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.Transvection.Basic | {
"line": 481,
"column": 8
} | {
"line": 481,
"column": 50
} | [
{
"pp": "case h.right\nV : Type u_2\ninst✝³ : AddCommGroup V\nK : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nx✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1\nhe : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1\nhe' : e.fixedReduce = 1\nhe_one : ¬e = 1\nhefixed_... | simp_all [v, LinearMap.transvection.apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 301,
"column": 2
} | {
"line": 305,
"column": 15
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx : CliffordAlgebra Q\n⊢ (changeForm ⋯) x = x",
"usedConstants": [
"Eq.mpr",
"LinearMap.zero_apply",
"CliffordAlgebra.ι",
"Algebra.to_smulCommClass",
"Sem... | induction x using CliffordAlgebra.left_induction with
| algebraMap => simp_rw [changeForm_algebraMap]
| add _ _ hx hy => rw [map_add, hx, hy]
| ι_mul _ _ hx => rw [changeForm_ι_mul, hx, LinearMap.zero_apply, map_zero, LinearMap.zero_apply,
sub_zero] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 301,
"column": 2
} | {
"line": 305,
"column": 15
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx : CliffordAlgebra Q\n⊢ (changeForm ⋯) x = x",
"usedConstants": [
"Eq.mpr",
"LinearMap.zero_apply",
"CliffordAlgebra.ι",
"Algebra.to_smulCommClass",
"Sem... | induction x using CliffordAlgebra.left_induction with
| algebraMap => simp_rw [changeForm_algebraMap]
| add _ _ hx hy => rw [map_add, hx, hy]
| ι_mul _ _ hx => rw [changeForm_ι_mul, hx, LinearMap.zero_apply, map_zero, LinearMap.zero_apply,
sub_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 301,
"column": 2
} | {
"line": 305,
"column": 15
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx : CliffordAlgebra Q\n⊢ (changeForm ⋯) x = x",
"usedConstants": [
"Eq.mpr",
"LinearMap.zero_apply",
"CliffordAlgebra.ι",
"Algebra.to_smulCommClass",
"Sem... | induction x using CliffordAlgebra.left_induction with
| algebraMap => simp_rw [changeForm_algebraMap]
| add _ _ hx hy => rw [map_add, hx, hy]
| ι_mul _ _ hx => rw [changeForm_ι_mul, hx, LinearMap.zero_apply, map_zero, LinearMap.zero_apply,
sub_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 147,
"column": 14
} | {
"line": 147,
"column": 44
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nm₁ : M₁\n⊢ (GradedTensorProduct.lift (evenOdd Q₁) (evenOdd Q₂) (map (QuadraticMap.Isometry.in... | GradedTensorProduct.lift_tmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 152,
"column": 14
} | {
"line": 152,
"column": 44
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nm₂ : M₂\n⊢ (GradedTensorProduct.lift (evenOdd Q₁) (evenOdd Q₂) (map (QuadraticMap.Isometry.in... | GradedTensorProduct.lift_tmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.External | {
"line": 121,
"column": 28
} | {
"line": 121,
"column": 44
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝⁷ : CommSemiring ι\ninst✝⁶ : Module ι (Additive ℤˣ)\ninst✝⁵ : DecidableEq ι\n𝒜 : ι → Type u_3\nℬ : ι → Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝² : (i : ι) → AddCommGroup (ℬ i)\ninst✝¹ : (i : ι) → Module R (𝒜 i)\ninst✝ : (i : ι) → Mo... | ← Units.smul_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.ExteriorAlgebra.Grading | {
"line": 49,
"column": 2
} | {
"line": 50,
"column": 80
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\n⊢ (GradedAlgebra.ι R M) m * (GradedAlgebra.ι R M) m = 0",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submodule",
"Submodule.addSubmonoidClass",
"Semiring.toModule",
... | rw [GradedAlgebra.ι_apply, DirectSum.of_mul_of]
exact DFinsupp.single_eq_zero.mpr (Subtype.ext <| ExteriorAlgebra.ι_sq_zero _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.ExteriorAlgebra.Grading | {
"line": 49,
"column": 2
} | {
"line": 50,
"column": 80
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\n⊢ (GradedAlgebra.ι R M) m * (GradedAlgebra.ι R M) m = 0",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submodule",
"Submodule.addSubmonoidClass",
"Semiring.toModule",
... | rw [GradedAlgebra.ι_apply, DirectSum.of_mul_of]
exact DFinsupp.single_eq_zero.mpr (Subtype.ext <| ExteriorAlgebra.ι_sq_zero _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 44
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nx : Fin n → M\n⊢ (List.ofFn (⇑toDirectSum ∘ fun i ↦ (ι R) (x i))).prod =\n (DirectSum.of (fun i ↦ ⨂[R]^i M) n) ((PiTensorProduct.tprod R) x)",
"usedConstants": [
"PiTensorProduct.instM... | simp_rw [Function.comp_def, toDirectSum_ι] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 123,
"column": 4
} | {
"line": 125,
"column": 27
} | [
{
"pp": "case e_a\nι : Type u_1\nR : Type u_2\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Fintype ι\ninst✝ : Module R M\nN : Submodule R M\nbM : Basis ι R M\nbN : Basis (Fin n) R ↥N\nf : Fin n ↪ ι\na : Fin n → R\nsnf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)\nN' : Submodule R (ι... | have hf' : Function.Injective f' := fun i j hij ↦ by
