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.Analysis.BoxIntegral.Basic | {
"line": 274,
"column": 47
} | {
"line": 274,
"column": 80
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nh : HasIntegral I l f vol y\nh' : HasIntegral... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 39
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\n⊢ HasIntegral I l (fun x ↦ 0) vol 0",
"usedConstants": [
"Eq.mpr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 52
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny : F\nhf : HasIntegral I l f vol y\nc : ℝ\n⊢ HasIntegral... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 13
} | [
{
"pp": "case hnc\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\n_i : Fact (finrank ℝ E = 0)\nh : SameRay ℝ (AlternatingMap.constLinearEquivOfIsEmpty 1) (-AlternatingMap.constLinearEquivOfIsEmpty 1)\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 33
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nc : ℝ\nhf : Integrable I l (c • f) vol\nhc : c ≠ 0\n⊢ Int... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 35
} | [
{
"pp": "ι : Type u\nE : Type v\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nI : Box ι\ninst✝¹ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nc : ℝ\nhc : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ c\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ ‖integral I l f μ.toBoxAdditive.toSMul‖ ≤ μ.real ↑I * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 248,
"column": 55
} | {
"line": 248,
"column": 66
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\nv : Fin (n + 1) → E\nthis : FiniteDimensional ℝ E\n⊢ finrank ℝ E = Fintype.card (Fin n.succ)",
"usedConstants": [
"Eq.mpr",
"InnerProduc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 453,
"column": 2
} | {
"line": 453,
"column": 39
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nc₁ c₂ : ℝ≥0\nε₁ ε₂ : ℝ\nπ₁ π₂ : TaggedPrepartition I\nh :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 270,
"column": 60
} | {
"line": 270,
"column": 71
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\nv : Fin (n + 1) → E\nhv : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0\nthis : FiniteDimensional ℝ E\n⊢ finrank ℝ E = Fintype.card (Fin n.succ)",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 40
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n)\no : Orientation ℝ E (Fin n)\nv : OrthonormalBasis (Fin n) ℝ E\n⊢ |o.volumeForm ⇑v| = 1",
"usedConstants": [
"AlternatingMap",
"Eq.mpr",
"InnerProductSpace.toNormedSpace"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumOverResidueClass | {
"line": 42,
"column": 55
} | {
"line": 42,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝² : AddCommGroup R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalAddGroup R\nm✝ : ℕ\nhm : NeZero m✝\nk : ℕ\nf : ℕ → R\ng : ℕ → ℕ := fun n ↦ m✝ * n + k\nm n : ℕ\nhmn : g m = g n\n⊢ m = n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 664,
"column": 4
} | {
"line": 664,
"column": 75
} | [
{
"pp": "ι : Type u\nE : Type v\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : Fintype ι\nl : IntegrationParams\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 701,
"column": 6
} | {
"line": 701,
"column": 75
} | [
{
"pp": "case hbc\nι : Type u\nE : Type v\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : Fintype ι\nl : IntegrationParams\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 88,
"column": 95
} | {
"line": 101,
"column": 8
} | [
{
"pp": "M : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedAddMonoid M\nf : ℕ → M\nu : ℕ → ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nhu : Monotone u\nn : ℕ\n⊢ ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k",... | by
induction n with
| zero => simp
| succ n ihn =>
suffices (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k by
rw [sum_range_succ, ← sum_Ico_consecutive]
exacts [add_le_add ihn this,
(add_le_add_left (hu n.zero_le) _ : u 0 + 1 ≤ u n + 1),
add_le_add_lef... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.PSeries | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 13
} | [
{
"pp": "u : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\nn : ℕ\n⊢ ∑ a ∈ range n.succ, (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤\n (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑ x ∈ Ico ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 13
} | [
{
