module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 416,
"column": 52
} | {
"line": 416,
"column": 66
} | {
"line": 416,
"column": 66
} | [
{
"pp": "⊢ Complex.re 1 = 1",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"id",
"Complex.re",
"Real.instOne",
"One.toOfNat1",
"Complex",
"OfNat.ofNat",
"Complex.one_re",
"Eq",
"Complex.i... | [
"⊢ 1 = 1"
] | Complex.one_re | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 450,
"column": 2
} | {
"line": 451,
"column": 31
} | {
"line": 452,
"column": 2
} | [
{
"pp": "s : ℝ\nhs : 0 < s\nthis : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0\n⊢ 0 < volume ((Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0)",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"Set.Io... | [
"case hf\ns : ℝ\nhs : 0 < s\nthis : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0\n⊢ 0 ≤ᵐ[volume.restrict (Ioi 0)] fun x ↦ rexp (-x) * x ^ (s - 1)",
"case hfi\ns : ℝ\nhs : 0 < s\nthis : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0\n⊢ IntegrableOn (fun x ↦ rexp (-x) * x ^... | · rw [this, volume_Ioi, ← ENNReal.ofReal_zero]
exact ENNReal.ofReal_lt_top | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 63
} | {
"line": 133,
"column": 0
} | [
{
"pp": "b : ℝ\nhb : 0 < b\ns : ℝ\nhs : -1 < s\nthis : MeasurableSet (Ioi 0)\nx : ℝ\nhx : x ∈ Ioi 0\nh'x : 0 ≤ x\n⊢ |(-x) ^ s| ≤ x ^ s",
"ppTerm": "?m.289",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"abs_neg",
"Real.instPow",
... | [] | simpa [abs_of_nonneg h'x] using abs_rpow_le_abs_rpow (-x) s | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 336,
"column": 2
} | {
"line": 342,
"column": 8
} | {
"line": 344,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nb : ℝ\nhb : 0 < b\n⊢ ∫ (v : V), rexp (-b * ‖v‖ ^ 2) = (π / b) ^ (↑(Module.finrank ℝ V) / 2)",
"ppTerm": "?m.66",
"assigned": true,
"... | [] | rw [← ofReal_inj]
convert! integral_cexp_neg_mul_sq_norm (show 0 < (b : ℂ).re from hb) (V := V)
· change ofRealLI (∫ (v : V), rexp (-b * ‖v‖ ^ 2)) = ∫ (v : V), cexp (-↑b * ↑‖v‖ ^ 2)
rw [← ofRealLI.integral_comp_comm]
simp [ofRealLI]
· rw [← ofReal_div, ofReal_cpow (by positivity)]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 336,
"column": 2
} | {
"line": 342,
"column": 8
} | {
"line": 344,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nb : ℝ\nhb : 0 < b\n⊢ ∫ (v : V), rexp (-b * ‖v‖ ^ 2) = (π / b) ^ (↑(Module.finrank ℝ V) / 2)",
"ppTerm": "?m.66",
"assigned": true,
"... | [] | rw [← ofReal_inj]
convert! integral_cexp_neg_mul_sq_norm (show 0 < (b : ℂ).re from hb) (V := V)
· change ofRealLI (∫ (v : V), rexp (-b * ‖v‖ ^ 2)) = ∫ (v : V), cexp (-↑b * ↑‖v‖ ^ 2)
rw [← ofRealLI.integral_comp_comm]
simp [ofRealLI]
· rw [← ofReal_div, ofReal_cpow (by positivity)]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.Convolution | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 55
} | {
"line": 63,
"column": 2
} | [
{
"pp": "case e_a\n𝕜 : Type u_1\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F₁\ninst✝⁸ : NormedAddCommGroup F₂\ninst✝⁷ : NormedAddCommGroup F₃\ninst✝⁶ : InnerProductSpace ℝ E\ninst✝⁵ : FiniteDimens... | [
"case e'_2.e'_7\n𝕜 : Type u_1\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F₁\ninst✝⁸ : NormedAddCommGroup F₂\ninst✝⁷ : NormedAddCommGroup F₃\ninst✝⁶ : InnerProductSpace ℝ E\ninst✝⁵ : FiniteDimensional ... | convert! integral_sub_right_eq_self _ x (μ := volume) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 39
} | {
"line": 264,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\n⊢ n > 0 → ∏ i ∈ n.divisors, f i = g n ↔\n n > 0 → (∏ i ∈ n.divisors, if h : 0 < i then Units.mk0 (f i) ⋯ else 1) = if h : 0 < n then Units.mk0 (g n) ⋯ else ... | [
"case refine_1\nR : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : n > 0\n⊢ ∏ i ∈ n.divisors, f i = g n ↔\n (∏ i ∈ n.divisors, if h : 0 < i then Units.mk0 (f i) ⋯ else 1) = if h : 0 < n then Units.mk0 (g n) ⋯ else 1"
] | refine imp_congr_right fun hn => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 39
} | {
"line": 264,
"column": 2
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\n⊢ (n > 0 →\n ∏ x ∈ n.divisorsAntidiagonal, (if h : 0 < x.2 then Units.mk0 (g x.2) ⋯ else 1) ^ μ x.1 =\n if h : 0 < n then Units.mk0 (f n) ⋯ else 1)... | [
"case refine_2\nR : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : n > 0\n⊢ (∏ x ∈ n.divisorsAntidiagonal, (if h : 0 < x.2 then Units.mk0 (g x.2) ⋯ else 1) ^ μ x.1 =\n if h : 0 < n then Units.mk0 (f n) ⋯ else 1) ↔\n ∏ x ∈ n.d... | refine imp_congr_right fun hn => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 347,
"column": 4
} | {
"line": 347,
"column": 39
} | {
"line": 348,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommGroupWithZero R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nf g : ℕ → R\nhf : ∀ n > 0, f n ≠ 0\nhg : ∀ n > 0, g n ≠ 0\nn : ℕ\n⊢ n > 0 → n ∈ s → ∏ i ∈ n.divisors, f i = g n ↔\n n > 0 →\n n ∈ s →\n (∏ i ∈ n.divisors, if h : 0 < i then Units... | [
"case refine_1\nR : Type u_1\ninst✝ : CommGroupWithZero R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nf g : ℕ → R\nhf : ∀ n > 0, f n ≠ 0\nhg : ∀ n > 0, g n ≠ 0\nn : ℕ\nhn : n > 0\n⊢ n ∈ s → ∏ i ∈ n.divisors, f i = g n ↔\n n ∈ s → (∏ i ∈ n.divisors, if h : 0 < i then Units.mk0 (f i) ⋯ else 1) = if h : 0 ... | refine imp_congr_right fun hn => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 347,
"column": 4
} | {
"line": 347,
"column": 39
} | {
"line": 348,
"column": 2
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : CommGroupWithZero R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nf g : ℕ → R\nhf : ∀ n > 0, f n ≠ 0\nhg : ∀ n > 0, g n ≠ 0\nn : ℕ\n⊢ (n > 0 →\n n ∈ s →\n ∏ x ∈ n.divisorsAntidiagonal, (if h : 0 < x.2 then Units.mk0 (g x.2) ⋯ else 1) ^ μ x.1 =\n ... | [
"case refine_2\nR : Type u_1\ninst✝ : CommGroupWithZero R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nf g : ℕ → R\nhf : ∀ n > 0, f n ≠ 0\nhg : ∀ n > 0, g n ≠ 0\nn : ℕ\nhn : n > 0\n⊢ (n ∈ s →\n ∏ x ∈ n.divisorsAntidiagonal, (if h : 0 < x.2 then Units.mk0 (g x.2) ⋯ else 1) ^ μ x.1 =\n if h : 0 < ... | refine imp_congr_right fun hn => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Fourier.PoissonSummation | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 38
} | {
"line": 170,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x ↦ |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a ↦ -a, continuous_toFun := ⋯ }) =O[atTop] fun x ↦ |x| ^ (-b)\nh2 :\n ((fun x ↦ ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun... | [
"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x ↦ |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a ↦ -a, continuous_toFun := ⋯ }) =O[atTop] fun x ↦ |x| ^ (-b)\nh2 :\n ((fun x ↦ ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a ↦ -a, con... | rw [ContinuousMap.restrict_apply_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 432,
"column": 2
} | {
"line": 433,
"column": 48
} | {
"line": 435,
"column": 0
} | [
{
"pp": "R : Type u_4\nk : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nhζ : IsPrimitiveRoot ζ k\nhk : 1 < k\n⊢ ∑ i ∈ range k, ζ ^ i = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
