module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 35
} | {
"line": 255,
"column": 2
} | [
{
"pp": "case inl\nK : Type u_1\nK' : Type u_2\ninst✝³ : Field K\ninst✝² : Field K'\ninst✝¹ : Algebra K K'\ninst✝ : Finite K'\n⊢ ∃ a, (Algebra.norm K) a = 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"MonoidHom.instFunLike",
"PrincipalIdealRing.isNoetherianRing",
"Mon... | [] | exact ⟨0, Algebra.norm_zero ..⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 35
} | {
"line": 255,
"column": 2
} | [
{
"pp": "case inl\nK : Type u_1\nK' : Type u_2\ninst✝³ : Field K\ninst✝² : Field K'\ninst✝¹ : Algebra K K'\ninst✝ : Finite K'\n⊢ ∃ a, (Algebra.norm K) a = 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"MonoidHom.instFunLike",
"PrincipalIdealRing.isNoetherianRing",
"Mon... | [] | exact ⟨0, Algebra.norm_zero ..⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 35
} | {
"line": 255,
"column": 2
} | [
{
"pp": "case inl\nK : Type u_1\nK' : Type u_2\ninst✝³ : Field K\ninst✝² : Field K'\ninst✝¹ : Algebra K K'\ninst✝ : Finite K'\n⊢ ∃ a, (Algebra.norm K) a = 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"MonoidHom.instFunLike",
"PrincipalIdealRing.isNoetherianRing",
"Mon... | [] | exact ⟨0, Algebra.norm_zero ..⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 294,
"column": 50
} | {
"line": 295,
"column": 65
} | {
"line": 297,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝³ : Fact (Nat.Prime p)\nk : Type u_3\ninst✝² : AddCommGroup k\ninst✝¹ : Finite k\ninst✝ : Module (ZMod p) k\n⊢ p ^ Module.finrank (ZMod p) k = Nat.card k",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrincipalIdealRing.isNoetherianRing",
... | [] | by
rw [Module.natCard_eq_pow_finrank (K := ZMod p), Nat.card_zmod] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 300,
"column": 6
} | {
"line": 300,
"column": 29
} | {
"line": 300,
"column": 30
} | [
{
"pp": "p : ℕ\ninst✝³ : Fact (Nat.Prime p)\nk : Type u_3\ninst✝² : AddCommGroup k\ninst✝¹ : Fintype k\ninst✝ : Module (ZMod p) k\n⊢ p ^ Module.finrank (ZMod p) k = Fintype.card k",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddCommGroup.toAddCom... | [
"p : ℕ\ninst✝³ : Fact (Nat.Prime p)\nk : Type u_3\ninst✝² : AddCommGroup k\ninst✝¹ : Fintype k\ninst✝ : Module (ZMod p) k\n⊢ Nat.card k = Fintype.card k"
] | pow_finrank_eq_natCard, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 686,
"column": 2
} | {
"line": 687,
"column": 78
} | {
"line": 689,
"column": 0
} | [
{
"pp": "case refine_2\nn : ℕ\ninst✝⁷ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : NeZero ↑n\nthis✝¹ : NeZero ↑n\nthis✝ : DecidableEq (CyclotomicField n K... | [] | · rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 691,
"column": 20
} | {
"line": 698,
"column": 63
} | {
"line": 700,
"column": 0
} | [
{
"pp": "n : ℕ\ninst✝⁶ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : CyclotomicField 0 K\n⊢ x ∈ adjoin K {b | ∃ n ∈ {0}, n ≠ 0 ∧ b ^ n = 1}",
"ppTerm": "?m.... | [] | by
have finrank : Module.finrank K (CyclotomicField 0 K) = 1 := by
have : Polynomial.IsSplittingField K K (Polynomial.cyclotomic 0 K) :=
Polynomial.isSplittingField_C 1
let e : K ≃ₗ[K] (CyclotomicField 0 K) :=
(Polynomial.IsSplittingField.algEquiv K (Polynomial.cyclotomic 0 K)).toLinearE... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 158,
"column": 24
} | {
"line": 158,
"column": 77
} | {
"line": 159,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nf : Rˣ →* R'ˣ\n⊢ R → R'",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Units.val",
"MonoidHom.instFunLike",
"MonoidHom",
"IsUnit",
"Classical.propDecidable",
"E... | [] | exact fun x => if hx : IsUnit x then f hx.unit else 0 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 158,
"column": 24
} | {
"line": 158,
"column": 77
} | {
"line": 159,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nf : Rˣ →* R'ˣ\n⊢ R → R'",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Units.val",
"MonoidHom.instFunLike",
"MonoidHom",
"IsUnit",
"Classical.propDecidable",
"E... | [] | exact fun x => if hx : IsUnit x then f hx.unit else 0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 158,
"column": 24
} | {
"line": 158,
"column": 77
} | {
"line": 159,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nf : Rˣ →* R'ˣ\n⊢ R → R'",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Units.val",
"MonoidHom.instFunLike",
"MonoidHom",
"IsUnit",
"Classical.propDecidable",
"E... | [] | exact fun x => if hx : IsUnit x then f hx.unit else 0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 808,
"column": 41
} | {
"line": 808,
"column": 53
} | {
"line": 808,
"column": 53
} | [
{
"pp": "n : ℕ\ninst✝¹⁰ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra A K\ninst✝² : IsFractionRing A K\ninst✝¹ : IsDomain A\ninst✝ : NeZero ↑n\nx ... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 10
} | {
"line": 208,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nhn : n = 1\n⊢ χ = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"ZMod.commRing",
"MulChar.hasOne",
"instOfNatNat",
"ZMod",
"Nat",
"DirichletCharacter",
"Eq.n... | [
"R : Type u_1\ninst✝ : CommMonoidWithZero R\nχ : DirichletCharacter R 1\n⊢ χ = 1"
] | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.NumberTheory.MulChar.Lemmas | {
"line": 184,
"column": 49
} | {
"line": 190,
"column": 26
} | {
"line": 192,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\ninst✝¹ : Finite F\nR : Type u_2\ninst✝ : CommRing R\nχ : MulChar F R\na : F\nha : a ≠ 0\n⊢ ∃ ζ ∈ rootsOfUnity (orderOf χ) R, ↑ζ = χ a",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"mem_rootsOfUnity",
"Units.val",
... | [] | by
have hu : IsUnit (χ a) := ha.isUnit.map χ
refine ⟨hu.unit, ?_, hu.unit_spec⟩
rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one,
IsUnit.unit_spec, ← χ.pow_apply' χ.orderOf_pos.ne', pow_orderOf_eq_one,
show a = (isUnit_iff_ne_zero.mpr ha).unit by simp only [IsUnit.unit_spec],
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 325,
"column": 4
} | {
"line": 325,
"column": 86
} | {
"line": 326,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\ninst✝ : NeZero n\nthis : NeZero (conductor 1)\n⊢ (primitiveCharacter 1).conductor = 1",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"DirichletCharacter.conductor",
"Eq.mpr",
"DirichletCharacter.isPrimitive_def... | [
"R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\ninst✝ : NeZero n\nthis : NeZero (conductor 1)\n⊢ conductor 1 = 1"
] | (isPrimitive_def _).1 (1 : DirichletCharacter R n).primitiveCharacter_isPrimitive, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 454,