"pp": "f : ℕ → ℝ≥0∞\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nn : ℕ\n⊢ ∑ a ∈ range n.succ, 2 ^ a * f (2 ^ a) ≤ f 1 + 2 • ∑ x ∈ Ico 2 (2 ^ n + 1), f x",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ENNReal.instAdd",
"instHSMul",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 64
} | [
{
"pp": "case mp\nC : ℕ\nu : ℕ → ℕ\nf : ℕ → ℝ≥0\nh_pos : ∀ (n : ℕ), 0 < u n\nhu_strict : StrictMono u\nhC_nonzero : C ≠ 0\nh_succ_diff : SuccDiffBounded C u\nh : ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞\nhf : ∀ (m n : ℕ), 1 < m → m ≤ n → ↑(f n) ≤ ↑(f m)\nh_nonneg : ∀ (n : ℕ), 0 ≤ ↑(f n)\nhC : ∑' (k... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace | {
"line": 231,
"column": 31
} | {
"line": 231,
"column": 42
} | [
{
"pp": "E : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\ninst✝ : FiniteDimensional ℝ E\nh : finrank ℝ E = 1\nv : E\nhv : v ≠ 0\n⊢ ‖v‖⁻¹ ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 198,
"column": 4
} | {
"line": 199,
"column": 38
} | [
{
"pp": "case mpr\nC : ℕ\nu : ℕ → ℕ\nf : ℕ → ℝ≥0\nhf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nhu_strict : StrictMono u\nhC_nonzero : C ≠ 0\nh_succ_diff : SuccDiffBounded C u\nh : ∑' (b : ℕ), ↑(f b) = ∞\n⊢ ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞",
"usedConstants": [... | replace hf : ∀ m n, 0 < m → m ≤ n → (f n : ℝ≥0∞) ≤ f m := fun m n hm hmn =>
ENNReal.coe_le_coe.2 (hf hm hmn) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Analysis.PSeries | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 37
} | [
{
"pp": "case mpr\nC : ℕ\nu : ℕ → ℕ\nf : ℕ → ℝ≥0\nh_pos : ∀ (n : ℕ), 0 < u n\nhu_strict : StrictMono u\nhC_nonzero : C ≠ 0\nh_succ_diff : SuccDiffBounded C u\nh : ∑' (b : ℕ), ↑(f b) = ∞\nhf : ∀ (m n : ℕ), 0 < m → m ≤ n → ↑(f n) ≤ ↑(f m)\nthis : ∑ k ∈ range (u 0), ↑(f k) ≠ ∞\n⊢ ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 756,
"column": 6
} | {
"line": 756,
"column": 69
} | [
{
"pp": "ι : Type u\nE : Type v\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : Fintype ι\nl : IntegrationParams\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nhc : ContinuousOn f (Box.Icc I)\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nC : ℝ\nhC : f '' Box.Icc I ⊆ C • Metr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 305,
"column": 4
} | {
"line": 306,
"column": 27
} | [
{
"pp": "case inr\np : ℝ\nhp : p < 0\nh : Summable fun n ↦ (↑n ^ p)⁻¹\nk : ℕ\nhk₁ : (↑k ^ p)⁻¹ < 1\nhk₀ : 0 < ↑k\n⊢ k = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 35
} | [
{
"pp": "⊢ ¬Summable fun n ↦ 1 / ↑n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 50,
"column": 31
} | {
"line": 50,
"column": 42
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝¹ : DiscreteTopology ↥L\nι : Type u_2\nb : Basis ι ℤ ↥L\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {L : Submod... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 400,
"column": 29
} | {
"line": 400,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ 0 < ↑n",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"NonUnitalCommRing.toNonUnitalNonA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 426,
"column": 6
} | {
"line": 426,
"column": 17
} | [
{
"pp": "case bc.hdb\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nk n : ℕ\nA : 1 ≤ ↑k + 1\n⊢ ↑k + 1 ≤ (↑k + 1) ^ 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 33
} | [
{
"pp": "E✝ : Type u_1\ninst✝⁴ : NormedAddCommGroup E✝\nL✝ : Submodule ℤ E✝\nι✝ : Type u_2\nb✝ : Basis ι✝ ℤ ↥L✝\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nι : Type u_2\nb : Basis ι ℤ ↥L\nH : IsZLattice ℝ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 468,
"column": 6
} | {
"line": 468,
"column": 17
} | [
{
"pp": "a s b c : ℝ\nh : Summable fun n ↦ 1 / |↑n + b| ^ s\n⊢ Tendsto (fun x ↦ 1 + (b - c) / x) atTop (𝓝 1)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 487,
"column": 2
} | {
"line": 489,
"column": 11
} | [
{
"pp": "case h\nα : Type u_1\nx : α\ninst✝ : RCLike α\nq k : ℕ\nhq : 1 < q\n⊢ Summable fun x ↦ 1 / ‖(↑x + ↑k) ^ q‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 127,
"column": 57
} | {
"line": 127,
"column": 72
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\nι : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι ℤ ↥L\nn : ℕ\nr : ℝ\ns : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 129,
"column": 35
} | {
"line": 129,
"column": 69
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\nι : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι ℤ ↥L\nn : ℕ\nr : ℝ\ns : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 64
