module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Star.Pointwise | {
"line": 120,
"column": 78
} | {
"line": 122,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Group α\ninst✝ : StarMul α\ns : Set α\n⊢ s⁻¹⋆ = s⋆⁻¹",
"usedConstants": [
"Set.ext",
"_private.Mathlib.Algebra.Star.Pointwise.0.Set.star_inv._simp_1_1",
"Set.star",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
... | by
ext
simp only [mem_star, mem_inv, star_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 166,
"column": 8
} | {
"line": 166,
"column": 19
} | [
{
"pp": "case pos\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 : ¬IsUnit (s • 1 - a)\n⊢ r • (s • 1 - a)⁻¹ʳ = ((r⁻¹ • s) • 1 - r⁻¹ • a)⁻¹ʳ",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul"... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 85
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : Nontrivial A\nk : 𝕜\nhk : k ≠ 0\n⊢ k ∈ resolventSet 𝕜 0",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Units.instMulAction",
"Algebra.to_smulCommClass",
"instHSMul",
... | have : IsUnit (Units.mk0 k hk • (1 : A)) := IsUnit.smul (Units.mk0 k hk) isUnit_one | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 346,
"column": 2
} | {
"line": 346,
"column": 62
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : Nontrivial A\nk : 𝕜\nhk : k ≠ 0\nthis : IsUnit (Units.mk0 k hk • 1)\n⊢ k ∈ resolventSet 𝕜 0",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"R... | simpa [mem_resolventSet_iff, Algebra.algebraMap_eq_smul_one] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 509,
"column": 15
} | {
"line": 509,
"column": 77
} | [
{
"pp": "R : Type uR\nR' : Type u_1\nS : Type uS\nT : Type u_2\nA : Type uA\nB : Type uB\nC : Type uC\nD : Type uD\nE : Type uE\nF : Type uF\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\na : A\nb : B\nx : A ⊗[R] B\nx✝¹ : A\nx✝ : B\n⊢ a • b • x... | by simp [Algebra.smul_def, right_algebraMap_apply, smul_tmul'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Unitization | {
"line": 632,
"column": 10
} | {
"line": 632,
"column": 18
} | [
{
"pp": "case inl_add_inr\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁸ : CommSemiring S\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : IsScalarTower R A A\ninst✝³ : SMulCommClass R A A\ninst✝² : Algebra S R\ninst✝¹ : DistribMulAction S A\ninst✝ : IsScalarTower S R A\n... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Unitization | {
"line": 637,
"column": 10
} | {
"line": 637,
"column": 18
} | [
{
"pp": "case inl_add_inr\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁸ : CommSemiring S\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : IsScalarTower R A A\ninst✝³ : SMulCommClass R A A\ninst✝² : Algebra S R\ninst✝¹ : DistribMulAction S A\ninst✝ : IsScalarTower S R A\n... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Unitization | {
"line": 741,
"column": 21
} | {
"line": 741,
"column": 29
} | [
{
"pp": "case inl_add_inr.inl_add_inr\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : A... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.StrictPositivity | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 58
} | [
{
"pp": "A : Type u_1\n𝕜 : Type u_2\ninst✝⁵ : Ring A\ninst✝⁴ : PartialOrder A\ninst✝³ : CommSemiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : Algebra 𝕜 A\ninst✝ : NonnegSpectrumClass 𝕜 A\na : A\nha : IsStrictlyPositive a\nx : 𝕜\nhx : x ∈ spectrum 𝕜 a\nh₁ : 0 ≤ x\n⊢ 0 < x",
"usedConstants": [
"CommS... | have h₂ : x ≠ 0 := by grind [= spectrum.zero_notMem_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 99,
"column": 36
} | {
"line": 99,
"column": 44
} | [
{
"pp": "case h.mpr.add\nR : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\ny z : ↥(centralizer R S) ⊗[R] B\nhy :\n includeLeft x * (Algebra.TensorProduct... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 214,
"column": 12
} | {
"line": 214,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : a ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nc : R\n⊢ 0 ≤ star c * 0 * c",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 221,
"column": 22
} | {
"line": 221,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : a ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nc x y : R\nx✝¹ : x ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nx✝ : y ∈ AddSubmonoid.closure (range fun s ↦ star s... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 410,
"column": 4
} | {
"line": 410,
"column": 63
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : PartialOrder R\ninst✝⁹ : StarRing R\ninst✝⁸ : StarOrderedRing R\ninst✝⁷ : NonUnitalSemiring A\ninst✝⁶ : StarRing A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : Module R A\ninst✝² : StarModule R A\ninst✝¹ : IsScalarTower R A A... | exact ⟨r • a, smul_mem_closure_star_mul hr ha, add_smul ..⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Algebra.Subalgebra.Directed | {
"line": 47,
"column": 59
} | {
"line": 77,
"column": 35
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Subalgebra R A\nι : Type u_4\ninst✝ : Nonempty ι\nK : ι → Subalgebra R A\ndir : Directed (fun x1 x2 ↦ x1 ≤ x2) K\nf : (i : ι) → ↥(K i) →ₐ[R] B\nhf... | by
