module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1092,
"column": 11
} | {
"line": 1092,
"column": 20
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\nι₁ : Type u_6\nι₂ : Type u_7\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Fintype ι₁\ninst✝¹ : Fintype ι₂\ninst✝ : DecidableEq ι₁\nb₁ : Basis ι₁ R M₁\nb₂ : Basis ι₂ R M₂\n... | ite_smul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1092,
"column": 87
} | {
"line": 1092,
"column": 94
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\nι₁ : Type u_6\nι₂ : Type u_7\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Fintype ι₁\ninst✝¹ : Fintype ι₂\ninst✝ : DecidableEq ι₁\nb₁ : Basis ι₁ R M₁\nb₂ : Basis ι₂ R M₂\n... | if_true | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Adjoin.Basic | {
"line": 157,
"column": 51
} | {
"line": 157,
"column": 72
} | [
{
"pp": "case h\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ninst✝³ : CommSemiring B\ninst✝² : Algebra R B\ninst✝¹ : Algebra A B\ninst✝ : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r • s ⊆ ↑B'\nhr : (algebraMap A B) r ∈ B'\... | smul_smul (r ^ n₁ a), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 108,
"column": 12
} | {
"line": 114,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : FaithfulSMul R K\nsurj : ∀ (z : K), ∃ x y, z = (algebraMap R K) x / (algebraMap R K) y\ninj : Function.Injective ⇑(algebraMap R K)\nthis✝ : NoZeroDivisors R\nthis : Nontrivial R\nz : K\n⊢ ∃ x, z * (algebraM... | by
have ⟨x, y, eq⟩ := surj z
obtain rfl | hy := eq_or_ne y 0
· obtain rfl : z = 0 := by simpa using eq
exact ⟨(0, 1), by simp⟩
exact ⟨⟨x, y, mem_nonZeroDivisors_iff_ne_zero.mpr hy⟩,
(eq_div_iff_mul_eq <| (map_ne_zero_iff _ inj).mpr hy).mp eq⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 173,
"column": 42
} | {
"line": 173,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ R⁰ = IsUnit.submonoid R ↔ R⁰ ≤ IsUnit.submonoid R",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"Preorder.toLE",
"IsUnit.submonoid",
"nonZeroDivisors",
"id",
... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Star.StarProjection | {
"line": 129,
"column": 11
} | {
"line": 129,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝² : NonUnitalRing R\ninst✝¹ : StarRing R\ninst✝ : IsAddTorsionFree R\np q : R\nhp : IsStarProjection p\nhq : IsStarProjection q\n⊢ p = p * q ∧ p = p * q ↔ p = p * q",
"usedConstants": [
"HMul.hMul",
"congrArg",
"and_self",
"NonUnitalRing.toNonUnitalNonAsso... | and_self | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 196,
"column": 60
} | {
"line": 196,
"column": 76
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\na : Aˣ\nh : IsUnit (↑r • 1 - ↑a)\n⊢ IsUnit (↑r⁻¹ • 1 - ↑a⁻¹)",
"usedConstants": [
"Units.val",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"AddGroupWithOne.toAddGroup",... | ← Units.smul_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 372,
"column": 19
} | {
"line": 372,
"column": 27
} | [
{
"pp": "case a.add\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nC : Type u_3\ninst✝ : Semiring C\nf g : A ⊗[R] B →+* C\nh₁ : f.comp includeLeftRingHom = g.comp includeLeftRingHom\nh₂ : f.comp includeRight.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 372,
"column": 19
} | {
"line": 372,
"column": 27
} | [
{
"pp": "case a.add\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nC : Type u_3\ninst✝ : Semiring C\nf g : A ⊗[R] B →+* C\nh₁ : f.comp includeLeftRingHom = g.comp includeLeftRingHom\nh₂ : f.comp includeRight.... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 372,
"column": 19
} | {
"line": 372,
"column": 27
} | [
{
"pp": "case a.add\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nC : Type u_3\ninst✝ : Semiring C\nf g : A ⊗[R] B →+* C\nh₁ : f.comp includeLeftRingHom = g.comp includeLeftRingHom\nh₂ : f.comp includeRight.... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 39
} | [
{
"pp": "case h.mp\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\nℬ : Module.Basis (Module.Free.ChooseBasisIndex R B) R B := Module.Free.chooseBasis R B\nb : Module.Free... | rw [Subalgebra.mem_centralizer_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 79
} | [
{
"pp": "case h.mp\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\nℬ : Module.Basis (Module.Free.ChooseBasisIndex R B) R B := Module.Free.chooseBasis R B\nb : Module.Free... | suffices x • b = b.mapRange (· * x) (by simp) from Finsupp.ext_iff.1 this j | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 39
} | [
{
"pp": "case h.mpr\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\nw : ↥(centralizer R S) ⊗[R] B\n⊢ (Algebra.TensorProduct.map (centralizer R S).val (AlgHom.id R B)).toR... | rw [Subalgebra.mem_centralizer_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 304,
