module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1215,
"column": 2
} | {
"line": 1216,
"column": 92
} | {
"line": 1217,
"column": 2
} | [
{
"pp": "ι✝ : Type u_1\ninst✝¹⁰ : Fintype ι✝\ninst✝⁹ : DecidableEq ι✝\nR : Type u_2\ninst✝⁸ : CommSemiring R\nA : Type u_3\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\nM : Type u_4\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\nι : Type u_5\ninst✝¹ : Finite ι\n... | [
"ι✝ : Type u_1\ninst✝¹⁰ : Fintype ι✝\ninst✝⁹ : DecidableEq ι✝\nR : Type u_2\ninst✝⁸ : CommSemiring R\nA : Type u_3\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\nM : Type u_4\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\nι : Type u_5\ninst✝¹ : Finite ι\ninst✝ : IsSt... | classical rw [← MulOpposite.isStablyFiniteRing_iff,
← RingEquiv.isStablyFiniteRing_iff (matrixRingEquivEndVecMulOpposite (ι := ι) (A := A))] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.Localization.Defs | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 89
} | {
"line": 146,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nm : ↥M\n⊢ Bijective fun s ↦ m • s",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
... | [] | simpa only [Submonoid.smul_def, Algebra.smul_def] using! (map_units S m).smul_bijective | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.Localization.Defs | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 89
} | {
"line": 146,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nm : ↥M\n⊢ Bijective fun s ↦ m • s",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
... | [] | simpa only [Submonoid.smul_def, Algebra.smul_def] using! (map_units S m).smul_bijective | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Defs | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 89
} | {
"line": 146,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nm : ↥M\n⊢ Bijective fun s ↦ m • s",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
... | [] | simpa only [Submonoid.smul_def, Algebra.smul_def] using! (map_units S m).smul_bijective | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Defs | {
"line": 766,
"column": 6
} | {
"line": 766,
"column": 34
} | {
"line": 766,
"column": 34
} | [
{
"pp": "case e'_6\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_3\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra\nH : IsLocalization (Submonoid.map (↑h) M) S\nr : R... | [
"case e'_6\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_3\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra\nH : IsLocalization (Submonoid.map (↑h) M) S\nr : R\n⊢ (algebra... | RingHom.algebraMap_toAlgebra | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Defs | {
"line": 937,
"column": 6
} | {
"line": 937,
"column": 20
} | {
"line": 937,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nT : Type u_4\ninst✝ : CommRing T\nf : S →+* T\nh : ∀ (x : R), f ((algebraMap R S) x) = 0 → (algebraMap R S) x = 0\nx y : R\nhz : f ((algebraMap R S) x) = f ((algebra... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nT : Type u_4\ninst✝ : CommRing T\nf : S →+* T\nh : ∀ (x : R), f ((algebraMap R S) x) = 0 → (algebraMap R S) x = 0\nx y : R\nhz : f ((algebraMap R S) x) - f ((algebraMap R S) y) ... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Basic | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 35
} | {
"line": 481,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx : R\ny : ↥M\n⊢ (algEquiv M S).symm (mk' S x y) = mk x y",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Localiz... | [] | rw [mk_eq_mk', algEquiv_symm_mk'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Localization.Basic | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 35
} | {
"line": 481,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx : R\ny : ↥M\n⊢ (algEquiv M S).symm (mk' S x y) = mk x y",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Localiz... | [] | rw [mk_eq_mk', algEquiv_symm_mk'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Basic | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 35
} | {
"line": 481,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx : R\ny : ↥M\n⊢ (algEquiv M S).symm (mk' S x y) = mk x y",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Localiz... | [] | rw [mk_eq_mk', algEquiv_symm_mk'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Basic | {
"line": 569,
"column": 2
} | {
"line": 573,
"column": 77
} | {
"line": 575,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝¹⁰ : CommSemiring S\ninst✝⁹ : Algebra R S\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁸ : CommSemiring Rₘ\ninst✝⁷ : CommSemiring Sₘ\ninst✝⁶ : Algebra R Rₘ\ninst✝⁵ : IsLocalization M Rₘ\ninst✝⁴ : Algebra S Sₘ\ni : IsLocalization (Algebr... | [] | rw [IsLocalization.eq_mk'_iff_mul_eq, Subtype.coe_mk, ← IsScalarTower.algebraMap_apply, ←
IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply R Rₘ Sₘ,
IsScalarTower.algebraMap_apply R Rₘ Sₘ, ← map_mul, mul_comm,
IsLocalization.mul_mk'_eq_mk'_of_mul]
exact congr_arg (algebraMap Rₘ Sₘ) (IsLocaliz... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Basic | {
"line": 569,
"column": 2
} | {
"line": 573,
"column": 77
} | {
"line": 575,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝¹⁰ : CommSemiring S\ninst✝⁹ : Algebra R S\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁸ : CommSemiring Rₘ\ninst✝⁷ : CommSemiring Sₘ\ninst✝⁶ : Algebra R Rₘ\ninst✝⁵ : IsLocalization M Rₘ\ninst✝⁴ : Algebra S Sₘ\ni : IsLocalization (Algebr... | [] | rw [IsLocalization.eq_mk'_iff_mul_eq, Subtype.coe_mk, ← IsScalarTower.algebraMap_apply, ←
IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply R Rₘ Sₘ,
IsScalarTower.algebraMap_apply R Rₘ Sₘ, ← map_mul, mul_comm,
IsLocalization.mul_mk'_eq_mk'_of_mul]
exact congr_arg (algebraMap Rₘ Sₘ) (IsLocaliz... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Star.Pointwise | {
"line": 105,
"column": 25
} | {
"line": 105,
"column": 38
} | {
"line": 105,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Mul α\ninst✝ : StarMul α\ns t : Set α\n⊢ star '' (s * t) = star '' t * star '' s",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"id",
"StarMul.toInvolutiveStar",
"Set.image",
"InvolutiveStar.toStar",
"Set.mul... | [
"α : Type u_1\ninst✝¹ : Mul α\ninst✝ : StarMul α\ns t : Set α\n⊢ star '' image2 (fun x1 x2 ↦ x1 * x2) s t = image2 (fun x1 x2 ↦ x1 * x2) (star '' t) (star '' s)"
] | ← image2_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Ring.TransferInstance | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 49
} | {
"line": 88,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : α ≃ β\ninst✝ : NonAssocSemiring β\nmul : Mul α := e.mul\nadd_monoid_with_one : AddMonoidWithOne α := e.addMonoidWithOne\n⊢ NonAssocSemiring α",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"AddMonoid.toAddSemigrou... | [
