module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Integral.Prod | {
"line": 549,
"column": 2
} | {
"line": 549,
"column": 13
} | {
"line": 549,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝³ : NormedAddCommGroup E\ninst✝² : SFinite ν\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\nf : β → E\n⊢ ∫ (z : α × β), f z.2 ∂μ.prod ν = μ.real univ • ∫ (y : β), f y ∂ν",... | [
"α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝³ : NormedAddCommGroup E\ninst✝² : SFinite ν\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\nf : β → E\n⊢ ∫ (z : α × β), f z.2 ∂μ.prod ν = μ.real univ • ∫ (y : β), f y ∂ν"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 338,
"column": 21
} | {
"line": 338,
"column": 63
} | {
"line": 338,
"column": 64
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nf : ℂ → E\ns : Set ℂ\na : ℝ\nh0 : 0 < rexp a\nb : ℝ\nhle : a ≤ b\nhd : ∀ z ∈ (ball c (rexp b) \\ closedBall c (rexp a)) \\ s, DifferentiableAt ℂ f z\nA : Set ℂ := closedBall c (rexp b) \\ ball c (rexp a)\nhc : ContinuousOn f A\n... | [
"E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nf : ℂ → E\ns : Set ℂ\na : ℝ\nh0 : 0 < rexp a\nb : ℝ\nhle : a ≤ b\nhd : ∀ z ∈ (ball c (rexp b) \\ closedBall c (rexp a)) \\ s, DifferentiableAt ℂ f z\nA : Set ℂ := closedBall c (rexp b) \\ ball c (rexp a)\nhc : ContinuousOn f A\nR : Set ℂ :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 343,
"column": 4
} | {
"line": 343,
"column": 53
} | {
"line": 343,
"column": 54
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nf : ℂ → E\ns : Set ℂ\na : ℝ\nh0 : 0 < rexp a\nb : ℝ\nhle : a ≤ b\nhd : ∀ z ∈ (ball c (rexp b) \\ closedBall c (rexp a)) \\ s, DifferentiableAt ℂ f z\nA : Set ℂ := closedBall c (rexp b) \\ ball c (rexp a)\nR : Set ℂ := [[a, b]] ×... | [
"E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nf : ℂ → E\ns : Set ℂ\na : ℝ\nh0 : 0 < rexp a\nb : ℝ\nhle : a ≤ b\nhd : ∀ z ∈ (ball c (rexp b) \\ closedBall c (rexp a)) \\ s, DifferentiableAt ℂ f z\nA : Set ℂ := closedBall c (rexp b) \\ ball c (rexp a)\nR : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 68
} | {
"line": 345,
"column": 4
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nf : ℂ → E\ns : Set ℂ\na : ℝ\nh0 : 0 < rexp a\nb : ℝ\nhle : a ≤ b\nA : Set ℂ := closedBall c (rexp b) \\ ball c (rexp a)\nR : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]\ng : ℂ → ℂ := fun x ↦ c + cexp x\nhdg : Differentiable ℂ g\nhs : (g ⁻... | [
"E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nf : ℂ → E\ns : Set ℂ\na : ℝ\nh0 : 0 < rexp a\nb : ℝ\nhle : a ≤ b\nA : Set ℂ := closedBall c (rexp b) \\ ball c (rexp a)\nR : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]\ng : ℂ → ℂ := fun x ↦ c + cexp x\nhdg : Differentiable ℂ g\nhs : (g ⁻¹' s).Counta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Liouville | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 13
} | {
"line": 69,
"column": 14
} | [
{
"pp": "F : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : CompleteSpace F\nc : ℂ\nR C : ℝ\nf : ℂ → F\nhR : 0 < R\nhf : DiffContOnCl ℂ f (ball c R)\nhC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C\n⊢ ‖deriv f c‖ ≤ C / R",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"us... | [
"F : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : CompleteSpace F\nc : ℂ\nR C : ℝ\nf : ℂ → F\nhR : 0 < R\nhf : DiffContOnCl ℂ f (ball c R)\nhC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C\n⊢ ‖deriv f c‖ ≤ C / R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Liouville | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 13
} | {
"line": 146,
"column": 2
} | [
{
"pp": "E : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : Nontrivial E\nf : E → F\nhf : Differentiable ℂ f\nc : F\nhb : Tendsto f (cocompact E) (𝓝 c)\nh_bdd : Bornology.IsBounded (range f)\nc' : F\nhc' : f = const ... | [
"E : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : Nontrivial E\nf : E → F\nhf : Differentiable ℂ f\nc : F\nhb : Tendsto f (cocompact E) (𝓝 c)\nh_bdd : Bornology.IsBounded (range f)\nc' : F\nhc' : f = const E c'\n⊢ c = ... | convert hc' | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 130,
"column": 41
} | {
"line": 141,
"column": 92
} | {
"line": 143,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\np : F[X]\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nh : Fact (map (algebraMap F E) p).Splits\n⊢ Function.Bijective (mapRoots p E)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Iff.mpr",
"Polynomia... | [] | by
constructor
· exact fun _ _ h => Subtype.ext (RingHom.injective _ (Subtype.ext_iff.mp h))
· intro y
-- this is just an equality of two different ways to write the roots of `p` as an `E`-polynomial
have key := (IsSplittingField.splits p.SplittingField p).roots_map
(IsScalarTower.toAlgHom F p.Split... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 232,
"column": 2
} | {
"line": 235,
"column": 73
} | {
"line": 237,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nhpq : p ∣ q\nhq : q ≠ 0\n⊢ Function.Surjective ⇑(restrictDvd hpq)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Polynomial.SplittingField.instCommRing",
"Dvd.dvd",
"MonoidHom.instFunLi... | [] | classical
haveI := Fact.mk <|
(SplittingField.splits q).of_dvd (map_ne_zero hq) ((map_dvd_map' _).mpr hpq)
simpa only [restrictDvd_def, dif_neg hq] using! restrict_surjective _ _ | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 232,
"column": 2
} | {
"line": 235,
"column": 73
} | {
"line": 237,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nhpq : p ∣ q\nhq : q ≠ 0\n⊢ Function.Surjective ⇑(restrictDvd hpq)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Polynomial.SplittingField.instCommRing",
"Dvd.dvd",
"MonoidHom.instFunLi... | [] | classical
haveI := Fact.mk <|
(SplittingField.splits q).of_dvd (map_ne_zero hq) ((map_dvd_map' _).mpr hpq)
simpa only [restrictDvd_def, dif_neg hq] using! restrict_surjective _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 232,
"column": 2
} | {
"line": 235,
"column": 73
} | {
"line": 237,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nhpq : p ∣ q\nhq : q ≠ 0\n⊢ Function.Surjective ⇑(restrictDvd hpq)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Polynomial.SplittingField.instCommRing",
"Dvd.dvd",
"MonoidHom.instFunLi... | [] | classical
haveI := Fact.mk <|
(SplittingField.splits q).of_dvd (map_ne_zero hq) ((map_dvd_map' _).mpr hpq)
simpa only [restrictDvd_def, dif_neg hq] using! restrict_surjective _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 158,
"column": 15
} | {
"line": 158,
"column": 26
} | {
"line": 158,
"column": 27