rw [Subtype.ext_iff] at hij
exact f.injective hij | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.Determinant.Misc | {
"line": 65,
"column": 50
} | {
"line": 65,
"column": 63
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nM : Matrix (Fin n) (Fin (n + 1)) R\nhv : ∀ (i : Fin n), ∑ j, M i j = 0\nj₁ j₂ : Fin (n + 1)\nx✝ : Fin n\n⊢ 0 = 0 x✝",
"usedConstants": [
"CommSemiring.toSemiring",
"CommRing.toCommSemiring",
"eq_self",
"of_eq_true",
"Zer... | Pi.zero_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 108,
"column": 63
} | {
"line": 108,
"column": 71
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\nA : Matrix n n R\ninst✝ : Nontrivial n\nh_irr : A.IsIrreducible\ni : n\nthis : Quiver n := A.toQuiver\nh_row : ¬∃ j, 0 < A i j\nno_out : ∀ (j : n), IsEmpty (i ⟶ j)\nj : n\nhij : ∃ y, j ≠ y\np : Path i j\nhp_pos : 0 < p.length\nh_le : ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 108,
"column": 63
} | {
"line": 108,
"column": 71
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\nA : Matrix n n R\ninst✝ : Nontrivial n\nh_irr : A.IsIrreducible\ni : n\nthis : Quiver n := A.toQuiver\nh_row : ¬∃ j, 0 < A i j\nno_out : ∀ (j : n), IsEmpty (i ⟶ j)\nj : n\nhij : ∃ y, j ≠ y\np : Path i j\nhp_pos : 0 < p.length\nh_le : ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 108,
"column": 63
} | {
"line": 108,
"column": 71
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\nA : Matrix n n R\ninst✝ : Nontrivial n\nh_irr : A.IsIrreducible\ni : n\nthis : Quiver n := A.toQuiver\nh_row : ¬∃ j, 0 < A i j\nno_out : ∀ (j : n), IsEmpty (i ⟶ j)\nj : n\nhij : ∃ y, j ≠ y\np : Path i j\nhp_pos : 0 < p.length\nh_le : ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 126,
"column": 8
} | {
"line": 126,
"column": 16
} | [
{
"pp": "case zero.refine_1.inr\nn : Type u_1\nR : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\nA : Matrix n n R\ninst✝⁴ : Fintype n\ninst✝³ : IsOrderedRing R\ninst✝² : PosMulStrictMono R\ninst✝¹ : Nontrivial R\ninst✝ : DecidableEq n\nhA : ∀ (i j : n), 0 ≤ A i j\nthis : Quiver n := A.toQuiver\ni j : n\nh_... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 126,
"column": 8
} | {
"line": 126,
"column": 16
} | [
{
"pp": "case zero.refine_1.inr\nn : Type u_1\nR : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\nA : Matrix n n R\ninst✝⁴ : Fintype n\ninst✝³ : IsOrderedRing R\ninst✝² : PosMulStrictMono R\ninst✝¹ : Nontrivial R\ninst✝ : DecidableEq n\nhA : ∀ (i j : n), 0 ≤ A i j\nthis : Quiver n := A.toQuiver\ni j : n\nh_... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 126,
"column": 8
} | {
"line": 126,
"column": 16
} | [
{
"pp": "case zero.refine_1.inr\nn : Type u_1\nR : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\nA : Matrix n n R\ninst✝⁴ : Fintype n\ninst✝³ : IsOrderedRing R\ninst✝² : PosMulStrictMono R\ninst✝¹ : Nontrivial R\ninst✝ : DecidableEq n\nhA : ∀ (i j : n), 0 ≤ A i j\nthis : Quiver n := A.toQuiver\ni j : n\nh_... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 12
} | [
{
"pp": "case «1».«0»\nm : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nh10 : ↑A 1 0 = 0\nh00 : 0 < ↑A 0 0\nh01 : 0 ≤ ↑A 0 1\nh11 : |↑A 0 1| < |↑A 1 1|\nh1 : 0 < |↑A 1 1|\nh2 : 0 < |↑A 0 0|\n⊢ |↑A ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)| ≤ |m|",
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 12
} | [
{
"pp": "case «1».«0»\nm : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nh10 : ↑A 1 0 = 0\nh00 : 0 < ↑A 0 0\nh01 : 0 ≤ ↑A 0 1\nh11 : |↑A 0 1| < |↑A 1 1|\nh1 : 0 < |↑A 1 1|\nh2 : 0 < |↑A 0 0|\n⊢ |↑A ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)| ≤ |m|",
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtr... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 12
} | [
{
"pp": "case «1».«0»\nm : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nh10 : ↑A 1 0 = 0\nh00 : 0 < ↑A 0 0\nh01 : 0 ≤ ↑A 0 1\nh11 : |↑A 0 1| < |↑A 1 1|\nh1 : 0 < |↑A 1 1|\nh2 : 0 < |↑A 0 0|\n⊢ |↑A ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)| ≤ |m|",
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtr... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 156,
"column": 4
} | {
"line": 157,
"column": 20
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nI J : RingCon R\neq : matrix n I = matrix n J\nr s : R\n⊢ I r s ↔ J r s",
"usedConstants": [
"DFunLike.congr_fun",
"Matrix.add",
"Equiv.instEquivLike",
"RingCon.instFunL... | have := congr_fun (DFunLike.congr_fun eq (Matrix.of fun _ _ ↦ r)) (Matrix.of fun _ _ ↦ s)
simpa using this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 156,
"column": 4
} | {
"line": 157,
"column": 20
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nI J : RingCon R\neq : matrix n I = matrix n J\nr s : R\n⊢ I r s ↔ J r s",
"usedConstants": [
"DFunLike.congr_fun",
