module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.EssentialFiniteness | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 47
} | {
"line": 109,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nM : Submonoid R\ninst✝ : IsLocalization M S\ns : S\n⊢ ∃ a, IsUnit ((algebraMap R S) a) ∧ ∃ y, (algebraMap R S) y = s * (algebraMap R S) a",
"ppTerm": "?h",
"assigned": true,
"usedConstants": ... | [
"case h\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nM : Submonoid R\ninst✝ : IsLocalization M S\ns : S\nx : R\nt : ↥M\ne : s * (algebraMap R S) ↑(x, t).2 = (algebraMap R S) (x, t).1\n⊢ ∃ a, IsUnit ((algebraMap R S) a) ∧ ∃ y, (algebraMap R S) y = s * (algebraMap R S) ... | obtain ⟨⟨x, t⟩, e⟩ := IsLocalization.surj M s | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 64,
"column": 8
} | {
"line": 64,
"column": 35
} | {
"line": 64,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nhze : 0 = 1\n⊢ False",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne"... | [
"R : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nhze : 0 = (algebraMap R S) 1\n⊢ False"
] | ← (algebraMap R S).map_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 120,
"column": 39
} | {
"line": 120,
"column": 44
} | {
"line": 122,
"column": 0
} | [
{
"pp": "S : Type u_2\ninst✝⁴ : CommSemiring S\nR : Type u_4\ninst✝³ : CommRing R\ninst✝² : NoZeroDivisors R\ninst✝¹ : Algebra R S\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nx✝ : ↥P.primeCompl\nval✝ : R\nh : val✝ ∈ P.primeCompl\n⊢ ↑⟨val✝, h⟩ ≠ 0",
"ppTerm": "?m.46",
"assigned": tr... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 120,
"column": 39
} | {
"line": 120,
"column": 44
} | {
"line": 122,
"column": 0
} | [
{
"pp": "S : Type u_2\ninst✝⁴ : CommSemiring S\nR : Type u_4\ninst✝³ : CommRing R\ninst✝² : NoZeroDivisors R\ninst✝¹ : Algebra R S\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nx✝ : ↥P.primeCompl\nval✝ : R\nh : val✝ ∈ P.primeCompl\n⊢ ↑⟨val✝, h⟩ ≠ 0",
"ppTerm": "?m.46",
"assigned": tr... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 120,
"column": 39
} | {
"line": 120,
"column": 44
} | {
"line": 122,
"column": 0
} | [
{
"pp": "S : Type u_2\ninst✝⁴ : CommSemiring S\nR : Type u_4\ninst✝³ : CommRing R\ninst✝² : NoZeroDivisors R\ninst✝¹ : Algebra R S\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nx✝ : ↥P.primeCompl\nval✝ : R\nh : val✝ ∈ P.primeCompl\n⊢ ↑⟨val✝, h⟩ ≠ 0",
"ppTerm": "?m.46",
"assigned": tr... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 259,
"column": 56
} | {
"line": 259,
"column": 61
} | {
"line": 259,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\nI : Ideal R\nhI : I.IsPrime\nJ : Ideal P\ninst✝ : J.IsPrime\nf : R →+* P\nhIJ : I = Ideal.comap f J\nx : R\ny : ↥I.primeCompl\n⊢ f ↑y ∈ J.primeCompl",
"ppTerm":... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 259,
"column": 56
} | {
"line": 259,
"column": 61
} | {
"line": 259,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\nI : Ideal R\nhI : I.IsPrime\nJ : Ideal P\ninst✝ : J.IsPrime\nf : R →+* P\nhIJ : I = Ideal.comap f J\nx : R\ny : ↥I.primeCompl\n⊢ f ↑y ∈ J.primeCompl",
"ppTerm":... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 259,
"column": 56
} | {
"line": 259,
"column": 61
} | {
"line": 259,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\nI : Ideal R\nhI : I.IsPrime\nJ : Ideal P\ninst✝ : J.IsPrime\nf : R →+* P\nhIJ : I = Ideal.comap f J\nx : R\ny : ↥I.primeCompl\n⊢ f ↑y ∈ J.primeCompl",
"ppTerm":... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 340,
"column": 50
} | {
"line": 340,
"column": 72
} | {
"line": 340,
"column": 72
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Ideal S\ninst✝² : P.IsPrime\ns : Subalgebra R S\nH : s.saturation (P.primeCompl ⊓ s.toSubmonoid) ⋯ = ⊤\np : Ideal ↥s\ninst✝¹ : p.IsPrime\ninst✝ : P.LiesOver p\na : S\na✝⁵ : a ∈ s\nb : S\na✝⁴ : b ∈ s\... | [] | by simp [mul_assoc, e] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 333,
"column": 2
} | {
"line": 340,
"column": 73
} | {
"line": 341,
"column": 2
} | [
{
"pp": "case left\nR : Type u_1\ninst✝⁵ : CommSemiring R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Ideal S\ninst✝² : P.IsPrime\ns : Subalgebra R S\nH : s.saturation (P.primeCompl ⊓ s.toSubmonoid) ⋯ = ⊤\np : Ideal ↥s\ninst✝¹ : p.IsPrime\ninst✝ : P.LiesOver p\n⊢ Function.Injective ⇑(local... | [
"case right\nR : Type u_1\ninst✝⁵ : CommSemiring R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Ideal S\ninst✝² : P.IsPrime\ns : Subalgebra R S\nH : s.saturation (P.primeCompl ⊓ s.toSubmonoid) ⋯ = ⊤\np : Ideal ↥s\ninst✝¹ : p.IsPrime\ninst✝ : P.LiesOver p\n⊢ Function.Surjective ⇑(localRingHom p ... | · suffices ∀ a ∈ s, ∀ b ∈ s, b ∉ P → ∀ c ∈ s, ∀ d ∈ s, d ∉ P → ∀ x ∉ P,
x * (a * d) = x * (c * b) → ∃ a_6 ∉ P, a_6 ∈ s ∧ a_6 * (a * d) = a_6 * (c * b) by
simpa [Function.Injective, (IsLocalization.mk'_surjective p.primeCompl).forall, P.over_def p,
Localization.localRingHom_mk', IsLocalization.mk'_... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 590,
"column": 2
} | {
"line": 593,
"column": 33
} | {
"line": 595,
"column": 0
} | [
{
"pp": "R : Type u_7\ninst✝⁵ : CommRing R\np : Ideal R\ninst✝⁴ : p.IsMaximal\nRₚ : Type u_8\ninst✝³ : CommRing Rₚ\ninst✝² : Algebra R Rₚ\ninst✝¹ : IsLocalization.AtPrime Rₚ p\ninst✝ : IsLocalRing Rₚ\nx : R\ns : ↥p.primeCompl\nh₁ : (Ideal.Quotient.mk p) ↑s ≠ 0\nh₂ : (equivQuotMaximalIdeal p Rₚ) ((Ideal.Quotient... | [] | rw [RingEquiv.symm_apply_eq, ← mul_left_inj' h₂, map_mul, mul_assoc, ← map_mul,
inv_mul_cancel₀ h₁, map_one, mul_one, equivQuotMaximalIdeal_apply_mk, ← map_mul,
mk'_spec, Ideal.Quotient.mk_algebraMap, equivQuotMaximalIdeal_apply_mk,
Ideal.Quotient.mk_algebraMap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.LocalProperties.Basic | {
"line": 198,
"column": 4
} | {
"line": 198,
"column": 20
} | {
"line": 199,
"column": 4
} | [
{
"pp": "case mpr\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\na✝ b✝ : Type u\nc✝ : CommRing a✝\nd✝ : CommRing b✝\ne✝ : a✝ →+* b✝\n⊢ (∀ (s : Finset a✝), Ideal.span ↑s = ⊤ → (∀ (r : ↥s), P (Localization.awayMap e✝ ↑r)) → P e✝) →\n ∀ (s : Set a✝), Ideal.span s = ⊤ → (∀ ... | [
"case mpr\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\na✝ b✝ : Type u\nc✝ : CommRing a✝\nd✝ : CommRing b✝\ne✝ : a✝ →+* b✝\nh : ∀ (s : Finset a✝), Ideal.span ↑s = ⊤ → (∀ (r : ↥s), P (Localization.awayMap e✝ ↑r)) → P e✝\ns : Set a✝\nhs : Ideal.span s = ⊤\nhs' : ∀ (r : ↑s), P (... | intro h s hs hs' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.LocalProperties.Basic | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 20