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\ninst✝ : CommSemiring R\nφ : MvPolynomial σ R\ne : σ ≃ τ\nh : ((rename ⇑e) φ).IsSymmetric\n⊢ φ.IsSymmetric",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"σ : Type u_1\nτ : Type u_2\nR : Type u_3\ninst✝ : CommSemiring R\nφ : MvPolynomial σ R\ne : σ ≃ τ\nh : ((rename ⇑e) φ).IsSymmetric\n⊢ φ.IsSymmetric"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 50
} | {
"line": 242,
"column": 4
} | [
{
"pp": "τ : Type u_2\nσ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : Fintype σ\ninst✝ : Fintype τ\nn : ℕ\ne : σ ≃ τ\n⊢ (rename ⇑e) (esymm σ R n) = ∑ x ∈ powersetCard n univ, ∏ i ∈ x, X (e i)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
... | [] | simp_rw [esymm, map_sum, map_prod, rename_X] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 50
} | {
"line": 242,
"column": 4
} | [
{
"pp": "τ : Type u_2\nσ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : Fintype σ\ninst✝ : Fintype τ\nn : ℕ\ne : σ ≃ τ\n⊢ (rename ⇑e) (esymm σ R n) = ∑ x ∈ powersetCard n univ, ∏ i ∈ x, X (e i)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
... | [] | simp_rw [esymm, map_sum, map_prod, rename_X] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 50
} | {
"line": 242,
"column": 4
} | [
{
"pp": "τ : Type u_2\nσ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : Fintype σ\ninst✝ : Fintype τ\nn : ℕ\ne : σ ≃ τ\n⊢ (rename ⇑e) (esymm σ R n) = ∑ x ∈ powersetCard n univ, ∏ i ∈ x, X (e i)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
... | [] | simp_rw [esymm, map_sum, map_prod, rename_X] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 276,
"column": 6
} | {
"line": 276,
"column": 20
} | {
"line": 276,
"column": 20
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝³ : CommSemiring R\ninst✝² : Fintype σ\ninst✝¹ : DecidableEq σ\ninst✝ : Nontrivial R\nn : ℕ\n⊢ (esymm σ R n).support = image (fun t ↦ ∑ i ∈ t, Finsupp.single i 1) (powersetCard n univ)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"σ : Type u_5\nR : Type u_6\ninst✝³ : CommSemiring R\ninst✝² : Fintype σ\ninst✝¹ : DecidableEq σ\ninst✝ : Nontrivial R\nn : ℕ\n⊢ ((powersetCard n univ).biUnion fun t ↦ {∑ i ∈ t, Finsupp.single i 1}) =\n image (fun t ↦ ∑ i ∈ t, Finsupp.single i 1) (powersetCard n univ)"
] | support_esymm' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 313,
"column": 2
} | {
"line": 325,
"column": 27
} | {
"line": 326,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nhq : q.natDegree ≠ 0\nP : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits\nkey1 : ∀ {r : F[X]}, Irreducible r → P r\n⊢ (map (algebraMap F (p.comp q).SplittingField) p).Splits",
"ppTerm": "?m.45",
"assigned": true,
"us... | [
"F : Type u_1\ninst✝ : Field F\np q : F[X]\nhq : q.natDegree ≠ 0\nP : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits\nkey1 : ∀ {r : F[X]}, Irreducible r → P r\nkey2 : ∀ {p₁ p₂ : F[X]}, P p₁ → P p₂ → P (p₁ * p₂)\n⊢ (map (algebraMap F (p.comp q).SplittingField) p).Splits"
] | have key2 : ∀ {p₁ p₂ : F[X]}, P p₁ → P p₂ → P (p₁ * p₂) := by
intro p₁ p₂ hp₁ hp₂
by_cases h₁ : p₁.comp q = 0
· rcases comp_eq_zero_iff.mp h₁ with h | h
· rw [h, zero_mul]
simp [P]
· exact False.elim (hq (by rw [h.2, natDegree_C]))
by_cases h₂ : p₂.comp q = 0
· rcases comp_eq_zer... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 350,
"column": 54
} | {
"line": 350,
"column": 65
} | {
"line": 352,
"column": 0
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\n⊢ psum σ R 0 = ↑(Fintype.card σ)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Nat.instMulZeroCl... | [] | simp [psum] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 350,
"column": 54
} | {
"line": 350,
"column": 65
} | {
"line": 352,
"column": 0
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\n⊢ psum σ R 0 = ↑(Fintype.card σ)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Nat.instMulZeroCl... | [] | simp [psum] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 350,
"column": 54
} | {
"line": 350,
"column": 65
} | {
"line": 352,
"column": 0
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\n⊢ psum σ R 0 = ↑(Fintype.card σ)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Nat.instMulZeroCl... | [] | simp [psum] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 353,
"column": 47
} | {
"line": 353,
"column": 58
} | {
"line": 355,
"column": 0
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\n⊢ psum σ R 1 = ∑ i, X i",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Finset.univ",
"congrArg",
"CommSemiring.toSemiring",
... | [] | simp [psum] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 353,
"column": 47
} | {
"line": 353,
"column": 58
} | {
"line": 355,
"column": 0
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\n⊢ psum σ R 1 = ∑ i, X i",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Finset.univ",
"congrArg",
"CommSemiring.toSemiring",
... | [] | simp [psum] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 353,
"column": 47
} | {
"line": 353,
"column": 58
} | {
"line": 355,
"column": 0
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\n⊢ psum σ R 1 = ∑ i, X i",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Finset.univ",
"congrArg",
"CommSemiring.toSemiring",
... | [] | simp [psum] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 558,
"column": 4
} | {
"line": 558,
"column": 48
} | {
"line": 558,
"column": 49
} | [
{
"pp": "case refine_1\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\nhf : DiffContOnCl ℂ f (ball c R)\nhw : w ∈ ball c R\nhR : 0 < R\n⊢ ContinuousOn f (closedBall c R)",
"ppTerm": "?refine_1",
"assigned": false,
"usedConstan... | [
"case refine_1\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\nhf : DiffContOnCl ℂ f (ball c R)\nhw : w ∈ ball c R\nhR : 0 < R\n⊢ ContinuousOn f (closedBall c R)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 559,
"column": 4
} | {
"line": 559,
"column": 34
} | {
"line": 559,
"column": 35
} | [
{
"pp": "case refine_2\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\nhf : DiffContOnCl ℂ f (ball c R)\nhw : w ∈ ball c R\nhR : 0 < R\n⊢ ∀ x ∈ ball c R \\ ∅, DifferentiableAt ℂ f x",
"ppTerm": "?refine_2",
"assigned": true,
"... | [
"case refine_2\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\nhf : DiffContOnCl ℂ f (ball c R)\nhw : w ∈ ball c R\nhR : 0 < R\n⊢ ∀ x ∈ ball c R, DifferentiableAt ℂ f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 576,
"column": 2
} | {
"line": 576,
"column": 48
} | {
"line": 577,
"column": 4
} | [
{
"pp": "R : ℝ\nc w : ℂ\ns : Set ℂ\nhs : s.Countable\nhw : w ∈ ball c R\nf : ℂ → ℂ\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ z ∈ ball c R \\ s, DifferentiableAt ℂ f z\n⊢ ∮ (z : ℂ) in C(c, R), f z / (z - w) = 2 * ↑π * I * f w",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"R : ℝ\nc w : ℂ\ns : Set ℂ\nhs : s.Countable\nhw : w ∈ ball c R\nf : ℂ → ℂ\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ z ∈ ball c R \\ s, DifferentiableAt ℂ f z\n⊢ ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ * f z = 2 * ↑π * I * f w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 597,
