module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 35
} | {
"line": 258,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝⁶ : LinearOrder K\ninst✝⁵ : Field K\ninst✝⁴ : IsOrderedRing K\nR : Type u_2\ninst✝³ : LinearOrder R\ninst✝² : CommRing R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nf : R →+*o K\nr : R\nhr : r ≠ 0\n⊢ ArchimedeanClass.mk ↑(FiniteElement.mk (f r) ⋯) = 0",
"ppTerm": "?m.... | [] | exact mk_map_of_archimedean' f hr | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 329,
"column": 6
} | {
"line": 329,
"column": 11
} | {
"line": 330,
"column": 2
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx : K\nhx : mk x = 0\nhx' : 0 ≤ mk x⁻¹\n⊢ x ≠ 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"IsDomain.to_noZeroDivisors",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Derivation.MapCoeffs | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 43
} | {
"line": 166,
"column": 2
} | [
{
"pp": "case h\nA : Type u_1\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Differential A\nR : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : Differential R\ninst✝⁷ : Algebra A R\ninst✝⁶ : DifferentialAlgebra A R\nR' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : Differential R'\ninst✝³ : Algebra A R'\ninst✝² : DifferentialAlgebra A ... | [
"case h\nA : Type u_1\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Differential A\nR : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : Differential R\ninst✝⁷ : Algebra A R\ninst✝⁶ : DifferentialAlgebra A R\nR' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : Differential R'\ninst✝³ : Algebra A R'\ninst✝² : DifferentialAlgebra A R'\ninst✝¹ :... | conv => lhs; rw [Polynomial.aeval_algHom] | Lean.Elab.Tactic.Conv.evalConv | Lean.Parser.Tactic.Conv.conv |
Mathlib.Algebra.Polynomial.CoeffMem | {
"line": 44,
"column": 6
} | {
"line": 44,
"column": 22
} | {
"line": 45,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_2\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\np q : S[X]\nhq : q.Monic\ni : ℕ\nH₀ : ∀ (i : ℕ), p.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p)\nhpq : ¬(q.degree ≤ p.degree ∧ p ≠ 0)\n⊢ (0, p).1.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p) ∧ (0, p... | [] | simpa using H₀ _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Polynomial.CoeffMem | {
"line": 44,
"column": 6
} | {
"line": 44,
"column": 22
} | {
"line": 45,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_2\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\np q : S[X]\nhq : q.Monic\ni : ℕ\nH₀ : ∀ (i : ℕ), p.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p)\nhpq : ¬(q.degree ≤ p.degree ∧ p ≠ 0)\n⊢ (0, p).1.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p) ∧ (0, p... | [] | simpa using H₀ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.CoeffMem | {
"line": 44,
"column": 6
} | {
"line": 44,
"column": 22
} | {
"line": 45,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_2\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\np q : S[X]\nhq : q.Monic\ni : ℕ\nH₀ : ∀ (i : ℕ), p.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p)\nhpq : ¬(q.degree ≤ p.degree ∧ p ≠ 0)\n⊢ (0, p).1.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p) ∧ (0, p... | [] | simpa using H₀ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 273,
"column": 11
} | {
"line": 273,
"column": 16
} | {
"line": 274,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\ninst✝ : CommSemiring R\np✝ q✝ : R[X]\na✝¹ : swap (map C p✝) = C p✝\na✝ : swap (map C q✝) = C q✝\n⊢ swap (map C (p✝ + q✝)) = C (p✝ + q✝)",
"ppTerm": "?add",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Polynomial.eval",
"NonAssocSemiring.toA... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 273,
"column": 11
} | {
"line": 273,
"column": 16
} | {
"line": 274,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\ninst✝ : CommSemiring R\np✝ q✝ : R[X]\na✝¹ : swap (map C p✝) = C p✝\na✝ : swap (map C q✝) = C q✝\n⊢ swap (map C (p✝ + q✝)) = C (p✝ + q✝)",
"ppTerm": "?add",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Polynomial.eval",
"NonAssocSemiring.toA... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 273,
"column": 11
} | {
"line": 273,
"column": 16
} | {
"line": 274,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\ninst✝ : CommSemiring R\np✝ q✝ : R[X]\na✝¹ : swap (map C p✝) = C p✝\na✝ : swap (map C q✝) = C q✝\n⊢ swap (map C (p✝ + q✝)) = C (p✝ + q✝)",
"ppTerm": "?add",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Polynomial.eval",
"NonAssocSemiring.toA... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 289,
"column": 11
} | {
"line": 289,
"column": 16
} | {
"line": 290,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx y : A\np✝ q✝ : R[X][Y]\na✝¹ : (aevalAeval x y) (swap p✝) = (aevalAeval y x) p✝\na✝ : (aevalAeval x y) (swap q✝) = (aevalAeval y x) q✝\n⊢ (aevalAeval x y) (swap (p✝ + q✝)) = (aevalAeval y x) (p... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 289,
"column": 11
} | {
"line": 289,
"column": 16
} | {
"line": 290,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx y : A\np✝ q✝ : R[X][Y]\na✝¹ : (aevalAeval x y) (swap p✝) = (aevalAeval y x) p✝\na✝ : (aevalAeval x y) (swap q✝) = (aevalAeval y x) q✝\n⊢ (aevalAeval x y) (swap (p✝ + q✝)) = (aevalAeval y x) (p... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 289,
"column": 11
} | {
"line": 289,
"column": 16
} | {
"line": 290,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx y : A\np✝ q✝ : R[X][Y]\na✝¹ : (aevalAeval x y) (swap p✝) = (aevalAeval y x) p✝\na✝ : (aevalAeval x y) (swap q✝) = (aevalAeval y x) q✝\n⊢ (aevalAeval x y) (swap (p✝ + q✝)) = (aevalAeval y x) (p... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 298,
"column": 11
} | {
"line": 298,
"column": 16
} | {
"line": 299,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\np✝ q✝ : R[X][Y]\na✝¹ : (aeval (C x)) p✝ = (mapAlgHom (aeval x)) (swap p✝)\na✝ : (aeval (C x)) q✝ = (mapAlgHom (aeval x)) (swap q✝)\n⊢ (aeval (C x)) (p✝ + q✝) = (mapAlgHom (aeval x)) (swap... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 298,
"column": 11
} | {
"line": 298,
"column": 16
} | {
"line": 299,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\np✝ q✝ : R[X][Y]\na✝¹ : (aeval (C x)) p✝ = (mapAlgHom (aeval x)) (swap p✝)\na✝ : (aeval (C x)) q✝ = (mapAlgHom (aeval x)) (swap q✝)\n⊢ (aeval (C x)) (p✝ + q✝) = (mapAlgHom (aeval x)) (swap... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 298,