"column": 6
} | {
"line": 304,
"column": 29
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.Finite R M\ninst✝ : StrongRankCondition R\nt : Finset M\nh : finrank R M < #t\ng : ↥t → R\nsum : ∑ i, g i • ↑i = 0\nz : ↥t\nnonzero : g z ≠ 0\n⊢ ∑ e ∈ t, extend Subtype.val g 0 e • e = 0",
"us... | ← Finset.sum_finset_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 585,
"column": 6
} | {
"line": 585,
"column": 93
} | [
{
"pp": "case hsp\nR✝ : Type u\nS : Type u'\nM : Type v\nM' : Type v'\nM₁ : Type v\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type u_1\ninst✝⁸ : Semiring R✝\ninst✝⁷ : CommSemiring S\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : Module R✝ M\nR : Type u_2\nV : Typ... | show Set.range (fun i ↦ (bW i : V)) = W.subtype '' (Set.range (fun i ↦ bW i)) by aesop, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 318,
"column": 42
} | {
"line": 318,
"column": 50
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module K V\ninst✝² : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\ninst✝¹ : Nontrivial V\ninst✝ : Nontrivial V'\nv : V\nhv : v ≠ 0\nw : V'\nhw ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 318,
"column": 42
} | {
"line": 318,
"column": 50
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module K V\ninst✝² : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\ninst✝¹ : Nontrivial V\ninst✝ : Nontrivial V'\nv : V\nhv : v ≠ 0\nw : V'\nhw ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 318,
"column": 42
} | {
"line": 318,
"column": 50
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module K V\ninst✝² : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\ninst✝¹ : Nontrivial V\ninst✝ : Nontrivial V'\nv : V\nhv : v ≠ 0\nw : V'\nhw ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 401,
"column": 4
} | {
"line": 401,
"column": 12
} | [
{
"pp": "case h\nK : Type u_6\nV : Type u_7\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : V →ₗ[K] K\nv : ↥f.ker\nhv : ↑v ≠ 0\nthis : LinearIndepOn K _root_.id {v}\nb₁ : Basis (↑(this.extend ⋯)) K ↥f.ker := ⋯\nw : V\nhw : ¬f w = 0 w\n⊢ f ((f w)⁻¹ • w) = 1",
"usedConstants": [
"Gro... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 1095,
"column": 8
} | {
"line": 1096,
"column": 26
} | [
{
"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 : NonUnitalAlgHomCl... | apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => star x)
(fun _ _ => rfl) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 573,
"column": 4
} | {
"line": 573,
"column": 35
} | [
{
"pp": "case h\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nh : finrank K V = 1\nx : V\nhx : x ≠ 0\ny : V\nthis : FiniteDimensional K V\n⊢ finrank K ↥(K ∙ x) = 1",
"usedConstants": [
"finrank_span_singleton"
]
}
] | exact finrank_span_singleton hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 330,
"column": 2
} | {
"line": 333,
"column": 7
} | [
{
"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\nf g : E →ₛₗ.[σ] F\nh : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y\n⊢ f ≤ f.sup g h",
"usedC... | refine ⟨le_sup_left, fun z₁ z₂ hz => ?_⟩
rw [← add_zero (f _), ← g.map_zero]
refine (sup_apply h _ _ _ ?_).symm
simpa | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 330,
"column": 2
} | {
"line": 333,
"column": 7
} | [
{
"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\nf g : E →ₛₗ.[σ] F\nh : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y\n⊢ f ≤ f.sup g h",
"usedC... | refine ⟨le_sup_left, fun z₁ z₂ hz => ?_⟩
rw [← add_zero (f _), ← g.map_zero]
refine (sup_apply h _ _ _ ?_).symm
simpa | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 80,
"column": 17
} | {
"line": 80,
"column": 33
} | [
{
"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\na₁ a₂ : R... | Finsupp.mul_sum, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Nat.Prime.Defs | {
"line": 139,
"column": 38
} | {
"line": 139,
"column": 46
} | [
{
"pp": "p : ℕ\nhp✝ : 2 ≤ p\nm : ℕ\nh : m < p\nn : ℕ\nhp : p = m * n\n⊢ m = 1 → p ≤ n",
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"Lean.Grind.instIsPreorderNat",
"congrArg",
"PartialOrder.toPreorder",
"Std.instReflLeOfIsPreorder",
"Nat.instMulOneClass",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Nat.Prime.Defs | {
"line": 139,
"column": 38
} | {
"line": 139,
"column": 46
} | [
{
"pp": "p : ℕ\nhp✝ : 2 ≤ p\nm : ℕ\nh : m < p\nn : ℕ\nhp : p = m * n\n⊢ m = 1 → p ≤ n",
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"Lean.Grind.instIsPreorderNat",
"congrArg",
"PartialOrder.toPreorder",
"Std.instReflLeOfIsPreorder",
"Nat.instMulOneClass",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Defs | {
"line": 139,
"column": 38
} | {
