\n (∑ x in antidiagonal (succ n), bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(Nat.add x.snd 0)!)⁻¹) =\n if succ (succ n) = 1 then 1 else 0",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ↑(algebraMap ℚ A) (bernoulli (n + 1) / ((↑n + 1) * ↑(Nat.add n 0)!)) * (↑(algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +\n ↑(algebraMap ℚ A) (∑ x in antidiagonal n, bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(Nat.add x.snd 0)!)⁻¹) =\n if succ n = 1 then 1 else 0",
"tactic": "cases' n with n"
},
{
"state_after": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ↑(algebraMap ℚ A) (bernoulli (succ n + 1) / ((↑(succ n) + 1) * ↑(Nat.add (succ n) 0)!)) *\n (↑(algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +\n ↑(algebraMap ℚ A)\n (∑ x in antidiagonal (succ n), bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(Nat.add x.snd 0)!)⁻¹) =\n 0",
"state_before": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ↑(algebraMap ℚ A) (bernoulli (succ n + 1) / ((↑(succ n) + 1) * ↑(Nat.add (succ n) 0)!)) *\n (↑(algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +\n ↑(algebraMap ℚ A)\n (∑ x in antidiagonal (succ n), bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(Nat.add x.snd 0)!)⁻¹) =\n if succ (succ n) = 1 then 1 else 0",
"tactic": "rw [if_neg n.succ_succ_ne_one]"
},
{
"state_after": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\n⊢ ↑(algebraMap ℚ A) (bernoulli (succ n + 1) / ((↑(succ n) + 1) * ↑(Nat.add (succ n) 0)!)) *\n (↑(algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +\n ↑(algebraMap ℚ A)\n (∑ x in antidiagonal (succ n), bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(Nat.add x.snd 0)!)⁻¹) =\n 0",
"state_before": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ↑(algebraMap ℚ A) (bernoulli (succ n + 1) / ((↑(succ n) + 1) * ↑(Nat.add (succ n) 0)!)) *\n (↑(algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +\n ↑(algebraMap ℚ A)\n (∑ x in antidiagonal (succ n), bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(Nat.add x.snd 0)!)⁻¹) =\n 0",
"tactic": "have hfact : ∀ m, (m ! : ℚ) ≠ 0 := fun m => by exact_mod_cast factorial_ne_zero m"
},
{
"state_after": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\n⊢ ↑(algebraMap ℚ A) (bernoulli (succ n + 1) / ((↑(succ n) + 1) * ↑(Nat.add (succ n) 0)!)) *\n (↑(algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +\n ↑(algebraMap ℚ A)\n (∑ x in antidiagonal (succ n), bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(Nat.add x.snd 0)!)⁻¹) =\n 0",
"state_before": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\n⊢ ↑(algebraMap ℚ A) (bernoulli (succ n + 1) / ((↑(succ n) + 1) * ↑(Nat.add (succ n) 0)!)) *\n (↑(algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +\n ↑(algebraMap ℚ A)\n (∑ x in antidiagonal (succ n), bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(Nat.add x.snd 0)!)⁻¹) =\n 0",
"tactic": "have hite2 : ite (n.succ = 0) 1 0 = (0 : ℚ) := if_neg n.succ_ne_zero"
},
{
"state_after": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\n⊢ ↑(algebraMap ℚ A) (∑ x in antidiagonal (succ n), bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(x.snd + 0)!)⁻¹) = 0",
"state_before": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\n⊢ ↑(algebraMap ℚ A) (bernoulli (succ n + 1) / ((↑(succ n) + 1) * ↑(Nat.add (succ n) 0)!)) *\n (↑(algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +\n ↑(algebraMap ℚ A)\n (∑ x in antidiagonal (succ n), bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(Nat.add x.snd 0)!)⁻¹) =\n 0",
"tactic": "simp only [CharP.cast_eq_zero, zero_add, inv_one, map_one, sub_self, mul_zero, add_eq]"
},
{
"state_after": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\n⊢ ↑(algebraMap ℚ A) (∑ x in antidiagonal (succ n), bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(x.snd + 0)!)⁻¹) =\n ↑(algebraMap ℚ A)\n (∑ x in antidiagonal (succ n), ↑(Nat.choose (x.fst + x.snd) x.snd) / (↑x.snd + 1) * bernoulli x.fst / ↑(succ n)!)",
"state_before": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\n⊢ ↑(algebraMap ℚ A) (∑ x in antidiagonal (succ n), bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(x.snd + 0)!)⁻¹) = 0",
"tactic": "rw [← map_zero (algebraMap ℚ A), ← zero_div (n.succ ! : ℚ), ← hite2, ← bernoulli_spec', sum_div]"
},
{
"state_after": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x ∈ antidiagonal (succ n)\n⊢ bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(x.snd + 0)!)⁻¹ * ↑(succ n)! =\n ↑(Nat.choose (x.fst + x.snd) x.snd) / (↑x.snd + 1) * bernoulli x.fst",
"state_before": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\n⊢ ↑(algebraMap ℚ A) (∑ x in antidiagonal (succ n), bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(x.snd + 0)!)⁻¹) =\n ↑(algebraMap ℚ A)\n (∑ x in antidiagonal (succ n), ↑(Nat.choose (x.fst + x.snd) x.snd) / (↑x.snd + 1) * bernoulli x.fst / ↑(succ n)!)",
"tactic": "refine' congr_arg (algebraMap ℚ A) (sum_congr rfl fun x h => eq_div_of_mul_eq (hfact n.succ) _)"
},
{
"state_after": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\n⊢ bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(x.snd + 0)!)⁻¹ * ↑(succ n)! =\n ↑(Nat.choose (x.fst + x.snd) x.snd) / (↑x.snd + 1) * bernoulli x.fst",
"state_before": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x ∈ antidiagonal (succ n)\n⊢ bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(x.snd + 0)!)⁻¹ * ↑(succ n)! =\n ↑(Nat.choose (x.fst + x.snd) x.snd) / (↑x.snd + 1) * bernoulli x.fst",
"tactic": "rw [mem_antidiagonal] at h"
},
{
"state_after": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\nhj : ↑x.snd + 1 ≠ 0\n⊢ bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(x.snd + 0)!)⁻¹ * ↑(succ n)! =\n ↑(Nat.choose (x.fst + x.snd) x.snd) / (↑x.snd + 1) * bernoulli x.fst",
"state_before": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\n⊢ bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(x.snd + 0)!)⁻¹ * ↑(succ n)! =\n ↑(Nat.choose (x.fst + x.snd) x.snd) / (↑x.snd + 1) * bernoulli x.fst",
"tactic": "have hj : (x.2 + 1 : ℚ) ≠ 0 := by norm_cast; exact succ_ne_zero _"
},
{
"state_after": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\nhj : ↑x.snd + 1 ≠ 0\n⊢ bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(x.snd + 0)!)⁻¹ * ↑(x.fst + x.snd)! =\n ↑(x.fst + x.snd)! / ↑(x.fst ! * x.snd !) / (↑x.snd + 1) * bernoulli x.fst",
"state_before": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\nhj : ↑x.snd + 1 ≠ 0\n⊢ bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(x.snd + 0)!)⁻¹ * ↑(succ n)! =\n ↑(Nat.choose (x.fst + x.snd) x.snd) / (↑x.snd + 1) * bernoulli x.fst",
"tactic": "rw [← h, add_choose, cast_div_charZero (factorial_mul_factorial_dvd_factorial_add _ _)]"
},
{
"state_after": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\nhj : ↑x.snd + 1 ≠ 0\n⊢ ↑x.snd ! * (↑x.snd + 1) = (↑x.snd + 1) * ↑x.snd ! ∨ bernoulli x.fst = 0",
"state_before": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\nhj : ↑x.snd + 1 ≠ 0\n⊢ bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(x.snd + 0)!)⁻¹ * ↑(x.fst + x.snd)! =\n ↑(x.fst + x.snd)! / ↑(x.fst ! * x.snd !) / (↑x.snd + 1) * bernoulli x.fst",
"tactic": "field_simp [mul_ne_zero hj (hfact x.2), hfact x.1, mul_comm _ (bernoulli x.1), mul_assoc,\n Nat.factorial_ne_zero, hj]"
},
{
"state_after": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\nhj : ↑x.snd + 1 ≠ 0\n⊢ ↑x.snd ! * (↑x.snd + 1) = (↑x.snd + 1) * ↑x.snd ! ∨ bernoulli x.fst = 0",
"state_before": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\nhj : ↑x.snd + 1 ≠ 0\n⊢ ↑x.snd ! * (↑x.snd + 1) = (↑x.snd + 1) * ↑x.snd ! ∨ bernoulli x.fst = 0",
"tactic": "simp only"
},
{
"state_after": "case h.succ.succ.h\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\nhj : ↑x.snd + 1 ≠ 0\n⊢ ↑x.snd ! * (↑x.snd + 1) = (↑x.snd + 1) * ↑x.snd !",
"state_before": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\nhj : ↑x.snd + 1 ≠ 0\n⊢ ↑x.snd ! * (↑x.snd + 1) = (↑x.snd + 1) * ↑x.snd ! ∨ bernoulli x.fst = 0",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case h.succ.succ.h\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\nhj : ↑x.snd + 1 ≠ 0\n⊢ ↑x.snd ! * (↑x.snd + 1) = (↑x.snd + 1) * ↑x.snd !",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "case h.zero\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ ↑(coeff A zero) (bernoulliPowerSeries A * (exp A - 1)) = ↑(coeff A zero) X",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h.succ.zero\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ ↑(algebraMap ℚ A) (bernoulli (zero + 1) / ((↑zero + 1) * ↑(Nat.add zero 0)!)) * (↑(algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +\n ↑(algebraMap ℚ A)\n (∑ x in antidiagonal zero, bernoulli x.fst / ↑x.fst ! * ((↑x.snd + 1) * ↑(Nat.add x.snd 0)!)⁻¹) =\n if succ zero = 1 then 1 else 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn m : ℕ\n⊢ ↑m ! ≠ 0",