let compat :
∀ (i j) (x : A) (hxi : x ∈ (K i : Set A)) (hxj : x ∈ (K j : Set A)),
f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩ := by
intro i j x hxi hxj
rcases dir i j with ⟨k, hik, hjk⟩
simp [hf i k hik, hf j k hjk]
let liftSup : ((iSup K : Subalgebra R A)) →ₐ[R] B :=
{ toFun :=
Set.iUnionL... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 120,
"column": 25
} | {
"line": 125,
"column": 32
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Free R M\nn : ℕ\nh : Module.rank R M = ↑n\n⊢ Module.Finite R M",
"usedConstants": [
"Nontrivial",
"Preorder.toLT",
"LE.le.trans_eq",
"Cardinal",
"Finite",
"Partial... | by
nontriviality R
obtain ⟨⟨ι, b⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
have := mk_lt_aleph0_iff.mp <|
b.linearIndependent.cardinal_le_rank |>.trans_eq h |>.trans_lt natCast_lt_aleph0
exact Module.Finite.of_basis b | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 58
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : Cardinal.{v}\nhn : n < Module.rank R M\ns : Set M\nhs : LinearIndepOn R id s\nhs' : n < #↑↑⟨s, hs⟩\n⊢ ∃ s, #↑s = n ∧ LinearIndepOn R id s",
"usedConstants": [
"Cardinal.le_mk_iff_exists_subset",
... | obtain ⟨t, ht, ht'⟩ := le_mk_iff_exists_subset.mp hs'.le | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 178,
"column": 25
} | {
"line": 178,
"column": 72
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : StrongRankCondition R\ninst✝ : Free R M\nι : Type w\nfst✝ : Type v\nbs : Basis fst✝ R M\n⊢ Module.rank R (ι →₀ M) = lift.{v, w} #ι * lift.{w, v} #fst✝",
"usedConstants": [
"Eq.mpr",
"Fin... | ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 597,
"column": 64
} | {
"line": 599,
"column": 58
} | [
{
"pp": "R : Type u_2\nV : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nW : Submodule R V\nm : Type u_4\nn : Type u_5\nbW : Basis m R ↥W\nbQ : Basis n R (V ⧸ W)\nj : n\n⊢ Submodule.Quotient.mk ((bW.sumQuot bQ) (Sum.inr j)) = bQ j",
"usedConstants": [
"Eq.mpr",
"Sub... | by
simpa only [sumQuot, Basis.coe_mk, Sum.elim_inr, Function.comp_apply, ← W.mkQ_apply]
using Function.rightInverse_surjInv W.mkQ_surjective _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 138,
"column": 8
} | {
"line": 138,
"column": 34
} | [
{
"pp": "case a\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type u_4\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nx : E\ny : F\nH✝ : ∀ (c : R), c • x = 0 → c • y = 0\nH : ∀ (c₁ c₂ : R), c₁ • x = c₂ •... | simp only [mul_smul, this] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Ring R\nE : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_3\ninst✝¹ : AddCommGroup F\ninst✝ : Module R F\nf g : E →ₗ.[R] F\nh : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y\n⊢ f ≤ f.sup g h",
"usedConstants": [
"Submodule",
"Lattic... | refine ⟨le_sup_left, fun z₁ z₂ hz => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.FreeAlgebra | {
"line": 290,
"column": 8
} | {
"line": 290,
"column": 19
} | [
{
"pp": "case e_a\nR✝ : Type u_1\nX : Type u_2\ninst✝⁷ : CommSemiring R✝\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : CommSemiring A\ninst✝³ : SMul R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nr : R\ns : S\nx : FreeAlgebra... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 447,
"column": 6
} | {
"line": 447,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ (p * (X + ↑n)).comp q = p.comp q * (q + ↑n)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Polynomial.instAdd",
"Nat.cast",
"Polynomial",
"instHAdd"... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 68,
"column": 19
} | {
"line": 68,
"column": 30
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\nM : Type u_7\nN : Type u_8\nO : Type u_9\ninst✝⁵ : Semiring R\ninst✝⁴ : Mul M\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nf : M →ₙ* A\nt' : R\na... | smul_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Nat.Prime.Defs | {
"line": 352,
"column": 8
} | {
"line": 352,
"column": 59
} | [
{
"pp": "n : ℕ\npos : 0 < n\nnp : n < 2\nh1 : n = n.minFac\n⊢ n.minFac ≤ n / n.minFac",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg",
"Nat.succ_le_of_lt",
"id",
"Nat.minFac",
"HDiv.hDiv",
"instOfNatNat",
"LE.le",
"instLENat",
"le_antis... | le_antisymm (le_of_lt_succ np) (succ_le_of_lt pos), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 111,
"column": 2
} | {
"line": 113,
"column": 79
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ (p * q).coeff n = ∑ x ∈ antidiagonal n, p.coeff x.1 * q.coeff x.2",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Finset",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"Nat.... | rcases p with ⟨p⟩; rcases q with ⟨q⟩
simp_rw [← ofFinsupp_mul, coeff]
exact AddMonoidAlgebra.mul_apply_antidiagonal p q n _ Finset.mem_antidiagonal | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 111,
"column": 2
} | {
"line": 113,
"column": 79
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ (p * q).coeff n = ∑ x ∈ antidiagonal n, p.coeff x.1 * q.coeff x.2",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Finset",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"Nat.... | rcases p with ⟨p⟩; rcases q with ⟨q⟩