"column": 17
} | {
"line": 454,
"column": 37
} | {
"line": 455,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\ninst✝ : NeZero n\nd : ℕ\n⊢ 1 ∈ {χ | d.Coprime χ.conductor}",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"DirichletCharacter.conductor",
"Nat.Coprime",
"MulOne.toOne",
"ZMod.c... | [] | simp [conductor_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 454,
"column": 17
} | {
"line": 454,
"column": 37
} | {
"line": 455,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\ninst✝ : NeZero n\nd : ℕ\n⊢ 1 ∈ {χ | d.Coprime χ.conductor}",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"DirichletCharacter.conductor",
"Nat.Coprime",
"MulOne.toOne",
"ZMod.c... | [] | simp [conductor_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 454,
"column": 17
} | {
"line": 454,
"column": 37
} | {
"line": 455,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\ninst✝ : NeZero n\nd : ℕ\n⊢ 1 ∈ {χ | d.Coprime χ.conductor}",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"DirichletCharacter.conductor",
"Nat.Coprime",
"MulOne.toOne",
"ZMod.c... | [] | simp [conductor_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 855,
"column": 15
} | {
"line": 855,
"column": 81
} | {
"line": 855,
"column": 81
} | [
{
"pp": "S : Set ℕ\nK : Type w\ninst✝¹ : Field K\ninst✝ : IsSepClosed K\nh : ∀ a ∈ S, a ≠ 0 → NeZero ↑a\na : ℕ\nha : a ∈ S\nha' : a ≠ 0\nthis : NeZero ↑a\nr : K\nhr : (aeval r) (cyclotomic a K) = 0\n⊢ IsPrimitiveRoot r a",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Polynomial.eva... | [] | rwa [coe_aeval_eq_eval, ← IsRoot.def, isRoot_cyclotomic_iff] at hr | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 855,
"column": 15
} | {
"line": 855,
"column": 81
} | {
"line": 855,
"column": 81
} | [
{
"pp": "S : Set ℕ\nK : Type w\ninst✝¹ : Field K\ninst✝ : IsSepClosed K\nh : ∀ a ∈ S, a ≠ 0 → NeZero ↑a\na : ℕ\nha : a ∈ S\nha' : a ≠ 0\nthis : NeZero ↑a\nr : K\nhr : (aeval r) (cyclotomic a K) = 0\n⊢ IsPrimitiveRoot r a",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Polynomial.eva... | [] | rwa [coe_aeval_eq_eval, ← IsRoot.def, isRoot_cyclotomic_iff] at hr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 855,
"column": 15
} | {
"line": 855,
"column": 81
} | {
"line": 855,
"column": 81
} | [
{
"pp": "S : Set ℕ\nK : Type w\ninst✝¹ : Field K\ninst✝ : IsSepClosed K\nh : ∀ a ∈ S, a ≠ 0 → NeZero ↑a\na : ℕ\nha : a ∈ S\nha' : a ≠ 0\nthis : NeZero ↑a\nr : K\nhr : (aeval r) (cyclotomic a K) = 0\n⊢ IsPrimitiveRoot r a",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Polynomial.eva... | [] | rwa [coe_aeval_eq_eval, ← IsRoot.def, isRoot_cyclotomic_iff] at hr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.GaussSum | {
"line": 313,
"column": 24
} | {
"line": 313,
"column": 36
} | {
"line": 313,
"column": 37
} | [
{
"pp": "F : Type u_1\ninst✝³ : Field F\ninst✝² : Fintype F\nF' : Type u_2\ninst✝¹ : Field F'\ninst✝ : Fintype F'\nχ : MulChar F F'\nhχ₁ : χ ≠ 1\nhχ₂ : χ.IsQuadratic\nhch₁ : ringChar F' ≠ ringChar F\nhch₂ : ringChar F' ≠ 2\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nn' : ℕ+\nhp'... | [
"F : Type u_1\ninst✝³ : Field F\ninst✝² : Fintype F\nF' : Type u_2\ninst✝¹ : Field F'\ninst✝ : Fintype F'\nχ : MulChar F F'\nhχ₁ : χ ≠ 1\nhχ₂ : χ.IsQuadratic\nhch₁ : ringChar F' ≠ ringChar F\nhch₂ : ringChar F' ≠ 2\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nn' : ℕ+\nhp' : Nat.Prime... | map_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.GaussSum | {
"line": 317,
"column": 47
} | {
"line": 317,
"column": 73
} | {
"line": 317,
"column": 74
} | [
{
"pp": "F : Type u_1\ninst✝³ : Field F\ninst✝² : Fintype F\nF' : Type u_2\ninst✝¹ : Field F'\ninst✝ : Fintype F'\nχ : MulChar F F'\nhχ₁ : χ ≠ 1\nhχ₂ : χ.IsQuadratic\nhch₁✝ : ringChar F' ≠ ringChar F\nhch₁ : ¬ringChar F' ∣ ringChar F\nhch₂ : ringChar F' ≠ 2\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.car... | [
"F : Type u_1\ninst✝³ : Field F\ninst✝² : Fintype F\nF' : Type u_2\ninst✝¹ : Field F'\ninst✝ : Fintype F'\nχ : MulChar F F'\nhχ₁ : χ ≠ 1\nhχ₂ : χ.IsQuadratic\nhch₁✝ : ringChar F' ≠ ringChar F\nhch₁ : IsUnit ↑(ringChar F')\nhch₂ : ringChar F' ≠ 2\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F... | ← isUnit_iff_not_dvd_char, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 135,
"column": 4
} | {
"line": 138,
"column": 34
} | {
"line": 139,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : Coalgebra 𝕜 E\nx : E\n⊢ (adjoint comul) (x ⊗ₜ[𝕜] (adjoint counit) One.one) = x",
"ppTerm": "?m.203",
"assigned": true,
"usedConstants":... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : Coalgebra 𝕜 E\nx : E\n⊢ ↑((commIsometry 𝕜 E 𝕜).trans (lidIsometry 𝕜 E)).symm.symm.toLinearEquiv (x ⊗ₜ[𝕜] One.one) = x"
] | rw [← lTensor_tmul, ← comp_apply, ← adjoint_lTensor, ← adjoint_comp, lTensor_counit_comp_comul,
← toLinearMap_symm_rid, ← comm_trans_lid, ← toLinearEquiv_commIsometry,
← toLinearEquiv_lidIsometry, ← toLinearEquiv_trans, ← toLinearEquiv_symm,
adjoint_toLinearMap_eq_symm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 358,
"column": 2
} | {
"line": 358,
"column": 58
} | {
"line": 359,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : T.IsPositive\nS : E →L[𝕜] F\nx : F\n⊢ 0... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : T.IsPositive\nS : E →L[𝕜] F\nx : F\n⊢ 0 ≤ re ⟪(T ∘S... | rw [reApplyInnerSelf, comp_apply, ← adjoint_inner_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 362,
"column": 28
} | {
"line": 363,
"column": 44
} | {
"line": 365,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nS : E →L[𝕜] F\n⊢ (S ∘SL adjoint S).IsPositive",
"ppTerm"... | [] | by
simpa using! isPositive_one.conj_adjoint S | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.SingularValues | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 10
} | {
"line": 117,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : FiniteDimensional 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\nn : ℕ\nhn : finrank 𝕜 E = n\ni... | [
"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : FiniteDimensional 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\ni : Fin (finrank 𝕜 E)\n⊢ T.singularValues ... | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 187,
"column": 4
} | {
"line": 188,
"column": 29
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n ... | [] | rw [htop, add_top, ← htop]
exact μ.mono sdiff_subset | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 187,
"column": 4
} | {
"line": 188,
"column": 29
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n ... | [] | rw [htop, add_top, ← htop]