... | [] | refine eq_zero_of_ne_zero_of_mul_left_eq_zero (sub_ne_zero_of_ne (hζ.ne_one hk).symm) ?_
rw [mul_neg_geom_sum, hζ.pow_eq_one, sub_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 432,
"column": 2
} | {
"line": 433,
"column": 48
} | {
"line": 435,
"column": 0
} | [
{
"pp": "R : Type u_4\nk : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nhζ : IsPrimitiveRoot ζ k\nhk : 1 < k\n⊢ ∑ i ∈ range k, ζ ^ i = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
... | [] | refine eq_zero_of_ne_zero_of_mul_left_eq_zero (sub_ne_zero_of_ne (hζ.ne_one hk).symm) ?_
rw [mul_neg_geom_sum, hζ.pow_eq_one, sub_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 600,
"column": 4
} | {
"line": 600,
"column": 79
} | {
"line": 602,
"column": 0
} | [
{
"pp": "case neg.a\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nζ : R\nhζ : IsPrimitiveRoot ζ n\nα a : R\ne : α ^ n = a\nhn : n > 0\nhα : ¬α = 0\n⊢ (nthRoots n a).card ≤ (Multiset.map (fun x ↦ ζ ^ x * α) (Multiset.range n)).card",
"ppTerm": "?neg.a✝",
"assigned": true,
"usedConsta... | [] | simpa only [Multiset.card_map, Multiset.card_range] using card_nthRoots n a | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 600,
"column": 4
} | {
"line": 600,
"column": 79
} | {
"line": 602,
"column": 0
} | [
{
"pp": "case neg.a\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nζ : R\nhζ : IsPrimitiveRoot ζ n\nα a : R\ne : α ^ n = a\nhn : n > 0\nhα : ¬α = 0\n⊢ (nthRoots n a).card ≤ (Multiset.map (fun x ↦ ζ ^ x * α) (Multiset.range n)).card",
"ppTerm": "?neg.a✝",
"assigned": true,
"usedConsta... | [] | simpa only [Multiset.card_map, Multiset.card_range] using card_nthRoots n a | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 600,
"column": 4
} | {
"line": 600,
"column": 79
} | {
"line": 602,
"column": 0
} | [
{
"pp": "case neg.a\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nζ : R\nhζ : IsPrimitiveRoot ζ n\nα a : R\ne : α ^ n = a\nhn : n > 0\nhα : ¬α = 0\n⊢ (nthRoots n a).card ≤ (Multiset.map (fun x ↦ ζ ^ x * α) (Multiset.range n)).card",
"ppTerm": "?neg.a✝",
"assigned": true,
"usedConsta... | [] | simpa only [Multiset.card_map, Multiset.card_range] using card_nthRoots n a | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 484,
"column": 4
} | {
"line": 484,
"column": 53
} | {
"line": 485,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nhk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic m K = cyclotomic' m K\nζ : K\nhz : IsPrimitiveRoot ζ k\nhpos : k > 0\ni : ℕ\nhi : i ∈ k.properDivisors\n⊢ cyclotomic i K = cyclotomic' i K",
"ppTerm": "?m.73",
"assigned": tru... | [
"K : Type u_1\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nhk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic m K = cyclotomic' m K\nζ : K\nhz : IsPrimitiveRoot ζ k\nhpos : k > 0\ni : ℕ\nhi : i ∈ k.properDivisors\nd : ℕ\nhd : k = i * d\n⊢ cyclotomic i K = cyclotomic' i K"
] | obtain ⟨d, hd⟩ := (Nat.mem_properDivisors.1 hi).1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 134,
"column": 2
} | {
"line": 151,
"column": 53
} | {
"line": 153,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\n⊢ Function.Injective fun n ↦ cyclotomic n R",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"IsPrimitiveRoot.eq_orderOf",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"F... | [] | intro n m hnm
simp only at hnm
rcases eq_or_ne n 0 with (rfl | hzero)
· rw [cyclotomic_zero] at hnm
replace hnm := congr_arg natDegree hnm
rwa [natDegree_one, natDegree_cyclotomic, eq_comm, Nat.totient_eq_zero, eq_comm] at hnm
· haveI := NeZero.mk hzero
rw [← map_cyclotomic_int _ R, ← map_cyclotomic... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 134,
"column": 2
} | {
"line": 151,
"column": 53
} | {
"line": 153,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\n⊢ Function.Injective fun n ↦ cyclotomic n R",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"IsPrimitiveRoot.eq_orderOf",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"F... | [] | intro n m hnm
simp only at hnm
rcases eq_or_ne n 0 with (rfl | hzero)
· rw [cyclotomic_zero] at hnm
replace hnm := congr_arg natDegree hnm
rwa [natDegree_one, natDegree_cyclotomic, eq_comm, Nat.totient_eq_zero, eq_comm] at hnm
· haveI := NeZero.mk hzero
rw [← map_cyclotomic_int _ R, ← map_cyclotomic... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.RiemannLebesgueLemma | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 64
} | {
"line": 264,
"column": 2
} | [
{
"pp": "E : Type u_1\nV : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℂ E\nf : V → E\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : TopologicalSpace V\ninst✝⁷ : IsTopologicalAddGroup V\ninst✝⁶ : T2Space V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : Module ℝ V\ninst✝² : ContinuousSMul ℝ... | [
"E : Type u_1\nV : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℂ E\nf : V → E\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : TopologicalSpace V\ninst✝⁷ : IsTopologicalAddGroup V\ninst✝⁶ : T2Space V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : Module ℝ V\ninst✝² : ContinuousSMul ℝ V\ninst✝¹ :... | have : (μ.map Aₘ).IsAddHaarMeasure := A.isAddHaarMeasure_map _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 40
} | {
"line": 104,
"column": 4
} | [
{
"pp": "case hb\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ ran... | [
"case hb.ha\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^... | apply mul_nonpos_of_nonneg_of_nonpos | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 41
} | {
"line": 113,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Nat.Prime p\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn m : ℕ\nhmn : m ≤ n\nh : Irreducible (cyclotomic (p ^ n) R)\n⊢ Irreducible (cyclotomic (p ^ m) R)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Nat.instMonoid",
"Polynomial.cyclotomic",... | [
"case inl\np : ℕ\nhp : Nat.Prime p\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nh : Irreducible (cyclotomic (p ^ n) R)\nhmn : 0 ≤ n\n⊢ Irreducible (cyclotomic (p ^ 0) R)",
"case inr\np : ℕ\nhp : Nat.Prime p\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn m : ℕ\nhmn : m ≤ n\nh : Irreduc... | rcases m.eq_zero_or_pos with (rfl | hm) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 150,
"column": 8
} | {
"line": 150,
"column": 24
} | {
"line": 151,
"column": 8
} | [
{
"pp": "p n : ℕ\ninst✝¹² : NeZero n\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC✝ : Type w\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsCyclotomicExtension {n} A B\ninst✝⁷ : Field K\ninst✝⁶ : CommRing L\ninst✝⁵ : IsDomain L\ninst✝⁴ : Algebra K L\ninst✝³ : IsCyclotomicExte... | [
"p n : ℕ\ninst✝¹² : NeZero n\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC✝ : Type w\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsCyclotomicExtension {n} A B\ninst✝⁷ : Field K\ninst✝⁶ : CommRing L\ninst✝⁵ : IsDomain L\ninst✝⁴ : Algebra K L\ninst✝³ : IsCyclotomicExtension {n} K ... | refine ⟨x.1, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 158,