"column": 6
} | {
"line": 598,
"column": 46
} | {
"line": 598,
"column": 47
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ≥0\nc : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z\nhR : 0 < R\nw : ℂ\nhw : w ∈ eball 0 ↑R\n⊢ c + w ∈ ball c ↑R",
... | [
"E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ≥0\nc : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z\nhR : 0 < R\nw : ℂ\nhw : w ∈ eball 0 ↑R\n⊢ ‖w‖₊ < R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 349,
"column": 8
} | {
"line": 350,
"column": 71
} | {
"line": 350,
"column": 72
} | [
{
"pp": "case refine_2\nE : Type u\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nn : ℕ\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : Preorder F\ninst✝¹ : MeasureSpace F\ninst✝ : BorelSpace F\neL : F ≃L[ℝ] Fin (n + 1) → ℝ\nhe_ord : ∀ (x y : F), eL x ≤ eL y ↔ x ≤ y\n... | [
"case refine_2\nE : Type u\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nn : ℕ\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : Preorder F\ninst✝¹ : MeasureSpace F\ninst✝ : BorelSpace F\neL : F ≃L[ℝ] Fin (n + 1) → ℝ\nhe_ord : ∀ (x y : F), eL x ≤ eL y ↔ x ≤ y\nhe_vol : Mea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 739,
"column": 2
} | {
"line": 739,
"column": 13
} | {
"line": 739,
"column": 14
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nf : ℂ → E\nc : ℂ\ns : Set ℂ\nh0 : 0 < R\nhs : s.Countable\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ z ∈ ball c R \\ s, DifferentiableAt ℂ f z\n⊢ ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ 2) • f z = (2 * ... | [
"E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nf : ℂ → E\nc : ℂ\ns : Set ℂ\nh0 : 0 < R\nhs : s.Countable\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ z ∈ ball c R \\ s, DifferentiableAt ℂ f z\n⊢ ∮ (z : ℂ) in C(c, R), ((z - c) ^ 2)⁻¹ • f z = (2 * ↑π * I) • deri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 756,
"column": 2
} | {
"line": 756,
"column": 13
} | {
"line": 756,
"column": 14
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nf : ℂ → E\nc : ℂ\nh0 : 0 < R\nhc : DiffContOnCl ℂ f (ball c R)\n⊢ ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ 2) • f z = (2 * ↑π * I) • deriv f c",
"ppTerm": "?m.79",
"assigned": true,
"usedConstan... | [
"E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nf : ℂ → E\nc : ℂ\nh0 : 0 < R\nhc : DiffContOnCl ℂ f (ball c R)\n⊢ ∮ (z : ℂ) in C(c, R), ((z - c) ^ 2)⁻¹ • f z = (2 * ↑π * I) • deriv f c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 771,
"column": 2
} | {
"line": 771,
"column": 13
} | {
"line": 771,
"column": 14
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nf : ℂ → E\nc : ℂ\nh0 : 0 < R\nhc : DifferentiableOn ℂ f (closedBall c R)\n⊢ ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ 2) • f z = (2 * ↑π * I) • deriv f c",
"ppTerm": "?m.81",
"assigned": true,
"u... | [
"E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nf : ℂ → E\nc : ℂ\nh0 : 0 < R\nhc : DifferentiableOn ℂ f (closedBall c R)\n⊢ ∮ (z : ℂ) in C(c, R), ((z - c) ^ 2)⁻¹ • f z = (2 * ↑π * I) • deriv f c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Polynomial | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 29
} | {
"line": 106,
"column": 30
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Ring S\ninst✝³ : Field k\ninst✝² : LinearOrder k\ninst✝¹ : IsStrictOrderedRing k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\na✝ : R\n... | [
"case refine_1\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Ring S\ninst✝³ : Field k\ninst✝² : LinearOrder k\ninst✝¹ : IsStrictOrderedRing k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\na✝ : R\nhc : f a✝ ≠ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Polynomial | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 29
} | {
"line": 109,
"column": 30
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Ring S\ninst✝³ : Field k\ninst✝² : LinearOrder k\ninst✝¹ : IsStrictOrderedRing k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X... | [
"case refine_2\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Ring S\ninst✝³ : Field k\ninst✝² : LinearOrder k\ninst✝¹ : IsStrictOrderedRing k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na✝ : 0 < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 44,
"column": 8
} | {
"line": 45,
"column": 15
} | {
"line": 45,
"column": 16
} | [
{
"pp": "f : ℂ[X]\nhf : 0 < f.degree\nhf' : ∀ (z : ℂ), ¬f.IsRoot z\nz : ℂ\n⊢ Filter.Tendsto (fun x ↦ eval x f) (cobounded ℂ) (cobounded ℂ)",
"ppTerm": "?m.62",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℂ[X]\nhf : 0 < f.degree\nhf' : ∀ (z : ℂ), ¬f.IsRoot z\nz : ℂ\n⊢ Filter.Tendsto (fun x ↦ eval x f) (cobounded ℂ) (cobounded ℂ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Polynomial | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 15
} | {
"line": 114,
"column": 16
} | [
{
"pp": "case refine_3\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Ring S\ninst✝³ : Field k\ninst✝² : LinearOrder k\ninst✝¹ : IsStrictOrderedRing k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X... | [
"case refine_3\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Ring S\ninst✝³ : Field k\ninst✝² : LinearOrder k\ninst✝¹ : IsStrictOrderedRing k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na : R\nhd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 36
} | {
"line": 411,
"column": 2
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : s.Countable\nHc : ContinuousOn f [[a, b]]\nHd : ∀ x ∈ Set.Ioo (min a b) (max a b) \\ s, HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\n⊢ ∫ (x : ℝ) in a..b,... | [
"case inl\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : s.Countable\nHc : ContinuousOn f [[a, b]]\nHd : ∀ x ∈ Set.Ioo (min a b) (max a b) \\ s, HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\nhab : a ≤ b\n⊢ ∫ (x : ... | rcases le_total a b with hab | hab | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Algebra.Polynomial | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 26
} | {
"line": 154,
"column": 27
} | [
{
"pp": "R : Type u_2\ninst✝² : NormedRing R\ninst✝¹ : IsAbsoluteValue norm\ninst✝ : ProperSpace R\nn : ℕ\n⊢ IsClosedMap fun x ↦ x ^ n",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_2\ninst✝² : NormedRing R\ninst✝¹ : IsAbsoluteValue norm\ninst✝ : ProperSpace R\nn : ℕ\n⊢ IsClosedMap fun x ↦ x ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 410,
"column": 2
} | {
"line": 415,
"column": 81
} | {
"line": 417,
"column": 0