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nh✝ : Subsingleton ↥L\nr : ℝ\nhr✝ : r < -↑(finrank ℤ ↥L)\ns : Finset ↥L\nhr : r ≠ 0\n⊢ ∑ z ∈ s, ‖z‖ ^ r ≤ 1 ^ r * ∑' (k : ℕ), ↑k ^ (↑(finrank ℤ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Ideal | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 15
} | [
{
"pp": "case mp\nk : Type u_1\nG : Type u_3\ninst✝¹ : Monoid G\ninst✝ : Semiring k\ns : Set G\nx : k[G]\nRHS : Ideal k[G] :=\n { carrier := {p | ∀ m ∈ p.support, ∃ m' ∈ s, ∃ d, m = d * m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\ni : G\nhi : i ∈ s\nm : G\nhm : m ∈ ((of k G) i).support\n⊢ m = 1 * i",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Ideal | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 15
} | [
{
"pp": "k : Type u_1\nG : Type u_3\ninst✝¹ : Monoid G\ninst✝ : Semiring k\ns : Set G\nx : k[G]\nRHS : Ideal k[G] :=\n { carrier := {p | ∀ m ∈ p.support, ∃ m' ∈ s, ∃ d, m = d * m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nhx : x ∈ RHS\nd : G\nhd : d ∈ s\nd2 : G\nhi : d2 * d ∈ x.support\n⊢ Finsupp.sing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.AddTorsor | {
"line": 130,
"column": 2
} | {
"line": 135,
"column": 46
} | [
{
"pp": "G : Type u_1\nP : Type u_2\ninst✝³ : LE G\ninst✝² : Preorder P\ninst✝¹ : SMul G P\ninst✝ : IsOrderedCancelSMul G P\na b : G\nc d : P\nh₁ : a ≤ b\nh₂ : c < d\n⊢ a • c < b • d",
"usedConstants": [
"lt_of_le_of_lt",
"False",
"instHSMul",
"Preorder.toLT",
"congrArg",
... | refine lt_of_le_of_lt (IsOrderedSMul.smul_le_smul_right a b h₁ c) ?_
refine lt_of_le_not_ge (IsOrderedSMul.smul_le_smul_left c d (le_of_lt h₂) b) ?_
by_contra hbdc
have h : d ≤ c := IsOrderedCancelSMul.le_of_smul_le_smul_left b d c hbdc
rw [@lt_iff_le_not_ge] at h₂
simp_all only [not_true_eq_false, and_false] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.AddTorsor | {
"line": 130,
"column": 2
} | {
"line": 135,
"column": 46
} | [
{
"pp": "G : Type u_1\nP : Type u_2\ninst✝³ : LE G\ninst✝² : Preorder P\ninst✝¹ : SMul G P\ninst✝ : IsOrderedCancelSMul G P\na b : G\nc d : P\nh₁ : a ≤ b\nh₂ : c < d\n⊢ a • c < b • d",
"usedConstants": [
"lt_of_le_of_lt",
"False",
"instHSMul",
"Preorder.toLT",
"congrArg",
... | refine lt_of_le_of_lt (IsOrderedSMul.smul_le_smul_right a b h₁ c) ?_
refine lt_of_le_not_ge (IsOrderedSMul.smul_le_smul_left c d (le_of_lt h₂) b) ?_
by_contra hbdc
have h : d ≤ c := IsOrderedCancelSMul.le_of_smul_le_smul_left b d c hbdc
rw [@lt_iff_le_not_ge] at h₂
simp_all only [not_true_eq_false, and_false] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 182,
"column": 12
} | {
"line": 182,
"column": 23
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nh✝ : Nontrivial ↥L\nI : Type u_1 := Free.ChooseBasisIndex ℤ ↥L\nthis : Fintype I\nb : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L\nd : ℕ := Fintype.card I\nhd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 39
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nh✝ : Nontrivial ↥L\nI : Type u_1 := Free.ChooseBasisIndex ℤ ↥L\nthis : Fintype I\nb : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L\nd : ℕ := Fintype.card I\nhd... | refine ⟨⌊r⌋.toNat, fun x hx ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 253,
"column": 24
} | {
"line": 253,
"column": 35
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nr : ℝ\nhr : r < -↑(finrank ℤ ↥L)\nx : E\nh✝ : Nontrivial ↥L\nH : IsClosed ↑L\nt : ↥L\nht : t ∈ {x_1 | (fun i ↦ ‖‖↑i - x‖ ^ r‖ ≤ (1 / 2) ^ r * ‖i‖ ^ r) x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 255,
"column": 25
} | {
"line": 255,
"column": 50
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nr : ℝ\nhr : r < -↑(finrank ℤ ↥L)\nx : E\nh✝ : Nontrivial ↥L\nH : IsClosed ↑L\nt : ↥L\nht : t ∈ {x_1 | (fun i ↦ ‖‖↑i - x‖ ^ r‖ ≤ (1 / 2) ^ r * ‖i‖ ^ r) x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 70
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nr : ℝ\nhr : r < -↑(finrank ℤ ↥L)\nx : E\nh✝ : Nontrivial ↥L\nH : IsClosed ↑L\nt : ↥L\nht : t ∈ {x_1 | (fun i ↦ ‖‖↑i - x‖ ^ r‖ ≤ (1 / 2) ^ r * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 85
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nr : ℝ\nhr : r < -↑(finrank ℤ ↥L)\nx : E\nh✝ : Nontrivial ↥L\nH : IsClosed ↑L\nt : ↥L\nht : t ∈ {x_1 | (fun i ↦ ‖‖↑i - x‖ ^ r‖ ≤ (1 / 2) ^ r * ‖i‖ ^ r) x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 13