simp_rw [← ofFinsupp_mul, coeff]
exact AddMonoidAlgebra.mul_apply_antidiagonal p q n _ Finset.mem_antidiagonal | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 199,
"column": 73
} | {
"line": 199,
"column": 82
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nk m : ℕ\nhkm : k ≠ m\nx y : R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ x + y * 0 ≠ 0 ∧ x * 0 + y ≠ 0",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"id",
"Distrib.toAdd",
"Ne",
"Mu... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 208,
"column": 75
} | {
"line": 208,
"column": 84
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nx y z : R\nhx : x ≠ 0\nhy : y ≠ 0\nhz : z ≠ 0\n⊢ x + y * 0 + z * 0 ≠ 0 ∧ x * 0 + y + z * 0 ≠ 0 ∧ x * 0 + y * 0 + z ≠ 0",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"NonUnitalNonAssocSemiring.toMu... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 135,
"column": 12
} | {
"line": 135,
"column": 23
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Semiring R\np q : R[X]\nh : q.coeff p.natDegree ≠ 0\nhp : p = 0\n⊢ degree 0 ≤ q.degree",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"congrArg",
"Preorder.toLE",
"id",
"Bot.bot",
"Polynomial.degree",
... | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 256,
"column": 6
} | {
"line": 256,
"column": 23
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ (X * p).coeff (n + 1) = p.coeff n",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"instOfNatNat",
"Polynomial",
"Polynomial.coeff",
"instHAdd",
"Polynomial.commute_X",
"HAdd.... | (commute_X p).eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 203,
"column": 53
} | {
"line": 203,
"column": 64
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nn : ℕ\na : R\nthis : DecidableEq R := Classical.decEq R\nh : a = 0\n⊢ degree 0 ≤ ↑n",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"congrArg",
"AddMonoid.toAddZeroClass",
"Preorder.toLE",
"AddZeroClass.toAd... | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 36
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nh : p ≠ 0\n⊢ (↑p.natDegree).succ = p.natDegree + 1",
"usedConstants": [
"WithBot.succ_coe",
"Nat.instSuccOrder",
"Nat.instPreorder",
"Nat",
"Polynomial.natDegree",
"Nat.instOrderBot"
]
}
] | exact WithBot.succ_coe p.natDegree | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 313,
"column": 2
} | {
"line": 315,
"column": 23
} | [
{
"pp": "case mp\nR : Type u\ninst✝ : Semiring R\nr : R\nφ : R[X]\n⊢ C r ∣ φ → ∀ (i : ℕ), r ∣ φ.coeff i",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Semigroup.toMul",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"semigroupDvd",
"RingHom",
"Polynomial.coeff_C_m... | · rintro ⟨φ, rfl⟩ c
rw [coeff_C_mul]
apply dvd_mul_right | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 415,
"column": 10
} | {
"line": 415,
"column": 24
} | [
{
"pp": "case succ\nR : Type u\ninst✝ : Semiring R\np : R[X]\na : WithBot ℕ\nhp : p.degree ≤ a\nn : ℕ\nhn : (p ^ n).degree ≤ ↑n * a\n⊢ (p ^ (n + 1)).degree ≤ ↑(n + 1) * a",
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.WithBot | {
"line": 29,
"column": 2
} | {
"line": 33,
"column": 26
} | [
{
"pp": "n m : WithBot ℕ\n⊢ n + m = 0 ↔ n = 0 ∧ m = 0",
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"WithBot.some",
"WithBot",
"congrArg",
"Nat.add_eq_zero_iff._simp_1",
"WithBot.zero",
"false_and",
"WithBot.add_bot",
"WithBot.bot_ne_zero._... | cases n
· simp [WithBot.bot_add]
cases m
· simp [WithBot.add_bot]
simp [← WithBot.coe_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.WithBot | {
"line": 29,
"column": 2
} | {
"line": 33,
"column": 26
} | [
{
"pp": "n m : WithBot ℕ\n⊢ n + m = 0 ↔ n = 0 ∧ m = 0",
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"WithBot.some",
"WithBot",
"congrArg",
"Nat.add_eq_zero_iff._simp_1",
"WithBot.zero",
"false_and",
"WithBot.add_bot",
"WithBot.bot_ne_zero._... | cases n
· simp [WithBot.bot_add]
cases m
· simp [WithBot.add_bot]
simp [← WithBot.coe_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 54,
"column": 29
} | {
"line": 54,
"column": 40
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\np : R[X]\ninst✝ : Subsingleton R\n⊢ degree 0 = ⊥",
"usedConstants": [
"Eq.mpr",
"WithBot",
"congrArg",
"id",
"Bot.bot",
"Polynomial.degree",
"Polynomial.degree_zero",
"Polynomial",
"Nat",
"Zero.toOfNat0",
... | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 15
} | [
{
"pp": "R : Type u\nn : ℕ\ninst✝ : Semiring R\nh : coeff 0 n ≠ 0\n⊢ False",
"usedConstants": [
"Polynomial",
"Polynomial.coeff",
"Zero.toOfNat0",
"Polynomial.instZero",
"OfNat.ofNat",
"rfl"
]
}
] | exact h rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 232,
"column": 2
} | {
"line": 237,
"column": 23
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nH : (p + q).natDegree < p.natDegree\n⊢ p.natDegree = q.natDegree",
"usedConstants": [
"False",
"congrArg",
"Nat.lt_or_lt_of_ne",
"lt_asymm",
"Eq.mp",
"Polynomial.natDegree_add_eq_left_of_natDegree_lt",
"Polynomial... | by_contra h
