module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 241,
"column": 6
} | {
"line": 242,
"column": 16
} | {
"line": 243,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nS : Type u_2\nR : Type u_3\nM : Type u_4\ninst✝¹¹ : SeminormedCommRing S\ninst✝¹⁰ : SeminormedRing R\ninst✝⁹ : SeminormedAddCommGroup M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsBoundedSMul S R\ninst✝⁵ : IsBoundedSMul S M\ninst✝⁴ : Module R M\ninst✝³ : IsBoundedSMul R M\nins... | [] | apply le_add_of_nonneg_right
positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 62
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case h.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy0 : f y = 0 y\n⊢ (fun y ↦ f (x * y) / f y) y ≤ c * f x",
"ppTerm": "?h.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero... | [] | simpa [hy0] using seminormFromBounded_aux f_nonneg f_mul x | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 62
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case h.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy0 : f y = 0 y\n⊢ (fun y ↦ f (x * y) / f y) y ≤ c * f x",
"ppTerm": "?h.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero... | [] | simpa [hy0] using seminormFromBounded_aux f_nonneg f_mul x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 62
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case h.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy0 : f y = 0 y\n⊢ (fun y ↦ f (x * y) / f y) y ≤ c * f x",
"ppTerm": "?h.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero... | [] | simpa [hy0] using seminormFromBounded_aux f_nonneg f_mul x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 15
} | {
"line": 130,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy : 0 y < f y\n⊢ f (x * y) ≤ c * f x * f y",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"f_mul",
"x",
... | [] | apply f_mul | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 156,
"column": 46
} | {
"line": 164,
"column": 67
} | {
"line": 166,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\n⊢ seminormFromBounded' f x = 0 ↔ f x = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real.instLE",
"Real",
... | [] | by
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· have hf := seminormFromBounded_ge f_nonneg f_mul x
rw [h, mul_zero] at hf
exact hf.antisymm (f_nonneg _)
· have hf : seminormFromBounded' f x ≤ c * f x :=
seminormFromBounded_le f_nonneg f_mul x
rw [h, mul_zero] at hf
exact hf.antisymm (seminormFromBounde... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Order.LiminfLimsup | {
"line": 210,
"column": 2
} | {
"line": 211,
"column": 98
} | {
"line": 213,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_1\nR : Type u_4\ninst✝⁷ : ConditionallyCompleteLinearOrder R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : OrderTopology R\nF : Filter ι\ninst✝⁴ : AddCommSemigroup R\ninst✝³ : Sub R\ninst✝² : ContinuousSub R\ninst✝¹ : OrderedSub R\ninst✝ : AddLeftMono R\nf : ι → R\nc : R\ncobdd : IsCoboun... | [] | · exact (Antitone.map_limsInf_of_continuousAt (F := F.map f) (f := fun (x : R) ↦ c - x)
(fun _ _ h ↦ tsub_le_tsub_left h c) (continuous_sub_left c).continuousAt cobdd bdd_below).symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 105,
"column": 71
} | {
"line": 116,
"column": 61
} | {
"line": 118,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nx : R\nhx : μ x = 0\n⊢ Tendsto (smoothingSeminormSeq μ x) atTop (𝓝 (smoothingFun μ x))",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"PNat.val",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"P... | [] | by
have h0 (n : ℕ) (hn : 1 ≤ n) : μ (x ^ n) ^ (1 / (n : ℝ)) = 0 := by
have hμn : μ (x ^ n) = 0 := by
apply le_antisymm _ (apply_nonneg μ _)
rw [← zero_pow (pos_iff_ne_zero.mp hn), ← hx]
exact map_pow_le_pow _ x (one_le_iff_ne_zero.mp hn)
rw [hμn, zero_rpow (one_div_cast_ne_zero (one_le_iff_n... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 99,
"column": 31
} | {
"line": 99,
"column": 55
} | {
"line": 100,
"column": 4
} | [
{
"pp": "R : 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⊢ n - m = 0",
"ppTerm": "?m.135",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"HSub.hSub"... | [] | rw [heq, Nat.sub_self n] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 90,
"column": 91
} | {
"line": 108,
"column": 89
} | {
"line": 110,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\n⊢ Antitone (seminormFromConst_seq c f x)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
"NonUnitalNonAss... | [] | by
intro m n hmn
simp only [seminormFromConst_seq]
nth_rw 1 [← Nat.add_sub_of_le hmn]
rw [pow_add, ← mul_assoc]
have hc_pos : 0 < f c := lt_of_le_of_ne (apply_nonneg f _) hc.symm
apply le_trans ((div_le_div_iff_of_pos_right (pow_pos hc_pos _)).mpr (map_mul_le_mul f _ _))
cases hmn.eq_or_lt with
| inl he... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 28
} | {
"line": 218,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhx : ∀ (y : R), f (x * y) = f x * f y\nhseq : seminormFromConst_seq c f x = fun _n ↦ f x\n⊢ Tendsto (fun _n ↦ f x) atTop (𝓝 (f x))",
"ppTerm": "?m.82",
"assigned": true,
"use... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 243,
"column": 4
} | {
"line": 243,
"column": 28
} | {
"line": 244,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nhseq : seminormFromConst_seq c f c = fun _n ↦ f c\n⊢ Tendsto (fun _n ↦ f c) atTop (𝓝 (f c))",
"ppTerm": "?m.106",
"assigned": true,
"usedConstants": [
"Real",
"NonUnital... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Unbundled.FiniteExtension | {
"line": 221,
"column": 55
} | {
"line": 221,
"column": 87
} | {
"line": 221,
"column": 87
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nhfd : FiniteDimensional K L\nhna : IsNonarchimedean norm\nh1 : LinearIndepOn K id {1}\nι : Type u_2 := { x // x ∈ h1.extend ⋯ }\nB : Basis ι K L := Basis.extend h1\nhfin : Fintype ι := FiniteDimensional.fintypeBa... | [] | by rw [hF'_1]; exact one_ne_zero | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 46
} | {
"line": 247,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : μ x = 0\n⊢ Tendsto (smoothingSeminormSeq μ x) atTop (𝓝 (smoothingFun μ x))",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"tendsto_smoothingFun_of_eq_zero"
],
"usedFVars": [
... | [
"case neg\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : ¬μ x = 0\n⊢ Tendsto (smoothingSeminormSeq μ x) atTop (𝓝 (smoothingFun μ x))"
] | · exact tendsto_smoothingFun_of_eq_zero μ hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 57,