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nr : ℝ\nhr : r < -↑(finrank ℤ ↥L)\nx : E\nh✝ : Nontrivial ↥L\nH : IsClosed ↑L\nt : ↥L\nht : t ∈ {x_1 | (fun i ↦ ‖‖↑i - x‖ ^ r‖ ≤ (1 / 2) ^ r * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nn : ℤ\nhn : n < -↑(finrank ℤ ↥L)\n⊢ Summable fun z ↦ ‖z‖ ^ n",
"usedConstants": [
"Norm.norm",
"Submodule",
"Real",
"AddComm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nn : ℕ\nhn : finrank ℤ ↥L < n\nx : E\n⊢ Summable fun z ↦ ‖↑z - x‖⁻¹ ^ n",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Submodule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nn : ℕ\nhn : finrank ℤ ↥L < n\n⊢ Summable fun z ↦ ‖z‖⁻¹ ^ n",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Submodule",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 869,
"column": 26
} | {
"line": 869,
"column": 60
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nhl : l ≤ Henstock\nB : ι →ᵇᵃ[↑I] ℝ\nhB0 : ∀ (J : Box ι), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 889,
"column": 4
} | {
"line": 889,
"column": 92
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nB : ι →ᵇᵃ[↑I] ℝ\nhB0 : ∀ (J : Box ι), 0 ≤ B J\ng : ι →ᵇᵃ[↑I] F\nH :\n ∀ (x : ℝ≥... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.ToFinsupp | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 39
} | [
{
"pp": "case h.inl.h\nR : Type u_2\ninst✝³ : AddZeroClass R\nl₁ l₂ : List R\ninst✝² : DecidablePred fun x ↦ (l₁ ++ l₂).getD x 0 ≠ 0\ninst✝¹ : DecidablePred fun x ↦ l₁.getD x 0 ≠ 0\ninst✝ : DecidablePred fun x ↦ l₂.getD x 0 ≠ 0\nn : ℕ\nh : n < l₁.length\n⊢ n ∉ Set.range ⇑(addLeftEmbedding l₁.length)",
"used... | rintro ⟨k, rfl : length l₁ + k = n⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 57
} | [
{
"pp": "α : Type u_1\nf : α → ℕ\na b : α\ninst✝ : DecidableEq α\nhab : a ≠ b\n⊢ (f a)! * (f b)! * multinomial {a, b} f = (f a + f b)!",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 13
} | [
{
"pp": "case a\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : IsLeftCancelAdd R\ni : σ\np q r : MvPolynomial σ R\nh : p = X i * q + r\nhr : ∀ n ∈ r.support, n i = 0\nn : σ →₀ ℕ\nhn : Finsupp.single i 1 + n ∈ r.support\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 13
} | [
{
"pp": "case a\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : IsLeftCancelAdd R\nf g h : MvPolynomial σ R\nH : (fun x ↦ f + x) g = (fun x ↦ f + x) h\nd : σ →₀ ℕ\n⊢ coeff d g = coeff d h",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 293,
"column": 8
} | {
"line": 293,
"column": 46
} | [
{
"pp": "case mp.h.inl.h\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\ni : σ\np q : MvPolynomial σ R\ninst✝ : IsCancelMulZero R\na✝ : Nontrivial R\nx✝ : NoZeroDivisors (MvPolynomial σ R)\nh : X i ∣ p * q\nhp : p.modMonomial (Finsupp.single i 1) + X i * p.divMonomial (Finsupp.single i 1) = p\nhq : q.modMonom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 295,
"column": 8
} | {
"line": 295,
"column": 46
} | [
{
"pp": "case mp.h.inr.h\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\ni : σ\np q : MvPolynomial σ R\ninst✝ : IsCancelMulZero R\na✝ : Nontrivial R\nx✝ : NoZeroDivisors (MvPolynomial σ R)\nh : X i ∣ p * q\nhp : p.modMonomial (Finsupp.single i 1) + X i * p.divMonomial (Finsupp.single i 1) = p\nhq : q.modMonom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 292,
"column": 10
} | {
"line": 292,
"column": 31
} | [
{
"pp": "α : Type u_1\nR : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Semiring R\nf : α → R\na : α\ns : Finset α\nhas : a ∉ s\nih :\n ∀ (hc : (↑s).Pairwise (Commute on f)) (n : ℕ),\n (∑ i ∈ s, f i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f i ^ k i) ⋯\nhc : (↑(cons a s has)).Pa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Nilpotent | {
"line": 50,
"column": 37
} | {
"line": 50,
"column": 48
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\nP : MvPolynomial (Fin n) R\nf : Fin n ↪ σ\nH : ∀ (i : σ →₀ ℕ), IsNilpotent (coeff i ((rename ⇑f) P))\ni : Fin n →₀ ℕ\n⊢ IsNilpotent (coeff i P)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 327,
"column": 29
} | {
"line": 327,
"column": 50
} | [
{
"pp": "case mp.g\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\ni : σ\np q : MvPolynomial σ R\ninst✝ : IsCancelMulZero R\nr : MvPolynomial σ R\nhp : X i * q = X i * (p.divMonomial (Finsupp.single i 1) * r)\nthis : X i ∣ p ∨ X i ∣ r\nhip : p.modMonomial (Finsupp.single i 1) = 0\n⊢ p.divMonomial (Finsupp.sin... | X_mul_cancel_left_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Nilpotent | {
"line": 62,
"column": 40
} | {
"line": 62,
"column": 69