cases Nat.lt_or_lt_of_ne h with
| inl h => exact lt_asymm h (by rwa [natDegree_add_eq_right_of_natDegree_lt h] at H)
| inr h =>
rw [natDegree_add_eq_left_of_natDegree_lt h] at H
exact LT.lt.false H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 232,
"column": 2
} | {
"line": 237,
"column": 23
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nH : (p + q).natDegree < p.natDegree\n⊢ p.natDegree = q.natDegree",
"usedConstants": [
"False",
"congrArg",
"Nat.lt_or_lt_of_ne",
"lt_asymm",
"Eq.mp",
"Polynomial.natDegree_add_eq_left_of_natDegree_lt",
"Polynomial... | by_contra h
cases Nat.lt_or_lt_of_ne h with
| inl h => exact lt_asymm h (by rwa [natDegree_add_eq_right_of_natDegree_lt h] at H)
| inr h =>
rw [natDegree_add_eq_left_of_natDegree_lt h] at H
exact LT.lt.false H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 405,
"column": 45
} | {
"line": 405,
"column": 63
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝³ : Semiring R\ninst✝² : LinearOrder B\ninst✝¹ : OrderBot B\np q : R[A]\nD : A → B\ninst✝ : AddZeroClass A\nh : supDegree D q < supDegree D p\na : A\nhe : D a = supDegree D p\n⊢ (p + q) (Function.invFun D (supDegree D p)) = leadingCoeff D p",
"usedCons... | Finsupp.add_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 292,
"column": 12
} | {
"line": 292,
"column": 28
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝ : Semiring R\np q : R[X]\ni j : ℕ\nh₁ : (i, j) ∈ antidiagonal (p.natDegree + q.natDegree)\nh₂ : (i, j) ≠ (p.natDegree, q.natDegree)\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0",
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
"Finset",
... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 311,
"column": 12
} | {
"line": 311,
"column": 28
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝ : Semiring R\np q : R[X]\nH : (p.natDegree, q.natDegree) ∉ antidiagonal (p.natDegree + q.natDegree)\n⊢ (p.natDegree, q.natDegree) ∈ antidiagonal (p.natDegree + q.natDegree)",
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"congrArg",
... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 363,
"column": 65
} | {
"line": 363,
"column": 76
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nh : 0 ≠ 0\nthis : DecidableEq R := Classical.decEq R\nhp0 : ¬p = 0\nhpn0 : p ^ n = 0\nh1 : p.leadingCoeff ^ n ≠ 0\n⊢ False",
"usedConstants": [
"NonUnitalNonAssocSemiring.toMulZeroClass",
"NonAssocSemiring.toNonUnitalNonAssocSemiring",
... | exact h rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 524,
"column": 61
} | {
"line": 524,
"column": 70
} | [
{
"pp": "case inl\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) ... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 449,
"column": 8
} | {
"line": 449,
"column": 24
} | [
{
"pp": "case h\nR : Type u\ninst✝ : Semiring R\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\n⊢ (df, dg) ∈ antidiagonal (df + dg)",
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"congrArg",
"Finset",
"Nat.instAddMonoid",
"Membership.me... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 533,
"column": 32
} | {
"line": 533,
"column": 41
} | [
{
"pp": "case inr.inl\nR : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝⁷ : Semiring R\ninst✝⁶ : LinearOrder B\ninst✝⁵ : OrderBot B\np : R[A]\nD : A → B\ninst✝⁴ : AddZeroClass A\ninst✝³ : Add B\ninst✝² : AddLeftStrictMono B\ninst✝¹ : AddRightStrictMono B\ninst✝ : NoZeroDivisors R\nhD : Function.Injective D\nhadd ... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 461,
"column": 2
} | {
"line": 461,
"column": 56
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nS : Type u_2\ninst✝ : SMulZeroClass S R\na : S\np : R[X]\n⊢ (a • p).degree ≤ p.degree",
"usedConstants": [
"Iff.mpr",
"WithBot.instPreorder",
"WithBot",
"instHSMul",
"Preorder.toLT",
"NonUnitalNonAssocSemiring.toMulZeroClass",
... | refine (degree_le_iff_coeff_zero _ _).2 fun m hm => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 622,
"column": 44
} | {
"line": 622,
"column": 56
} | [
{
"pp": "R : Type u\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nhn : 0 < n\na : R\nthis : (C a).degree < (X ^ n).degree\n⊢ (X ^ n).degree = ↑n",
"usedConstants": [
"Eq.mpr",
"WithBot",
"congrArg",
"id",
"Polynomial.degree",
"instNatCastNat",
"Nat.cast",
... | degree_X_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Monomial | {
"line": 49,
"column": 4
} | {
"line": 52,
"column": 31
} | [
{
"pp": "case mpr\nR : Type u\ninst✝ : Semiring R\nf : R[X]\n⊢ (∃ n a, f = (monomial n) a) → f.support.card ≤ 1",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"Finset",
"Finset.card_le_card",
"LinearMap.instFunLike",
"Exists",
"Polynomial.mono... | rintro ⟨n, a, rfl⟩
rw [← Finset.card_singleton n]
apply Finset.card_le_card
exact support_monomial' _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Monomial | {
"line": 49,
"column": 4
} | {
"line": 52,
"column": 31
} | [
{
"pp": "case mpr\nR : Type u\ninst✝ : Semiring R\nf : R[X]\n⊢ (∃ n a, f = (monomial n) a) → f.support.card ≤ 1",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"Finset",
"Finset.card_le_card",
"LinearMap.instFunLike",