"tactic": "exact_mod_cast factorial_ne_zero m"
},
{
"state_after": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\n⊢ ¬x.snd + 1 = 0",
"state_before": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\n⊢ ↑x.snd + 1 ≠ 0",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if succ n = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x.fst + x.snd = succ n\n⊢ ¬x.snd + 1 = 0",
"tactic": "exact succ_ne_zero _"
}
] |
[
305,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.map_eq_comap_of_inverse
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.599712\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.599721\ninst✝¹ : AddGroup A\nN : Type u_2\ninst✝ : Group N\nf✝ f : G →* N\ng : N →* G\nhl : LeftInverse ↑g ↑f\nhr : Function.RightInverse ↑g ↑f\nH : Subgroup G\n⊢ ↑(map f H) = ↑(comap g H)",
"tactic": "rw [coe_map, coe_comap, Set.image_eq_preimage_of_inverse hl hr]"
}
] |
[
3122,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3120,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.count_singleton
|
[] |
[
2318,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2317,
1
] |
Mathlib/RingTheory/WittVector/StructurePolynomial.lean
|
wittStructureRat_vars
|
[
{
"state_after": "p : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn : ℕ\n⊢ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n)) ⊆\n Finset.univ ×ˢ Finset.range (n + 1)",
"state_before": "p : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn : ℕ\n⊢ vars (wittStructureRat p Φ n) ⊆ Finset.univ ×ˢ Finset.range (n + 1)",
"tactic": "rw [wittStructureRat]"
},
{
"state_after": "p : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn : ℕ\nx : idx × ℕ\nhx : x ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\n⊢ x ∈ Finset.univ ×ˢ Finset.range (n + 1)",
"state_before": "p : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn : ℕ\n⊢ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n)) ⊆\n Finset.univ ×ˢ Finset.range (n + 1)",
"tactic": "intro x hx"
},
{
"state_after": "p : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn : ℕ\nx : idx × ℕ\nhx : x ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\n⊢ x.snd < n + 1",
"state_before": "p : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn : ℕ\nx : idx × ℕ\nhx : x ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\n⊢ x ∈ Finset.univ ×ˢ Finset.range (n + 1)",
"tactic": "simp only [Finset.mem_product, true_and_iff, Finset.mem_univ, Finset.mem_range]"
},
{
"state_after": "case intro.intro\np : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn : ℕ\nx : idx × ℕ\nhx : x ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\nk : ℕ\nhk : k ∈ vars (xInTermsOfW p ℚ n)\nhx' : x ∈ vars (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ)\n⊢ x.snd < n + 1",
"state_before": "p : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn : ℕ\nx : idx × ℕ\nhx : x ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\n⊢ x.snd < n + 1",
"tactic": "obtain ⟨k, hk, hx'⟩ := mem_vars_bind₁ _ _ hx"
},
{
"state_after": "case intro.intro.intro.intro\np : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn : ℕ\nx : idx × ℕ\nhx : x ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\nk : ℕ\nhk : k ∈ vars (xInTermsOfW p ℚ n)\nhx' : x ∈ vars (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ)\ni : idx\nhx'' : x ∈ vars (↑(rename (Prod.mk i)) (W_ ℚ k))\n⊢ x.snd < n + 1",
"state_before": "case intro.intro\np : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn : ℕ\nx : idx × ℕ\nhx : x ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\nk : ℕ\nhk : k ∈ vars (xInTermsOfW p ℚ n)\nhx' : x ∈ vars (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ)\n⊢ x.snd < n + 1",
"tactic": "obtain ⟨i, -, hx''⟩ := mem_vars_bind₁ _ _ hx'"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\np : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn k : ℕ\nhk : k ∈ vars (xInTermsOfW p ℚ n)\ni : idx\nj : ℕ\nhj : j ∈ vars (W_ ℚ k)\nhx : (i, j) ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\nhx' : (i, j) ∈ vars (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ)\nhx'' : (i, j) ∈ vars (↑(rename (Prod.mk i)) (W_ ℚ k))\n⊢ (i, j).snd < n + 1",
"state_before": "case intro.intro.intro.intro\np : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn : ℕ\nx : idx × ℕ\nhx : x ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\nk : ℕ\nhk : k ∈ vars (xInTermsOfW p ℚ n)\nhx' : x ∈ vars (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ)\ni : idx\nhx'' : x ∈ vars (↑(rename (Prod.mk i)) (W_ ℚ k))\n⊢ x.snd < n + 1",
"tactic": "obtain ⟨j, hj, rfl⟩ := mem_vars_rename _ _ hx''"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\np : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn k : ℕ\nhk : k ∈ vars (xInTermsOfW p ℚ n)\ni : idx\nj : ℕ\nhj : j < k + 1\nhx : (i, j) ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\nhx' : (i, j) ∈ vars (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ)\nhx'' : (i, j) ∈ vars (↑(rename (Prod.mk i)) (W_ ℚ k))\n⊢ (i, j).snd < n + 1",
"state_before": "case intro.intro.intro.intro.intro.intro\np : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn k : ℕ\nhk : k ∈ vars (xInTermsOfW p ℚ n)\ni : idx\nj : ℕ\nhj : j ∈ vars (W_ ℚ k)\nhx : (i, j) ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\nhx' : (i, j) ∈ vars (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ)\nhx'' : (i, j) ∈ vars (↑(rename (Prod.mk i)) (W_ ℚ k))\n⊢ (i, j).snd < n + 1",
"tactic": "rw [wittPolynomial_vars, Finset.mem_range] at hj"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\np : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn k : ℕ\ni : idx\nj : ℕ\nhj : j < k + 1\nhx : (i, j) ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\nhx' : (i, j) ∈ vars (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ)\nhx'' : (i, j) ∈ vars (↑(rename (Prod.mk i)) (W_ ℚ k))\nhk : k ∈ Finset.range (n + 1)\n⊢ (i, j).snd < n + 1",
"state_before": "case intro.intro.intro.intro.intro.intro\np : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn k : ℕ\nhk : k ∈ vars (xInTermsOfW p ℚ n)\ni : idx\nj : ℕ\nhj : j < k + 1\nhx : (i, j) ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\nhx' : (i, j) ∈ vars (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ)\nhx'' : (i, j) ∈ vars (↑(rename (Prod.mk i)) (W_ ℚ k))\n⊢ (i, j).snd < n + 1",
"tactic": "replace hk := xInTermsOfW_vars_subset p _ hk"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\np : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn k : ℕ\ni : idx\nj : ℕ\nhj : j < k + 1\nhx : (i, j) ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\nhx' : (i, j) ∈ vars (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ)\nhx'' : (i, j) ∈ vars (↑(rename (Prod.mk i)) (W_ ℚ k))\nhk : k < n + 1\n⊢ (i, j).snd < n + 1",
"state_before": "case intro.intro.intro.intro.intro.intro\np : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn k : ℕ\ni : idx\nj : ℕ\nhj : j < k + 1\nhx : (i, j) ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\nhx' : (i, j) ∈ vars (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ)\nhx'' : (i, j) ∈ vars (↑(rename (Prod.mk i)) (W_ ℚ k))\nhk : k ∈ Finset.range (n + 1)\n⊢ (i, j).snd < n + 1",
"tactic": "rw [Finset.mem_range] at hk"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\np : ℕ\nR : Type ?u.2115631\nidx : Type u_1\ninst✝¹ : CommRing R\nhp : Fact (Nat.Prime p)\ninst✝ : Fintype idx\nΦ : MvPolynomial idx ℚ\nn k : ℕ\ni : idx\nj : ℕ\nhj : j < k + 1\nhx : (i, j) ∈ vars (↑(bind₁ fun k => ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n))\nhx' : (i, j) ∈ vars (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ k)) Φ)\nhx'' : (i, j) ∈ vars (↑(rename (Prod.mk i)) (W_ ℚ k))\nhk : k < n + 1\n⊢ (i, j).snd < n + 1",
"tactic": "exact lt_of_lt_of_le hj hk"
}
] |
[
405,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
LinearMap.toMatrix_mulVec_repr
|
[
{
"state_after": "case h\nR : Type u_1\ninst✝⁹ : CommSemiring R\nl : Type ?u.2016127\nm : Type u_4\nn : Type u_5\ninst✝⁸ : Fintype n\ninst✝⁷ : Fintype m\ninst✝⁶ : DecidableEq n\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₁\ninst✝² : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type ?u.2016656\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Module R M₃\nv₃ : Basis l R M₃\nf : M₁ →ₗ[R] M₂\nx : M₁\ni : m\n⊢ mulVec (↑(toMatrix v₁ v₂) f) (↑(↑v₁.repr x)) i = ↑(↑v₂.repr (↑f x)) i",