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\nP : MvPolynomial σ R\nH : IsUnit P\nn : σ →₀ ℕ\nhn : n ≠ 0\n⊢ ∃ i, n i ≠ 0",
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroClass",
"Exists",
"id",
"Ne",
"instOfNatNat",
"Nat",
"OfNat.ofNat",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Nilpotent | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 71
} | [
{
"pp": "case refine_1\nσ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\nP : MvPolynomial σ R\nH✝ : IsUnit P\nn : σ →₀ ℕ\nhn : n ≠ 0\ni : σ\nhi : n i ≠ 0\ne : Polynomial (MvPolynomial { b // b ≠ i } R) ≃ₐ[R] MvPolynomial σ R :=\n (optionEquivLeft R { b // b ≠ i }).symm.trans (renameEquiv R (Equiv.optionSubtypeN... | convert! ← H (n.equivMapDomain (Equiv.optionSubtypeNe i).symm).some | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Algebra.MvPolynomial.Nilpotent | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 15
} | [
{
"pp": "case refine_2\nσ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\nP : MvPolynomial σ R\nx✝ : IsUnit (coeff 0 P) ∧ ∀ (i : σ →₀ ℕ), i ≠ 0 → IsNilpotent (coeff i P)\nh₁ : IsUnit (coeff 0 P)\nh₂ : ∀ (i : σ →₀ ℕ), i ≠ 0 → IsNilpotent (coeff i P)\nthis : IsNilpotent (P - C (coeff 0 P))\n⊢ IsUnit P",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 376,
"column": 10
} | {
"line": 376,
"column": 29
} | [
{
"pp": "case left\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np q✝ : MvPolynomial σ R\ninst✝ : IsCancelMulZero R\nn✝ : σ →₀ ℕ\nhR : Nontrivial R\nd : ℕ\nhd :\n ∀ (n : σ →₀ ℕ),\n Finsupp.degree n = d →\n ∀ (p q : MvPolynomial σ R), p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 402,
"column": 34
} | {
"line": 402,
"column": 49
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : DecidableEq α\nm : Fin (n + 1)\ns : Sym α (n - ↑m)\nx : α\nhx : x ∉ ↑s\nj : α\nhj : j ∈ ↑s\nh : x = j\n⊢ x ∈ ↑s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"HSub.hSub",
"Membership.mem",
"Multiset",
"id",
"instSubNat",
"ins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 461,
"column": 50
} | {
"line": 461,
"column": 70
} | [
{
"pp": "case swap\nm : Multiset ℕ\nl✝ l' : List ℕ\nx y : ℕ\nl : List ℕ\n⊢ (y + (x + l.sum)).choose (y + (x + l.sum) - y) * (x + l.sum).choose x * l.multinomial =\n (x + (y + l.sum)).choose x * (y + l.sum).choose y * l.multinomial",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"... | add_tsub_cancel_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 462,
"column": 62
} | {
"line": 462,
"column": 82
} | [
{
"pp": "case swap\nm : Multiset ℕ\nl✝ l' : List ℕ\nx y : ℕ\nl : List ℕ\n⊢ (x + (y + l.sum)).choose x * (x + (y + l.sum) - x).choose (x + l.sum - x) * l.multinomial =\n (x + (y + l.sum)).choose x * (y + l.sum).choose y * l.multinomial",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
... | add_tsub_cancel_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Expand | {
"line": 169,
"column": 18
} | {
"line": 169,
"column": 26
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝¹ : CommSemiring R\np : ℕ\nφ : MvPolynomial σ R\ninst✝ : DecidableEq σ\n| ((expand p) φ).support",
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Semiring.toModule",
"AddMonoidAlgebra.addAddCommMonoid",
"congrA... | φ.as_sum | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Algebra.MvPolynomial.Funext | {
"line": 49,
"column": 36
} | {
"line": 49,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih :\n ∀ {p : MvPolynomial (Fin n) R} (s : Fin n → Set R),\n (∀ (i : Fin n), (s i).Infinite) → (∀ x ∈ Set.univ.pi s, (eval x) p = 0) → p = 0\np : MvPolynomial (Fin (n + 1)) R\ns : Fin (n + 1) → Set R\nhs : ∀ (i : Fin (n + 1)), (s i).Infi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Funext | {
"line": 49,
"column": 56
} | {
"line": 49,
"column": 67
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih :\n ∀ {p : MvPolynomial (Fin n) R} (s : Fin n → Set R),\n (∀ (i : Fin n), (s i).Infinite) → (∀ x ∈ Set.univ.pi s, (eval x) p = 0) → p = 0\np : MvPolynomial (Fin (n + 1)) R\ns : Fin (n + 1) → Set R\nhs : ∀ (i : Fin (n + 1)), (s i).Infi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Funext | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 39
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nσ : Type u_2\np✝ q : MvPolynomial σ R\ns : σ → Set R\nhs : ∀ (i : σ), (s i).Infinite\nh✝ : ∀ x ∈ Set.univ.pi s, (eval x) p✝ = (eval x) q\nn : ℕ\nf : Fin n → σ\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : ∀ x ∈ Set.univ.pi s... | rw [hf.extend_apply]; exact hx _ ⟨⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Funext | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 39