"case zero\nα : Type u_1\nβ : Type u_2\ne : α ≃ β\ninst✝ : NonAssocSemiring β\nmul : Mul α := ⋯\nadd_monoid_with_one : AddMonoidWithOne α := ⋯\n⊢ e 0 = 0",
"case one\nα : Type u_1\nβ : Type u_2\ne : α ≃ β\ninst✝ : NonAssocSemiring β\nmul : Mul α := ⋯\nadd_monoid_with_one : AddMonoidWithOne α := ⋯\n⊢ e 1 = 1",
"... | apply e.injective.nonAssocSemiring _ <;> intros | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 194,
"column": 60
} | {
"line": 194,
"column": 78
} | {
"line": 194,
"column": 79
} | [
{
"pp": "case neg\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\ns : R\na : A\nh : s ∉ σ a\nh' : IsUnit (r • ↑ₐ (r⁻¹ • s) - a)\n⊢ r • (↑⋯.unit)⁻¹ʳ = (↑h'.subInvSMul)⁻¹ʳ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Units.val",
... | [
"case neg\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\ns : R\na : A\nh : s ∉ σ a\nh' : IsUnit (r • ↑ₐ (r⁻¹ • s) - a)\n⊢ r • ↑⋯.unit⁻¹ = (↑h'.subInvSMul)⁻¹ʳ"
] | Ring.inverse_unit, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 194,
"column": 79
} | {
"line": 194,
"column": 97
} | {
"line": 195,
"column": 6
} | [
{
"pp": "case neg\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\ns : R\na : A\nh : s ∉ σ a\nh' : IsUnit (r • ↑ₐ (r⁻¹ • s) - a)\n⊢ r • ↑⋯.unit⁻¹ = (↑h'.subInvSMul)⁻¹ʳ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Units.val",
"E... | [
"case neg\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\ns : R\na : A\nh : s ∉ σ a\nh' : IsUnit (r • ↑ₐ (r⁻¹ • s) - a)\n⊢ r • ↑⋯.unit⁻¹ = ↑h'.subInvSMul⁻¹"
] | Ring.inverse_unit, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 552,
"column": 2
} | {
"line": 553,
"column": 63
} | {
"line": 554,
"column": 2
} | [
{
"pp": "R : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns : Set B\nhs : adjoin R s = ⊤\n⊢ adjoin A ((fun x ↦ 1 ⊗ₜ[R] x) '' s) = ⊤",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
... | [
"R : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns : Set B\nhs : adjoin R s = ⊤\n⊢ adjoin A ↑(Subalgebra.map includeRight ⊤) = ⊤"
] | suffices h : adjoin A ((⊤ : Subalgebra R B).map (includeRight (A := A)) : Set (A ⊗[R] B)) = ⊤ by
simp [← h, ← hs, AlgHom.map_adjoin, adjoin_adjoin_of_tower] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 68
} | {
"line": 365,
"column": 2
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\nr : R\n⊢ r ∈ quasispectrum R a ↔ r ... | [
"R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\nr : R\nthis : {0} ⊆ spectrum R ↑a\n⊢ r ∈ quasis... | have := Set.singleton_subset_iff.mpr (zero_mem_spectrum_inr R S a) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 87
} | {
"line": 366,
"column": 2
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\nr : R\nthis : {0} ⊆ spectrum R ↑a\n... | [
"R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\nr : R\nthis : {0} ⊆ spectrum R ↑a\n⊢ r ∈ quasis... | rw [← Set.union_eq_self_of_subset_right this, ← quasispectrum_eq_spectrum_union_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Star.Basic | {
"line": 383,
"column": 2
} | {
"line": 383,
"column": 70
} | {
"line": 384,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nu x : R\nhu : IsUnit u\nh : 0 ≤ u * x * star u\nv : R\nhv : v * u = 1\n⊢ 0 ≤ x",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
... | [
"R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nu x : R\nhu : IsUnit u\nh : 0 ≤ u * x * star u\nv : R\nhv : v * u = 1\nthis : 0 ≤ v * u * x * star u * star v\n⊢ 0 ≤ x"
] | have := by simpa [← mul_assoc] using star_right_conjugate_nonneg h v | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 15
} | {
"line": 64,
"column": 4
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na✝ : Nontrivial R\n⊢ Module.rank R M = 0 → ∀ (x : M), ∃ a, a ≠ 0 ∧ a • x = 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Mathlib.T... | [
"case mp\nR : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na✝ : Nontrivial R\n⊢ (∃ x, ∀ (a : R), a ≠ 0 → a • x ≠ 0) → Module.rank R M ≠ 0"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 70,
"column": 2
} | {
"line": 78,
"column": 98
} | {
"line": 80,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na✝ : Nontrivial R\n⊢ (∀ (x : M), ∃ a, a ≠ 0 ∧ a • x = 0) → Module.rank R M = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Module.rank_def",
"Finsupp.instFunLike",
... | [] | · intro h
rw [← nonpos_iff_eq_zero, Module.rank_def]
apply ciSup_le'
intro ⟨s, hs⟩
rw [nonpos_iff_eq_zero, Cardinal.mk_eq_zero_iff, ← not_nonempty_iff]
rintro ⟨i : s⟩
obtain ⟨a, ha, ha'⟩ := h i
apply ha
simpa using DFunLike.congr_fun (linearIndependent_iff.mp hs (Finsupp.single i a) (by ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 13
} | {
"line": 96,
"column": 13
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\n⊢ 0 < Module.rank R M ↔ ∃ x, x ≠ 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.m... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\n⊢ Module.rank R M ≤ 0 ↔ ∀ (x : M), x = 0"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 198,
"column": 24
} | {
"line": 198,
"column": 42
} | {
"line": 198,
"column": 42
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nhn : ↑n ≤ Module.rank R M\nh : ↑n = Module.rank R M\ns : Set M\nhs : LinearIndepOn R id s\nhs' : #↑↑⟨s, hs⟩ = ↑n\nthis : Finite ↑s\nval✝ : Fintype ↑s\n⊢ s.toFinset.card = n",
"ppTerm": "?m.117",
"a... | [] | by simpa using hs' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 965,
"column": 30
} | {
"line": 965,
"column": 87
} | {
"line": 965,
"column": 87
} | [
{
"pp": "F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB✝ : Type w\nC✝ : Type w'\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : NonUnitalSemiring A\ninst✝¹⁵ : StarRing A\ninst✝¹⁴ : Module R A\ninst✝¹³ : NonUnitalSemiring B✝\ninst✝¹² : StarRing B✝\ninst✝¹¹ : Module R B✝\ninst✝¹⁰ : FunLike F A B✝\ninst✝⁹ : NonUnitalAl... | [] | simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 965,
"column": 30
} | {
"line": 965,
"column": 87
} | {
"line": 965,
"column": 87
} | [
{
"pp": "F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB✝ : Type w\nC✝ : Type w'\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : NonUnitalSemiring A\ninst✝¹⁵ : StarRing A\ninst✝¹⁴ : Module R A\ninst✝¹³ : NonUnitalSemiring B✝\ninst✝¹² : StarRing B✝\ninst✝¹¹ : Module R B✝\ninst✝¹⁰ : FunLike F A B✝\ninst✝⁹ : NonUnitalAl... | [] | simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 965,
"column": 30
} | {
"line": 965,
"column": 87
} | {
"line": 965,
"column": 87
} | [
{
"pp": "F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB✝ : Type w\nC✝ : Type w'\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : NonUnitalSemiring A\ninst✝¹⁵ : StarRing A\ninst✝¹⁴ : Module R A\ninst✝¹³ : NonUnitalSemiring B✝\ninst✝¹² : StarRing B✝\ninst✝¹¹ : Module R B✝\ninst✝¹⁰ : FunLike F A B✝\ninst✝⁹ : NonUnitalAl... | [] | simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 457,
"column": 2
} | {
"line": 457,
"column": 13
} | {
"line": 457,
"column": 13