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : s.Countable\nHc : ContinuousOn f [[a, b]]\nHd : ∀ x ∈ Set.Ioo (min a b) (max a b) \\ s, HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\n⊢ ∫ (x : ℝ) in a..b,... | [] | rcases le_total a b with hab | hab
· simp only [uIcc_of_le hab, min_eq_left hab, max_eq_right hab] at *
exact integral_eq_of_hasDerivAt_off_countable_of_le f f' hab hs Hc Hd Hi
· simp only [uIcc_of_ge hab, min_eq_right hab, max_eq_left hab] at *
rw [intervalIntegral.integral_symm, neg_eq_iff_eq_neg, neg_sub... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 410,
"column": 2
} | {
"line": 415,
"column": 81
} | {
"line": 417,
"column": 0
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : s.Countable\nHc : ContinuousOn f [[a, b]]\nHd : ∀ x ∈ Set.Ioo (min a b) (max a b) \\ s, HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\n⊢ ∫ (x : ℝ) in a..b,... | [] | rcases le_total a b with hab | hab
· simp only [uIcc_of_le hab, min_eq_left hab, max_eq_right hab] at *
exact integral_eq_of_hasDerivAt_off_countable_of_le f f' hab hs Hc Hd Hi
· simp only [uIcc_of_ge hab, min_eq_right hab, max_eq_left hab] at *
rw [intervalIntegral.integral_symm, neg_eq_iff_eq_neg, neg_sub... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 237,
"column": 36
} | {
"line": 237,
"column": 47
} | {
"line": 237,
"column": 48
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nk : Type u_2\ninst✝ : Field k\nf : k →+* K\nφ : K →+* ℂ\nhφ : IsReal φ\nx : k\n⊢ (star (φ.comp f)) x = (φ.comp f) x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"RingHom",
"id",
"Field.toSemifield",
"RingHom.comp",
"... | [
"K : Type u_1\ninst✝¹ : Field K\nk : Type u_2\ninst✝ : Field k\nf : k →+* K\nφ : K →+* ℂ\nhφ : IsReal φ\nx : k\n⊢ (starRingEnd ℂ) (φ (f x)) = φ (f x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 26
} | {
"line": 43,
"column": 27
} | [
{
"pp": "M : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedAddMonoid M\ninst✝ : One M\nu v : ℚ\nx y : M\nhu : u.num • 1 < u.den • x\nhv : v.num • 1 < v.den • y\n⊢ (u.num * ↑v.den) • 1 < (↑u.den * ↑v.den) • x",
"ppTerm": "?m.72",
"assigned": false,
"usedConstants": [],
... | [
"M : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedAddMonoid M\ninst✝ : One M\nu v : ℚ\nx y : M\nhu : u.num • 1 < u.den • x\nhv : v.num • 1 < v.den • y\n⊢ (u.num * ↑v.den) • 1 < (↑u.den * ↑v.den) • x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 50,
"column": 39
} | {
"line": 50,
"column": 50
} | {
"line": 50,
"column": 51
} | [
{
"pp": "M : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedAddMonoid M\ninst✝ : One M\nu v : ℚ\nx y : M\nhu : u.num • 1 < u.den • x\nhv : v.num • 1 < v.den • y\nhu' : (u.num * ↑v.den) • 1 < (↑u.den * ↑v.den) • x\n⊢ 0 < ↑(u + v).den",
"ppTerm": "?m.221",
"assigned": true,
... | [
"M : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedAddMonoid M\ninst✝ : One M\nu v : ℚ\nx y : M\nhu : u.num • 1 < u.den • x\nhv : v.num • 1 < v.den • y\nhu' : (u.num * ↑v.den) • 1 < (↑u.den * ↑v.den) • x\n⊢ 0 < (u + v).den"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 286,
"column": 46
} | {
"line": 286,
"column": 57
} | {
"line": 286,
"column": 58
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nk : Type u_2\ninst✝¹ : Field k\ninst✝ : Algebra k K\nφ : K →+* ℂ\nσ : Gal(K/k)\nhσ : IsConj φ σ\nx : K\n⊢ (conjugate φ) x = (φ.comp ↑σ.symm) x",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"NumberField.ComplexEmbedding.conjugate",
"Alg... | [
"K : Type u_1\ninst✝² : Field K\nk : Type u_2\ninst✝¹ : Field k\ninst✝ : Algebra k K\nφ : K →+* ℂ\nσ : Gal(K/k)\nhσ : IsConj φ σ\nx : K\n⊢ (starRingEnd ℂ) (φ x) = φ (σ.symm x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 302,
"column": 36
} | {
"line": 302,
"column": 47
} | {
"line": 302,
"column": 48
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nk : Type u_2\ninst✝¹ : Field k\ninst✝ : Algebra k K\nφ : K →+* ℂ\nσ : Gal(K/k)\nhσ : IsConj φ σ\nhσ' : σ ≠ 1\na✝ : K\n⊢ (σ ^ 2) a✝ = 1 a✝",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneCla... | [
"K : Type u_1\ninst✝² : Field K\nk : Type u_2\ninst✝¹ : Field k\ninst✝ : Algebra k K\nφ : K →+* ℂ\nσ : Gal(K/k)\nhσ : IsConj φ σ\nhσ' : σ ≠ 1\na✝ : K\n⊢ σ (σ a✝) = a✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 13
} | {
"line": 63,
"column": 14
} | [
{
"pp": "M : Type u_1\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : LinearOrder M\ninst✝³ : IsOrderedAddMonoid M\ninst✝² : One M\ninst✝¹ : ZeroLEOneClass M\ninst✝ : NeZero 1\nnum : ℤ\nden : ℕ\nx : M\nh : num • 1 ≤ den • x\nn : ℤ\nhn : x ≤ n • 1\n⊢ den • x ≤ ↑den • n • 1",
"ppTerm": "?m.66",
"assigned": true,
"... | [
"M : Type u_1\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : LinearOrder M\ninst✝³ : IsOrderedAddMonoid M\ninst✝² : One M\ninst✝¹ : ZeroLEOneClass M\ninst✝ : NeZero 1\nnum : ℤ\nden : ℕ\nx : M\nh : num • 1 ≤ den • x\nn : ℤ\nhn : x ≤ n • 1\n⊢ den • x ≤ den • n • 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 94,
"column": 48
} | {
"line": 94,
"column": 59
} | {
"line": 94,
"column": 60
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nn : ℕ\nhn : x ≤ n • 1\nnum : ℤ\nden : ℕ\nden_nz✝ : den ≠ 0\nreduced✝ : num.natAbs.Coprime den\nthis : num • 1 < den •... | [
"M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nn : ℕ\nhn : x ≤ n • 1\nnum : ℤ\nden : ℕ\nden_nz✝ : den ≠ 0\nreduced✝ : num.natAbs.Coprime den\nthis : num • 1 < den • x → num ≤ ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 96,
"column": 36
} | {
"line": 96,
"column": 47
} | {
"line": 96,
"column": 48
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nn : ℕ\nhn : x ≤ n • 1\nnum : ℤ\nden : ℕ\nden_nz✝ : den ≠ 0\nreduced✝ : num.natAbs.Coprime den\nh : num • 1 < den • x\... | [
"M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nn : ℕ\nhn : x ≤ n • 1\nnum : ℤ\nden : ℕ\nden_nz✝ : den ≠ 0\nreduced✝ : num.natAbs.Coprime den\nh : num • 1 < den • x\n⊢ x ≤ n • 1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 102,
"column": 29
} | {
"line": 102,
"column": 40
} | {
"line": 102,
"column": 41
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nhneg : x < 0\nn : ℕ\nhn : -x - x ≤ n • 1\nthis : -(n • 1) < x\n⊢ Rat.ofInt (-↑n) ∈ ratLt x",
"ppTerm": "?m.99",
... | [
"M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nhneg : x < 0\nn : ℕ\nhn : -x - x ≤ n • 1\nthis : -(n • 1) < x\n⊢ -(n • 1) < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 103,
"column": 41
} | {
"line": 103,
"column": 52
} | {
"line": 103,
"column": 53
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nhneg : x < 0\nn : ℕ\nhn : -x - x ≤ n • 1\n⊢ -x < -x - x",
"ppTerm": "?m.121",
"assigned": true,
"usedCons... | [
"M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nhneg : x < 0\nn : ℕ\nhn : -x - x ≤ n • 1\n⊢ x < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 15
} | {
"line": 107,
"column": 16
} | [
{