"column": 8
} | {
"line": 158,
"column": 24
} | {
"line": 159,
"column": 8
} | [
{
"pp": "p n : ℕ\ninst✝¹² : NeZero n\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC✝ : Type w\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsCyclotomicExtension {n} A B\ninst✝⁷ : Field K\ninst✝⁶ : CommRing L\ninst✝⁵ : IsDomain L\ninst✝⁴ : Algebra K L\ninst✝³ : IsCyclotomicExte... | [
"p n : ℕ\ninst✝¹² : NeZero n\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC✝ : Type w\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsCyclotomicExtension {n} A B\ninst✝⁷ : Field K\ninst✝⁶ : CommRing L\ninst✝⁵ : IsDomain L\ninst✝⁴ : Algebra K L\ninst✝³ : IsCyclotomicExtension {n} K ... | refine ⟨x.1, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 107,
"column": 2
} | {
"line": 108,
"column": 39
} | {
"line": 109,
"column": 2
} | [
{
"pp": "case succ\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n\nnat_degr... | [
"case succ.refine_1\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n\nnat_degree_... | refine Subring.closure_induction ?_ ?_ ?_ ?_ ?_ ?_ hx
<;> simp_rw [mem_rootSet_of_ne aux] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 126,
"column": 4
} | {
"line": 128,
"column": 9
} | {
"line": 128,
"column": 9
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nz : ↥(rootsOfUnity n ℂ)\n⊢ (rootsOfUnitytoCircle n).toHomUnits z ∈ rootsOfUnity n Circle",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"Subgroup.instSubg... | [] | rw [mem_rootsOfUnity', MonoidHom.coe_toHomUnits, ← map_pow, ← (rootsOfUnitytoCircle n).map_one]
congr
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 126,
"column": 4
} | {
"line": 128,
"column": 9
} | {
"line": 128,
"column": 9
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nz : ↥(rootsOfUnity n ℂ)\n⊢ (rootsOfUnitytoCircle n).toHomUnits z ∈ rootsOfUnity n Circle",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"Subgroup.instSubg... | [] | rw [mem_rootsOfUnity', MonoidHom.coe_toHomUnits, ← map_pow, ← (rootsOfUnitytoCircle n).map_one]
congr
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 553,
"column": 2
} | {
"line": 562,
"column": 36
} | {
"line": 564,
"column": 0
} | [
{
"pp": "M : Type u_4\nR : Type u_5\ninst✝³ : CommMonoid M\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nχ : MulChar M R\n⊢ χ.IsQuadratic ↔ χ ^ 2 = 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"one_pow",
"sq_eq_one_iff",
"Units.val",
"E... | [] | refine ⟨fun h ↦ ext (fun x ↦ ?_), fun h x ↦ ?_⟩
· rw [one_apply_coe, χ.pow_apply_coe]
rcases h x with H | H | H
· exact (not_isUnit_zero <| H ▸ IsUnit.map χ <| x.isUnit).elim
· simp only [H, one_pow]
· simp only [H, even_two, Even.neg_pow, one_pow]
· by_cases hx : IsUnit x
· refine .inr <| sq_eq... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 553,
"column": 2
} | {
"line": 562,
"column": 36
} | {
"line": 564,
"column": 0
} | [
{
"pp": "M : Type u_4\nR : Type u_5\ninst✝³ : CommMonoid M\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nχ : MulChar M R\n⊢ χ.IsQuadratic ↔ χ ^ 2 = 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"one_pow",
"sq_eq_one_iff",
"Units.val",
"E... | [] | refine ⟨fun h ↦ ext (fun x ↦ ?_), fun h x ↦ ?_⟩
· rw [one_apply_coe, χ.pow_apply_coe]
rcases h x with H | H | H
· exact (not_isUnit_zero <| H ▸ IsUnit.map χ <| x.isUnit).elim
· simp only [H, one_pow]
· simp only [H, even_two, Even.neg_pow, one_pow]
· by_cases hx : IsUnit x
· refine .inr <| sq_eq... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 589,
"column": 2
} | {
"line": 589,
"column": 78
} | {
"line": 591,
"column": 0
} | [
{
"pp": "case intro\nM : Type u_1\ninst✝² : CommMonoid M\nR : Type u_2\ninst✝¹ : CommMonoidWithZero R\ninst✝ : Finite Mˣ\nχ : MulChar M R\nval✝ : Fintype Mˣ\n⊢ IsOfFinOrder χ",
"ppTerm": "?intro",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass... | [] | exact isOfFinOrder_iff_pow_eq_one.2 ⟨_, Fintype.card_pos, χ.pow_card_eq_one⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 549,
"column": 2
} | {
"line": 564,
"column": 45
} | {
"line": 566,
"column": 0
} | [
{
"pp": "S : Set ℕ\nK : Type w\nL : Type z\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : IsCyclotomicExtension S K L\nM : Type u_1\ninst✝² : Field M\ninst✝¹ : Algebra K M\ninst✝ : IsSepClosed M\n⊢ Nonempty (L ≃ₐ[K] ↥(IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}))",
"ppTe... | [] | have := isSeparable S K L
let i : L →ₐ[K] M := IsSepClosed.lift
refine ⟨(show L ≃ₐ[K] i.fieldRange from AlgEquiv.ofInjectiveField i).trans
(IntermediateField.equivOfEq ?_)⟩
have htop : IntermediateField.adjoin K {x : L | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1} = ⊤ :=
IntermediateField.adjoin_eq_top_of_algebra K _ ((i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 549,
"column": 2
} | {
"line": 564,
"column": 45
} | {
"line": 566,
"column": 0
} | [
{
"pp": "S : Set ℕ\nK : Type w\nL : Type z\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : IsCyclotomicExtension S K L\nM : Type u_1\ninst✝² : Field M\ninst✝¹ : Algebra K M\ninst✝ : IsSepClosed M\n⊢ Nonempty (L ≃ₐ[K] ↥(IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}))",
"ppTe... | [] | have := isSeparable S K L
let i : L →ₐ[K] M := IsSepClosed.lift
refine ⟨(show L ≃ₐ[K] i.fieldRange from AlgEquiv.ofInjectiveField i).trans
(IntermediateField.equivOfEq ?_)⟩
have htop : IntermediateField.adjoin K {x : L | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1} = ⊤ :=
IntermediateField.adjoin_eq_top_of_algebra K _ ((i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 589,
"column": 4
} | {
"line": 589,
"column": 16
} | {
"line": 590,
"column": 4
} | [
{
"pp": "case right.mul\nS : Set ℕ\nK : Type w\nL : Type z\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension S K L\ni : L ≃ₐ[K] ↥(IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1})\nf : AlgebraicClosure K →ₐ[K] AlgebraicClosure K\ny✝ x y : AlgebraicClosure K\nhx ... | [
"case right.mul\nS : Set ℕ\nK : Type w\nL : Type z\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension S K L\ni : L ≃ₐ[K] ↥(IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1})\nf : AlgebraicClosure K →ₐ[K] AlgebraicClosure K\ny✝ x y : AlgebraicClosure K\nhx : x ∈ Interm... | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 273,
"column": 4
} | {
"line": 274,
"column": 48
} | {
"line": 275,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\n⊢ χ.conductor = 0 → n = 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"DirichletCharacter.conductor",
"Nat.instMulZeroClass",
"NeZero.mk",
"id",
"Ne"... | [] | contrapose!
exact fun h ↦ @conductor_ne_zero _ _ _ χ ⟨h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 273,
"column": 4
} | {
"line": 274,
"column": 48
} | {
"line": 275,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\n⊢ χ.conductor = 0 → n = 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"DirichletCharacter.conductor",
"Nat.instMulZeroClass",
"NeZero.mk",
"id",
"Ne"... | [] | contrapose!
