module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 942,
"column": 4
} | {
"line": 942,
"column": 37
} | [
{
"pp": "case refine_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw : ι → k\ns : Finset ι\nhw : ∑ i ∈ s, w i = 1\ni₁... | simp_all only [sub_pos, sign_pos] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.NhdsKer | {
"line": 128,
"column": 32
} | {
"line": 128,
"column": 52
} | [
{
"pp": "X : Type u_2\ninst✝¹ : TopologicalSpace X\nY : Type u_3\ninst✝ : TopologicalSpace Y\ns : Set X\nt : Set Y\n⊢ nhdsKer (⋃ i ∈ s, {i}) ×ˢ nhdsKer (⋃ i ∈ t, {i}) = nhdsKer s ×ˢ nhdsKer t",
"usedConstants": [
"Set.instSProd",
"SProd.sprod",
"congrArg",
"Membership.mem",
"Se... | biUnion_of_singleton | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.NhdsKer | {
"line": 143,
"column": 32
} | {
"line": 143,
"column": 52
} | [
{
"pp": "ι : Type u_3\nX : ι → Type u_4\ninst✝ : (i : ι) → TopologicalSpace (X i)\ns : (i : ι) → Set (X i)\n⊢ (univ.pi fun i ↦ nhdsKer (⋃ i_1 ∈ s i, {i_1})) = univ.pi fun i ↦ nhdsKer (s i)",
"usedConstants": [
"congrArg",
"Set.univ",
"Membership.mem",
"Set.biUnion_of_singleton",
... | biUnion_of_singleton | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 704,
"column": 30
} | {
"line": 704,
"column": 44
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring k\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module k V\ninst✝⁴ : AffineSpace V P\ninst✝³ : LinearOrder k\ninst✝² : IsOrderedAddMonoid k\ninst✝¹ : ZeroLEOneClass k\nn : ℕ\ninst✝ : NeZero n\ns : Simplex k P n\nw : Fin (n + 1) → k\nhw1 : ∑ i, w i = 1\nhp : ... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 704,
"column": 30
} | {
"line": 704,
"column": 44
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring k\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module k V\ninst✝⁴ : AffineSpace V P\ninst✝³ : LinearOrder k\ninst✝² : IsOrderedAddMonoid k\ninst✝¹ : ZeroLEOneClass k\nn : ℕ\ninst✝ : NeZero n\ns : Simplex k P n\nw : Fin (n + 1) → k\nhw1 : ∑ i, w i = 1\nhp : ... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 704,
"column": 30
} | {
"line": 704,
"column": 44
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring k\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module k V\ninst✝⁴ : AffineSpace V P\ninst✝³ : LinearOrder k\ninst✝² : IsOrderedAddMonoid k\ninst✝¹ : ZeroLEOneClass k\nn : ℕ\ninst✝ : NeZero n\ns : Simplex k P n\nw : Fin (n + 1) → k\nhw1 : ∑ i, w i = 1\nhp : ... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Topology | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 52
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\ninst✝ : ContinuousSMul 𝕜 E\ns... | have h := AffineMap.lineMap_apply_one (k := 𝕜) x y | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.Module.LocallyConvex | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 42
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : Module 𝕜 E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 F\nt : TopologicalSpace F\ninst✝ : LocallyConvexSpace 𝕜 F\nf : E →ₗ[𝕜] F\n⊢ LocallyConvexSpace 𝕜 E",
"usedConsta... | letI : TopologicalSpace E := t.induced f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.Topology.Algebra.Module.LocallyConvex | {
"line": 229,
"column": 4
} | {
"line": 235,
"column": 24
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝⁴ : TopologicalSpace R\ninst✝³ : Semiring R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : OrderTopology R\nx : R\nhl : IsBot x\n⊢ (𝓝 x).HasBasis (fun s ↦ s ∈ 𝓝 x ∧ Convex R s) id",
"usedConstants": [
"_private.Mathlib.Topology.Algebra.Module.L... | · refine hl.rec ?_ _
intro
refine nhds_bot_basis.to_hasBasis' ?_ ?_
· intros
refine ⟨Set.Iio _, ?_, .rfl⟩
simp_all [Iio_mem_nhds, convex_Iio]
· simp +contextual | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Combination | {
"line": 568,
"column": 2
} | {
"line": 604,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : Finite ι\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\ns : Set ι\nt : (i : ι) → Set (E i)\nx : (i : ι) → E i\nh : ∀ i ∈ s, x i ∈ (convexHull ... | classical
cases nonempty_fintype ι
wlog hs : s = Set.univ generalizing s t
· rw [← pi_univ_ite]
refine this (fun i _ ↦ ?_) rfl
split_ifs with hi
· exact h i hi
· simp
subst hs
simp only [Set.mem_univ, mem_convexHull_iff_exists_fintype, true_implies] at h
choose κ _ w f hw₀ hw₁ hft hf using h... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Analysis.Convex.Combination | {
"line": 568,
"column": 2
} | {
"line": 604,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : Finite ι\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\ns : Set ι\nt : (i : ι) → Set (E i)\nx : (i : ι) → E i\nh : ∀ i ∈ s, x i ∈ (convexHull ... | classical
cases nonempty_fintype ι
wlog hs : s = Set.univ generalizing s t
· rw [← pi_univ_ite]
refine this (fun i _ ↦ ?_) rfl
split_ifs with hi
· exact h i hi
· simp
subst hs
simp only [Set.mem_univ, mem_convexHull_iff_exists_fintype, true_implies] at h
choose κ _ w f hw₀ hw₁ hft hf using h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Combination | {
"line": 568,
"column": 2
} | {
"line": 604,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : Finite ι\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\ns : Set ι\nt : (i : ι) → Set (E i)\nx : (i : ι) → E i\nh : ∀ i ∈ s, x i ∈ (convexHull ... | classical
cases nonempty_fintype ι
wlog hs : s = Set.univ generalizing s t
· rw [← pi_univ_ite]
refine this (fun i _ ↦ ?_) rfl
split_ifs with hi
· exact h i hi
· simp
subst hs
simp only [Set.mem_univ, mem_convexHull_iff_exists_fintype, true_implies] at h
choose κ _ w f hw₀ hw₁ hft hf using h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 364,
"column": 48
} | {
"line": 364,
"column": 63