"Matrix.add",
"Equiv.instEquivLike",
"RingCon.instFunL... | have := congr_fun (DFunLike.congr_fun eq (Matrix.of fun _ _ ↦ r)) (Matrix.of fun _ _ ↦ s)
simpa using this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 194,
"column": 2
} | {
"line": 196,
"column": 49
} | [
{
"pp": "case H.mp\nR : Type u_1\nn : Type u_2\ninst✝³ : NonUnitalNonAssocSemiring R\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : Nonempty n\nc : RingCon R\nx y : R\n⊢ (matrix n c).ofMatrix x y → c x y",
"usedConstants": [
"Inhabited.default",
"RingCon.ofMatrix",
"RingCon.instFunLi... | · intro h
inhabit n
simpa using h default default default default | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 235,
"column": 35
} | {
"line": 235,
"column": 48
} | [
{
"pp": "case pred\nm✝ : ℤ\nC : FixedDetMatrix (Fin 2) ℤ m✝ → Prop\nhS : ∀ (B : FixedDetMatrix (Fin 2) ℤ m✝), C B → C (S • B)\nhT : ∀ (B : FixedDetMatrix (Fin 2) ℤ m✝), C B → C (T • B)\nB : FixedDetMatrix (Fin 2) ℤ m✝\nm : ℕ\nhm : C (T ^ (-↑m) • B) ↔ C B\n⊢ C ((T ^ (-1) * T ^ (-↑m)) • B) ↔ C B",
"usedConsta... | zpow_neg_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Multilinear.DirectSum | {
"line": 80,
"column": 89
} | {
"line": 83,
"column": 14
} | [
{
"pp": "R : Type u_1\nι : Type u_2\nM' : Type u_3\nκ : ι → Type u_4\nM : (i : ι) → κ i → Type u_5\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → (j : κ i) → AddCommMonoid (M i j)\ninst✝⁵ : (i : ι) → (j : κ i) → Module R (M i j)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : Module R M'\ninst✝² : DecidableEq ι\ninst✝¹ : Fi... | by
haveI : Fintype ι := Fintype.ofFinite ι
simp_rw [fromDirectSumEquiv, DirectSum.lof, ← fromDFinsuppEquiv_symm_apply]
convert! rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.WithConv | {
"line": 108,
"column": 31
} | {
"line": 108,
"column": 39
} | [
{
"pp": "case a.inl\nm : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝² : Semiring α\ninst✝¹ : IsLeftCancelMulZero α\ninst✝ : StarRing α\nf : WithConv (Matrix m n α)\nhf : ∀ (i : m) (j : n), f.ofConv i j = 0 ∨ f.ofConv i j = 1\ni : m\nj : n\nh : f.ofConv i j = 0\n⊢ (star f).ofConv i j = f.ofConv i j",
"usedCo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Matrix.WithConv | {
"line": 108,
"column": 31
} | {
"line": 108,
"column": 39
} | [
{
"pp": "case a.inr\nm : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝² : Semiring α\ninst✝¹ : IsLeftCancelMulZero α\ninst✝ : StarRing α\nf : WithConv (Matrix m n α)\nhf : ∀ (i : m) (j : n), f.ofConv i j = 0 ∨ f.ofConv i j = 1\ni : m\nj : n\nh : f.ofConv i j = 1\n⊢ (star f).ofConv i j = f.ofConv i j",
"usedCo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.PiTensorProduct | {
"line": 73,
"column": 18
} | {
"line": 73,
"column": 34
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : ι → Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → NonUnitalNonAssocSemiring (A i)\ninst✝² : (i : ι) → Module R (A i)\ninst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)\na₁ a₂ a₃ : (i : ι) → A i\nha : SemiconjBy a₁ a₂ ... | tprod_mul_tprod, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PiTensorProduct | {
"line": 73,
"column": 35
} | {
"line": 73,
"column": 51
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : ι → Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → NonUnitalNonAssocSemiring (A i)\ninst✝² : (i : ι) → Module R (A i)\ninst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)\na₁ a₂ a₃ : (i : ι) → A i\nha : SemiconjBy a₁ a₂ ... | tprod_mul_tprod, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PiTensorProduct | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 22
} | [
{
"pp": "ι : Type u_1\nR' : Type u_2\nR : Type u_3\nA : ι → Type u_4\ninst✝⁶ : CommSemiring R'\ninst✝⁵ : CommSemiring R\ninst✝⁴ : (i : ι) → Semiring (A i)\ninst✝³ : Algebra R' R\ninst✝² : (i : ι) → Algebra R (A i)\ninst✝¹ : (i : ι) → Algebra R' (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R' R (A i)\nr : R'\nx : ⨂[R... | change _ = mul _ _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.PiTensorProduct | {
"line": 286,
"column": 17
} | {
"line": 286,
"column": 33
} | [
{
"pp": "case H.h.H.h\nι : Type u_1\nR' : Type u_2\nR : Type u_3\nA : ι → Type u_4\ninst✝³ : CommSemiring R\ninst✝² : (i : ι) → CommSemiring (A i)\ninst✝¹ : (i : ι) → Algebra R (A i)\ninst✝ : Fintype ι\ntoFun : (⨂[R] (i : ι), R) →ₗ[R] R := lift (MultilinearMap.mkPiAlgebra R ι R)\nthis✝ : IsScalarTower R (⨂[R] (... | tprod_mul_tprod, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Projectivization.Subspace | {
"line": 194,
"column": 4
} | {
"line": 195,
"column": 17
} | [
{
"pp": "case carrier.h.refine_1\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nS : Set (ℙ K V)\nx : ℙ K V\nhx : ∀ (W : Subspace K V), S ⊆ ↑W → x ∈ W\n⊢ x ∈ sInf {W | S ⊆ ↑W}",
"usedConstants": [
"setOf",
"Membership.mem",
"HasSubset.Subs... | rintro W ⟨T, hT, rfl⟩