exact fun h ↦ @conductor_ne_zero _ _ _ χ ⟨h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Hofer | {
"line": 67,
"column": 48
} | {
"line": 70,
"column": 22
} | {
"line": 71,
"column": 10
} | [
{
"pp": "X : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nx : X\nε : ℝ\nε_pos : 0 < ε\nϕ : X → ℝ\ncont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ\nnonneg : ∀ (y : X), 0 ≤ ϕ y\nreformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'\nthis : No... | [] | by
rw [Finset.sum_mul]
simp
field_simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 29
} | {
"line": 233,
"column": 2
} | [
{
"pp": "case e'_4\nι : Type u_1\nA : ι → Type u_2\ninst✝³ : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝² : DecidableEq ι\np : ℝ\ninst✝¹ : Fintype ι\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nhp₀ : 0 ≤ p\nhp : (↑#ι - 1) * p ≤ 1\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\ns : Finset ι\ni ... | [] | exact mem_insert_self i s | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 237,
"column": 4
} | {
"line": 237,
"column": 46
} | {
"line": 238,
"column": 4
} | [
{
"pp": "case hbc\nι : Type u_1\nA : ι → Type u_2\ninst✝³ : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝² : DecidableEq ι\np : ℝ\ninst✝¹ : Fintype ι\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nhp₀ : 0 ≤ p\nhp : (↑#ι - 1) * p ≤ 1\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\ns : Finset ι\ni :... | [
"case hbc\nι : Type u_1\nA : ι → Type u_2\ninst✝³ : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝² : DecidableEq ι\np : ℝ\ninst✝¹ : Fintype ι\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nhp₀ : 0 ≤ p\nhp : (↑#ι - 1) * p ≤ 1\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\ns : Finset ι\ni : ι\nhi : i ∉... | suffices (s.card : ℝ) + 1 ≤ #ι by linarith | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.InnerProductSpace.LinearPMap | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 69
} | {
"line": 334,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\nT : E →ₗ.[𝕜] F\ninst✝ : CompleteSpace E\nhT : Dense ↑T.domain\n⊢ T†.IsClosed",
"ppTerm": "?m.47",
"assi... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\nT : E →ₗ.[𝕜] F\ninst✝ : CompleteSpace E\nhT : Dense ↑T.domain\n⊢ _root_.IsClosed\n ↑(Submodule.map (↑(WithLp.linearEquiv... | rw [IsClosed, adjoint_graph_eq_graph_adjoint hT, Submodule.adjoint] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 160,
"column": 25
} | {
"line": 160,
"column": 38
} | {
"line": 160,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx x' : E\ny y' : F\n⊢ ‖x ⊗ₜ[𝕜] y - x' ⊗ₜ[𝕜] y'‖₊ ≤ ‖x‖₊ * ‖y‖₊ + ‖x'‖₊ * ‖y'‖₊",
"ppTerm": "?m.55",
"as... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx x' : E\ny y' : F\n⊢ ‖x ⊗ₜ[𝕜] y‖₊ + ‖x' ⊗ₜ[𝕜] y'‖₊ ≤ ‖x‖₊ * ‖y‖₊ + ‖x'‖₊ * ‖y'‖₊"
] | nnnorm_sub_le | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.OfNorm | {
"line": 186,
"column": 37
} | {
"line": 186,
"column": 77
} | {
"line": 187,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nhI : ¬I = 0\nx y : E\nhI' : I * I = -1\nh₁ : ‖-x - y‖ = ‖x + y‖\n⊢ ‖-x + y‖ = ‖x - y‖",
"ppTerm": "?m.134",
"assigned": true,
"usedConstants": [
... | [] | rw [← neg_sub, norm_neg, sub_eq_neg_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Reproducing | {
"line": 256,
"column": 19
} | {
"line": 258,
"column": 80
} | {
"line": 260,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace 𝕜 V\nH : Type u_4\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : RKHS 𝕜 H X V\ninst✝² : CompleteSpace H\ninst✝¹ : CompleteSpace V\nK : Matrix X X (V →L[�... | [] | by
refine UniformSpace.Completion.denseRange_coe.eq_of_inner_left 𝕜 fun f ↦ ?_
simp [inner_smul_left, inner_H₀_def, Finsupp.mul_sum, ← mul_assoc, mul_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 711,
"column": 2
} | {
"line": 711,
"column": 62
} | {
"line": 712,
"column": 2
} | [
{
"pp": "F : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\nμ : Measure E\ninst✝¹ : μ.IsAddHaarMeasure\ninst✝ : FiniteDimensional ℝ F\nu :... | [
"F : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\nμ : Measure E\ninst✝¹ : μ.IsAddHaarMeasure\ninst✝ : FiniteDimensional ℝ F\nu : E → F\ns : ... | refine eLpNorm_le_eLpNorm_fderiv_of_le μ hu h2u hp h2p ?_ hs | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 138,
"column": 32
} | {
"line": 138,
"column": 77
} | {
"line": 138,
"column": 77
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nh : ⟪x, y⟫ = 0\n⊢ |o.volumeForm ![x, y]| = ‖x‖ * ‖y‖",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"AlternatingMap",
"No... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nh : ⟪x, y⟫ = 0\n⊢ ∏ i, ‖![x, y] i‖ = ‖x‖ * ‖y‖",
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Ori... | o.abs_volumeForm_apply_of_pairwise_orthogonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 214,
"column": 50
} | {
"line": 214,
"column": 62
} | {
"line": 214,
"column": 62
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx : E\nK : Submodule ℝ E := ℝ ∙ x\nthis : Nontrivial ↥Kᗮ\nw : ↥Kᗮ\nhw₀ : w ≠ 0\nhw' : ⟪x, ↑w⟫ = 0\nhw : ↑w ≠ 0\n⊢ 0 < ‖↑w‖",
"ppTerm": "?m.281",
"assigned": ... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx : E\nK : Submodule ℝ E := ℝ ∙ x\nthis : Nontrivial ↥Kᗮ\nw : ↥Kᗮ\nhw₀ : w ≠ 0\nhw' : ⟪x, ↑w⟫ = 0\nhw : ↑w ≠ 0\n⊢ ↑w ≠ 0"
] | norm_pos_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 97
} | {
"line": 77,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : TopologicalSpace E\ninst✝⁶... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : IsTopolog... | rw [← Set.image_subset_image_iff (toWeakSpace 𝕜 F).injective, h_convex.toWeakSpace_closure 𝕜] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 123,
"column": 4
} | {
"line": 125,
"column": 37
} | {
"line": 125,
"column": 37
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝²⁵ : RCLike 𝕜\ninst✝²⁴ : AddCommGroup E\ninst✝²³ : Module 𝕜 E\ninst✝²² : AddCommGroup F\ninst✝²¹ : Module 𝕜 F\ninst✝²⁰ : Module ℝ E\ninst✝¹⁹ : IsScalarTower ℝ 𝕜 E\ninst✝¹⁸ : Module ℝ F\ninst✝¹⁷ : IsScalarTower ℝ 𝕜 F\ninst✝¹⁶ : TopologicalSpace E\nins... | [] | by_contra hne
obtain ⟨f, hf⟩ := SeparatingDual.exists_separating_of_ne (R := R) hne
exact hf (DFunLike.congr_fun h f) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 123,
"column": 4
} | {
"line": 125,
"column": 37
} | {
"line": 125,
"column": 37
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝²⁵ : RCLike 𝕜\ninst✝²⁴ : AddCommGroup E\ninst✝²³ : Module 𝕜 E\ninst✝²² : AddCommGroup F\ninst✝²¹ : Module 𝕜 F\ninst✝²⁰ : Module ℝ E\ninst✝¹⁹ : IsScalarTower ℝ 𝕜 E\ninst✝¹⁸ : Module ℝ F\ninst✝¹⁷ : IsScalarTower ℝ 𝕜 F\ninst✝¹⁶ : TopologicalSpace E\nins... | [] | by_contra hne
obtain ⟨f, hf⟩ := SeparatingDual.exists_separating_of_ne (R := R) hne
exact hf (DFunLike.congr_fun h f) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.WeakOperatorTopology | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 60
} | {
"line": 335,
"column": 0
} | [
{