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\ninst✝³ : NormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nh : ∀ (x : E), x ≠ 0 → ∃ i, (p i) x ≠ 0\nthis : IsTopologicalAddGroup E\nx : E\nhx : x ≠ 0\ni : ι\npi_nonzero :... | map_neg_eq_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 618,
"column": 6
} | {
"line": 619,
"column": 15
} | [
{
"pp": "case refine_3.h.h.refine_2.inl\n𝕜 : Type u_2\nE : Type u_6\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nh : p.ball 0 1 ∈ 𝓝 0\nh' : IsVonNBounded ... | simp only [ball, Finset.sup_empty, sub_zero, coe_bot, Pi.zero_apply, r_pos, setOf_true] at hw
simp [hw] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 618,
"column": 6
} | {
"line": 619,
"column": 15
} | [
{
"pp": "case refine_3.h.h.refine_2.inl\n𝕜 : Type u_2\nE : Type u_6\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nh : p.ball 0 1 ∈ 𝓝 0\nh' : IsVonNBounded ... | simp only [ball, Finset.sup_empty, sub_zero, coe_bot, Pi.zero_apply, r_pos, setOf_true] at hw
simp [hw] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 29
} | [
{
"pp": "R : Type u_1\n𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : Semiring R\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : NormedField 𝕜₃\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module R E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : Topol... | rw [f.map_zero, zero_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 29
} | [
{
"pp": "R : Type u_1\n𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : Semiring R\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : NormedField 𝕜₃\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module R E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : Topol... | rw [f.map_zero, zero_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 29
} | [
{
"pp": "R : Type u_1\n𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : Semiring R\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : NormedField 𝕜₃\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module R E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : Topol... | rw [f.map_zero, zero_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 207,
"column": 42
} | {
"line": 208,
"column": 74
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : SeminormedAddCommGroup F\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NontriviallyNormedField 𝕜₂\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\nf : E →SL[σ₁₂] F\nM : ℝ\nhM... | by
simp only [h, mul_zero, norm_image_of_norm_eq_zero f f.2 h, le_refl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 732,
"column": 6
} | {
"line": 732,
"column": 53
} | [
{
"pp": "𝕜 : Type u_2\n𝕜₂ : Type u_3\nE : Type u_6\nF : Type u_7\nι' : Type u_10\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝³ : RingHomIsometric σ₁₂\nκ : Type u_11\nq : Se... | q.withSeminorms_iff_uniformSpace_eq_iInf.mp hq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 734,
"column": 2
} | {
"line": 734,
"column": 47
} | [
{
"pp": "𝕜 : Type u_2\n𝕜₂ : Type u_3\nE : Type u_6\nF : Type u_7\nι' : Type u_10\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝³ : RingHomIsometric σ₁₂\nκ : Type u_11\nq : Se... | refine forall_tfae [_, _, _, _, _] fun i ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap | {
"line": 384,
"column": 76
} | {
"line": 387,
"column": 66
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹³ : AddCommGroup E\ninst✝¹² : TopologicalSpace E\ninst✝¹¹ : Module 𝕜 E\ninst✝¹⁰ : ContinuousSMul 𝕜 E\nF : Type u_3\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : UniformSpace F\ninst✝⁷ : IsUniformAddGroup F\ninst✝⁶ : Module 𝕜 F\n𝕜' : Type... | by
rw [← isUniformEmbedding_toUniformOnFun.of_comp_iff]
convert! isUniformEmbedding_toUniformOnFun using 4 with s
exact ⟨fun h ↦ h.extend_scalars _, fun h ↦ h.restrict_scalars _⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 799,
"column": 2
} | {
"line": 805,
"column": 32
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\nι' : Type u_10\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nq : SeminormFamily 𝕜 E ι'\nt : TopologicalSpace E\nhp : WithSeminorms p\nhpq : Seminorm.IsBounded p q LinearMap.id\nhqp : Seminorm.IsBounded q p ... | constructor
rw [hp.topology_eq_withSeminorms]
clear hp t
refine le_antisymm ?_ ?_ <;>
rw [← continuous_id_iff_le] <;>
refine continuous_of_isBounded (.mk (topology := _) rfl) (.mk (topology := _) rfl)
LinearMap.id (by assumption) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 799,
"column": 2
} | {
"line": 805,
"column": 32
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\nι' : Type u_10\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nq : SeminormFamily 𝕜 E ι'\nt : TopologicalSpace E\nhp : WithSeminorms p\nhpq : Seminorm.IsBounded p q LinearMap.id\nhqp : Seminorm.IsBounded q p ... | constructor
rw [hp.topology_eq_withSeminorms]
clear hp t
refine le_antisymm ?_ ?_ <;>
rw [← continuous_id_iff_le] <;>
refine continuous_of_isBounded (.mk (topology := _) rfl) (.mk (topology := _) rfl)
LinearMap.id (by assumption) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 396,
"column": 52
} | {
"line": 398,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_7\ninst✝¹¹ : SeminormedAddCommGroup E\ninst✝¹⁰ : SeminormedAddCommGroup F\ninst✝⁹ : SeminormedAddCommGroup G\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NontriviallyNormedField... | by
rw [mul_assoc]
exact h.le_opNorm_of_le (f.le_opNorm x) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 798,
"column": 69
} | {
"line": 805,
"column": 32
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\nι' : Type u_10\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nq : SeminormFamily 𝕜 E ι'\nt : TopologicalSpace E\nhp : WithSeminorms p\nhpq : Seminorm.IsBounded p q LinearMap.id\nhqp : Seminorm.IsBounded q p ... | by