} | {
"line": 211,
"column": 4
} | [
{
"pp": "case mpr\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\na✝ b✝ : Type u\nc✝ : CommRing a✝\nd✝ : CommRing b✝\ne✝ : a✝ →+* b✝\n⊢ (∀ (s : Finset b✝), Ideal.span ↑s = ⊤ → (∀ (r : ↥s), P ((algebraMap b✝ (Localization.Away ↑r)).comp e✝)) → P e✝) →\n ∀ (s : Set b✝), Id... | [
"case mpr\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\na✝ b✝ : Type u\nc✝ : CommRing a✝\nd✝ : CommRing b✝\ne✝ : a✝ →+* b✝\nh : ∀ (s : Finset b✝), Ideal.span ↑s = ⊤ → (∀ (r : ↥s), P ((algebraMap b✝ (Localization.Away ↑r)).comp e✝)) → P e✝\ns : Set b✝\nhs : Ideal.span s = ⊤\nh... | intro h s hs hs' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.LocalProperties.Basic | {
"line": 291,
"column": 2
} | {
"line": 292,
"column": 48
} | {
"line": 293,
"column": 2
} | [
{
"pp": "R S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nH : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] ↦ P\nhf : Function.Bijective ⇑f\nthis✝ : Algebra R S := f.toAlgebra\nthis : IsLocaliz... | [
"R S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nH : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] ↦ P\nhf : Function.Bijective ⇑f\nthis✝¹ : Algebra R S := f.toAlgebra\nthis✝ : IsLocalization (Sub... | have := IsLocalization.isLocalization_of_algEquiv (.powers (1 : R))
(AlgEquiv.ofBijective (Algebra.ofId R S) hf) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.LocalProperties.Basic | {
"line": 559,
"column": 2
} | {
"line": 559,
"column": 55
} | {
"line": 560,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nx : R\nhx : x ∈ ⨅ p, ⨅ (x : p.IsPrime), ⨅ (_ : I ≤ p), RingHom.ker (algebraMap R (Localization.AtPrime p))\nm : Ideal R\nhm : m.IsMaximal\n⊢ (algebraMap R (Localization.AtPrime m)) x ∈ map (algebraMap R (Localization.AtPrime m)) I",
"ppTerm": "?m.5... | [
"R : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nx : R\nm : Ideal R\nhm : m.IsMaximal\nhx : ∀ (i : Ideal R) (i_1 : i.IsPrime), I ≤ i → (algebraMap R (Localization.AtPrime i)) x = 0\n⊢ (algebraMap R (Localization.AtPrime m)) x ∈ map (algebraMap R (Localization.AtPrime m)) I"
] | simp only [Submodule.mem_iInf, RingHom.mem_ker] at hx | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 148,
"column": 24
} | {
"line": 148,
"column": 29
} | {
"line": 150,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nt : R\nn : ℤ\n⊢ (Finsupp.single 0 t) n = if n = 0 then t else 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"False",
"Decidable.casesOn",
"Finsupp.single_eq_same",
"eq_false"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 346,
"column": 6
} | {
"line": 346,
"column": 48
} | {
"line": 347,
"column": 4
} | [
{
"pp": "case refine_2.ofNat\nR : Type u_1\ninst✝ : Semiring R\na : R\nn : ℕ\n⊢ ∃ n_1 f', toLaurent f' = C a * T (Int.ofNat n) * T ↑n_1",
"ppTerm": "?refine_2.ofNat",
"assigned": true,
"usedConstants": [
"LaurentPolynomial.T",
"CharP.cast_eq_zero",
"Polynomial.C",
"AddMonoidA... | [] | exact ⟨0, Polynomial.C a * X ^ n, by simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 346,
"column": 6
} | {
"line": 346,
"column": 48
} | {
"line": 347,
"column": 4
} | [
{
"pp": "case refine_2.ofNat\nR : Type u_1\ninst✝ : Semiring R\na : R\nn : ℕ\n⊢ ∃ n_1 f', toLaurent f' = C a * T (Int.ofNat n) * T ↑n_1",
"ppTerm": "?refine_2.ofNat",
"assigned": true,
"usedConstants": [
"LaurentPolynomial.T",
"CharP.cast_eq_zero",
"Polynomial.C",
"AddMonoidA... | [] | exact ⟨0, Polynomial.C a * X ^ n, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 346,
"column": 6
} | {
"line": 346,
"column": 48
} | {
"line": 347,
"column": 4
} | [
{
"pp": "case refine_2.ofNat\nR : Type u_1\ninst✝ : Semiring R\na : R\nn : ℕ\n⊢ ∃ n_1 f', toLaurent f' = C a * T (Int.ofNat n) * T ↑n_1",
"ppTerm": "?refine_2.ofNat",
"assigned": true,
"usedConstants": [
"LaurentPolynomial.T",
"CharP.cast_eq_zero",
"Polynomial.C",
"AddMonoidA... | [] | exact ⟨0, Polynomial.C a * X ^ n, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 215,
"column": 27
} | {
"line": 215,
"column": 44
} | {
"line": 215,
"column": 44
} | [
{
"pp": "case pos.e_f.e_t\nn : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\nσ : Perm n\na✝ : σ ∈ univ\nj' : n\nthis : σ j' = σ j' ↔ j' = j'\n⊢ 1 = Pi.single (σ j') 1 (σ j')",
"ppTerm": "?pos.e_f.e_t✝",
"assigned": true,
"usedConstants": [
... | [
"case pos.e_f.e_t\nn : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\nσ : Perm n\na✝ : σ ∈ univ\nj' : n\nthis : σ j' = σ j' ↔ j' = j'\n⊢ 1 = 1"
] | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 554,
"column": 23
} | {
"line": 554,
"column": 32
} | {
"line": 554,
"column": 33
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommSemiring R\nS : Type u_3\ninst✝ : CommSemiring S\nf : R →+* S\nx : Sˣ\nm : ℕ\nhn : -↑m < 0\n⊢ ↑x⁻¹ ^ m = ↑(x ^ (-↑m))",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"DivisionCommMonoid.toDivisionMonoid"... | [
"case neg\nR : Type u_1\ninst✝¹ : CommSemiring R\nS : Type u_3\ninst✝ : CommSemiring S\nf : R →+* S\nx : Sˣ\nm : ℕ\nhn : -↑m < 0\n⊢ ↑x⁻¹ ^ m = ↑(x ^ ↑m)⁻¹"
] | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Invertible | {
"line": 174,
"column": 89
} | {
"line": 175,
"column": 79
} | {
"line": 176,
"column": 4
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝⁷ : Fintype n\ninst✝⁶ : DecidableEq n\ninst✝⁵ : Fintype m\ninst✝⁴ : DecidableEq m\ninst✝³ : Ring α\nA : Matrix n n α\nU : Matrix n m α\nC : Matrix m m α\nV : Matrix m n α\ninst✝² : Invertible A\ninst✝¹ : Invertible C\ninst✝ : Invertible (⅟C + V * ⅟A * U)\n... | [] | by
simp_rw [add_sub_assoc, _root_.mul_add, _root_.sub_mul, Matrix.mul_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Trace | {
"line": 122,
"column": 51
} | {
"line": 123,
"column": 49
} | {
"line": 125,
"column": 0
} | [
{
"pp": "p : Type u_4\nR : Type u_6\ninst✝³ : Fintype p\ninst✝² : AddCommMonoid R\ninst✝¹ : DecidableEq p\nm : p → Type u_8\ninst✝ : (i : p) → Fintype (m i)\nM : (i : p) → Matrix (m i) (m i) R\n⊢ (blockDiagonal' M).trace = ∑ i, (M i).trace",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
... | [] | by
simp [blockDiagonal', trace, Finset.sum_sigma'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Kronecker | {
"line": 257,
"column": 49
} | {
"line": 257,
"column": 65
} | {
"line": 257,
"column": 65