"column": 11
} | {
"line": 298,
"column": 16
} | {
"line": 299,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\np✝ q✝ : R[X][Y]\na✝¹ : (aeval (C x)) p✝ = (mapAlgHom (aeval x)) (swap p✝)\na✝ : (aeval (C x)) q✝ = (mapAlgHom (aeval x)) (swap q✝)\n⊢ (aeval (C x)) (p✝ + q✝) = (mapAlgHom (aeval x)) (swap... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 348,
"column": 19
} | {
"line": 348,
"column": 24
} | {
"line": 349,
"column": 2
} | [
{
"pp": "case add\nR : Type u_2\ninst✝ : CommRing R\np q : R[X][Y]\na✝¹ :\n (MvPolynomial.pderiv 0) ((equivMvPolynomial R) p) =\n (equivMvPolynomial R) (PolynomialModule.equivPolynomialSelf (derivative'.mapCoeffs p))\na✝ :\n (MvPolynomial.pderiv 0) ((equivMvPolynomial R) q) =\n (equivMvPolynomial R) (Po... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 348,
"column": 19
} | {
"line": 348,
"column": 24
} | {
"line": 349,
"column": 2
} | [
{
"pp": "case add\nR : Type u_2\ninst✝ : CommRing R\np q : R[X][Y]\na✝¹ :\n (MvPolynomial.pderiv 0) ((equivMvPolynomial R) p) =\n (equivMvPolynomial R) (PolynomialModule.equivPolynomialSelf (derivative'.mapCoeffs p))\na✝ :\n (MvPolynomial.pderiv 0) ((equivMvPolynomial R) q) =\n (equivMvPolynomial R) (Po... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 348,
"column": 19
} | {
"line": 348,
"column": 24
} | {
"line": 349,
"column": 2
} | [
{
"pp": "case add\nR : Type u_2\ninst✝ : CommRing R\np q : R[X][Y]\na✝¹ :\n (MvPolynomial.pderiv 0) ((equivMvPolynomial R) p) =\n (equivMvPolynomial R) (PolynomialModule.equivPolynomialSelf (derivative'.mapCoeffs p))\na✝ :\n (MvPolynomial.pderiv 0) ((equivMvPolynomial R) q) =\n (equivMvPolynomial R) (Po... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 351,
"column": 19
} | {
"line": 351,
"column": 24
} | {
"line": 352,
"column": 2
} | [
{
"pp": "case monomial.add\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\np q : R[X]\na✝¹ :\n (MvPolynomial.pderiv 0) ((equivMvPolynomial R) ((monomial n) p)) =\n (equivMvPolynomial R) (PolynomialModule.equivPolynomialSelf (derivative'.mapCoeffs ((monomial n) p)))\na✝ :\n (MvPolynomial.pderiv 0) ((equivMvPolyno... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 351,
"column": 19
} | {
"line": 351,
"column": 24
} | {
"line": 352,
"column": 2
} | [
{
"pp": "case monomial.add\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\np q : R[X]\na✝¹ :\n (MvPolynomial.pderiv 0) ((equivMvPolynomial R) ((monomial n) p)) =\n (equivMvPolynomial R) (PolynomialModule.equivPolynomialSelf (derivative'.mapCoeffs ((monomial n) p)))\na✝ :\n (MvPolynomial.pderiv 0) ((equivMvPolyno... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 351,
"column": 19
} | {
"line": 351,
"column": 24
} | {
"line": 352,
"column": 2
} | [
{
"pp": "case monomial.add\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\np q : R[X]\na✝¹ :\n (MvPolynomial.pderiv 0) ((equivMvPolynomial R) ((monomial n) p)) =\n (equivMvPolynomial R) (PolynomialModule.equivPolynomialSelf (derivative'.mapCoeffs ((monomial n) p)))\na✝ :\n (MvPolynomial.pderiv 0) ((equivMvPolyno... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 363,
"column": 19
} | {
"line": 363,
"column": 24
} | {
"line": 364,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\ninst✝ : CommSemiring R\np q : R[X][Y]\na✝¹ : (MvPolynomial.pderiv 1) ((equivMvPolynomial R) p) = (equivMvPolynomial R) (derivative p)\na✝ : (MvPolynomial.pderiv 1) ((equivMvPolynomial R) q) = (equivMvPolynomial R) (derivative q)\n⊢ (MvPolynomial.pderiv 1) ((equivMvPolynomial R) ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 363,
"column": 19
} | {
"line": 363,
"column": 24
} | {
"line": 364,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\ninst✝ : CommSemiring R\np q : R[X][Y]\na✝¹ : (MvPolynomial.pderiv 1) ((equivMvPolynomial R) p) = (equivMvPolynomial R) (derivative p)\na✝ : (MvPolynomial.pderiv 1) ((equivMvPolynomial R) q) = (equivMvPolynomial R) (derivative q)\n⊢ (MvPolynomial.pderiv 1) ((equivMvPolynomial R) ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 363,
"column": 19
} | {
"line": 363,
"column": 24
} | {
"line": 364,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\ninst✝ : CommSemiring R\np q : R[X][Y]\na✝¹ : (MvPolynomial.pderiv 1) ((equivMvPolynomial R) p) = (equivMvPolynomial R) (derivative p)\na✝ : (MvPolynomial.pderiv 1) ((equivMvPolynomial R) q) = (equivMvPolynomial R) (derivative q)\n⊢ (MvPolynomial.pderiv 1) ((equivMvPolynomial R) ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 366,
"column": 19
} | {
"line": 366,
"column": 24
} | {
"line": 367,
"column": 2
} | [
{
"pp": "case monomial.add\nR : Type u_1\ninst✝ : CommSemiring R\nn : ℕ\np q : R[X]\na✝¹ :\n (MvPolynomial.pderiv 1) ((equivMvPolynomial R) ((monomial n) p)) = (equivMvPolynomial R) (derivative ((monomial n) p))\na✝ :\n (MvPolynomial.pderiv 1) ((equivMvPolynomial R) ((monomial n) q)) = (equivMvPolynomial R) (... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 366,
"column": 19
} | {
"line": 366,
"column": 24
} | {
"line": 367,
"column": 2
} | [
{
"pp": "case monomial.add\nR : Type u_1\ninst✝ : CommSemiring R\nn : ℕ\np q : R[X]\na✝¹ :\n (MvPolynomial.pderiv 1) ((equivMvPolynomial R) ((monomial n) p)) = (equivMvPolynomial R) (derivative ((monomial n) p))\na✝ :\n (MvPolynomial.pderiv 1) ((equivMvPolynomial R) ((monomial n) q)) = (equivMvPolynomial R) (... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 366,
"column": 19
} | {
"line": 366,
"column": 24
} | {
"line": 367,
"column": 2
} | [
{
"pp": "case monomial.add\nR : Type u_1\ninst✝ : CommSemiring R\nn : ℕ\np q : R[X]\na✝¹ :\n (MvPolynomial.pderiv 1) ((equivMvPolynomial R) ((monomial n) p)) = (equivMvPolynomial R) (derivative ((monomial n) p))\na✝ :\n (MvPolynomial.pderiv 1) ((equivMvPolynomial R) ((monomial n) q)) = (equivMvPolynomial R) (... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 74,
"column": 39
} | {
"line": 82,
"column": 60
} | {
"line": 84,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np q : R[X]\nm n : ℕ\nhp : p.IsMonicOfDegree m\nhq : q.IsMonicOfDegree n\n⊢ (p * q).IsMonicOfDegree (m + n)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"Polynomial.IsMonicOfDegree.natDegree_eq",
"... | [] | by
rcases subsingleton_or_nontrivial R with H | H