constructor
rw [hp.topology_eq_withSeminorms]
clear hp t
refine le_antisymm ?_ ?_ <;>
rw [← continuous_id_iff_le] <;>
refine continuous_of_isBounded (.mk (topology := _) rfl) (.mk (topology := _) rfl)
LinearMap.id (by assumption) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 20
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : DenselyNormedField 𝕜\ninst✝³ : NontriviallyNormedField 𝕜₂\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\nf : E... | refine csSup_eq_of_forall_le_of_forall_lt_exists_gt ((nonempty_ball.mpr zero_lt_one).image _) ?_
fun ub hub => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 157,
"column": 2
} | {
"line": 158,
"column": 76
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : DenselyNormedField 𝕜\ninst✝³ : NontriviallyNormedField 𝕜₂\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsome... | · rintro - ⟨x, hx, rfl⟩
simpa only [mul_one] using f.le_opNorm_of_le (mem_ball_zero_iff.1 hx).le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.MeasurableSpace.Pi | {
"line": 55,
"column": 2
} | {
"line": 62,
"column": 35
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝ : Finite ι\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), IsCountablySpanning (C i)\n⊢ IsCountablySpanning (univ.pi '' univ.pi C)",
"usedConstants": [
"Eq.mpr",
"Inhabited.default",
"congrArg",
"Set.univ",
"Option.getD",
"in... | choose s h1s h2s using hC
cases nonempty_encodable (ι → ℕ)
let e : ℕ → ι → ℕ := fun n => (@decode (ι → ℕ) _ n).getD default
refine ⟨fun n => Set.pi univ fun i => s i (e n i), fun n =>
mem_image_of_mem _ fun i _ => h1s i _, ?_⟩
simp_rw [e,
(surjective_decode_getD (ι → ℕ) default).iUnion_comp fun x => Set... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Pi | {
"line": 55,
"column": 2
} | {
"line": 62,
"column": 35
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝ : Finite ι\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), IsCountablySpanning (C i)\n⊢ IsCountablySpanning (univ.pi '' univ.pi C)",
"usedConstants": [
"Eq.mpr",
"Inhabited.default",
"congrArg",
"Set.univ",
"Option.getD",
"in... | choose s h1s h2s using hC
cases nonempty_encodable (ι → ℕ)
let e : ℕ → ι → ℕ := fun n => (@decode (ι → ℕ) _ n).getD default
refine ⟨fun n => Set.pi univ fun i => s i (e n i), fun n =>
mem_image_of_mem _ fun i _ => h1s i _, ?_⟩
simp_rw [e,
(surjective_decode_getD (ι → ℕ) default).iUnion_comp fun x => Set... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Pi | {
"line": 78,
"column": 8
} | {
"line": 78,
"column": 31
} | [
{
"pp": "case intro.a.h\nι : Type u_1\nα : ι → Type u_2\ninst✝ : Finite ι\nC : (i : ι) → Set (Set (α i))\nval✝ : Encodable ι\ni : ι\ns : Set (α i)\nhs : s ∈ C i\nt : (i : ι) → ℕ → Set (α i)\nh1t : ∀ (i : ι) (n : ℕ), t i n ∈ C i\nh2t : ∀ (i : ι), ⋃ n, t i n = univ\n⊢ MeasurableSet (univ.pi (update (fun x ↦ ⋃ n, ... | ← @iUnion_const _ ℕ _ s | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.ConditionalProbability | {
"line": 203,
"column": 43
} | {
"line": 203,
"column": 54
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\n⊢ μ[|∅] = 0",
"usedConstants": [
"instHSMul",
"MeasureTheory.Measure",
"Semiring.toModule",
"instSMulOfMul",
"MeasureTheory.Measure.restrict_empty",
"IsScalarTower.right",
"ENNReal.instAddCommMonoid",
... | simp [cond] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.ConditionalProbability | {
"line": 203,
"column": 43
} | {
"line": 203,
"column": 54
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\n⊢ μ[|∅] = 0",
"usedConstants": [
"instHSMul",
"MeasureTheory.Measure",
"Semiring.toModule",
"instSMulOfMul",
"MeasureTheory.Measure.restrict_empty",
"IsScalarTower.right",
"ENNReal.instAddCommMonoid",
... | simp [cond] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ConditionalProbability | {
"line": 203,
"column": 43
} | {
"line": 203,
"column": 54
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\n⊢ μ[|∅] = 0",
"usedConstants": [
"instHSMul",
"MeasureTheory.Measure",
"Semiring.toModule",
"instSMulOfMul",
"MeasureTheory.Measure.restrict_empty",
"IsScalarTower.right",
"ENNReal.instAddCommMonoid",
... | simp [cond] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ConditionalProbability | {
"line": 209,
"column": 65
} | {
"line": 209,
"column": 76
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\ns : Set Ω\n⊢ μ[|s] = 0 ↔ μ s = ∞ ∨ μ s = 0",
"usedConstants": [
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul",
"IsScalarTower.right",
"congrArg",
"CommSemiring.toSemiring",
"ENNReal.inv_eq_zero._si... | simp [cond] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.ConditionalProbability | {
"line": 209,
"column": 65
} | {
"line": 209,
"column": 76
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\ns : Set Ω\n⊢ μ[|s] = 0 ↔ μ s = ∞ ∨ μ s = 0",
"usedConstants": [
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul",
"IsScalarTower.right",
"congrArg",
"CommSemiring.toSemiring",
"ENNReal.inv_eq_zero._si... | simp [cond] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ConditionalProbability | {
"line": 209,
"column": 65
} | {
"line": 209,
"column": 76
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\ns : Set Ω\n⊢ μ[|s] = 0 ↔ μ s = ∞ ∨ μ s = 0",
"usedConstants": [
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul",
"IsScalarTower.right",
"congrArg",
"CommSemiring.toSemiring",
"ENNReal.inv_eq_zero._si... | simp [cond] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.ENNReal | {
"line": 211,
"column": 24
} | {
"line": 211,
"column": 38
} | [
{
"pp": "α : Type u_1\nf : Filter α\ninst✝ : CountableInterFilter f\nu : α → ℝ≥0∞\na : ℝ≥0∞\nha_top : a = ∞\nhu : ¬∀ᶠ (x : α) in f, u x = 0 x\n⊢ ∃ᶠ (x : α) in f, ∞ ≤ if u x = 0 then 0 else ∞",