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : IsDomain R\ninst✝² : IsTorsionFree R M\ninst✝¹ : StrongRankCondition R\nS : Submodule R M\ninst✝ : Module.Finite R ↥S\n⊢ 1 ≤ finrank R ↥S ↔ S ≠ ⊥",
"ppTerm": "?m.26",
"assigned": true,
"usedC... | [
"R : Type u_1\nM : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : IsDomain R\ninst✝² : IsTorsionFree R M\ninst✝¹ : StrongRankCondition R\nS : Submodule R M\ninst✝ : Module.Finite R ↥S\n⊢ finrank R ↥S < 1 ↔ S = ⊥"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 138,
"column": 10
} | {
"line": 138,
"column": 24
} | {
"line": 138,
"column": 25
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : Ring R\ninst✝⁷ : Ring S\ninst✝⁶ : Ring T\nσ : R →+* S\nτ : S →+* T\nE : Type u_4\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_5\ninst✝³ : AddCommGroup F\ninst✝² : Module S F\nG : Type u_6\ninst✝¹ : AddCommGroup G\ninst✝ : Module T G\nx : E... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : Ring R\ninst✝⁷ : Ring S\ninst✝⁶ : Ring T\nσ : R →+* S\nτ : S →+* T\nE : Type u_4\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_5\ninst✝³ : AddCommGroup F\ninst✝² : Module S F\nG : Type u_6\ninst✝¹ : AddCommGroup G\ninst✝ : Module T G\nx : E\ny : F\nH :... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 161,
"column": 6
} | {
"line": 161,
"column": 20
} | {
"line": 161,
"column": 21
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : Ring R\ninst✝⁴ : Ring S\nσ : R →+* S\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module S F\nx : E\ny : F\nH : ∀ (c : R), c • x = 0 → σ c • y = 0\nc : R\nh : c • x ∈ (mkSpanSingleton' x y H).domain\n⊢ σ... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : Ring R\ninst✝⁴ : Ring S\nσ : R →+* S\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module S F\nx : E\ny : F\nH : ∀ (c : R), c • x = 0 → σ c • y = 0\nc : R\nh : c • x ∈ (mkSpanSingleton' x y H).domain\n⊢ σ (Classical.... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 50
} | {
"line": 41,
"column": 4
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝⁵ : Ring K\ninst✝⁴ : StrongRankCondition K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Free K V\nι : Type u_1\ninst✝ : IsEmpty ι\nhV : Module.rank K V = 0\nfst✝ : Type v\nb : Basis fst✝ K V\n⊢ Subsingleton V",
"ppTerm": "?m.45",
"assigned": true,
"us... | [
"K : Type u\nV : Type v\ninst✝⁵ : Ring K\ninst✝⁴ : StrongRankCondition K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Free K V\nι : Type u_1\ninst✝ : IsEmpty ι\nhV : Module.rank K V = 0\nfst✝ : Type v\nb : Basis fst✝ K V\nthis : IsEmpty fst✝\n⊢ Subsingleton V"
] | have := mk_eq_zero_iff.1 (hV ▸ b.mk_eq_rank'') | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 298,
"column": 2
} | {
"line": 298,
"column": 62
} | {
"line": 299,
"column": 2
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : V →ₗ[K] V\nhinj : Injective ⇑f\nh : Module.rank K ↥f.range = Module.rank K V\n⊢ Surjective ⇑f",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Is... | [
"K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : V →ₗ[K] V\nhinj : Injective ⇑f\nh : finrank K ↥f.range = finrank K V\n⊢ Surjective ⇑f"
] | rw [← finrank_eq_rank, ← finrank_eq_rank, Nat.cast_inj] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 173,
"column": 2
} | {
"line": 174,
"column": 56
} | {
"line": 176,
"column": 0
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : StrongRankCondition K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Free K V\n⊢ Module.rank K V ≤ 1 ↔ ⊤.IsPrincipal",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"LinearEquiv.symm... | [] | have := Module.Free.of_equiv (topEquiv (R := K) (M := V)).symm
rw [← Submodule.rank_le_one_iff_isPrincipal, rank_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 173,
"column": 2
} | {
"line": 174,
"column": 56
} | {
"line": 176,
"column": 0
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : StrongRankCondition K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Free K V\n⊢ Module.rank K V ≤ 1 ↔ ⊤.IsPrincipal",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"LinearEquiv.symm... | [] | have := Module.Free.of_equiv (topEquiv (R := K) (M := V)).symm
rw [← Submodule.rank_le_one_iff_isPrincipal, rank_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 220,
"column": 45
} | {
"line": 220,
"column": 54
} | {
"line": 220,
"column": 55
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝⁵ : Ring K\ninst✝⁴ : StrongRankCondition K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Free K V\ninst✝ : Module.Finite K V\nthis✝¹ : Nontrivial K\ns : Type v\nhs : Basis s K V\nthis✝ : Finite s\nthis : lift.{max u v, v} #V = lift.{v, max u v} #(s → K)\n⊢ lift.{u... | [
"K : Type u\nV : Type v\ninst✝⁵ : Ring K\ninst✝⁴ : StrongRankCondition K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Free K V\ninst✝ : Module.Finite K V\nthis✝¹ : Nontrivial K\ns : Type v\nhs : Basis s K V\nthis✝ : Finite s\nthis : lift.{max u v, v} #V = lift.{v, max u v} (lift.{v, u} #K ^ lift.{u, v} #... | mk_arrow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 578,
"column": 17
} | {
"line": 581,
"column": 44
} | {
"line": 583,
"column": 0
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nS : Submodule K V\nw : V\nhS : finrank K ↥S = 1\nhw : w ∈ S\nhw0 : w ≠ 0\n⊢ S = K ∙ w",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsNoetherianRing.strongRankCon... | [] | by
have : FiniteDimensional K S := Module.finite_of_finrank_pos (by lia)
exact Eq.symm <| eq_of_le_of_finrank_le (by simpa)
(by rw [hS, finrank_span_singleton hw0]) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 459,
"column": 72
} | {
"line": 459,
"column": 86
} | {
"line": 459,
"column": 87
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ p.comp q * q + ↑n * p.comp q = p.comp q * (q + ↑n)",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Nat.cast_comm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
... | [
"R : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ p.comp q * q + p.comp q * ↑n = p.comp q * (q + ↑n)"
] | Nat.cast_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 791,
"column": 72
} | {
"line": 791,
"column": 86
} | {
"line": 791,
"column": 87
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np q : R[X]\nn : ℕ\n⊢ p.comp q * q - ↑n * p.comp q = p.comp q * (q - ↑n)",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Nat.cast_comm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"H... | [
"R : Type u\ninst✝ : Ring R\np q : R[X]\nn : ℕ\n⊢ p.comp q * q - p.comp q * ↑n = p.comp q * (q - ↑n)"
] | Nat.cast_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.CharP.Defs | {
"line": 195,
"column": 26
} | {
"line": 198,
"column": 76
} | {
"line": 200,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : AddGroupWithOne R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\n⊢ -1 ≠ 1",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Dvd.dvd",