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nα : Type u_3\nβ : Type u_5\nγ : Type u_7\nm : Type u_10\nn : Type u_11\ninst✝¹³ : Semiring S\ninst✝¹² : Semiring R\ninst✝¹¹ : Fintype m\ninst✝¹⁰ : Fintype n\ninst✝⁹ : DecidableEq m\ninst✝⁸ : DecidableEq n\ninst✝⁷ : NonAssocSemiring α\ninst✝⁶ : NonAssocSemiring β\ninst✝⁵ : Co... | [
"R : Type u_1\nS : Type u_2\nα : Type u_3\nβ : Type u_5\nγ : Type u_7\nm : Type u_10\nn : Type u_11\ninst✝¹³ : Semiring S\ninst✝¹² : Semiring R\ninst✝¹¹ : Fintype m\ninst✝¹⁰ : Fintype n\ninst✝⁹ : DecidableEq m\ninst✝⁸ : DecidableEq n\ninst✝⁷ : NonAssocSemiring α\ninst✝⁶ : NonAssocSemiring β\ninst✝⁵ : CommRing γ\nin... | det_reindex_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 540,
"column": 2
} | {
"line": 550,
"column": 71
} | {
"line": 552,
"column": 0
} | [
{
"pp": "𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\n⊢ ∃ L L', ((List.map toMatrix L).prod * M * (List.map toMatrix L').prod).IsTwoBlockDiagonal",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Uni... | [] | let L : List (TransvectionStruct (Fin r ⊕ Unit) 𝕜) :=
List.ofFn fun i : Fin r =>
⟨inl i, inr unit, by simp, -M (inl i) (inr unit) / M (inr unit) (inr unit)⟩
let L' : List (TransvectionStruct (Fin r ⊕ Unit) 𝕜) :=
List.ofFn fun i : Fin r =>
⟨inr unit, inl i, by simp, -M (inr unit) (inl i) / M (inr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 540,
"column": 2
} | {
"line": 550,
"column": 71
} | {
"line": 552,
"column": 0
} | [
{
"pp": "𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\n⊢ ∃ L L', ((List.map toMatrix L).prod * M * (List.map toMatrix L').prod).IsTwoBlockDiagonal",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Uni... | [] | let L : List (TransvectionStruct (Fin r ⊕ Unit) 𝕜) :=
List.ofFn fun i : Fin r =>
⟨inl i, inr unit, by simp, -M (inl i) (inr unit) / M (inr unit) (inr unit)⟩
let L' : List (TransvectionStruct (Fin r ⊕ Unit) 𝕜) :=
List.ofFn fun i : Fin r =>
⟨inr unit, inl i, by simp, -M (inr unit) (inl i) / M (inr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 665,
"column": 33
} | {
"line": 665,
"column": 50
} | {
"line": 665,
"column": 51
} | [
{
"pp": "n : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\nh : IsUnit A.det\n⊢ A.det • b ᵥ* A⁻¹ᵀᵀ = Aᵀ.cramer b",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"instHSMul",... | [
"n : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\nh : IsUnit A.det\n⊢ A.det • A⁻¹ᵀ *ᵥ b = Aᵀ.cramer b"
] | vecMul_transpose, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MatrixPolynomialAlgebra | {
"line": 120,
"column": 2
} | {
"line": 124,
"column": 20
} | {
"line": 126,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\np : R[X]\n⊢ matPolyEquiv (p • 1) = Polynomial.map (algebraMap R (Matrix n n R)) p",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.smul",
"Polynomial.in... | [] | ext m i j
simp only [matPolyEquiv_coeff_apply, Matrix.smul_apply, Matrix.one_apply, smul_eq_mul, mul_ite,
mul_one, mul_zero, coeff_map, algebraMap_matrix_apply, Algebra.algebraMap_self,
RingHom.id_apply]
split_ifs <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MatrixPolynomialAlgebra | {
"line": 120,
"column": 2
} | {
"line": 124,
"column": 20
} | {
"line": 126,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\np : R[X]\n⊢ matPolyEquiv (p • 1) = Polynomial.map (algebraMap R (Matrix n n R)) p",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.smul",
"Polynomial.in... | [] | ext m i j
simp only [matPolyEquiv_coeff_apply, Matrix.smul_apply, Matrix.one_apply, smul_eq_mul, mul_ite,
mul_one, mul_zero, coeff_map, algebraMap_matrix_apply, Algebra.algebraMap_self,
RingHom.id_apply]
split_ifs <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Charpoly.Basic | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 32
} | {
"line": 100,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝² : CommRing R\nn : Type u_4\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\nk : ℕ\ni : n\n⊢ X.coeff k - (C (M i i)).coeff k = (X.coeff k - (C M).coeff k) i i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomi... | [
"case pos\nR : Type u_1\ninst✝² : CommRing R\nn : Type u_4\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\nk : ℕ\ni : n\n⊢ ((if 1 = k then 1 else 0) - if k = 0 then M i i else 0) = ((if 1 = k then 1 else 0) - if k = 0 then M else 0) i i"
] | simp only [coeff_X, coeff_C] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Nilpotent.Basic | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 27
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : Ring R\nr : R\nn : ℕ\nhn : r ^ n = 0\n⊢ (r - 1) * -∑ i ∈ Finset.range n, r ^ i = 1",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"MulOne.toOne",
"HMul.hMul",
"Ring.toNonAssocRing",
"Mono... | [
"case refine_2\nR : Type u_1\ninst✝ : Ring R\nr : R\nn : ℕ\nhn : r ^ n = 0\n⊢ (-∑ i ∈ Finset.range n, r ^ i) * (r - 1) = 1"
] | · simp [mul_geom_sum, hn] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MatrixAlgebra | {
"line": 224,
"column": 61
} | {
"line": 224,
"column": 66
} | {
"line": 225,
"column": 2
} | [
{
"pp": "n : Type u_3\nR : Type u_5\nA : Type u_7\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\ni j : n\nx : A\n⊢ ∀ (p : n × n), i = p.1 ∧ j = p.2 ↔ p = (i, j)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.MatrixAlgebra | {
"line": 307,
"column": 8
} | {
"line": 307,
"column": 61
} | {
"line": 307,
"column": 61
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nR : Type u_5\nS : Type u_6\nA : Type u_7\nB : Type u_8\nM : Type u_9\nN : Type u_10\ninst✝¹³ : CommSemiring R\ninst✝¹² : Semiring A\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra R B\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq n\nins... | [] | simp [star_eq_conjTranspose, conjTranspose_kronecker] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MatrixAlgebra | {
"line": 307,
"column": 8
} | {
"line": 307,
"column": 61
} | {
"line": 307,
"column": 61
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nR : Type u_5\nS : Type u_6\nA : Type u_7\nB : Type u_8\nM : Type u_9\nN : Type u_10\ninst✝¹³ : CommSemiring R\ninst✝¹² : Semiring A\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra R B\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq n\nins... | [] | simp [star_eq_conjTranspose, conjTranspose_kronecker] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MatrixAlgebra | {
"line": 307,
"column": 8
} | {
"line": 307,
"column": 61
} | {
"line": 307,
"column": 61