"pp": "case h\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nhxpos : 0 < x\nn : ℕ\nhn : 1 ≤ n • x\n⊢ { num := 1, den := n + 1, den_nz := ⋯, reduced := ⋯ } ∈ ratLt x",
... | [
"case h\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nhxpos : 0 < x\nn : ℕ\nhn : 1 ≤ n • x\n⊢ 1 < (n + 1) • x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 134,
"column": 47
} | {
"line": 134,
"column": 63
} | {
"line": 134,
"column": 63
} | [
{
"pp": "p : ℚ[X]\np_irr : Irreducible p\np_deg : Nat.Prime p.natDegree\np_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2\n⊢ Fintype.card ↥(p.aroots ℂ).toFinset = p.natDegree",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
... | [
"p : ℚ[X]\np_irr : Irreducible p\np_deg : Nat.Prime p.natDegree\np_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2\n⊢ (p.aroots ℂ).toFinset.card = p.natDegree"
] | Fintype.card_coe | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Real.Embedding | {
"line": 131,
"column": 6
} | {
"line": 131,
"column": 17
} | {
"line": 131,
"column": 18
} | [
{
"pp": "case mp.refine_1\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx y : M\na : ℚ\nh : a.num • 1 < a.den • (x + y)\nk : ℕ\nhk : 1 + 1 ≤ k • (a.den • (x + y) - a.num • 1)\nhk... | [
"case mp.refine_1\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx y : M\na : ℚ\nh : a.num • 1 < a.den • (x + y)\nk : ℕ\nhk : 1 + 1 ≤ k • (a.den • (x + y) - a.num • 1)\nhk0 : k ≠ 0\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 146,
"column": 4
} | {
"line": 148,
"column": 11
} | {
"line": 148,
"column": 12
} | [
{
"pp": "case h1\np : ℚ[X]\np_irr : Irreducible p\np_deg : Nat.Prime p.natDegree\np_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2\nh1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree\nconj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)\nx : Equiv.Perm ↑(p.rootSet ℂ)\n⊢ p.natDe... | [
"case h1\np : ℚ[X]\np_irr : Irreducible p\np_deg : Nat.Prime p.natDegree\np_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2\nh1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree\nconj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)\nx : Equiv.Perm ↑(p.rootSet ℂ)\n⊢ p.natDegree ∣ Nat.c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 85,
"column": 6
} | {
"line": 85,
"column": 63
} | {
"line": 85,
"column": 64
} | [
{
"pp": "R : Type u_1\nΓ₀ : Type u_2\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\ninst✝ : v.IsNontrivial\na✝ : MulArchimedean (ofClass v).ValueGroup₀\nf : Additive (ofClass v).ValueGroup₀ˣ →+o ℝ\nhf : Injective ⇑f\ne : (ofClass v).ValueGroup₀ˣ →* Multiplicative ℝ := AddMonoi... | [
"R : Type u_1\nΓ₀ : Type u_2\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\ninst✝ : v.IsNontrivial\na✝ : MulArchimedean (ofClass v).ValueGroup₀\nf : Additive (ofClass v).ValueGroup₀ˣ →+o ℝ\nhf : Injective ⇑f\ne : (ofClass v).ValueGroup₀ˣ →* Multiplicative ℝ := AddMonoidHom.toMulti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 138,
"column": 62
} | {
"line": 138,
"column": 73
} | {
"line": 138,
"column": 74
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx y : M\na : ℚ\nh : a.num • 1 < a.den • (x + y)\nk : ℕ\nhk : 1 + 1 ≤ k • (a.den • (x + y) - a.num • 1)\nhk0 : k ≠ 0\nhka0 : ... | [
"M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx y : M\na : ℚ\nh : a.num • 1 < a.den • (x + y)\nk : ℕ\nhk : 1 + 1 ≤ k • (a.den • (x + y) - a.num • 1)\nhk0 : k ≠ 0\nhka0 : k * a.den ≠ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 149,
"column": 2
} | {
"line": 161,
"column": 38
} | {
"line": 163,
"column": 0
} | [
{
"pp": "Γ₀ : Type u_2\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\nK : Type u_3\ninst✝¹ : DivisionRing K\nv : Valuation K Γ₀\ninst✝ : v.RankOne\nγ : ℝ≥0\nhγ : γ ≠ 0\n⊢ ∃ x, x ≠ 0 ∧ (hom v) (v.restrict x) < γ",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"WithZero.instNontrivial",
... | [] | have hγ_pos : 0 < γ := pos_iff_ne_zero.mpr hγ
obtain ⟨x, h⟩ := NNReal.exists_lt_of_strictMono (RankOne.strictMono v.restrict) hγ_pos
obtain ⟨k, hk⟩ := ValueGroup₀.restrict₀_surjective _ x.val
refine ⟨k, ?_, ?_⟩
· simp only [restrict₀_apply, MonoidWithZeroHom.coe_ofClass, restrict_def, map_eq_zero,
dite_eq... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 204,
"column": 70
} | {
"line": 204,
"column": 81
} | {
"line": 204,
"column": 82
} | [
{
"pp": "p : ℝ[X]\nhp : Irreducible p\nz : ℂ\nhz : (aeval z) p = 0\n⊢ C p.leadingCoeff⁻¹ ≠ 0",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"inv_eq_zero._simp_1",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"RingHom.instRingHomClass",
"R... | [
"p : ℝ[X]\nhp : Irreducible p\nz : ℂ\nhz : (aeval z) p = 0\n⊢ ¬p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 149,
"column": 2
} | {
"line": 161,
"column": 38
} | {
"line": 163,
"column": 0
} | [
{
"pp": "Γ₀ : Type u_2\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\nK : Type u_3\ninst✝¹ : DivisionRing K\nv : Valuation K Γ₀\ninst✝ : v.RankOne\nγ : ℝ≥0\nhγ : γ ≠ 0\n⊢ ∃ x, x ≠ 0 ∧ (hom v) (v.restrict x) < γ",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"WithZero.instNontrivial",
... | [] | have hγ_pos : 0 < γ := pos_iff_ne_zero.mpr hγ
obtain ⟨x, h⟩ := NNReal.exists_lt_of_strictMono (RankOne.strictMono v.restrict) hγ_pos
obtain ⟨k, hk⟩ := ValueGroup₀.restrict₀_surjective _ x.val
refine ⟨k, ?_, ?_⟩
· simp only [restrict₀_apply, MonoidWithZeroHom.coe_ofClass, restrict_def, map_eq_zero,
dite_eq... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Real.Embedding | {
"line": 164,
"column": 63
} | {
"line": 164,
"column": 74
} | {
"line": 164,
"column": 75
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nthis : ∀ (y : ℚ), y.num • 1 < 0 → ↑y = x → x ≤ 0\n⊢ x ∈ ⇑(Rat.castHom ℝ) '' {r | r.num • 1 < r.den • 0} → x ≤ 0",
... | [
"M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nthis : ∀ (y : ℚ), y.num • 1 < 0 → ↑y = x → x ≤ 0\n⊢ ∀ (x_1 : ℚ), x_1.num • 1 < 0 → ↑x_1 = x → x ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 171,
"column": 65
} | {
"line": 171,
"column": 76
} | {
"line": 171,
"column": 77
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nthis : (∀ (y : ℚ), y.num • 1 < 0 → ↑y ≤ x) → 0 ≤ x\n⊢ (∀ x_1 ∈ ratLt' 0, x_1 ≤ x) → 0 ≤ x",
"ppTerm": "?m.154",
... | [
"M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nthis : (∀ (y : ℚ), y.num • 1 < 0 → ↑y ≤ x) → 0 ≤ x\n⊢ (∀ (a : ℚ), a.num • 1 < 0 → ↑a ≤ x) → 0 ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 174,
"column": 49
} | {
"line": 174,
"column": 60
} | {
"line": 174,
"column": 61
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nh : ∀ (y : ℚ), y.num • 1 < 0 → ↑y ≤ x\ny : ℚ\nhy : y < 0\n⊢ y.num < 0",
"ppTerm": "?m.188",
"assigned": true,... | [