"pp": "𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedField 𝕜₁\ninst✝⁸ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : AddCommGroup F\ninst✝³ : TopologicalSpace F\ninst✝² : Module 𝕜₂ F\ninst✝¹ : IsTopologic... | [] | refine (continuous_pi_iff.mp continuous_inducingFn) ⟨x, y⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.LocallyConvex.WeakOperatorTopology | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 60
} | {
"line": 335,
"column": 0
} | [
{
"pp": "𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedField 𝕜₁\ninst✝⁸ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : AddCommGroup F\ninst✝³ : TopologicalSpace F\ninst✝² : Module 𝕜₂ F\ninst✝¹ : IsTopologic... | [] | refine (continuous_pi_iff.mp continuous_inducingFn) ⟨x, y⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.WeakOperatorTopology | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 60
} | {
"line": 335,
"column": 0
} | [
{
"pp": "𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedField 𝕜₁\ninst✝⁸ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : AddCommGroup F\ninst✝³ : TopologicalSpace F\ninst✝² : Module 𝕜₂ F\ninst✝¹ : IsTopologic... | [] | refine (continuous_pi_iff.mp continuous_inducingFn) ⟨x, y⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MellinTransform | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 57
} | {
"line": 154,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\ns : ℂ\na : ℝ\nha : 0 < a\n⊢ mellin (fun t ↦ f (t * a)) s = ↑a ^ (-s) • mellin f s",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
... | [] | simpa only [mul_comm] using mellin_comp_mul_left f s ha | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.MellinTransform | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 57
} | {
"line": 154,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\ns : ℂ\na : ℝ\nha : 0 < a\n⊢ mellin (fun t ↦ f (t * a)) s = ↑a ^ (-s) • mellin f s",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
... | [] | simpa only [mul_comm] using mellin_comp_mul_left f s ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MellinTransform | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 57
} | {
"line": 154,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\ns : ℂ\na : ℝ\nha : 0 < a\n⊢ mellin (fun t ↦ f (t * a)) s = ↑a ^ (-s) • mellin f s",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
... | [] | simpa only [mul_comm] using mellin_comp_mul_left f s ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 102,
"column": 10
} | {
"line": 102,
"column": 63
} | {
"line": 103,
"column": 8
} | [
{
"pp": "z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nx : ℝ\na✝ : x ∈ uIcc 0 (π / 2)\nc : HasDerivAt ((fun x ↦ x ^ (n - 1)) ∘ Complex.cos) (↑(n - 1) * Complex.cos ↑x ^ (n - 1 - 1) * -Complex.sin ↑x) ↑x\n⊢ ↑(n - 1) = ↑n - 1",
"ppTerm": "?m.459",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Na... | [] | rw [Nat.cast_sub (one_le_two.trans hn), Nat.cast_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MellinTransform | {
"line": 377,
"column": 8
} | {
"line": 380,
"column": 60
} | {
"line": 381,
"column": 6
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x ↦ x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\nhs_bot : b < s.re\nF : ℂ → ℝ → E := fun z t ↦ ↑t... | [] | simp_rw [mul_comm]
refine hfc.norm.mul_continuousOn ?_ isOpen_Ioi.isLocallyClosed
refine Continuous.comp_continuousOn _root_.continuous_abs (continuousOn_log.mono ?_)
exact subset_compl_singleton_iff.mpr self_notMem_Ioi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MellinTransform | {
"line": 377,
"column": 8
} | {
"line": 380,
"column": 60
} | {
"line": 381,
"column": 6
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x ↦ x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\nhs_bot : b < s.re\nF : ℂ → ℝ → E := fun z t ↦ ↑t... | [] | simp_rw [mul_comm]
refine hfc.norm.mul_continuousOn ?_ isOpen_Ioi.isLocallyClosed
refine Continuous.comp_continuousOn _root_.continuous_abs (continuousOn_log.mono ?_)
exact subset_compl_singleton_iff.mpr self_notMem_Ioi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 109,
"column": 80
} | {
"line": 109,
"column": 93
} | {
"line": 109,
"column": 93
} | [
{
"pp": "s t a b : ℝ\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))\ne : (1 / a).HolderConjugate (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x\nposf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 109,
"column": 80
} | {
"line": 109,
"column": 93
} | {
"line": 109,
"column": 93
} | [
{
"pp": "s t a b : ℝ\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))\ne : (1 / a).HolderConjugate (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x\nposf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 22
} | {
"line": 122,
"column": 0
} | [
{
"pp": "case hx\nx y a : ℝ\nha : 0 < a\nhb : 0 < 1 - a\nhab : a + (1 - a) = 1\nhx : 0 < x\nhy : 0 < y\n⊢ Γ x ^ a ≠ 0",
"ppTerm": "?hx✝",
"assigned": true,
"usedConstants": [
"Real.Gamma_pos_of_pos",
"Real.instPow",
"Real.partialOrder",
"Real.rpow_pos_of_pos",
"Real",
... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 161,
"column": 4
} | {
"line": 165,
"column": 50
} | {
"line": 167,
"column": 0
} | [
{
"pp": "case succ\nf : ℝ → ℝ\nx : ℝ\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nn : ℕ\nhn : f (x + ↑n) = f x + ∑ m ∈ Finset.range n, log (x + ↑m)\n⊢ f (x + ↑(n + 1)) = f x + ∑ m ∈ Finset.range (n + 1), log (x + ↑m)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
... | [] | have : x + n.succ = x + n + 1 := by push_cast; ring
rw [this, hf_feq, hn]
· rw [Finset.range_add_one, Finset.sum_insert Finset.notMem_range_self]
abel
· linarith [(Nat.cast_nonneg n : 0 ≤ (n : ℝ))] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 161,
"column": 4
} | {
"line": 165,
"column": 50
} | {
"line": 167,
"column": 0
} | [
{
"pp": "case succ\nf : ℝ → ℝ\nx : ℝ\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nn : ℕ\nhn : f (x + ↑n) = f x + ∑ m ∈ Finset.range n, log (x + ↑m)\n⊢ f (x + ↑(n + 1)) = f x + ∑ m ∈ Finset.range (n + 1), log (x + ↑m)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
... | [] | have : x + n.succ = x + n + 1 := by push_cast; ring
rw [this, hf_feq, hn]
· rw [Finset.range_add_one, Finset.sum_insert Finset.notMem_range_self]
abel
· linarith [(Nat.cast_nonneg n : 0 ≤ (n : ℝ))] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 158,
"column": 2
} | {
"line": 165,
"column": 50
} | {
"line": 167,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\nx : ℝ\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nn : ℕ\n⊢ f (x + ↑n) = f x + ∑ m ∈ Finset.range n, log (x + ↑m)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.... | [] | induction n with
| zero => simp
| succ n hn =>
have : x + n.succ = x + n + 1 := by push_cast; ring
rw [this, hf_feq, hn]
· rw [Finset.range_add_one, Finset.sum_insert Finset.notMem_range_self]
abel
· linarith [(Nat.cast_nonneg n : 0 ≤ (n : ℝ))] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 158,
"column": 2
} | {
"line": 165,