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nR : Type u_5\nS : Type u_6\nA : Type u_7\nB : Type u_8\nM : Type u_9\nN : Type u_10\ninst✝¹³ : CommSemiring R\ninst✝¹² : Semiring A\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra R B\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq n\nins... | [] | simp [star_eq_conjTranspose, conjTranspose_kronecker] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.SchurComplement | {
"line": 122,
"column": 6
} | {
"line": 123,
"column": 40
} | {
"line": 125,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix m m α\nB : Matrix m n α\nD : Matrix n n α\ninst✝ : Invertible (fromBlocks A B 0 D)\... | [] | simpa only [Matrix.toBlocks_fromBlocks₂₂, Matrix.zero_mul, zero_add, ← fromBlocks_one] using
congr_arg Matrix.toBlocks₂₂ this | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 125,
"column": 22
} | {
"line": 125,
"column": 27
} | {
"line": 125,
"column": 27
} | [
{
"pp": "case neg.refine_1\nR : Type u_1\ninst✝ : CommRing R\nk : ℕ\nhind :\n ∀ m < k, ∀ {P : R[X]}, IsUnit (P.coeff 0) → (∀ (i : ℕ), i ≠ 0 → IsNilpotent (P.coeff i)) → P.natDegree = m → IsUnit P\nP : R[X]\nhunit : IsUnit (P.coeff 0)\nhnil : ∀ (i : ℕ), i ≠ 0 → IsNilpotent (P.coeff i)\nh : P.natDegree = k\nhdeg... | [] | hdeg₂ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.SchurComplement | {
"line": 292,
"column": 2
} | {
"line": 293,
"column": 44
} | {
"line": 295,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq m\ninst✝⁴ : DecidableEq n\ninst✝³ : CommRing α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\ninst✝² : Invertible A\ninst✝¹ : Invertible (D - C * ⅟A * B)\ninst✝ : Invertible ... | [] | letI := fromBlocks₁₁Invertible A B C D
convert! (rfl : ⅟(fromBlocks A B C D) = _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.SchurComplement | {
"line": 292,
"column": 2
} | {
"line": 293,
"column": 44
} | {
"line": 295,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq m\ninst✝⁴ : DecidableEq n\ninst✝³ : CommRing α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\ninst✝² : Invertible A\ninst✝¹ : Invertible (D - C * ⅟A * B)\ninst✝ : Invertible ... | [] | letI := fromBlocks₁₁Invertible A B C D
convert! (rfl : ⅟(fromBlocks A B C D) = _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Lifts | {
"line": 149,
"column": 55
} | {
"line": 159,
"column": 20
} | {
"line": 161,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\n⊢ ∃ q, map f q = p ∧ q.support = p.support",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Exists.choose_spec",
... | [] | by
rw [lifts_iff_coeff_lifts] at hlifts
let g : ℕ → R := fun k ↦ (hlifts k).choose
have hg : ∀ k, f (g k) = p.coeff k := fun k ↦ (hlifts k).choose_spec
let q : R[X] := ∑ k ∈ p.support, monomial k (g k)
have hq : map f q = p := by simp_rw [q, Polynomial.map_sum, map_monomial, hg, ← as_sum_support]
have hq' :... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.HasseDeriv | {
"line": 117,
"column": 28
} | {
"line": 117,
"column": 48
} | {
"line": 117,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nr : R\nhk : 0 < k\n⊢ (hasseDeriv k) ((monomial 0) r) = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.choose",
"Semiring.toModule",
"HMul.hMul",
... | [
"R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nr : R\nhk : 0 < k\n⊢ (monomial (0 - k)) (↑(choose 0 k) * r) = 0"
] | hasseDeriv_monomial, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.HasseDeriv | {
"line": 124,
"column": 31
} | {
"line": 124,
"column": 51
} | {
"line": 124,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nhk : 1 < k\n⊢ (hasseDeriv k) ((monomial 1) 1) = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.choose",
"Semiring.toModule",
"HMul.hMul",
"cong... | [
"R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nhk : 1 < k\n⊢ (monomial (1 - k)) (↑(choose 1 k) * 1) = 0"
] | hasseDeriv_monomial, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.HasseDeriv | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 25
} | {
"line": 140,
"column": 2
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : Semiring R\nk✝ k : ℕ\nih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]\nf : R[X]\nn : ℕ\n⊢ ↑((k + 1) * k !) * (↑((n + k + 1).choose (k + 1)) * f.coeff (n + k + 1)) =\n ↑k ! * (↑((n + k + 1).choose (n + 1)) * (↑(n + 1) * f.coeff (n + k + 1)))",
"ppTerm": "?succ",
... | [
"case succ\nR : Type u_1\ninst✝ : Semiring R\nk✝ k : ℕ\nih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]\nf : R[X]\nn : ℕ\n⊢ ↑((k + 1) * k !) * ↑((n + k + 1).choose (k + 1)) * f.coeff (n + k + 1) =\n ↑k ! * ↑((n + k + 1).choose (n + 1)) * ↑(n + 1) * f.coeff (n + k + 1)"
] | simp only [← mul_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Splits | {
"line": 86,
"column": 53
} | {
"line": 86,
"column": 58
} | {
"line": 88,
"column": 0
} | [
{
"pp": "case mem\nR : Type u_1\ninst✝¹ : Semiring R\nf : R[X]\nS : Type u_2\ninst✝ : Semiring S\ni : R →+* S\nx✝ : R[X]\nh✝ : x✝ ∈ {x | ∃ a, C a = x} ∪ {x | ∃ a, X + C a = x}\n⊢ (map i x✝).Splits",
"ppTerm": "?mem",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Polynomial.Splits... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 86,
"column": 53
} | {
"line": 86,
"column": 58
} | {
"line": 88,
"column": 0
} | [
{
"pp": "case one\nR : Type u_1\ninst✝¹ : Semiring R\nf : R[X]\nS : Type u_2\ninst✝ : Semiring S\ni : R →+* S\n⊢ (map i 1).Splits",
"ppTerm": "?one",
"assigned": true,
"usedConstants": [
"Polynomial.map_one",
"MulOne.toOne",
"Polynomial.instOne",
"congrArg",
"Polynomial... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 86,
"column": 53
} | {
"line": 86,
"column": 58
} | {
"line": 88,
"column": 0
} | [
{
"pp": "case mul\nR : Type u_1\ninst✝¹ : Semiring R\nf : R[X]\nS : Type u_2\ninst✝ : Semiring S\ni : R →+* S\nx✝ y✝ : R[X]\nhx✝ : x✝ ∈ Submonoid.closure ({x | ∃ a, C a = x} ∪ {x | ∃ a, X + C a = x})\nhy✝ : y✝ ∈ Submonoid.closure ({x | ∃ a, C a = x} ∪ {x | ∃ a, X + C a = x})\na✝¹ : (map i x✝).Splits\na✝ : (map ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 90,
"column": 48
} | {
"line": 90,
"column": 53
} | {
"line": 92,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nhf : f.natDegree = 0\n⊢ (C ⋯.choose).Splits",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Polynomial.Splits.C._simp_1",
"RingHom",
"Exists",
"instOfNatNat",
"Polynomial",
"Ring... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 157,
"column": 53
} | {
"line": 157,
"column": 58
} | {
"line": 159,
"column": 0
} | [
{