exact hx T hT | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Projectivization.Subspace | {
"line": 194,
"column": 4
} | {
"line": 195,
"column": 17
} | [
{
"pp": "case carrier.h.refine_1\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nS : Set (ℙ K V)\nx : ℙ K V\nhx : ∀ (W : Subspace K V), S ⊆ ↑W → x ∈ W\n⊢ x ∈ sInf {W | S ⊆ ↑W}",
"usedConstants": [
"setOf",
"Membership.mem",
"HasSubset.Subs... | rintro W ⟨T, hT, rfl⟩
exact hx T hT | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Projectivization.Action | {
"line": 133,
"column": 20
} | {
"line": 133,
"column": 31
} | [
{
"pp": "case pos\nK : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field K\ninst✝ : Module K V\nthis✝ : ∀ {a b c d : ℙ K V}, a ≠ b → c ≠ d → ∃ g, g • a = c ∧ g • b = d\nD D' E E' : ℙ K V\nhD : LinearIndependent K ![D.rep, D'.rep]\nhE : E ≠ E'\ng : V ≃ₗ[K] V\ngD : g • D = E\ngE : g • D' = E'\nhV : ... | ← mk_rep D' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 269,
"column": 4
} | {
"line": 270,
"column": 46
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\nW : Type u_3\nX : Type u_4\ninst✝³ : AddCommGroup W\ninst✝² : Module R W\ninst✝¹ : AddCommGroup X\ninst✝ : Module R X\ne : V ≃ₗ[R] W\ng : SpecialLinearGroup R W\n⊢ LinearEquiv.det (e ≪≫ₗ ↑g ≪≫ₗ e.symm) = 1",
... | nth_rewrite 1 [← LinearEquiv.symm_symm e]
rw [LinearEquiv.det_conj g e.symm, g.prop] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 269,
"column": 4
} | {
"line": 270,
"column": 46
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\nW : Type u_3\nX : Type u_4\ninst✝³ : AddCommGroup W\ninst✝² : Module R W\ninst✝¹ : AddCommGroup X\ninst✝ : Module R X\ne : V ≃ₗ[R] W\ng : SpecialLinearGroup R W\n⊢ LinearEquiv.det (e ≪≫ₗ ↑g ≪≫ₗ e.symm) = 1",
... | nth_rewrite 1 [← LinearEquiv.symm_symm e]
rw [LinearEquiv.det_conj g e.symm, g.prop] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basis | {
"line": 101,
"column": 72
} | {
"line": 101,
"column": 79
} | [
{
"pp": "case H\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nQ : QuadraticMap R M N\nbm : Basis ι R M\nx : M\n⊢ ((∑ x_1 ∈ (bm.repr x).support,\n if True then... | if_true | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.QuadraticForm.Real | {
"line": 71,
"column": 40
} | {
"line": 71,
"column": 48
} | [
{
"pp": "case zero\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module ℝ M\ninst✝ : FiniteDimensional ℝ M\nQ : QuadraticForm ℝ M\nhQ : LinearMap.SeparatingLeft (associated Q)\nw : Fin (finrank ℝ M) → SignType\nhw₀ : ∀ (i : Fin (finrank ℝ M)), w i ≠ 0\nhw : Equivalent Q (weightedSumSquares ℝ fun i ↦ ↑(w i))\... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.QuadraticForm.Real | {
"line": 71,
"column": 40
} | {
"line": 71,
"column": 48
} | [
{
"pp": "case neg\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module ℝ M\ninst✝ : FiniteDimensional ℝ M\nQ : QuadraticForm ℝ M\nhQ : LinearMap.SeparatingLeft (associated Q)\nw : Fin (finrank ℝ M) → SignType\nhw₀ : ∀ (i : Fin (finrank ℝ M)), w i ≠ 0\nhw : Equivalent Q (weightedSumSquares ℝ fun i ↦ ↑(w i))\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.QuadraticForm.Real | {
"line": 71,
"column": 40
} | {
"line": 71,
"column": 48
} | [
{
"pp": "case pos\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module ℝ M\ninst✝ : FiniteDimensional ℝ M\nQ : QuadraticForm ℝ M\nhQ : LinearMap.SeparatingLeft (associated Q)\nw : Fin (finrank ℝ M) → SignType\nhw₀ : ∀ (i : Fin (finrank ℝ M)), w i ≠ 0\nhw : Equivalent Q (weightedSumSquares ℝ fun i ↦ ↑(w i))\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 14
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb : P.Base\ninst✝ : DecidableEq ι\n⊢ span R (range h) ≤ (Matrix.diagonalLinear... | rw [span_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 118,
"column": 15
} | {
"line": 118,
"column": 18
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb : P.Base\ninst✝ : DecidableEq ι\nd : ι → R\ng : Matrix ι ι R →ₗ[R] Matrix (↥... | h₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 267,
"column": 62
} | {
"line": 267,
"column": 70
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.IsCrystallographic\nb : P.Base\ninst✝³ : Finite ι\ninst✝² : IsDomain R\ninst✝¹ : CharZero R\ninst✝... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 267,
"column": 62
} | {
"line": 267,
"column": 70
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.IsCrystallographic\nb : P.Base\ninst✝³ : Finite ι\ninst✝² : IsDomain R\ninst✝¹ : CharZero R\ninst✝... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 267,
"column": 62
} | {
"line": 267,
"column": 70
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.IsCrystallographic\nb : P.Base\ninst✝³ : Finite ι\ninst✝² : IsDomain R\ninst✝¹ : CharZero R\ninst✝... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 225,
"column": 49
} | {
"line": 225,
"column": 62
} | [
{