"line": 139,
"column": 46
} | [
{
"pp": "p : ℕ\nhp✝ : 2 ≤ p\nm : ℕ\nh : m < p\nn : ℕ\nhp : p = m * n\n⊢ m = 1 → p ≤ n",
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"Lean.Grind.instIsPreorderNat",
"congrArg",
"PartialOrder.toPreorder",
"Std.instReflLeOfIsPreorder",
"Nat.instMulOneClass",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Prime.Defs | {
"line": 390,
"column": 29
} | {
"line": 390,
"column": 37
} | [
{
"pp": "n : ℕ\nh : 2 ∣ n\nub : n.minFac ≤ 2\nlb : 0 < n.minFac\nh' : n.minFac < 2\nthis : n.minFac = 1\n⊢ False",
"usedConstants": [
"False",
"Dvd.dvd",
"congrArg",
"False.elim",
"Eq.mp",
"id",
"Nat.minFac",
"Nat.minFac_eq_one_iff._simp_1",
"instOfNatNa... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Nat.Prime.Defs | {
"line": 390,
"column": 29
} | {
"line": 390,
"column": 37
} | [
{
"pp": "n : ℕ\nh : 2 ∣ n\nub : n.minFac ≤ 2\nlb : 0 < n.minFac\nh' : n.minFac < 2\nthis : n.minFac = 1\n⊢ False",
"usedConstants": [
"False",
"Dvd.dvd",
"congrArg",
"False.elim",
"Eq.mp",
"id",
"Nat.minFac",
"Nat.minFac_eq_one_iff._simp_1",
"instOfNatNa... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Defs | {
"line": 390,
"column": 29
} | {
"line": 390,
"column": 37
} | [
{
"pp": "n : ℕ\nh : 2 ∣ n\nub : n.minFac ≤ 2\nlb : 0 < n.minFac\nh' : n.minFac < 2\nthis : n.minFac = 1\n⊢ False",
"usedConstants": [
"False",
"Dvd.dvd",
"congrArg",
"False.elim",
"Eq.mp",
"id",
"Nat.minFac",
"Nat.minFac_eq_one_iff._simp_1",
"instOfNatNa... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 881,
"column": 2
} | {
"line": 881,
"column": 12
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Ring R\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module R F\nf g : E →ₗ.[R] F\nh : f.graph = g.graph\n⊢ f = g",
"usedConstants": [
"LinearPMap.dExt",
"RingHom.id",
"Semiring.toNonAssocSemiring... | apply dExt | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 1008,
"column": 2
} | {
"line": 1009,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Ring R\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module R F\nf : E →ₗ.[R] F\n⊢ f.inverse.domain = f.toFun.range",
"usedConstants": [
"LinearMap.fst",
"Eq.mpr",
"Submodule",
"RingHomSurje... | rw [inverse, Submodule.toLinearPMap_domain, ← graph_map_snd_eq_range,
← LinearEquiv.fst_comp_prodComm, Submodule.map_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 1008,
"column": 2
} | {
"line": 1009,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Ring R\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module R F\nf : E →ₗ.[R] F\n⊢ f.inverse.domain = f.toFun.range",
"usedConstants": [
"LinearMap.fst",
"Eq.mpr",
"Submodule",
"RingHomSurje... | rw [inverse, Submodule.toLinearPMap_domain, ← graph_map_snd_eq_range,
← LinearEquiv.fst_comp_prodComm, Submodule.map_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 1008,
"column": 2
} | {
"line": 1009,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Ring R\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module R F\nf : E →ₗ.[R] F\n⊢ f.inverse.domain = f.toFun.range",
"usedConstants": [
"LinearMap.fst",
"Eq.mpr",
"Submodule",
"RingHomSurje... | rw [inverse, Submodule.toLinearPMap_domain, ← graph_map_snd_eq_range,
← LinearEquiv.fst_comp_prodComm, Submodule.map_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 308,
"column": 58
} | {
"line": 308,
"column": 66
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ ¬p.natDegree = 0 → p.coeff (p.natDegree - 1) = 0 ↔ ¬p.natDegree = 0 → ¬p.natDegree = 0 ∧ p.coeff (p.natDegree - 1) = 0",
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"eq_false",
"congrArg",
"HSub.hSub",
"instSubN... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 32
} | [
{
"pp": "R : Type u\nn : ℕ\ninst✝ : Semiring R\np : R[X]\nh : p.coeff n ≠ 0\n⊢ ↑n ≤ p.degree",
"usedConstants": [
"Polynomial.le_degree_of_ne_zero"
]
},
{
"pp": "R : Type u\nn : ℕ\ninst✝ : Semiring R\np : R[X]\nh : p.coeff n ≠ 0\n⊢ p ≠ 0",
"usedConstants": []
}
] | · exact le_degree_of_ne_zero h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 394,
"column": 18
} | {
"line": 394,
"column": 59
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nι : Type u_1\ns✝ : Finset ι\nf : ι → R[X]\na : ι\ns : Finset ι\nhas : a ∉ s\nih : (∑ i ∈ s, f i).degree ≤ s.sup fun b ↦ (f b).degree\n⊢ max (f a).degree (∑ i ∈ s, f i).degree ≤ (cons a s has).sup fun b ↦ (f b).degree",
"usedConstants": [
"WithBot.instPreorder",... | rw [sup_cons]; exact max_le_max le_rfl ih | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 394,
"column": 18
} | {
"line": 394,
"column": 59
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nι : Type u_1\ns✝ : Finset ι\nf : ι → R[X]\na : ι\ns : Finset ι\nhas : a ∉ s\nih : (∑ i ∈ s, f i).degree ≤ s.sup fun b ↦ (f b).degree\n⊢ max (f a).degree (∑ i ∈ s, f i).degree ≤ (cons a s has).sup fun b ↦ (f b).degree",