"Exists",
"Polynomial.mono... | rintro ⟨n, a, rfl⟩
rw [← Finset.card_singleton n]
apply Finset.card_le_card
exact support_monomial' _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Adjoin.Polynomial.Basic | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 26
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np : R[X]\n_hp : p ∈ ⊤\nS : Subalgebra R R[X] := R[ X]\n⊢ p ∈ S",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.algebraOfAlgebra",
"Polynomia... | rw [← sum_monomial_eq p] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 80,
"column": 22
} | {
"line": 80,
"column": 30
} | [
{
"pp": "S : Type v\ninst✝ : CommRing S\nd : ℕ\ny : S\ncast_succ : ↑d + 1 = ↑d.succ\n⊢ ↑d.succ * (∑ x ∈ range d, ↑(d.choose x) * y ^ x + ↑(d.choose d) * y ^ d) - ↑d.succ * y ^ d =\n ∑ x_1 ∈ range (d + 1), ↑((d + 1).choose x_1) * (↑x_1 * y ^ (x_1 - 1))",
"usedConstants": [
"NonUnitalNonAssocCommRing... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 81,
"column": 55
} | {
"line": 81,
"column": 64
} | [
{
"pp": "S : Type v\ninst✝ : CommRing S\nd : ℕ\ny : S\ncast_succ : ↑d + 1 = ↑d.succ\n⊢ ∑ i ∈ range d, ↑d.succ * (↑(d.choose i) * y ^ i) =\n ∑ k ∈ range d, ↑((d + 1).choose (k + 1)) * (↑(k + 1) * y ^ (k + 1 - 1)) + ↑((d + 1).choose 0) * 0",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNo... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 110,
"column": 13
} | {
"line": 110,
"column": 30
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nr : R\nn : ℕ\n⊢ (∑ x ∈ range (p.natDegree + 1), C (p.coeff x) * (C (r ^ x) * X ^ x)).coeff n = p.coeff n * r ^ n",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"RingHom",
... | finset_sum_coeff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 141,
"column": 29
} | {
"line": 141,
"column": 65
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nhp : p.Monic\nh : ∀ (x : R), f x = 0\nn : ℕ\n⊢ (map f p).coeff n = coeff 0 n",
"usedConstants": [
"Polynomial.coeff_map",
"congrArg",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"RingHom"... | simp only [h, coeff_map, coeff_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 141,
"column": 29
} | {
"line": 141,
"column": 65
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nhp : p.Monic\nh : ∀ (x : R), f x = 0\nn : ℕ\n⊢ (map f p).coeff n = coeff 0 n",
"usedConstants": [
"Polynomial.coeff_map",
"congrArg",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"RingHom"... | simp only [h, coeff_map, coeff_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 141,
"column": 29
} | {
"line": 141,
"column": 65
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nhp : p.Monic\nh : ∀ (x : R), f x = 0\nn : ℕ\n⊢ (map f p).coeff n = coeff 0 n",
"usedConstants": [
"Polynomial.coeff_map",
"congrArg",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"RingHom"... | simp only [h, coeff_map, coeff_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 56
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\n⊢ (map f p).degree ≤ p.degree",
"usedConstants": [
"Iff.mpr",
"WithBot.instPreorder",
"WithBot",
"Preorder.toLT",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"Preorder.toLE",
... | refine (degree_le_iff_coeff_zero _ _).2 fun m hm => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 157,
"column": 34
} | {
"line": 157,
"column": 45
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nhp : map f p = 0\nhp₀ : p ≠ 0\nhpq : (map f p).degree = p.degree\n⊢ degree 0 = ⊥",
"usedConstants": [
"Eq.mpr",
"WithBot",
"congrArg",
"id",
"Bot.bot",
"Polynomial.degree",
... | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 611,
"column": 4
} | {
"line": 611,
"column": 90
} | [
{
"pp": "case pos\nS : Type v\ninst✝ : CommRing S\nz p : S\nf : S[X]\ni : ℕ\ndvd_eval : p ∣ ∑ n ∈ f.support, (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",
"usedConstants": [
"Dvd.dvd",
"HMul.hMul",
"C... | rw [← Finset.insert_erase hi, Finset.sum_insert (Finset.notMem_erase _ _)] at dvd_eval | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 465,
"column": 57
} | {
"line": 467,
"column": 5
} | [
{
"pp": "R : Type u\nσ : Type u_1\na : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nh : Decidable (a = 0)\n⊢ ((monomial s) a).support = if a = 0 then ∅ else {s}",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Semiring.toModule",
"instSubsingletonDecidable",
"AddMonoidAlgebra.... | by
rw [← Subsingleton.elim (Classical.decEq R a 0) h]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 311,
"column": 2
} | {
"line": 316,
"column": 13
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nS : Type u_2\nτ : Type u_3\nx : τ → S\ninst✝ : CommSemiring S\nf : R →+* MvPolynomial τ S\ng : σ → MvPolynomial τ S\np : MvPolynomial σ R\n⊢ (eval x) (eval₂ f g p) = eval₂ ((eval x).comp f) (fun s ↦ (eval x) (g s)) p",
"usedConstants": [
"Fin... | apply induction_on p
· simp
· intro p q hp hq
simp [hp, hq]
· intro p n hp
simp [hp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 311,
"column": 2
} | {
"line": 316,
"column": 13