"column": 10
} | {
"line": 57,
"column": 56
} | {
"line": 57,
"column": 56
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [] | simpa using degree_ne_of_natDegree_ne fnatdeg0 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 57,
"column": 10
} | {
"line": 57,
"column": 56
} | {
"line": 57,
"column": 56
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [] | simpa using degree_ne_of_natDegree_ne fnatdeg0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 57,
"column": 10
} | {
"line": 57,
"column": 56
} | {
"line": 57,
"column": 56
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [] | simpa using degree_ne_of_natDegree_ne fnatdeg0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 61,
"column": 10
} | {
"line": 61,
"column": 56
} | {
"line": 61,
"column": 56
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [] | simpa using degree_ne_of_natDegree_ne fnatdeg0 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 61,
"column": 10
} | {
"line": 61,
"column": 56
} | {
"line": 61,
"column": 56
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [] | simpa using degree_ne_of_natDegree_ne fnatdeg0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 61,
"column": 10
} | {
"line": 61,
"column": 56
} | {
"line": 61,
"column": 56
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [] | simpa using degree_ne_of_natDegree_ne fnatdeg0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 63
} | {
"line": 71,
"column": 4
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"case pos\nK : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDe... | by_cases hS : S.Nonempty <;> simp only [hS, ↓reduceDIte, δ] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 63
} | {
"line": 84,
"column": 4
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"case pos\nK : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDe... | by_cases hS : S.Nonempty <;> simp only [hS, ↓reduceDIte, δ] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 416,
"column": 4
} | {
"line": 419,
"column": 60
} | {
"line": 421,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\ns : ℕ → ℕ\nhs_le : ∀ (n : ℕ), s n ≤ n\nx : R\na : ℝ\na_in : a ∈ Set.Icc 0 1\nψ : ℕ → ℕ\nhψ_mono : StrictMono ψ\nhψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 a)\nha : ¬a = 0\nha_pos : 0 < a\nh_eq :\n (fun n ↦ (μ (x... | [] | exact ((tendsto_smoothingFun_of_map_one_le_one μ hμ1 x |>.comp <|
tendsto_natCast_atTop_iff.mp <| (tendsto_natCast_atTop_atTop.comp
hψ_mono.tendsto_atTop).num ha_pos hψ_lim).rpow
hψ_lim <| .inr ha_pos).congr' h_eq |>.limsup_eq.le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 453,
"column": 4
} | {
"line": 453,
"column": 14
} | {
"line": 454,
"column": 4
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nhna : IsNonarchimedean ⇑μ\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nmu : ℕ → ℕ := ⋯\nnu : ℕ → ℕ := ⋯\nhnu : nu = fun n ↦ n - mu n\nhmu_le : ∀ (n : ℕ), mu n ≤ ... | [
"case h\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nhna : IsNonarchimedean ⇑μ\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nmu : ℕ → ℕ := fun n ↦ _root_.mu μ hn n\nnu : ℕ → ℕ := fun n ↦ n - mu n\nhnu : nu = fun n ↦ n - mu n\nh... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 265,
"column": 10
} | {
"line": 265,
"column": 99
} | {
"line": 266,
"column": 8
} | [
{
"pp": "case pos\nK : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\np : K[X]\nhp : p.Monic\nx : L\nhx : (aeval x) p = 0\nhx0 : ¬f x = 0\nh_ge : ¬f x ≤ spectralValue p\nhn_lt : ∀ n < p.natDegree, ‖p.c... | [] | exact (mul_le_of_le_one_right (norm_nonneg _) hf_pm.map_one_le_one).trans_lt (hn_lt n hn) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 316,
"column": 46
} | {
"line": 316,
"column": 62
} | {
"line": 316,
"column": 63
} | [
{
"pp": "K : Type u_2\ninst✝³ : NormedField K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : DecidableEq L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\nhf1 : f 1 = 1\np : K[X]\ns : Multiset L\nhp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod\nh_le : 0 ≤ ⨆ ... | [
"K : Type u_2\ninst✝³ : NormedField K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : DecidableEq L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\nhf1 : f 1 = 1\np : K[X]\ns : Multiset L\nhp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod\nh_le : 0 ≤ ⨆ x, if x ∈ s ... | ← mapAlg_eq_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 708,
"column": 21
} | {
"line": 710,
"column": 34
} | {
"line": 711,
"column": 6
} | [
{
"pp": "K : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nx : L\nE : Type v := id ↥K⟮x⟯\nthis✝ : Field E :=\n id\n (have this := i... | [] | by
simp only [← spectralAlgNorm_def]
exact map_add_le_add _ _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 964,
"column": 2
} | {
"line": 968,
"column": 74
} | {
"line": 969,
"column": 2
} | [
{
"pp": "K : Type u\ninst✝⁹ : NontriviallyNormedField K\nL : Type v\ninst✝⁸ : Field L\ninst✝⁷ : Algebra K L\nhu : IsUltrametricDist K\ninst✝⁶ : CompleteSpace K\nx : L\nE : Type u_2\ninst✝⁵ : Field E\ninst✝⁴ : Algebra K E\ninst✝³ : Algebra L E\ninst✝² : IsScalarTower K L E\ninst✝¹ : IsSplittingField L E ((mapAlg... | [
"K : Type u\ninst✝⁹ : NontriviallyNormedField K\nL : Type v\ninst✝⁸ : Field L\ninst✝⁷ : Algebra K L\nhu : IsUltrametricDist K\ninst✝⁶ : CompleteSpace K\nx : L\nE : Type u_2\ninst✝⁵ : Field E\ninst✝⁴ : Algebra K E\ninst✝³ : Algebra L E\ninst✝² : IsScalarTower K L E\ninst✝¹ : IsSplittingField L E ((mapAlg K L) (minpo... | have h_deg : (minpoly K x).natDegree = Multiset.card ((mapAlg K E) (minpoly K x)).roots := by
trans (mapAlg K E (minpoly K x)).natDegree
· rw [mapAlg_eq_map, natDegree_map]
· rw [eq_comm, ← splits_iff_card_roots]
exact IsSplittingField.IsScalarTower.splits (K := L) E (minpoly K x) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Module.DoubleDual | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 80
} | {
"line": 152,
"column": 2
} | [
{
"pp": "𝕜 : Type u_3\ninst✝² : RCLike 𝕜\nX : Type u_4\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nS : Set (WeakSpace 𝕜 X)\nhb : IsBounded (⇑(toWeakSpace 𝕜 X) ⁻¹' S)\nhrange : closure (⇑(inclusionInDoubleDualWeak 𝕜 X) '' S) ⊆ Set.range ⇑(inclusionInDoubleDualWeak 𝕜 X)\n⊢ IsCompact (⇑(inclusi... | [
"𝕜 : Type u_3\ninst✝² : RCLike 𝕜\nX : Type u_4\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nS : Set (WeakSpace 𝕜 X)\nhb : IsBounded (⇑(toWeakSpace 𝕜 X) ⁻¹' S)\nhrange : closure (⇑(inclusionInDoubleDualWeak 𝕜 X) '' S) ⊆ Set.range ⇑(inclusionInDoubleDualWeak 𝕜 X)\n⊢ IsCompact (closure (⇑(inclusionI... | apply (isEmbedding_inclusionInDoubleDualWeak 𝕜 X).isCompact_preimage' _ hrange | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 103,