"state_before": "R : Type u_1\ninst✝⁹ : CommSemiring R\nl : Type ?u.2016127\nm : Type u_4\nn : Type u_5\ninst✝⁸ : Fintype n\ninst✝⁷ : Fintype m\ninst✝⁶ : DecidableEq n\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₁\ninst✝² : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type ?u.2016656\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Module R M₃\nv₃ : Basis l R M₃\nf : M₁ →ₗ[R] M₂\nx : M₁\n⊢ mulVec (↑(toMatrix v₁ v₂) f) ↑(↑v₁.repr x) = ↑(↑v₂.repr (↑f x))",
"tactic": "ext i"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁹ : CommSemiring R\nl : Type ?u.2016127\nm : Type u_4\nn : Type u_5\ninst✝⁸ : Fintype n\ninst✝⁷ : Fintype m\ninst✝⁶ : DecidableEq n\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₁\ninst✝² : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type ?u.2016656\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Module R M₃\nv₃ : Basis l R M₃\nf : M₁ →ₗ[R] M₂\nx : M₁\ni : m\n⊢ ↑(↑v₂.repr (↑f (↑(LinearEquiv.symm (Basis.equivFun v₁)) ↑(↑v₁.repr x)))) i = ↑(↑v₂.repr (↑f x)) i",
"state_before": "case h\nR : Type u_1\ninst✝⁹ : CommSemiring R\nl : Type ?u.2016127\nm : Type u_4\nn : Type u_5\ninst✝⁸ : Fintype n\ninst✝⁷ : Fintype m\ninst✝⁶ : DecidableEq n\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₁\ninst✝² : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type ?u.2016656\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Module R M₃\nv₃ : Basis l R M₃\nf : M₁ →ₗ[R] M₂\nx : M₁\ni : m\n⊢ mulVec (↑(toMatrix v₁ v₂) f) (↑(↑v₁.repr x)) i = ↑(↑v₂.repr (↑f x)) i",
"tactic": "rw [← Matrix.toLin'_apply, LinearMap.toMatrix, LinearEquiv.trans_apply, Matrix.toLin'_toMatrix',\n LinearEquiv.arrowCongr_apply, v₂.equivFun_apply]"
},
{
"state_after": "case h.e_a.h.e_6.h.h.e_6.h\nR : Type u_1\ninst✝⁹ : CommSemiring R\nl : Type ?u.2016127\nm : Type u_4\nn : Type u_5\ninst✝⁸ : Fintype n\ninst✝⁷ : Fintype m\ninst✝⁶ : DecidableEq n\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₁\ninst✝² : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type ?u.2016656\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Module R M₃\nv₃ : Basis l R M₃\nf : M₁ →ₗ[R] M₂\nx : M₁\ni : m\n⊢ ↑(LinearEquiv.symm (Basis.equivFun v₁)) ↑(↑v₁.repr x) = x",
"state_before": "case h\nR : Type u_1\ninst✝⁹ : CommSemiring R\nl : Type ?u.2016127\nm : Type u_4\nn : Type u_5\ninst✝⁸ : Fintype n\ninst✝⁷ : Fintype m\ninst✝⁶ : DecidableEq n\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₁\ninst✝² : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type ?u.2016656\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Module R M₃\nv₃ : Basis l R M₃\nf : M₁ →ₗ[R] M₂\nx : M₁\ni : m\n⊢ ↑(↑v₂.repr (↑f (↑(LinearEquiv.symm (Basis.equivFun v₁)) ↑(↑v₁.repr x)))) i = ↑(↑v₂.repr (↑f x)) i",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case h.e_a.h.e_6.h.h.e_6.h\nR : Type u_1\ninst✝⁹ : CommSemiring R\nl : Type ?u.2016127\nm : Type u_4\nn : Type u_5\ninst✝⁸ : Fintype n\ninst✝⁷ : Fintype m\ninst✝⁶ : DecidableEq n\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₁\ninst✝² : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type ?u.2016656\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Module R M₃\nv₃ : Basis l R M₃\nf : M₁ →ₗ[R] M₂\nx : M₁\ni : m\n⊢ ↑(LinearEquiv.symm (Basis.equivFun v₁)) ↑(↑v₁.repr x) = x",
"tactic": "exact v₁.equivFun.symm_apply_apply x"
}
] |
[
670,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
664,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.ofDual_apply_top
|
[] |
[
678,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
677,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.cauchySeq_iff'
|
[] |
[
1508,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1506,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.SimpleFunc.nnnorm_approxOn_le
|
[
{
"state_after": "α : Type ?u.8872\nβ : Type u_2\nι : Type ?u.8878\nE : Type u_1\nF : Type ?u.8884\n𝕜 : Type ?u.8887\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\nq : ℝ\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\nthis : edist (↑(approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist y₀ (f x)\n⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊",
"state_before": "α : Type ?u.8872\nβ : Type u_2\nι : Type ?u.8878\nE : Type u_1\nF : Type ?u.8884\n𝕜 : Type ?u.8887\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\nq : ℝ\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\n⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊",
"tactic": "have := edist_approxOn_le hf h₀ x n"
},
{
"state_after": "α : Type ?u.8872\nβ : Type u_2\nι : Type ?u.8878\nE : Type u_1\nF : Type ?u.8884\n𝕜 : Type ?u.8887\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\nq : ℝ\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\nthis : edist (↑(approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist (f x) y₀\n⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊",
"state_before": "α : Type ?u.8872\nβ : Type u_2\nι : Type ?u.8878\nE : Type u_1\nF : Type ?u.8884\n𝕜 : Type ?u.8887\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\nq : ℝ\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\nthis : edist (↑(approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist y₀ (f x)\n⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊",
"tactic": "rw [edist_comm y₀] at this"
},
{
"state_after": "α : Type ?u.8872\nβ : Type u_2\nι : Type ?u.8878\nE : Type u_1\nF : Type ?u.8884\n𝕜 : Type ?u.8887\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\nq : ℝ\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\nthis : ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ↑‖f x - y₀‖₊\n⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊",
"state_before": "α : Type ?u.8872\nβ : Type u_2\nι : Type ?u.8878\nE : Type u_1\nF : Type ?u.8884\n𝕜 : Type ?u.8887\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\nq : ℝ\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\nthis : edist (↑(approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist (f x) y₀\n⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊",
"tactic": "simp only [edist_nndist, nndist_eq_nnnorm] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.8872\nβ : Type u_2\nι : Type ?u.8878\nE : Type u_1\nF : Type ?u.8884\n𝕜 : Type ?u.8887\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\nq : ℝ\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\nthis : ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ↑‖f x - y₀‖₊\n⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊",
"tactic": "exact_mod_cast this"
}
] |
[
79,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Set.finite_Ioo
|
[] |
[
661,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
660,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.preimage_eq_empty_iff
|
[] |
[
134,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Topology/GDelta.lean
|
isGδ_biInter
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.5171\nγ : Type ?u.5174\nι : Type u_1\ninst✝ : TopologicalSpace α\ns : Set ι\nhs : Set.Countable s\nt : (i : ι) → i ∈ s → Set α\nht : ∀ (i : ι) (hi : i ∈ s), IsGδ (t i hi)\n⊢ IsGδ (⋂ (x : ↑s), t ↑x (_ : ↑x ∈ s))",
"state_before": "α : Type u_2\nβ : Type ?u.5171\nγ : Type ?u.5174\nι : Type u_1\ninst✝ : TopologicalSpace α\ns : Set ι\nhs : Set.Countable s\nt : (i : ι) → i ∈ s → Set α\nht : ∀ (i : ι) (hi : i ∈ s), IsGδ (t i hi)\n⊢ IsGδ (⋂ (i : ι) (h : i ∈ s), t i h)",
"tactic": "rw [biInter_eq_iInter]"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.5171\nγ : Type ?u.5174\nι : Type u_1\ninst✝ : TopologicalSpace α\ns : Set ι\nhs : Set.Countable s\nt : (i : ι) → i ∈ s → Set α\nht : ∀ (i : ι) (hi : i ∈ s), IsGδ (t i hi)\nthis : Encodable ↑s\n⊢ IsGδ (⋂ (x : ↑s), t ↑x (_ : ↑x ∈ s))",
"state_before": "α : Type u_2\nβ : Type ?u.5171\nγ : Type ?u.5174\nι : Type u_1\ninst✝ : TopologicalSpace α\ns : Set ι\nhs : Set.Countable s\nt : (i : ι) → i ∈ s → Set α\nht : ∀ (i : ι) (hi : i ∈ s), IsGδ (t i hi)\n⊢ IsGδ (⋂ (x : ↑s), t ↑x (_ : ↑x ∈ s))",
"tactic": "haveI := hs.toEncodable"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.5171\nγ : Type ?u.5174\nι : Type u_1\ninst✝ : TopologicalSpace α\ns : Set ι\nhs : Set.Countable s\nt : (i : ι) → i ∈ s → Set α\nht : ∀ (i : ι) (hi : i ∈ s), IsGδ (t i hi)\nthis : Encodable ↑s\n⊢ IsGδ (⋂ (x : ↑s), t ↑x (_ : ↑x ∈ s))",
"tactic": "exact isGδ_iInter fun x => ht x x.2"
}
] |
[
95,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Analysis/Convex/Side.lean
|
AffineSubspace.sSameSide_vadd_left_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.135022\nP : Type u_3\nP' : Type ?u.135028\ninst✝⁶ : StrictOrderedCommRing R\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'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v ∈ direction s\n⊢ SSameSide s (v +ᵥ x) y ↔ SSameSide s x y",
"tactic": "rw [SSameSide, SSameSide, wSameSide_vadd_left_iff hv, vadd_mem_iff_mem_of_mem_direction hv]"