"pp": "case h.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : CharZero R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb : P... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 256,
"column": 49
} | {
"line": 256,
"column": 62
} | [
{
"pp": "case h.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : CharZero R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb : P... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 63
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.EmbeddedG2\ninst✝² : Finite ι\ninst✝¹ : CharZero R\ninst✝ : IsDomain R\ni : ι\nthis : Fintype ι\nB... | apply mul_right_cancel₀ (B.ne_zero <| threeShortAddTwoLong P) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 564,
"column": 13
} | {
"line": 564,
"column": 21
} | [
{
"pp": "case h.add\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsI... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 564,
"column": 13
} | {
"line": 564,
"column": 21
} | [
{
"pp": "case h.add\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsI... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 564,
"column": 13
} | {
"line": 564,
"column": 21
} | [
{
"pp": "case h.add\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsI... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 564,
"column": 4
} | {
"line": 564,
"column": 12
} | [
{
"pp": "case h.add\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsI... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 565,
"column": 14
} | {
"line": 565,
"column": 22
} | [
{
"pp": "case h.smul\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.Is... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 565,
"column": 14
} | {
"line": 565,
"column": 22
} | [
{
"pp": "case h.smul\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.Is... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 565,
"column": 14
} | {
"line": 565,
"column": 22
} | [
{
"pp": "case h.smul\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.Is... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorPower.Symmetric | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 12
} | [
{
"pp": "case trans\nR ι : Type u\ninst✝² : CommSemiring R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nx y x✝ y✝ z✝ : ⨂[R] (x : ι), M\na✝¹ : AddConGen.Rel (Rel R ι M) x✝ y✝\na✝ : AddConGen.Rel (Rel R ι M) y✝ z✝\na_ih✝¹ : (addConGen (Rel R ι M)) (r • x✝) (r • y✝)\na_ih✝ : (addConGen (Rel R ... | | trans => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.TensorPower.Symmetric | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 10
} | [
{
"pp": "case add\nR ι : Type u\ninst✝² : CommSemiring R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nx y w✝ x✝ y✝ z✝ : ⨂[R] (x : ι), M\na✝¹ : AddConGen.Rel (Rel R ι M) w✝ x✝\na✝ : AddConGen.Rel (Rel R ι M) y✝ z✝\na_ih✝¹ : (addConGen (Rel R ι M)) (r • w✝) (r • x✝)\na_ih✝ : (addConGen (Rel R... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 144,
"column": 49
} | {
"line": 144,
"column": 67
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\ninst✝² : P.IsReduced\nb : P.Base\ni... | by simp [this, hn] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Logic.Hydra | {
"line": 69,
"column": 4
} | {
"line": 71,
"column": 14
} | [
{
"pp": "case refine_1\nα : Type u_1\nr : α → α → Prop\ninst✝¹ : DecidableEq α\ninst✝ : Std.Irrefl r\ns t u : Multiset α\na : α\nhe : s + {a} = t + u\nhr : ∀ (a' : α), ¬r a' a → a' ∉ u\nb : α\nh : (rᶜ ⊓ fun x1 x2 ↦ x1 ≠ x2) b a\n⊢ count b s = count b t",
"usedConstants": [
"Iff.mpr",
"congrArg",... | apply_fun count b at he
simpa only [count_add, count_singleton, if_neg h.2, add_zero, count_eq_zero.2 (hr b h.1)]
using he | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Hydra | {
"line": 69,
"column": 4
} | {
"line": 71,
"column": 14
} | [
{
"pp": "case refine_1\nα : Type u_1\nr : α → α → Prop\ninst✝¹ : DecidableEq α\ninst✝ : Std.Irrefl r\ns t u : Multiset α\na : α\nhe : s + {a} = t + u\nhr : ∀ (a' : α), ¬r a' a → a' ∉ u\nb : α\nh : (rᶜ ⊓ fun x1 x2 ↦ x1 ≠ x2) b a\n⊢ count b s = count b t",
"usedConstants": [
"Iff.mpr",
"congrArg",... | apply_fun count b at he
simpa only [count_add, count_singleton, if_neg h.2, add_zero, count_eq_zero.2 (hr b h.1)]
using he | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.ClosedCompactCylinders | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 60
} | [
{
"pp": "case refine_1\nι : Type u_1\nX : ι → Type u_2\ninst✝³ : (i : ι) → TopologicalSpace (X i)\nt : Set ((i : ι) → X i)\ninst✝² : (i : ι) → MeasurableSpace (X i)\ninst✝¹ : ∀ (i : ι), SecondCountableTopology (X i)\ninst✝ : ∀ (i : ι), OpensMeasurableSpace (X i)\nht : t ∈ closedCompactCylinders X\n⊢ MeasurableS... | · exact (closedCompactCylinders.isClosed ht).measurableSet | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.Projective | {
"line": 87,
"column": 6
} | {
"line": 88,
"column": 17