· simp only [isMonicOfDegree_iff_of_subsingleton, Nat.add_eq_zero_iff] at hp hq ⊢
exact ⟨hp, hq⟩
refine ⟨?_, hp.monic.mul hq.monic⟩
have : p.leadingCoeff * q.leadingCoeff ≠ 0 := by
rw [hp.leadingCoeff_eq, hq.leadingCoeff_eq, one_mul]
exact one_ne_zer... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 85,
"column": 6
} | {
"line": 85,
"column": 20
} | {
"line": 85,
"column": 21
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nh2 : n ≤ p.natDegree\nh3 : p.natTrailingDegree ≤ n\n⊢ p.reverse.coeff (n - p.natTrailingDegree) = p.coeff (p.natDegree - (n - p.natTrailingDegree))",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"P... | [
"case pos\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nh2 : n ≤ p.natDegree\nh3 : p.natTrailingDegree ≤ n\n⊢ p.coeff ((revAt p.natDegree) (n - p.natTrailingDegree)) = p.coeff (p.natDegree - (n - p.natTrailingDegree))"
] | coeff_reverse, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 204,
"column": 6
} | {
"line": 204,
"column": 14
} | {
"line": 205,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\nf : R[X]\nh1 : ¬IsUnit f\nh2 : ∀ (k : R[X]), f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror\nh3 : IsRelPrime f f.mirror\ng h : R[X]\nfgh : f = g * h\nk : R[X] := g * h.mirror\nkey : f * f.mirror = k * k.mirror\n⊢ ... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\nf : R[X]\nh1 : ¬IsUnit f\nh2 : ∀ (k : R[X]), f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror\nh3 : IsRelPrime f f.mirror\ng h : R[X]\nfgh : f = g * h\nk : R[X] := g * h.mirror\nkey : f * f.mirror = k * k.mirror\n⊢ g ∣ g * h"
] | rw [fgh] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 207,
"column": 6
} | {
"line": 207,
"column": 14
} | {
"line": 208,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\nf : R[X]\nh1 : ¬IsUnit f\nh2 : ∀ (k : R[X]), f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror\nh3 : IsRelPrime f f.mirror\ng h : R[X]\nfgh : f = g * h\nk : R[X] := g * h.mirror\nkey : f * f.mirror = k * k.mirror\ng_... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\nf : R[X]\nh1 : ¬IsUnit f\nh2 : ∀ (k : R[X]), f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror\nh3 : IsRelPrime f f.mirror\ng h : R[X]\nfgh : f = g * h\nk : R[X] := g * h.mirror\nkey : f * f.mirror = k * k.mirror\ng_dvd_f : g ∣ ... | rw [fgh] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Homogenize | {
"line": 120,
"column": 6
} | {
"line": 120,
"column": 11
} | {
"line": 121,
"column": 4
} | [
{
"pp": "case monomial.inl\nR : Type u_1\ninst✝ : CommSemiring R\nn : ℕ\nm : Fin 2 →₀ ℕ\nk : ℕ\nc : R\nhkn : k ≤ n\nthis : (fun₀ | 0 => m 0 | 1 => m 1) = m\n⊢ (if (fun₀ | 0 => k | 1 => n - k) = m then c else 0) = if m 0 + m 1 = n then if k = m 0 then c else 0 else 0",
"ppTerm": "?monomial.inl",
"assigne... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Homogenize | {
"line": 194,
"column": 4
} | {
"line": 194,
"column": 40
} | {
"line": 195,
"column": 4
} | [
{
"pp": "case cons\nR : Type u_1\ninst✝ : CommSemiring R\nι : Type u_2\np : ι → R[X]\nn : ι → ℕ\ni : ι\ns : Finset ι\nhi : i ∉ s\nihs : (∀ i ∈ s, (p i).natDegree ≤ n i) → (∏ i ∈ s, p i).homogenize (∑ i ∈ s, n i) = ∏ i ∈ s, (p i).homogenize (n i)\nh : (p i).natDegree ≤ n i ∧ ∀ x ∈ s, (p x).natDegree ≤ n x\n⊢ (p ... | [
"case cons\nR : Type u_1\ninst✝ : CommSemiring R\nι : Type u_2\np : ι → R[X]\nn : ι → ℕ\ni : ι\ns : Finset ι\nhi : i ∉ s\nihs : (∀ i ∈ s, (p i).natDegree ≤ n i) → (∏ i ∈ s, p i).homogenize (∑ i ∈ s, n i) = ∏ i ∈ s, (p i).homogenize (n i)\nh : (p i).natDegree ≤ n i ∧ ∀ x ∈ s, (p x).natDegree ≤ n x\n⊢ (∏ i ∈ s, p i).... | rw [homogenize_mul _ _ h.1, ihs h.2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Homogenize | {
"line": 268,
"column": 66
} | {
"line": 269,
"column": 28
} | {
"line": 271,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\np : R[X]\n⊢ (MvPolynomial.aeval ![X, 1]) (p.toTupleMvPolynomial 0) = p * (MvPolynomial.aeval ![X, 1]) (p.toTupleMvPolynomial 1)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"one_pow",
"Finsupp.instAddZeroClass",
"MulOne.to... | [] | by
simp [toTupleMvPolynomial] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 199,
"column": 2
} | {
"line": 205,
"column": 18
} | {
"line": 207,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Semiring R\np : R[X]\nS : Type u_2\ninst✝³ : NonAssocSemiring S\ninst✝² : Module R S\ninst✝¹ : Pow S ℕ\ninst✝ : NatPowAssoc S\n⊢ p.smeval 0 = p.coeff 0 • 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocS... | [] | induction p using Polynomial.induction_on' with
| add p q ph qh => simp_all only [smeval_add, coeff_add, add_smul]
| monomial n a =>
cases n with
| zero => simp only [monomial_zero_left, smeval_C, npow_zero, coeff_C_zero]
| succ n => rw [coeff_monomial_succ, smeval_monomial, npow_add, npow_one, mul_zero... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 199,
"column": 2
} | {
"line": 205,
"column": 18
} | {
"line": 207,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Semiring R\np : R[X]\nS : Type u_2\ninst✝³ : NonAssocSemiring S\ninst✝² : Module R S\ninst✝¹ : Pow S ℕ\ninst✝ : NatPowAssoc S\n⊢ p.smeval 0 = p.coeff 0 • 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocS... | [] | induction p using Polynomial.induction_on' with
| add p q ph qh => simp_all only [smeval_add, coeff_add, add_smul]
| monomial n a =>
cases n with
| zero => simp only [monomial_zero_left, smeval_C, npow_zero, coeff_C_zero]
| succ n => rw [coeff_monomial_succ, smeval_monomial, npow_add, npow_one, mul_zero... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 199,
"column": 2
} | {
"line": 205,
"column": 18
} | {
"line": 207,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Semiring R\np : R[X]\nS : Type u_2\ninst✝³ : NonAssocSemiring S\ninst✝² : Module R S\ninst✝¹ : Pow S ℕ\ninst✝ : NatPowAssoc S\n⊢ p.smeval 0 = p.coeff 0 • 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocS... | [] | induction p using Polynomial.induction_on' with
| add p q ph qh => simp_all only [smeval_add, coeff_add, add_smul]
| monomial n a =>
cases n with
| zero => simp only [monomial_zero_left, smeval_C, npow_zero, coeff_C_zero]
| succ n => rw [coeff_monomial_succ, smeval_monomial, npow_add, npow_one, mul_zero... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 34
} | {