"column": 81
} | {
"line": 125,
"column": 65
} | {
"line": 126,
"column": 2
} | [
{
"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 : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nR : M\nh₃ : IsLprojection X R\n⊢ P * R = R * P * R",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [] | by
refine @eq_of_smul_eq_smul _ X _ _ _ _ fun x => by
rw [← norm_sub_eq_zero_iff]
have e1 : ‖R • x‖ ≥ ‖R • x‖ + 2 • ‖(P * R) • x - (R * P * R) • x‖ :=
calc
‖R • x‖ = ‖R • P • R • x‖ + ‖(1 - R) • P • R • x‖ +
(‖(R * R) • x - R • P • R • x‖ + ‖(1 - R) • (1 - P) • R • x‖) :=... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 256,
"column": 16
} | {
"line": 256,
"column": 24
} | {
"line": 256,
"column": 25
} | [
{
"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 : { P // IsLprojection X P }\n⊢ ↑P = ↑(P ⊓ (P ⊔ Q))",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"IsLproje... | [
"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 : { P // IsLprojection X P }\n⊢ ↑P = ↑P * ↑(P ⊔ Q)"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 259,
"column": 16
} | {
"line": 259,
"column": 24
} | {
"line": 259,
"column": 25
} | [
{
"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 : { P // IsLprojection X P }\n⊢ ↑Q = ↑(Q ⊓ (P ⊔ Q))",
"ppTerm": "?m.123",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"IsLproj... | [
"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 : { P // IsLprojection X P }\n⊢ ↑Q = ↑Q * ↑(P ⊔ Q)"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 262,
"column": 32
} | {
"line": 262,
"column": 40
} | {
"line": 262,
"column": 41
} | [
{
"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 }\n⊢ ↑P = ↑(P ⊓ R) → ↑Q = ↑(Q ⊓ R) → ↑(P ⊔ Q) = ↑((P ⊔ Q) ⊓ R)",
"ppTerm": "?m.229",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"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 }\n⊢ ↑P = ↑P * ↑R → ↑Q = ↑(Q ⊓ R) → ↑(P ⊔ Q) = ↑((P ⊔ Q) ⊓ R)"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 262,
"column": 41
} | {
"line": 262,
"column": 49
} | {
"line": 262,
"column": 50
} | [
{
"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 }\n⊢ ↑P = ↑P * ↑R → ↑Q = ↑(Q ⊓ R) → ↑(P ⊔ Q) = ↑((P ⊔ Q) ⊓ R)",
"ppTerm": "?m.238",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"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 }\n⊢ ↑P = ↑P * ↑R → ↑Q = ↑Q * ↑R → ↑(P ⊔ Q) = ↑((P ⊔ Q) ⊓ R)"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 262,
"column": 59
} | {
"line": 262,
"column": 67
} | {
"line": 262,
"column": 68
} | [
{
"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 }\n⊢ ↑P = ↑P * ↑R → ↑Q = ↑Q * ↑R → ↑P + ↑Q - ↑P * ↑Q = ↑((P ⊔ Q) ⊓ R)",
"ppTerm": "?m.256",
"assigned": true,
"usedConstants": [
... | [
"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 }\n⊢ ↑P = ↑P * ↑R → ↑Q = ↑Q * ↑R → ↑P + ↑Q - ↑P * ↑Q = ↑(P ⊔ Q) * ↑R"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 265,
"column": 4
} | {
"line": 265,
"column": 19
} | {
"line": 266,
"column": 2
} | [
{
"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 }\nh₁ : ↑P = ↑P * ↑R\nh₂ : ↑Q = ↑Q * ↑R\n⊢ ↑P + (↑Q - ↑P * ↑Q) = ↑P * ↑R + (↑Q * ↑R - ↑P * (↑Q * ↑R))",
"ppTerm": "?m.303",
"assigned": tru... | [] | rw [← h₂, ← h₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 267,
"column": 16
} | {
"line": 267,
"column": 24
} | {
"line": 267,
"column": 25
} | [
{
"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 : { P // IsLprojection X P }\n⊢ ↑(P ⊓ Q) = ↑(P ⊓ Q ⊓ P)",
"ppTerm": "?m.316",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"HMu... | [
"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 : { P // IsLprojection X P }\n⊢ ↑P * ↑Q = ↑(P ⊓ Q ⊓ P)"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 267,
"column": 25
} | {
"line": 267,
"column": 33
} | {
"line": 267,
"column": 34
} | [
{
"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 : { P // IsLprojection X P }\n⊢ ↑P * ↑Q = ↑(P ⊓ Q ⊓ P)",
"ppTerm": "?m.325",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"HMul... | [
"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 : { P // IsLprojection X P }\n⊢ ↑P * ↑Q = ↑(P ⊓ Q) * ↑P"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 267,
"column": 34
} | {
"line": 267,
"column": 42
} | {
"line": 267,
"column": 43
} | [
{
"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 : { P // IsLprojection X P }\n⊢ ↑P * ↑Q = ↑(P ⊓ Q) * ↑P",
"ppTerm": "?m.334",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"HMu... | [
"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 : { P // IsLprojection X P }\n⊢ ↑P * ↑Q = ↑P * ↑Q * ↑P"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 269,
"column": 37
} | {
"line": 269,
"column": 45
} | {
"line": 269,
"column": 46
} | [
{
"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 : { P // IsLprojection X P }\n⊢ ↑(P ⊓ Q) = ↑(P ⊓ Q ⊓ Q)",
"ppTerm": "?m.394",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"HMu... | [
"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 : { P // IsLprojection X P }\n⊢ ↑P * ↑Q = ↑(P ⊓ Q ⊓ Q)"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 269,
"column": 46
} | {
"line": 269,
"column": 54
} | {
"line": 269,
"column": 55
} | [
{
"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 : { P // IsLprojection X P }\n⊢ ↑P * ↑Q = ↑(P ⊓ Q ⊓ Q)",
"ppTerm": "?m.403",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"HMul... | [
"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 : { P // IsLprojection X P }\n⊢ ↑P * ↑Q = ↑(P ⊓ Q) * ↑Q"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 269,
"column": 55
} | {
"line": 269,
"column": 63
} | {
"line": 269,
"column": 64
} | [
{
"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 : { P // IsLprojection X P }\n⊢ ↑P * ↑Q = ↑(P ⊓ Q) * ↑Q",
"ppTerm": "?m.412",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"HMu... | [
"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 : { P // IsLprojection X P }\n⊢ ↑P * ↑Q = ↑P * ↑Q * ↑Q"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 271,
"column": 32
} | {
"line": 271,
"column": 40
} | {
"line": 271,
"column": 41
} | [
{
"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 }\n⊢ ↑P = ↑(P ⊓ Q) → ↑P = ↑(P ⊓ R) → ↑P = ↑(P ⊓ (Q ⊓ R))",
"ppTerm": "?m.467",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"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 }\n⊢ ↑P = ↑P * ↑Q → ↑P = ↑(P ⊓ R) → ↑P = ↑(P ⊓ (Q ⊓ R))"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 271,