}
] |
[
295,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Std/Data/Array/Init/Lemmas.lean
|
Array.foldrM_eq_reverse_foldlM_data.aux
|
[
{
"state_after": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝ : Monad m\nf : α → β → m β\narr : Array α\ninit : β\ni : Nat\nh : i ≤ size arr\n⊢ List.foldlM (fun x y => f y x) init (List.reverse (List.take i arr.data)) =\n if (i == 0) = true then pure init\n else\n match i, h with\n | 0, x => pure init\n | Nat.succ i, h =>\n let_fun this := (_ : i < size arr);\n do\n let __do_lift ← f arr[i] init\n foldrM.fold f arr 0 i (_ : i ≤ size arr) __do_lift",
"state_before": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝ : Monad m\nf : α → β → m β\narr : Array α\ninit : β\ni : Nat\nh : i ≤ size arr\n⊢ List.foldlM (fun x y => f y x) init (List.reverse (List.take i arr.data)) = foldrM.fold f arr 0 i h init",
"tactic": "unfold foldrM.fold"
},
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝ : Monad m\nf : α → β → m β\narr : Array α\ninit : β\ni : Nat\nh : 0 ≤ size arr\n⊢ List.foldlM (fun x y => f y x) init (List.reverse (List.take 0 arr.data)) =\n if (0 == 0) = true then pure init\n else\n match 0, h with\n | 0, x => pure init\n | Nat.succ i, h =>\n let_fun this := (_ : i < size arr);\n do\n let __do_lift ← f arr[i] init\n foldrM.fold f arr 0 i (_ : i ≤ size arr) __do_lift",
"tactic": "simp [List.foldlM, List.take]"
},
{
"state_after": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝ : Monad m\nf : α → β → m β\narr : Array α\ninit : β\ni✝ i : Nat\nh : i + 1 ≤ size arr\n⊢ List.foldlM (fun x y => f y x) init (List.reverse (List.concat (List.take i arr.data) arr.data[i])) =\n if (i + 1 == 0) = true then pure init\n else\n match i + 1, h with\n | 0, x => pure init\n | Nat.succ i, h =>\n let_fun this := (_ : i < size arr);\n do\n let __do_lift ← f arr[i] init\n foldrM.fold f arr 0 i (_ : i ≤ size arr) __do_lift",
"state_before": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝ : Monad m\nf : α → β → m β\narr : Array α\ninit : β\ni✝ i : Nat\nh : i + 1 ≤ size arr\n⊢ List.foldlM (fun x y => f y x) init (List.reverse (List.take (i + 1) arr.data)) =\n if (i + 1 == 0) = true then pure init\n else\n match i + 1, h with\n | 0, x => pure init\n | Nat.succ i, h =>\n let_fun this := (_ : i < size arr);\n do\n let __do_lift ← f arr[i] init\n foldrM.fold f arr 0 i (_ : i ≤ size arr) __do_lift",
"tactic": "rw [← List.take_concat_get _ _ h]"
},
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝ : Monad m\nf : α → β → m β\narr : Array α\ninit : β\ni✝ i : Nat\nh : i + 1 ≤ size arr\n⊢ (do\n let s' ← f arr.data[i] init\n List.foldrM (fun x y => f x y) s' (List.take i arr.data)) =\n if i + 1 = 0 then pure init\n else do\n let s' ← f arr[i] init\n List.foldrM (fun x y => f x y) s' (List.take i arr.data)",
"tactic": "rfl"
}
] |
[
57,
76
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
51,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.aecover_Ioo_of_Ioo
|
[] |
[
200,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Topology/Algebra/Polynomial.lean
|
Polynomial.tendsto_norm_atTop
|
[] |
[
138,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Order/Basic.lean
|
Subtype.mk_le_mk
|
[] |
[
1134,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1132,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
UniformContinuous.mul
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : Group α\ninst✝¹ : UniformGroup α\ninst✝ : UniformSpace β\nf g : β → α\nhf : UniformContinuous f\nhg : UniformContinuous g\nthis : UniformContinuous fun x => f x / (g x)⁻¹\n⊢ UniformContinuous fun x => f x * g x",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : Group α\ninst✝¹ : UniformGroup α\ninst✝ : UniformSpace β\nf g : β → α\nhf : UniformContinuous f\nhg : UniformContinuous g\n⊢ UniformContinuous fun x => f x * g x",
"tactic": "have : UniformContinuous fun x => f x / (g x)⁻¹ := hf.div hg.inv"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : Group α\ninst✝¹ : UniformGroup α\ninst✝ : UniformSpace β\nf g : β → α\nhf : UniformContinuous f\nhg : UniformContinuous g\nthis : UniformContinuous fun x => f x / (g x)⁻¹\n⊢ UniformContinuous fun x => f x * g x",
"tactic": "simp_all"
}
] |
[
107,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/SetTheory/Cardinal/Continuum.lean
|
Cardinal.continuum_toNat
|
[] |
[
112,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometry.preimage_ball
|
[] |
[
310,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.subtypeDomain_sum
|
[] |
[
2127,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2124,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
DifferentiableAt.mul
|
[] |
[
330,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/Combinatorics/SetFamily/Compression/UV.lean
|
UV.mem_of_mem_compression
|
[
{
"state_after": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nha : a ∈ s ∧ compress u v a ∈ s ∨ ¬a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = a\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\n⊢ a ∈ s",
"state_before": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nha : a ∈ 𝓒 u v s\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\n⊢ a ∈ s",
"tactic": "rw [mem_compression] at ha"
},
{
"state_after": "case inl\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nha : a ∈ s ∧ compress u v a ∈ s\n⊢ a ∈ s\n\ncase inr.intro.intro.intro\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : ¬a ∈ s\nb : α\nhb : b ∈ s\nh : compress u v b = a\n⊢ a ∈ s",
"state_before": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nha : a ∈ s ∧ compress u v a ∈ s ∨ ¬a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = a\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\n⊢ a ∈ s",
"tactic": "obtain ha | ⟨_, b, hb, h⟩ := ha"
},
{
"state_after": "case inr.intro.intro.intro\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : ¬a ∈ s\nb : α\nhb : b ∈ s\nh : (if Disjoint u b ∧ v ≤ b then (b ⊔ u) \\ v else b) = a\n⊢ a ∈ s",
"state_before": "case inr.intro.intro.intro\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : ¬a ∈ s\nb : α\nhb : b ∈ s\nh : compress u v b = a\n⊢ a ∈ s",
"tactic": "unfold compress at h"
},
{
"state_after": "case inr.intro.intro.intro.inl\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : ¬a ∈ s\nb : α\nhb : b ∈ s\nh✝ : Disjoint u b ∧ v ≤ b\nh : (b ⊔ u) \\ v = a\n⊢ a ∈ s\n\ncase inr.intro.intro.intro.inr\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : ¬a ∈ s\nb : α\nhb : b ∈ s\nh✝ : ¬(Disjoint u b ∧ v ≤ b)\nh : b = a\n⊢ a ∈ s",
"state_before": "case inr.intro.intro.intro\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : ¬a ∈ s\nb : α\nhb : b ∈ s\nh : (if Disjoint u b ∧ v ≤ b then (b ⊔ u) \\ v else b) = a\n⊢ a ∈ s",
"tactic": "split_ifs at h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nha : a ∈ s ∧ compress u v a ∈ s\n⊢ a ∈ s",
"tactic": "exact ha.1"
},
{
"state_after": "case inr.intro.intro.intro.inl\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nhvu : v = ⊥ → u = ⊥\nleft✝ : ¬a ∈ s\nb : α\nhva : v = ⊥\nhb : b ∈ s\nh✝ : Disjoint u b ∧ v ≤ b\nh : (b ⊔ u) \\ v = a\n⊢ a ∈ s",
"state_before": "case inr.intro.intro.intro.inl\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : ¬a ∈ s\nb : α\nhb : b ∈ s\nh✝ : Disjoint u b ∧ v ≤ b\nh : (b ⊔ u) \\ v = a\n⊢ a ∈ s",
"tactic": "rw [← h, le_sdiff_iff] at hva"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro.inl\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nhvu : v = ⊥ → u = ⊥\nleft✝ : ¬a ∈ s\nb : α\nhva : v = ⊥\nhb : b ∈ s\nh✝ : Disjoint u b ∧ v ≤ b\nh : (b ⊔ u) \\ v = a\n⊢ a ∈ s",
"tactic": "rwa [← h, hvu hva, hva, sup_bot_eq, sdiff_bot]"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro.inr\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : ¬a ∈ s\nb : α\nhb : b ∈ s\nh✝ : ¬(Disjoint u b ∧ v ≤ b)\nh : b = a\n⊢ a ∈ s",
"tactic": "rwa [← h]"
}
] |
[
291,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/Data/TwoPointing.lean
|
TwoPointing.snd_ne_fst
|
[] |
[
52,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
Subgroup.mem_pointwise_smul_iff_inv_smul_mem
|
[] |
[
345,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.ne_iff_vne
|
[] |
[
199,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
fderivWithin_snd
|
[] |
[
326,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/RingTheory/Subsemiring/Pointwise.lean
|
Subsemiring.smul_sup
|
[] |
[
86,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Analysis/Complex/Circle.lean
|
mem_circle_iff_normSq
|
[
{
"state_after": "no goals",
"state_before": "z : ℂ\n⊢ z ∈ circle ↔ ↑normSq z = 1",
"tactic": "simp [Complex.abs]"
}
] |
[
65,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.coeff_mul
|
[] |
[