"AddMonoid.toAddSemigroup",
... | [] | by
rw [ne_comm, ← sub_ne_zero, sub_neg_eq_add, one_add_one_eq_two, ← Nat.cast_two, Ne,
CharP.cast_eq_zero_iff R p 2]
exact fun h ↦ (Fact.out : 2 < p).not_ge <| Nat.le_of_dvd Nat.zero_lt_two h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.CharP.Defs | {
"line": 426,
"column": 65
} | {
"line": 428,
"column": 37
} | {
"line": 430,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsDomain R\n⊢ ∃ q, ExpChar R q",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Fact.casesOn",
"IsDomain.to_noZeroDivisors",
"Nat.Prime",
"ExpChar.zero",
"Ring.toNonAssocRing",
"AddGroupWithOne.toAddMonoidWi... | [] | by
obtain _ | ⟨p, ⟨hp⟩, _⟩ := CharP.exists' R
exacts [⟨1, .zero⟩, ⟨p, .prime hp⟩] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Extr | {
"line": 760,
"column": 2
} | {
"line": 760,
"column": 23
} | {
"line": 761,
"column": 2
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : SemilatticeSup β\ninst✝¹ : OrderBot β\nD : α → β\ns : Finset α\ninst✝ : Nonempty α\nb : β\nhb : b ∈ range D\nhmem : Function.invFun D b ∈ s\nhmax : ∀ a ∈ s, D a ≤ b\n⊢ s.sup D = b",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Function.invF... | [
"α : Type u\nβ : Type v\ninst✝² : SemilatticeSup β\ninst✝¹ : OrderBot β\nD : α → β\ns : Finset α\ninst✝ : Nonempty α\na : α\nhmem : Function.invFun D (D a) ∈ s\nhmax : ∀ a_1 ∈ s, D a_1 ≤ D a\n⊢ s.sup D = D a"
] | obtain ⟨a, rfl⟩ := hb | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 246,
"column": 2
} | {
"line": 248,
"column": 87
} | {
"line": 250,
"column": 0
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : Semiring R\np : R[X]\nn d : ℕ\nh : ¬n ≤ d\n⊢ (p * X ^ n).coeff d = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"not_le",
"Eq.mpr",
"le_of_add_le_right",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemiring.toAddCommMo... | [] | · refine (coeff_mul _ _ _).trans (Finset.sum_eq_zero fun x hx => ?_)
rw [coeff_X_pow, if_neg, mul_zero]
exact ((le_of_add_le_right (mem_antidiagonal.mp hx).le).trans_lt <| not_le.mp h).ne | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 277,
"column": 38
} | {
"line": 278,
"column": 22
} | {
"line": 280,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝² : Semiring R\ninst✝¹ : SemilatticeSup B\ninst✝ : OrderBot B\nD : A → B\na : A\nr : R\nhr : r ≠ 0\n⊢ supDegree D (single a r) = D a",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"False",
"Finsupp.support_single",
... | [] | by
simp [supDegree, hr] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 330,
"column": 8
} | {
"line": 330,
"column": 15
} | {
"line": 331,
"column": 6
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Semiring R\nr : R\nφ : R[X]\nc : ℕ → R\nhc : ∀ (i : ℕ), φ.coeff i = r * c i\nc' : ℕ → R := fun i ↦ if i ∈ φ.support then c i else 0\nψ : R[X] := ∑ i ∈ φ.support, (monomial i) (c' i)\ni : ℕ\nhi : φ.coeff i ≠ 0\n⊢ φ.coeff i = r * c i",
"ppTerm": "?pos✝",
"assigned": ... | [] | rw [hc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 330,
"column": 8
} | {
"line": 330,
"column": 15
} | {
"line": 331,
"column": 6
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Semiring R\nr : R\nφ : R[X]\nc : ℕ → R\nhc : ∀ (i : ℕ), φ.coeff i = r * c i\nc' : ℕ → R := fun i ↦ if i ∈ φ.support then c i else 0\nψ : R[X] := ∑ i ∈ φ.support, (monomial i) (c' i)\ni : ℕ\nhi : φ.coeff i ≠ 0\n⊢ φ.coeff i = r * c i",
"ppTerm": "?pos✝",
"assigned": ... | [] | rw [hc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 330,
"column": 8
} | {
"line": 330,
"column": 15
} | {
"line": 331,
"column": 6
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Semiring R\nr : R\nφ : R[X]\nc : ℕ → R\nhc : ∀ (i : ℕ), φ.coeff i = r * c i\nc' : ℕ → R := fun i ↦ if i ∈ φ.support then c i else 0\nψ : R[X] := ∑ i ∈ φ.support, (monomial i) (c' i)\ni : ℕ\nhi : φ.coeff i ≠ 0\n⊢ φ.coeff i = r * c i",
"ppTerm": "?pos✝",
"assigned": ... | [] | rw [hc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 391,
"column": 40
} | {
"line": 391,
"column": 51
} | {
"line": 391,
"column": 51
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝⁴ : Semiring R\ninst✝³ : LinearOrder B\ninst✝² : OrderBot B\np : R[A]\nD : A → B\ninst✝¹ : Nonempty A\ninst✝ : Nontrivial R\nh : p = 0\nhp : 0 = 1\n⊢ False",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddC... | [] | zero_ne_one | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Degree.Domain | {
"line": 66,
"column": 2
} | {
"line": 67,
"column": 33
} | {
"line": 69,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nh1 : p ∣ q\nh2 : q ≠ 0\n⊢ p.degree ≤ q.degree",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Semigroup.toMul",
"WithBot",
"Dvd.dvd",
"HMul.hMul",
"M... | [] | rcases h1 with ⟨q, rfl⟩; rw [mul_ne_zero_iff] at h2
exact degree_le_mul_left p h2.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Domain | {
"line": 66,
"column": 2
} | {
"line": 67,
"column": 33
} | {
"line": 69,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nh1 : p ∣ q\nh2 : q ≠ 0\n⊢ p.degree ≤ q.degree",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Semigroup.toMul",
"WithBot",
"Dvd.dvd",
"HMul.hMul",
"M... | [] | rcases h1 with ⟨q, rfl⟩; rw [mul_ne_zero_iff] at h2
exact degree_le_mul_left p h2.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 459,
"column": 8
} | {
"line": 459,
"column": 22
} | {
"line": 459,
"column": 23
} | [
{
"pp": "case refine_2\nA : Type u_3\nB : Type u_5\ninst✝³ : LinearOrder B\ninst✝² : OrderBot B\nD : A → B\ninst✝¹ : AddZeroClass A\nhD : Function.Injective D\nR : Type u_8\ninst✝ : Ring R\np q : R[A]\nhd : supDegree D p = supDegree D q\nhc : leadingCoeff D p = leadingCoeff D q\nhe : ¬p = q\n⊢ supDegree D (p - ... | [
"case refine_2\nA : Type u_3\nB : Type u_5\ninst✝³ : LinearOrder B\ninst✝² : OrderBot B\nD : A → B\ninst✝¹ : AddZeroClass A\nhD : Function.Injective D\nR : Type u_8\ninst✝ : Ring R\np q : R[A]\nhd : supDegree D p = supDegree D q\nhc : leadingCoeff D p = leadingCoeff D q\nhe : ¬p - q = 0\n⊢ supDegree D (p - q) ≠ sup... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 380,
"column": 6
} | {
"line": 380,
"column": 36
} | {
"line": 382,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nH : p.leadingCoeff ≠ 0\n⊢ p.leadingCoeff * q.leadingCoeff ≠ 0",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | rwa [hq.leadingCoeff, mul_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 57
} | {
"line": 509,
"column": 2
} | [
{
"pp": "case inr.inr\nR : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder B\ninst✝⁴ : OrderBot B\np q : R[A]\nD : A → B\ninst✝³ : AddZeroClass A\ninst✝² : Add B\ninst✝¹ : AddLeftStrictMono B\ninst✝ : AddRightStrictMono B\nhD : Function.Injective D\nhadd : ∀ (a1 a2 : A), D (a1 + ... | [