"usedConstants": [
"WithBot.instPreorder",... | rw [sup_cons]; exact max_le_max le_rfl ih | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 245,
"column": 33
} | {
"line": 245,
"column": 75
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Semiring R\np q : R[X]\nH : (p + q).natDegree = 0\nh₁ : p.natDegree = 0\n⊢ p.natDegree = q.natDegree",
"usedConstants": [
"instOfNatNat",
"dite",
"Nat",
"instDecidableEqNat",
"Polynomial.natDegree",
"OfNat.ofNat",
"Eq",
"... | on_goal 1 => by_cases h₂ : natDegree q = 0 | Batteries.Tactic.«_aux_Batteries_Tactic_PermuteGoals___elabRules_Batteries_Tactic_tacticOn_goal-_=>__1» | Batteries.Tactic.«tacticOn_goal-_=>_» |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 419,
"column": 42
} | {
"line": 419,
"column": 52
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : Semiring R\na : R\nn : ℕ\nha : ¬a = 0\n⊢ ((monomial n) a).coeff (if a = 0 then 0 else n) = a",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"LinearMap.instFunLike",
"Classical.propDecidable",
"Polynomial.monomial",
... | if_neg ha, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 419,
"column": 4
} | {
"line": 419,
"column": 68
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : Semiring R\na : R\nn : ℕ\nha : ¬a = 0\n⊢ ((monomial n) a).leadingCoeff = a",
"usedConstants": [
"Eq.mpr",
"Polynomial.leadingCoeff.eq_1",
"Semiring.toModule",
"congrArg",
"LinearMap.instFunLike",
"Classical.propDecidable",
"Pol... | rw [leadingCoeff, natDegree_monomial, if_neg ha, coeff_monomial] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 261,
"column": 45
} | {
"line": 264,
"column": 24
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nh : p.degree < q.degree\n⊢ (p + q).leadingCoeff = q.leadingCoeff",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.degree_add_eq_right_of_degree_lt",
"congrArg",
"AddMonoid.toAddZeroClass",
"Distrib.... | by
have : coeff p (natDegree q) = 0 := coeff_natDegree_eq_zero_of_degree_lt h
simp only [leadingCoeff, natDegree_eq_of_degree_eq (degree_add_eq_right_of_degree_lt h), this,
coeff_add, zero_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 490,
"column": 28
} | {
"line": 490,
"column": 55
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ p.natDegree ≤ 0 ↔ p.degree ≤ 0",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Nat.instMulZeroClass",
"WithBot",
"congrArg",
"PartialOrder.toPreorder",
"WithBot.zero",
"Preorder.toLE",
"id",
... | natDegree_le_iff_degree_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Domain | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 61
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np₁ p₂ q₁ q₂ : R[X]\nhp₁ : p₁ ≠ 0\nhq₁ : q₁ ≠ 0\nhp₂ : p₂ ≠ 0\nhq₂ : q₂ ≠ 0\nh_eq : p₁ * q₂ = p₂ * q₁\n⊢ p₁.natDegree + q₂.natDegree = p₂.natDegree + q₁.natDegree",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Polynomial.natD... | rw [← natDegree_mul hp₁ hq₂, ← natDegree_mul hp₂ hq₁, h_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 296,
"column": 12
} | {
"line": 296,
"column": 20
} | [
{
"pp": "case neg.inl\nR : Type u\ninst✝ : Semiring R\np q : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natDegree + q.natDegree\nh₂ : (i, j) ≠ (p.natDegree, q.natDegree)\nH : p.natDegree = i\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0",
"usedConstants": [
"False",
"HMul.hMul",
"AddMonoid.t... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 296,
"column": 12
} | {
"line": 296,
"column": 20
} | [
{
"pp": "case neg.inl\nR : Type u\ninst✝ : Semiring R\np q : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natDegree + q.natDegree\nh₂ : (i, j) ≠ (p.natDegree, q.natDegree)\nH : p.natDegree = i\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0",
"usedConstants": [
"False",
"HMul.hMul",
"AddMonoid.t... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 296,
"column": 12
} | {
"line": 296,
"column": 20
} | [
{
"pp": "case neg.inl\nR : Type u\ninst✝ : Semiring R\np q : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natDegree + q.natDegree\nh₂ : (i, j) ≠ (p.natDegree, q.natDegree)\nH : p.natDegree = i\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0",
"usedConstants": [
"False",
"HMul.hMul",
"AddMonoid.t... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.SmallDegree | {
"line": 147,
"column": 2
} | {
"line": 149,
"column": 29
} | [
{
"pp": "R : Type u\na b c d : R\ninst✝ : Semiring R\nha : a ≠ 0\n⊢ (C a * X ^ 3 + C b * X ^ 2 + C c * X + C d).leadingCoeff = a",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrArg",
... | rw [add_assoc, add_assoc, ← add_assoc (C b * X ^ 2), add_comm,