exact μ.mono sdiff_subset | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 525,
"column": 68
} | {
"line": 525,
"column": 83
} | {
"line": 525,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\ng : E →L[𝕜] F\n⊢ ‖rTensor G g‖ ≤ ‖... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\ng : E →L[𝕜] F\n⊢ ‖g‖ ≤ ‖g‖"
] | norm_rTensor_le | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 218,
"column": 8
} | {
"line": 218,
"column": 27
} | {
"line": 218,
"column": 28
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreS... | [
"X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreSeparated (S ... | ENNReal.inv_lt_inv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 285,
"column": 55
} | {
"line": 285,
"column": 65
} | {
"line": 285,
"column": 65
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nC r : ℝ≥0\nf : X → Y\ns : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis✝³ : MeasurableSpace X := borel X\nthis✝² : BorelSpace X\nthis✝¹ : MeasurableSpace Y := borel Y\nthis✝ : BorelSpace Y\nd : ℝ≥0\nhd : μH[↑d] (f '' s) = ... | [
"X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nC r : ℝ≥0\nf : X → Y\ns : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis✝³ : MeasurableSpace X := borel X\nthis✝² : BorelSpace X\nthis✝¹ : MeasurableSpace Y := borel Y\nthis✝ : BorelSpace Y\nd : ℝ≥0\nhd : μH[↑d] (f '' s) = ∞\nthis : ↑(... | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 378,
"column": 10
} | {
"line": 378,
"column": 20
} | {
"line": 378,
"column": 20
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nK : ℝ≥0\nf : X → Y\nhf : AntilipschitzWith K f\ns : Set Y\nthis✝³ : MeasurableSpace X := borel X\nthis✝² : BorelSpace X\nthis✝¹ : MeasurableSpace Y := borel Y\nthis✝ : BorelSpace Y\nd : ℝ≥0\nhd : μH[↑d] (f ⁻¹' s) = ∞\nthis : ∞... | [
"X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nK : ℝ≥0\nf : X → Y\nhf : AntilipschitzWith K f\ns : Set Y\nthis✝³ : MeasurableSpace X := borel X\nthis✝² : BorelSpace X\nthis✝¹ : MeasurableSpace Y := borel Y\nthis✝ : BorelSpace Y\nd : ℝ≥0\nhd : μH[↑d] (f ⁻¹' s) = ∞\nthis : ↑K ^ ↑d * μH[... | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 375,
"column": 2
} | {
"line": 380,
"column": 41
} | {
"line": 382,
"column": 0
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nK : ℝ≥0\nf : X → Y\nhf : AntilipschitzWith K f\ns : Set Y\n⊢ dimH (f ⁻¹' s) ≤ dimH s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"NNReal.not_toReal_neg._simp_... | [] | borelize X Y
refine dimH_le fun d hd => le_dimH_of_hausdorffMeasure_eq_top ?_
have := hf.hausdorffMeasure_preimage_le d.coe_nonneg s
rw [hd, top_le_iff] at this
contrapose! this
exact ENNReal.mul_ne_top (by simp) this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 375,
"column": 2
} | {
"line": 380,
"column": 41
} | {
"line": 382,
"column": 0
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nK : ℝ≥0\nf : X → Y\nhf : AntilipschitzWith K f\ns : Set Y\n⊢ dimH (f ⁻¹' s) ≤ dimH s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"NNReal.not_toReal_neg._simp_... | [] | borelize X Y
refine dimH_le fun d hd => le_dimH_of_hausdorffMeasure_eq_top ?_
have := hf.hausdorffMeasure_preimage_le d.coe_nonneg s
rw [hd, top_le_iff] at this
contrapose! this
exact ENNReal.mul_ne_top (by simp) this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.OfNorm | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 95
} | {
"line": 134,
"column": 2
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx y : E\nhI : ¬I = 0\nhI' : I * I = -1\n⊢ 4⁻¹ *\n (↑‖x + y‖ * ↑‖x + y‖ - ↑‖x - y‖ * ↑‖x - y‖ + -I * ↑‖I • y + x‖ * ↑‖I • y + x‖ -\n -I * ↑‖I • y - x‖ * ↑‖I • y - x‖) =\n 4⁻... | [
"case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx y : E\nhI : ¬I = 0\nhI' : I * I = -1\nI_smul : ∀ (v : E), ‖I • v‖ = ‖v‖\n⊢ 4⁻¹ *\n (↑‖x + y‖ * ↑‖x + y‖ - ↑‖x - y‖ * ↑‖x - y‖ + -I * ↑‖I • y + x‖ * ↑‖I • y + x‖ -\n -I * ↑‖I • y - x‖ * ... | have I_smul (v : E) : ‖(I : 𝕜) • v‖ = ‖v‖ := by rw [norm_smul, norm_I_of_ne_zero hI, one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 67
} | {
"line": 124,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\ng : U ≃ₗ[𝕜] ↥f.range := LinearEquiv.ofBijective ... | [] | simpa [toMatrix_adjoint, Matrix.det_conjTranspose] using this | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 67
} | {
"line": 124,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\ng : U ≃ₗ[𝕜] ↥f.range := LinearEquiv.ofBijective ... | [] | simpa [toMatrix_adjoint, Matrix.det_conjTranspose] using this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 67
} | {
"line": 124,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\ng : U ≃ₗ[𝕜] ↥f.range := LinearEquiv.ofBijective ... | [] | simpa [toMatrix_adjoint, Matrix.det_conjTranspose] using this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 626,
"column": 4
} | {
"line": 632,
"column": 9
} | {
"line": 633,
"column": 4
} | [
{
"pp": "case a\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nx : X\n⊢ 1 ≤\n ⨆ r,\n ⨆ (_ : 0 < r),\n ⨅ t,\n ⨅ (_ : {x} ⊆ ⋃ n, t n),\n ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ediam (t n) ^ 0",
"ppTe... | [
"case a\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nx : X\n⊢ 1 ≤\n ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ediam (t n) ^ 0"
] | suffices
(1 : ℝ≥0∞) ≤
⨅ (t : ℕ → Set X) (_ : {x} ⊆ ⋃ n, t n) (_ : ∀ n, ediam (t n) ≤ 1),
∑' n, ⨆ _ : (t n).Nonempty, ediam (t n) ^ (0 : ℝ) by
apply le_trans this _
convert! le_iSup₂ (α := ℝ≥0∞) (1 : ℝ≥0∞) zero_lt_one
rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 59
} | {
"line": 236,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup U\ninst✝⁴ : InnerProductSpace 𝕜 U\ninst✝³ : FiniteDimensional 𝕜 U\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : Subsingleton U\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\nhrank : finrank 𝕜 U = 0... | [] | simp [normDet_eq_norm_det_toMatrix_rangeRestrict f bu bv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Reproducing | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 40
} | {
"line": 144,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : InnerProductSpace 𝕜 V\nH : Type u_4\ninst✝⁴ : NormedAddCommGroup H\ninst✝³ : InnerProductSpace 𝕜 H\ninst✝² : RKHS 𝕜 H X V\ninst✝¹ : CompleteSpace H\ninst✝ : CompleteSpace V\nx y : X\n⊢ ‖adjoint (ke... | [] | simp [norm_kerFun_eq_sqrt_norm_kernel] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Reproducing | {
"line": 151,
"column": 46
} | {
"line": 151,
"column": 78
} | {
"line": 151,
"column": 79