"case inr.inr\nR : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder B\ninst✝⁴ : OrderBot B\np q : R[A]\nD : A → B\ninst✝³ : AddZeroClass A\ninst✝² : Add B\ninst✝¹ : AddLeftStrictMono B\ninst✝ : AddRightStrictMono B\nhD : Function.Injective D\nhadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 +... | obtain ⟨ap, -, hp⟩ := exists_supDegree_mem_support D hp | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 59
} | {
"line": 556,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder B\ninst✝⁴ : OrderBot B\np q : R[A]\nD : A → B\ninst✝³ : AddZeroClass A\ninst✝² : Add B\ninst✝¹ : AddLeftStrictMono B\ninst✝ : AddRightStrictMono B\nhD : Function.Injective D\nhadd : ∀ (a1 a2 : A), D (a1 + a2) ... | [] | exact hq.supDegree_mul_of_ne_zero_left hD hadd hp.ne_zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 760,
"column": 6
} | {
"line": 760,
"column": 22
} | {
"line": 760,
"column": 23
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nx : R\n⊢ (X - C x).natDegree = 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"HSub.hSub",
"RingHom",
"id",
"instOfNatNat",
"Polynomial",
... | [
"R : Type u\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nx : R\n⊢ X.natDegree = 1"
] | natDegree_sub_C, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.Antidiagonal | {
"line": 64,
"column": 2
} | {
"line": 73,
"column": 33
} | {
"line": 74,
"column": 2
} | [
{
"pp": "case mp\nα : Type u\ninst✝ : DecidableEq α\na : α\nn : ℕ\nx y : α →₀ ℕ\n⊢ x + y = single a n → ∃ a_2 b, a_2 + b = n ∧ single a a_2 = x ∧ single a b = y",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"And.imp",
"Eq.mpr",
"Nat.instMulZer... | [
"case mpr\nα : Type u\ninst✝ : DecidableEq α\na : α\nn : ℕ\nx y : α →₀ ℕ\n⊢ (∃ a_1 b, a_1 + b = n ∧ single a a_1 = x ∧ single a b = y) → x + y = single a n"
] | · intro h
refine ⟨x a, y a, DFunLike.congr_fun h a |>.trans single_eq_same, ?_⟩
simp_rw [DFunLike.ext_iff, ← forall_and]
intro i
replace h := DFunLike.congr_fun h i
simp_rw [single_apply, Finsupp.add_apply] at h ⊢
obtain rfl | hai := Decidable.eq_or_ne a i
· exact ⟨if_pos rfl, if_pos rfl⟩
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Finsupp.Antidiagonal | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 33
} | {
"line": 77,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u\ninst✝ : DecidableEq α\na✝ : α\na b : ℕ\n⊢ single a✝ a + single a✝ b = single a✝ (a + b)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddZeroClass",
"Nat.instAddMonoid",
"AddZeroClass.toAddZero",
"AddZero.toZero",
"... | [] | exact (single_add _ _ _).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finsupp.Antidiagonal | {
"line": 102,
"column": 8
} | {
"line": 102,
"column": 13
} | {
"line": 102,
"column": 14
} | [
{
"pp": "case refine_1\nα : Type u\ninst✝¹ : DecidableEq α\nβ : Type u_1\ninst✝ : DecidableEq β\nx : α →₀ ℕ\ny : β →₀ ℕ\nu v : α ⊕ β →₀ ℕ\nx✝ : ∃ a b a_1 b_1, (a + b = x ∧ a_1 + b_1 = y) ∧ a.sumElim a_1 = u ∧ b.sumElim b_1 = v\na b : α →₀ ℕ\na' b' : β →₀ ℕ\nh1 : a + b = x ∧ a' + b' = y\nh2 : a.sumElim a' = u\nh... | [
"case refine_1\nα : Type u\ninst✝¹ : DecidableEq α\nβ : Type u_1\ninst✝ : DecidableEq β\nx : α →₀ ℕ\ny : β →₀ ℕ\nu v : α ⊕ β →₀ ℕ\nx✝ : ∃ a b a_1 b_1, (a + b = x ∧ a_1 + b_1 = y) ∧ a.sumElim a_1 = u ∧ b.sumElim b_1 = v\na b : α →₀ ℕ\na' b' : β →₀ ℕ\nh1 : a + b = x ∧ a' + b' = y\nh2 : a.sumElim a' = u\nh3 : b.sumEli... | ← h2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 613,
"column": 34
} | {
"line": 613,
"column": 77
} | {
"line": 613,
"column": 77
} | [
{
"pp": "case pos\nS : Type v\ninst✝ : CommRing S\nz p : S\nf : S[X]\ni : ℕ\ndvd_eval : p ∣ ∑ n ∈ insert i (f.support.erase i), (RingHom.id S) (f.coeff n) * z ^ n\ndvd_terms : ∀ (j : ℕ), j ≠ i → p ∣ f.coeff j * z ^ j\nhi : i ∈ f.support\n⊢ p ∣ f.coeff i * z ^ i",
"ppTerm": "?pos✝",
"assigned": true,
... | [
"case pos\nS : Type v\ninst✝ : CommRing S\nz p : S\nf : S[X]\ni : ℕ\ndvd_eval : p ∣ (RingHom.id S) (f.coeff i) * z ^ i + ∑ x ∈ f.support.erase i, (RingHom.id S) (f.coeff x) * z ^ x\ndvd_terms : ∀ (j : ℕ), j ≠ i → p ∣ f.coeff j * z ^ j\nhi : i ∈ f.support\n⊢ p ∣ f.coeff i * z ^ i"
] | Finset.sum_insert (Finset.notMem_erase _ _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 76
} | {
"line": 315,
"column": 2
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nx : σ → ℕ\nt : Finset σ\n⊢ ∏ y ∈ t, X y ^ x y = ∏ x_1 ∈ t, X x_1 ^ (Finsupp.indicator t fun i x_2 ↦ x i) x_1",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Finsupp.indicator",
"instDecidab... | [
"R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nx : σ → ℕ\nt : Finset σ\n⊢ ∀ x_1 ∈ t, x_1 ∉ (Finsupp.indicator t fun i x_2 ↦ x i).support → X x_1 ^ (Finsupp.indicator t fun i x_2 ↦ x i) x_1 = 1"
] | · exact Finset.prod_congr rfl (fun _ hi ↦ by simp [Finsupp.indicator, hi]) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 317,
"column": 8
} | {
"line": 317,
"column": 12
} | {
"line": 317,
"column": 13
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nx : σ → ℕ\nt : Finset σ\ni : σ\nhi : i ∈ t\nhi' : (Finsupp.indicator t fun i x_1 ↦ x i) i = 0\n⊢ X i ^ (Finsupp.indicator t fun i x_1 ↦ x i) i = 1",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
... | [
"R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nx : σ → ℕ\nt : Finset σ\ni : σ\nhi : i ∈ t\nhi' : (Finsupp.indicator t fun i x_1 ↦ x i) i = 0\n⊢ X i ^ 0 = 1"
] | hi', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 526,
"column": 2
} | {
"line": 526,
"column": 36
} | {
"line": 528,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nA : Type u_2\ninst✝ : AddCommMonoid A\np : MvPolynomial σ R\nb : (σ →₀ ℕ) → R → A\n⊢ p.coeff.sum b = ∑ m ∈ p.support, b m (coeff m p)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZ... | [] | simp [support, Finsupp.sum, coeff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 526,
"column": 2
} | {
"line": 526,
"column": 36
} | {
"line": 528,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nA : Type u_2\ninst✝ : AddCommMonoid A\np : MvPolynomial σ R\nb : (σ →₀ ℕ) → R → A\n⊢ p.coeff.sum b = ∑ m ∈ p.support, b m (coeff m p)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZ... | [] | simp [support, Finsupp.sum, coeff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 526,
"column": 2
} | {
"line": 526,
"column": 36
} | {
"line": 528,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nA : Type u_2\ninst✝ : AddCommMonoid A\np : MvPolynomial σ R\nb : (σ →₀ ℕ) → R → A\n⊢ p.coeff.sum b = ∑ m ∈ p.support, b m (coeff m p)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZ... | [] | simp [support, Finsupp.sum, coeff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.DFinsupp.Lex | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 38
} | {
"line": 77,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → PartialOrder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt : x < y\n⊢ Pi.Lex r (fun {i} x1 x2 ↦ x1 < x2) ⇑x ⇑y",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"... | [] | simp_rw [Pi.Lex, le_antisymm_iff]