leadingCoeff_add_of_degree_lt <| degree_quadratic_lt_degree_C_mul_X_cb ha,
leadingCoeff_C_mul_X_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Degree.SmallDegree | {
"line": 147,
"column": 2
} | {
"line": 149,
"column": 29
} | [
{
"pp": "R : Type u\na b c d : R\ninst✝ : Semiring R\nha : a ≠ 0\n⊢ (C a * X ^ 3 + C b * X ^ 2 + C c * X + C d).leadingCoeff = a",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrArg",
... | rw [add_assoc, add_assoc, ← add_assoc (C b * X ^ 2), add_comm,
leadingCoeff_add_of_degree_lt <| degree_quadratic_lt_degree_C_mul_X_cb ha,
leadingCoeff_C_mul_X_pow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.SmallDegree | {
"line": 147,
"column": 2
} | {
"line": 149,
"column": 29
} | [
{
"pp": "R : Type u\na b c d : R\ninst✝ : Semiring R\nha : a ≠ 0\n⊢ (C a * X ^ 3 + C b * X ^ 2 + C c * X + C d).leadingCoeff = a",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrArg",
... | rw [add_assoc, add_assoc, ← add_assoc (C b * X ^ 2), add_comm,
leadingCoeff_add_of_degree_lt <| degree_quadratic_lt_degree_C_mul_X_cb ha,
leadingCoeff_C_mul_X_pow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 445,
"column": 60
} | {
"line": 453,
"column": 55
} | [
{
"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\nD : A → B\nι : Type u_7\ns : Finset ι\ni : ι\nf : ι → R[A]\ninst✝ : AddZeroClass A\nhi : i ∈ s\nhmax : ∀ j ∈ s, j ≠ i → supDegree D (f j) < supDegree D (f i)\n⊢ supDegree D (∑ j ∈ s, f j) = supDe... | by
classical
rw [← s.add_sum_erase _ hi]
by_cases! hs : s.erase i = ∅
· rw [hs, Finset.sum_empty, add_zero]; exact ⟨rfl, rfl⟩
suffices _ from ⟨supDegree_add_eq_left this, leadingCoeff_add_eq_left this⟩
refine supDegree_sum_lt ?_ (fun j hj => ?_)
· exact hs
· rw [Finset.mem_erase] at hj; exact hmax j hj.... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Eval.Coeff | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 10
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\np : R[X]\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\n⊢ ∑ n_1 ∈ p.support, ((C.comp f) (p.coeff n_1) * X ^ n_1).coeff n = f (p.coeff n)",
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClas... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Adjoin.Polynomial.Basic | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 10
} | [
{
"pp": "R : Type u\nA : Type z\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx a : A\nh : a ∈ (aeval x).range\n⊢ ∃ p, (aeval x) p = a",
"usedConstants": [
"Subalgebra.instSetLike",
"CommSemiring.toSemiring",
"AlgHom",
"AlgHom.funLike",
"Polynomial.algebra... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Adjoin.Polynomial.Basic | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 10
} | [
{
"pp": "R : Type u\nA : Type z\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A\na : ↥R[x]\ny : A := ↑a\nthis : y ∈ (aeval x).range\nh : y = ↑a\n⊢ ∃ p, (aeval x) p = y",
"usedConstants": [
"Subalgebra.instSetLike",
"congrArg",
"CommSemiring.toSemiring",
"Alg... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 27
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nt t' : R\n⊢ algEquivAevalXAddC t = algEquivAevalXAddC t' ↔ t = t'",
"usedConstants": [
"Polynomial.C",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.algEquivAevalXAddC._proof_2",
"... | simp [algEquivAevalXAddC] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 27
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nt t' : R\n⊢ algEquivAevalXAddC t = algEquivAevalXAddC t' ↔ t = t'",
"usedConstants": [
"Polynomial.C",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.algEquivAevalXAddC._proof_2",
"... | simp [algEquivAevalXAddC] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 27
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nt t' : R\n⊢ algEquivAevalXAddC t = algEquivAevalXAddC t' ↔ t = t'",
"usedConstants": [
"Polynomial.C",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.algEquivAevalXAddC._proof_2",
"... | simp [algEquivAevalXAddC] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 528,
"column": 2
} | {
"line": 530,
"column": 33
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\np : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhf : Function.Injective ⇑f\nr : R\n⊢ eval₂ f (f r) p = 0 → p.IsRoot r",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Polynomial.eval₂_hom",
"congrArg",
"CommSemiring.toSemir... | intro h
apply hf
rw [← eval₂_hom, h, f.map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 528,
"column": 2
} | {
"line": 530,
"column": 33
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\np : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhf : Function.Injective ⇑f\nr : R\n⊢ eval₂ f (f r) p = 0 → p.IsRoot r",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Polynomial.eval₂_hom",