"column": 41
} | {
"line": 271,
"column": 49
} | {
"line": 271,
"column": 50
} | [
{
"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 }\n⊢ ↑P = ↑P * ↑Q → ↑P = ↑(P ⊓ R) → ↑P = ↑(P ⊓ (Q ⊓ R))",
"ppTerm": "?m.476",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"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 }\n⊢ ↑P = ↑P * ↑Q → ↑P = ↑P * ↑R → ↑P = ↑(P ⊓ (Q ⊓ R))"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 271,
"column": 50
} | {
"line": 271,
"column": 58
} | {
"line": 271,
"column": 59
} | [
{
"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 }\n⊢ ↑P = ↑P * ↑Q → ↑P = ↑P * ↑R → ↑P = ↑(P ⊓ (Q ⊓ R))",
"ppTerm": "?m.485",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"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 }\n⊢ ↑P = ↑P * ↑Q → ↑P = ↑P * ↑R → ↑P = ↑P * ↑(Q ⊓ R)"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 271,
"column": 59
} | {
"line": 271,
"column": 67
} | {
"line": 271,
"column": 68
} | [
{
"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 }\n⊢ ↑P = ↑P * ↑Q → ↑P = ↑P * ↑R → ↑P = ↑P * ↑(Q ⊓ R)",
"ppTerm": "?m.494",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"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 }\n⊢ ↑P = ↑P * ↑Q → ↑P = ↑P * ↑R → ↑P = ↑P * (↑Q * ↑R)"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 279,
"column": 10
} | {
"line": 279,
"column": 18
} | {
"line": 279,
"column": 19
} | [
{
"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 }\n⊢ ↑((P ⊔ Q) ⊓ (P ⊔ R)) = ↑P + ↑Q * ↑R * ↑Pᶜ",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLproj... | [
"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 }\n⊢ ↑(P ⊔ Q) * ↑(P ⊔ R) = ↑P + ↑Q * ↑R * ↑Pᶜ"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 285,
"column": 10
} | {
"line": 285,
"column": 18
} | {
"line": 285,
"column": 19
} | [
{
"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... | [
"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"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 286,
"column": 67
} | {
"line": 286,
"column": 75
} | {
"line": 286,
"column": 76
} | [
{
"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 + ↑Pᶜ * ↑Q) * (↑P + ↑Pᶜ * ↑R) * (↑P + ↑(Q ⊓ R) * ↑Pᶜ) = ↑P + ↑Q * ↑R * ↑Pᶜ",
"ppTerm":... | [
"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 + ↑Pᶜ * ↑Q) * (↑P + ↑Pᶜ * ↑R) * (↑P + ↑Q * ↑R * ↑Pᶜ) = ↑P + ↑Q * ↑R * ↑Pᶜ"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 89
} | {
"line": 91,
"column": 2
} | [
{
"pp": "ι : Type u_1\nf : ι → ℝ\nhf : Summable f\n⊢ (fun i ↦ Complex.log (1 + ↑(f i))) =ᶠ[cofinite] fun x ↦ ↑(log (1 + f x))",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Filter.Tendsto.eventually_const_le",
"NegZeroClass.toNeg",
"Complex.log",
"instClosedIicTop... | [
"ι : Type u_1\nf : ι → ℝ\nhf : Summable f\ni : ι\nhi : -1 ≤ f i\n⊢ Complex.log (1 + ↑(f i)) = ↑(log (1 + f i))"
] | filter_upwards [hf.tendsto_cofinite_zero.eventually_const_le neg_one_lt_zero] with i hi | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 288,
"column": 20
} | {
"line": 288,
"column": 28
} | {
"line": 288,
"column": 29
} | [
{
"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\ne₂ : ↑((P ⊔ Q) ⊓ (P ⊔ R)) * ↑(P ⊔ Q ⊓ R) = ↑P + ↑Q * ↑R * ↑Pᶜ\n⊢ ↑P + ↑Q * ↑R * ↑Pᶜ = ↑((P ⊔ Q) ... | [
"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\ne₂ : ↑((P ⊔ Q) ⊓ (P ⊔ R)) * ↑(P ⊔ Q ⊓ R) = ↑P + ↑Q * ↑R * ↑Pᶜ\n⊢ ↑P + ↑Q * ↑R * ↑Pᶜ = ↑((P ⊔ Q) ⊓ (P ⊔ R)) *... | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 277,
"column": 22
} | {
"line": 288,
"column": 32
} | {
"line": 290,
"column": 0
} | [
{
"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 }\n⊢ (P ⊔ Q) ⊓ (P ⊔ R) ≤ P ⊔ Q ⊓ R",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"add_mul",
"Distrib.leftDistri... | [] | by
have e₁ : ↑((P ⊔ Q) ⊓ (P ⊔ R)) = ↑P + ↑Q * (R : M) * ↑Pᶜ := by
rw [coe_inf, coe_sup, coe_sup, ← add_sub, ← add_sub, ← compl_mul, ← compl_mul, add_mul,
mul_add, (Pᶜ.prop.commute Q.prop).eq, mul_add, ← mul_assoc, mul_assoc (Q : M),
(Pᶜ.prop.commute P.prop).eq, mul_compl_self, zero_mul, mul_ze... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 296,
"column": 27
} | {
"line": 296,
"column": 35
} | {
"line": 296,
"column": 36
} | [
{
"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 : { P // IsLprojection X P }\n⊢ ↑(P ⊓ Pᶜ) = ↑⊥",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"Lattice.toSemil... | [
"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 : { P // IsLprojection X P }\n⊢ ↑P * ↑Pᶜ = ↑⊥"
] | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 174,
"column": 4
} | {
"line": 175,
"column": 39
} | {
"line": 176,
"column": 2
} | [
{
"pp": "case inr.inr.inl\nι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : (i : ι) → NormedAddCommGroup (F i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝¹ : (... | [] | simp_all only [inv_top, zero_add, inv_inj]
exact he.bilin_of_top_left B hBK hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 174,
"column": 4
} | {
"line": 175,
"column": 39
} | {
"line": 176,
"column": 2
} | [
{
"pp": "case inr.inr.inl\nι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : (i : ι) → NormedAddCommGroup (F i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝¹ : (... | [] | simp_all only [inv_top, zero_add, inv_inj]
exact he.bilin_of_top_left B hBK hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.ODE.Gronwall | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 32
} | {
"line": 119,
"column": 4
} | [
{
"pp": "f f' : ℝ → ℝ\nδ K ε a b : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, (z - x)⁻¹ * (f z - f x) < r\nha : f a ≤ δ\nbound : ∀ x ∈ Ico a b, f' x ≤ K * f x + ε\n⊢ ∀ x ∈ Icc a b, ∀ ε' ∈ Ioi ε, f x ≤ gronwallBound δ K ε' (x - a)",
"ppTerm": "?m.89",
... | [
"f f' : ℝ → ℝ\nδ K ε a b : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, (z - x)⁻¹ * (f z - f x) < r\nha : f a ≤ δ\nbound : ∀ x ∈ Ico a b, f' x ≤ K * f x + ε\nx : ℝ\nhx : x ∈ Icc a b\nε' : ℝ\nhε' : ε < ε'\n⊢ f x ≤ gronwallBound δ K ε' (x - a)"
] | intro x hx ε' (hε' : ε < ε') | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.ODE.DiscreteGronwall | {
"line": 86,
"column": 19
} | {
"line": 86,
"column": 70
} | {
"line": 87,
"column": 8
} | [
{
"pp": "case ha\nu b c : ℕ → ℝ\nn₀ : ℕ\nhun₀ : 0 ≤ u n₀\nhu : ∀ n ≥ n₀, u (n + 1) ≤ (1 + c n) * u n + b n\nhc : ∀ n ≥ n₀, 0 ≤ c n\nhb : ∀ n ≥ n₀, 0 ≤ b n\nn : ℕ\nhn : n₀ ≤ n\n⊢ 0 ≤ u n₀ + ∑ k ∈ Ico n₀ n, b k",
"ppTerm": "?ha",
"assigned": true,
"usedConstants": [
"Real",