"M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nh : ∀ (y : ℚ), y.num • 1 < 0 → ↑y ≤ x\ny : ℚ\nhy : y < 0\n⊢ y < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 177,
"column": 17
} | {
"line": 177,
"column": 28
} | {
"line": 177,
"column": 29
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nh : ∀ (y : ℚ), y.num • 1 < 0 → ↑y ≤ x\nh' : x < 0\ny : ℚ\nhxy : x < ↑y\nhy : ↑y < 0\n⊢ y < 0",
"ppTerm": "?m.236"... | [
"M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nh : ∀ (y : ℚ), y.num • 1 < 0 → ↑y ≤ x\nh' : x < 0\ny : ℚ\nhxy : x < ↑y\nhy : ↑y < 0\n⊢ y < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 272,
"column": 2
} | {
"line": 274,
"column": 71
} | {
"line": 276,
"column": 0
} | [
{
"pp": "Γ₀ : Type u_2\ninst✝⁴ : LinearOrderedCommGroupWithZero Γ₀\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : MulArchimedean Γ₀\nv : Valuation R Γ₀\ninst✝ : v.Compatible\n⊢ IsRankLeOne R",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toComm... | [] | rw [isRankLeOne_iff_mulArchimedean]
exact MulArchimedean.comap (embedding.toMonoidHom.comp (ValueGroupWithZero.embed v).toMonoidHom)
(embedding_strictMono.comp (ValueGroupWithZero.embed_strictMono v)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 272,
"column": 2
} | {
"line": 274,
"column": 71
} | {
"line": 276,
"column": 0
} | [
{
"pp": "Γ₀ : Type u_2\ninst✝⁴ : LinearOrderedCommGroupWithZero Γ₀\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : MulArchimedean Γ₀\nv : Valuation R Γ₀\ninst✝ : v.Compatible\n⊢ IsRankLeOne R",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toComm... | [] | rw [isRankLeOne_iff_mulArchimedean]
exact MulArchimedean.comap (embedding.toMonoidHom.comp (ValueGroupWithZero.embed v).toMonoidHom)
(embedding_strictMono.comp (ValueGroupWithZero.embed_strictMono v)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Ultra.Basic | {
"line": 63,
"column": 8
} | {
"line": 63,
"column": 25
} | {
"line": 63,
"column": 25
} | [
{
"pp": "case inl\nX : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y z : X\nh✝ : dist x y ≠ dist y z\nh : dist x y < dist y z\n⊢ max (dist x y) (dist y z) ≤ dist x z",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"... | [
"case inl\nX : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y z : X\nh✝ : dist x y ≠ dist y z\nh : dist x y < dist y z\n⊢ dist y z ≤ dist x z"
] | max_eq_right h.le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Ultra.Basic | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 44
} | {
"line": 65,
"column": 45
} | [
{
"pp": "case inl\nX : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y z : X\nh✝ : dist x y ≠ dist y z\nh : dist x y < dist y z\n⊢ ¬dist y z ≤ dist y x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"_priva... | [
"case inl\nX : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y z : X\nh✝ : dist x y ≠ dist y z\nh : dist x y < dist y z\n⊢ dist y x < dist y z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Ultra.Basic | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 44
} | {
"line": 68,
"column": 45
} | [
{
"pp": "case inr\nX : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y z : X\nh✝ : dist x y ≠ dist y z\nh : dist y z < dist x y\n⊢ ¬dist y x ≤ dist y z",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"_priva... | [
"case inr\nX : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y z : X\nh✝ : dist x y ≠ dist y z\nh : dist y z < dist x y\n⊢ dist y z < dist y x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Ultra.Basic | {
"line": 71,
"column": 18
} | {
"line": 71,
"column": 47
} | {
"line": 71,
"column": 48
} | [
{
"pp": "X : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y z : X\nr s : ℝ\np : X → Prop\nx✝² x✝¹ x✝ : Subtype p\n⊢ dist x✝² x✝ ≤ max (dist x✝² x✝¹) (dist x✝¹ x✝)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
... | [
"X : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y z : X\nr s : ℝ\np : X → Prop\nx✝² x✝¹ x✝ : Subtype p\n⊢ dist ↑x✝² ↑x✝ ≤ dist ↑x✝² ↑x✝¹ ∨ dist ↑x✝² ↑x✝ ≤ dist ↑x✝¹ ↑x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 15
} | {
"line": 192,
"column": 16
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx y : M\nh : x < y\nhyz : 0 < y - x\nhy : y = y - x + x\nn : ℕ\nhn : 1 ≤ n • (y - x)\n⊢ ↑{ num := 1, den := n + 1, den_nz :=... | [
"M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx y : M\nh : x < y\nhyz : 0 < y - x\nhy : y = y - x + x\nn : ℕ\nhn : 1 ≤ n • (y - x)\n⊢ 1 < (n + 1) • (y - x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 215,
"column": 81
} | {
"line": 215,
"column": 92
} | {
"line": 215,
"column": 93
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nthis : ∀ (x : ℚ), x.num • 1 < ↑x.den • 1 → ↑x ≤ 1\n⊢ ∀ b ∈ ratLt' 1, b ≤ 1",
"ppTerm": "?m.76",
"assigned": true,
... | [
"M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nthis : ∀ (x : ℚ), x.num • 1 < ↑x.den • 1 → ↑x ≤ 1\n⊢ ∀ (a : ℚ), a.num • 1 < a.den • 1 → ↑a ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 28
} | {
"line": 218,
"column": 29
} | [
{
"pp": "case a\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℚ\nhx : x.num • 1 < ↑x.den • 1\n⊢ x ≤ 1",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
... | [
"case a\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℚ\nhx : x.num • 1 < ↑x.den • 1\n⊢ x.num ≤ ↑x.den"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 17
} | {
"line": 222,
"column": 18
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nthis : ∀ (x : ℝ), (∀ (y : ℚ), y.num • 1 < ↑y.den • 1 → ↑y ≤ x) → 1 ≤ x\n⊢ ∀ (b : ℝ), (∀ x ∈ ratLt' 1, x ≤ b) → 1 ≤ b",
"... | [
"M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nthis : ∀ (x : ℝ), (∀ (y : ℚ), y.num • 1 < ↑y.den • 1 → ↑y ≤ x) → 1 ≤ x\n⊢ ∀ (b : ℝ), (∀ (a : ℚ), a.num • 1 < a.den • 1 → ↑a ≤ b) → 1 ≤ b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 225,
"column": 63
} | {
"line": 225,
"column": 74
} | {
"line": 225,
"column": 75
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nh : ∀ (y : ℚ), y.num • 1 < ↑y.den • 1 → ↑y ≤ x\ny : ℚ\nhy : y < 1\n⊢ y.num < ↑y.den",
"ppTerm": "?m.218",
"as... | [
"M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nh : ∀ (y : ℚ), y.num • 1 < ↑y.den • 1 → ↑y ≤ x\ny : ℚ\nhy : y < 1\n⊢ y.num < ↑y.den"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 40
} | {
"line": 230,
"column": 0
} | [
{
"pp": "case a\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nh : ∀ (y : ℚ), y.num • 1 < ↑y.den • 1 → ↑y ≤ x\nh' : x < 1\ny : ℚ\nhxy : x < ↑y\nhy : ↑y < 1\n⊢ ∃ y < 1, x < ... | [] | exact ⟨y, (by norm_cast at hy), hxy⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 48
} | {
"line": 49,
"column": 49
} | [
{
"pp": "S : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx y : S\n⊢ ‖x * y‖ ≤ max ‖x‖ ‖y‖",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"Monoid.toMulOneClass",
"Sem... | [