"congrArg",
"CommSemiring.toSemir... | intro h
apply hf
rw [← eval₂_hom, h, f.map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 596,
"column": 2
} | {
"line": 596,
"column": 15
} | [
{
"pp": "case tmul\nR : Type u\nS : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nk : ℕ\ns : S\nn : N\n⊢ X ^ k • (rTensor N (↑R (monomial 0))) (s ⊗ₜ[R] n) = (rTensor N (↑R (monomial k))) (s ⊗ₜ[R] n)",
"usedConstant... | | tmul s n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Order.PiLex | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 24
} | [
{
"pp": "ι : Type u_1\nβ : ι → Type u_2\ninst✝² : LinearOrder ι\nx : (i : ι) → β i\ninst✝¹ : (i : ι) → PartialOrder (β i)\ninst✝ : WellFoundedLT ι\nj : ι\na : β j\nhj : ∀ (j_1 : ι), (fun x1 x2 ↦ x1 < x2) j_1 j → toLex x j_1 = toLex (update x j a) j_1\nh : x j < update x j a j\n⊢ x j < a",
"usedConstants": [... | rwa [update_self] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Order.PiLex | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 24
} | [
{
"pp": "ι : Type u_1\nβ : ι → Type u_2\ninst✝² : LinearOrder ι\nx : (i : ι) → β i\ninst✝¹ : (i : ι) → PartialOrder (β i)\ninst✝ : WellFoundedLT ι\nj : ι\na : β j\nhj : ∀ (j_1 : ι), (fun x1 x2 ↦ x1 < x2) j_1 j → toLex (update x j a) j_1 = toLex x j_1\nh : update x j a j < x j\n⊢ a < x j",
"usedConstants": [... | rwa [update_self] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Data.DFinsupp.NeLocus | {
"line": 69,
"column": 46
} | {
"line": 69,
"column": 59
} | [
{
"pp": "case h\nα : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → Zero (N a)\nf : Π₀ (a : α), N a\na✝ : α\n⊢ f a✝ ≠ 0 a✝ ↔ f a✝ ≠ 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Pi.zero_apply",
"DFinsupp.instDFunLike",
... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 316,
"column": 4
} | {
"line": 316,
"column": 18
} | [
{
"pp": "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",
"usedConstants": []
}
] | intro i hi hi' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 890,
"column": 36
} | {
"line": 890,
"column": 44
} | [
{
"pp": "case pos\nR : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq R\np q : MvPolynomial σ R\nh✝ : Disjoint p.support q.support\nr : R\nhl : ∀ (n : σ →₀ ℕ), coeff n p ≠ 0 → coeff n q = 0\nhr : ∀ (n : σ →₀ ℕ), coeff n q ≠ 0 → coeff n p = 0\nn : σ →₀ ℕ\nh : ¬coeff n p + coeff n q = 0\nhp : ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 890,
"column": 36
} | {
"line": 890,
"column": 44
} | [
{
"pp": "case neg\nR : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq R\np q : MvPolynomial σ R\nh✝ : Disjoint p.support q.support\nr : R\nhl : ∀ (n : σ →₀ ℕ), coeff n p ≠ 0 → coeff n q = 0\nhr : ∀ (n : σ →₀ ℕ), coeff n q ≠ 0 → coeff n p = 0\nn : σ →₀ ℕ\nh : ¬coeff n p + coeff n q = 0\nhp : ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 373,
"column": 50
} | {
"line": 373,
"column": 58
} | [
{
"pp": "case C\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nι : Type u_2\nf : R →+* S₁\nh : ι → MvPolynomial σ S₁\na✝ : R\n⊢ eval₂ ((map f).comp C) h (C a✝) = eval₂ C h ((map f) (C a✝))",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Nat.instMulZer... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 373,
"column": 50
} | {
"line": 373,
"column": 58
} | [
{
"pp": "case add\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nι : Type u_2\nf : R →+* S₁\nh : ι → MvPolynomial σ S₁\np✝ q✝ : MvPolynomial ι R\na✝¹ : eval₂ ((map f).comp C) h p✝ = eval₂ C h ((map f) p✝)\na✝ : eval₂ ((map f).comp C) h q✝ = eval₂ C h ((map f) q✝)\n⊢ ev... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 373,
"column": 50
} | {
"line": 373,
"column": 58
} | [
{
"pp": "case mul_X\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nι : Type u_2\nf : R →+* S₁\nh : ι → MvPolynomial σ S₁\np✝ : MvPolynomial ι R\nn✝ : ι\na✝ : eval₂ ((map f).comp C) h p✝ = eval₂ C h ((map f) p✝)\n⊢ eval₂ ((map f).comp C) h (p✝ * X n✝) = eval₂ C h ((map ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.MvPolynomial.Rename | {
"line": 155,
"column": 57
} | {
"line": 155,
"column": 74
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\nf : σ → τ\nhf : Injective f\ni : σ\n⊢ (if h : f i ∈ range f then X ((Equiv.ofInjective f hf).symm ⟨f i, h⟩) else 0) = X i",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Eq... | dif_pos ⟨i, rfl⟩, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 508,
"column": 6
} | {
"line": 513,
"column": 28
} | [
{
"pp": "case refine_2.C\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\na : S₁\nhx : ↑(C a).coeffs ⊆ range ⇑f\n⊢ C a ∈ range ⇑(map f)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"RingHom.instRingHomClass",