"usedConstants": [
"congrArg",
"Filter.Eventually",
"Eq.mp",
"Pi.instZero",
"Filter.F... | not_eventually | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.ConditionalProbability | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 47
} | [
{
"pp": "case inl\nΩ : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\ns : Set Ω\nhms : MeasurableSet s\nhcs' : μ s ≠ ∞\nt : Set Ω\nhcs : μ s = 0\n⊢ μ[t | s] * μ s = μ (s ∩ t)",
"usedConstants": [
"MeasureTheory.Measure",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"ENN... | · simp [hcs, measure_inter_null_of_null_left] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Operator.Bilinear | {
"line": 160,
"column": 69
} | {
"line": 162,
"column": 5
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nF : Type u_6\nG : Type u_8\ninst✝¹⁰ : SeminormedAddCommGroup E\ninst✝⁹ : SeminormedAddCommGroup F\ninst✝⁸ : SeminormedAddCommGroup G\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NontriviallyNormedField 𝕜₂\ninst✝⁵ : NontriviallyNormedField ... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.EssSup | {
"line": 245,
"column": 55
} | {
"line": 247,
"column": 44
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder β\nm : MeasurableSpace α\nμ : Measure α\nf : α → β\n⊢ essInf f μ = sSup {a | μ {x | f x < a} = 0}",
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheory.Measure",
"Preorder.toLT",
"_private.Mathlib.Measure... | by
dsimp [essInf, liminf, limsInf]
simp only [eventually_map, ae_iff, not_le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Holder | {
"line": 134,
"column": 24
} | {
"line": 136,
"column": 22
} | [
{
"pp": "⊢ 2⁻¹ + 2⁻¹ = 1⁻¹",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ENNReal.instAdd",
"False",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"eq_false",
"inv_one",
"congrArg",
"CommSemiri... | by
rw [← two_mul, ENNReal.mul_inv_cancel]
all_goals norm_num | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 515,
"column": 22
} | {
"line": 515,
"column": 47
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ninst✝⁴ : Fintype ι\ninst✝³ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝² : ∀ (i : ι), SigmaFinite (μ i)\ninst✝¹ : (i : ι) → PartialOrder (α i)\ninst✝ : ∀ (i : ι), NoAtoms (μ i)\nf g : (i : ι) → α i\n⊢ (univ.pi fun i ↦ Ico (f i) (g i)) =ᶠ[ae (Meas... | exact pi_Ico_ae_eq_pi_Icc | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 570,
"column": 2
} | {
"line": 570,
"column": 47
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝⁵ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝⁴ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝³ : ∀ (i : ι), SigmaFinite (μ i)\ninst✝² : (i : ι) → Group (α i)\ninst✝¹ : ∀ (i : ι), MeasurableMul (α i)\ninst✝ : ∀ (i : ι), ... | refine ⟨fun v => (pi_eq fun s hs => ?_).symm⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 584,
"column": 2
} | {
"line": 584,
"column": 47
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝⁵ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝⁴ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝³ : ∀ (i : ι), SigmaFinite (μ i)\ninst✝² : (i : ι) → Group (α i)\ninst✝¹ : ∀ (i : ι), MeasurableMul (α i)\ninst✝ : ∀ (i : ι), ... | refine ⟨fun v => (pi_eq fun s hs => ?_).symm⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 848,
"column": 2
} | {
"line": 848,
"column": 81
} | [
{
"pp": "α : Fin 2 → Type u\nm : (i : Fin 2) → MeasurableSpace (α i)\nμ : (i : Fin 2) → Measure (α i)\ninst✝ : ∀ (i : Fin 2), SigmaFinite (μ i)\n⊢ MeasurePreserving (⇑(MeasurableEquiv.piFinTwo α)) (Measure.pi μ) ((μ 0).prod (μ 1))",
"usedConstants": [
"MeasurableEquiv.instEquivLike",
"MeasureThe... | refine ⟨MeasurableEquiv.measurable _, (Measure.prod_eq fun s t _ _ => ?_).symm⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.LpSeminorm.Indicator | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 56
} | [
{
"pp": "case inr\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nβ : Type u_7\ninst✝ : NormedAddCommGroup β\nhp_top : p ≠ ∞\nf : α → β\nhf : MemLp f p μ\nε : ℝ≥0∞\nhε✝ : ε ≠ 0\nhp₀ : p ≠ 0\ns : Set α\nhsm : MeasurableSet s\nhs : μ s < ∞\nhε : ∫⁻ (a : α) in sᶜ, ‖f a‖ₑ ^ p.toReal ∂μ < ε ^ p.toRea... | eLpNorm_eq_lintegral_rpow_enorm_toReal hp₀ hp_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 294,
"column": 39
} | {
"line": 296,
"column": 66
} | [
{
"pp": "α : Type u_1\nF : Type u_5\nm0 : MeasurableSpace α\nq : ℝ\nμ : Measure α\ninst✝ : NormedAddCommGroup F\nf g : α → F\nhfg : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ = ‖g x‖₊\n⊢ eLpNorm' f q μ = eLpNorm' g q μ",
"usedConstants": [
"MeasureTheory.ae",
"Real",
"ENNReal.ofNNReal",
"MeasureTheory.Me... | by
have : (‖f ·‖ₑ ^ q) =ᵐ[μ] (‖g ·‖ₑ ^ q) := hfg.mono fun x hx ↦ by simp [enorm, hx]
simp only [eLpNorm'_eq_lintegral_enorm, lintegral_congr_ae this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 21
} | [
{
"pp": "α : Type u_1\nε : Type u_2\nε' : Type u_3\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : ENorm ε\ninst✝ : ENorm ε'\nf : α → ε\ng : α → ε'\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ‖g x‖ₑ\n⊢ eLpNorm f p μ ≤ eLpNorm g p μ",
"usedConstants": [
"id",
"LE.le",
"ENNReal.instLE",
"E... | simp only [eLpNorm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 188,
"column": 58
} | {
"line": 188,
"column": 78
} | [
{
"pp": "p : ℝ\nhp : 1 < p\nx y z : ℝ\nhx : 0 ≤ x\nhz : 0 ≤ z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nq : 0 < y - x\n⊢ 1 - p * y ^ (p - 1 - p) * (y - x) < (1 + (x / y - 1)) ^ p",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instPow",
"Real.partialOrder",
... | sub_sub_cancel_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 199,
"column": 73
} | {