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\nP : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)\nI J : Finset ι\nhP : IsProjectiveMeasureFamily P\nS : Set ((i : ↥I) → α ↑i)\nT : Set ((i : ↥J) → α ↑i)\nhT : MeasurableSet T\nh_eq : cylinder I S = cylinder J T\nhJI : J ⊆ I\nh : IsE... | simp only [Measure.measure_univ_eq_zero] at this
simp [this] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.Projective | {
"line": 87,
"column": 6
} | {
"line": 88,
"column": 17
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\nP : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)\nI J : Finset ι\nhP : IsProjectiveMeasureFamily P\nS : Set ((i : ↥I) → α ↑i)\nT : Set ((i : ↥J) → α ↑i)\nhT : MeasurableSet T\nh_eq : cylinder I S = cylinder J T\nhJI : J ⊆ I\nh : IsE... | simp only [Measure.measure_univ_eq_zero] at this
simp [this] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 431,
"column": 55
} | {
"line": 439,
"column": 59
} | [
{
"pp": "ι : Type u_2\nX : ι → Type u_3\nm : (i : ι) → MeasurableSpace (X i)\nΔ : Set ι\ni : ι\ninst✝ : DecidableEq ι\n⊢ Measurable fun p ↦ update p.1 i p.2",
"usedConstants": [
"Eq.mpr",
"MeasurableSpace.prod",
"Function.update",
"congrArg",
"Measurable",
"Membership.mem... | by
rw [measurable_cylinderEvents_iff]
intro j hj
dsimp [update]
split_ifs with h
· subst h
dsimp
exact measurable_snd
· exact measurable_cylinderEvents_iff.1 measurable_fst hj | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 347,
"column": 4
} | {
"line": 347,
"column": 16
} | [
{
"pp": "case refine_2\nι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module K M\ninst✝² : AddCommGroup N\ninst✝¹ : Module K N\nP : RootPairing ι K M N\ninst✝ : P.IsCrystallographic\nb ... | rw [span_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 336,
"column": 85
} | {
"line": 351,
"column": 87
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module K M\ninst✝² : AddCommGroup N\ninst✝¹ : Module K N\nP : RootPairing ι K M N\ninst✝ : P.IsCrystallographic\nb : P.Base\n⊢ ↑(g... | by
refine le_antisymm (fun w hw ↦ Pi.mem_span_range_single_inl_iff.mpr fun i ↦ ?_) ?_
· replace hw : ∀ (x) (hx : x ∈ lieAlgebra b), ⟨x, hx⟩ ∈ H →
∃ k, (x.toLin' ^ k) w = 0 := by simpa [mem_genWeightSpace] using hw
obtain ⟨j, hj⟩ : ∃ j : b.support, P.pairingIn ℤ i j ≠ 0 := by
obtain ⟨j, hj, hj₀⟩ :=... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.SetSemiring | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 66
} | [
{
"pp": "case cons\nα : Type u_1\nC : Set (Set α)\nJ✝ : Finset (Set α)\nhC : IsSetSemiring C\ns : Set α\nJ : Finset (Set α)\nhJ : s ∉ J\nhind :\n ↑J ⊆ C →\n ∃ K,\n (↑J).PairwiseDisjoint K ∧\n (∀ i ∈ J, ↑(K i) ⊆ C) ∧\n (⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧\n (∀ j ∈ J, ⋃₀ ↑(K... | have hK1_of_ne t (ht : t ≠ s) : K1 t = K t := by simp [K1, ht] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 97,
"column": 8
} | {
"line": 97,
"column": 21
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nhs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.WithTop | {
"line": 35,
"column": 2
} | {
"line": 178,
"column": 19
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\n⊢ SecondCountableTopology (WithTop ι)",
"usedConstants": [
"Preorder.topology",
"dite_cond_eq_true",
"Iff.mpr",
"Eq.mpr",
"False",
"Exists.choose_... | rcases isEmpty_or_nonempty ι with hι | ⟨⟨x₀⟩⟩
· infer_instance
/- Let `c` be a countable set in `ι` such that the topology is generated by the sets `Iio a`
and `Ioi a` for `a ∈ c`, by second-countability. Let `c'` be a dense set in `ι`, again by
second-countability. Let `d` in `WithTop ι` be obtained from `c ∪ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.WithTop | {
"line": 35,
"column": 2
} | {
"line": 178,
"column": 19
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\n⊢ SecondCountableTopology (WithTop ι)",
"usedConstants": [
"Preorder.topology",
"dite_cond_eq_true",
"Iff.mpr",
"Eq.mpr",
"False",
"Exists.choose_... | rcases isEmpty_or_nonempty ι with hι | ⟨⟨x₀⟩⟩
· infer_instance
/- Let `c` be a countable set in `ι` such that the topology is generated by the sets `Iio a`
and `Ioi a` for `a ∈ c`, by second-countability. Let `c'` be a dense set in `ι`, again by
second-countability. Let `d` in `WithTop ι` be obtained from `c ∪ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 269,
"column": 4
} | {
"line": 272,
"column": 11
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns t : Set α\nI✝ : Finset (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm✝ m' m : AddContent G C\nhC : IsSetSemiring C\nI : Finset (Set α)\nhI : ↑I ⊆ _root_.supClosure C\nh'I : (↑I).PairwiseDisjoint id\nhh'I : ⋃₀ ↑I ∈ _root_.supClosure C\nJ : Set α → Finset (Set α)\nhJC ... | · simp only [K, coe_biUnion]
refine (h'I.mono_on ?_).biUnion hJdisj
simp
grind | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 10
} | [
{
"pp": "case h\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 10