"Real.instAddMo... | [] | try exact add_nonneg hun₀ <| sum_nonneg <| by grind | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.Analysis.ODE.DiscreteGronwall | {
"line": 86,
"column": 19
} | {
"line": 86,
"column": 70
} | {
"line": 87,
"column": 8
} | [
{
"pp": "case hbc\nu b c : ℕ → ℝ\nn₀ : ℕ\nhun₀ : 0 ≤ u n₀\nhu : ∀ n ≥ n₀, u (n + 1) ≤ (1 + c n) * u n + b n\nhc : ∀ n ≥ n₀, 0 ≤ c n\nhb : ∀ n ≥ n₀, 0 ≤ b n\nn : ℕ\nhn : n₀ ≤ n\n⊢ ∏ i ∈ Ico n₀ n, (1 + c i) ≤ rexp (∑ i ∈ Ico n₀ n, c i)",
"ppTerm": "?hbc",
"assigned": false,
"usedConstants": [],
"u... | [
"case hbc\nu b c : ℕ → ℝ\nn₀ : ℕ\nhun₀ : 0 ≤ u n₀\nhu : ∀ n ≥ n₀, u (n + 1) ≤ (1 + c n) * u n + b n\nhc : ∀ n ≥ n₀, 0 ≤ c n\nhb : ∀ n ≥ n₀, 0 ≤ b n\nn : ℕ\nhn : n₀ ≤ n\n⊢ ∏ i ∈ Ico n₀ n, (1 + c i) ≤ rexp (∑ i ∈ Ico n₀ n, c i)"
] | try exact add_nonneg hun₀ <| sum_nonneg <| by grind | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.Analysis.ODE.Gronwall | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 27
} | {
"line": 149,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → E\nK a b : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x\nha : f a = 0\nbound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖\nx : ℝ\nhx : x ∈ Icc a b\n⊢ f x = 0",
"ppTerm": "?m.55",
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → E\nK a b : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x\nha : f a = 0\nbound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖\nx : ℝ\nhx : x ∈ Icc a b\n⊢ ‖f x‖ ≤ 0"
] | apply norm_le_zero_iff.mp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.ODE.Transform | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 26
} | {
"line": 121,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\ns : Set ℝ\na : ℝ\nha : a ≠ 0\nheq : a⁻¹ • s = {t | t * a ∈ s}\nhγ : IsIntegralCurveOn (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (a⁻¹ • s)\n⊢ IsIntegralCurveOn γ v s",
"ppTerm": "?m.86",
"assigned":... | [
"case e'_4\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\ns : Set ℝ\na : ℝ\nha : a ≠ 0\nheq : a⁻¹ • s = {t | t * a ∈ s}\nhγ : IsIntegralCurveOn (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (a⁻¹ • s)\n⊢ γ = (γ ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹",
"case e'_5\nE : Typ... | convert! hγ.comp_mul a⁻¹ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.ODE.Transform | {
"line": 123,
"column": 4
} | {
"line": 125,
"column": 26
} | {
"line": 126,
"column": 2
} | [
{
"pp": "case e'_5\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\ns : Set ℝ\na : ℝ\nha : a ≠ 0\nheq : a⁻¹ • s = {t | t * a ∈ s}\nhγ : IsIntegralCurveOn (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (a⁻¹ • s)\n⊢ v = a⁻¹ • (a • v ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹",... | [] | ext t
simp only [comp_apply, Pi.smul_apply, mul_assoc, inv_mul_eq_div, div_self ha, mul_one,
smul_smul, one_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.Transform | {
"line": 123,
"column": 4
} | {
"line": 125,
"column": 26
} | {
"line": 126,
"column": 2
} | [
{
"pp": "case e'_5\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\ns : Set ℝ\na : ℝ\nha : a ≠ 0\nheq : a⁻¹ • s = {t | t * a ∈ s}\nhγ : IsIntegralCurveOn (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (a⁻¹ • s)\n⊢ v = a⁻¹ • (a • v ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹",... | [] | ext t
simp only [comp_apply, Pi.smul_apply, mul_assoc, inv_mul_eq_div, div_self ha, mul_one,
smul_smul, one_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.ODE.Transform | {
"line": 145,
"column": 4
} | {
"line": 147,
"column": 26
} | {
"line": 148,
"column": 2
} | [
{
"pp": "case e'_5\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\nt₀ a : ℝ\nha : a ≠ 0\nhγ : IsIntegralCurveAt (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (t₀ / a)\n⊢ v = a⁻¹ • (a • v ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹",
"ppTerm": "?e'_5",
"assigned": tr... | [] | ext t
simp only [comp_apply, Pi.smul_apply, mul_assoc, inv_mul_eq_div, div_self ha, mul_one,
smul_smul, one_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.Transform | {
"line": 145,
"column": 4
} | {
"line": 147,
"column": 26
} | {
"line": 148,
"column": 2
} | [
{
"pp": "case e'_5\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\nt₀ a : ℝ\nha : a ≠ 0\nhγ : IsIntegralCurveAt (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (t₀ / a)\n⊢ v = a⁻¹ • (a • v ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹",
"ppTerm": "?e'_5",
"assigned": tr... | [] | ext t
simp only [comp_apply, Pi.smul_apply, mul_assoc, inv_mul_eq_div, div_self ha, mul_one,
smul_smul, one_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.ODE.Transform | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 26
} | {
"line": 159,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\na : ℝ\nha : a ≠ 0\nhγ : IsIntegralCurve (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a)\n⊢ IsIntegralCurve γ v",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"IsIntegralCurve.comp_mu... | [
"case e'_4\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\na : ℝ\nha : a ≠ 0\nhγ : IsIntegralCurve (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a)\n⊢ γ = (γ ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹",
"case e'_5\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpac... | convert! hγ.comp_mul a⁻¹ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.ODE.Transform | {
"line": 161,
"column": 4
} | {
"line": 163,
"column": 26
} | {
"line": 165,
"column": 0
} | [
{
"pp": "case e'_5\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\na : ℝ\nha : a ≠ 0\nhγ : IsIntegralCurve (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a)\n⊢ v = a⁻¹ • (a • v ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹",
"ppTerm": "?e'_5",
"assigned": true,
"usedC... | [] | ext t
simp only [comp_apply, Pi.smul_apply, mul_assoc, inv_mul_eq_div, div_self ha, mul_one,
smul_smul, one_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.Transform | {
"line": 161,
"column": 4
} | {
"line": 163,
"column": 26
} | {
"line": 165,
"column": 0
} | [
{
"pp": "case e'_5\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\na : ℝ\nha : a ≠ 0\nhγ : IsIntegralCurve (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a)\n⊢ v = a⁻¹ • (a • v ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹",
"ppTerm": "?e'_5",
"assigned": true,
"usedC... | [] | ext t
simp only [comp_apply, Pi.smul_apply, mul_assoc, inv_mul_eq_div, div_self ha, mul_one,
smul_smul, one_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.ODE.ExistUnique | {