"line": 199,
"column": 93
} | [
{
"pp": "p : ℝ\nhp : 1 < p\nx y z : ℝ\nhx : 0 ≤ x\nhz : 0 ≤ z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nq : 0 < z - y\n⊢ 1 + p * y ^ (p - 1 - p) * (z - y) < (1 + (z / y - 1)) ^ p",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instPow",
"Real.partialOrder",
... | sub_sub_cancel_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 482,
"column": 47
} | {
"line": 482,
"column": 91
} | [
{
"pp": "α : Type u_1\nF : Type u_5\nm0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ : Measure α\ninst✝ : NormedAddCommGroup F\nf : α → F\nhq_pos : 0 < q\n⊢ ∀ᵐ (x : α) ∂?m.48, 0 ≤ (fun x ↦ ‖f x‖ ^ q) x",
"usedConstants": [
"MeasureTheory.ae",
"Norm.norm",
"Real.instPow",
"Real.instLE",
... | by filter_upwards with x using by positivity | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 212,
"column": 2
} | {
"line": 214,
"column": 55
} | [
{
"pp": "p a b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhp : 0 ≤ p\nhp1 : p ≤ 1\n⊢ (a + b) ^ p ≤ a ^ p + b ^ p",
"usedConstants": [
"Real.instPow",
"Real.instLE",
"Real",
"NNReal.canLift",
"Real.instZero",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal.coe_le_coe._si... | lift a to NNReal using ha
lift b to NNReal using hb
exact_mod_cast NNReal.rpow_add_le_add_rpow a b hp hp1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 212,
"column": 2
} | {
"line": 214,
"column": 55
} | [
{
"pp": "p a b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhp : 0 ≤ p\nhp1 : p ≤ 1\n⊢ (a + b) ^ p ≤ a ^ p + b ^ p",
"usedConstants": [
"Real.instPow",
"Real.instLE",
"Real",
"NNReal.canLift",
"Real.instZero",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal.coe_le_coe._si... | lift a to NNReal using ha
lift b to NNReal using hb
exact_mod_cast NNReal.rpow_add_le_add_rpow a b hp hp1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 431,
"column": 2
} | {
"line": 433,
"column": 67
} | [
{
"pp": "case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"False",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
... | · refine le_trans le_top (le_of_eq ?_)
repeat rw [div_eq_mul_inv]
rcases h with h | h <;> rw [h] <;> simp [hpq.pos, hpq.symm.pos] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 331,
"column": 46
} | {
"line": 334,
"column": 23
} | [
{
"pp": "⊢ LpAddConst 0 = 1",
"usedConstants": [
"Set.decidableMemIoo",
"Eq.mpr",
"Real",
"Preorder.toLT",
"instHDiv",
"lt_irrefl",
"congrArg",
"ENNReal.instPowReal",
"ENNReal.LpAddConst.eq_1",
"Real.instDivInvMonoid",
"Real.instSub",
"... | by
rw [LpAddConst, if_neg]
intro h
exact lt_irrefl _ h.1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Mul | {
"line": 53,
"column": 2
} | {
"line": 55,
"column": 87
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝¹⁷ : CommRing 𝕜\ninst✝¹⁶ : LinearOrder 𝕜\ninst✝¹⁵ : IsStrictOrderedRing 𝕜\ninst✝¹⁴ : CommRing E\ninst✝¹³ : LinearOrder E\ninst✝¹² : IsStrictOrderedRing E\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : LinearOrder F\ninst✝⁹ : IsOrderedAddMonoid F\ni... | rw [← smul_smul_smul_comm a, ← smul_smul_smul_comm b, ← smul_smul_smul_comm a b,
← smul_smul_smul_comm b b, smul_eq_mul, smul_eq_mul, smul_eq_mul, smul_eq_mul, mul_comm b,
add_comm _ ((b * b) • f y • g y), add_add_add_comm, add_comm ((a * b) • f y • g x)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 630,
"column": 2
} | {
"line": 630,
"column": 60
} | [
{
"pp": "case inr\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\nq : ℝ := p / (p - 1)\nhpq : p.HolderConjugate q\n⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p",
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"Real.instDivInvMonoid",
"R... | have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 344,
"column": 50
} | {
"line": 348,
"column": 88
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\n⊢ ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ ≤\n ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^... | by
rw [add_mul]
gcongr
· exact lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hf (hf.add hg)
· exact lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hg (hf.add hg) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 246,
"column": 53
} | {
"line": 246,
"column": 75
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np q r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nc : ... | by simpa using hq₁.ne' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.MeanInequalities | {
"line": 863,
"column": 2
} | {
"line": 871,
"column": 12
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ\np q r : ℝ\nhpqr : p.HolderTriple q r\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\nhf_sum : Summable fun i ↦ f i ^ p\nhg_sum : Summable fun i ↦ g i ^ q\n⊢ (∑' (i : ι), (f i * g i) ^ r) ^ (1 / r) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"use... | have hf' : 0 ≤ ∑' i, f i ^ p := tsum_nonneg fun i ↦ rpow_nonneg (hf i) p
have hg' : 0 ≤ ∑' i, g i ^ q := tsum_nonneg fun i ↦ rpow_nonneg (hg i) q
have hr := hpqr.pos'
convert!
rpow_le_rpow_iff (tsum_nonneg fun i ↦ by positivity [hf i, hg i]) (by positivity)
(inv_eq_one_div r ▸ inv_pos.mpr hr) |>.mpr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalities | {
"line": 863,
"column": 2
} | {
"line": 871,
"column": 12
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ\np q r : ℝ\nhpqr : p.HolderTriple q r\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\nhf_sum : Summable fun i ↦ f i ^ p\nhg_sum : Summable fun i ↦ g i ^ q\n⊢ (∑' (i : ι), (f i * g i) ^ r) ^ (1 / r) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"use... | have hf' : 0 ≤ ∑' i, f i ^ p := tsum_nonneg fun i ↦ rpow_nonneg (hf i) p
have hg' : 0 ≤ ∑' i, g i ^ q := tsum_nonneg fun i ↦ rpow_nonneg (hg i) q
have hr := hpqr.pos'
convert!