"S : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx y : S\n⊢ ‖x * y‖ ≤ ‖x‖ ∨ ‖x * y‖ ≤ ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 38
} | {
"line": 55,
"column": 39
} | [
{
"pp": "S' : Type u_2\ninst✝ : SeminormedGroup S'\nh : ∀ (x y : S'), ‖x * y‖ ≤ max ‖x‖ ‖y‖\nx y z : S'\n⊢ dist x z ≤ max (dist x y) (dist y z)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"Di... | [
"S' : Type u_2\ninst✝ : SeminormedGroup S'\nh : ∀ (x y : S'), ‖x * y‖ ≤ max ‖x‖ ‖y‖\nx y z : S'\n⊢ ‖x⁻¹ * z‖ ≤ ‖x⁻¹ * y‖ ∨ ‖x⁻¹ * z‖ ≤ ‖y⁻¹ * z‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 13
} | {
"line": 88,
"column": 14
} | [
{
"pp": "R : Type u_4\ninst✝¹ : SeminormedAddCommGroup R\ninst✝ : IsUltrametricDist R\n⊢ IsNonarchimedean fun x ↦ ↑‖x‖₊",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Real",
"AddCommGroup.toAddCommMonoid",
"SeminormedAddGroup.toNNNorm",
"NNNorm.nnnorm",
"id"... | [
"R : Type u_4\ninst✝¹ : SeminormedAddCommGroup R\ninst✝ : IsUltrametricDist R\n⊢ IsNonarchimedean fun x ↦ ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 39
} | {
"line": 122,
"column": 40
} | [
{
"pp": "S : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx y z : S\nh : ‖x / y‖ ≠ ‖y / z‖\n⊢ ‖x / z‖ = max ‖x / y‖ ‖y / z‖",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"S : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx y z : S\nh : ‖x / y‖ ≠ ‖y / z‖\n⊢ ‖x / z‖ = max ‖x / y‖ ‖y / z‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 136,
"column": 17
} | {
"line": 136,
"column": 42
} | {
"line": 136,
"column": 43
} | [
{
"pp": "case succ\nS : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx : S\nn : ℕ\nhn : ‖x ^ n‖₊ ≤ ‖x‖₊\n⊢ ‖x ^ (n + 1)‖₊ ≤ ‖x‖₊",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"N... | [
"case succ\nS : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx : S\nn : ℕ\nhn : ‖x ^ n‖₊ ≤ ‖x‖₊\n⊢ ‖x ^ n * x‖₊ ≤ ‖x‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 13
} | {
"line": 147,
"column": 14
} | [
{
"pp": "case ofNat\nS : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx : S\na✝ : ℕ\n⊢ ‖x ^ Int.ofNat a✝‖₊ ≤ ‖x‖₊",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg",
"NNNorm.nnnorm",
"SeminormedGroup.toG... | [
"case ofNat\nS : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx : S\na✝ : ℕ\n⊢ ‖x ^ a✝‖₊ ≤ ‖x‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 13
} | {
"line": 147,
"column": 14
} | [
{
"pp": "case negSucc\nS : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx : S\na✝ : ℕ\n⊢ ‖x ^ Int.negSucc a✝‖₊ ≤ ‖x‖₊",
"ppTerm": "?negSucc",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"congrArg",
"zpow_negSucc",
"NNNorm.n... | [
"case negSucc\nS : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx : S\na✝ : ℕ\n⊢ ‖x ^ (a✝ + 1)‖₊ ≤ ‖x‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 198,
"column": 26
} | {
"line": 199,
"column": 9
} | {
"line": 199,
"column": 10
} | [
{
"pp": "M : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\n⊢ ∀ U ∈ nhds 1, ∃ V, ↑V ⊆ U",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"OpenSubgroup",
"Real",
"InvOneClass.toOne",
"... | [
"M : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\n⊢ ∀ (U : Set M), (∃ ε > 0, ball 1 ε ⊆ U) → ∃ V, ↑V ⊆ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 215,
"column": 6
} | {
"line": 215,
"column": 60
} | {
"line": 216,
"column": 6
} | [
{
"pp": "case cons.refine_1\nM : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\ns : Finset ι\nf : ι → M\nj : ι\nt : Finset ι\nhj : j ∉ t\nhs✝ : t.Nonempty\nIH : ∃ b ∈ t, ‖∏ i ∈ t, f i‖ ≤ ‖f b‖\nh : ‖∏ i ∈ t, f i‖ ≤ ‖f j‖\n⊢ ‖f j * ∏ i ∈ t, f i‖ ≤ ‖f j‖",
"ppTerm": "?con... | [
"case cons.refine_2\nM : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\ns : Finset ι\nf : ι → M\nj : ι\nt : Finset ι\nhj : j ∉ t\nhs✝ : t.Nonempty\nIH : ∃ b ∈ t, ‖∏ i ∈ t, f i‖ ≤ ‖f b‖\nh : ‖f j‖ ≤ ‖∏ i ∈ t, f i‖\n⊢ ∃ a ∈ t, ‖f j * ∏ i ∈ t, f i‖ ≤ ‖f a‖"
] | · exact (norm_mul_le_max _ _).trans (max_eq_left h).le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 78,
"column": 40
} | {
"line": 80,
"column": 14
} | {
"line": 81,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder σ\ninst✝ : WellFoundedGT σ\nφ : MvPowerSeries σ R\nd : σ →₀ ℕ\nh : ↑(toLex d) = φ.lexOrder\nhφ : φ ≠ 0\nne : (⇑toLex '' Function.support φ).Nonempty\nhφ' : toLex d = ⋯.min (⇑toLex '' Function.support φ) ne\nthis : toLex d ∈ ⇑toLex ''... | [] | by
simp only [Set.mem_image_equiv, toLex_symm_eq, ofLex_toLex, Function.mem_support, ne_eq] at this
apply this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 91
} | {
"line": 131,
"column": 92
} | [
{
"pp": "L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation L Γ₀\nhv : v.RankOne\nx : L\nhx : v.norm x = 0\n⊢ x = 0",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation L Γ₀\nhv : v.RankOne\nx : L\nhx : v.norm x = 0\n⊢ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 63
} | {
"line": 339,
"column": 64
} | [
{
"pp": "M : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\nf : ι → M\n⊢ ‖∏' (i : ι), f i‖₊ ≤ ⨆ i, ‖f i‖₊",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"NNReal.coe_iSup",
... | [
"M : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\nf : ι → M\n⊢ ‖∏' (i : ι), f i‖ ≤ ⨆ i, ‖f i‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 350,
"column": 4
} | {
"line": 350,
"column": 45
} | {
"line": 350,
"column": 46
} | [
{
"pp": "case inl\nM : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\nf : ι → M\nC : ℝ\nhC : 0 ≤ C\nh : ∀ (i : ι), ‖f i‖ ≤ C\nh✝ : IsEmpty ι\n⊢ ‖∏' (i : ι), f i‖ ≤ C",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
... | [
"case inl\nM : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\nf : ι → M\nC : ℝ\nhC : 0 ≤ C\nh : ∀ (i : ι), ‖f i‖ ≤ C\nh✝ : IsEmpty ι\n⊢ 0 ≤ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 368,
"column": 6
} | {
"line": 368,
"column": 22
} | {
"line": 368,
"column": 23
} | [
{
"pp": "M : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\nf : ι → M\na : ι\ns : Finset ι\nha : a ∉ s\nIH : ‖∏ i ∈ s, f i‖₊ = s.sup fun i ↦ ‖f i‖₊\nhs : (↑(Finset.cons a s ha)).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊\nhs' : s.Nonempty\n⊢ ∃ j ∈ s, ‖∏ i ∈ s, f i‖₊ = ‖f j‖₊",
... | [
"M : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\nf : ι → M\na : ι\ns : Finset ι\nha : a ∉ s\nIH : ‖∏ i ∈ s, f i‖₊ = s.sup fun i ↦ ‖f i‖₊\nhs : (↑(Finset.cons a s ha)).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊\nhs' : s.Nonempty\n⊢ ∃ j ∈ s, (s.sup fun i ↦ ‖f i‖₊) = ‖f j‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 34
} | {
"line": 379,
"column": 35
} | [
{
"pp": "M : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\ns : Finset ι\nf : ι → M\nhs' : s.Nonempty\nhs : (↑s).Pairwise fun i j ↦ ‖f i‖ ≠ ‖f j‖\n⊢ (↑s).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Norm.nor... | [