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : InnerProductSpace 𝕜 V\nH : Type u_4\ninst✝⁴ : NormedAddCommGroup H\ninst✝³ : InnerProductSpace 𝕜 H\ninst✝² : RKHS 𝕜 H X V\ninst✝¹ : CompleteSpace H\ninst✝ : CompleteSpace V\nf : H\nx : X\n⊢ ‖kerFun... | [
"𝕜 : Type u_1\ninst✝⁷ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : InnerProductSpace 𝕜 V\nH : Type u_4\ninst✝⁴ : NormedAddCommGroup H\ninst✝³ : InnerProductSpace 𝕜 H\ninst✝² : RKHS 𝕜 H X V\ninst✝¹ : CompleteSpace H\ninst✝ : CompleteSpace V\nf : H\nx : X\n⊢ √‖kernel H x x‖ * ‖... | norm_kerFun_eq_sqrt_norm_kernel, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 408,
"column": 4
} | {
"line": 410,
"column": 88
} | {
"line": 411,
"column": 4
} | [
{
"pp": "case pos\nU : Type u_5\nV : Type u_6\ninst✝⁸ : NormedAddCommGroup U\ninst✝⁷ : InnerProductSpace ℝ U\ninst✝⁶ : FiniteDimensional ℝ U\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MeasurableSpace U\ninst✝² : BorelSpace U\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nf : U ... | [
"case pos\nU : Type u_5\nV : Type u_6\ninst✝⁸ : NormedAddCommGroup U\ninst✝⁷ : InnerProductSpace ℝ U\ninst✝⁶ : FiniteDimensional ℝ U\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MeasurableSpace U\ninst✝² : BorelSpace U\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nf : U →ₗ[ℝ] V\ns :... | rw [(LinearIsometry.isometry _).hausdorffMeasure_image (by simp),
addHaar_image_linearMap μH[finrank ℝ U], ← normDet_eq_abs_det,
normDet_comp_of_finrank_eq _ _ hrank.symm, g.symm.toLinearIsometry.normDet_eq_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.LocallyConvex.ContinuousOfBounded | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 78
} | {
"line": 113,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : No... | [
"𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : NormedField 𝕜... | simp only [← Set.range_comp', LinearMap.map_smulₛₗ, map_inv₀, map_pow] at hf | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 92
} | {
"line": 134,
"column": 0
} | [
{
"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\n⊢ |(o.areaForm x) y| ≤ ‖x‖ * ‖y‖",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"AlternatingMap",
"Norm.norm",
"Eq.... | [] | simpa [areaForm_to_volumeForm, Fin.prod_univ_succ] using o.abs_volumeForm_apply_le ![x, y] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 92
} | {
"line": 134,
"column": 0
} | [
{
"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\n⊢ |(o.areaForm x) y| ≤ ‖x‖ * ‖y‖",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"AlternatingMap",
"Norm.norm",
"Eq.... | [] | simpa [areaForm_to_volumeForm, Fin.prod_univ_succ] using o.abs_volumeForm_apply_le ![x, y] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 92
} | {
"line": 134,
"column": 0
} | [
{
"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\n⊢ |(o.areaForm x) y| ≤ ‖x‖ * ‖y‖",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"AlternatingMap",
"Norm.norm",
"Eq.... | [] | simpa [areaForm_to_volumeForm, Fin.prod_univ_succ] using o.abs_volumeForm_apply_le ![x, y] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 240,
"column": 2
} | {
"line": 241,
"column": 47
} | {
"line": 243,
"column": 0
} | [
{
"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\n⊢ ⟪o.rightAngleRotation x, y⟫ = (o.areaForm x) y",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"LinearIsometryEquiv.instEquivL... | [] | rw [rightAngleRotation]
exact o.inner_rightAngleRotationAux₁_left x y | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 240,
"column": 2
} | {
"line": 241,
"column": 47
} | {
"line": 243,
"column": 0
} | [
{
"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\n⊢ ⟪o.rightAngleRotation x, y⟫ = (o.areaForm x) y",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"LinearIsometryEquiv.instEquivL... | [] | rw [rightAngleRotation]
exact o.inner_rightAngleRotationAux₁_left x y | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 73
} | {
"line": 125,
"column": 4
} | [
{
"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... | [
"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\ninst✝¹⁵ : IsTop... | obtain ⟨f, hf⟩ := SeparatingDual.exists_separating_of_ne (R := R) hne | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.MellinInversion | {
"line": 44,
"column": 29
} | {
"line": 44,
"column": 53
} | {
"line": 45,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nx : ℝ\ns : ℂ\nf : E\n| cexp (-↑x)",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"congrArg",
"Complex.instPow",
"Complex.ofReal",
"Complex.cpow_one",
"HPow.hPow",
"Complex.ex... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nx : ℝ\ns : ℂ\nf : E\n| cexp (-↑x) ^ 1"
] | rw [← cpow_one (cexp _)] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Analysis.LocallyConvex.WeakOperatorTopology | {
"line": 516,
"column": 24
} | {
"line": 516,
"column": 85
} | {
"line": 517,
"column": 2
} | [
{
"pp": "𝕜₁ : Type u_5\n𝕜₂ : Type u_6\n𝕜₃ : Type u_7\nE : Type u_9\nF : Type u_10\nG : Type u_11\ninst✝¹⁸ : NormedField 𝕜₁\ninst✝¹⁷ : NormedField 𝕜₂\ninst✝¹⁶ : NormedField 𝕜₃\nσ₁₂ : 𝕜₁ →+* 𝕜₂\nσ₁₃ : 𝕜₁ →+* 𝕜₃\nσ₂₃ : 𝕜₂ →+* 𝕜₃\ninst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹⁴ : AddCommGroup E\ninst✝¹... | [
"𝕜₁ : Type u_5\n𝕜₂ : Type u_6\n𝕜₃ : Type u_7\nE : Type u_9\nF : Type u_10\nG : Type u_11\ninst✝¹⁸ : NormedField 𝕜₁\ninst✝¹⁷ : NormedField 𝕜₂\ninst✝¹⁶ : NormedField 𝕜₃\nσ₁₂ : 𝕜₁ →+* 𝕜₂\nσ₁₃ : 𝕜₁ →+* 𝕜₃\nσ₂₃ : 𝕜₂ →+* 𝕜₃\ninst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹⁴ : AddCommGroup E\ninst✝¹³ : Topologi... | rw [← σ_li.apply_symm_apply (z _), comp_apply, ← toCLM_apply] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 101,
"column": 2
} | {
"line": 107,
"column": 8
} | {
"line": 108,
"column": 2
} | [
{
"pp": "case e'_3\ns 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)... | [
"case e'_4\ns 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)... | · refine setIntegral_congr_fun measurableSet_Ioi fun x hx => ?_
have A : exp (-x) = exp (-a * x) * exp (-b * x) := by
rw [← exp_add, ← add_mul, ← neg_add, hab, neg_one_mul]
have B : x ^ (a * s + b * t - 1) = x ^ (a * (s - 1)) * x ^ (b * (t - 1)) := by
rw [← rpow_add hx, hab']; congr 1; ring
rw [... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 161,
"column": 2
} | {
"line": 178,
"column": 27
} | {
"line": 180,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ ∫ (x : ℝ) in 0..π / 2, cos x ^ n = 1 / 2 * ∫ (x : ℝ) in 0..π, sin x ^ n",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Mathlib.Tactic.Ring.Common.neg... | [] | rw [mul_comm (1 / 2 : ℝ), ← div_eq_iff (one_div_ne_zero (two_ne_zero' ℝ)), ← div_mul, div_one,