"pp": "case mem\nR : Type u_1\ninst✝ : CommSemiring R\np : R[X]\nr : R\nthis : ∀ (i : R), (X + C r + C i).Splits\nx✝ : R[X]\nh✝ : x✝ ∈ {x | ∃ a, C a = x} ∪ {x | ∃ a, X + C a = x}\n⊢ ((Polynomial.taylor r) x✝).Splits",
"ppTerm": "?mem",
"assigned": true,
"usedConstants": [
"Polynomial.C",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 157,
"column": 53
} | {
"line": 157,
"column": 58
} | {
"line": 159,
"column": 0
} | [
{
"pp": "case one\nR : Type u_1\ninst✝ : CommSemiring R\np : R[X]\nr : R\nthis : ∀ (i : R), (X + C r + C i).Splits\n⊢ ((Polynomial.taylor r) 1).Splits",
"ppTerm": "?one",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRi... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 157,
"column": 53
} | {
"line": 157,
"column": 58
} | {
"line": 159,
"column": 0
} | [
{
"pp": "case mul\nR : Type u_1\ninst✝ : CommSemiring R\np : R[X]\nr : R\nthis : ∀ (i : R), (X + C r + C i).Splits\nx✝ y✝ : R[X]\nhx✝ : x✝ ∈ Submonoid.closure ({x | ∃ a, C a = x} ∪ {x | ∃ a, X + C a = x})\nhy✝ : y✝ ∈ Submonoid.closure ({x | ∃ a, C a = x} ∪ {x | ∃ a, X + C a = x})\na✝¹ : ((Polynomial.taylor r) x... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 29
} | {
"line": 354,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nr : R\nhr : IsUnit r\na : R\nthis : Function.Bijective fun x ↦ r * x\n⊢ Multiset.count a (p.scaleRoots r).roots = Multiset.count a (Multiset.map (fun x ↦ r * x) p.roots)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": ... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nr : R\nhr : IsUnit r\nthis : Function.Bijective fun x ↦ r * x\na : R\n⊢ Multiset.count ((fun x ↦ r * x) a) (p.scaleRoots r).roots =\n Multiset.count ((fun x ↦ r * x) a) (Multiset.map (fun x ↦ r * x) p.roots)"
] | obtain ⟨a, rfl⟩ := this.2 a | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Polynomial.Subring | {
"line": 50,
"column": 4
} | {
"line": 51,
"column": 7
} | {
"line": 52,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Ring R\np : R[X]\nT : Subring R\nhp : ↑p.coeffs ⊆ ↑T\nn : ℕ\nh : p.coeff n = 0\n⊢ ↑0 = p.coeff n",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subring.instSetLike",
"Ring.toNonAssocRing",
"congrArg",
"AddMo... | [] | rw [h]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Subring | {
"line": 50,
"column": 4
} | {
"line": 51,
"column": 7
} | {
"line": 52,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Ring R\np : R[X]\nT : Subring R\nhp : ↑p.coeffs ⊆ ↑T\nn : ℕ\nh : p.coeff n = 0\n⊢ ↑0 = p.coeff n",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subring.instSetLike",
"Ring.toNonAssocRing",
"congrArg",
"AddMo... | [] | rw [h]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 185,
"column": 4
} | {
"line": 192,
"column": 8
} | {
"line": 194,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nhf : f.Splits\nh : Irreducible f\na✝ : Nontrivial R\nm : Multiset R\nhm : f = C f.leadingCoeff * (Multiset.map (fun x ↦ X + C x) m).prod\na : R\nha : a ∈ m\n⊢ f.natDegree ≤ 1",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨m, rfl⟩ := Multiset.exists_cons_of_mem ha
rw [Multiset.map_cons, Multiset.prod_cons] at hm
rw [hm] at h
simp only [irreducible_mul_iff, IsUnit.mul_iff, not_isUnit_X_add_C, false_and, and_false,
or_false, false_or, ← Multiset.prod_toList, List.prod_isUnit_iff] at h
have : m = 0 := by simpa ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 185,
"column": 4
} | {
"line": 192,
"column": 8
} | {
"line": 194,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nhf : f.Splits\nh : Irreducible f\na✝ : Nontrivial R\nm : Multiset R\nhm : f = C f.leadingCoeff * (Multiset.map (fun x ↦ X + C x) m).prod\na : R\nha : a ∈ m\n⊢ f.natDegree ≤ 1",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨m, rfl⟩ := Multiset.exists_cons_of_mem ha
rw [Multiset.map_cons, Multiset.prod_cons] at hm
rw [hm] at h
simp only [irreducible_mul_iff, IsUnit.mul_iff, not_isUnit_X_add_C, false_and, and_false,
or_false, false_or, ← Multiset.prod_toList, List.prod_isUnit_iff] at h
have : m = 0 := by simpa ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 82
} | {
"line": 352,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nf : R[X]\nS : Type u_4\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\ni : R →+* S\nhi : Function.Injective ⇑i\nhf : (map i f).Splits\nj : (a : S) → a ∈ (map i f).roots → R\nhj : ∀ (a : S) (a_1 : a ∈ (map i f).roots), i (j a a_1) = a\n⊢ C (i f.leadingCoeff) * (Multiset.map ... | [] | simp [Multiset.pmap_eq_map, hj, Multiset.map_pmap, Polynomial.map_multiset_prod] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Splits | {
"line": 390,
"column": 64
} | {
"line": 390,
"column": 69
} | {
"line": 390,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nS : Type u_4\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nf : R[X]\nφ : R →+* S\nx y : R[X]\nhx✝ : x ∈ Submonoid.closure ({x | ∃ a, C a = x} ∪ {x | ∃ a, X + C a = x})\nhy✝ : y ∈ Submonoid.closure ({x | ∃ a, C a = x} ∪ {x | ∃ a, X + C a = x})\nhx : ma... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 390,
"column": 64
} | {
"line": 390,
"column": 69
} | {
"line": 390,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nS : Type u_4\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nf : R[X]\nφ : R →+* S\nx y : R[X]\nhx✝ : x ∈ Submonoid.closure ({x | ∃ a, C a = x} ∪ {x | ∃ a, X + C a = x})\nhy✝ : y ∈ Submonoid.closure ({x | ∃ a, C a = x} ∪ {x | ∃ a, X + C a = x})\nhx : ma... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 390,
"column": 64
} | {
"line": 390,
"column": 69
} | {
"line": 390,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nS : Type u_4\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nf : R[X]\nφ : R →+* S\nx y : R[X]\nhx✝ : x ∈ Submonoid.closure ({x | ∃ a, C a = x} ∪ {x | ∃ a, X + C a = x})\nhy✝ : y ∈ Submonoid.closure ({x | ∃ a, C a = x} ∪ {x | ∃ a, X + C a = x})\nhx : ma... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 431,
"column": 4
} | {
"line": 431,
"column": 9
} | {
"line": 432,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\na : R\nhf : ((X - C a) * f).Splits\nhf₀ : f = 0\n⊢ f.Splits",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"HMul.hMul",
"CommSemiring.toSemiring",
"HSub.hSub",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 431,
"column": 4
} | {
"line": 431,
"column": 9
} | {
"line": 432,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\na : R\nhf : ((X - C a) * f).Splits\nhf₀ : f = 0\n⊢ f.Splits",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"HMul.hMul",
"CommSemiring.toSemiring",