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nS : Type u_2\nτ : Type u_3\nx : τ → S\ninst✝ : CommSemiring S\nf : R →+* MvPolynomial τ S\ng : σ → MvPolynomial τ S\np : MvPolynomial σ R\n⊢ (eval x) (eval₂ f g p) = eval₂ ((eval x).comp f) (fun s ↦ (eval x) (g s)) p",
"usedConstants": [
"Fin... | apply induction_on p
· simp
· intro p q hp hq
simp [hp, hq]
· intro p n hp
simp [hp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 786,
"column": 2
} | {
"line": 788,
"column": 23
} | [
{
"pp": "case mp\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nr : R\nφ : MvPolynomial σ R\n⊢ C r ∣ φ → ∀ (i : σ →₀ ℕ), r ∣ coeff i φ",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"Semigroup.toMul",
"Dvd.dvd",
"HMul.hMul",
"... | · rintro ⟨φ, rfl⟩ c
rw [coeff_C_mul]
apply dvd_mul_right | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.DFinsupp.Lex | {
"line": 198,
"column": 2
} | {
"line": 200,
"column": 58
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : LinearOrder ι\ninst✝ : (i : ι) → PartialOrder (α i)\na b : Π₀ (i : ι), α i\nh : a ≤ b\nhne : ¬toLex a = toLex b\n⊢ toLex a < toLex b",
"usedConstants": [
"Finset.min'",
"Iff.mpr",
"Preorder.toLT",
"Equiv... | exact ⟨Finset.min' _ (nonempty_neLocus_iff.2 hne),
fun j hj ↦ notMem_neLocus.1 fun h ↦ (Finset.min'_le _ _ h).not_gt hj,
(h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.DFinsupp.Lex | {
"line": 198,
"column": 2
} | {
"line": 200,
"column": 58
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : LinearOrder ι\ninst✝ : (i : ι) → PartialOrder (α i)\na b : Π₀ (i : ι), α i\nh : a ≤ b\nhne : ¬toLex a = toLex b\n⊢ toLex a < toLex b",
"usedConstants": [
"Finset.min'",
"Iff.mpr",
"Preorder.toLT",
"Equiv... | exact ⟨Finset.min' _ (nonempty_neLocus_iff.2 hne),
fun j hj ↦ notMem_neLocus.1 fun h ↦ (Finset.min'_le _ _ h).not_gt hj,
(h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.DFinsupp.Lex | {
"line": 198,
"column": 2
} | {
"line": 200,
"column": 58
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : LinearOrder ι\ninst✝ : (i : ι) → PartialOrder (α i)\na b : Π₀ (i : ι), α i\nh : a ≤ b\nhne : ¬toLex a = toLex b\n⊢ toLex a < toLex b",
"usedConstants": [
"Finset.min'",
"Iff.mpr",
"Preorder.toLT",
"Equiv... | exact ⟨Finset.min' _ (nonempty_neLocus_iff.2 hne),
fun j hj ↦ notMem_neLocus.1 fun h ↦ (Finset.min'_le _ _ h).not_gt hj,
(h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 87,
"column": 12
} | {
"line": 87,
"column": 60
} | [
{
"pp": "R : Type u\nσ : Type u_1\nr : R\ninst✝ : CommSemiring R\n⊢ (C r).vars = ∅",
"usedConstants": [
"Multiset.toFinset",
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"CommSemiring.toSemiring",
"Finset",
"AddMonoid.toAddZeroClas... | rw [vars_def, degrees_C, Multiset.toFinset_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 87,
"column": 12
} | {
"line": 87,
"column": 60
} | [
{
"pp": "R : Type u\nσ : Type u_1\nr : R\ninst✝ : CommSemiring R\n⊢ (C r).vars = ∅",
"usedConstants": [
"Multiset.toFinset",
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"CommSemiring.toSemiring",
"Finset",
"AddMonoid.toAddZeroClas... | rw [vars_def, degrees_C, Multiset.toFinset_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 87,
"column": 12
} | {
"line": 87,
"column": 60
} | [
{
"pp": "R : Type u\nσ : Type u_1\nr : R\ninst✝ : CommSemiring R\n⊢ (C r).vars = ∅",
"usedConstants": [
"Multiset.toFinset",
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"CommSemiring.toSemiring",
"Finset",
"AddMonoid.toAddZeroClas... | rw [vars_def, degrees_C, Multiset.toFinset_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 471,
"column": 2
} | {
"line": 474,
"column": 30
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\np : MvPolynomial σ R\n⊢ ((map f) p).support ⊆ p.support",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"RingHom.instRingHomClass",
"Nat.instMulZeroClass",
... | simp only [Finset.subset_iff, mem_support_iff]
intro x hx
contrapose! hx
rw [coeff_map, hx, map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 471,
"column": 2
} | {
"line": 474,
"column": 30
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\np : MvPolynomial σ R\n⊢ ((map f) p).support ⊆ p.support",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"RingHom.instRingHomClass",
"Nat.instMulZeroClass",
... | simp only [Finset.subset_iff, mem_support_iff]
intro x hx
contrapose! hx
rw [coeff_map, hx, map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 59
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\ns : σ →₀ ℕ\nc : R\nhc : c ≠ 0\n⊢ ((monomial s) c).totalDegree = s.sum fun x e ↦ e",
"usedConstants": [
"Nat.instMulZeroClass",
"Nat.instLattice",
"Lattice.toSemilatticeSup",
"MvPolynomial.support_monomial",
"Semiring.to... | classical simp [totalDegree, support_monomial, if_neg hc] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 59
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\ns : σ →₀ ℕ\nc : R\nhc : c ≠ 0\n⊢ ((monomial s) c).totalDegree = s.sum fun x e ↦ e",
"usedConstants": [
"Nat.instMulZeroClass",
"Nat.instLattice",
"Lattice.toSemilatticeSup",
"MvPolynomial.support_monomial",
"Semiring.to... | classical simp [totalDegree, support_monomial, if_neg hc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 59