} | [
{
"pp": "case h\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 74
} | [
{
"pp": "case refine_3\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nhs_m : MeasurableSet s\nhf_int : Integrable f μ\nthis : SigmaFinite ... | exact (stronglyMeasurable_condExp.indicator hs_m).aestronglyMeasurable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 74
} | [
{
"pp": "case refine_3\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nhs_m : MeasurableSet s\nhf_int : Integrable f μ\nthis : SigmaFinite ... | exact (stronglyMeasurable_condExp.indicator hs_m).aestronglyMeasurable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 74
} | [
{
"pp": "case refine_3\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nhs_m : MeasurableSet s\nhf_int : Integrable f μ\nthis : SigmaFinite ... | exact (stronglyMeasurable_condExp.indicator hs_m).aestronglyMeasurable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 235,
"column": 4
} | {
"line": 240,
"column": 21
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nhr... | rw [eventuallyLE_congr (blimsup_cthickening_mul_ae_eq μ p s (one_half_pos (α := ℝ)) r hr).symm
EventuallyEq.rfl]
apply HasSubset.Subset.eventuallyLE
change _ ≤ _
refine mono_blimsup' (hr'.mono fun i hi pi => cthickening_subset_thickening' (hi pi) ?_ (s i))
nlinarith [hi pi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 235,
"column": 4
} | {
"line": 240,
"column": 21
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)\nhr... | rw [eventuallyLE_congr (blimsup_cthickening_mul_ae_eq μ p s (one_half_pos (α := ℝ)) r hr).symm
EventuallyEq.rfl]
apply HasSubset.Subset.eventuallyLE
change _ ≤ _
refine mono_blimsup' (hr'.mono fun i hi pi => cthickening_subset_thickening' (hi pi) ?_ (s i))
nlinarith [hi pi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 565,
"column": 58
} | {
"line": 565,
"column": 87
} | [
{
"pp": "case succ\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nhC : IsSetRing C\ns : ℕ → Set α\nhs_disj : Pairwise (Disjoint on s)\nhsC : ∀ (i : ℕ), s i ∈ C\nn : ℕ\nhn : m (accumulate s n) = ∑ i ∈ Finset.range (n + 1), m (s i)\n⊢ m (accumulate s n ∪ s (n + 1)) = m ... | addContent_union hC _ (hsC _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 650,
"column": 15
} | {
"line": 650,
"column": 17
} | [
{
"pp": "case refine_2\nα : Type u_1\nC : Set (Set α)\nhC : IsSetRing C\nm : AddContent ℝ≥0∞ C\nhm_ne_top : ∀ s ∈ C, m s ≠ ∞\nhm_tendsto : ∀ ⦃s : ℕ → Set α⦄, (∀ (n : ℕ), s n ∈ C) → Antitone s → ⋂ n, s n = ∅ → Tendsto (fun n ↦ m (s n)) atTop (𝓝 0)\nf : ℕ → Set α\nhf : ∀ (i : ℕ), f i ∈ C\nhUf : ⋃ i, f i ∈ C\nh_d... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym | {
"line": 139,
"column": 61
} | {
"line": 139,
"column": 77
} | [
{
"pp": "𝓧 : Type u_1\nm m𝓧 : MeasurableSpace 𝓧\nμ ν : Measure 𝓧\nhm : m ≤ m𝓧\ninst✝ : IsFiniteMeasure μ\nhsf : SigmaFinite (ν.trim hm)\nhμν : μ ≪ ν\n⊢ SigmaFinite (Measure.map id ν)",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"MeasureTheory.trim_eq_map",
"congrArg",... | ← trim_eq_map hm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 16
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : ↑s i < 0\nhn : ¬∀ (n : ℕ), ¬s ≤[i \\ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0\nh : ¬s ≤[i] 0\n⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ ↑s j < 0",
"usedConstants": [
"Mathli... | push Not at hn | Mathlib.Tactic.Push._aux_Mathlib_Tactic_Push___elabRules_Mathlib_Tactic_Push_pushStx_1 | Mathlib.Tactic.Push.pushStx |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 267,
"column": 4
} | {
"line": 267,
"column": 52
} | [
{
"pp": "case neg.h\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : ↑s i < 0\nh : ¬s ≤[i] 0\nhn : ∃ n, s ≤[i \\ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0\nk : ℕ := Nat.find hn\nhk₂ : s ≤[i \\ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0\nhmeas : Meas... | simp only [and_imp, exists_prop, Set.mem_iUnion] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 155,
"column": 28
} | {
"line": 155,
"column": 41
} | [
{
"pp": "case h\nα : Type u_1\ninst✝ : MeasurableSpace α\ni : Set α\nhi : MeasurableSet i\n⊢ 0 = 0 i",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
"Pi.zero_apply",
"id",
"Pi.instZero",
"Real.inst... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : 0 ≤[u] -s\nhw₁ : ↑s w = 0\nhw₂ : w ⊆ u\nhwt : v ⊆ w\n⊢ ↑s v = 0",
"usedConstants": [
"Real",
"Real.instZero",
"AddCommGroup.toAddCom... | have := subset_positive_null_set hu hv hw hsu | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 43
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\nε : ℝ\nhε : 0 < ε\ni : ι\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α), ... | refine ⟨δ, hδpos, fun j s hs hμs => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 535,
"column": 8
} | {
"line": 535,
"column": 50