"False",
"Nat.in... | by_cases h : a = 0
· subst h
exact ⟨0, by simp⟩
· simp only [coeffs_C, h, reduceIte, Finset.coe_singleton, Set.singleton_subset_iff] at hx
obtain ⟨b, rfl⟩ := hx
exact ⟨C b, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 508,
"column": 6
} | {
"line": 513,
"column": 28
} | [
{
"pp": "case refine_2.C\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\na : S₁\nhx : ↑(C a).coeffs ⊆ range ⇑f\n⊢ C a ∈ range ⇑(map f)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"RingHom.instRingHomClass",
"False",
"Nat.in... | by_cases h : a = 0
· subst h
exact ⟨0, by simp⟩
· simp only [coeffs_C, h, reduceIte, Finset.coe_singleton, Set.singleton_subset_iff] at hx
obtain ⟨b, rfl⟩ := hx
exact ⟨C b, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 10
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\nh : p.degrees = 0\n⊢ p.degrees.card = 0",
"usedConstants": [
"congrArg",
"Multiset",
"instOfNatNat",
"Nat",
"True",
"eq_self",
"of_eq_true",
"Zero.toOfNat0",
"Multiset.card"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 241,
"column": 2
} | {
"line": 265,
"column": 21
} | [
{
"pp": "R : Type u\nS : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\ng : σ → S\np : MvPolynomial σ R\nhp : ∀ i ∈ p.vars, g i = 0\n⊢ (eval₂Hom f g) p = f (constantCoeff p)",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Finsupp.mem_support_iff",
... | conv_lhs => rw [p.as_sum]
simp only [map_sum, eval₂Hom_monomial]
by_cases h0 : constantCoeff p = 0
on_goal 1 =>
rw [h0, f.map_zero, Finset.sum_eq_zero]
intro d hd
on_goal 2 =>
rw [Finset.sum_eq_single (0 : σ →₀ ℕ)]
· rw [Finsupp.prod_zero_index, mul_one]
rfl
on_goal 1 => intro d hd hd0... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 241,
"column": 2
} | {
"line": 265,
"column": 21
} | [
{
"pp": "R : Type u\nS : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\ng : σ → S\np : MvPolynomial σ R\nhp : ∀ i ∈ p.vars, g i = 0\n⊢ (eval₂Hom f g) p = f (constantCoeff p)",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Finsupp.mem_support_iff",
... | conv_lhs => rw [p.as_sum]
simp only [map_sum, eval₂Hom_monomial]
by_cases h0 : constantCoeff p = 0
on_goal 1 =>
rw [h0, f.map_zero, Finset.sum_eq_zero]
intro d hd
on_goal 2 =>
rw [Finset.sum_eq_single (0 : σ →₀ ℕ)]
· rw [Finsupp.prod_zero_index, mul_one]
rfl
on_goal 1 => intro d hd hd0... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.CommRing | {
"line": 120,
"column": 2
} | {
"line": 121,
"column": 69
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommRing R\np q : MvPolynomial σ R\ninst✝ : DecidableEq σ\nhpq : Disjoint p.vars q.vars\n⊢ (p - q).vars = p.vars ∪ q.vars",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Finse... | rw [← vars_neg q] at hpq
convert! vars_add_of_disjoint hpq using 2 <;> simp [sub_eq_add_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.CommRing | {
"line": 120,
"column": 2
} | {
"line": 121,
"column": 69
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommRing R\np q : MvPolynomial σ R\ninst✝ : DecidableEq σ\nhpq : Disjoint p.vars q.vars\n⊢ (p - q).vars = p.vars ∪ q.vars",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Finse... | rw [← vars_neg q] at hpq
convert! vars_add_of_disjoint hpq using 2 <;> simp [sub_eq_add_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Fin | {
"line": 83,
"column": 44
} | {
"line": 83,
"column": 57
} | [
{
"pp": "n : ℕ\nM : Type u_1\ninst✝ : Zero M\ny : M\ns : Fin n →₀ M\nc : cons y s = 0\n⊢ 0 0 = 0",
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
"congrArg",
"Pi.zero_apply",
"id",
"Fin.instOfNat",
"Pi.instZero",
"instOfNatNat",
"instHAdd",
"H... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Rename | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 10
} | [
{
"pp": "case add\nσ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\nf : σ → τ\nhf : Injective f\nd : σ →₀ ℕ\np✝ q✝ : MvPolynomial σ R\na✝¹ : coeff (Finsupp.mapDomain f d) ((rename f) p✝) = coeff d p✝\na✝ : coeff (Finsupp.mapDomain f d) ((rename f) q✝) = coeff d q✝\n⊢ coeff (Finsupp.mapDomain f ... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 636,
"column": 6
} | {
"line": 636,
"column": 22
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\ns t : Multiset σ\n⊢ degreesLE R σ (s + t) = degreesLE R σ s * degreesLE R σ t",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Semiring.toModule",
"HMul.hMul",
... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Monomial | {
"line": 37,
"column": 4
} | {
"line": 37,
"column": 12
} | [
{
"pp": "case succ\nR : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nhf : p.natDegree ≤ n + 1\nhn : p = 0\nh : p.natDegree = n.succ\n⊢ False",
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"False.elim",