mul_two]
have L : IntervalIntegrable _ volume 0 (π / 2) :=
(continuous_sin.fun_pow n).intervalIntegrable _ _
have R : IntervalIntegrable _ volume (π / 2) π :=
(continuous_sin.fun_pow n).intervalIntegrable _ _
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 161,
"column": 2
} | {
"line": 178,
"column": 27
} | {
"line": 180,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ ∫ (x : ℝ) in 0..π / 2, cos x ^ n = 1 / 2 * ∫ (x : ℝ) in 0..π, sin x ^ n",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Mathlib.Tactic.Ring.Common.neg... | [] | rw [mul_comm (1 / 2 : ℝ), ← div_eq_iff (one_div_ne_zero (two_ne_zero' ℝ)), ← div_mul, div_one,
mul_two]
have L : IntervalIntegrable _ volume 0 (π / 2) :=
(continuous_sin.fun_pow n).intervalIntegrable _ _
have R : IntervalIntegrable _ volume (π / 2) π :=
(continuous_sin.fun_pow n).intervalIntegrable _ _
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 211,
"column": 68
} | {
"line": 211,
"column": 81
} | {
"line": 211,
"column": 82
} | [
{
"pp": "z : ℂ\nhz : z ≠ 0\nn : ℕ\nA : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)\nB : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)\nC : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)\nhn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C\naux' : 2 * n.succ = 2 * n + 2\n⊢ (↑(2 * n... | [
"z : ℂ\nhz : z ≠ 0\nn : ℕ\nA : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)\nB : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)\nC : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)\nhn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C\naux' : 2 * n.succ = 2 * n + 2\n⊢ (↑2 * ↑n + 1) / (↑2 ... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 170,
"column": 35
} | {
"line": 175,
"column": 95
} | {
"line": 177,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\nx : ℝ\nn : ℕ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhn : n ≠ 0\nhx : 0 < x\nhx' : x ≤ 1\n⊢ f (↑n + x) ≤ f ↑n + x * log ↑n",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
... | [] | by
have hn' : 0 < (n : ℝ) := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn)
have : f n + x * log n = (1 - x) * f n + x * f (n + 1) := by rw [hf_feq hn']; ring
rw [this, (by ring : (n : ℝ) + x = (1 - x) * n + x * (n + 1))]
simpa only [smul_eq_mul] using
hf_conv.2 hn' (by linarith : 0 < (n + 1 : ℝ)) (by linarith : ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.MellinTransform | {
"line": 377,
"column": 6
} | {
"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... | [
"case refine_2\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 ^ (z - 1) •... | · 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.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 194,
"column": 6
} | {
"line": 194,
"column": 19
} | {
"line": 194,
"column": 20
} | [
{
"pp": "x : ℝ\nn : ℕ\n⊢ (x + 1) * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + 1 + ↑m) =\n x * log ↑(n + 1) + log ↑((n + 1) * n !) - (∑ k ∈ Finset.range (n + 1), log (x + ↑(k + 1)) + log x) + log x -\n (x + 1) * (log (↑n + 1) - log ↑n)",
"ppTerm": "?m.60",
"assigned": true,
"use... | [
"x : ℝ\nn : ℕ\n⊢ (x + 1) * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + 1 + ↑m) =\n x * log ↑(n + 1) + log (↑(n + 1) * ↑n !) - (∑ k ∈ Finset.range (n + 1), log (x + ↑(k + 1)) + log x) + log x -\n (x + 1) * (log (↑n + 1) - log ↑n)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 212,
"column": 28
} | {
"line": 212,
"column": 41
} | {
"line": 212,
"column": 42
} | [
{
"pp": "case succ.refine_2\nn : ℕ\nIH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)\nu : ℂ\nhu : 0 < u.re\nthis : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u\n⊢ (↑n + 1) * (↑n ! / ∏ j ∈ Finset.range (n + 1), (u + 1 + ↑j)) / u =\n ... | [
"case succ.refine_2\nn : ℕ\nIH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)\nu : ℂ\nhu : 0 < u.re\nthis : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u\n⊢ (↑n + 1) * (↑n ! / ∏ j ∈ Finset.range (n + 1), (u + 1 + ↑j)) / u =\n ↑(n + 1) * ↑... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 379,
"column": 2
} | {
"line": 379,
"column": 59
} | {
"line": 380,
"column": 2
} | [
{
"pp": "z : ℂ\nn : ℕ\nhn : n ≠ 0\naux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)\n⊢ z.GammaSeq n * (1 - z).GammaSeq n = ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))",
"ppTerm": "?m.143",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHD... | [
"z : ℂ\nn : ℕ\nhn : n ≠ 0\naux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)\n⊢ ↑n ^ z * ↑n ^ (1 - z) * ↑n ! ^ 2 /\n ((∏ j ∈ Finset.range (n + 1), (z + ↑j)) * ∏ j ∈ Finset.range (n + 1), (1 - z + ↑j)) =\n ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))"
] | rw [GammaSeq, GammaSeq, div_mul_div_comm, aux, ← pow_two] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 405,
"column": 72
} | {
"line": 405,
"column": 81
} | {
"line": 406,
"column": 6
} | [
{
"pp": "case pos\nz : ℂ\npi_ne : ↑π ≠ 0\nk : ℤ\nhk : -z = ↑k ∨ False\n⊢ Gamma z * Gamma (1 - z) = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Int.cast",
"False",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"Complex.instMul",
"Eq.... | [
"case pos\nz : ℂ\npi_ne : ↑π ≠ 0\nk : ℤ\nhk : -z = ↑k\n⊢ Gamma z * Gamma (1 - z) = 0"
] | or_false, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 409,
"column": 6
} | {
"line": 409,
"column": 90
} | {
"line": 410,
"column": 4
} | [
{
"pp": "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",
"ppTerm": "?pos.ofNat✝",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Int.cast_n... | [] | rw [Int.ofNat_eq_natCast, Int.cast_natCast, Complex.Gamma_neg_nat_eq_zero, zero_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 409,
"column": 6
} | {
"line": 409,
"column": 90
} | {
"line": 410,
"column": 4
} | [
{
"pp": "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",
"ppTerm": "?pos.ofNat✝",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Int.cast_n... | [] | rw [Int.ofNat_eq_natCast, Int.cast_natCast, Complex.Gamma_neg_nat_eq_zero, zero_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 409,
"column": 6
} | {
"line": 409,
"column": 90
} | {
"line": 410,
"column": 4
} | [
{
"pp": "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",
"ppTerm": "?pos.ofNat✝",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Int.cast_n... | [] | rw [Int.ofNat_eq_natCast, Int.cast_natCast, Complex.Gamma_neg_nat_eq_zero, zero_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 435,
"column": 4
} | {
"line": 435,
"column": 17
} | {
"line": 436,
"column": 4
} | [
{