216,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_zero_fun
|
[] |
[
175,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.coe_zero
|
[] |
[
1701,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1700,
1
] |
src/lean/Init/Data/List/Control.lean
|
List.forIn_nil
|
[] |
[
154,
6
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
153,
9
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.union_div
|
[] |
[
708,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
707,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.b_of_eq
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.286658\nF : Type ?u.286661\nK : Type ?u.286664\nP Q : Cubic R\na b c d a' b' c' d' : R\ninst✝ : Semiring R\nh : toPoly P = toPoly Q\n⊢ P.b = Q.b",
"tactic": "rw [← coeff_eq_b, h, coeff_eq_b]"
}
] |
[
127,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/LinearAlgebra/Matrix/Hermitian.lean
|
Matrix.isHermitian_transpose_iff
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.3927\nm : Type ?u.3930\nn : Type u_1\nA✝ : Matrix n n α\ninst✝¹ : Star α\ninst✝ : Star β\nA : Matrix n n α\nh : IsHermitian Aᵀ\n⊢ IsHermitian A",
"state_before": "α : Type u_2\nβ : Type ?u.3927\nm : Type ?u.3930\nn : Type u_1\nA✝ : Matrix n n α\ninst✝¹ : Star α\ninst✝ : Star β\nA : Matrix n n α\n⊢ IsHermitian Aᵀ → IsHermitian A",
"tactic": "intro h"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.3927\nm : Type ?u.3930\nn : Type u_1\nA✝ : Matrix n n α\ninst✝¹ : Star α\ninst✝ : Star β\nA : Matrix n n α\nh : IsHermitian Aᵀ\n⊢ IsHermitian Aᵀᵀ",
"state_before": "α : Type u_2\nβ : Type ?u.3927\nm : Type ?u.3930\nn : Type u_1\nA✝ : Matrix n n α\ninst✝¹ : Star α\ninst✝ : Star β\nA : Matrix n n α\nh : IsHermitian Aᵀ\n⊢ IsHermitian A",
"tactic": "rw [← transpose_transpose A]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.3927\nm : Type ?u.3930\nn : Type u_1\nA✝ : Matrix n n α\ninst✝¹ : Star α\ninst✝ : Star β\nA : Matrix n n α\nh : IsHermitian Aᵀ\n⊢ IsHermitian Aᵀᵀ",
"tactic": "exact IsHermitian.transpose h"
}
] |
[
81,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.foldr_eq
|
[
{
"state_after": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nf : α → β → β\nl₁ l₂ : List α\nlcomm : LeftCommutative f\np : l₁ ~ l₂\nx : α\nt₁ t₂ : List α\n_p : t₁ ~ t₂\nr : ∀ (b : β), foldr f b t₁ = foldr f b t₂\nb : β\n⊢ f x (foldr f b t₁) = f x (foldr f b t₂)",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nf : α → β → β\nl₁ l₂ : List α\nlcomm : LeftCommutative f\np : l₁ ~ l₂\nx : α\nt₁ t₂ : List α\n_p : t₁ ~ t₂\nr : ∀ (b : β), foldr f b t₁ = foldr f b t₂\nb : β\n⊢ foldr f b (x :: t₁) = foldr f b (x :: t₂)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nf : α → β → β\nl₁ l₂ : List α\nlcomm : LeftCommutative f\np : l₁ ~ l₂\nx : α\nt₁ t₂ : List α\n_p : t₁ ~ t₂\nr : ∀ (b : β), foldr f b t₁ = foldr f b t₂\nb : β\n⊢ f x (foldr f b t₁) = f x (foldr f b t₂)",
"tactic": "rw [r b]"
},
{
"state_after": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nf : α → β → β\nl₁ l₂ : List α\nlcomm : LeftCommutative f\np : l₁ ~ l₂\nx y : α\nt₁ t₂ : List α\n_p : t₁ ~ t₂\nr : ∀ (b : β), foldr f b t₁ = foldr f b t₂\nb : β\n⊢ f y (f x (foldr f b t₁)) = f x (f y (foldr f b t₂))",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nf : α → β → β\nl₁ l₂ : List α\nlcomm : LeftCommutative f\np : l₁ ~ l₂\nx y : α\nt₁ t₂ : List α\n_p : t₁ ~ t₂\nr : ∀ (b : β), foldr f b t₁ = foldr f b t₂\nb : β\n⊢ foldr f b (y :: x :: t₁) = foldr f b (x :: y :: t₂)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nf : α → β → β\nl₁ l₂ : List α\nlcomm : LeftCommutative f\np : l₁ ~ l₂\nx y : α\nt₁ t₂ : List α\n_p : t₁ ~ t₂\nr : ∀ (b : β), foldr f b t₁ = foldr f b t₂\nb : β\n⊢ f y (f x (foldr f b t₁)) = f x (f y (foldr f b t₂))",
"tactic": "rw [lcomm, r b]"
}
] |
[
529,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
525,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean
|
ContDiffBump.integral_normed
|
[
{
"state_after": "E : Type u_1\nX : Type ?u.1858193\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup X\ninst✝⁶ : NormedSpace ℝ X\ninst✝⁵ : HasContDiffBump E\nc : E\nf : ContDiffBump c\nx : E\nn : ℕ∞\ninst✝⁴ : MeasurableSpace E\nμ : MeasureTheory.Measure E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : Measure.IsOpenPosMeasure μ\n⊢ ((∫ (x : E), ↑f x ∂μ)⁻¹ • ∫ (x : E), ↑f x ∂μ) = 1",
"state_before": "E : Type u_1\nX : Type ?u.1858193\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup X\ninst✝⁶ : NormedSpace ℝ X\ninst✝⁵ : HasContDiffBump E\nc : E\nf : ContDiffBump c\nx : E\nn : ℕ∞\ninst✝⁴ : MeasurableSpace E\nμ : MeasureTheory.Measure E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : Measure.IsOpenPosMeasure μ\n⊢ (∫ (x : E), ContDiffBump.normed f μ x ∂μ) = 1",
"tactic": "simp_rw [ContDiffBump.normed, div_eq_mul_inv, mul_comm (f _), ← smul_eq_mul, integral_smul]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nX : Type ?u.1858193\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup X\ninst✝⁶ : NormedSpace ℝ X\ninst✝⁵ : HasContDiffBump E\nc : E\nf : ContDiffBump c\nx : E\nn : ℕ∞\ninst✝⁴ : MeasurableSpace E\nμ : MeasureTheory.Measure E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : Measure.IsOpenPosMeasure μ\n⊢ ((∫ (x : E), ↑f x ∂μ)⁻¹ • ∫ (x : E), ↑f x ∂μ) = 1",
"tactic": "exact inv_mul_cancel f.integral_pos.ne'"
}
] |
[
532,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
530,
1
] |
Mathlib/CategoryTheory/Types.lean
|
equivEquivIso_hom
|
[] |
[
412,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
411,
1
] |
Std/Data/List/Basic.lean
|
List.inits_eq_initsTR
|
[
{
"state_after": "case h.h\nα : Type u_1\nl : List α\n⊢ inits l = initsTR l",
"state_before": "⊢ @inits = @initsTR",
"tactic": "funext α l"
},
{
"state_after": "case h.h\nα : Type u_1\nl : List α\n⊢ inits l = reverse (foldr (fun a arrs => Array.push (Array.map (fun t => a :: t) arrs) []) #[[]] l).data",
"state_before": "case h.h\nα : Type u_1\nl : List α\n⊢ inits l = initsTR l",
"tactic": "simp [initsTR]"
},
{
"state_after": "no goals",
"state_before": "case h.h\nα : Type u_1\nl : List α\n⊢ inits l = reverse (foldr (fun a arrs => Array.push (Array.map (fun t => a :: t) arrs) []) #[[]] l).data",
"tactic": "induction l <;> simp [*, reverse_map]"
}
] |
[
796,
68
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
795,
10
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.smul_finset_singleton
|
[] |
[
1657,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1656,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.lift'_le
|
[] |
[
279,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.map_mono
|
[
{
"state_after": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂✝ : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\nh : ∀ ⦃x : L⦄, x ∈ I₁ → x ∈ I₂\n⊢ map f I₁ ≤ map f I₂",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂✝ : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\nh : I₁ ≤ I₂\n⊢ map f I₁ ≤ map f I₂",
"tactic": "rw [SetLike.le_def] at h"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI I₂✝ : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\nh : ∀ ⦃x : L⦄, x ∈ I₁ → x ∈ I₂\n⊢ map f I₁ ≤ map f I₂",
"tactic": "apply LieSubmodule.lieSpan_mono (Set.image_subset (⇑f) h)"
}
] |
[
862,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
860,
1
] |
Mathlib/Order/Category/LatCat.lean
|
LatCat.coe_of
|
[] |
[
54,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.val_lt_last
|
[] |
[
553,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
552,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean
|
Nat.factorial_inj
|
[
{
"state_after": "m n : ℕ\nhn : 1 < n !\nh : n ! = m !\n⊢ n = m",
"state_before": "m n : ℕ\nhn : 1 < n !\n⊢ n ! = m ! ↔ n = m",
"tactic": "refine' ⟨fun h => _, congr_arg _⟩"
},
{
"state_after": "case inl\nm n : ℕ\nhn : 1 < n !\nh : n ! = m !\nhnm : n < m\n⊢ n = m\n\ncase inr.inl\nn : ℕ\nhn : 1 < n !\nh : n ! = n !\n⊢ n = n\n\ncase inr.inr\nm n : ℕ\nhn : 1 < n !\nh : n ! = m !\nhnm : m < n\n⊢ n = m",
"state_before": "m n : ℕ\nhn : 1 < n !\nh : n ! = m !\n⊢ n = m",
"tactic": "obtain hnm | rfl | hnm := lt_trichotomy n m"
},
{
"state_after": "case inr.inr\nm n : ℕ\nhn : 1 < m\nh : n ! = m !\nhnm : m < n\n⊢ n = m",
"state_before": "case inr.inr\nm n : ℕ\nhn : 1 < n !\nh : n ! = m !\nhnm : m < n\n⊢ n = m",
"tactic": "rw [h, one_lt_factorial] at hn"
},
{
"state_after": "case inr.inr\nm n : ℕ\nhn : 1 < m\nh : n ! = m !\nhnm : m ! < m !\n⊢ n = m",
"state_before": "case inr.inr\nm n : ℕ\nhn : 1 < m\nh : n ! = m !\nhnm : m < n\n⊢ n = m",
"tactic": "rw [← factorial_lt (lt_trans one_pos hn), h] at hnm"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nm n : ℕ\nhn : 1 < m\nh : n ! = m !\nhnm : m ! < m !\n⊢ n = m",