"column": 50
} | {
"line": 167,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\nx : ℝ\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nn : ℕ\n⊢ f (x + ↑n) = f x + ∑ m ∈ Finset.range n, log (x + ↑m)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.... | [] | induction n with
| zero => simp
| succ n hn =>
have : x + n.succ = x + n + 1 := by push_cast; ring
rw [this, hf_feq, hn]
· rw [Finset.range_add_one, Finset.sum_insert Finset.notMem_range_self]
abel
· linarith [(Nat.cast_nonneg n : 0 ≤ (n : ℝ))] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 158,
"column": 2
} | {
"line": 165,
"column": 50
} | {
"line": 167,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\nx : ℝ\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nn : ℕ\n⊢ f (x + ↑n) = f x + ∑ m ∈ Finset.range n, log (x + ↑m)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.... | [] | induction n with
| zero => simp
| succ n hn =>
have : x + n.succ = x + n + 1 := by push_cast; ring
rw [this, hf_feq, hn]
· rw [Finset.range_add_one, Finset.sum_insert Finset.notMem_range_self]
abel
· linarith [(Nat.cast_nonneg n : 0 ≤ (n : ℝ))] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 256,
"column": 26
} | {
"line": 256,
"column": 36
} | {
"line": 256,
"column": 37
} | [
{
"pp": "s : ℂ\nhs : 0 < s.re\nn : ℕ\nhn : n ≠ 0\nthis✝ : ∀ (x : ℝ), x = x / ↑n * ↑n\nthis :\n ∫ (x : ℝ) in 0..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =\n ↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)\n⊢ ↑n ^ s * ∫ (x : ℝ) in 0..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (↑n + 1 - 1) ... | [
"s : ℂ\nhs : 0 < s.re\nn : ℕ\nhn : n ≠ 0\nthis✝ : ∀ (x : ℝ), x = x / ↑n * ↑n\nthis :\n ∫ (x : ℝ) in 0..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =\n ↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)\n⊢ ↑n ^ s * ∫ (x : ℝ) in 0..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (↑n + 1 - 1) =\n ↑↑n *... | real_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 33
} | {
"line": 253,
"column": 2
} | [
{
"pp": "f : ℝ → ℝ\nx : ℝ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nm : ℕ\n⊢ ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))",
"ppTerm": "?m.264",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_p... | [] | induction m generalizing x with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 35
} | {
"line": 411,
"column": 36
} | [
{
"pp": "s : ℝ\nhs : s ∈ Ioi 0\nh1 : √π ≠ 0\nh2 : Γ (s / 2) ≠ 0\nh3 : Γ (s / 2 + 1 / 2) ≠ 0\nh4 : 2 ^ (s - 1) ≠ 0\n⊢ log (Γ (s / 2) * Γ (s / 2 + 1 / 2) * 2 ^ (s - 1)) - log √π =\n (fun s ↦ log (Γ (s / 2)) + log (Γ (s / 2 + 1 / 2)) + s * log 2 - log (2 * √π)) s",
"ppTerm": "?m.203",
"assigned": true,
... | [
"s : ℝ\nhs : s ∈ Ioi 0\nh1 : √π ≠ 0\nh2 : Γ (s / 2) ≠ 0\nh3 : Γ (s / 2 + 1 / 2) ≠ 0\nh4 : 2 ^ (s - 1) ≠ 0\n⊢ log (Γ (s / 2) * Γ (s / 2 + 1 / 2)) + log (2 ^ (s - 1)) - log √π =\n (fun s ↦ log (Γ (s / 2)) + log (Γ (s / 2 + 1 / 2)) + s * log 2 - log (2 * √π)) s"
] | log_mul (mul_ne_zero h2 h3) h4, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Affine.MazurUlam | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 20
} | {
"line": 57,
"column": 4
} | [
{
"pp": "E : Type u_1\nPE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : MetricSpace PE\ninst✝ : NormedAddTorsor E PE\nx y : PE\nz : PE := midpoint ℝ x y\ns : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}\nthis : Nonempty ↑s\n⊢ ∀ (a : PE ≃ᵢ PE) (b : a ∈ s), dist (↑⟨a, b⟩ z) z ≤ dist x... | [
"E : Type u_1\nPE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : MetricSpace PE\ninst✝ : NormedAddTorsor E PE\nx y : PE\nz : PE := midpoint ℝ x y\ns : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}\nthis : Nonempty ↑s\ne : PE ≃ᵢ PE\nhx : e x = x\nright✝ : e y = y\n⊢ dist (↑⟨e, ⋯⟩ z) z ≤ di... | rintro e ⟨hx, _⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 408,
"column": 4
} | {
"line": 408,
"column": 11
} | {
"line": 409,
"column": 4
} | [
{
"pp": "case pos\nz : ℂ\npi_ne : ↑π ≠ 0\nk : ℤ\nhk : z = -↑k\n⊢ Gamma (-↑k) * Gamma (1 - -↑k) = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Int.cast",
"GroupWithZero.toMonoidWithZero",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"HSub... | [
"case pos.ofNat\nz : ℂ\npi_ne : ↑π ≠ 0\na✝ : ℕ\nhk : z = -↑(Int.ofNat a✝)\n⊢ Gamma (-↑(Int.ofNat a✝)) * Gamma (1 - -↑(Int.ofNat a✝)) = 0",
"case pos.negSucc\nz : ℂ\npi_ne : ↑π ≠ 0\na✝ : ℕ\nhk : z = -↑(Int.negSucc a✝)\n⊢ Gamma (-↑(Int.negSucc a✝)) * Gamma (1 - -↑(Int.negSucc a✝)) = 0"
] | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 468,
"column": 86
} | {
"line": 475,
"column": 64
} | {
"line": 477,
"column": 0
} | [
{
"pp": "s : ℝ\n⊢ Tendsto s.GammaSeq atTop (𝓝 (Gamma s))",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instPow",
"Complex.GammaS... | [] | by
suffices Tendsto ((↑) ∘ GammaSeq s : ℕ → ℂ) atTop (𝓝 <| Complex.Gamma s) by
exact (Complex.continuous_re.tendsto (Complex.Gamma ↑s)).comp this
convert! Complex.GammaSeq_tendsto_Gamma s
ext1 n
dsimp only [GammaSeq, Function.comp_apply, Complex.GammaSeq]
push_cast
rw [Complex.ofReal_cpow n.cast_nonneg... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 96,
"column": 40
} | {
"line": 96,
"column": 58
} | {
"line": 96,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R) (m : ℕ), ‖x + 1‖ ^ m ≤ (m + 1) • max 1 (‖x‖ ^ m)\nx : R\na : ℝ\nha : max 1 ‖x‖ < a\nha' : 1 < a\nm : ℕ\nhm : (m + 1) • max 1 ‖x‖ ^ m < a ^ m\n⊢ 1 ∈ {x | 0 ≤ x}",
"ppTerm": "?m.186",
"assigned": true,
"usedConstants": [
"Real",
... | [] | simp [zero_le_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 96,
"column": 40
} | {
"line": 96,
"column": 58
} | {
"line": 96,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R) (m : ℕ), ‖x + 1‖ ^ m ≤ (m + 1) • max 1 (‖x‖ ^ m)\nx : R\na : ℝ\nha : max 1 ‖x‖ < a\nha' : 1 < a\nm : ℕ\nhm : (m + 1) • max 1 ‖x‖ ^ m < a ^ m\n⊢ 1 ∈ {x | 0 ≤ x}",
"ppTerm": "?m.186",
"assigned": true,
"usedConstants": [
"Real",
... | [] | simp [zero_le_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 96,
"column": 40
} | {
"line": 96,
"column": 58
} | {
"line": 96,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R) (m : ℕ), ‖x + 1‖ ^ m ≤ (m + 1) • max 1 (‖x‖ ^ m)\nx : R\na : ℝ\nha : max 1 ‖x‖ < a\nha' : 1 < a\nm : ℕ\nhm : (m + 1) • max 1 ‖x‖ ^ m < a ^ m\n⊢ 1 ∈ {x | 0 ≤ x}",
"ppTerm": "?m.186",
"assigned": true,
"usedConstants": [
"Real",
... | [] | simp [zero_le_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 109,
"column": 6
} | {
"line": 109,
"column": 26
} | {
"line": 111,
"column": 0
} | [
{
"pp": "case a0\nK : Type u_1\ninst✝ : NormedField K\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : K\nha : eval a f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : g.Splits\ni : ℕ\nhi : i < f.natDegree + 1\n⊢ 0 ≤ ‖g.coeff i - f.coeff i‖",
"pp... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Topology.Algebra.Order.LiminfLimsup | {
"line": 218,
"column": 2
} | {
"line": 229,
"column": 49
} | {