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nσ : Type u_2\np✝ q : MvPolynomial σ R\ns : σ → Set R\nhs : ∀ (i : σ), (s i).Infinite\nh✝ : ∀ x ∈ Set.univ.pi s, (eval x) p✝ = (eval x) q\nn : ℕ\nf : Fin n → σ\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : ∀ x ∈ Set.univ.pi s... | rw [hf.extend_apply]; exact hx _ ⟨⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.WellFounded | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 41
} | [
{
"pp": "α : Type u_1\nN : Type u_2\ninst✝ : Zero N\nr : α → α → Prop\ns : N → N → Prop\nhbot : ∀ ⦃n : N⦄, ¬s n 0\nhs : WellFounded s\nx : α →₀ N\nh : ∀ a ∈ x.support, Acc (rᶜ ⊓ fun x1 x2 ↦ x1 ≠ x2) a\n⊢ ∀ i ∈ x.toDFinsupp.support, Acc (rᶜ ⊓ fun x1 x2 ↦ x1 ≠ x2) i",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.GameAdd | {
"line": 215,
"column": 8
} | {
"line": 215,
"column": 40
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nrα : α → α → Prop\nrβ : β → β → Prop\na✝ : α\nb✝ : β\nC : α → α → Sort u_3\nhr : WellFounded rα\nIH : (a₁ b₁ : α) → ((a₂ b₂ : α) → GameAdd rα s(a₂, b₂) s(a₁, b₁) → C a₂ b₂) → C a₁ b₁\na b : α\n⊢ WellFounded fun x y ↦ Prod.GameAdd rα rα x y ∨ Prod.GameAdd rα rα x.swap y",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.WellFounded | {
"line": 100,
"column": 8
} | {
"line": 100,
"column": 24
} | [
{
"pp": "case pos.hnc\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j\nhp : i ∉ p\nhs : s i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.WellFounded | {
"line": 128,
"column": 45
} | {
"line": 128,
"column": 67
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝² : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\nhbot : ∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬s i a 0\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → (x : α i) → Decidable (x ≠ 0)\nb : ι\nt : Finset ι\nhb : b ∉ t\nih :\n ∀ (x : Π₀ (i : ι), α i), x.support = t... | Finset.erase_insert hb | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.MonomialOrder | {
"line": 132,
"column": 32
} | {
"line": 132,
"column": 43
} | [
{
"pp": "α : Type u_1\nN : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : AddCommMonoid N\ninst✝¹ : PartialOrder N\ninst✝ : IsOrderedCancelAddMonoid N\na b : Lex (α →₀ N)\nh : a ≤ b\nc : Lex (α →₀ N)\n⊢ a + c ≤ b + c",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Finsupp.instIsRightCancelAdd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.MonomialOrder | {
"line": 131,
"column": 38
} | {
"line": 131,
"column": 76
} | [
{
"pp": "α : Type u_1\nN : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : AddCommMonoid N\ninst✝¹ : PartialOrder N\ninst✝ : IsOrderedCancelAddMonoid N\na b c : Lex (α →₀ N)\nh : a + b ≤ a + c\n⊢ b ≤ c",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.WellFounded | {
"line": 140,
"column": 6
} | {
"line": 140,
"column": 18
} | [
{
"pp": "case intro\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\nhbot : ∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬s i a 0\nhs✝ : ∀ (i : ι), WellFounded (s i)\ninst✝ : DecidableEq ι\ni✝ i : ι\nh✝ : ∀ (y : ι), (rᶜ ⊓ fun x1 x2 ↦ x1 ≠ x2) y i → Acc (rᶜ ⊓ fun x1 x2 ↦ ... | single_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.MonomialOrder.DegLex | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b : DegLex (α →₀ ℕ)\nh :\n degree (ofDegLex a) < degree (ofDegLex b) ∨\n degree (ofDegLex a) = degree (ofDegLex b) ∧ toLex (ofDegLex a) ≤ toLex (ofDegLex b)\nc : DegLex (α →₀ ℕ)\n⊢ degree (ofDegLex (a + c)) < degree (ofDegLex (b + c)) ∨\n degree (ofDegLex (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.MonomialOrder.DegLex | {
"line": 147,
"column": 4
} | {
"line": 148,
"column": 32
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b c : DegLex (α →₀ ℕ)\nh :\n degree (ofDegLex (a + b)) < degree (ofDegLex (a + c)) ∨\n degree (ofDegLex (a + b)) = degree (ofDegLex (a + c)) ∧ toLex (ofDegLex (a + b)) ≤ toLex (ofDegLex (a + c))\n⊢ degree (ofDegLex b) < degree (ofDegLex c) ∨\n degree (ofDeg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 102,
"column": 6
} | {
"line": 102,
"column": 17
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nf : MvPolynomial σ R\na : R\nha : a ≠ 0\nhf : ∀ (i : σ →₀ ℕ), coeff 0 f ∣ coeff i (C a)\nthis : f = C (coeff 0 f)\n⊢ coeff 0 f ∣ a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 15
} | [
{