rpow_le_rpow_iff (tsum_nonneg fun i ↦ by positivity [hf i, hg i]) (by positivity)
(inv_eq_one_div r ▸ inv_pos.mpr hr) |>.mpr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 99,
"column": 2
} | {
"line": 104,
"column": 75
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : PseudoMetricSpace E\nf : ι → α → E\nl : Filter ι\ng : α → E\n⊢ TendstoInMeasure μ f l g ↔ ∀ (ε : ℝ), 0 < ε → Tendsto (fun i ↦ μ {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0)",
"usedConstants": [
"Iff.mpr",
... | refine ⟨fun h ε hε ↦ ?_, fun h ↦ ?_⟩
· convert! h (ENNReal.ofReal ε) (ENNReal.ofReal_pos.mpr hε) with i a
rw [edist_dist, ENNReal.ofReal_le_ofReal_iff (by positivity)]
· refine tendstoInMeasure_of_ne_top fun ε hε hε_top ↦ ?_
convert! h ε.toReal (ENNReal.toReal_pos hε.ne' hε_top) with i a
rw [edist_dist,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 99,
"column": 2
} | {
"line": 104,
"column": 75
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : PseudoMetricSpace E\nf : ι → α → E\nl : Filter ι\ng : α → E\n⊢ TendstoInMeasure μ f l g ↔ ∀ (ε : ℝ), 0 < ε → Tendsto (fun i ↦ μ {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0)",
"usedConstants": [
"Iff.mpr",
... | refine ⟨fun h ε hε ↦ ?_, fun h ↦ ?_⟩
· convert! h (ENNReal.ofReal ε) (ENNReal.ofReal_pos.mpr hε) with i a
rw [edist_dist, ENNReal.ofReal_le_ofReal_iff (by positivity)]
· refine tendstoInMeasure_of_ne_top fun ε hε hε_top ↦ ?_
convert! h ε.toReal (ENNReal.toReal_pos hε.ne' hε_top) with i a
rw [edist_dist,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 1036,
"column": 2
} | {
"line": 1036,
"column": 60
} | [
{
"pp": "case inr\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0∞\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\nq : ℝ := p / (p - 1)\nhpq : p.HolderConjugate q\n⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p",
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"Real.instDivInvMonoid",
"... | have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.LpSpace.Complete | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 19
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_3\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nh : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p\n... | rw [h_liminf_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 333,
"column": 2
} | {
"line": 334,
"column": 63
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nf : ↥(Lp E p μ)\ng : ↥(Lp F p μ)\nh : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ ‖↑↑g x‖\n⊢ ‖f‖ ≤ ‖g‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"R... | rw [norm_def, norm_def]
exact ENNReal.toReal_mono (by finiteness) (eLpNorm_mono_ae h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 333,
"column": 2
} | {
"line": 334,
"column": 63
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nf : ↥(Lp E p μ)\ng : ↥(Lp F p μ)\nh : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ ‖↑↑g x‖\n⊢ ‖f‖ ≤ ‖g‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"R... | rw [norm_def, norm_def]
exact ENNReal.toReal_mono (by finiteness) (eLpNorm_mono_ae h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Multilinear.Basic | {
"line": 687,
"column": 6
} | {
"line": 687,
"column": 41
} | [
{
"pp": "R : Type u\nι : Type v\nM : Type u_1\ninst✝⁷ : Fintype ι\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : TopologicalSpace R\ninst✝² : TopologicalSpace M\ninst✝¹ : ContinuousMul R\ninst✝ : ContinuousSMul R M\nz₁ z₂ : M\n⊢ ContinuousMultilinearMap.mkPiRing R ι z₁ = Continuou... | ← toMultilinearMap_injective.eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 698,
"column": 87
} | {
"line": 698,
"column": 90
} | [
{
"pp": "case h\nα : Type u_1\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\nf f' : ↥(Lp E p μ)\na : α\nha1 : ↑↑(hg.compLp g0 f) a = (g ∘ ↑↑f) a\nha2 : ↑↑(hg.compL... | ha3 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 750,
"column": 21
} | {
"line": 755,
"column": 47
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nK : Type u_8\ninst✝ : RCLike K\nf : α → K\n⊢ MemLp (fun x ↦ RCLike.re (f x)) p μ ∧ MemLp (fun x ↦ RCLike.im (f x)) p μ ↔ MemLp f p μ",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NonUnitalNonAssocCommRing.toNonUnitalNo... | by
refine ⟨?_, fun hf => ⟨hf.re, hf.im⟩⟩
rintro ⟨hre, him⟩
convert! MeasureTheory.MemLp.add (ε := K) hre.ofReal (him.ofReal.const_mul RCLike.I)
ext1 x
rw [Pi.add_apply, mul_comm, RCLike.re_add_im] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 784,
"column": 20
} | {
"line": 784,
"column": 23
} | [
{
"pp": "case h\nα : Type u_1\n𝕜✝ : Type u_2\n𝕜'✝ : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ng✝ : E → F\nc : ℝ≥0\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : Nontrivially... | ha3 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 789,
"column": 39
} | {
"line": 789,
"column": 42
} | [
{
"pp": "case h\nα : Type u_1\n𝕜✝ : Type u_2\n𝕜'✝ : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ng : E → F\nc✝ : ℝ≥0\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : Nontrivially... | ha3 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.LpSpace.Indicator | {
"line": 51,
"column": 6
} | {
"line": 51,
"column": 81
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nhp : p ≠ ∞\nc : E\nε : ℝ≥0∞\nhε : ε ≠ 0\nh'p : p ≠ 0\nhp₀ : 0 < p\nhp₀' : 0 ≤ 1 / p.toReal\nhp₀'' : 0 < p.toReal\n⊢ Tendsto (fun x ↦ ‖c‖₊ * x ^ (1 / p.toReal)) (𝓝 0) (𝓝 0)",
"usedConstants": ... | convert! (NNReal.continuousAt_rpow_const (Or.inr hp₀')).tendsto.const_mul _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 83
} | [
{
"pp": "α : Type u_1\nε : Type u_3\nmα : MeasurableSpace α\nf : α → ε\ns : Set α\nμ ν : Measure α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\ninst✝ : PseudoMetrizableSpace ε\nhμ : IntegrableOn f s μ\nhν : IntegrableOn f s ν\n⊢ IntegrableOn f s (μ + ν)",