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\ns : σ →₀ ℕ\nc : R\nhc : c ≠ 0\n⊢ ((monomial s) c).totalDegree = s.sum fun x e ↦ e",
"usedConstants": [
"Nat.instMulZeroClass",
"Nat.instLattice",
"Lattice.toSemilatticeSup",
"MvPolynomial.support_monomial",
"Semiring.to... | classical simp [totalDegree, support_monomial, if_neg hc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Finsupp.Fin | {
"line": 32,
"column": 6
} | {
"line": 32,
"column": 44
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nn : ℕ\nσ : Fin n →₀ M\ni : M\nf : Fin (n + 1) → M → N\nh : ∀ (x : Fin (n + 1)), f x 0 = 0\n⊢ (cons i σ).sum f = f 0 i + σ.sum (Fin.tail f)",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"instNeZeroNatHAdd... | sum_fintype _ _ (fun _ => by apply h), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Monomial | {
"line": 36,
"column": 4
} | {
"line": 36,
"column": 80
} | [
{
"pp": "case succ\nR : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nhf : p.natDegree ≤ n + 1\nhn : p.coeff (n + 1) = 0\nh : p.natDegree = n.succ\n⊢ False",
"usedConstants": [
"Nat.succ_eq_add_one",
"congrArg",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"Eq.mp",
"instOfNatNat"... | rw [← Nat.succ_eq_add_one, ← h, coeff_natDegree, leadingCoeff_eq_zero] at hn | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 155,
"column": 21
} | {
"line": 155,
"column": 62
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nh : p.coeff 0 ≠ 0\n⊢ p.coeff p.natTrailingDegree = p.coeff 0",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"congrArg",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"Polynomial.natTrailingDegree_eq_zero",
"id",
"Ne",
... | (natTrailingDegree_eq_zero.mpr <| .inr h) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 56,
"column": 68
} | {
"line": 56,
"column": 96
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nthis : DecidableEq R := Classical.decEq R\nh0 : ¬p.comp q = 0\nn : ℕ\nhn : n ∈ p.support\n⊢ ↑(C (p.coeff n)).natDegree + n • q.degree ≤ ↑(C (p.coeff n)).natDegree + n • ↑q.natDegree",
"usedConstants": [
"WithBot.instPreorder",
"Polynomial.C",
... | by grw [degree_le_natDegree] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 58
} | [
{
"pp": "R : Type u\na : R\ninst✝ : Semiring R\nn : ℕ\nha : a ≠ 0\n⊢ (C a * X ^ n).trailingDegree = ↑n",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Semiring.toModule",
"HMul.hMul",
"ENat.instNatCast",
"congrArg",
"LinearMap.instFunLike",
"Polynomial.C_mul_X... | rw [C_mul_X_pow_eq_monomial, trailingDegree_monomial ha] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 58
} | [
{
"pp": "R : Type u\na : R\ninst✝ : Semiring R\nn : ℕ\nha : a ≠ 0\n⊢ (C a * X ^ n).trailingDegree = ↑n",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Semiring.toModule",
"HMul.hMul",
"ENat.instNatCast",
"congrArg",
"LinearMap.instFunLike",
"Polynomial.C_mul_X... | rw [C_mul_X_pow_eq_monomial, trailingDegree_monomial ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 58
} | [
{
"pp": "R : Type u\na : R\ninst✝ : Semiring R\nn : ℕ\nha : a ≠ 0\n⊢ (C a * X ^ n).trailingDegree = ↑n",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Semiring.toModule",
"HMul.hMul",
"ENat.instNatCast",
"congrArg",
"LinearMap.instFunLike",
"Polynomial.C_mul_X... | rw [C_mul_X_pow_eq_monomial, trailingDegree_monomial ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 317,
"column": 6
} | {
"line": 317,
"column": 22
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\ni j : ℕ\nh₁ : (i, j) ∈ antidiagonal (p.natTrailingDegree + q.natTrailingDegree)\nh₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0",
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 75
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nh : p.trailingCoeff * q.trailingCoeff ≠ 0\nhp : p ≠ 0\n⊢ (p * q).trailingDegree = p.trailingDegree + q.trailingDegree",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"NonUnitalNonAssocSemiring.to... | have hq : q ≠ 0 := fun hq => h (by rw [hq, trailingCoeff_zero, mul_zero]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 75
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nh : p.trailingCoeff * q.trailingCoeff ≠ 0\nhp : p ≠ 0\n⊢ (p * q).natTrailingDegree = p.natTrailingDegree + q.natTrailingDegree",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"NonUnitalNonAssocSe... | have hq : q ≠ 0 := fun hq => h (by rw [hq, trailingCoeff_zero, mul_zero]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 431,
"column": 12
} | {
"line": 431,
"column": 24
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ∞\nh : p.trailingDegree < n\nh₀ : p = 0\n⊢ n ≤ p.trailingDegree",
"usedConstants": [
"instTopENat",
"congrArg",
"le_top._simp_2",
"Preorder.toLE",
"instPreorderENat",
"LE.le",
"Polynomial",
"ENat",
... | by simp [h₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 162,
"column": 30
} | {
"line": 162,
"column": 39