"tactic": "cases lt_irrefl _ hnm"
},
{
"state_after": "case inl\nm n : ℕ\nhn : 1 < n !\nh : n ! = m !\nhnm : m ! < m !\n⊢ n = m",
"state_before": "case inl\nm n : ℕ\nhn : 1 < n !\nh : n ! = m !\nhnm : n < m\n⊢ n = m",
"tactic": "rw [← factorial_lt <| pos_of_gt <| one_lt_factorial.mp hn, h] at hnm"
},
{
"state_after": "no goals",
"state_before": "case inl\nm n : ℕ\nhn : 1 < n !\nh : n ! = m !\nhnm : m ! < m !\n⊢ n = m",
"tactic": "cases lt_irrefl _ hnm"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nn : ℕ\nhn : 1 < n !\nh : n ! = n !\n⊢ n = n",
"tactic": "rfl"
}
] |
[
142,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_le_of_subset
|
[] |
[
64,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.biUnion_assoc
|
[
{
"state_after": "case h\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ : Box ι\n⊢ (J ∈ biUnion π fun J => biUnion (πi J) (πi' J)) ↔ J ∈ biUnion (biUnion π πi) fun J => πi' (biUnionIndex π πi J) J",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\n⊢ (biUnion π fun J => biUnion (πi J) (πi' J)) = biUnion (biUnion π πi) fun J => πi' (biUnionIndex π πi J) J",
"tactic": "ext J"
},
{
"state_after": "case h\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ : Box ι\n⊢ (∃ J', J' ∈ π ∧ ∃ J'_1, J'_1 ∈ πi J' ∧ J ∈ πi' J' J'_1) ↔\n ∃ J', (∃ J'_1, J'_1 ∈ π ∧ J' ∈ πi J'_1) ∧ J ∈ πi' (biUnionIndex π πi J') J'",
"state_before": "case h\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ : Box ι\n⊢ (J ∈ biUnion π fun J => biUnion (πi J) (πi' J)) ↔ J ∈ biUnion (biUnion π πi) fun J => πi' (biUnionIndex π πi J) J",
"tactic": "simp only [mem_biUnion, exists_prop]"
},
{
"state_after": "case h.mp\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ : Box ι\n⊢ (∃ J', J' ∈ π ∧ ∃ J'_1, J'_1 ∈ πi J' ∧ J ∈ πi' J' J'_1) →\n ∃ J', (∃ J'_1, J'_1 ∈ π ∧ J' ∈ πi J'_1) ∧ J ∈ πi' (biUnionIndex π πi J') J'\n\ncase h.mpr\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ : Box ι\n⊢ (∃ J', (∃ J'_1, J'_1 ∈ π ∧ J' ∈ πi J'_1) ∧ J ∈ πi' (biUnionIndex π πi J') J') →\n ∃ J', J' ∈ π ∧ ∃ J'_1, J'_1 ∈ πi J' ∧ J ∈ πi' J' J'_1",
"state_before": "case h\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ : Box ι\n⊢ (∃ J', J' ∈ π ∧ ∃ J'_1, J'_1 ∈ πi J' ∧ J ∈ πi' J' J'_1) ↔\n ∃ J', (∃ J'_1, J'_1 ∈ π ∧ J' ∈ πi J'_1) ∧ J ∈ πi' (biUnionIndex π πi J') J'",
"tactic": "constructor"
},
{
"state_after": "case h.mp.intro.intro.intro.intro\nι : Type u_1\nI J✝ J₁✝ J₂✝ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ J₁ : Box ι\nhJ₁ : J₁ ∈ π\nJ₂ : Box ι\nhJ₂ : J₂ ∈ πi J₁\nhJ : J ∈ πi' J₁ J₂\n⊢ ∃ J', (∃ J'_1, J'_1 ∈ π ∧ J' ∈ πi J'_1) ∧ J ∈ πi' (biUnionIndex π πi J') J'",
"state_before": "case h.mp\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ : Box ι\n⊢ (∃ J', J' ∈ π ∧ ∃ J'_1, J'_1 ∈ πi J' ∧ J ∈ πi' J' J'_1) →\n ∃ J', (∃ J'_1, J'_1 ∈ π ∧ J' ∈ πi J'_1) ∧ J ∈ πi' (biUnionIndex π πi J') J'",
"tactic": "rintro ⟨J₁, hJ₁, J₂, hJ₂, hJ⟩"
},
{
"state_after": "case h.mp.intro.intro.intro.intro\nι : Type u_1\nI J✝ J₁✝ J₂✝ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ J₁ : Box ι\nhJ₁ : J₁ ∈ π\nJ₂ : Box ι\nhJ₂ : J₂ ∈ πi J₁\nhJ : J ∈ πi' J₁ J₂\n⊢ J ∈ πi' (biUnionIndex π πi J₂) J₂",
"state_before": "case h.mp.intro.intro.intro.intro\nι : Type u_1\nI J✝ J₁✝ J₂✝ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ J₁ : Box ι\nhJ₁ : J₁ ∈ π\nJ₂ : Box ι\nhJ₂ : J₂ ∈ πi J₁\nhJ : J ∈ πi' J₁ J₂\n⊢ ∃ J', (∃ J'_1, J'_1 ∈ π ∧ J' ∈ πi J'_1) ∧ J ∈ πi' (biUnionIndex π πi J') J'",
"tactic": "refine' ⟨J₂, ⟨J₁, hJ₁, hJ₂⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro.intro.intro.intro\nι : Type u_1\nI J✝ J₁✝ J₂✝ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ J₁ : Box ι\nhJ₁ : J₁ ∈ π\nJ₂ : Box ι\nhJ₂ : J₂ ∈ πi J₁\nhJ : J ∈ πi' J₁ J₂\n⊢ J ∈ πi' (biUnionIndex π πi J₂) J₂",
"tactic": "rwa [π.biUnionIndex_of_mem hJ₁ hJ₂]"
},
{
"state_after": "case h.mpr.intro.intro.intro.intro\nι : Type u_1\nI J✝ J₁✝ J₂✝ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ J₁ : Box ι\nhJ : J ∈ πi' (biUnionIndex π πi J₁) J₁\nJ₂ : Box ι\nhJ₂ : J₂ ∈ π\nhJ₁ : J₁ ∈ πi J₂\n⊢ ∃ J', J' ∈ π ∧ ∃ J'_1, J'_1 ∈ πi J' ∧ J ∈ πi' J' J'_1",
"state_before": "case h.mpr\nι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ : Box ι\n⊢ (∃ J', (∃ J'_1, J'_1 ∈ π ∧ J' ∈ πi J'_1) ∧ J ∈ πi' (biUnionIndex π πi J') J') →\n ∃ J', J' ∈ π ∧ ∃ J'_1, J'_1 ∈ πi J' ∧ J ∈ πi' J' J'_1",
"tactic": "rintro ⟨J₁, ⟨J₂, hJ₂, hJ₁⟩, hJ⟩"
},
{
"state_after": "case h.mpr.intro.intro.intro.intro\nι : Type u_1\nI J✝ J₁✝ J₂✝ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ J₁ : Box ι\nhJ : J ∈ πi' (biUnionIndex π πi J₁) J₁\nJ₂ : Box ι\nhJ₂ : J₂ ∈ π\nhJ₁ : J₁ ∈ πi J₂\n⊢ J ∈ πi' J₂ J₁",
"state_before": "case h.mpr.intro.intro.intro.intro\nι : Type u_1\nI J✝ J₁✝ J₂✝ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ J₁ : Box ι\nhJ : J ∈ πi' (biUnionIndex π πi J₁) J₁\nJ₂ : Box ι\nhJ₂ : J₂ ∈ π\nhJ₁ : J₁ ∈ πi J₂\n⊢ ∃ J', J' ∈ π ∧ ∃ J'_1, J'_1 ∈ πi J' ∧ J ∈ πi' J' J'_1",
"tactic": "refine' ⟨J₂, hJ₂, J₁, hJ₁, _⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro.intro.intro.intro\nι : Type u_1\nI J✝ J₁✝ J₂✝ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ J₁ : Box ι\nhJ : J ∈ πi' (biUnionIndex π πi J₁) J₁\nJ₂ : Box ι\nhJ₂ : J₂ ∈ π\nhJ₁ : J₁ ∈ πi J₂\n⊢ J ∈ πi' J₂ J₁",
"tactic": "rwa [π.biUnionIndex_of_mem hJ₂ hJ₁] at hJ"
}
] |
[
399,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
388,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
zsmul_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.84932\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✝ : AddGroupWithOne A\nn : ℤ\n⊢ n • 1 = ↑n",
"tactic": "cases n <;> simp"
}
] |
[
131,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Topology/MetricSpace/MetricSeparated.lean
|
IsMetricSeparated.empty_right
|
[] |
[
55,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
PosNum.bit0_of_bit0
|
[
{
"state_after": "α : Type ?u.80889\np : PosNum\n⊢ bit0 (succ (bit0 p)) = bit0 (bit1 p)",
"state_before": "α : Type ?u.80889\np : PosNum\n⊢ bit0 (succ (_root_.bit0 p)) = bit0 (bit1 p)",
"tactic": "rw [bit0_of_bit0 p]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.80889\np : PosNum\n⊢ bit0 (succ (bit0 p)) = bit0 (bit1 p)",
"tactic": "rfl"
}
] |
[
118,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean
|
Set.Finite.infsep_zero_iff_subsingleton
|
[] |
[
555,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
553,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
tendsto_inv
|
[] |
[
223,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.add_re
|
[] |
[
122,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
IntervalIntegrable.comp_mul_left
|
[
{
"state_after": "case inl\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntervalIntegrable f volume a b\nc : ℝ\nhc : c = 0\n⊢ IntervalIntegrable (fun x => f (c * x)) volume (a / c) (b / c)\n\ncase inr\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntervalIntegrable f volume a b\nc : ℝ\nhc : c ≠ 0\n⊢ IntervalIntegrable (fun x => f (c * x)) volume (a / c) (b / c)",
"state_before": "ι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntervalIntegrable f volume a b\nc : ℝ\n⊢ IntervalIntegrable (fun x => f (c * x)) volume (a / c) (b / c)",
"tactic": "rcases eq_or_ne c 0 with (hc | hc)"
},
{
"state_after": "case inr\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\n⊢ IntegrableOn (fun x => f (c * x)) [[a / c, b / c]]",
"state_before": "case inr\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntervalIntegrable f volume a b\nc : ℝ\nhc : c ≠ 0\n⊢ IntervalIntegrable (fun x => f (c * x)) volume (a / c) (b / c)",
"tactic": "rw [intervalIntegrable_iff'] at hf ⊢"
},
{
"state_after": "case inr\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\n⊢ IntegrableOn (fun x => f (c * x)) [[a / c, b / c]]",
"state_before": "case inr\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\n⊢ IntegrableOn (fun x => f (c * x)) [[a / c, b / c]]",
"tactic": "have A : MeasurableEmbedding fun x => x * c⁻¹ :=\n (Homeomorph.mulRight₀ _ (inv_ne_zero hc)).closedEmbedding.measurableEmbedding"
},
{
"state_after": "case inr\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\n⊢ IntegrableOn ((fun x => f (c * x)) ∘ fun x => x * c⁻¹) ((fun x => x * c⁻¹) ⁻¹' [[a / c, b / c]])",
"state_before": "case inr\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\n⊢ IntegrableOn (fun x => f (c * x)) [[a / c, b / c]]",
"tactic": "rw [← Real.smul_map_volume_mul_right (inv_ne_zero hc), IntegrableOn, Measure.restrict_smul,\n integrable_smul_measure (by simpa : ENNReal.ofReal (|c⁻¹|) ≠ 0) ENNReal.ofReal_ne_top,\n ← IntegrableOn, MeasurableEmbedding.integrableOn_map_iff A]"
},
{