"pp": "case h.h\nR : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nf : MvPolynomial σ R\na : R\nha : a ≠ 0\nhf : f ∣ C a\n⊢ f.totalDegree ≤ 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"id"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 123,
"column": 59
} | {
"line": 123,
"column": 70
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝ : CommSemiring R\np : MvPolynomial σ R\nj : σ\nc : R\nhc : c ∈ R⁰\nhp : ¬p = 0\nh : ((optionEquivLeft R { b // b ≠ j }) ((rename ⇑(optionSubtypeNe j).symm) p)).leadingCoeff = 0\n⊢ (rename ⇑?m.78) p = (rename ⇑?m.78) 0",
"usedConstants": [
"Nat.instMulZeroClas... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 15
} | [
{
"pp": "case neg.a.a\nR : Type u_1\nσ : Type u_2\ninst✝ : CommSemiring R\np : MvPolynomial σ R\nj : σ\nc : R\nhc : c ∈ R⁰\nhp : ¬p = 0\nhp' : C c * ((optionEquivLeft R { b // ¬b = j }) ((rename ⇑(optionSubtypeNe j).symm) p)).leadingCoeff = 0\nm : { b // b ≠ j } →₀ ℕ\n⊢ c * coeff m ((optionEquivLeft R { b // b ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Sym.Card | {
"line": 135,
"column": 12
} | {
"line": 135,
"column": 33
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\na b : α\nha : a ∈ s\nhb : b ∈ s\nhab : a ≠ b ∨ b ≠ a\n⊢ (a, b) ∉ {(b, a)}",
"usedConstants": [
"Eq.mpr",
"congrArg",
"and_self",
"_private.Mathlib.Data.Sym.Card.0.Sym2.two_mul_card_image_offDiag._simp_1_1",
"Finset",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Antidiag.FinsuppEquiv | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 66
} | [
{
"pp": "ι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\ns : Finset ι\nn : μ\nf : ↥(s.finsuppAntidiag n)\nhf : ((↑f).sum fun x x_1 ↦ x_1) = n ∧ (↑f).support ⊆ s\n⊢ ((subtypeDomain (fun x ↦ x ∈ s) ↑f).sum fun x ↦ id) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Antidiag.FinsuppEquiv | {
"line": 51,
"column": 8
} | {
"line": 51,
"column": 25
} | [
{
"pp": "ι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\ns : Finset ι\nn : μ\nf : { P // (P.sum fun x ↦ id) = n }\n⊢ ((↑f).extendDomain.sum fun x x_1 ↦ x_1) = n",
"usedConstants": [
"dite_cond_eq_true",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Nat | {
"line": 37,
"column": 4
} | {
"line": 37,
"column": 46
} | [
{
"pp": "case succ.succ\na n✝ : ℕ\nh : DvdNotUnit (a + 1) (n✝ + 1)\n⊢ (fun x ↦ if x = 0 then ⊤ else ↑x) (a + 1) < (fun x ↦ if x = 0 then ⊤ else ↑x) (n✝ + 1)",
"usedConstants": [
"Iff.mpr",
"Dvd.dvd",
"Nat.instIsCancelMulZero",
"semigroupDvd",
"dvd_and_not_dvd_iff",
"Semig... | obtain ⟨h1, h2⟩ := dvd_and_not_dvd_iff.2 h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.Nat.Squarefree | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 50
} | [
{
"pp": "case pos\nn p : ℕ\nhn : ∀ (x : ℕ), emultiplicity x n ≤ 1 ∨ IsUnit x\nhn' : n ≠ 0\nhp : Prime p\nthis : emultiplicity p n ≤ 1\n⊢ multiplicity p n ≤ 1",
"usedConstants": [
"Nat.instMonoid",
"multiplicity_le_of_emultiplicity_le",
"instOfNatNat",
"Nat",
"OfNat.ofNat"
]... | exact multiplicity_le_of_emultiplicity_le this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 336,
"column": 4
} | {
"line": 336,
"column": 63
} | [
{
"pp": "case neg.h\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf g : MvPolynomial σ R\nb : σ →₀ ℕ\nhb : b ∈ (f + g).support\nhf : coeff b f = 0\n⊢ b ∈ g.support",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"CommSemiring.toSemiring",
"Finset",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 43
} | [
{
"pp": "case pos\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf g : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : m.toSyn (m.degree f + m.degree g) < m.toSyn c\nd e : σ →₀ ℕ\nhde : d + e = c\nhd : m.toSyn (m.degree f) < m.toSyn d\n⊢ coeff d f * coeff e g = 0",
"usedConstants": [
"Eq... | rw [m.coeff_eq_zero_of_lt hd, zero_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 43
} | [
{
"pp": "case pos\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf g : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : m.toSyn (m.degree f + m.degree g) < m.toSyn c\nd e : σ →₀ ℕ\nhde : d + e = c\nhd : m.toSyn (m.degree f) < m.toSyn d\n⊢ coeff d f * coeff e g = 0",
"usedConstants": [
"Eq... | rw [m.coeff_eq_zero_of_lt hd, zero_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 43
} | [
{
"pp": "case pos\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf g : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : m.toSyn (m.degree f + m.degree g) < m.toSyn c\nd e : σ →₀ ℕ\nhde : d + e = c\nhd : m.toSyn (m.degree f) < m.toSyn d\n⊢ coeff d f * coeff e g = 0",