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nN O : ℕ\nf : R[X]\nCf : #f.support ≤ Nat.succ 0\nNf : f.natDegree ≤ N\ncg : ℕ\nhcg : ∀ (g : R[X]), #g.support ≤ cg.succ → g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g\ng : R[X]\nCg : #g.support ≤ (cg + 1).succ\nOg : g.natDegree ≤ O\ng... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 162,
"column": 40
} | {
"line": 162,
"column": 49
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nN O : ℕ\nf : R[X]\nCf : #f.support ≤ Nat.succ 0\nNf : f.natDegree ≤ N\ncg : ℕ\nhcg : ∀ (g : R[X]), #g.support ≤ cg.succ → g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g\ng : R[X]\nCg : #g.support ≤ (cg + 1).succ\nOg : g.natDegree ≤ O\ng... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 163,
"column": 41
} | {
"line": 163,
"column": 49
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\nN O : ℕ\nf : R[X]\nCf : #f.support ≤ Nat.succ 0\nNf : f.natDegree ≤ N\ncg : ℕ\nhcg : ∀ (g : R[X]), #g.support ≤ cg.succ → g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g\ng : R[X]\nCg : #g.support ≤ (cg + 1).succ\nOg : g.natDegree ≤ O\ng... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 163,
"column": 76
} | {
"line": 163,
"column": 84
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\nN O : ℕ\nf : R[X]\nCf : #f.support ≤ Nat.succ 0\nNf : f.natDegree ≤ N\ncg : ℕ\nhcg : ∀ (g : R[X]), #g.support ≤ cg.succ → g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g\ng : R[X]\nCg : #g.support ≤ (cg + 1).succ\nOg : g.natDegree ≤ O\ng... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 38
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\nhf : Function.Injective ⇑f\np : R[X]\n⊢ (map f p).degree = p.degree",
"usedConstants": [
"RingHom.instRingHomClass",
"WithBot",
"congrArg",
"_private.Mathlib.Algebra.Polynomial.Degree.Lemmas.0.Poly... | simp [hf, map_ne_zero_iff, ne_or_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 38
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\nhf : Function.Injective ⇑f\np : R[X]\n⊢ (map f p).degree = p.degree",
"usedConstants": [
"RingHom.instRingHomClass",
"WithBot",
"congrArg",
"_private.Mathlib.Algebra.Polynomial.Degree.Lemmas.0.Poly... | simp [hf, map_ne_zero_iff, ne_or_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 38
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\nhf : Function.Injective ⇑f\np : R[X]\n⊢ (map f p).degree = p.degree",
"usedConstants": [
"RingHom.instRingHomClass",
"WithBot",
"congrArg",
"_private.Mathlib.Algebra.Polynomial.Degree.Lemmas.0.Poly... | simp [hf, map_ne_zero_iff, ne_or_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 425,
"column": 2
} | {
"line": 426,
"column": 59
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nh : p.comp q = 0\n⊢ p = 0 ∨ eval (q.coeff 0) p = 0 ∧ q = C (q.coeff 0)",
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.instMulZeroClass",
"HMul.hMul",
"MulZeroClass.toMul",
... | have key : p.natDegree = 0 ∨ q.natDegree = 0 := by
rw [← mul_eq_zero, ← natDegree_comp, h, natDegree_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 283,
"column": 2
} | {
"line": 284,
"column": 74
} | [
{
"pp": "case neg\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nx : R\nP : R[X]\nhx : x ≠ 0\nh : P.nextCoeff = 0\nhp : ¬P = 0\nhe : ¬P.eraseLead = 0\n⊢ ((X - C x) * P).eraseLead.eraseLead = (X - C x) * P.eraseLead",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
... | have h₁ : ((X - C x) * P).natDegree = P.natDegree + 1 := by
rw [natDegree_mul (X_sub_C_ne_zero x) hp, natDegree_X_sub_C, add_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 305,
"column": 25
} | {
"line": 305,
"column": 34
} | [
{
"pp": "case neg.a.inl.succ\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nx : R\nP : R[X]\nhx : x ≠ 0\nh : P.nextCoeff = 0\nhp : ¬P = 0\nhe : ¬P.eraseLead = 0\nh₁ : ((X - C x) * P).natDegree = P.natDegree + 1\ndP : ℕ\nhdP : P.natDegree = dP + 2\nh₂ : ((X - C x) * P).nextCoeff... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 283,
"column": 12
} | {
"line": 283,
"column": 21
} | [
{
"pp": "case pos\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nf g : R[X]\nf0 : ¬f = 0\ng0 : g = 0\n⊢ (f * 0).reverse = f.reverse * reverse 0",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"NonUnitalNonAssocSemiring.toMulZeroClass... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Monic | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 99
} | [
{
"pp": "case refine_2\nR : Type u\nι : Type y\ninst✝ : CommSemiring R\nt✝ : Multiset ι\nf : ι → R[X]\na : ι\nt : Multiset ι\nih : (∀ i ∈ t, (f i).Monic) → (Multiset.map f t).prod.Monic\nht : ∀ i ∈ a ::ₘ t, (f i).Monic\n⊢ (f a * (Multiset.map f t).prod).Monic",
"usedConstants": [
"Multiset.map",
... | exact (ht _ (Multiset.mem_cons_self _ _)).mul (ih fun _ hi => ht _ (Multiset.mem_cons_of_mem hi)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.Monic | {
"line": 305,
"column": 4
} | {
"line": 305,
"column": 28
} | [
{
"pp": "case refine_1\nR : Type u\nι : Type y\ninst✝ : CommSemiring R\nt : Multiset ι\nf : ι → R[X]\n⊢ (C 1).nextCoeff = 0",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZer... | rw [nextCoeff_C_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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