"HSub.hSub",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 431,
"column": 4
} | {
"line": 431,
"column": 9
} | {
"line": 432,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\na : R\nhf : ((X - C a) * f).Splits\nhf₀ : f = 0\n⊢ f.Splits",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"HMul.hMul",
"CommSemiring.toSemiring",
"HSub.hSub",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 437,
"column": 2
} | {
"line": 437,
"column": 7
} | {
"line": 439,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\na : R\nhf : ((X - C a) * f).Splits\nhf₀ : ¬f = 0\nthis : (X - C a) * f = (X - C a) * (C f.leadingCoeff * (Multiset.map (fun x ↦ X - C x) f.roots).prod)\n⊢ (C f.leadingCoeff * (Multiset.map (fun x ↦ X - C x) f.roots).prod).Splits... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 457,
"column": 26
} | {
"line": 457,
"column": 31
} | {
"line": 457,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih : ∀ {f g : R[X]}, f ≠ 0 → g ≠ 0 → ∀ (p : R[X]), f * g = p → p.Splits → p.natDegree = n → f.Splits ∧ g.Splits\ng : R[X]\nhg₀ : g ≠ 0\na : R\nf : R[X]\nhf₀ : (X - C a) * f ≠ 0\nthis : X - C a ∣ (X - C a) * f ∨ X - C a ∣ g\nh : ((X - C a) * ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 457,
"column": 26
} | {
"line": 457,
"column": 31
} | {
"line": 457,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih : ∀ {f g : R[X]}, f ≠ 0 → g ≠ 0 → ∀ (p : R[X]), f * g = p → p.Splits → p.natDegree = n → f.Splits ∧ g.Splits\ng : R[X]\nhg₀ : g ≠ 0\na : R\nf : R[X]\nhf₀ : (X - C a) * f ≠ 0\nthis : X - C a ∣ (X - C a) * f ∨ X - C a ∣ g\nh : ((X - C a) * ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 457,
"column": 26
} | {
"line": 457,
"column": 31
} | {
"line": 457,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih : ∀ {f g : R[X]}, f ≠ 0 → g ≠ 0 → ∀ (p : R[X]), f * g = p → p.Splits → p.natDegree = n → f.Splits ∧ g.Splits\ng : R[X]\nhg₀ : g ≠ 0\na : R\nf : R[X]\nhf₀ : (X - C a) * f ≠ 0\nthis : X - C a ∣ (X - C a) * f ∨ X - C a ∣ g\nh : ((X - C a) * ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 457,
"column": 37
} | {
"line": 457,
"column": 42
} | {
"line": 457,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih : ∀ {f g : R[X]}, f ≠ 0 → g ≠ 0 → ∀ (p : R[X]), f * g = p → p.Splits → p.natDegree = n → f.Splits ∧ g.Splits\ng : R[X]\nhg₀ : g ≠ 0\na : R\nf : R[X]\nhf₀ : (X - C a) * f ≠ 0\nthis : X - C a ∣ (X - C a) * f ∨ X - C a ∣ g\nh : ((X - C a) * ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 457,
"column": 37
} | {
"line": 457,
"column": 42
} | {
"line": 457,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih : ∀ {f g : R[X]}, f ≠ 0 → g ≠ 0 → ∀ (p : R[X]), f * g = p → p.Splits → p.natDegree = n → f.Splits ∧ g.Splits\ng : R[X]\nhg₀ : g ≠ 0\na : R\nf : R[X]\nhf₀ : (X - C a) * f ≠ 0\nthis : X - C a ∣ (X - C a) * f ∨ X - C a ∣ g\nh : ((X - C a) * ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 457,
"column": 37
} | {
"line": 457,
"column": 42
} | {
"line": 457,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih : ∀ {f g : R[X]}, f ≠ 0 → g ≠ 0 → ∀ (p : R[X]), f * g = p → p.Splits → p.natDegree = n → f.Splits ∧ g.Splits\ng : R[X]\nhg₀ : g ≠ 0\na : R\nf : R[X]\nhf₀ : (X - C a) * f ≠ 0\nthis : X - C a ∣ (X - C a) * f ∨ X - C a ∣ g\nh : ((X - C a) * ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 458,
"column": 19
} | {
"line": 458,
"column": 24
} | {
"line": 458,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih : ∀ {f g : R[X]}, f ≠ 0 → g ≠ 0 → ∀ (p : R[X]), f * g = p → p.Splits → p.natDegree = n → f.Splits ∧ g.Splits\ng : R[X]\nhg₀ : g ≠ 0\na : R\nf : R[X]\nhf₀ : (X - C a) * f ≠ 0\nthis : X - C a ∣ (X - C a) * f ∨ X - C a ∣ g\nh : ((X - C a) * ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 458,
"column": 19
} | {
"line": 458,
"column": 24
} | {
"line": 458,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih : ∀ {f g : R[X]}, f ≠ 0 → g ≠ 0 → ∀ (p : R[X]), f * g = p → p.Splits → p.natDegree = n → f.Splits ∧ g.Splits\ng : R[X]\nhg₀ : g ≠ 0\na : R\nf : R[X]\nhf₀ : (X - C a) * f ≠ 0\nthis : X - C a ∣ (X - C a) * f ∨ X - C a ∣ g\nh : ((X - C a) * ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 458,
"column": 19
} | {
"line": 458,
"column": 24
} | {
"line": 458,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih : ∀ {f g : R[X]}, f ≠ 0 → g ≠ 0 → ∀ (p : R[X]), f * g = p → p.Splits → p.natDegree = n → f.Splits ∧ g.Splits\ng : R[X]\nhg₀ : g ≠ 0\na : R\nf : R[X]\nhf₀ : (X - C a) * f ≠ 0\nthis : X - C a ∣ (X - C a) * f ∨ X - C a ∣ g\nh : ((X - C a) * ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 459,
"column": 4
} | {
"line": 459,
"column": 9
} | {
"line": 461,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih : ∀ {f g : R[X]}, f ≠ 0 → g ≠ 0 → ∀ (p : R[X]), f * g = p → p.Splits → p.natDegree = n → f.Splits ∧ g.Splits\ng : R[X]\nhg₀ : g ≠ 0\na : R\nf : R[X]\nhf₀ : (X - C a) * f ≠ 0\nthis✝ : X - C a ∣ (X - C a) * f ∨ X - C a ∣ g\nh : ((X - C a) *... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 504,
"column": 2
} | {
"line": 504,
"column": 18
} | {
"line": 505,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ns : Multiset R\ng : R[X]\nhg : g.Monic\nhg' : g.Splits\nthis : g = (Multiset.map (fun x ↦ X - C x) g.roots).prod\n⊢ (Multiset.map (fun x ↦ (Multiset.map (fun x_1 ↦ x - x_1) g.roots).prod) s).prod =\n (Multiset.map (fun x ↦ (Multiset.map (eval x ... | [
"case a.e'_3.a\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ns : Multiset R\ng : R[X]\nhg : g.Monic\nhg' : g.Splits\nthis : g = (Multiset.map (fun x ↦ X - C x) g.roots).prod\nx : R\nhx : x ∈ s\nx✝ : R\na✝ : x✝ ∈ g.roots\n⊢ HSub.hSub x = eval x ∘ fun x ↦ X - C x"
] | congr! with x hx | Congr!._aux_Mathlib_Tactic_CongrExclamation___elabRules_Congr!_congr!_1 | Congr!.congr! |
Mathlib.Algebra.Polynomial.Splits | {
"line": 675,
"column": 80
} | {
"line": 675,
"column": 85
} | {
"line": 675,
"column": 85
} | [
{
"pp": "R : Type u_1\ninst✝ : Field R\nf : R[X]\nx : R\nh₁ : f.natDegree = 2\nh₂ : eval x f = 0\nh : (f /ₘ (X - C x)).natDegree = 1\n⊢ f /ₘ (X - C x) ≠ 0",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"False",
"Nat.instMulZeroClass",
"Nat.instOne",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 675,
"column": 80
} | {
"line": 675,
"column": 85
} | {
"line": 675,
"column": 85
} | [
{
"pp": "R : Type u_1\ninst✝ : Field R\nf : R[X]\nx : R\nh₁ : f.natDegree = 2\nh₂ : eval x f = 0\nh : (f /ₘ (X - C x)).natDegree = 1\n⊢ f /ₘ (X - C x) ≠ 0",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"False",