"pp": "case pos\ns : ℂ\nh_im : s.im = 0\nthis : s = ↑s.re\nn : ℕ\nhs : s ≠ -↑n\n⊢ s.re ≠ -↑n",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Real",
"Mathlib.Tactic.Contrapose.contrapose₄",
"Complex.instNatCast",
"Nat.cast",
"Complex.re",
"Real.instNeg... | [
"case pos\ns : ℂ\nh_im : s.im = 0\nthis : s = ↑s.re\nn : ℕ\nhs : s.re = -↑n\n⊢ s = -↑n"
] | contrapose hs | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 17
} | {
"line": 90,
"column": 18
} | [
{
"pp": "𝕜 : Type u_1\nR : Type u_3\nM : Type u_4\ninst✝¹⁶ : Field 𝕜\ninst✝¹⁵ : CharZero 𝕜\ninst✝¹⁴ : Ring R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra 𝕜 R\ninst✝¹¹ : Module 𝕜 M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁷ : IsScalarTower 𝕜 R M\ninst✝⁶ : IsScala... | [
"𝕜 : Type u_1\nR : Type u_3\nM : Type u_4\ninst✝¹⁶ : Field 𝕜\ninst✝¹⁵ : CharZero 𝕜\ninst✝¹⁴ : Ring R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra 𝕜 R\ninst✝¹¹ : Module 𝕜 M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁷ : IsScalarTower 𝕜 R M\ninst✝⁶ : IsScalarTower 𝕜 Rᵐ... | Nat.cast_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 324,
"column": 37
} | {
"line": 324,
"column": 61
} | {
"line": 326,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝³ : NormedRing F\ninst✝² : NormedAlgebra ℝ F\ninst✝¹ : NormOneClass F\ninst✝ : NormMulClass F\nx : F\nz : ℝ × ℝ\nh : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z\nw✝ : ℝ × ℝ\nM : ℝ := ‖φ x z‖\nH : M ≠ 0\nhM : M = ‖φ x z‖\nhM₀ : 0 < M\nw u : ℝ × ℝ\nhw : ‖φ x w‖ = M\nn : ℕ\nhn : n > 0\nq :... | [] | simp [q, aeval_eq_φ, hw] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 324,
"column": 37
} | {
"line": 324,
"column": 61
} | {
"line": 326,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝³ : NormedRing F\ninst✝² : NormedAlgebra ℝ F\ninst✝¹ : NormOneClass F\ninst✝ : NormMulClass F\nx : F\nz : ℝ × ℝ\nh : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z\nw✝ : ℝ × ℝ\nM : ℝ := ‖φ x z‖\nH : M ≠ 0\nhM : M = ‖φ x z‖\nhM₀ : 0 < M\nw u : ℝ × ℝ\nhw : ‖φ x w‖ = M\nn : ℕ\nhn : n > 0\nq :... | [] | simp [q, aeval_eq_φ, hw] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 324,
"column": 37
} | {
"line": 324,
"column": 61
} | {
"line": 326,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝³ : NormedRing F\ninst✝² : NormedAlgebra ℝ F\ninst✝¹ : NormOneClass F\ninst✝ : NormMulClass F\nx : F\nz : ℝ × ℝ\nh : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z\nw✝ : ℝ × ℝ\nM : ℝ := ‖φ x z‖\nH : M ≠ 0\nhM : M = ‖φ x z‖\nhM₀ : 0 < M\nw u : ℝ × ℝ\nhw : ‖φ x w‖ = M\nn : ℕ\nhn : n > 0\nq :... | [] | simp [q, aeval_eq_φ, hw] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Order.LiminfLimsup | {
"line": 207,
"column": 6
} | {
"line": 207,
"column": 46
} | {
"line": 208,
"column": 4
} | [
{
"pp": "case inl.refine_1\nι : Type u_1\nR : Type u_4\ninst✝⁷ : ConditionallyCompleteLinearOrder R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : AddCommSemigroup R\ninst✝³ : Sub R\ninst✝² : ContinuousSub R\ninst✝¹ : OrderedSub R\ninst✝ : AddLeftMono R\nf : ι → R\nc : R\nbdd_below : IsBounded... | [] | exact tsub_le_iff_tsub_le.1 (hx (x - y)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 135,
"column": 58
} | {
"line": 162,
"column": 76
} | {
"line": 164,
"column": 0
} | [
{
"pp": "K : 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\n⊢ ∃ g, g.Monic ∧ f.natDegree = g.natDegree ∧ ∀ (n : ℕ), ‖(map (algebraMap K L) g).coeff n - f.coeff n‖ < ε",
"ppTerm": "?m.45",
... | [] | by
by_cases h : f.natDegree = 0
· 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
choose c hc using fun i ↦ Metric.denseRange_iff.mp hd (f.coeff i) ε hε
have hdeg : (C 1 * X ^ f.natDegree +... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 204,
"column": 30
} | {
"line": 204,
"column": 43
} | {
"line": 204,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : μ x ≠ 0\nL : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)\nhL0 : 0 ≤ L\nε : ℝ\nhε : ε > 0\nm1 : ℕ+\nhm1 : μ (x ^ ↑m1) < (L + ε / 2) ^ ↑↑m1\nm2 : ℕ\nhm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ... | [
"R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : μ x ≠ 0\nL : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)\nhL0 : 0 ≤ L\nε : ℝ\nhε : ε > 0\nm1 : ℕ+\nhm1 : μ (x ^ ↑m1) < (L + ε / 2) ^ ↑↑m1\nm2 : ℕ\nhm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ≤ ε / (2 * (... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Field.Instances | {
"line": 32,
"column": 4
} | {
"line": 32,
"column": 52
} | {
"line": 33,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝ : NormedField F\nf : Filter F\nhc : f.NeBot ∧ Tendsto (fun p ↦ p.2 - p.1) (f ×ˢ f) (nhds 0)\nhn : nhds 0 ⊓ f = ⊥\nδ : ℝ\nδ_pos : δ > 0\nhδ : ∀ᶠ (y : F) in f, δ ≤ ‖y‖\nf_bdd : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun x ↦ ‖x⁻¹‖\nh₀ : ∀ᶠ (y : F) in f, y ≠ 0\nthis : ∀ᶠ (p : F × F) in ... | [
"F : Type u_1\ninst✝ : NormedField F\nf : Filter F\nhc : f.NeBot ∧ Tendsto (fun p ↦ p.2 - p.1) (f ×ˢ f) (nhds 0)\nhn : nhds 0 ⊓ f = ⊥\nδ : ℝ\nδ_pos : δ > 0\nhδ : ∀ᶠ (y : F) in f, δ ≤ ‖y‖\nf_bdd : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun x ↦ ‖x⁻¹‖\nh₀ : ∀ᶠ (y : F) in f, y ≠ 0\nthis : ∀ᶠ (p : F × F) in f ×ˢ f, p.1⁻... | rw [cauchy_map_iff_tendsto, tendsto_congr' this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 151,
"column": 10
} | {
"line": 151,
"column": 20
} | {
"line": 151,
"column": 21
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : SeminormedRing R\ninst✝ : Nontrivial R\nr : R\nthis : (⨆ n, if n = 0 then ‖r‖ else 0) = ‖r‖\nn : ℕ\nhn : n = 0\n⊢ (if n = 0 then ‖X.coeff n - (C r).coeff n‖ ^ (1 - ↑n)⁻¹ else 0) = if n = 0 then ‖r‖ else 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants... | [
"case pos\nR : Type u_1\ninst✝¹ : SeminormedRing R\ninst✝ : Nontrivial R\nr : R\nthis : (⨆ n, if n = 0 then ‖r‖ else 0) = ‖r‖\nn : ℕ\nhn : n = 0\n⊢ ‖X.coeff n - (C r).coeff n‖ ^ (1 - ↑n)⁻¹ = if n = 0 then ‖r‖ else 0"
] | if_pos hn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 151,
"column": 21
} | {
"line": 151,
"column": 31
} | {
"line": 151,
"column": 32
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : SeminormedRing R\ninst✝ : Nontrivial R\nr : R\nthis : (⨆ n, if n = 0 then ‖r‖ else 0) = ‖r‖\nn : ℕ\nhn : n = 0\n⊢ ‖X.coeff n - (C r).coeff n‖ ^ (1 - ↑n)⁻¹ = if n = 0 then ‖r‖ else 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Norm.norm",... | [
"case pos\nR : Type u_1\ninst✝¹ : SeminormedRing R\ninst✝ : Nontrivial R\nr : R\nthis : (⨆ n, if n = 0 then ‖r‖ else 0) = ‖r‖\nn : ℕ\nhn : n = 0\n⊢ ‖X.coeff n - (C r).coeff n‖ ^ (1 - ↑n)⁻¹ = ‖r‖"