"state_after": "case h.e'_5\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\n⊢ ((fun x => f (c * x)) ∘ fun x => x * c⁻¹) = f\n\ncase h.e'_6\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\n⊢ (fun x => x * c⁻¹) ⁻¹' [[a / c, b / c]] = [[a, b]]",
"state_before": "case inr\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\n⊢ IntegrableOn ((fun x => f (c * x)) ∘ fun x => x * c⁻¹) ((fun x => x * c⁻¹) ⁻¹' [[a / c, b / c]])",
"tactic": "convert hf using 1"
},
{
"state_after": "case inl\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntervalIntegrable f volume a b\nc : ℝ\nhc : c = 0\n⊢ IntervalIntegrable (fun x => f (0 * x)) volume (a / 0) (b / 0)",
"state_before": "case inl\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntervalIntegrable f volume a b\nc : ℝ\nhc : c = 0\n⊢ IntervalIntegrable (fun x => f (c * x)) volume (a / c) (b / c)",
"tactic": "rw [hc]"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntervalIntegrable f volume a b\nc : ℝ\nhc : c = 0\n⊢ IntervalIntegrable (fun x => f (0 * x)) volume (a / 0) (b / 0)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\n⊢ ENNReal.ofReal (Abs.abs c⁻¹) ≠ 0",
"tactic": "simpa"
},
{
"state_after": "case h.e'_5.h\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\nx✝ : ℝ\n⊢ ((fun x => f (c * x)) ∘ fun x => x * c⁻¹) x✝ = f x✝",
"state_before": "case h.e'_5\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\n⊢ ((fun x => f (c * x)) ∘ fun x => x * c⁻¹) = f",
"tactic": "ext"
},
{
"state_after": "case h.e'_5.h\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\nx✝ : ℝ\n⊢ f (c * (x✝ * c⁻¹)) = f x✝",
"state_before": "case h.e'_5.h\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\nx✝ : ℝ\n⊢ ((fun x => f (c * x)) ∘ fun x => x * c⁻¹) x✝ = f x✝",
"tactic": "simp only [comp_apply]"
},
{
"state_after": "case h.e'_5.h.e_a\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\nx✝ : ℝ\n⊢ c * (x✝ * c⁻¹) = x✝",
"state_before": "case h.e'_5.h\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\nx✝ : ℝ\n⊢ f (c * (x✝ * c⁻¹)) = f x✝",
"tactic": "congr 1"
},
{
"state_after": "case h.e'_5.h.e_a\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\nx✝ : ℝ\n⊢ c * x✝ = x✝ * c",
"state_before": "case h.e'_5.h.e_a\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\nx✝ : ℝ\n⊢ c * (x✝ * c⁻¹) = x✝",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.e_a\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\nx✝ : ℝ\n⊢ c * x✝ = x✝ * c",
"tactic": "ring"
},
{
"state_after": "case h.e'_6\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\n⊢ [[a / c / c⁻¹, b / c / c⁻¹]] = [[a, b]]",
"state_before": "case h.e'_6\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\n⊢ (fun x => x * c⁻¹) ⁻¹' [[a / c, b / c]] = [[a, b]]",
"tactic": "rw [preimage_mul_const_uIcc (inv_ne_zero hc)]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_6\nι : Type ?u.6953593\n𝕜 : Type ?u.6953596\nE : Type u_1\nF : Type ?u.6953602\nA✝ : Type ?u.6953605\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedRing A✝\nf g : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : IntegrableOn f [[a, b]]\nc : ℝ\nhc : c ≠ 0\nA : MeasurableEmbedding fun x => x * c⁻¹\n⊢ [[a / c / c⁻¹, b / c / c⁻¹]] = [[a, b]]",
"tactic": "field_simp [hc]"
}
] |
[
305,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Tactic/CategoryTheory/Elementwise.lean
|
Tactic.Elementwise.hom_elementwise
|
[
{
"state_after": "no goals",
"state_before": "C : Type u_1\ninst✝¹ : Category C\ninst✝ : ConcreteCategory C\nX Y : C\nf g : X ⟶ Y\nh : f = g\nx : (forget C).obj X\n⊢ (forget C).map f x = (forget C).map g x",
"tactic": "rw [h]"
}
] |
[
51,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/CategoryTheory/Monoidal/End.lean
|
CategoryTheory.ε_inv_hom_app
|
[] |
[
115,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Algebra/Homology/HomotopyCategory.lean
|
HomotopyCategory.homologyFactors_hom_app
|
[] |
[
162,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Combinatorics/Quiver/SingleObj.lean
|
Quiver.SingleObj.pathEquivList_nil
|
[] |
[
158,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_out
|
[] |
[
155,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Algebra/DirectSum/Internal.lean
|
DirectSum.coe_mul_of_apply_aux
|
[
{
"state_after": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (Dfinsupp.sum (↑(of (fun i => { x // x ∈ A i }) i) r') fun i₂ x₂ =>\n Dfinsupp.sum r fun i₁ x₁ => if i₁ + i₂ = n then ↑x₁ * ↑x₂ else 0) =\n ↑(↑r j) * ↑r'",
"state_before": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ ↑(↑(r * ↑(of (fun i => { x // x ∈ A i }) i) r') n) = ↑(↑r j) * ↑r'",
"tactic": "rw [coe_mul_apply_eq_dfinsupp_sum, Dfinsupp.sum_comm]"
},
{
"state_after": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (Dfinsupp.sum r fun i₁ x₁ => if i₁ + i = n then ↑x₁ * ↑r' else 0) = ↑(↑r j) * ↑r'\n\nι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (Dfinsupp.sum r fun i₁ x₁ => if i₁ + i = n then ↑x₁ * ↑0 else 0) = 0",
"state_before": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (Dfinsupp.sum (↑(of (fun i => { x // x ∈ A i }) i) r') fun i₂ x₂ =>\n Dfinsupp.sum r fun i₁ x₁ => if i₁ + i₂ = n then ↑x₁ * ↑x₂ else 0) =\n ↑(↑r j) * ↑r'",
"tactic": "apply (Dfinsupp.sum_single_index _).trans"
},
{
"state_after": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (Dfinsupp.sum r fun i₁ x₁ => if i₁ + i = n then ↑x₁ * ↑0 else 0) = 0\n\nι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (Dfinsupp.sum r fun i₁ x₁ => if i₁ + i = n then ↑x₁ * ↑r' else 0) = ↑(↑r j) * ↑r'",
"state_before": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (Dfinsupp.sum r fun i₁ x₁ => if i₁ + i = n then ↑x₁ * ↑r' else 0) = ↑(↑r j) * ↑r'\n\nι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (Dfinsupp.sum r fun i₁ x₁ => if i₁ + i = n then ↑x₁ * ↑0 else 0) = 0",
"tactic": "swap"
},
{
"state_after": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (if j ∈ Dfinsupp.support r then ↑(↑r j) * ↑r' else 0) = ↑(↑r j) * ↑r'",
"state_before": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (Dfinsupp.sum r fun i₁ x₁ => if i₁ + i = n then ↑x₁ * ↑r' else 0) = ↑(↑r j) * ↑r'",
"tactic": "simp_rw [Dfinsupp.sum, H, Finset.sum_ite_eq']"
},
{
"state_after": "case inl\nι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\nh : j ∈ Dfinsupp.support r\n⊢ ↑(↑r j) * ↑r' = ↑(↑r j) * ↑r'\n\ncase inr\nι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\nh : ¬j ∈ Dfinsupp.support r\n⊢ 0 = ↑(↑r j) * ↑r'",
"state_before": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (if j ∈ Dfinsupp.support r then ↑(↑r j) * ↑r' else 0) = ↑(↑r j) * ↑r'",
"tactic": "split_ifs with h"
},
{
"state_after": "case inr\nι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\nh : ¬j ∈ Dfinsupp.support r\n⊢ 0 = ↑(↑r j) * ↑r'",
"state_before": "case inl\nι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\nh : j ∈ Dfinsupp.support r\n⊢ ↑(↑r j) * ↑r' = ↑(↑r j) * ↑r'\n\ncase inr\nι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\nh : ¬j ∈ Dfinsupp.support r\n⊢ 0 = ↑(↑r j) * ↑r'",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\nh : ¬j ∈ Dfinsupp.support r\n⊢ 0 = ↑(↑r j) * ↑r'",
"tactic": "rw [Dfinsupp.not_mem_support_iff.mp h, ZeroMemClass.coe_zero, MulZeroClass.zero_mul]"
},
{
"state_after": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (Dfinsupp.sum r fun i₁ x₁ => 0) = 0",
"state_before": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (Dfinsupp.sum r fun i₁ x₁ => if i₁ + i = n then ↑x₁ * ↑0 else 0) = 0",
"tactic": "simp_rw [ZeroMemClass.coe_zero, MulZeroClass.mul_zero, ite_self]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.170111\nR : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), { x // x ∈ A i }\ni : ι\nr' : { x // x ∈ A i }\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (Dfinsupp.sum r fun i₁ x₁ => 0) = 0",
"tactic": "exact Dfinsupp.sum_zero"
}
] |
[
209,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.Integrable.add_measure
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.839059\nδ : Type ?u.839062\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhμ : Memℒp f 1\nhν : Memℒp f 1\n⊢ Memℒp f 1",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.839059\nδ : Type ?u.839062\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhμ : Integrable f\nhν : Integrable f\n⊢ Integrable f",
"tactic": "simp_rw [← memℒp_one_iff_integrable] at hμ hν⊢"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.839059\nδ : Type ?u.839062\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhμ : Memℒp f 1\nhν : Memℒp f 1\n⊢ snorm f 1 (μ + ν) < ⊤",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.839059\nδ : Type ?u.839062\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhμ : Memℒp f 1\nhν : Memℒp f 1\n⊢ Memℒp f 1",