"Nat.instMulZeroClass",
"Nat.instOne",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 675,
"column": 80
} | {
"line": 675,
"column": 85
} | {
"line": 675,
"column": 85
} | [
{
"pp": "R : Type u_1\ninst✝ : Field R\nf : R[X]\nx : R\nh₁ : f.natDegree = 2\nh₂ : eval x f = 0\nh : (f /ₘ (X - C x)).natDegree = 1\n⊢ f /ₘ (X - C x) ≠ 0",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"False",
"Nat.instMulZeroClass",
"Nat.instOne",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 688,
"column": 4
} | {
"line": 688,
"column": 9
} | {
"line": 689,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : Field R\n⊢ Splits 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Polynomial.Splits.zero._simp_1",
"Field.toSemifield",
"Polynomial",
"Semifield.toDivisionSemiring",
"DivisionSemiring.toSemiring",
"of_eq_true",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 688,
"column": 4
} | {
"line": 688,
"column": 9
} | {
"line": 689,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : Field R\n⊢ Splits 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Polynomial.Splits.zero._simp_1",
"Field.toSemifield",
"Polynomial",
"Semifield.toDivisionSemiring",
"DivisionSemiring.toSemiring",
"of_eq_true",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 688,
"column": 4
} | {
"line": 688,
"column": 9
} | {
"line": 689,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : Field R\n⊢ Splits 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Polynomial.Splits.zero._simp_1",
"Field.toSemifield",
"Polynomial",
"Semifield.toDivisionSemiring",
"DivisionSemiring.toSemiring",
"of_eq_true",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.GoingUp | {
"line": 41,
"column": 6
} | {
"line": 41,
"column": 25
} | {
"line": 41,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nS : Type u_2\ninst✝ : CommRing S\nf : R →+* S\nI : Ideal S\nr : S\nhr : r ∈ I\np : R[X]\nhp : eval₂ f r p ∈ I\n⊢ p.coeff 0 ∈ comap f I",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Semiring.toModule",
"HMul.hM... | [
"R : Type u_1\ninst✝¹ : CommRing R\nS : Type u_2\ninst✝ : CommRing S\nf : R →+* S\nI : Ideal S\nr : S\nhr : r ∈ I\np : R[X]\nhp : eval₂ f r (p.divX * X + C (p.coeff 0)) ∈ I\n⊢ p.coeff 0 ∈ comap f I"
] | ← p.divX_mul_X_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Integral | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 34
} | {
"line": 239,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Submonoid R\nRₘ : Type u_3\ninst✝² : CommRing Rₘ\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\np : Rₘ[X]\nhp : p.leadingCoeff ∈ (algebraMap R Rₘ).range\nn : ℕ\n⊢ (p.scaleRoots ((algebraMap R Rₘ) ↑(commonDenom M p.support p.coeff))).coeff n ∈ Set.range ⇑(alg... | [
"R : Type u_1\ninst✝³ : CommRing R\nM : Submonoid R\nRₘ : Type u_3\ninst✝² : CommRing Rₘ\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\np : Rₘ[X]\nhp : p.leadingCoeff ∈ (algebraMap R Rₘ).range\nn : ℕ\n⊢ p.coeff n * (algebraMap R Rₘ) ↑(commonDenom M p.support p.coeff) ^ (p.natDegree - n) ∈ Set.range ⇑(algebraM... | rw [Polynomial.coeff_scaleRoots] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic | {
"line": 480,
"column": 22
} | {
"line": 498,
"column": 32
} | {
"line": 500,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra R B\ninst✝² : Algebra R A\ninst✝¹ : IsScalarTower R A B\ninst✝ : Algebra.IsIntegral R A\nx : B\nhx : IsIntegral A x\n⊢ IsIntegral R x",
"ppTerm": "?m.25",
... | [] | by
rcases hx with ⟨p, pmonic, hp⟩
let S := adjoin R (p.coeffs : Set A)
have : Module.Finite R S := ⟨(Subalgebra.toSubmodule S).fg_top.mpr <|
fg_adjoin_of_finite p.coeffs.finite_toSet fun a _ ↦ Algebra.IsIntegral.isIntegral a⟩
let p' : S[X] := p.toSubring S.toSubring subset_adjoin
have hSx : IsIntegral S x... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 97,
"column": 2
} | {
"line": 98,
"column": 17
} | {
"line": 99,
"column": 2
} | [
{
"pp": "case pos\nR : Type u\nM : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nfl : FaithfullyFlat R M\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Nontrivial N\nn : N\nhn : n ≠ 0\nI : Ideal R := (R ∙ n).annihilator\nI_ne_top : I = ⊤\n⊢ Nontrivial (N ⊗[R] M... | [
"case neg\nR : Type u\nM : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nfl : FaithfullyFlat R M\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Nontrivial N\nn : N\nhn : n ≠ 0\nI : Ideal R := (R ∙ n).annihilator\nI_ne_top : ¬I = ⊤\n⊢ Nontrivial (N ⊗[R] M)"
] | · rw [Ideal.eq_top_iff_one, Submodule.mem_annihilator_span_singleton, one_smul] at I_ne_top
contradiction | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.GoingUp | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 91
} | {
"line": 204,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝⁵ : CommRing R\nS : Type u_2\ninst✝⁴ : CommRing S\nI : Ideal S\ninst✝³ : Algebra R S\ninst✝² : Nontrivial R\ninst✝¹ : IsDomain S\ninst✝ : Algebra.IsIntegral R S\nhI : comap (algebraMap R S) I = ⊥\nx : S\nhx : x ∈ I\nhx0 : ¬x = 0\n⊢ x ∈ ⊥",
"ppTerm": "?neg✝",
"assign... | [] | exact absurd hI (comap_ne_bot_of_integral_mem hx0 hx (Algebra.IsIntegral.isIntegral x)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.GoingUp | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 91
} | {
"line": 204,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝⁵ : CommRing R\nS : Type u_2\ninst✝⁴ : CommRing S\nI : Ideal S\ninst✝³ : Algebra R S\ninst✝² : Nontrivial R\ninst✝¹ : IsDomain S\ninst✝ : Algebra.IsIntegral R S\nhI : comap (algebraMap R S) I = ⊥\nx : S\nhx : x ∈ I\nhx0 : ¬x = 0\n⊢ x ∈ ⊥",
"ppTerm": "?neg✝",
"assign... | [] | exact absurd hI (comap_ne_bot_of_integral_mem hx0 hx (Algebra.IsIntegral.isIntegral x)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.GoingUp | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 91
} | {
"line": 204,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝⁵ : CommRing R\nS : Type u_2\ninst✝⁴ : CommRing S\nI : Ideal S\ninst✝³ : Algebra R S\ninst✝² : Nontrivial R\ninst✝¹ : IsDomain S\ninst✝ : Algebra.IsIntegral R S\nhI : comap (algebraMap R S) I = ⊥\nx : S\nhx : x ∈ I\nhx0 : ¬x = 0\n⊢ x ∈ ⊥",
"ppTerm": "?neg✝",
"assign... | [] | exact absurd hI (comap_ne_bot_of_integral_mem hx0 hx (Algebra.IsIntegral.isIntegral x)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 267,
"column": 31
} | {
"line": 267,
"column": 56
} | {
"line": 267,
"column": 57