"line": 260,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : Semiring R\np : R[X]\nS : Type u_2\ninst✝⁴ : NonAssocSemiring S\ninst✝³ : Module R S\ninst✝² : Pow S ℕ\nx : S\ninst✝¹ : NatPowAssoc S\ninst✝ : IsScalarTower R S S\n⊢ (p * X ^ 0).smeval x = p.smeval x * x ^ 0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
... | [] | simp only [npow_zero, mul_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 34
} | {
"line": 260,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : Semiring R\np : R[X]\nS : Type u_2\ninst✝⁴ : NonAssocSemiring S\ninst✝³ : Module R S\ninst✝² : Pow S ℕ\nx : S\ninst✝¹ : NatPowAssoc S\ninst✝ : IsScalarTower R S S\n⊢ (p * X ^ 0).smeval x = p.smeval x * x ^ 0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
... | [] | simp only [npow_zero, mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 34
} | {
"line": 260,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : Semiring R\np : R[X]\nS : Type u_2\ninst✝⁴ : NonAssocSemiring S\ninst✝³ : Module R S\ninst✝² : Pow S ℕ\nx : S\ninst✝¹ : NatPowAssoc S\ninst✝ : IsScalarTower R S S\n⊢ (p * X ^ 0).smeval x = p.smeval x * x ^ 0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
... | [] | simp only [npow_zero, mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 131,
"column": 2
} | {
"line": 135,
"column": 70
} | {
"line": 137,
"column": 0
} | [
{
"pp": "p : ℤ[X]\nhp : p.IsUnitTrinomial\n⊢ ¬IsUnit p",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Units.val",
"Int.instIsStrictOrderedRing",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Polynomial.trinomial",
"Int.ins... | [] | obtain ⟨k, m, n, hkm, hmn, u, v, w, rfl⟩ := hp
exact fun h =>
ne_zero_of_lt hmn
((trinomial_natDegree hkm hmn w.ne_zero).symm.trans
(natDegree_eq_of_degree_eq_some (degree_eq_zero_of_isUnit h))) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 131,
"column": 2
} | {
"line": 135,
"column": 70
} | {
"line": 137,
"column": 0
} | [
{
"pp": "p : ℤ[X]\nhp : p.IsUnitTrinomial\n⊢ ¬IsUnit p",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Units.val",
"Int.instIsStrictOrderedRing",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Polynomial.trinomial",
"Int.ins... | [] | obtain ⟨k, m, n, hkm, hmn, u, v, w, rfl⟩ := hp
exact fun h =>
ne_zero_of_lt hmn
((trinomial_natDegree hkm hmn w.ne_zero).symm.trans
(natDegree_eq_of_degree_eq_some (degree_eq_zero_of_isUnit h))) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 76
} | {
"line": 160,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\np : R[X]\nq k : ℕ\nhk : q ≤ k\n⊢ ∃ gp, gp.natDegree ≤ p.natDegree - k ∧ ∀ (r : A), (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) gp",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\np : R[X]\nq k : ℕ\nhk : q ≤ k\np' : R[X]\np'_le : p'.natDegree ≤ p.natDegree - k\nhp' : (⇑derivative)^[k] p = k ! • p'\n⊢ ∃ gp, gp.natDegree ≤ p.natDegree - k ∧ ∀ (r : A), (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r... | obtain ⟨p', p'_le, hp'⟩ := exists_iterate_derivative_eq_factorial_smul p k | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 217,
"column": 2
} | {
"line": 219,
"column": 49
} | {
"line": 220,
"column": 2
} | [
{
"pp": "p : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ C ↑v * (monomial (m + n)) ↑u + C ↑v * (monomial (n - m + k + n)) ↑w =\n {\n toFinsupp :=\n AddMonoidAlgebra.ofCoeff\n (0 + 0 + 0 + (0 + 0 + Finsupp.single (m + n)... | [
"case h\np : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ n + n ∉ Set.Ioo (k + n) (n + n)",
"case h\np : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ n + (n - m + k) ∈ Set.Ioo (k... | · simp only [add_zero, zero_add, AddMonoidAlgebra.ofCoeff_add, ofFinsupp_add,
AddMonoidAlgebra.ofCoeff_single, ofFinsupp_single, C_mul_monomial, C_mul_monomial,
mul_comm (v : ℤ) w, add_comm (n - m + k) n] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 252,
"column": 28
} | {
"line": 252,
"column": 35
} | {
"line": 252,
"column": 35
} | [
{
"pp": "case inr.refine_2\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial A\ninst✝ : NoZeroDivisors A\np : R[X]\nq : ℕ\nhq : 0 < q\ninj_amap : Function.Injective ⇑(algebraMap R A)\np0 : p ≠ 0\nc : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choo... | [
"case inr.refine_2\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial A\ninst✝ : NoZeroDivisors A\np : R[X]\nq : ℕ\nhq : 0 < q\ninj_amap : Function.Injective ⇑(algebraMap R A)\np0 : p ≠ 0\nc : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0\nc... | mem_Ico | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.PresentedMonoid.Basic | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 18
} | {
"line": 110,
"column": 2
} | [
{
"pp": "case h\nα : Type u_2\nrels : FreeMonoid α → FreeMonoid α → Prop\na : FreeMonoid α\n⊢ (mk rels) a ∈ closure (range (of rels))",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"PresentedMonoid",
"CancelMonoid.toRightCancelMonoid",
"FreeMonoid",
"MonoidHom.instFun... | [] | induction a with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 239,
"column": 10
} | {
"line": 239,
"column": 30
} | {
"line": 239,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nP : R[X]\nη : R\nh₁ : 0 < P.leadingCoeff\nh₃ : P ≠ 0\nh₄ : P.eraseLead.natDegree + 1 = P.natDegree\nh₅ : X - C η ≠ 0\nh₆ : P.eraseLead ≠ 0\nd : ℕ\nhd : P.natDegree = 0 + d + 1\nh₂ : P.eraseLead.leadingCoeff = P.nextCo... | [
"R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nP : R[X]\nη : R\nh₁ : 0 < P.leadingCoeff\nh₃ : P ≠ 0\nh₄ : P.eraseLead.natDegree + 1 = P.natDegree\nh₅ : X - C η ≠ 0\nh₆ : P.eraseLead ≠ 0\nd : ℕ\nhd : P.natDegree = 0 + d + 1\nh₂ : P.eraseLead.leadingCoeff = P.nextCoeff\nn : ℕ\n... | natDegree_mul h₅ h₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 323,
"column": 6
} | {
"line": 323,
"column": 11
} | {
"line": 324,
"column": 4
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝ : Field K\na b : K\nHab : Fact (∀ (r : K), r ^ 2 ≠ a + b * r)\nz : QuadraticAlgebra K a b\nh : z.im = 0\nhz : z.re = 0\n⊢ z = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"QuadraticAlgebra.re",
"QuadraticAlgebra.ext",
"Quadratic... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 325,
"column": 34
} | {
"line": 325,
"column": 53
} | {
"line": 325,
"column": 53
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝ : Field K\na b : K\nHab : Fact (∀ (r : K), r ^ 2 ≠ a + b * r)\nz : QuadraticAlgebra K a b\nhz : z.re ^ 2 + b * z.re * z.im = a * z.im * z.im\nh : ¬z.im = 0\n⊢ False",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"QuadraticAlgebra.re",