"usedConstants": [
"Eq... | rw [m.coeff_eq_zero_of_lt hd, zero_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 421,
"column": 8
} | {
"line": 421,
"column": 41
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf g : MvPolynomial σ R\na b : σ →₀ ℕ\nha : m.toSyn (m.degree f) ≤ m.toSyn a\nhb : m.toSyn (m.degree g) ≤ m.toSyn b\nc d : σ →₀ ℕ\nh : (c, d) ≠ (a, b)\nhcd : c + d = a + b\nhf : m.toSyn c ≤ m.toSyn (m.degree f)\nhf' : m.toSyn d ≤ m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Squarefree | {
"line": 361,
"column": 2
} | {
"line": 361,
"column": 49
} | [
{
"pp": "m n : ℕ\nhm : Squarefree m\nhn : n ≠ 0\nthis : (m / m.gcd n).Coprime (m.gcd n)\n⊢ (m / m.gcd n).Coprime n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.SchwartzZippel | {
"line": 164,
"column": 12
} | {
"line": 164,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\nhp : p ≠ 0\nS : Fin (n + 1) → Finset R\np' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p\nhp' : p' = (finSuccEquiv R n) p\nk : ℕ := p'.natDegree\nhk : k = p'.natDegree... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 489,
"column": 68
} | {
"line": 492,
"column": 89
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf g : MvPolynomial σ R\nhf : IsRegular (m.leadingCoeff f)\n⊢ m.leadingCoeff (f * g) = m.leadingCoeff f * m.leadingCoeff g",
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"HMul.hMul",... | by
by_cases hg : g = 0
· simp [hg]
· simp only [leadingCoeff, degree_mul_of_isRegular_left hf hg, coeff_mul_of_degree_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 632,
"column": 2
} | {
"line": 637,
"column": 31
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nι : Type u_3\nP : ι → MvPolynomial σ R\ns : Finset ι\n⊢ coeff (∑ i ∈ s, m.degree (P i)) (∏ i ∈ s, P i) = ∏ i ∈ s, m.leadingCoeff (P i)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | induction s using Finset.induction_on with
| empty => simp
| insert a s has hrec =>
simp only [Finset.prod_insert has, Finset.sum_insert has]
rw [coeff_mul_of_add_of_degree_le (le_of_eq rfl) degree_prod_le]
exact congr_arg₂ _ rfl hrec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 632,
"column": 2
} | {
"line": 637,
"column": 31
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nι : Type u_3\nP : ι → MvPolynomial σ R\ns : Finset ι\n⊢ coeff (∑ i ∈ s, m.degree (P i)) (∏ i ∈ s, P i) = ∏ i ∈ s, m.leadingCoeff (P i)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | induction s using Finset.induction_on with
| empty => simp
| insert a s has hrec =>
simp only [Finset.prod_insert has, Finset.sum_insert has]
rw [coeff_mul_of_add_of_degree_le (le_of_eq rfl) degree_prod_le]
exact congr_arg₂ _ rfl hrec | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 632,
"column": 2
} | {
"line": 637,
"column": 31
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nι : Type u_3\nP : ι → MvPolynomial σ R\ns : Finset ι\n⊢ coeff (∑ i ∈ s, m.degree (P i)) (∏ i ∈ s, P i) = ∏ i ∈ s, m.leadingCoeff (P i)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | induction s using Finset.induction_on with
| empty => simp
| insert a s has hrec =>
simp only [Finset.prod_insert has, Finset.sum_insert has]
rw [coeff_mul_of_add_of_degree_le (le_of_eq rfl) degree_prod_le]
exact congr_arg₂ _ rfl hrec | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 630,
"column": 86
} | {
"line": 637,
"column": 31
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nι : Type u_3\nP : ι → MvPolynomial σ R\ns : Finset ι\n⊢ coeff (∑ i ∈ s, m.degree (P i)) (∏ i ∈ s, P i) = ∏ i ∈ s, m.leadingCoeff (P i)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | by
classical
induction s using Finset.induction_on with
| empty => simp
| insert a s has hrec =>
simp only [Finset.prod_insert has, Finset.sum_insert has]
rw [coeff_mul_of_add_of_degree_le (le_of_eq rfl) degree_prod_le]
exact congr_arg₂ _ rfl hrec | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 691,
"column": 69
} | {
"line": 692,
"column": 75
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nι : Type u_3\nP : ι → MvPolynomial σ R\ns : Finset ι\nH : ∀ i ∈ s, IsRegular (m.leadingCoeff (P i))\n⊢ m.leadingCoeff (∏ i ∈ s, P i) = ∏ i ∈ s, m.leadingCoeff (P i)",
"usedConstants": [
"Nat.instMulZeroClass",
"con... | by
simp only [leadingCoeff, degree_prod_of_regular H, coeff_prod_sum_degree] | [anonymous] | Lean.Parser.Term.byTactic |
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