"usedConstants": [
"Eq.mpr",
... | delta IntegrableOn; rw [Measure.restrict_add]; exact hμ.integrable.add_measure hν | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 83
} | [
{
"pp": "α : Type u_1\nε : Type u_3\nmα : MeasurableSpace α\nf : α → ε\ns : Set α\nμ ν : Measure α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\ninst✝ : PseudoMetrizableSpace ε\nhμ : IntegrableOn f s μ\nhν : IntegrableOn f s ν\n⊢ IntegrableOn f s (μ + ν)",
"usedConstants": [
"Eq.mpr",
... | delta IntegrableOn; rw [Measure.restrict_add]; exact hμ.integrable.add_measure hν | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 238,
"column": 37
} | {
"line": 238,
"column": 89
} | [
{
"pp": "case inr.inr\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\ninst✝ : IsFiniteMeasure μ\nf : α → β\nhf : AEStronglyMeasurable f μ\np q : ℝ\nhp✝ : 0 ≤ p\nhq✝ : 0 ≤ q\nhpq : p ≤ q\nhint : Integrable (fun x ↦ ‖f x‖ ^ (ENNReal.ofReal q).toReal) μ\nhp : 0 < p... | integrable_norm_rpow_iff hf (by simp [hq]) (by simp) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 743,
"column": 2
} | {
"line": 744,
"column": 28
} | [
{
"pp": "case refine_1\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nn : ℕ\nA : Type u_1\ninst✝¹ : SeminormedRing A\ninst✝ : NormedAlgebra 𝕜 A\nm : Fin n.succ → A\n⊢ List.map m (List.finRange n.succ) ≠ []",
"usedConstants": [
"Eq.mpr",
"congrArg",
"List.map",
"List.map_eq_nil_i... | · rw [Ne, List.map_eq_nil_iff, List.finRange_eq_nil_iff]
exact Nat.succ_ne_zero _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 756,
"column": 4
} | {
"line": 756,
"column": 93
} | [
{
"pp": "case refine_2\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nA : Type u_1\ninst✝¹ : SeminormedRing A\ninst✝ : NormedAlgebra 𝕜 A\n⊢ ‖1‖ ≤ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real.partialOr... | convert! ratio_le_opNorm (ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A) fun _ => (1 : A) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1030,
"column": 48
} | {
"line": 1030,
"column": 80
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE : ι → Type wE\nE₁ : ι → Type wE₁\nG : Type wG\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝² : S... | rw [prod_mul_distrib, mul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1030,
"column": 48
} | {
"line": 1030,
"column": 80
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE : ι → Type wE\nE₁ : ι → Type wE₁\nG : Type wG\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝² : S... | rw [prod_mul_distrib, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1030,
"column": 48
} | {
"line": 1030,
"column": 80
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE : ι → Type wE\nE₁ : ι → Type wE₁\nG : Type wG\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝² : S... | rw [prod_mul_distrib, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1263,
"column": 2
} | {
"line": 1263,
"column": 36
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nG : Type wG\ninst✝⁵ : Fintype ι\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : Subsingleton ι\ninst✝ : Nontrivial G\ni : ι\n⊢ ‖(ofSubsingleton 𝕜 G G i) (ContinuousLinearMap.id 𝕜 G)‖ = 1",
"usedConstants": [
"... | simp [ContinuousLinearMap.norm_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1263,
"column": 2
} | {
"line": 1263,
"column": 36
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nG : Type wG\ninst✝⁵ : Fintype ι\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : Subsingleton ι\ninst✝ : Nontrivial G\ni : ι\n⊢ ‖(ofSubsingleton 𝕜 G G i) (ContinuousLinearMap.id 𝕜 G)‖ = 1",
"usedConstants": [
"... | simp [ContinuousLinearMap.norm_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1263,
"column": 2
} | {
"line": 1263,
"column": 36
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nG : Type wG\ninst✝⁵ : Fintype ι\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : Subsingleton ι\ninst✝ : Nontrivial G\ni : ι\n⊢ ‖(ofSubsingleton 𝕜 G G i) (ContinuousLinearMap.id 𝕜 G)‖ = 1",
"usedConstants": [
"... | simp [ContinuousLinearMap.norm_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 552,
"column": 2
} | {
"line": 552,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : CompleteSpace 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : ContinuousSMul 𝕜 E\ns t : Submodule 𝕜 E\nhs : IsClosed[inst✝³] ↑s\nht : FiniteDimensional ... | rw [← comap_map_mkQ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 435,
"column": 4
} | {
"line": 435,
"column": 50
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF' : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm : MeasurableSpace α\nμ : Measure α\nT T' : Set α → E →L[ℝ] F'\nc : ℝ\nh_smul : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T' s = c • T s\nf : α... | rw [this x hx, ContinuousLinearMap.smul_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 78,
"column": 63
} | {
"line": 80,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v : α → β\nl : Filter α\nh : u ~[l] v\n⊢ v =O[l] u",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"HEq.refl",
"Asymptotics.IsBigO",
"HSub.hSub",
"AddCommGroup.toAddGroup... | by
convert! h.isLittleO.right_isBigO_add
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 339,
"column": 2
} | {
"line": 340,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : NormedField β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\nu v : α → β\nl : Filter α\ninst✝ : ClosedIicTopology β\nh : u ~[l] v\n⊢ ∃ φ, (∀ᶠ (x : α) in l, 0 < φ x) ∧ u =ᶠ[l] φ * v",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Normed... | obtain ⟨φ, hφ, h_eq⟩ := h.exists_eq_mul