"HM... | [
"case neg\nK : Type u_1\ninst✝ : Field K\na b : K\nHab : Fact (∀ (r : K), r ^ 2 ≠ a + b * r)\nz : QuadraticAlgebra K a b\nhz : z.re ^ 2 = a * z.im * z.im - b * z.re * z.im\nh : ¬z.im = 0\n⊢ False"
] | ← eq_sub_iff_add_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 291,
"column": 32
} | {
"line": 291,
"column": 54
} | {
"line": 291,
"column": 54
} | [
{
"pp": "case isTrue.isTrue.cons\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nP : R[X]\nη : R\nh : 0 < P.leadingCoeff\nh₂ : 0 < P.nextCoeff\nc₀ : R\ncs : List R\nhcs✝ : ((X - C η) * P).coeffList = P.leadingCoeff :: c₀ :: cs\nhecs : ((X - C η) * P.eraseLead).coeffList = ... | [
"case isTrue.isTrue.cons\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nP : R[X]\nη : R\nh : 0 < P.leadingCoeff\nh₂ : 0 < P.nextCoeff\nc₀ : R\ncs : List R\nhcs✝ : ((X - C η) * P).coeffList = P.leadingCoeff :: c₀ :: cs\nhecs : ((X - C η) * P.eraseLead).coeffList = P.nextCoeff ... | ← List.destutter_cons' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.QuaternionBasis | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 25
} | {
"line": 134,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nc₁ c₂ c₃ : R\nq : Basis A c₁ c₂ c₃\nx y : ℍ[R,c₁,c₂,c₃]\n⊢ (x * y).re • 1 + (x * y).imI • q.i + (x * y).imJ • q.j + (x * y).imK • q.k =\n (x.re * y.re) • 1 + (x.re * y.imI) • q.i + (x.re * y.imJ) • q.j + (x.re * y... | [
"R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nc₁ c₂ c₃ : R\nq : Basis A c₁ c₂ c₃\nx y : ℍ[R,c₁,c₂,c₃]\n⊢ (x * y).re • 1 + (x * y).imI • q.i + (x * y).imJ • q.j + (x * y).imK • q.k =\n (x.re * y.re) • 1 + (x.re * y.imI) • q.i + (x.re * y.imJ) • q.j + (x.re * y.imK) • q.k ... | simp only [← mul_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 51,
"column": 18
} | {
"line": 51,
"column": 23
} | {
"line": 53,
"column": 0
} | [
{
"pp": "case sq\nR : Type u_1\ninst✝¹ : AddMonoid R\ninst✝ : Mul R\ns a✝ : R\nha✝ : a✝ ≠ 0\n⊢ IsSumSq (a✝ * a✝)",
"ppTerm": "?sq",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
"AddZero.toZero",
"of_eq_true",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 51,
"column": 18
} | {
"line": 51,
"column": 23
} | {
"line": 53,
"column": 0
} | [
{
"pp": "case sq_add\nR : Type u_1\ninst✝¹ : AddMonoid R\ninst✝ : Mul R\ns a✝ s✝ : R\nha✝ : a✝ ≠ 0\nhs✝ : IsSumNonzeroSq s✝\nhs_ih✝ : IsSumSq s✝\n⊢ IsSumSq (a✝ * a✝ + s✝)",
"ppTerm": "?sq_add",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"AddMonoid.toAddZeroClass",
"AddZeroCl... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 85,
"column": 41
} | {
"line": 85,
"column": 46
} | {
"line": 85,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddMonoid R\ninst✝ : Mul R\nx : R\nhx : x ∈ {x | ∃ x_1, x_1 ≠ 0 ∧ x_1 * x_1 = x}\n⊢ x ∈ ↑(sumNonzeroSq R)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsSumNonzeroSq",
"False",
"HMul.hMul",
"AddSubsemigroup.instSet... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 85,
"column": 41
} | {
"line": 85,
"column": 46
} | {
"line": 85,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddMonoid R\ninst✝ : Mul R\nx : R\nhx : x ∈ {x | ∃ x_1, x_1 ≠ 0 ∧ x_1 * x_1 = x}\n⊢ x ∈ ↑(sumNonzeroSq R)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsSumNonzeroSq",
"False",
"HMul.hMul",
"AddSubsemigroup.instSet... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 85,
"column": 41
} | {
"line": 85,
"column": 46
} | {
"line": 85,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddMonoid R\ninst✝ : Mul R\nx : R\nhx : x ∈ {x | ∃ x_1, x_1 ≠ 0 ∧ x_1 * x_1 = x}\n⊢ x ∈ ↑(sumNonzeroSq R)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsSumNonzeroSq",
"False",
"HMul.hMul",
"AddSubsemigroup.instSet... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 88,
"column": 13
} | {
"line": 88,
"column": 18
} | {
"line": 89,
"column": 2
} | [
{
"pp": "case sq\nR : Type u_1\ninst✝¹ : AddMonoid R\ninst✝ : Mul R\nx a✝ : R\nha : a✝ ≠ 0\n⊢ a✝ * a✝ ∈ closure {x | ∃ x_1, x_1 ≠ 0 ∧ x_1 * x_1 = x}",
"ppTerm": "?sq",
"assigned": true,
"usedConstants": [
"False",
"HMul.hMul",
"AddSubsemigroup.instSetLike",
"eq_false",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 88,
"column": 13
} | {
"line": 88,
"column": 18
} | {
"line": 89,
"column": 2
} | [
{
"pp": "case sq\nR : Type u_1\ninst✝¹ : AddMonoid R\ninst✝ : Mul R\nx a✝ : R\nha : a✝ ≠ 0\n⊢ a✝ * a✝ ∈ closure {x | ∃ x_1, x_1 ≠ 0 ∧ x_1 * x_1 = x}",
"ppTerm": "?sq",
"assigned": true,
"usedConstants": [
"False",
"HMul.hMul",
"AddSubsemigroup.instSetLike",
"eq_false",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 88,
"column": 13
} | {
"line": 88,
"column": 18
} | {
"line": 89,
"column": 2
} | [
{
"pp": "case sq\nR : Type u_1\ninst✝¹ : AddMonoid R\ninst✝ : Mul R\nx a✝ : R\nha : a✝ ≠ 0\n⊢ a✝ * a✝ ∈ closure {x | ∃ x_1, x_1 ≠ 0 ∧ x_1 * x_1 = x}",
"ppTerm": "?sq",
"assigned": true,
"usedConstants": [
"False",
"HMul.hMul",
"AddSubsemigroup.instSetLike",
"eq_false",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 93,
"column": 6
} | {
"line": 93,
"column": 11
} | {
"line": 94,
"column": 4
} | [
{
"pp": "case sq_add.a\nR : Type u_1\ninst✝¹ : AddMonoid R\ninst✝ : Mul R\nx a✝ s✝ : R\nha : a✝ ≠ 0\nhs : IsSumNonzeroSq s✝\nih : s✝ ∈ closure {x | ∃ x_1, x_1 ≠ 0 ∧ x_1 * x_1 = x}\n⊢ a✝ * a✝ ∈ {x | ∃ x_1, x_1 ≠ 0 ∧ x_1 * x_1 = x}",
"ppTerm": "?sq_add.a",
"assigned": true,
"usedConstants": [
"F... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 9
} | {
"line": 96,
"column": 0
} | [
{
"pp": "case sq_add.a\nR : Type u_1\ninst✝¹ : AddMonoid R\ninst✝ : Mul R\nx a✝ s✝ : R\nha : a✝ ≠ 0\nhs : IsSumNonzeroSq s✝\nih : s✝ ∈ closure {x | ∃ x_1, x_1 ≠ 0 ∧ x_1 * x_1 = x}\n⊢ s✝ ∈ closure {x | ∃ x_1, x_1 ≠ 0 ∧ x_1 * x_1 = x}",
"ppTerm": "?sq_add.a✝",
"assigned": true,
"usedConstants": [
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 84,
"column": 77
} | {
"line": 94,
"column": 9
} | {
"line": 96,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddMonoid R\ninst✝ : Mul R\n⊢ closure {x | ∃ x_1, x_1 ≠ 0 ∧ x_1 * x_1 = x} = sumNonzeroSq R",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsSumNonzeroSq",
"False",
"HMul.hMul",
"AddSubsemigroup.instSetLike",
"... | [] | by
refine closure_eq_of_le (fun x hx ↦ by aesop) (fun x hx ↦ ?_)
-- TODO : fix aesop timeout and change to `induction hx <;> aesop`
induction hx with
| sq ha => aesop
| sq_add ha hs ih =>