"line": 233,
"column": 4
} | {
"line": 235,
"column": 8
} | {
"line": 236,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ≥0\nf g : ℝ → E\na b : ℝ\nhv : ∀ t ∈ Ioc a b, LipschitzOnWith K (v t) (s t)\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ t ∈ Ioc a b, HasDerivWithinAt f (v t (f t)) (Iic t) t\nhfs : ∀ t ∈ Ioc a b, f t ∈ ... | [] | rw [eqOn_comp_right_iff] at this
convert this
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.ExistUnique | {
"line": 233,
"column": 4
} | {
"line": 235,
"column": 8
} | {
"line": 236,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ≥0\nf g : ℝ → E\na b : ℝ\nhv : ∀ t ∈ Ioc a b, LipschitzOnWith K (v t) (s t)\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ t ∈ Ioc a b, HasDerivWithinAt f (v t (f t)) (Iic t) t\nhfs : ∀ t ∈ Ioc a b, f t ∈ ... | [] | rw [eqOn_comp_right_iff] at this
convert this
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Polynomial.Basic | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 28
} | {
"line": 119,
"column": 0
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nc : 𝕜\nh : P.leadingCoeff = c ∧ P.degree ≤ 0\nthis : P.natDegree = 0\n⊢ Tendsto (fun x ↦ c) atTop (𝓝 c)",
"ppTerm": "?refine_2",
"assigned": tr... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 382,
"column": 2
} | {
"line": 388,
"column": 98
} | {
"line": 390,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\n⊢ ∃ n C, ∀ (x : E) (hx : x ∈ closedBall x₀ ↑r), ContractingWith C (next hf hx)^[n]",
"ppTerm": "?m.33",
"assig... | [] | obtain ⟨n, hn⟩ := FloorSemiring.tendsto_pow_div_factorial_atTop (K * max (tmax - t₀) (t₀ - tmin))
|>.eventually (gt_mem_nhds zero_lt_one) |>.exists
have : (0 : ℝ) ≤ (K * max (tmax - t₀) (t₀ - tmin)) ^ n / n ! := by
have : 0 ≤ max (tmax - t₀) (t₀ - tmin) := le_max_of_le_left <| sub_nonneg_of_le t₀.2.2
posi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 382,
"column": 2
} | {
"line": 388,
"column": 98
} | {
"line": 390,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\n⊢ ∃ n C, ∀ (x : E) (hx : x ∈ closedBall x₀ ↑r), ContractingWith C (next hf hx)^[n]",
"ppTerm": "?m.33",
"assig... | [] | obtain ⟨n, hn⟩ := FloorSemiring.tendsto_pow_div_factorial_atTop (K * max (tmax - t₀) (t₀ - tmin))
|>.eventually (gt_mem_nhds zero_lt_one) |>.exists
have : (0 : ℝ) ≤ (K * max (tmax - t₀) (t₀ - tmin)) ^ n / n ! := by
have : 0 ≤ max (tmax - t₀) (t₀ - tmin) := le_max_of_le_left <| sub_nonneg_of_le t₀.2.2
posi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Polynomial.CauchyBound | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 31
} | {
"line": 114,
"column": 6
} | [
{
"pp": "K : Type u_1\ninst✝ : NormedDivisionRing K\np : K[X]\nhp : p ≠ 0\na : K\nh : ‖p.leadingCoeff‖₊ * ‖a‖₊ ^ p.natDegree ≤ ‖∑ i ∈ range p.natDegree, p.coeff i * a ^ i‖₊\npld : ‖p.leadingCoeff‖₊ ≠ 0\n⊢ ‖a‖₊ ^ p.natDegree = ‖p.leadingCoeff‖₊ * ‖a‖₊ ^ p.natDegree / ‖p.leadingCoeff‖₊",
"ppTerm": "?m.684",
... | [
"case ha\nK : Type u_1\ninst✝ : NormedDivisionRing K\np : K[X]\nhp : p ≠ 0\na : K\nh : ‖p.leadingCoeff‖₊ * ‖a‖₊ ^ p.natDegree ≤ ‖∑ i ∈ range p.natDegree, p.coeff i * a ^ i‖₊\npld : ‖p.leadingCoeff‖₊ ≠ 0\n⊢ ‖p.leadingCoeff‖₊ ≠ 0"
] | rw [mul_div_cancel_left₀] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Polynomial.CauchyBound | {
"line": 122,
"column": 6
} | {
"line": 125,
"column": 44
} | {
"line": 126,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝ : NormedDivisionRing K\np : K[X]\nhp : p ≠ 0\na : K\nh : ‖p.leadingCoeff‖₊ * ‖a‖₊ ^ p.natDegree ≤ ‖∑ i ∈ range p.natDegree, p.coeff i * a ^ i‖₊\npld : ‖p.leadingCoeff‖₊ ≠ 0\n⊢ (∑ x ∈ range p.natDegree, ‖p.coeff x‖₊ * ‖a‖₊ ^ x) / ‖p.leadingCoeff‖₊ ≤\n (∑ x ∈ range p.natDegree, ‖p.... | [] | gcongr (∑ x ∈ _, ?_ * _) / _
rw [cauchyBound, add_tsub_cancel_right]
field_simp
apply le_sup (f := (‖p.coeff ·‖₊)) ‹_› | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Polynomial.CauchyBound | {
"line": 122,
"column": 6
} | {
"line": 125,
"column": 44
} | {
"line": 126,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝ : NormedDivisionRing K\np : K[X]\nhp : p ≠ 0\na : K\nh : ‖p.leadingCoeff‖₊ * ‖a‖₊ ^ p.natDegree ≤ ‖∑ i ∈ range p.natDegree, p.coeff i * a ^ i‖₊\npld : ‖p.leadingCoeff‖₊ ≠ 0\n⊢ (∑ x ∈ range p.natDegree, ‖p.coeff x‖₊ * ‖a‖₊ ^ x) / ‖p.leadingCoeff‖₊ ≤\n (∑ x ∈ range p.natDegree, ‖p.... | [] | gcongr (∑ x ∈ _, ?_ * _) / _
rw [cauchyBound, add_tsub_cancel_right]
field_simp
apply le_sup (f := (‖p.coeff ·‖₊)) ‹_› | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Polynomial.Basic | {
"line": 313,
"column": 2
} | {
"line": 313,
"column": 92
} | {
"line": 314,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : (P.comp (-X)).degree < (Q.comp (-X)).degree\n⊢ (fun x ↦ eval x P) =o[atBot] fun x ↦ eval x Q",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [... | [
"case e'_7\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : (P.comp (-X)).degree < (Q.comp (-X)).degree\nx✝ : 𝕜\n⊢ eval x✝ P = ((fun x ↦ eval x (P.comp (-X))) ∘ Neg.neg) x✝",
"case e'_8\n𝕜 : Type u_1\ninst✝³ : NormedFie... | convert! (isLittleO_atTop_of_degree_lt _ _ h).comp_tendsto tendsto_neg_atBot_atTop using 2 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 566,
"column": 4
} | {
"line": 574,
"column": 19
} | {
"line": 575,
"column": 2
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E → E\nα : ℝ → E\nu : Set E\ntmin tmax : ℝ\nn : ℕ∞\nhf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)\nhα : ∀ t ∈ Icc tmin tmax, HasDerivWithinAt α (f t (α t)) (Icc tmin tmax) t\nhmem... | [] | set t₀ := (tmin + tmax) / 2 with h
have ht₀ : t₀ ∈ Icc tmin tmax := ⟨by linarith, by linarith⟩
have : ∀ t ∈ Icc tmin tmax, α t = picard f t₀ (α t₀) α t := by
intro t ht
have : uIcc t₀ t ⊆ Icc tmin tmax := uIcc_subset_Icc ht₀ ht
rw [picard_eq_of_hasDerivAt (hf.continuousOn.mono (prod_subset_pro... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 566,
"column": 4
} | {
"line": 574,
"column": 19
} | {
"line": 575,
"column": 2
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E → E\nα : ℝ → E\nu : Set E\ntmin tmax : ℝ\nn : ℕ∞\nhf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)\nhα : ∀ t ∈ Icc tmin tmax, HasDerivWithinAt α (f t (α t)) (Icc tmin tmax) t\nhmem... | [] | set t₀ := (tmin + tmax) / 2 with h
have ht₀ : t₀ ∈ Icc tmin tmax := ⟨by linarith, by linarith⟩
have : ∀ t ∈ Icc tmin tmax, α t = picard f t₀ (α t₀) α t := by
intro t ht
have : uIcc t₀ t ⊆ Icc tmin tmax := uIcc_subset_Icc ht₀ ht
rw [picard_eq_of_hasDerivAt (hf.continuousOn.mono (prod_subset_pro... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 214,