] | if_pos hn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 302,
"column": 2
} | {
"line": 308,
"column": 51
} | {
"line": 309,
"column": 2
} | [
{
"pp": "case a\nK : 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 ... | [
"case a\nK : 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, i... | · apply ciSup_le (fun x ↦ ?_)
by_cases hx : x ∈ s
· have hx0 : aeval x p = 0 := aeval_root_of_mapAlg_eq_multiset_prod_X_sub_C s hx hp
rw [if_pos hx]
exact norm_root_le_spectralValue hf_pm hf_na
(monic_of_monic_mapAlg (hp ▸ monic_multisetProd_X_sub_C s)) hx0
· simp only [if_neg hx, spectr... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 156,
"column": 93
} | {
"line": 177,
"column": 65
} | {
"line": 179,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nC : ℝ\nh : f.SurjectiveOnWith f.range C\n⊢ ↑f.completion.ker = closure ↑(toCompl.comp (incl f.ker)).range",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
... | [] | by
refine le_antisymm ?_ (closure_minimal f.ker_le_ker_completion f.completion.isClosed_ker)
rintro hatg (hatg_in : f.completion hatg = 0)
rw [SeminormedAddCommGroup.mem_closure_iff]
intro ε ε_pos
rcases h.exists_pos with ⟨C', C'_pos, hC'⟩
rcases exists_pos_mul_lt ε_pos (1 + C' * ‖f‖) with ⟨δ, δ_pos, hδ⟩
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 388,
"column": 64
} | {
"line": 391,
"column": 30
} | {
"line": 393,
"column": 0
} | [
{
"pp": "K : Type u_2\ninst✝⁶ : NormedField K\nL : Type u_3\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\nE : Type u_4\ninst✝³ : Field E\ninst✝² : Algebra K E\ninst✝¹ : Algebra E L\ninst✝ : IsScalarTower K E L\nx : E\n⊢ spectralNorm K E x = spectralNorm K L ((algebraMap E L) x)",
"ppTerm": "?m.21",
"assigned... | [] | by
have hx : minpoly K (algebraMap E L x) = minpoly K x :=
minpoly.algebraMap_eq (algebraMap E L).injective x
simp only [spectralNorm, hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.SeparationQuotient | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 36
} | {
"line": 82,
"column": 2
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhf : ∀ (x : M), ‖x‖ = 0 → f x = 0\nx : SeparationQuotient M\n⊢ ‖(liftNormedAddGroupHom f hf) x‖ ≤ ‖f‖ * ‖x‖",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
... | [
"M : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhf : ∀ (x : M), ‖x‖ = 0 → f x = 0\nx : M\n⊢ ‖(liftNormedAddGroupHom f hf) (mk x)‖ ≤ ‖f‖ * ‖mk x‖"
] | obtain ⟨x, rfl⟩ := surjective_mk x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Normed.Group.SeparationQuotient | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 38
} | {
"line": 98,
"column": 4
} | [
{
"pp": "M : Type u_1\nN✝ : Type u_2\ninst✝² : SeminormedAddCommGroup M\ninst✝¹ : SeminormedAddCommGroup N✝\nN : Type u_3\ninst✝ : SeminormedAddCommGroup N\nx✝ : NormedAddGroupHom (SeparationQuotient M) N\nx : SeparationQuotient M\n⊢ ((fun f ↦ liftNormedAddGroupHom ↑f ⋯) ((fun g ↦ ⟨g.comp normedMk, ⋯⟩) x✝)) x =... | [
"M : Type u_1\nN✝ : Type u_2\ninst✝² : SeminormedAddCommGroup M\ninst✝¹ : SeminormedAddCommGroup N✝\nN : Type u_3\ninst✝ : SeminormedAddCommGroup N\nx✝ : NormedAddGroupHom (SeparationQuotient M) N\nx : M\n⊢ ((fun f ↦ liftNormedAddGroupHom ↑f ⋯) ((fun g ↦ ⟨g.comp normedMk, ⋯⟩) x✝)) (mk x) = x✝ (mk x)"
] | obtain ⟨x, rfl⟩ := surjective_mk x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 411,
"column": 37
} | {
"line": 411,
"column": 64
} | {
"line": 411,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nP : ℕ → X →L[𝕜] X\nhrank : ∀ (n : ℕ), Module.finrank 𝕜 ↥(↑(P n)).range = n\nhcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x\nn : ℕ\nU : Submodule 𝕜 X := (↑(succSu... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nP : ℕ → X →L[𝕜] X\nhrank : ∀ (n : ℕ), Module.finrank 𝕜 ↥(↑(P n)).range = n\nhcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x\nn : ℕ\nU : Submodule 𝕜 X := (↑(succSub P n)).rang... | ContinuousLinearMap.coe_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 24
} | {
"line": 148,
"column": 2
} | [
{
"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... | [
"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✝ : Module R... | simp only [comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 24
} | {
"line": 155,
"column": 2
} | [
{
"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... | [
"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✝ : Module R... | simp only [comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 313,
"column": 2
} | {
"line": 313,
"column": 24
} | {
"line": 314,
"column": 2
} | [
{
"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... | [
"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✝ : Module R... | simp only [comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Module.DoubleDual | {
"line": 152,
"column": 2
} | {
"line": 154,
"column": 20
} | {
"line": 156,
"column": 0
} | [
{
"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 (closure (... | [] | exact WeakDual.isCompact_of_bounded_of_closed
(WeakDual.isBounded_closure ((inclusionInDoubleDual 𝕜 X).lipschitz.isBounded_image hb))
isClosed_closure | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 50
} | {
"line": 219,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝³ : NormedCommRing R\ninst✝² : NormOneClass R\nf : ι → R\ninst✝¹ : CompleteSpace R\ninst✝ : NormMulClass R\nhf : ∀ (i : ι), 1 + f i ≠ 0\nhu : Summable fun x ↦ ‖f x‖\n⊢ ∏' (i : ι), (1 + f i) ≠ 0",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
... | [
"ι : Type u_1\nR : Type u_2\ninst✝³ : NormedCommRing R\ninst✝² : NormOneClass R\nf : ι → R\ninst✝¹ : CompleteSpace R\ninst✝ : NormMulClass R\nhf : ∀ (i : ι), 1 + f i ≠ 0\nhu : Summable fun x ↦ ‖f x‖\n⊢ ∏' (i : ι), ‖1 + f i‖ ≠ 0",
"ι : Type u_1\nR : Type u_2\ninst✝³ : NormedCommRing R\ninst✝² : NormOneClass R\nf :... | rw [← norm_ne_zero_iff, Multipliable.norm_tprod] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm | {
"line": 297,
"column": 59
} | {
"line": 299,
"column": 59
} | {
"line": 301,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝⁶ : Fintype ι\n𝕜 : Type u_2\nE : ι → Type u_3\ninst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E i)\nE' : ι → Type u_5\ninst✝² : (i : ι) → SeminormedAddCommGroup (E' i)\ninst✝¹ : (i : ι) → NormedSpace 𝕜 (E' i)... | [] | by
ext
simp [mapL_add_smul_aux, PiTensorProduct.map_update_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Dynamics.FixedPoints.Topology | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 68
} | {
"line": 38,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\nf : α → α\nx y : α\nhy : Tendsto (fun n ↦ f^[n] x) atTop (𝓝 y)\nhf : ContinuousAt f y\n⊢ IsFixedPt f y",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Nat.instLattice",