-- `aesop` times out
apply add_mem
· apply AddSubsemigroup.mem_closure_of_mem
aesop
aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.CentroidHom | {
"line": 514,
"column": 2
} | {
"line": 514,
"column": 7
} | {
"line": 516,
"column": 0
} | [
{
"pp": "α : Type u_5\ninst✝ : NonUnitalNonAssocCommSemiring α\na : α\n⊢ (L a = L a ∧ ∀ g ∈ Set.range ⇑L, g * L a = L a * g) ↔ ∀ (b : α), Commute (L b) (L a)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"AddMonoid.End.mulLeft",
"Eq.mpr",
"HMul.hMul",
"congrArg",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.SkewMonoidAlgebra.Lift | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 7
} | {
"line": 204,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nA : Type u_4\ninst✝⁵ : Monoid G\ninst✝⁴ : Semiring A\ninst✝³ : CommSemiring k\ninst✝² : Algebra k A\ninst✝¹ : MulSemiringAction G A\ninst✝ : SMulCommClass G k A\n⊢ ∀ (a : SkewMonoidAlgebra A G), (domCongrAlg k A ⋯) a = AlgEquiv.refl a",
"ppTerm": "?m.44",
"assigned":... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 123,
"column": 58
} | {
"line": 123,
"column": 75
} | {
"line": 125,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : SkewPolynomial R\n⊢ p.support = ∅ ↔ p = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Equiv.instEquivLike",
"congrArg",
"Finset",
"AddMonoid.toAddZeroClass",
"... | [] | by simp [support] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 213,
"column": 40
} | {
"line": 213,
"column": 45
} | {
"line": 215,
"column": 0
} | [
{
"pp": "case h0\nR : Type u_1\ninst✝ : Semiring R\nmotive : SkewPolynomial R → Prop\np : SkewPolynomial R\nh0 : motive 0\nha : ∀ (n : ℕ) (r : R) (q : SkewPolynomial R), n ∉ q.support → r ≠ 0 → motive q → motive ((monomial n) r + q)\n⊢ motive 0",
"ppTerm": "?h0",
"assigned": true,
"usedConstants": [... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 213,
"column": 40
} | {
"line": 213,
"column": 45
} | {
"line": 215,
"column": 0
} | [
{
"pp": "case ha\nR : Type u_1\ninst✝ : Semiring R\nmotive : SkewPolynomial R → Prop\np : SkewPolynomial R\nh0 : motive 0\nha : ∀ (n : ℕ) (r : R) (q : SkewPolynomial R), n ∉ q.support → r ≠ 0 → motive q → motive ((monomial n) r + q)\n⊢ ∀ (a : Multiplicative ℕ) (b : R) (f : SkewMonoidAlgebra R (Multiplicative ℕ)... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Star.CHSH | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 96
} | {
"line": 180,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : Algebra ℝ R\ninst✝¹ : IsOrderedModule ℝ R\ninst✝ : StarModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nM : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x\nP : R := (√2)⁻¹ • (A₁... | [
"R : Type u\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : Algebra ℝ R\ninst✝¹ : IsOrderedModule ℝ R\ninst✝ : StarModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nM : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x\nP : R := (√2)⁻¹ • (A₁ + A₀) - B₀\... | simp only [Algebra.mul_smul_comm, Algebra.smul_mul_assoc, ← mul_smul, sqrt_two_inv_mul_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Star.CHSH | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 60
} | {
"line": 182,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : Algebra ℝ R\ninst✝¹ : IsOrderedModule ℝ R\ninst✝ : StarModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nM : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x\nP : R := (√2)⁻¹ • (A₁... | [
"R : Type u\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : Algebra ℝ R\ninst✝¹ : IsOrderedModule ℝ R\ninst✝ : StarModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nM : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x\nP : R := (√2)⁻¹ • (A₁ + A₀) - B₀\... | simp only [← sq, T.A₀_inv, T.A₁_inv, T.B₀_inv, T.B₁_inv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 508,
"column": 38
} | {
"line": 513,
"column": 47
} | {
"line": 515,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\na : R\nn : ℕ\ninst✝ : MulSemiringAction (Multiplicative ℕ) R\n⊢ a • X ^ n = (monomial n) a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"Trans.trans",
... | [] | by
rw [eq_comm]
calc monomial n a = monomial n (a * 1) := by simp only [mul_one]
_ = monomial n (a • 1) := by simp [mul_one, smul_eq_mul]
_ = a • monomial n 1 := (SkewMonoidAlgebra.smul_single _ _ _).symm
_ = a • X ^ n := by rw [X_pow_eq_monomial] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 704,
"column": 2
} | {
"line": 704,
"column": 32
} | {
"line": 706,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : SkewPolynomial R\nn : ℕ\na : R\n⊢ (p.update n a).coeff = Function.update p.coeff n a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Function.update",
"Equiv.instEquivLike",
"cong... | [] | ext; simp [coeff, update]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 704,
"column": 2
} | {
"line": 704,
"column": 32
} | {
"line": 706,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : SkewPolynomial R\nn : ℕ\na : R\n⊢ (p.update n a).coeff = Function.update p.coeff n a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Function.update",
"Equiv.instEquivLike",
"cong... | [] | ext; simp [coeff, update]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 539,
"column": 47
} | {
"line": 541,
"column": 31
} | {
"line": 543,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : AddCommMonoid k\nG' : Type u_3\nf : G → G'\nv : SkewMonoidAlgebra k G\nR : Type u_5\ninst✝¹ : Monoid R\ninst✝ : DistribMulAction R k\nb : R\n⊢ (mapDomain f) (b • v) = b • (mapDomain f) v",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq... | [] | by
simp_rw [← coeff_inj, coeff_smul, coeff_mapDomain]
simp [Finsupp.mapDomain_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 918,
"column": 4
} | {
"line": 918,
"column": 9
} | {
"line": 919,
"column": 2
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\n⊢ (Function.support fun p ↦ f.coeff p.1 * p.1 • g.coeff p.2) ⊆ ↑(f.support.product g.support)",
"ppTerm": "?m.115",
"assigned": true,
"usedConstants": [
"Set... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 931,
"column": 50
} | {
"line": 931,
"column": 55
} | {
"line": 932,
"column": 4
} | [
{
"pp": "case h₁\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\nthis : ({p | p.1 * p.2 = x} ∩ Function.support fun p ↦ f.coeff p.1 * p.1 • g.coeff p.2).Finite\ns : Finset (G × G) := ⋯\nF : G × G → k := ⋯\n⊢ ∀ a ∈ {p ∈ f.support ×ˢ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 931,