exact ⟨φ, hφ.eventually_const_lt (zero_lt_one' β), h_eq⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 339,
"column": 2
} | {
"line": 340,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : NormedField β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\nu v : α → β\nl : Filter α\ninst✝ : ClosedIicTopology β\nh : u ~[l] v\n⊢ ∃ φ, (∀ᶠ (x : α) in l, 0 < φ x) ∧ u =ᶠ[l] φ * v",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Normed... | obtain ⟨φ, hφ, h_eq⟩ := h.exists_eq_mul
exact ⟨φ, hφ.eventually_const_lt (zero_lt_one' β), h_eq⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.ContinuousAffineMap | {
"line": 107,
"column": 79
} | {
"line": 109,
"column": 5
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nW : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : TopologicalSpace P\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup W\ninst✝² : Module R W\ninst✝¹ : TopologicalSpace Q\ninst✝ : AddTorsor W Q\nf : P →ᴬ[R] Q\nh : ... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.ContinuousAffineMap | {
"line": 591,
"column": 61
} | {
"line": 592,
"column": 85
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nV : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝¹⁵ : Ring S\ninst✝¹⁴ : Ring R\ninst✝¹³ : AddCommGroup V\ninst✝¹² : Module R V\ninst✝¹¹ : TopologicalSpace V\ninst✝¹⁰ : IsTopologicalAddGroup V\ninst✝⁹ : AddCommGroup W\ninst✝⁸ : Module R W\ninst✝⁷ : TopologicalSpace W\ninst✝⁶ : ... | by
rw [decompAffineEquiv, ← AffineEquiv.coe_symm_toEquiv, decompEquiv_symm_contLinear] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1181,
"column": 2
} | {
"line": 1194,
"column": 48
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ\nhf_nonneg : 0 ≤ᶠ[ae μ] f\nhg_nonneg : 0 ≤ᶠ[ae μ] g\nhf : MemLp f (ENNReal.ofReal p) μ\nhg : MemLp g (ENNReal.ofReal q) μ\n⊢ ∫ (a : α), f a * g a ∂μ ≤ (∫ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫ (a : α), g a ... | have h_left : ∫ a, f a * g a ∂μ = ∫ a, ‖f a‖ * ‖g a‖ ∂μ := by
refine integral_congr_ae ?_
filter_upwards [hf_nonneg, hg_nonneg] with x hxf hxg
rw [Real.norm_of_nonneg hxf, Real.norm_of_nonneg hxg]
have h_right_f : ∫ a, f a ^ p ∂μ = ∫ a, ‖f a‖ ^ p ∂μ := by
refine integral_congr_ae ?_
filter_upwards... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1181,
"column": 2
} | {
"line": 1194,
"column": 48
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ\nhf_nonneg : 0 ≤ᶠ[ae μ] f\nhg_nonneg : 0 ≤ᶠ[ae μ] g\nhf : MemLp f (ENNReal.ofReal p) μ\nhg : MemLp g (ENNReal.ofReal q) μ\n⊢ ∫ (a : α), f a * g a ∂μ ≤ (∫ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫ (a : α), g a ... | have h_left : ∫ a, f a * g a ∂μ = ∫ a, ‖f a‖ * ‖g a‖ ∂μ := by
refine integral_congr_ae ?_
filter_upwards [hf_nonneg, hg_nonneg] with x hxf hxg
rw [Real.norm_of_nonneg hxf, Real.norm_of_nonneg hxg]
have h_right_f : ∫ a, f a ^ p ∂μ = ∫ a, ‖f a‖ ^ p ∂μ := by
refine integral_congr_ae ?_
filter_upwards... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1294,
"column": 10
} | {
"line": 1294,
"column": 32
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nr : ℝ≥0\nhfint : Integrable f μ\nhfint' : 0 ≤ ∫ (x : α), f x ∂μ\nhf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r\nhr : r = 0\nthis : f =ᶠ[ae μ] 0\n⊢ eLpNorm f 1 μ ≤ 2 * μ univ * ↑r",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCo... | eLpNorm_congr_ae this, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1301,
"column": 6
} | {
"line": 1302,
"column": 58
} | [
{
"pp": "case pos.refine_2\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nr : ℝ≥0\nhfint : Integrable f μ\nhfint' : 0 ≤ ∫ (x : α), f x ∂μ\nhr : r = 0\nhf : ∀ᵐ (ω : α) ∂μ, f ω ≤ 0\nhnegf : ∫ (x : α), -f x ∂μ = 0\nthis : -f =ᶠ[ae μ] 0\n⊢ f =ᶠ[ae μ] 0",
"usedConstants": [
"MeasureTheory... | filter_upwards [this] with ω hω
rwa [Pi.neg_apply, Pi.zero_apply, neg_eq_zero] at hω | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1301,
"column": 6
} | {
"line": 1302,
"column": 58
} | [
{
"pp": "case pos.refine_2\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nr : ℝ≥0\nhfint : Integrable f μ\nhfint' : 0 ≤ ∫ (x : α), f x ∂μ\nhr : r = 0\nhf : ∀ᵐ (ω : α) ∂μ, f ω ≤ 0\nhnegf : ∫ (x : α), -f x ∂μ = 0\nthis : -f =ᶠ[ae μ] 0\n⊢ f =ᶠ[ae μ] 0",
"usedConstants": [
"MeasureTheory... | filter_upwards [this] with ω hω
rwa [Pi.neg_apply, Pi.zero_apply, neg_eq_zero] at hω | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Affine.Isometry | {
"line": 72,
"column": 78
} | {
"line": 74,
"column": 5
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nV₂ : Type u_5\nP : Type u_10\nP₂ : Type u_11\ninst✝⁸ : NormedField 𝕜\ninst✝⁷ : SeminormedAddCommGroup V\ninst✝⁶ : NormedSpace 𝕜 V\ninst✝⁵ : PseudoMetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ninst✝³ : SeminormedAddCommGroup V₂\ninst✝² : NormedSpace 𝕜 V₂\ninst✝¹ : PseudoMe... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Affine.Isometry | {
"line": 110,
"column": 83
} | {
"line": 112,
"column": 5
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nV₂ : Type u_5\ninst✝⁴ : NormedField 𝕜\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : NormedSpace 𝕜 V₂\nf : V →ₗᵢ[𝕜] V₂\n⊢ f.toAffineIsometry.linearIsometry = f",
"usedConstants": [
"LinearIsometry",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Affine.Isometry | {
"line": 310,
"column": 86
} | {
"line": 312,
"column": 5
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nV₂ : Type u_5\nP : Type u_10\nP₂ : Type u_11\ninst✝⁸ : NormedField 𝕜\ninst✝⁷ : SeminormedAddCommGroup V\ninst✝⁶ : NormedSpace 𝕜 V\ninst✝⁵ : PseudoMetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ninst✝³ : SeminormedAddCommGroup V₂\ninst✝² : NormedSpace 𝕜 V₂\ninst✝¹ : PseudoMe... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
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