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : NoZeroDivisors S\ninst✝ : Algebra.IsAlgebraic R S\na : A\nh : IsAlgebraic S a\np✝ ... | [
"case refine_2\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : NoZeroDivisors S\ninst✝ : Algebra.IsAlgebraic R S\na : A\nh : IsAlgebraic S a\np✝ : S[X]\nhp :... | Subalgebra.algebraMap_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 277,
"column": 4
} | {
"line": 278,
"column": 41
} | {
"line": 280,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\ninst✝ : NoZeroDivisors S\nalg : Algebra.IsAlgebraic R S\na : A\nh : IsIntegral S a\nh✝ : Nontriv... | [] | have := Module.nontrivial S A
exact h.isAlgebraic.restrictScalars _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 277,
"column": 4
} | {
"line": 278,
"column": 41
} | {
"line": 280,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\ninst✝ : NoZeroDivisors S\nalg : Algebra.IsAlgebraic R S\na : A\nh : IsIntegral S a\nh✝ : Nontriv... | [] | have := Module.nontrivial S A
exact h.isAlgebraic.restrictScalars _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 405,
"column": 2
} | {
"line": 408,
"column": 65
} | {
"line": 410,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx : PrimeSpectrum R\n⊢ IsMax x ↔ x.asIdeal.IsMaximal",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"PrimeSpectrum.mk",
"False",
"Eq.ge",
"Preorder.toLT",
"Semiring.toModule",
"PrimeSpectrum.isPrime",
"C... | [] | refine ⟨fun hx ↦ ⟨⟨x.2.ne_top, fun I hI ↦ ?_⟩⟩, fun hx y e ↦ (hx.eq_of_le y.2.ne_top e).ge⟩
by_contra e
obtain ⟨m, hm, hm'⟩ := Ideal.exists_le_maximal I e
exact hx.not_lt (show x < ⟨m, hm.isPrime⟩ from hI.trans_le hm') | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 405,
"column": 2
} | {
"line": 408,
"column": 65
} | {
"line": 410,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx : PrimeSpectrum R\n⊢ IsMax x ↔ x.asIdeal.IsMaximal",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"PrimeSpectrum.mk",
"False",
"Eq.ge",
"Preorder.toLT",
"Semiring.toModule",
"PrimeSpectrum.isPrime",
"C... | [] | refine ⟨fun hx ↦ ⟨⟨x.2.ne_top, fun I hI ↦ ?_⟩⟩, fun hx y e ↦ (hx.eq_of_le y.2.ne_top e).ge⟩
by_contra e
obtain ⟨m, hm, hm'⟩ := Ideal.exists_le_maximal I e
exact hx.not_lt (show x < ⟨m, hm.isPrime⟩ from hI.trans_le hm') | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 628,
"column": 6
} | {
"line": 628,
"column": 38
} | {
"line": 628,
"column": 38
} | [
{
"pp": "case neg\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nz : A\nz' : S\nalg : Algebra.IsAlgebraic R S\nσ : Type u_4\ninst✝ : NoZeroDivisors S\nh : ¬Funct... | [
"case neg\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nz : A\nz' : S\nalg : Algebra.IsAlgebraic R S\nσ : Type u_4\ninst✝ : NoZeroDivisors S\nh✝ : ¬Function.Injecti... | ← MvPolynomial.map_injective_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 393,
"column": 65
} | {
"line": 397,
"column": 6
} | {
"line": 399,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nN : Type u_4\nN' : Type u_5\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup N'\ninst✝² : Module R N\ninst✝¹ : Module R N'\nf : N →ₗ[R] N'\ninst✝ : FaithfullyFlat R M\n⊢ Function.Surjective ⇑(LinearMap.lTensor M f... | [] | by
rw [← LinearMap.exact_zero_iff_surjective (M ⊗[R] Unit),
← LinearMap.exact_zero_iff_surjective Unit]
conv_rhs => rw [← lTensor_exact_iff_exact R M]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 421,
"column": 6
} | {
"line": 422,
"column": 44
} | {
"line": 422,
"column": 44
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\niff_exact :\n ∀ {N1 : Type (max u v)} [inst : AddCommGroup N1] [inst_1 : Module R N1] {N2 : Type (max u v)}\n [inst_2 : AddCommGroup N2] [inst_3 : Module R N2] {N3 : Type (max u v)} [inst_4 : AddCommGroup N3]\... | [] | simpa [eq_comm] using (iff_exact (0 : PUnit →ₗ[R] N) (0 : N →ₗ[R] PUnit) |>.2 fun x => by
simpa using Subsingleton.elim _ _) y | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 421,
"column": 6
} | {
"line": 422,
"column": 44
} | {
"line": 422,
"column": 44
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\niff_exact :\n ∀ {N1 : Type (max u v)} [inst : AddCommGroup N1] [inst_1 : Module R N1] {N2 : Type (max u v)}\n [inst_2 : AddCommGroup N2] [inst_3 : Module R N2] {N3 : Type (max u v)} [inst_4 : AddCommGroup N3]\... | [] | simpa [eq_comm] using (iff_exact (0 : PUnit →ₗ[R] N) (0 : N →ₗ[R] PUnit) |>.2 fun x => by
simpa using Subsingleton.elim _ _) y | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 421,
"column": 6
} | {
"line": 422,
"column": 44
} | {
"line": 422,
"column": 44
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\niff_exact :\n ∀ {N1 : Type (max u v)} [inst : AddCommGroup N1] [inst_1 : Module R N1] {N2 : Type (max u v)}\n [inst_2 : AddCommGroup N2] [inst_3 : Module R N2] {N3 : Type (max u v)} [inst_4 : AddCommGroup N3]\... | [] | simpa [eq_comm] using (iff_exact (0 : PUnit →ₗ[R] N) (0 : N →ₗ[R] PUnit) |>.2 fun x => by
simpa using Subsingleton.elim _ _) y | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SurjectiveOnStalks | {
"line": 61,
"column": 4
} | {
"line": 65,
"column": 57
} | {
"line": 67,
"column": 0
} | [
{
"pp": "case mpr.refine_2\nR : Type u_1\ninst✝² : CommRing R\nS : Type u_2\ninst✝¹ : CommRing S\nf : R →+* S\nP : Ideal S\ninst✝ : P.IsPrime\nH : ∀ (s : S), ∃ x r, ∃ c ∉ P, f r ∉ P ∧ c * f r * s = c * f x\ny✝ : Localization.AtPrime P\nx✝ : S × ↥P.primeCompl\ny t : S\nh : t ∈ P.primeCompl\nyx ys : R\nyc : S\nhy... | [] | · simp only [Localization.mk_eq_mk', Localization.localRingHom_mk', map_mul f,
IsLocalization.mk'_eq_iff_eq, IsLocalization.eq_iff_exists P.primeCompl]
refine ⟨⟨yc, hyc⟩ * ⟨yt, hyt⟩, ?_⟩
simp only [Submonoid.coe_mul]
convert! congr($(ey.symm) * $(et)) using 1 <;> ring | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 549,
"column": 94
} | {
"line": 549,
"column": 99
} | {
"line": 549,
"column": 99
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\nS : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : FaithfullyFlat R S\ninst✝ : FaithfullyFlat S M\nN : Type (max u_1 u_3)\nx✝³ : AddCommGroup ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 549,
"column": 94
} | {
"line": 549,
"column": 99
} | {
"line": 549,
"column": 99
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\nS : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : FaithfullyFlat R S\ninst✝ : FaithfullyFlat S M\nN : Type (max u_1 u_3)\nx✝³ : AddCommGroup ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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