exact lex_lt_of_lt_of_preorder r hlt | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.DFinsupp.Lex | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 38
} | {
"line": 77,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → PartialOrder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt : x < y\n⊢ Pi.Lex r (fun {i} x1 x2 ↦ x1 < x2) ⇑x ⇑y",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"... | [] | simp_rw [Pi.Lex, le_antisymm_iff]
exact lex_lt_of_lt_of_preorder r hlt | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 831,
"column": 8
} | {
"line": 831,
"column": 15
} | {
"line": 832,
"column": 6
} | [
{
"pp": "case pos\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nr : R\nφ : MvPolynomial σ R\nC : (σ →₀ ℕ) → R\nhc : ∀ (i : σ →₀ ℕ), coeff i φ = r * C i\nc' : (σ →₀ ℕ) → R := fun i ↦ if i ∈ φ.support then C i else 0\nψ : MvPolynomial σ R := ∑ i ∈ φ.support, (monomial i) (c' i)\ni : σ →₀ ℕ\nhi : i ∈ φ.suppor... | [] | rw [hc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 831,
"column": 8
} | {
"line": 831,
"column": 15
} | {
"line": 832,
"column": 6
} | [
{
"pp": "case pos\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nr : R\nφ : MvPolynomial σ R\nC : (σ →₀ ℕ) → R\nhc : ∀ (i : σ →₀ ℕ), coeff i φ = r * C i\nc' : (σ →₀ ℕ) → R := fun i ↦ if i ∈ φ.support then C i else 0\nψ : MvPolynomial σ R := ∑ i ∈ φ.support, (monomial i) (c' i)\ni : σ →₀ ℕ\nhi : i ∈ φ.suppor... | [] | rw [hc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 831,
"column": 8
} | {
"line": 831,
"column": 15
} | {
"line": 832,
"column": 6
} | [
{
"pp": "case pos\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nr : R\nφ : MvPolynomial σ R\nC : (σ →₀ ℕ) → R\nhc : ∀ (i : σ →₀ ℕ), coeff i φ = r * C i\nc' : (σ →₀ ℕ) → R := fun i ↦ if i ∈ φ.support then C i else 0\nψ : MvPolynomial σ R := ∑ i ∈ φ.support, (monomial i) (c' i)\ni : σ →₀ ℕ\nhi : i ∈ φ.suppor... | [] | rw [hc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Rename | {
"line": 217,
"column": 28
} | {
"line": 217,
"column": 81
} | {
"line": 218,
"column": 4
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nα : Type u_3\nR : Type u_4\nS : Type u_5\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ ≃ τ\np : MvPolynomial σ R\n⊢ (rename ⇑f.symm) ((rename ⇑f) p) = p",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClas... | [] | rw [rename_rename, f.symm_comp_self, rename_id_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Rename | {
"line": 217,
"column": 28
} | {
"line": 217,
"column": 81
} | {
"line": 218,
"column": 4
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nα : Type u_3\nR : Type u_4\nS : Type u_5\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ ≃ τ\np : MvPolynomial σ R\n⊢ (rename ⇑f.symm) ((rename ⇑f) p) = p",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClas... | [] | rw [rename_rename, f.symm_comp_self, rename_id_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Rename | {
"line": 217,
"column": 28
} | {
"line": 217,
"column": 81
} | {
"line": 218,
"column": 4
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nα : Type u_3\nR : Type u_4\nS : Type u_5\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ ≃ τ\np : MvPolynomial σ R\n⊢ (rename ⇑f.symm) ((rename ⇑f) p) = p",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClas... | [] | rw [rename_rename, f.symm_comp_self, rename_id_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 496,
"column": 10
} | {
"line": 496,
"column": 29
} | {
"line": 496,
"column": 30
} | [
{
"pp": "case monomial_add\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\nf : R →+* S₁\ninst✝ : DecidableEq S₁\na : σ →₀ ℕ\ns : R\np : MvPolynomial σ R\nha : a ∉ p.coeff.support\nhs : s ≠ 0\nhp : ((map f) p).coeffs ⊆ Finset.image (⇑f) p.coeffs\nih : ((map f) ((monomia... | [
"case monomial_add\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\nf : R →+* S₁\ninst✝ : DecidableEq S₁\na : σ →₀ ℕ\ns : R\np : MvPolynomial σ R\nha : a ∉ p.coeff.support\nhs : s ≠ 0\nhp : ((map f) p).coeffs ⊆ Finset.image (⇑f) p.coeffs\nih : ((map f) ((monomial a) s)).coe... | Finset.image_union, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 202,
"column": 48
} | {
"line": 207,
"column": 41
} | {
"line": 209,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\nτ : Type u_2\ninst✝ : CommSemiring R\np : MvPolynomial σ R\nf : σ → τ\nh : Injective f\n⊢ ((rename f) p).degrees = Multiset.map f p.degrees",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMu... | [] | by
classical
simp only [degrees, Multiset.map_finset_sup p.support Finsupp.toMultiset f h,
support_rename_of_injective h, Finset.sup_image]
refine Finset.sup_congr rfl fun x _ => ?_
exact (Finsupp.toMultiset_map _ _).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 222,
"column": 43
} | {
"line": 222,
"column": 70
} | {
"line": 224,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq σ\nn : σ\np : MvPolynomial σ R\n⊢ degreeOf n p = Multiset.count n p.degrees",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Lean.Meta.instFastSubsingletonForall",
... | [] | rw [degreeOf]; convert! rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 222,
"column": 43
} | {
"line": 222,
"column": 70
} | {
"line": 224,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq σ\nn : σ\np : MvPolynomial σ R\n⊢ degreeOf n p = Multiset.count n p.degrees",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Lean.Meta.instFastSubsingletonForall",
... | [] | rw [degreeOf]; convert! rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.CommRing | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 68
} | {
"line": 101,
"column": 2
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommRing R\np : MvPolynomial σ R\ninst✝ : DecidableEq σ\nd d' : σ →₀ ℕ\nc : R\nhdd' : d ≠ d'\nhc : coeff d p = c\nx : σ →₀ ℕ\nhx : x ∈ (p - ((monomial d) c - (monomial d') c)).support\nhd_not : d ∉ (p - ((monomial d) c - (monomial d') c)).support\n⊢ x ∈ p.support.eras... | [
"case inl\nR : Type u\nσ : Type u_1\ninst✝¹ : CommRing R\np : MvPolynomial σ R\ninst✝ : DecidableEq σ\nd d' : σ →₀ ℕ\nc : R\nhdd' : d ≠ d'\nhc : coeff d p = c\nx : σ →₀ ℕ\nhx : x ∈ (p - ((monomial d) c - (monomial d') c)).support\nhd_not : d ∉ (p - ((monomial d) c - (monomial d') c)).support\nhp : x ∈ p.support\n⊢ ... | rcases Finset.mem_union.mp (support_sub σ p _ hx) with hp | hdelta | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 158,
"column": 4
} | {
"line": 160,
"column": 63
} | {
"line": 162,
"column": 0
} | [
{
"pp": "case insert\nR : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nι : Type u_3\ninst✝ : DecidableEq σ\nf : ι → MvPolynomial σ R\na✝ : ι\ns✝ : Finset ι\nhs : a✝ ∉ s✝\nhsub : (∏ i ∈ s✝, f i).vars ⊆ s✝.biUnion fun i ↦ (f i).vars\n⊢ (∏ i ∈ insert a✝ s✝, f i).vars ⊆ (insert a✝ s✝).biUnion fun i ↦ (f i).vars",... | [] | simp only [hs, Finset.biUnion_insert, Finset.prod_insert, not_false_iff]