"column": 6
} | {
"line": 218,
"column": 12
} | {
"line": 220,
"column": 0
} | [
{
"pp": "R : Type u_1\nF : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : FunLike F R ℝ\nv : F\nc : ℝ\ninst✝² : ZeroHomClass F R ℝ\ninst✝¹ : NonnegHomClass F R ℝ\ninst✝ : MulHomClass F R ℝ\nhna : IsNonarchimedean ⇑v\np q : R[X]\nhc : 0 ≤ c\nhpq : p * q ≠ 0\nh_supp_p : p.support.Nonempty\nh_supp_q : q.support.Nonempty\... | [] | have hp_le := p.le_gaussNorm v hc j
have hq_le := q.le_gaussNorm v hc (i - j)
have := p.gaussNorm_nonneg v hc
simp_all only [gaussNorm, ↓reduceDIte]
gcongr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 214,
"column": 6
} | {
"line": 218,
"column": 12
} | {
"line": 220,
"column": 0
} | [
{
"pp": "R : Type u_1\nF : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : FunLike F R ℝ\nv : F\nc : ℝ\ninst✝² : ZeroHomClass F R ℝ\ninst✝¹ : NonnegHomClass F R ℝ\ninst✝ : MulHomClass F R ℝ\nhna : IsNonarchimedean ⇑v\np q : R[X]\nhc : 0 ≤ c\nhpq : p * q ≠ 0\nh_supp_p : p.support.Nonempty\nh_supp_q : q.support.Nonempty\... | [] | have hp_le := p.le_gaussNorm v hc j
have hq_le := q.le_gaussNorm v hc (i - j)
have := p.gaussNorm_nonneg v hc
simp_all only [gaussNorm, ↓reduceDIte]
gcongr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rat.NatSqrt.Defs | {
"line": 39,
"column": 6
} | {
"line": 39,
"column": 73
} | {
"line": 39,
"column": 73
} | [
{
"pp": "x prec : ℕ\nh : 0 < prec\n⊢ ↑x < (↑(x * prec ^ 2).sqrt / ↑prec + 1 / ↑prec) ^ 2",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Rat.instOfNat",
"Eq.mpr",
"pow_pos",
"Rat.instMul",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",... | [
"x prec : ℕ\nh : 0 < prec\n⊢ ↑x * ↑prec ^ 2 < (↑(x * prec ^ 2).sqrt / ↑prec + 1 / ↑prec) ^ 2 * ↑prec ^ 2"
] | ← mul_lt_mul_iff_of_pos_right (a := (prec ^ 2 : ℚ)) (by positivity) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 172,
"column": 84
} | {
"line": 173,
"column": 36
} | {
"line": 175,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ (X + C z).logMahlerMeasure = log⁺ ‖z‖",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Polynomial.C",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"RingHom.instRingHomClass",
... | [] | by
simp [← sub_neg_eq_add, ← map_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 196,
"column": 79
} | {
"line": 197,
"column": 36
} | {
"line": 199,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ (X + C z).mahlerMeasure = max 1 ‖z‖",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Polynomial.C",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"RingHom.instRingHomClass",
... | [] | by
simp [← sub_neg_eq_add, ← map_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 270,
"column": 2
} | {
"line": 294,
"column": 8
} | {
"line": 296,
"column": 0
} | [
{
"pp": "p : ℂ[X]\n⊢ p.mahlerMeasure ≤ p.sum fun x a ↦ ‖a‖",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"MeasureTheory.ae",
"Polynomial.mahlerMeasure_def_of_ne_zero",
"Iff.mpr",
"one_pow",
"Set.Icc_sdiff_right",
"AddGrou... | [] | by_cases hp : p = 0
· simp [hp]
have : 0 < p.sum fun _ a ↦ ‖a‖ :=
Finset.sum_pos' (fun i _ ↦ norm_nonneg (p.coeff i)) ⟨p.natDegree, by simp [hp]⟩
rw [show (p.sum fun _ a ↦ ‖a‖) = rexp (circleAverage (fun _ ↦ log (p.sum fun _ a ↦ ‖a‖)) 0 1)
by simp [circleAverage_def, mul_assoc, exp_log this], mahlerMeasur... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 270,
"column": 2
} | {
"line": 294,
"column": 8
} | {
"line": 296,
"column": 0
} | [
{
"pp": "p : ℂ[X]\n⊢ p.mahlerMeasure ≤ p.sum fun x a ↦ ‖a‖",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"MeasureTheory.ae",
"Polynomial.mahlerMeasure_def_of_ne_zero",
"Iff.mpr",
"one_pow",
"Set.Icc_sdiff_right",
"AddGrou... | [] | by_cases hp : p = 0
· simp [hp]
have : 0 < p.sum fun _ a ↦ ‖a‖ :=
Finset.sum_pos' (fun i _ ↦ norm_nonneg (p.coeff i)) ⟨p.natDegree, by simp [hp]⟩
rw [show (p.sum fun _ a ↦ ‖a‖) = rexp (circleAverage (fun _ ↦ log (p.sum fun _ a ↦ ‖a‖)) 0 1)
by simp [circleAverage_def, mul_assoc, exp_log this], mahlerMeasur... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 402,
"column": 6
} | {
"line": 402,
"column": 24
} | {
"line": 403,
"column": 6
} | [
{
"pp": "case hbc\nn : ℕ\np : ℂ[X]\nhp : ¬p = 0\nhn : n ≤ p.natDegree\nS : Multiset (Multiset ℂ) := powersetCard (p.natDegree - n) p.roots\nthis : ∀ x ∈ S.toFinset, ∏ x_1 ∈ x.toFinset, ‖x_1‖ ^ count x_1 x ≤ ∏ m ∈ p.roots.toFinset, max 1 ‖m‖ ^ count m p.roots\nx : Multiset ℂ\nhx : x ∈ S.toFinset\n⊢ ∏ b ∈ x.toFin... | [
"case hbc\nn : ℕ\np : ℂ[X]\nhp : ¬p = 0\nhn : n ≤ p.natDegree\nS : Multiset (Multiset ℂ) := powersetCard (p.natDegree - n) p.roots\nthis : ∀ x ∈ S.toFinset, ∏ x_1 ∈ x.toFinset, ‖x_1‖ ^ count x_1 x ≤ ∏ m ∈ p.roots.toFinset, max 1 ‖m‖ ^ count m p.roots\nx : Multiset ℂ\nhx : x ∈ S.toFinset\n⊢ ∏ x_1 ∈ x.toFinset, ‖x_1‖... | simp_rw [norm_pow] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Real.OfDigits | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 63
} | {
"line": 59,
"column": 2
} | [
{
"pp": "b : ℕ\ndigits : ℕ → Fin b\n⊢ Summable fun x ↦ (↑b - 1) * (↑b ^ (x + 1))⁻¹",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"LE.le.eq_or_lt",
"Real",
"Preorder.toLT",
"HMul.hMul",
"Real.instInv",
"Real.instSub",
"PartialOrder.toPreorder",
... | [
"case inl\ndigits : ℕ → Fin 1\n⊢ Summable fun x ↦ (↑1 - 1) * (↑1 ^ (x + 1))⁻¹",
"case inr\nb : ℕ\ndigits : ℕ → Fin b\nhb : 1 < b\n⊢ Summable fun x ↦ (↑b - 1) * (↑b ^ (x + 1))⁻¹"
] | obtain rfl | hb := (Nat.one_le_of_lt (b_pos digits)).eq_or_lt | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Real.OfDigits | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 63
} | {
"line": 75,
"column": 2
} | [
{
"pp": "b : ℕ\ndigits : ℕ → Fin b\n⊢ ofDigits digits ≤ 1",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"LE.le.eq_or_lt",
"Real.instLE",
"Real",
"Preorder.toLT",
"PartialOrder.toPreorder",
"_private.Mathlib.Analysis.Real.OfDigits.0.Real.b_pos",
"i... | [
"case inl\ndigits : ℕ → Fin 1\n⊢ ofDigits digits ≤ 1",
"case inr\nb : ℕ\ndigits : ℕ → Fin b\nhb : 1 < b\n⊢ ofDigits digits ≤ 1"
] | obtain rfl | hb := (Nat.one_le_of_lt (b_pos digits)).eq_or_lt | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Real.OfDigits | {
"line": 149,
"column": 44
} | {
"line": 155,
"column": 45
} | {
"line": 157,
"column": 0
} | [
{
"pp": "x : ℝ\nb : ℕ\ninst✝ : NeZero b\nhb : 1 < b\nhx : x ∈ Set.Ico 0 1\n⊢ HasSum (ofDigitsTerm (x.digits b)) x",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Summable.abs",
"hasSum_iff_tendsto_nat_of_summable_norm",