"column": 50
} | {
"line": 931,
"column": 55
} | {
"line": 932,
"column": 4
} | [
{
"pp": "case h₂\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\nthis : ({p | p.1 * p.2 = x} ∩ Function.support fun p ↦ f.coeff p.1 * p.1 • g.coeff p.2).Finite\ns : Finset (G × G) := ⋯\nF : G × G → k := ⋯\n⊢ ∀ a ∈ {p ∈ s | p.1 ∈ f.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 931,
"column": 50
} | {
"line": 931,
"column": 55
} | {
"line": 932,
"column": 4
} | [
{
"pp": "case h\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\nthis : ({p | p.1 * p.2 = x} ∩ Function.support fun p ↦ f.coeff p.1 * p.1 • g.coeff p.2).Finite\ns : Finset (G × G) := ⋯\nF : G × G → k := ⋯\n⊢ ∀ a ∈ {p ∈ f.support ×ˢ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Tropical.Basic | {
"line": 356,
"column": 53
} | {
"line": 356,
"column": 72
} | {
"line": 356,
"column": 72
} | [
{
"pp": "R : Type u\ninst✝² : LinearOrder R\ninst✝¹ : OrderTop R\ninst✝ : Zero R\nn : ℕ\n⊢ untrop (if n + 1 = 0 then 0 else 1) = untrop ((if n = 0 then 0 else 1) + 1)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"False",
"AddMonoid.toAddSemigroup",
"Tropical.instAddCom... | [] | by cases n <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.RingedSpace.Stalks | {
"line": 92,
"column": 2
} | {
"line": 93,
"column": 43
} | {
"line": 94,
"column": 2
} | [
{
"pp": "case op\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nU : TopCat\nX : PresheafedSpace C\nf : U ⟶ ↑X\nh : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nx : ↑U\nV : OpenNhds ((ConcreteCategory.hom f) x)\n⊢ colimit.ι\n (((whiskeringLeft (OpenNhds ((ConcreteCategory.hom f) x))ᵒᵖ (Open... | [
"case op\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nU : TopCat\nX : PresheafedSpace C\nf : U ⟶ ↑X\nh : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nx : ↑U\nV : OpenNhds ((ConcreteCategory.hom f) x)\ni : (h.functorNhds x).obj ((OpenNhds.map f x).obj V) ⟶ V := homOfLE ⋯\n⊢ colimit.ι\n (((whi... | let i : (h.functorNhds x).obj ((OpenNhds.map f x).obj V) ⟶ V :=
homOfLE (Set.image_preimage_subset f _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Geometry.RingedSpace.Stalks | {
"line": 96,
"column": 2
} | {
"line": 97,
"column": 27
} | {
"line": 98,
"column": 2
} | [
{
"pp": "case op\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nU : TopCat\nX : PresheafedSpace C\nf : U ⟶ ↑X\nh : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nx : ↑U\nV : OpenNhds ((ConcreteCategory.hom f) x)\ni : (h.functorNhds x).obj ((OpenNhds.map f x).obj V) ⟶ V := homOfLE ⋯\n⊢ colimit.ι\n ... | [
"case op\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nU : TopCat\nX : PresheafedSpace C\nf : U ⟶ ↑X\nh : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nx : ↑U\nV : OpenNhds ((ConcreteCategory.hom f) x)\ni : (h.functorNhds x).obj ((OpenNhds.map f x).obj V) ⟶ V := homOfLE ⋯\n⊢ colimit.ι\n (((whisk... | erw [colimit.ι_pre ((OpenNhds.inclusion (f x)).op ⋙ X.presheaf)
(h.functorNhds x).op] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.Geometry.RingedSpace.SheafedSpace | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 21
} | {
"line": 254,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (Cate... | [
"C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.f... | obtain ⟨f, fc⟩ := f | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Geometry.RingedSpace.PresheafedSpace | {
"line": 182,
"column": 74
} | {
"line": 184,
"column": 6
} | {
"line": 186,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : PresheafedSpace C\nα β : X ⟶ Y\nh : α = β\nU : (Opens ↑↑Y)ᵒᵖ\n⊢ α.c.app U = β.c.app U ≫ X.presheaf.map (eqToHom ⋯)",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor.op",
"Opposite",
"Algebra... | [] | by
subst h
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 20
} | {
"line": 174,
"column": 4
} | [
{
"pp": "case mp\nX : RingedSpace\nU V : (Opens ↑↑X.toPresheafedSpace)ᵒᵖ\ni : U ⟶ V\nf : ↑(X.presheaf.obj U)\nx : ↑↑X.toPresheafedSpace\n⊢ x ∈ ↑(X.basicOpen ((ConcreteCategory.hom (X.presheaf.map i)) f)) → x ∈ ↑(unop V ⊓ X.basicOpen f)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"A... | [
"case mp\nX : RingedSpace\nU V : (Opens ↑↑X.toPresheafedSpace)ᵒᵖ\ni : U ⟶ V\nf : ↑(X.presheaf.obj U)\nx : ↑↑X.toPresheafedSpace\nhxV : x ∈ unop V\nhx : IsUnit ((ConcreteCategory.hom (X.presheaf.germ (unop V) x hxV)) ((ConcreteCategory.hom (X.presheaf.map i)) f))\n⊢ x ∈ ↑(unop V ⊓ X.basicOpen f)"
] | rintro ⟨hxV, hx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 207,
"column": 17
} | {
"line": 207,
"column": 39
} | {
"line": 209,
"column": 0
} | [
{
"pp": "case succ\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nn : ℕ\nhn : 0 < n + Nat.succ 0 → X.basicOpen (f ^ (n + Nat.succ 0)) = X.basicOpen f\nh : 0 < n + 1 + Nat.succ 0\n⊢ X.basicOpen (f ^ (n + 1 + Nat.succ 0)) = X.basicOpen f",
"ppTerm": "?succ",
"assigned": t... | [] | rw [pow_add]; simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 207,
"column": 17
} | {
"line": 207,
"column": 39
} | {
"line": 209,
"column": 0
} | [
{
"pp": "case succ\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nn : ℕ\nhn : 0 < n + Nat.succ 0 → X.basicOpen (f ^ (n + Nat.succ 0)) = X.basicOpen f\nh : 0 < n + 1 + Nat.succ 0\n⊢ X.basicOpen (f ^ (n + 1 + Nat.succ 0)) = X.basicOpen f",
"ppTerm": "?succ",
"assigned": t... | [] | rw [pow_add]; simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 26
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case a\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nhf : IsUnit f\n⊢ X.basicOpen f ≤ U",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.RingedSpace.basicOpen_le"
],
"usedFVars": [
"X",
"U",
"f"
... | [] | exact X.basicOpen_le f | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 26
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case a\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nhf : IsUnit f\n⊢ X.basicOpen f ≤ U",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.RingedSpace.basicOpen_le"
],
"usedFVars": [
"X",
"U",
"f"
... | [] | exact X.basicOpen_le f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 26
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case a\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nhf : IsUnit f\n⊢ X.basicOpen f ≤ U",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.RingedSpace.basicOpen_le"