"line": 231,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝⁶ : ConditionallyCompleteLinearOrder R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : OrderTopology R\nF : Filter ι\ninst✝³ : AddCommSemigroup R\ninst✝² : Sub R\ninst✝¹ : ContinuousSub R\ninst✝ : OrderedSub R\nf : ι → R\nc : R\nbdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F... | [] | rcases F.eq_or_neBot with rfl | _
· have {a : R} : sInf Set.univ ≤ a := by
apply csInf_le _ (Set.mem_univ a)
simp only [IsCoboundedUnder, IsCobounded, map_bot, eventually_bot, true_implies] at cobdd
rcases cobdd with ⟨x, hx⟩
refine ⟨x, mem_lowerBounds.2 fun y ↦ ?_⟩
simp only [Set.mem_uni... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Order.LiminfLimsup | {
"line": 218,
"column": 2
} | {
"line": 229,
"column": 49
} | {
"line": 231,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝⁶ : ConditionallyCompleteLinearOrder R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : OrderTopology R\nF : Filter ι\ninst✝³ : AddCommSemigroup R\ninst✝² : Sub R\ninst✝¹ : ContinuousSub R\ninst✝ : OrderedSub R\nf : ι → R\nc : R\nbdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F... | [] | rcases F.eq_or_neBot with rfl | _
· have {a : R} : sInf Set.univ ≤ a := by
apply csInf_le _ (Set.mem_univ a)
simp only [IsCoboundedUnder, IsCobounded, map_bot, eventually_bot, true_implies] at cobdd
rcases cobdd with ⟨x, hx⟩
refine ⟨x, mem_lowerBounds.2 fun y ↦ ?_⟩
simp only [Set.mem_uni... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 137,
"column": 2
} | {
"line": 141,
"column": 13
} | {
"line": 142,
"column": 2
} | [
{
"pp": "case pos\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : NormedField L\ninst✝ : Algebra K L\nhd : DenseRange ⇑(algebraMap K L)\nf : L[X]\nhf : f.Monic\nε : ℝ\nhε : ε > 0\nh : f.natDegree = 0\n⊢ ∃ g, g.Monic ∧ f.natDegree = g.natDegree ∧ ∀ (n : ℕ), ‖(map (algebraMap K L) g).coeff n - f.coeff n‖ ... | [
"case neg\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : NormedField L\ninst✝ : Algebra K L\nhd : DenseRange ⇑(algebraMap K L)\nf : L[X]\nhf : f.Monic\nε : ℝ\nhε : ε > 0\nh : ¬f.natDegree = 0\n⊢ ∃ g, g.Monic ∧ f.natDegree = g.natDegree ∧ ∀ (n : ℕ), ‖(map (algebraMap K L) g).coeff n - f.coeff n‖ < ε"
] | · use 1
rw [hf.natDegree_eq_zero.mp]
· simp only [monic_one, natDegree_one, Polynomial.map_one, sub_self, norm_zero, hε,
implies_true, and_self]
· exact h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 48
} | {
"line": 336,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\n⊢ seminormFromBounded' f ≠ 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"Exists",
"Pi.inst... | [
"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ seminormFromBounded' f ≠ 0"
] | obtain ⟨x, hx⟩ := Function.ne_iff.mp f_ne_zero | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 22
} | {
"line": 337,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ seminormFromBounded' f ≠ 0",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.in... | [
"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ ∃ a, seminormFromBounded' f a ≠ 0 a"
] | rw [Function.ne_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 55
} | {
"line": 100,
"column": 4
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm n : ℕ\nhmn : m ≤ n\nhc_pos : 0 < f c\nheq : m = n\n⊢ f (x * c ^ m) * f (c ^ (n - m)) / f c ^ n ≤ f (x * c ^ m) / f c ^ m",
"ppTerm": "?inl",
"assigned": true,
"use... | [
"case inl\nR : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm n : ℕ\nhmn : m ≤ n\nhc_pos : 0 < f c\nheq : m = n\nhnm : n - m = 0\n⊢ f (x * c ^ m) * f (c ^ (n - m)) / f c ^ n ≤ f (x * c ^ m) / f c ^ m"
] | have hnm : n - m = 0 := by rw [heq, Nat.sub_self n] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 106,
"column": 10
} | {
"line": 107,
"column": 63
} | {
"line": 108,
"column": 10
} | [
{
"pp": "case a\nK : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : Is... | [
"case a\nK : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral K y... | · apply IsConjRoot.of_isScalarTower (L := K⟮y⟯) xsep.isIntegral
simpa [z, y'] using IsConjRoot.add_algebraMap y' h1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Group.CocompactMap | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 67
} | {
"line": 51,
"column": 2
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : ProperSpace E\nf : E → F\nh : ∀ (ε : ℝ), ∃ r, ∀ (x : E), r < ‖x‖ → ε < ‖f x‖\ns : Set F\nhs : s ∈ cocompact F\n⊢ f ⁻¹' s ∈ cocompact E",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
... | [
"E : Type u_2\nF : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : ProperSpace E\nf : E → F\nh : ∀ (ε : ℝ), ∃ r, ∀ (x : E), r < ‖x‖ → ε < ‖f x‖\ns : Set F\nhs : s ∈ cocompact F\nε : ℝ\nhε : (closedBall 0 ε)ᶜ ⊆ s\n⊢ f ⁻¹' s ∈ cocompact E"
] | rcases closedBall_compl_subset_of_mem_cocompact hs 0 with ⟨ε, hε⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 345,
"column": 4
} | {
"line": 352,
"column": 71
} | {
"line": 354,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\ns : ℕ → ℕ\nhs_le : ∀ (n : ℕ), s n ≤ n\nx : R\nψ : ℕ → ℕ\nhμx : ¬μ x < 1\n⊢ {a | ∀ᶠ (n : ℝ) in map (fun n ↦ μ x ^ (↑(s (ψ n)) * (1 / ↑(ψ n)))) atTop, n ≤ a}.Nonempty",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [] | use μ x
simp only [eventually_map, eventually_atTop, Set.mem_setOf_eq]
use 0
intro b _
nth_rw 2 [← rpow_one (μ x)]
apply rpow_le_rpow_of_exponent_le (not_lt.mp hμx)
rw [mul_one_div]
exact div_le_one_of_le₀ (cast_le.mpr (hs_le (ψ b))) (cast_nonneg _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 345,
"column": 4
} | {
"line": 352,
"column": 71
} | {
"line": 354,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\ns : ℕ → ℕ\nhs_le : ∀ (n : ℕ), s n ≤ n\nx : R\nψ : ℕ → ℕ\nhμx : ¬μ x < 1\n⊢ {a | ∀ᶠ (n : ℝ) in map (fun n ↦ μ x ^ (↑(s (ψ n)) * (1 / ↑(ψ n)))) atTop, n ≤ a}.Nonempty",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [] | use μ x
simp only [eventually_map, eventually_atTop, Set.mem_setOf_eq]
use 0
intro b _
nth_rw 2 [← rpow_one (μ x)]
apply rpow_le_rpow_of_exponent_le (not_lt.mp hμx)
rw [mul_one_div]
exact div_le_one_of_le₀ (cast_le.mpr (hs_le (ψ b))) (cast_nonneg _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 19
} | {
"line": 221,
"column": 4
} | [
{
"pp": "case zero\nX Y Z : SemiNormedGrp\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\ne : explicitCokernel f ⟶ Z\nhe : explicitCokernelπ f ≫ e = g\n⊢ (cokernelCocone f).ι.app WalkingParallelPair.zero ≫ e = (Cofork.ofπ g ⋯).ι.app WalkingParallelPair.zero",
"ppTerm": "?zero",
"assigned": true,
"usedConstant... | [
"case e'_2\nX Y Z : SemiNormedGrp\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\ne : explicitCokernel f ⟶ Z\nhe : explicitCokernelπ f ≫ e = g\ne_1✝ : ((parallelPair f 0).obj WalkingParallelPair.zero ⟶ (Cofork.ofπ g ⋯).pt) = (X ⟶ Z)\n⊢ (cokernelCocone f).ι.app WalkingParallelPair.zero ≫ e = 0"
] | convert! w.symm | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 487,
"column": 4
} | {
"line": 487,
"column": 48
} | {
"line": 489,
"column": 0
} | [
{