"Eq.mpr",
"Gr... | [] | by
rw [hasSum_iff_tendsto_nat_of_summable_norm (by exact summable_ofDigitsTerm.abs)]
refine tendsto_of_tendsto_of_tendsto_of_le_of_le ?_ tendsto_const_nhds
(le_sum_ofDigitsTerm_digits hx) (sum_ofDigitsTerm_digits_le hx)
convert! tendsto_const_nhds.sub (tendsto_pow_atTop_nhds_zero_of_abs_lt_one _)
· simp
·... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Real.Irrational | {
"line": 94,
"column": 38
} | {
"line": 94,
"column": 55
} | {
"line": 94,
"column": 55
} | [
{
"pp": "case inr\nn p : ℕ\nhp : Fact (Nat.Prime p)\nhnpos : n > 0\ny : ℤ\nhm : y ^ n ≠ 0\nhv : n * multiplicity (↑p) y % n ≠ 0\nthis : y ≠ 0\n⊢ False",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"CommSemiring.toCommMonoidWithZero",
"Eq.mp... | [
"case inr\nn p : ℕ\nhp : Fact (Nat.Prime p)\nhnpos : n > 0\ny : ℤ\nhm : y ^ n ≠ 0\nhv : 0 ≠ 0\nthis : y ≠ 0\n⊢ False"
] | Nat.mul_mod_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 413,
"column": 6
} | {
"line": 413,
"column": 20
} | {
"line": 413,
"column": 21
} | [
{
"pp": "x : ℝ*\n⊢ x.Infinitesimal ↔ 0 < ArchimedeanClass.mk x",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Hyperreal.instField",
"Eq.mpr",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"ArchimedeanClass.instLinearOrder",
"PartialO... | [
"x : ℝ*\n⊢ x.IsSt 0 ↔ 0 < ArchimedeanClass.mk x"
] | Infinitesimal, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 699,
"column": 65
} | {
"line": 700,
"column": 77
} | {
"line": 702,
"column": 0
} | [
{
"pp": "x : ℝ*\n⊢ |x|.Infinite ↔ x.Infinite",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Hyperreal.instField",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
"abs",
"congrArg",
"PartialOrder.toPreorder",
... | [] | by
cases le_total 0 x <;> simp [*, abs_of_nonneg, abs_of_nonpos, infinite_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Arcosh | {
"line": 78,
"column": 14
} | {
"line": 78,
"column": 22
} | {
"line": 78,
"column": 23
} | [
{
"pp": "x : ℝ\nhx : 1 ≤ x\n⊢ cosh (log (x + √(x ^ 2 - 1))) = x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"Real.instSub",
"Nat.instAtLeastTwoHAddOfNat",
"HSub.hSub",
... | [
"x : ℝ\nhx : 1 ≤ x\n⊢ (rexp (log (x + √(x ^ 2 - 1))) + rexp (-log (x + √(x ^ 2 - 1)))) / 2 = x"
] | cosh_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Arcosh | {
"line": 86,
"column": 2
} | {
"line": 87,
"column": 6
} | {
"line": 89,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : 1 ≤ x\n⊢ sinh (arcosh x) = √(x ^ 2 - 1)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Real.sinh_eq",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClas... | [] | rw [arcosh, sinh_eq, exp_neg, exp_log (by positivity), add_sqrt_self_sq_sub_one_inv hx]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Arcosh | {
"line": 86,
"column": 2
} | {
"line": 87,
"column": 6
} | {
"line": 89,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : 1 ≤ x\n⊢ sinh (arcosh x) = √(x ^ 2 - 1)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Real.sinh_eq",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClas... | [] | rw [arcosh, sinh_eq, exp_neg, exp_log (by positivity), add_sqrt_self_sq_sub_one_inv hx]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 206,
"column": 76
} | {
"line": 208,
"column": 50
} | {
"line": 210,
"column": 0
} | [
{
"pp": "x y : ℝ≥0\nn : ℕ\n⊢ (x.agmSequences y n).1 ≤ x.agm y",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"NNReal.agmSequences",
"Eq.mpr",
"congrArg",
"iSup",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal.agm",
"NNReal.bddAbove_range... | [] | by
rw [agm_eq_ciSup]
exact le_ciSup bddAbove_range_agmSequences_fst _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 184,
"column": 2
} | {
"line": 186,
"column": 30
} | {
"line": 188,
"column": 0
} | [
{
"pp": "case refine_2\nh : DifferentiableAt ℝ (fun p ↦ p * log p⁻¹ + (1 - p) * log (1 - p)⁻¹) 1\n⊢ False",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.toSeminormedCommRing",
"False",
"DifferentiableAt.fun_inv",
... | [] | · rw [DifferentiableAt.fun_add_iff_right, differentiableAt_iff_comp_const_sub (b := 1)] at h
· simp [log_inv, mul_neg, ← neg_mul, ← negMulLog_def, differentiableAt_negMulLog_iff] at h
· fun_prop (disch := simp) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 52,
"column": 41
} | {
"line": 52,
"column": 53
} | {
"line": 53,
"column": 6
} | [
{
"pp": "n : ℕ\nf : ℝ → ℝ := log ∘ Gamma\nhc : ConvexOn ℝ (Ioi 0) f\nh_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x\nhder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x\nx : ℝ\nhx : 0 < x\n⊢ deriv (fun x ↦ f (x + 1)) x = deriv f x + x⁻¹",
"ppTerm": "?m.250",
"assigned": true,
"usedConstants": [
... | [
"n : ℕ\nf : ℝ → ℝ := log ∘ Gamma\nhc : ConvexOn ℝ (Ioi 0) f\nh_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x\nhder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x\nx : ℝ\nhx : 0 < x\n⊢ deriv (fun x ↦ f (x + 1)) x = deriv f x + deriv log x"
] | ← deriv_log, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder | {
"line": 44,
"column": 45
} | {
"line": 44,
"column": 78
} | {
"line": 45,
"column": 8
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nxpos : IsStrictlyPositive x\ny : A\nypos : IsStrictlyPositive y\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nz : A := (conjSqrt x⁻¹ʳ) y\nzpos : IsStrictlyPositive z\nxinvpos : IsStrictlyPositive x⁻¹ʳ... | [
"A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nxpos : IsStrictlyPositive x\ny : A\nypos : IsStrictlyPositive y\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nz : A := (conjSqrt x⁻¹ʳ) y\nzpos : IsStrictlyPositive z\nxinvpos : IsStrictlyPositive x⁻¹ʳ\nhsp : IsSt... | ← Algebra.algebraMap_eq_smul_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder | {
"line": 67,
"column": 32
} | {
"line": 67,
"column": 65
} | {
"line": 67,
"column": 66
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nxpos : IsStrictlyPositive x\ny : A\nypos : IsStrictlyPositive y\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nz : A := (conjSqrt x⁻¹ʳ) y\nzpos : IsStrictlyPositive z\nxinvpos : IsStrictlyPositive x⁻¹ʳ... | [
"A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nxpos : IsStrictlyPositive x\ny : A\nypos : IsStrictlyPositive y\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nz : A := (conjSqrt x⁻¹ʳ) y\nzpos : IsStrictlyPositive z\nxinvpos : IsStrictlyPositive x⁻¹ʳ\nhsp : IsSt... | ← Algebra.algebraMap_eq_smul_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
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