apply Finset.Subset.trans (vars_mul _ _)
exact Finset.union_subset_union (Finset.Subset.refl _) hsub | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 158,
"column": 4
} | {
"line": 160,
"column": 63
} | {
"line": 162,
"column": 0
} | [
{
"pp": "case insert\nR : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nι : Type u_3\ninst✝ : DecidableEq σ\nf : ι → MvPolynomial σ R\na✝ : ι\ns✝ : Finset ι\nhs : a✝ ∉ s✝\nhsub : (∏ i ∈ s✝, f i).vars ⊆ s✝.biUnion fun i ↦ (f i).vars\n⊢ (∏ i ∈ insert a✝ s✝, f i).vars ⊆ (insert a✝ s✝).biUnion fun i ↦ (f i).vars",... | [] | simp only [hs, Finset.biUnion_insert, Finset.prod_insert, not_false_iff]
apply Finset.Subset.trans (vars_mul _ _)
exact Finset.union_subset_union (Finset.Subset.refl _) hsub | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.CommRing | {
"line": 130,
"column": 88
} | {
"line": 131,
"column": 38
} | {
"line": 133,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommRing R\ni : σ\np : MvPolynomial σ R\n⊢ degreeOf i (-p) = degreeOf i p",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MvPolynomial.degrees_neg",
"Eq.mpr",
"NegZeroClass.toNeg",
"Nat.instMulZeroClass",
"congrArg",... | [] | by
rw [degreeOf, degreeOf, degrees_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.CommRing | {
"line": 190,
"column": 6
} | {
"line": 191,
"column": 32
} | {
"line": 191,
"column": 32
} | [
{
"pp": "S : Type v\ninst✝ : CommRing S\nR : Type u\nc : ℤ →+* S\nf : MvPolynomial R ℤ →+* S\nx p : MvPolynomial R ℤ\nn : R\nhp : eval₂ c (⇑f ∘ X) p = f p\n⊢ eval₂ c (⇑f ∘ X) (p * X n) = f (p * X n)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq... | [] | rw [eval₂_mul, eval₂_X, hp]
exact (f.map_mul _ _).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.CommRing | {
"line": 190,
"column": 6
} | {
"line": 191,
"column": 32
} | {
"line": 191,
"column": 32
} | [
{
"pp": "S : Type v\ninst✝ : CommRing S\nR : Type u\nc : ℤ →+* S\nf : MvPolynomial R ℤ →+* S\nx p : MvPolynomial R ℤ\nn : R\nhp : eval₂ c (⇑f ∘ X) p = f p\n⊢ eval₂ c (⇑f ∘ X) (p * X n) = f (p * X n)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq... | [] | rw [eval₂_mul, eval₂_X, hp]
exact (f.map_mul _ _).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 497,
"column": 4
} | {
"line": 497,
"column": 80
} | {
"line": 499,
"column": 0
} | [
{
"pp": "case neg\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nh : q.totalDegree < p.totalDegree\nhp : ¬p = 0\nb : σ →₀ ℕ\nhb₁ : b ∈ p.support\nhb₂ : (p.support.sup fun m ↦ (toMultiset m).card) = (toMultiset b).card\nhb : b ∉ q.support\nhbb : b ∈ (p + q).support\n⊢ (toMultiset b).c... | [] | exact Finset.le_sup (f := fun m => Multiset.card (Finsupp.toMultiset m)) hbb | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 590,
"column": 2
} | {
"line": 595,
"column": 25
} | {
"line": 597,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\n⊢ p.totalDegree = 0 ↔ p = C (coeff 0 p)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Finsupp.mem_support_iff",
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"E... | [] | constructor <;> intro h
· ext m; classical rw [coeff_C]; split_ifs with hm; · rw [← hm]
apply coeff_eq_zero_of_totalDegree_lt; rw [h]
exact Finset.sum_pos (fun i hi ↦ Nat.pos_of_ne_zero <| Finsupp.mem_support_iff.mp hi)
(Finsupp.support_nonempty_iff.mpr <| Ne.symm hm)
· rw [h, totalDegree_C] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 590,
"column": 2
} | {
"line": 595,
"column": 25
} | {
"line": 597,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\n⊢ p.totalDegree = 0 ↔ p = C (coeff 0 p)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Finsupp.mem_support_iff",
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"E... | [] | constructor <;> intro h
· ext m; classical rw [coeff_C]; split_ifs with hm; · rw [← hm]
apply coeff_eq_zero_of_totalDegree_lt; rw [h]
exact Finset.sum_pos (fun i hi ↦ Nat.pos_of_ne_zero <| Finsupp.mem_support_iff.mp hi)
(Finsupp.support_nonempty_iff.mpr <| Ne.symm hm)
· rw [h, totalDegree_C] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 20
} | {
"line": 290,
"column": 0
} | [
{
"pp": "case e_a\nR : Type u\nS : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ i ∈ p₁.vars, i ∈ p₁.vars → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ p₁.support\ni : σ\nhi : i ∈ d.support\nthis : i ∈ p₁.vars\n⊢ g₁ i ^ d i = g₂ i ^ d i",
... | [] | rw [h i this this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 331,
"column": 35
} | {
"line": 331,
"column": 59
} | {
"line": 331,
"column": 59
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nq : MvPolynomial σ R\ns : Set σ\nhs : ↑q.vars ⊆ s\ninst✝ : (i : σ) → Decidable (i ∈ s)\nu : σ →₀ ℕ\nhu : u ∈ q.support\n⊢ ((algebraMap R (MvPolynomial σ R)) (coeff u q) * u.prod fun i k ↦ (if i ∈ s then X i else 0) ^ k) =\n (monomial u) (coeff u q)"... | [
"R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nq : MvPolynomial σ R\ns : Set σ\nhs : ↑q.vars ⊆ s\ninst✝ : (i : σ) → Decidable (i ∈ s)\nu : σ →₀ ℕ\nhu : u ∈ q.support\n⊢ ((algebraMap R (MvPolynomial σ R)) (coeff u q) * u.prod fun i k ↦ (if i ∈ s then X i else 0) ^ k) =\n C (coeff u q) * u.prod fun n e ↦ X n ... | MvPolynomial.monomial_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 124,
"column": 91
} | {
"line": 132,
"column": 8
} | {
"line": 134,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ p.coeff p.natTrailingDegree = 0 ↔ p = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"False",
"LinearOrder.toDecidableEq",
"ENat.instNatCast",
"congrArg",
"Finset",
"Classical.byCon... | [] | by
constructor
· rintro h
by_contra hp
obtain ⟨n, hpn, hn⟩ := by simpa using min_mem_image_coe <| support_nonempty.2 hp
obtain rfl := (trailingDegree_eq_iff_natTrailingDegree_eq hp).1 hn.symm
exact hpn h
· rintro rfl
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 163,
"column": 72
} | {
"line": 163,
"column": 81
} | {
"line": 163,
"column": 81
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : q.degree < p.degree\nn : ℕ\nnd : ¬n = p.natDegree\n⊢ p.coeff n + q.coeff n = (if n = p.natDegree then 0 else p.coeff n) + q.coeff n",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : q.degree < p.degree\nn : ℕ\nnd : ¬n = p.natDegree\n⊢ p.coeff n + q.coeff n = p.coeff n + q.coeff n",
"case neg.hnc\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : q.degree < p.degree\nn : ℕ\nnd : ¬n = p.natDegree\n⊢ ¬n = (p + q).natDegree"
] | if_neg nd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 177,
"column": 72
} | {
"line": 177,
"column": 81
} | {
"line": 177,
"column": 81
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : p.degree < q.degree\nn : ℕ\nnd : ¬n = q.natDegree\n⊢ p.coeff n + q.coeff n = p.coeff n + if n = q.natDegree then 0 else q.coeff n",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"i... | [