],
"usedFVars": [
"X",
"U",
"f"
... | [] | exact X.basicOpen_le f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits | {
"line": 314,
"column": 4
} | {
"line": 315,
"column": 28
} | {
"line": 316,
"column": 4
} | [
{
"pp": "case app\nJ : Type u'\ninst✝⁵ : Category.{v', u'} J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasColimitsOfShape J TopCat\ninst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝¹ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ninst✝ : HasColimit F\nU : Opens ↑↑(Limits.colimit ... | [
"case app\nJ : Type u'\ninst✝⁵ : Category.{v', u'} J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasColimitsOfShape J TopCat\ninst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝¹ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ninst✝ : HasColimit F\nU : Opens ↑↑(Limits.colimit F)\nX : Jᵒᵖ\... | simp only [Functor.op_obj, op_inj_iff, Opens.map_coe, SetLike.ext'_iff,
Set.preimage_preimage] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits | {
"line": 326,
"column": 6
} | {
"line": 326,
"column": 42
} | {
"line": 326,
"column": 42
} | [
{
"pp": "case naturality\nJ : Type u'\ninst✝⁵ : Category.{v', u'} J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasColimitsOfShape J TopCat\ninst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝¹ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ninst✝ : HasColimit F\nU : Opens ↑↑(Limits.c... | [
"case naturality\nJ : Type u'\ninst✝⁵ : Category.{v', u'} J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasColimitsOfShape J TopCat\ninst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝¹ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ninst✝ : HasColimit F\nU : Opens ↑↑(Limits.colimit F)\nX... | ← (F.obj (unop Y)).presheaf.map_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Spec | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 11
} | {
"line": 174,
"column": 2
} | [
{
"pp": "case w\nX : RingedSpace\nR : CommRingCat\nα β : X ⟶ sheafedSpaceObj R\nw : α.hom.base = β.hom.base\nh :\n ∀ (r : ↑R),\n let U := PrimeSpectrum.basicOpen r;\n (CommRingCat.ofHom (algebraMap (↑R) ((structureSheafInType ↑R ↑R).obj.obj (op U))) ≫ α.hom.c.app (op U)) ≫\n X.presheaf.map (eqToHo... | [] | exact w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Spec | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 11
} | {
"line": 174,
"column": 2
} | [
{
"pp": "case w\nX : RingedSpace\nR : CommRingCat\nα β : X ⟶ sheafedSpaceObj R\nw : α.hom.base = β.hom.base\nh :\n ∀ (r : ↑R),\n let U := PrimeSpectrum.basicOpen r;\n (CommRingCat.ofHom (algebraMap (↑R) ((structureSheafInType ↑R ↑R).obj.obj (op U))) ≫ α.hom.c.app (op U)) ≫\n X.presheaf.map (eqToHo... | [] | exact w | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Spec | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 11
} | {
"line": 174,
"column": 2
} | [
{
"pp": "case w\nX : RingedSpace\nR : CommRingCat\nα β : X ⟶ sheafedSpaceObj R\nw : α.hom.base = β.hom.base\nh :\n ∀ (r : ↑R),\n let U := PrimeSpectrum.basicOpen r;\n (CommRingCat.ofHom (algebraMap (↑R) ((structureSheafInType ↑R ↑R).obj.obj (op U))) ≫ α.hom.c.app (op U)) ≫\n X.presheaf.map (eqToHo... | [] | exact w | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 398,
"column": 2
} | {
"line": 400,
"column": 5
} | {
"line": 402,
"column": 0
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\nU : Z.Opens\nV : Y.Opens\nW : X.Opens\ne₁ : V ≤ g ⁻¹ᵁ U\ne₂ : W ≤ f ⁻¹ᵁ V\n⊢ appLE g U V e₁ ≫ appLE f V W e₂ = appLE (f ≫ g) U W ⋯",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"Al... | [] | dsimp [Hom.appLE]
rw [Category.assoc, f.naturality_assoc, ← Functor.map_comp]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 398,
"column": 2
} | {
"line": 400,
"column": 5
} | {
"line": 402,
"column": 0
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\nU : Z.Opens\nV : Y.Opens\nW : X.Opens\ne₁ : V ≤ g ⁻¹ᵁ U\ne₂ : W ≤ f ⁻¹ᵁ V\n⊢ appLE g U V e₁ ≫ appLE f V W e₂ = appLE (f ≫ g) U W ⋯",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"Al... | [] | dsimp [Hom.appLE]
rw [Category.assoc, f.naturality_assoc, ← Functor.map_comp]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 7
} | {
"line": 416,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU V : Y.Opens\ne : U = V\n⊢ app f U = Y.presheaf.map (eqToHom ⋯).op ≫ app f V ≫ X.presheaf.map (eqToHom ⋯).op",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Opposite",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 7
} | {
"line": 416,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU V : Y.Opens\ne : U = V\n⊢ app f U = Y.presheaf.map (eqToHom ⋯).op ≫ app f V ≫ X.presheaf.map (eqToHom ⋯).op",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Opposite",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 7
} | {
"line": 416,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU V : Y.Opens\ne : U = V\n⊢ app f U = Y.presheaf.map (eqToHom ⋯).op ≫ app f V ≫ X.presheaf.map (eqToHom ⋯).op",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Opposite",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Sites.MorphismProperty | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 7
} | {
"line": 94,
"column": 0
} | [
{
"pp": "P : MorphismProperty Scheme\nX S : Scheme\nf : X ⟶ S\n⊢ ((∀ (x : ↥S), ∃ i, x ∈ Set.range ⇑f) ∧ ∀ (i : PUnit.{1}), P f) ↔ Function.Surjective ⇑f ∧ P f",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Cover.MorphismProperty | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 54
} | {
"line": 171,
"column": 4
} | [
{
"pp": "K : Precoverage Scheme\nX✝ Y✝ Z : Scheme\n𝒰✝ : Cover K X✝\nf✝ : X✝ ⟶ Z\ng : Y✝ ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.I₀), HasPullback (𝒰✝.f x ≫ f✝) g\nP Q : MorphismProperty Scheme\nX Y : Scheme\n𝒰 : Cover (precoverage P) X\nf : Y ⟶ X\nhf : P f\n⊢ (∀ (x : ↥X), ∃ i, x ∈ Set.range ⇑((𝒰.add f).f i)) ∧ ∀ (i : (𝒰.ad... | [
"K : Precoverage Scheme\nX✝ Y✝ Z : Scheme\n𝒰✝ : Cover K X✝\nf✝ : X✝ ⟶ Z\ng : Y✝ ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.I₀), HasPullback (𝒰✝.f x ≫ f✝) g\nP Q : MorphismProperty Scheme\nX Y : Scheme\n𝒰 : Cover (precoverage P) X\nf : Y ⟶ X\nhf : P f\n⊢ ∀ (i : (𝒰.add f).I₀), P ((𝒰.add f).f i)"
] | refine ⟨fun x ↦ ⟨some <| 𝒰.idx x, 𝒰.covers x⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
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