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : s.totalVariation u = 0\nhu₂ : μ.ennrealToMeasure uᶜ = 0\nt : Set α\nhtv : t ⊆ uᶜ\nhmt : MeasurableSet t\n⊢ ↑μ t = 0",
"usedConstants": [
"Eq.mpr",
... | ← VectorMeasure.ennrealToMeasure_apply hmt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 704,
"column": 78
} | {
"line": 708,
"column": 20
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : TopologicalSpace M\ni : Set α\n⊢ restrict 0 i = 0",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.VectorMeasure.restrict._proof_3",
"MeasurableSet",
"congrArg",
"MeasureTheory.Vecto... | by
by_cases hi : MeasurableSet i
· ext j hj
rw [restrict_apply 0 hi hj, zero_apply, zero_apply]
· exact dif_neg hi | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 83
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : SignedMeasure α\nf : α → ℝ\nhtμ : t ⟂ᵥ μ.toENNRealVectorMeasure\nhadd : s = t + μ.withDensityᵥ f\nhfi : Integrable f μ\n⊢ t = s.singularPart μ",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"P... | refine eq_singularPart' t hfi.1.measurable_mk (hfi.congr hfi.1.ae_eq_mk) htμ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 317,
"column": 6
} | {
"line": 317,
"column": 92
} | [
{
"pp": "case a.μ\nα : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nr : ℝ≥0\n| (r • s).toJordanDecomposition.posPart.singularPart μ",
"usedConstants": [
"MeasureTheory.JordanDecomposition.posPart",
"Real",
"instHSMul",
"MeasureTheory.Measure",
"NonUnital... | · rw [toJordanDecomposition_smul, JordanDecomposition.smul_posPart, singularPart_smul] | Lean.Parser.Tactic.Conv.«_aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_conv·__1» | Lean.Parser.Tactic.Conv.«conv·_» |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 102,
"column": 42
} | {
"line": 102,
"column": 76
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhm : m ≤ m0\nhfint : Integrable f μ\n⊢ (∫⁻ (a : α), ENNReal.ofReal |μ[f | m] a| ∂μ).toReal ≤ ∫ (x : α), |f x| ∂μ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instLE",
"Real... | integral_eq_lintegral_of_nonneg_ae | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 444,
"column": 2
} | {
"line": 444,
"column": 68
} | [
{
"pp": "case h\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nc : ComplexMeasure α\ninst✝ : c.HaveLebesgueDecomposition μ\ni : Set α\nhi : MeasurableSet i\n⊢ ↑(c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ)) i = ↑((re c).toComplexMeasure (im c)) i",
"usedConstants": [
"instInnerProductSpaceRea... | rw [VectorMeasure.add_apply, SignedMeasure.toComplexMeasure_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 450,
"column": 10
} | {
"line": 450,
"column": 56
} | [
{
"pp": "case h.a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nc : ComplexMeasure α\ninst✝ : c.HaveLebesgueDecomposition μ\ni : Set α\nhi : MeasurableSet i\n⊢ ↑((re c).singularPart μ + μ.withDensityᵥ ((re c).rnDeriv μ)) i = { re := ↑(re c) i, im := ↑(im c) i }.re",
"usedConstants": [
"Eq.mpr",... | c.re.singularPart_add_withDensity_rnDeriv_eq μ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 981,
"column": 2
} | {
"line": 981,
"column": 50
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ 0 ≤[i] v",
"usedConstants": [
"Eq.mpr",
"le_refl",
"congrArg",
"MeasureTheory.VectorMe... | rw [restrict_zero, restrict_not_measurable _ hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 981,
"column": 2
} | {
"line": 981,
"column": 50
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ 0 ≤[i] v",
"usedConstants": [
"Eq.mpr",
"le_refl",
"congrArg",
"MeasureTheory.VectorMe... | rw [restrict_zero, restrict_not_measurable _ hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 981,
"column": 2
} | {
"line": 981,
"column": 50
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ 0 ≤[i] v",
"usedConstants": [
"Eq.mpr",
"le_refl",
"congrArg",
"MeasureTheory.VectorMe... | rw [restrict_zero, restrict_not_measurable _ hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 984,
"column": 2
} | {
"line": 984,
"column": 50
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ v ≤[i] 0",
"usedConstants": [
"Eq.mpr",
"le_refl",
"congrArg",
"MeasureTheory.VectorMe... | rw [restrict_zero, restrict_not_measurable _ hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 984,
"column": 2
} | {
"line": 984,
"column": 50
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ v ≤[i] 0",
"usedConstants": [
"Eq.mpr",
"le_refl",
"congrArg",
"MeasureTheory.VectorMe... | rw [restrict_zero, restrict_not_measurable _ hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 984,
"column": 2
} | {
"line": 984,
"column": 50
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ v ≤[i] 0",
"usedConstants": [
"Eq.mpr",
"le_refl",
"congrArg",
"MeasureTheory.VectorMe... | rw [restrict_zero, restrict_not_measurable _ hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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