"case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : p.degree < q.degree\nn : ℕ\nnd : ¬n = q.natDegree\n⊢ p.coeff n + q.coeff n = p.coeff n + q.coeff n",
"case neg.hnc\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : p.degree < q.degree\nn : ℕ\nnd : ¬n = q.natDegree\n⊢ ¬n = (p + q).natDegree"
] | if_neg nd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 268,
"column": 84
} | {
"line": 271,
"column": 52
} | {
"line": 273,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\n⊢ (map f p).natDegree = p.natDegree ↔ f p.leadingCoeff ≠ 0 ∨ p.natDegree = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
"... | [] | by
rcases eq_or_ne (natDegree p) 0 with h | h
· simp_rw [h, ne_eq, or_true, iff_true, ← Nat.le_zero, ← h, natDegree_map_le]
simp_all [natDegree, WithBot.unbotD_eq_unbotD_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 168,
"column": 10
} | {
"line": 168,
"column": 13
} | {
"line": 168,
"column": 14
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\ncg N O : ℕ\ng : R[X]\nCg : #g.support ≤ cg.succ\nOg : g.natDegree ≤ O\ncf : ℕ\nhcf : ∀ (f : R[X]), #f.support ≤ cf.succ → f.natDegree ≤ N → reflect (N + O) (f * g) = reflect N f * reflect O g\nf : R[X]\nCf : #f.support ≤ (cf + 1).succ\nNf : f.natDegree ≤ N\nf... | [
"case pos\nR : Type u_1\ninst✝ : Semiring R\ncg N O : ℕ\ng : R[X]\nCg : #g.support ≤ cg.succ\nOg : g.natDegree ≤ O\ncf : ℕ\nhcf : ∀ (f : R[X]), #f.support ≤ cf.succ → f.natDegree ≤ N → reflect (N + O) (f * g) = reflect N f * reflect O g\nf : R[X]\nCf : #f.support ≤ (cf + 1).succ\nNf : f.natDegree ≤ N\nf0 : f = 0\n⊢... | f0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 33
} | {
"line": 342,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\na : R\nha : a ≠ 0\nc : R\n⊢ (C a * X + C c).nextCoeff = c",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.nextCoeff_of_natDegree_pos",
"HMul.hMul",
"congrArg",
"HSub.hSub",
... | [
"R : Type u\ninst✝ : Semiring R\na : R\nha : a ≠ 0\nc : R\n⊢ (C a * X + C c).coeff ((C a * X + C c).natDegree - 1) = c",
"R : Type u\ninst✝ : Semiring R\na : R\nha : a ≠ 0\nc : R\n⊢ 0 < (C a * X + C c).natDegree"
] | rw [nextCoeff_of_natDegree_pos] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 567,
"column": 48
} | {
"line": 568,
"column": 66
} | {
"line": 570,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nf : MvPolynomial (Option σ) R\nh : f ≠ 0\n⊢ ((optionEquivLeft R σ) f).support.Nonempty",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"AlgEquiv.instEquivLike",
"Nat.ins... | [] | by
rwa [Polynomial.support_nonempty, EmbeddingLike.map_ne_zero_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Monic | {
"line": 52,
"column": 46
} | {
"line": 52,
"column": 65
} | {
"line": 52,
"column": 65
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nhp : p.Monic\n| ∑ i ∈ range (p.natDegree + 1), C (p.coeff i) * X ^ i",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroClass",
"RingHom",
"Finse... | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\nhp : p.Monic\n| C (p.coeff p.natDegree) * X ^ p.natDegree + ∑ x ∈ range p.natDegree, C (p.coeff x) * X ^ x"
] | sum_range_succ_comm | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Algebra.Polynomial.Monic | {
"line": 107,
"column": 82
} | {
"line": 110,
"column": 33
} | {
"line": 112,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nhp : p.Monic\nhpq : (p * q).Monic\n⊢ q.Monic",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"Eq.mp",
"id",
"Polynomi... | [] | by
contrapose hpq
rw [Monic.def] at hpq ⊢
rwa [leadingCoeff_monic_mul hp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 803,
"column": 45
} | {
"line": 804,
"column": 66
} | {
"line": 806,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nh : f ≠ 0\n⊢ ((finSuccEquiv R n) f).support.Nonempty",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"AlgEquiv.instEquivLike",
"Nat.instMulZer... | [] | by
rwa [Polynomial.support_nonempty, EmbeddingLike.map_ne_zero_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Monic | {
"line": 347,
"column": 4
} | {
"line": 347,
"column": 56
} | {
"line": 348,
"column": 4
} | [
{
"pp": "case mpr.inl\nR : Type u\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nf g : R[X]\nhf : f.Monic\nhg : g.Monic\nhp : (f * g).Monic\nh : f.natDegree ≠ 0 ∧ g.natDegree ≠ 0\nhl : f.natDegree ≤ g.natDegree\n⊢ ∃ f_1 g_1,\n f_1.Monic ∧\n g_1.Monic ∧ f_1 * g_1 = f * g ∧ g_1.natDegree ≠ 0 ∧ g_1.na... | [
"case mpr.inr\nR : Type u\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nf g : R[X]\nhf : f.Monic\nhg : g.Monic\nhp : (f * g).Monic\nh : f.natDegree ≠ 0 ∧ g.natDegree ≠ 0\nhl : g.natDegree ≤ f.natDegree\n⊢ ∃ f_1 g_1,\n f_1.Monic ∧\n g_1.Monic ∧ f_1 * g_1 = f * g ∧ g_1.natDegree ≠ 0 ∧ g_1.natDegree + g_... | · exact ⟨g, f, hg, hf, mul_comm g f, h.1, by gcongr⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Monic | {
"line": 513,
"column": 8
} | {
"line": 513,
"column": 22
} | {
"line": 513,
"column": 23
} | [
{
"pp": "case left\nR : Type u_1\ninst✝ : Ring R\np : R[X]\nhp : p.Monic\nq r : R[X]\nh : p * q = p * r\n⊢ q = r",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"sub_eq_zero",
... | [
"case left\nR : Type u_1\ninst✝ : Ring R\np : R[X]\nhp : p.Monic\nq r : R[X]\nh : p * q = p * r\n⊢ q - r = 0"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Monic | {
"line": 516,
"column": 8
} | {
"line": 516,
"column": 22
} | {
"line": 516,
"column": 23
} | [
{
"pp": "case right\nR : Type u_1\ninst✝ : Ring R\np : R[X]\nhp : p.Monic\nq r : R[X]\nh : q * p = r * p\n⊢ q = r",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"sub_eq_zero"... | [
"case right\nR : Type u_1\ninst✝ : Ring R\np : R[X]\nhp : p.Monic\nq r : R[X]\nh : q * p = r * p\n⊢ q - r = 0"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 26
} | {
"line": 175,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\ns : Set R[X]\np : R[X]\nhs : s.Nonempty\nhp : p ∈ Submodule.span R s\nh : ∀ p' ∈ s, p'.degree < p.degree\n⊢ False",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"False",
"Polynomial.instDecidableEq",
"Classical.propDecidable",
... | [
"case pos\nR : Type u\ninst✝ : Semiring R\ns : Set R[X]\np : R[X]\nhs : s.Nonempty\nhp : p ∈ Submodule.span R s\nh : ∀ p' ∈ s, p'.degree < p.degree\nhp_zero : p = 0\n⊢ False",
"case neg\nR : Type u\ninst✝ : Semiring R\ns : Set R[X]\np : R[X]\nhs : s.Nonempty\nhp : p ∈ Submodule.span R s\nh : ∀ p' ∈ s, p'.degree <... | by_cases hp_zero : p = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 38
} | {
"line": 266,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nn : ℕ\nhn : n ≠ 0\na✝ : Nontrivial R\n⊢ 0 < X.natDegree",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Polynomial.natDegree_X",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"Nat.instZeroLEOneClass",
"instO... | [] | simp only [natDegree_X, zero_lt_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
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