"M : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\ns : Finset ι\nf : ι → M\nhs' : s.Nonempty\nhs : (↑s).Pairwise fun i j ↦ ‖f i‖ ≠ ‖f j‖\n⊢ (↑s).Pairwise fun i j ↦ ¬‖f i‖ = ‖f j‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 172,
"column": 10
} | {
"line": 181,
"column": 61
} | {
"line": 182,
"column": 8
} | [
{
"pp": "case refine_1\nL : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nthis : Nonempty { ε // ε > 0 }\nU : Set (L × L)\nx✝ : ∃ i, {p | v.restrict (p.2 - p.1) < ↑i} ⊆ U\nε : (ofClass v).ValueGroup₀ˣ\nhε : {p | v.restrict (p.2 - p.1) < ... | [] | set δ : ℝ≥0 := hv.hom _ ε with hδ
have hδ_pos : 0 < δ := by
rw [hδ, ← map_zero hv.hom]
exact hv.strictMono _ (Units.zero_lt ε)
use δ, hδ_pos
apply subset_trans _ hε
intro x hx
simp only [mem_setOf_eq, Valuation.norm, hδ, NNReal.coe_lt_coe] at hx
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 172,
"column": 10
} | {
"line": 181,
"column": 61
} | {
"line": 182,
"column": 8
} | [
{
"pp": "case refine_1\nL : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nthis : Nonempty { ε // ε > 0 }\nU : Set (L × L)\nx✝ : ∃ i, {p | v.restrict (p.2 - p.1) < ↑i} ⊆ U\nε : (ofClass v).ValueGroup₀ˣ\nhε : {p | v.restrict (p.2 - p.1) < ... | [] | set δ : ℝ≥0 := hv.hom _ ε with hδ
have hδ_pos : 0 < δ := by
rw [hδ, ← map_zero hv.hom]
exact hv.strictMono _ (Units.zero_lt ε)
use δ, hδ_pos
apply subset_trans _ hε
intro x hx
simp only [mem_setOf_eq, Valuation.norm, hδ, NNReal.coe_lt_coe] at hx
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 10
} | {
"line": 151,
"column": 11
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder σ\ninst✝ : WellFoundedGT σ\nφ ψ : MvPowerSeries σ R\nd : σ →₀ ℕ\nhd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder\nu v : σ →₀ ℕ\nh : u + v = d\n⊢ φ.lexOrder ≤ ↑(toLex u) → ¬ψ.lexOrder ≤ ↑(toLex v)",
"ppTerm": "?m.139",
"assigned": t... | [
"σ : Type u_1\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder σ\ninst✝ : WellFoundedGT σ\nφ ψ : MvPowerSeries σ R\nd : σ →₀ ℕ\nhd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder\nu v : σ →₀ ℕ\nh : u + v = d\nhu : φ.lexOrder ≤ ↑(toLex u)\n⊢ ¬ψ.lexOrder ≤ ↑(toLex v)"
] | intro hu | Lean.Elab.Tactic.evalIntro | null |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 15
} | {
"line": 108,
"column": 16
} | [
{
"pp": "case mp\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nr : R\nH : (monomial n) r ∈ nonZeroDivisorsLeft (MvPowerSeries σ R)\ns : R\nhrs : r * s = 0\nthis : C s = 0\n⊢ s = 0",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nr : R\nH : (monomial n) r ∈ nonZeroDivisorsLeft (MvPowerSeries σ R)\ns : R\nhrs : r * s = 0\nthis : C s = 0\n⊢ s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 15
} | {
"line": 113,
"column": 16
} | [
{
"pp": "case mpr\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nr : R\nH : r ∈ nonZeroDivisorsLeft R\np : MvPowerSeries σ R\nhrp : (monomial n) r * p = 0\ni : σ →₀ ℕ\nthis : r * (coeff i) p = (coeff (i + n)) 0\n⊢ (coeff i) p = (coeff i) 0",
"ppTerm": "?mpr",
"assigned": true,
"usedCon... | [
"case mpr\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nr : R\nH : r ∈ nonZeroDivisorsLeft R\np : MvPowerSeries σ R\nhrp : (monomial n) r * p = 0\ni : σ →₀ ℕ\nthis : r * (coeff i) p = (coeff (i + n)) 0\n⊢ (coeff i) p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 15
} | {
"line": 121,
"column": 16
} | [
{
"pp": "case mp\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nr : R\nH : (monomial n) r ∈ nonZeroDivisorsRight (MvPowerSeries σ R)\ns : R\nhrs : s * r = 0\nthis : C s = 0\n⊢ s = 0",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
... | [
"case mp\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nr : R\nH : (monomial n) r ∈ nonZeroDivisorsRight (MvPowerSeries σ R)\ns : R\nhrs : s * r = 0\nthis : C s = 0\n⊢ s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 15
} | {
"line": 126,
"column": 16
} | [
{
"pp": "case mpr\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nr : R\nH : r ∈ nonZeroDivisorsRight R\np : MvPowerSeries σ R\nhrp : p * (monomial n) r = 0\ni : σ →₀ ℕ\nthis : (coeff i) p * r = (coeff (i + n)) 0\n⊢ (coeff i) p = (coeff i) 0",
"ppTerm": "?mpr",
"assigned": true,
"usedCo... | [
"case mpr\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nr : R\nH : r ∈ nonZeroDivisorsRight R\np : MvPowerSeries σ R\nhrp : p * (monomial n) r = 0\ni : σ →₀ ℕ\nthis : (coeff i) p * r = (coeff (i + n)) 0\n⊢ (coeff i) p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 86
} | {
"line": 144,
"column": 87
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nφ ψ : MvPowerSeries σ R\nh : φ * ψ = 0\nw✝ : LinearOrder σ\nh✝ : WellFoundedGT σ\n⊢ φ = 0 ∨ ψ = 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"MvPower... | [
"σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nφ ψ : MvPowerSeries σ R\nh : φ * ψ = 0\nw✝ : LinearOrder σ\nh✝ : WellFoundedGT σ\n⊢ φ.lexOrder = ⊤ ∨ ψ.lexOrder = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.NoZeroDivisors | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 36
} | {
"line": 65,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nh : X = 0\n⊢ False",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nh : X = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 29
} | {
"line": 174,
"column": 4
} | [
{
"pp": "case neg\nσ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nw : σ → ℕ\nf g : MvPowerSeries σ R\nhf : ¬weightedOrder w f < ⊤\n⊢ weightedOrder w (f * g) ≤ weightedOrder w f + weightedOrder w g",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Preorder.t... | [
"case neg\nσ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nw : σ → ℕ\nf g : MvPowerSeries σ R\nhf : weightedOrder w f = ⊤\n⊢ weightedOrder w (f * g) ≤ weightedOrder w f + weightedOrder w g"
] | rw [not_lt_top_iff] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 158,
"column": 8
} | {
"line": 158,
"column": 19
} | {
"line": 158,
"column": 20
} | [
{
"pp": "case inl\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : ↑u = constantCoeff φ\n⊢ constantCoeff φ = ↑u",
"ppTerm": "?inl",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inl\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : ↑u = constantCoeff φ\n⊢ constantCoeff φ = ↑u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 158,
"column": 8
} | {
"line": 158,
"column": 19
} | {
"line": 158,
"column": 20
} | [
{
"pp": "case inr\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : ↑u = 1 - constantCoeff φ\n⊢ constantCoeff (1 - φ) = ↑u",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"NonAssocSemiring.toA... | [
"case inr\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : ↑u = 1 - constantCoeff φ\n⊢ 1 - constantCoeff φ = ↑u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.