"pp": "case neg\nK : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nh_fin : FiniteDimensional K L\nhn : Normal K L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\nhf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖\nx : L\nhf1 : f 1 = 1\np : K[X] :=... | [] | · exact iSup_nonneg fun σ ↦ apply_nonneg _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 524,
"column": 15
} | {
"line": 524,
"column": 81
} | {
"line": 525,
"column": 2
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝³ : NormedField K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\nE : IntermediateField K L\ny : L\nh_fin : FiniteDimensional K L\nhn : Normal K L\ninst✝ : IsUltrametricDist K\n⊢ ∀ (r : L), spectralNorm K L (-r) = spectralNorm K L r",
"ppTerm": "?m.68",
"... | [] | by rw [spectralNorm_eq_invariantExtension]; exact map_neg_eq_map _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 608,
"column": 2
} | {
"line": 608,
"column": 23
} | {
"line": 610,
"column": 0
} | [
{
"pp": "K : Type u_2\ninst✝³ : NormedField K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsUltrametricDist K\nk : K\ny : L\nhy : IsAlgebraic K y\nE : IntermediateField K L := K⟮y⟯\nh_finiteDimensional_E : FiniteDimensional K ↥E\ng : ↥K⟮y⟯ := AdjoinSimple.gen K y\nhgy : k • y = (algebraMap (↥... | [] | apply map_smul_eq_mul | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 19
} | {
"line": 150,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : Semiring R\nE : Type u_2\nF : Type u_4\nG : Type u_6\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommMonoid E\ninst✝⁶ : Module R E\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\nf : E →L[R] F\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommMonoid G\ninst... | [] | rw [hginv, hfinv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 285,
"column": 6
} | {
"line": 287,
"column": 81
} | {
"line": 288,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\ne₁ : ↑((P ⊔ Q) ⊓ (P ⊔ R)) = ↑P + ↑Q * ↑R * ↑Pᶜ\n⊢ ↑((P ⊔ Q) ⊓ (P ⊔ R)) * ↑(P ⊔ Q ⊓ R) = ↑P + ↑Q * ↑R * ↑Pᶜ",
"ppTerm": "?m.306",
"assign... | [] | rw [coe_inf, coe_sup, coe_sup, coe_sup, ← add_sub, ← add_sub, ← add_sub, ← compl_mul, ←
compl_mul, ← compl_mul, (Pᶜ.prop.commute (Q ⊓ R).prop).eq, coe_inf, mul_assoc,
distrib_lattice_lemma, (Q.prop.commute R.prop).eq, distrib_lattice_lemma] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn | {
"line": 50,
"column": 2
} | {
"line": 51,
"column": 75
} | {
"line": 53,
"column": 0
} | [
{
"pp": "α : Type u_1\nK : Set α\nf : ℕ → α → ℂ\nu : ℕ → ℝ\nhu : Summable u\nh : ∀ᶠ (n : ℕ) in atTop, ∀ x ∈ K, ‖f n x‖ ≤ u n\n⊢ TendstoUniformlyOn (fun n x ↦ ∑ m ∈ range n, log (1 + f m x)) (fun x ↦ ∑' (n : ℕ), log (1 + f n x)) atTop K",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
... | [] | rw [← Nat.cofinite_eq_atTop] at h
exact (hu.hasSumUniformlyOn_log_one_add h).tendstoUniformlyOn_finsetRange | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn | {
"line": 50,
"column": 2
} | {
"line": 51,
"column": 75
} | {
"line": 53,
"column": 0
} | [
{
"pp": "α : Type u_1\nK : Set α\nf : ℕ → α → ℂ\nu : ℕ → ℝ\nhu : Summable u\nh : ∀ᶠ (n : ℕ) in atTop, ∀ x ∈ K, ‖f n x‖ ≤ u n\n⊢ TendstoUniformlyOn (fun n x ↦ ∑ m ∈ range n, log (1 + f m x)) (fun x ↦ ∑' (n : ℕ), log (1 + f n x)) atTop K",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
... | [] | rw [← Nat.cofinite_eq_atTop] at h
exact (hu.hasSumUniformlyOn_log_one_add h).tendstoUniformlyOn_finsetRange | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Order.Hom.Basic | {
"line": 36,
"column": 83
} | {
"line": 36,
"column": 88
} | {
"line": 38,
"column": 0
} | [
{
"pp": "case e'_3\nF : Type u_1\nα : Type u_2\ninst✝² : FunLike F α ℝ\ninst✝¹ : Group α\ninst✝ : GroupSeminormClass F α ℝ\nf : F\nx y z : α\n⊢ x⁻¹ * z = x⁻¹ * y * (y⁻¹ * z)",
"ppTerm": "?e'_3",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"_private.Mathlib.Analysis.Normed.Order.... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.Topology.Algebra.Order.UpperLower | {
"line": 48,
"column": 57
} | {
"line": 50,
"column": 39
} | {
"line": 51,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : CommGroup α\ninst✝² : Preorder α\ninst✝¹ : IsOrderedMonoid α\ninst✝ : ContinuousConstSMul α α\ns : Set α\nh : IsUpperSet s\nx y : α\nhxy : x ≤ y\nhx : x ∈ closure s\n⊢ y ∈ closure ((y / x) • s)",
"ppTerm": "?m.35",
"assigned": true,
"usedC... | [] | by
rw [closure_smul]
exact ⟨x, hx, div_mul_cancel _ _⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Order.UpperLower | {
"line": 52,
"column": 57
} | {
"line": 54,
"column": 39
} | {
"line": 55,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : CommGroup α\ninst✝² : Preorder α\ninst✝¹ : IsOrderedMonoid α\ninst✝ : ContinuousConstSMul α α\ns : Set α\nh : IsLowerSet s\nx y : α\nhxy : y ≤ x\nhx : x ∈ closure s\n⊢ y ∈ closure ((y / x) • s)",
"ppTerm": "?m.63",
"assigned": true,
"usedC... | [] | by
rw [closure_smul]
exact ⟨x, hx, div_mul_cancel _ _⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 73
} | {
"line": 190,
"column": 2
} | [
{
"pp": "ι : Type uι\ninst✝⁵ : Fintype ι\n𝕜 : Type u𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\nF : Type uF\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : ContinuousMultilinearMap 𝕜 E F\n... | [
"ι : Type uι\ninst✝⁵ : Fintype ι\n𝕜 : Type u𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\nF : Type uF\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : ContinuousMultilinearMap 𝕜 E F\nx : ⨂[𝕜] (i... | refine le_trans ?_ (mul_le_mul_of_nonneg_left (hle x) (norm_nonneg f')) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 242,
"column": 12
} | {
"line": 242,
"column": 32
} | {
"line": 243,
"column": 10
} | [
{
"pp": "case hx\nι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁶ : (i : ι) → NormedAddCommGroup (F i)\ninst✝⁵ : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝⁴ : (i : ι) → ... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 243,
"column": 10
} | {
"line": 243,
"column": 30
} | {
"line": 245,
"column": 0
} | [
{
"pp": "case hx\nι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁶ : (i : ι) → NormedAddCommGroup (F i)\ninst✝⁵ : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝⁴ : (i : ι) → ... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 33
} | {
"line": 57,
"column": 4
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : SeminormedAddCommGroup W\ninst✝³ : NormedSpace 𝕜 V\ninst✝² : NormedSpace 𝕜 W\ninst✝¹ : SeparatingDual 𝕜 V\ninst✝ : SeparatingDual 𝕜 W\nf : (V →L[𝕜] V) ≃A[𝕜] W →L[�... | [
"case pos\n𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : SeminormedAddCommGroup W\ninst✝³ : NormedSpace 𝕜 V\ninst✝² : NormedSpace 𝕜 W\ninst✝¹ : SeparatingDual 𝕜 V\ninst✝ : SeparatingDual 𝕜 W\nf : (V →L[𝕜] V) ≃A[𝕜] W →L[𝕜] W\nhV✝ : ... | by_cases! hV : Subsingleton W | Mathlib.Tactic.ByCases._aux_Mathlib_Tactic_ByCases___macroRules_Mathlib_Tactic_ByCases_byCases!_1 | Mathlib.Tactic.ByCases.byCases! |
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