"Lattice.toSemilatticeSup",
"F... | [
"α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\nf : α → α\nx y : α\nhy : Tendsto (fun n ↦ f^[n] x) atTop (𝓝 y)\nhf : ContinuousAt f y\n⊢ Tendsto (fun n ↦ f^[n + 1] x) atTop (𝓝 (f y))"
] | refine tendsto_nhds_unique ((tendsto_add_atTop_iff_nat 1).1 ?_) hy | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.ODE.Gronwall | {
"line": 88,
"column": 4
} | {
"line": 89,
"column": 12
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case pos\nδ K x : ℝ\nhK : K = 0\n⊢ Continuous fun ε ↦ gronwallBound δ K ε x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Continuous",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSeparatelyContinuousAddOfContinuousAdd",
... | [] | simp only [gronwallBound_K0, hK]
fun_prop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.Gronwall | {
"line": 88,
"column": 4
} | {
"line": 89,
"column": 12
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case pos\nδ K x : ℝ\nhK : K = 0\n⊢ Continuous fun ε ↦ gronwallBound δ K ε x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Continuous",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSeparatelyContinuousAddOfContinuousAdd",
... | [] | simp only [gronwallBound_K0, hK]
fun_prop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.ODE.Transform | {
"line": 47,
"column": 2
} | {
"line": 53,
"column": 38
} | {
"line": 55,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\ns : Set ℝ\ndt : ℝ\n⊢ IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) (-dt +ᵥ s) ↔ IsIntegralCurveOn γ v s",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | refine ⟨fun hγ ↦ ?_, fun hγ ↦ hγ.comp_add _⟩
convert! hγ.comp_add (-dt)
· ext t
simp only [comp_apply, neg_add_cancel_right]
· ext t
simp only [comp_apply, neg_add_cancel_right]
· simp only [neg_neg, vadd_neg_vadd] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.Transform | {
"line": 47,
"column": 2
} | {
"line": 53,
"column": 38
} | {
"line": 55,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\ns : Set ℝ\ndt : ℝ\n⊢ IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) (-dt +ᵥ s) ↔ IsIntegralCurveOn γ v s",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | refine ⟨fun hγ ↦ ?_, fun hγ ↦ hγ.comp_add _⟩
convert! hγ.comp_add (-dt)
· ext t
simp only [comp_apply, neg_add_cancel_right]
· ext t
simp only [comp_apply, neg_add_cancel_right]
· simp only [neg_neg, vadd_neg_vadd] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Polynomial.Norm | {
"line": 74,
"column": 73
} | {
"line": 77,
"column": 52
} | {
"line": 79,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedRing A\nn : ℕ\na : A\n⊢ ((monomial n) a).supNorm = ‖a‖",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Polynomial.support_monomial",
"dite_cond_eq_true",
"one_pow",
"Norm.norm",
"MulOne.toOne",
"SeminormedRing.to... | [] | by
by_cases ha : a = 0
· simp [ha]
· simp [supNorm, gaussNorm, support_monomial n ha] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Rat.NatSqrt.Defs | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 43
} | {
"line": 33,
"column": 2
} | [
{
"pp": "x prec : ℕ\nh : 0 < prec\n⊢ (↑(x * prec ^ 2).sqrt / ↑prec) ^ 2 ≤ ↑x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"div_le_iff₀",
"pow_pos",
"Preorder.toLT",
"instHDiv",
"HMul.h... | [
"x prec : ℕ\nh : 0 < prec\n⊢ ↑(x * prec ^ 2).sqrt ^ 2 ≤ ↑x * ↑prec ^ 2"
] | rw [div_pow, div_le_iff₀ (by positivity)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 375,
"column": 4
} | {
"line": 375,
"column": 33
} | {
"line": 375,
"column": 33
} | [
{
"pp": "case neg.inr\nn : ℕ\np : ℂ[X]\nhp : ¬p = 0\nhn : n ≤ p.natDegree\n⊢ ‖(Multiset.map prod (powersetCard (p.natDegree - n) p.roots)).sum‖ ≤\n ↑(p.natDegree.choose n) * (Multiset.map (fun a ↦ max 1 ‖a‖) p.roots).prod",
"ppTerm": "?neg.inr✝",
"assigned": true,
"usedConstants": [
"Multis... | [
"case neg.inr\nn : ℕ\np : ℂ[X]\nhp : ¬p = 0\nhn : n ≤ p.natDegree\n⊢ ‖∑ m ∈ (powersetCard (p.natDegree - n) p.roots).toFinset, count m (powersetCard (p.natDegree - n) p.roots) • m.prod‖ ≤\n ↑(p.natDegree.choose n) * (Multiset.map (fun a ↦ max 1 ‖a‖) p.roots).prod"
] | Finset.sum_multiset_map_count | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 401,
"column": 70
} | {
"line": 401,
"column": 79
} | {
"line": 401,
"column": 79
} | [
{
"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⊢ ‖∏ m ∈ x.toFi... | [
"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.toFinset, ‖b ^ co... | norm_prod | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.OfDigits | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 28
} | {
"line": 161,
"column": 2
} | [
{
"pp": "b : ℕ\ninst✝ : NeZero b\nx : ℝ\nhb : 1 < b\nhx : x ∈ Set.Ico 0 1\n⊢ ∑' (n : ℕ), ofDigitsTerm (x.digits b) n = x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"EMetricSpace.metrizableSpace",
"PseudoMetricSpace.toUniformS... | [
"b : ℕ\ninst✝ : NeZero b\nx : ℝ\nhb : 1 < b\nhx : x ∈ Set.Ico 0 1\n⊢ HasSum (ofDigitsTerm (x.digits b)) x",
"b : ℕ\ninst✝ : NeZero b\nx : ℝ\nhb : 1 < b\nhx : x ∈ Set.Ico 0 1\n⊢ Summable (ofDigitsTerm (x.digits b))"
] | rw [← Summable.hasSum_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Real.Irrational | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 25
} | {
"line": 319,
"column": 2
} | [
{
"pp": "x : ℝ\nh : Irrational x\nm : ℤ\nhm : m ≠ 0\n⊢ Irrational (x * ↑↑m)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Int.cast",
"Rat",
"Rat.instIntCast",
"Irrational.mul_ratCast"
],
"usedFVars": [
"x",
"h",
"m"
],
"usedGoals... | [
"x : ℝ\nh : Irrational x\nm : ℤ\nhm : m ≠ 0\n⊢ ↑m ≠ 0"
] | refine h.mul_ratCast ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 35
} | {
"line": 75,
"column": 2
} | [
{
"pp": "θ : ℝ\nn : ℕ\nf : ℝ → ℝ := fun x ↦ 1 - x ^ 2\nu₁ : ℝ → ℝ := fun x ↦ f x ^ (n + 1)\nu₁' : ℝ → ℝ := fun x ↦ -(2 * (↑n + 1) * x * f x ^ n)\nv₁ : ℝ → ℝ := fun x ↦ sin (x * θ)\nv₁' : ℝ → ℝ := fun x ↦ cos (x * θ) * θ\nu₂ : ℝ → ℝ := fun x ↦ x * f x ^ n\nu₂' : ℝ → ℝ := fun x ↦ f x ^ n - 2 * ↑n * x ^ 2 * f x ^ ... | [
"θ : ℝ\nn : ℕ\nf : ℝ → ℝ := fun x ↦ 1 - x ^ 2\nu₁ : ℝ → ℝ := fun x ↦ f x ^ (n + 1)\nu₁' : ℝ → ℝ := fun x ↦ -(2 * (↑n + 1) * x * f x ^ n)\nv₁ : ℝ → ℝ := fun x ↦ sin (x * θ)\nv₁' : ℝ → ℝ := fun x ↦ cos (x * θ) * θ\nu₂ : ℝ → ℝ := fun x ↦ x * f x ^ n\nu₂' : ℝ → ℝ := fun x ↦ f x ^ n - 2 * ↑n * x ^ 2 * f x ^ (n - 1)\nv₂ ... | let v₂ (x : ℝ) : ℝ := cos (x * θ) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
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