"tactic": "refine' ⟨hμ.aestronglyMeasurable.add_measure hν.aestronglyMeasurable, _⟩"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.839059\nδ : Type ?u.839062\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhμ : Memℒp f 1\nhν : Memℒp f 1\n⊢ snorm f 1 μ < ⊤ ∧ snorm f 1 ν < ⊤",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.839059\nδ : Type ?u.839062\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhμ : Memℒp f 1\nhν : Memℒp f 1\n⊢ snorm f 1 (μ + ν) < ⊤",
"tactic": "rw [snorm_one_add_measure, ENNReal.add_lt_top]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.839059\nδ : Type ?u.839062\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhμ : Memℒp f 1\nhν : Memℒp f 1\n⊢ snorm f 1 μ < ⊤ ∧ snorm f 1 ν < ⊤",
"tactic": "exact ⟨hμ.snorm_lt_top, hν.snorm_lt_top⟩"
}
] |
[
532,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/Data/List/ToFinsupp.lean
|
List.toFinsupp_concat_eq_toFinsupp_add_single
|
[
{
"state_after": "no goals",
"state_before": "M : Type ?u.23867\ninst✝⁴ : Zero M\nl : List M\ninst✝³ : DecidablePred fun x => getD l x 0 ≠ 0\nn : ℕ\nR : Type u_1\ninst✝² : AddZeroClass R\nx : R\nxs : List R\ninst✝¹ : DecidablePred fun i => getD (xs ++ [x]) i 0 ≠ 0\ninst✝ : DecidablePred fun i => getD xs i 0 ≠ 0\n⊢ toFinsupp (xs ++ [x]) = toFinsupp xs + Finsupp.single (length xs) x",
"tactic": "classical rw [toFinsupp_append, toFinsupp_singleton, Finsupp.embDomain_single,\n addLeftEmbedding_apply, add_zero]"
},
{
"state_after": "no goals",
"state_before": "M : Type ?u.23867\ninst✝⁴ : Zero M\nl : List M\ninst✝³ : DecidablePred fun x => getD l x 0 ≠ 0\nn : ℕ\nR : Type u_1\ninst✝² : AddZeroClass R\nx : R\nxs : List R\ninst✝¹ : DecidablePred fun i => getD (xs ++ [x]) i 0 ≠ 0\ninst✝ : DecidablePred fun i => getD xs i 0 ≠ 0\n⊢ toFinsupp (xs ++ [x]) = toFinsupp xs + Finsupp.single (length xs) x",
"tactic": "rw [toFinsupp_append, toFinsupp_singleton, Finsupp.embDomain_single,\naddLeftEmbedding_apply, add_zero]"
}
] |
[
146,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.sInf_toSubalgebra
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set (IntermediateField F E)\n⊢ ↑(sInf S).toSubalgebra = ↑(sInf (toSubalgebra '' S))",
"tactic": "simp [Set.sUnion_image]"
}
] |
[
153,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
MultilinearMap.sub_apply
|
[] |
[
1117,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1116,
1
] |
Mathlib/Order/Ideal.lean
|
Order.Ideal.sup_mem_iff
|
[] |
[
353,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/MeasureTheory/Group/FundamentalDomain.lean
|
MeasureTheory.fundamentalFrontier_subset
|
[] |
[
570,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
569,
1
] |
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
StieltjesFunction.mono
|
[] |
[
226,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Topology/Homotopy/Path.lean
|
Path.Homotopy.eval_one
|
[
{
"state_after": "case a.h\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np₀ p₁ : Path x₀ x₁\nF : Homotopy p₀ p₁\nt : ↑I\n⊢ ↑(eval F 1) t = ↑p₁ t",
"state_before": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np₀ p₁ : Path x₀ x₁\nF : Homotopy p₀ p₁\n⊢ eval F 1 = p₁",
"tactic": "ext t"
},
{
"state_after": "no goals",
"state_before": "case a.h\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ x₂ x₃ : X\np₀ p₁ : Path x₀ x₁\nF : Homotopy p₀ p₁\nt : ↑I\n⊢ ↑(eval F 1) t = ↑p₁ t",
"tactic": "simp [eval]"
}
] |
[
95,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/CategoryTheory/Closed/Cartesian.lean
|
CategoryTheory.prod_map_pre_app_comp_ev
|
[] |
[
286,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.map_mapRange_eq_iff
|
[
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf✝ f : R →+* S₁\ng : S₁ → R\nhg : g 0 = 0\nφ : MvPolynomial σ S₁\n⊢ (∀ (m : σ →₀ ℕ), coeff m (↑(map f) (mapRange g hg φ)) = coeff m φ) ↔ ∀ (d : σ →₀ ℕ), ↑f (g (coeff d φ)) = coeff d φ",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf✝ f : R →+* S₁\ng : S₁ → R\nhg : g 0 = 0\nφ : MvPolynomial σ S₁\n⊢ ↑(map f) (mapRange g hg φ) = φ ↔ ∀ (d : σ →₀ ℕ), ↑f (g (coeff d φ)) = coeff d φ",
"tactic": "rw [MvPolynomial.ext_iff]"
},
{
"state_after": "case h\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf✝ f : R →+* S₁\ng : S₁ → R\nhg : g 0 = 0\nφ : MvPolynomial σ S₁\n⊢ ∀ (a : σ →₀ ℕ), coeff a (↑(map f) (mapRange g hg φ)) = coeff a φ ↔ ↑f (g (coeff a φ)) = coeff a φ",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf✝ f : R →+* S₁\ng : S₁ → R\nhg : g 0 = 0\nφ : MvPolynomial σ S₁\n⊢ (∀ (m : σ →₀ ℕ), coeff m (↑(map f) (mapRange g hg φ)) = coeff m φ) ↔ ∀ (d : σ →₀ ℕ), ↑f (g (coeff d φ)) = coeff d φ",
"tactic": "apply forall_congr'"
},
{
"state_after": "case h\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf✝ f : R →+* S₁\ng : S₁ → R\nhg : g 0 = 0\nφ : MvPolynomial σ S₁\nm : σ →₀ ℕ\n⊢ coeff m (↑(map f) (mapRange g hg φ)) = coeff m φ ↔ ↑f (g (coeff m φ)) = coeff m φ",
"state_before": "case h\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf✝ f : R →+* S₁\ng : S₁ → R\nhg : g 0 = 0\nφ : MvPolynomial σ S₁\n⊢ ∀ (a : σ →₀ ℕ), coeff a (↑(map f) (mapRange g hg φ)) = coeff a φ ↔ ↑f (g (coeff a φ)) = coeff a φ",
"tactic": "intro m"
},
{
"state_after": "case h\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf✝ f : R →+* S₁\ng : S₁ → R\nhg : g 0 = 0\nφ : MvPolynomial σ S₁\nm : σ →₀ ℕ\n⊢ ↑f (coeff m (mapRange g hg φ)) = coeff m φ ↔ ↑f (g (coeff m φ)) = coeff m φ",
"state_before": "case h\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf✝ f : R →+* S₁\ng : S₁ → R\nhg : g 0 = 0\nφ : MvPolynomial σ S₁\nm : σ →₀ ℕ\n⊢ coeff m (↑(map f) (mapRange g hg φ)) = coeff m φ ↔ ↑f (g (coeff m φ)) = coeff m φ",
"tactic": "rw [coeff_map]"
},
{
"state_after": "case h.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf✝ f : R →+* S₁\ng : S₁ → R\nhg : g 0 = 0\nφ : MvPolynomial σ S₁\nm : σ →₀ ℕ\n⊢ ↑f (coeff m (mapRange g hg φ)) = ↑f (g (coeff m φ))",
"state_before": "case h\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf✝ f : R →+* S₁\ng : S₁ → R\nhg : g 0 = 0\nφ : MvPolynomial σ S₁\nm : σ →₀ ℕ\n⊢ ↑f (coeff m (mapRange g hg φ)) = coeff m φ ↔ ↑f (g (coeff m φ)) = coeff m φ",
"tactic": "apply eq_iff_eq_cancel_right.mpr"
},
{
"state_after": "no goals",
"state_before": "case h.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf✝ f : R →+* S₁\ng : S₁ → R\nhg : g 0 = 0\nφ : MvPolynomial σ S₁\nm : σ →₀ ℕ\n⊢ ↑f (coeff m (mapRange g hg φ)) = ↑f (g (coeff m φ))",
"tactic": "rfl"
}
] |
[
1384,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1378,
1
] |
Mathlib/LinearAlgebra/PiTensorProduct.lean
|
PiTensorProduct.smul_tprodCoeff'
|
[] |
[
244,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
MeasurableSpace.map_comp
|
[] |
[
96,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snormEssSup_zero
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.920015\nF : Type u_2\nG : Type ?u.920021\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\n⊢ essSup (fun x => ⊥) μ = ⊥",
"state_before": "α : Type u_1\nE : Type ?u.920015\nF : Type u_2\nG : Type ?u.920021\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\n⊢ snormEssSup 0 μ = 0",
"tactic": "simp_rw [snormEssSup, Pi.zero_apply, nnnorm_zero, ENNReal.coe_zero, ← ENNReal.bot_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.920015\nF : Type u_2\nG : Type ?u.920021\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\n⊢ essSup (fun x => ⊥) μ = ⊥",
"tactic": "exact essSup_const_bot"
}
] |
[
200,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.val_zero
|
[] |
[
68,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.RingEquiv.bot_maximal_iff
|
[] |
[
1763,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1760,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
UniqueFactorizationMonoid.prime_of_factor
|
[
{
"state_after": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\na x : α\nhx : x ∈ factors a\nane0 : a ≠ 0\n⊢ Prime x",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\na x : α\nhx : x ∈ factors a\n⊢ Prime x",
"tactic": "have ane0 := ne_zero_of_mem_factors hx"
},
{
"state_after": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\na x : α\nane0 : a ≠ 0\nhx : x ∈ Classical.choose (_ : ∃ f, (∀ (b : α), b ∈ f → Prime b) ∧ Multiset.prod f ~ᵤ a)\n⊢ Prime x",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\na x : α\nhx : x ∈ factors a\nane0 : a ≠ 0\n⊢ Prime x",
"tactic": "rw [factors, dif_neg ane0] at hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\na x : α\nane0 : a ≠ 0\nhx : x ∈ Classical.choose (_ : ∃ f, (∀ (b : α), b ∈ f → Prime b) ∧ Multiset.prod f ~ᵤ a)\n⊢ Prime x",
"tactic": "exact (Classical.choose_spec (UniqueFactorizationMonoid.exists_prime_factors a ane0)).1 x hx"
}
] |
[
466,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
463,
1
] |
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