"Nat.add_eq_zero_iff._simp_1",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 79,
"column": 11
} | {
"line": 79,
"column": 38
} | [
{
"pp": "R : Type u\nn : ℕ\ninst✝ : Semiring R\np : R[X]\n⊢ p.natDegree ≤ n ↔ ∀ (N : ℕ), n < N → p.coeff N = 0",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"congrArg",
"Preorder.toLE",
"id",
"Polynomial.degree",
"LE.le",
"instLENat",... | natDegree_le_iff_degree_le, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 347,
"column": 4
} | {
"line": 347,
"column": 12
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝ : Semiring R\np : R[X]\nhp : p.natDegree = 1\nh : p = 0\n⊢ False",
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"False.elim",
"Eq.mp",
"instOfNatNat",
"Polynomial",
"zero_ne_one.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 238,
"column": 6
} | {
"line": 238,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\n⊢ f.reverse.natDegree ≤ f.natDegree",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"congrArg",
"Preorder.toLE",
"id",
"Polynomial.degree",
"LE.le",
"instLENat",
"instNatCastNat... | natDegree_le_iff_degree_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Monic | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 22
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\na✝ : Nontrivial R\n⊢ (p * q).natDegree = p.natDegree + q.natDegree",
"usedConstants": [
"Polynomial.natDegree_mul'"
]
}
] | apply natDegree_mul' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 351,
"column": 74
} | {
"line": 352,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nf : R[X]\nN : ℕ\n⊢ reflect N (-f) = -reflect N f",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Polynomial.instOne",
"Polynomial.C_1",
"Pol... | by
rw [neg_eq_neg_one_mul, ← C_1, ← C_neg, reflect_C_mul, C_neg, C_1, ← neg_eq_neg_one_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 10
} | [
{
"pp": "case h\nR : Type u\ninst✝ : CommSemiring R\nt : Multiset R[X]\nh : ∀ f ∈ t, f.Monic\na✝ : Nontrivial R\n⊢ (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0",
"usedConstants": [
"one_pow",
"Multiset.prod_replicate",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.Monic | {
"line": 356,
"column": 2
} | {
"line": 357,
"column": 63
} | [
{
"pp": "case mp\nR : Type u\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np : R[X]\nhp : p.Monic\nhp1 : p ≠ 1\n⊢ (∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → g.natDegree ∉ Ioc 0 (p.natDegree / 2)) →\n ∀ (q : R[X]), q.Monic → q.natDegree ∈ Ioc 0 (p.natDegree / 2) → ¬q ∣ p",
"usedConstants": [
... | · rintro h g hg hdg ⟨f, rfl⟩
exact h f g (hg.of_mul_monic_left hp) hg (mul_comm f g) hdg | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 378,
"column": 47
} | {
"line": 378,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\nS : Type u_2\nF : Type u_3\ninst✝² : Semiring S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : AddMonoidHomClass F R[X] S[X]\nφ : F\np : R[X]\nk : ℕ\nφ_k : ∀ (f : R[X]), f.natDegree < k → φ f = 0\nφ_mon : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = n - k\n⊢ ∀ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 378,
"column": 47
} | {
"line": 378,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\nS : Type u_2\nF : Type u_3\ninst✝² : Semiring S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : AddMonoidHomClass F R[X] S[X]\nφ : F\np : R[X]\nk : ℕ\nφ_k : ∀ (f : R[X]), f.natDegree < k → φ f = 0\nφ_mon : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = n - k\n⊢ ∀ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 378,
"column": 47
} | {
"line": 378,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\nS : Type u_2\nF : Type u_3\ninst✝² : Semiring S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : AddMonoidHomClass F R[X] S[X]\nφ : F\np : R[X]\nk : ℕ\nφ_k : ∀ (f : R[X]), f.natDegree < k → φ f = 0\nφ_mon : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = n - k\n⊢ ∀ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 423,
"column": 6
} | {
"line": 423,
"column": 27
} | [
{
"pp": "case succ.refine_1\nR : Type u_1\ninst✝ : Semiring R\nn : ℕ\nhn : ∀ {f : R[X]}, #f.support = n → ∃ k x, ∃ (_ : StrictMono k) (_ : ∀ (i : Fin n), x i ≠ 0), f = ∑ i, C (x i) * X ^ k i\nf : R[X]\nh : #f.support = n + 1\nk : Fin n → ℕ\nx : Fin n → R\nhk : StrictMono k\nhx : ∀ (i : Fin n), x i ≠ 0\nhf : f.e... | obtain ⟨i, rfl⟩ := hi | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 432,
"column": 8
} | {
"line": 432,
"column": 66
} | [
{
"pp": "case neg.h\nR : Type u_1\ninst✝ : Semiring R\nn : ℕ\nhn : ∀ {f : R[X]}, #f.support = n → ∃ k x, ∃ (_ : StrictMono k) (_ : ∀ (i : Fin n), x i ≠ 0), f = ∑ i, C (x i) * X ^ k i\nf : R[X]\nh : #f.support = n + 1\nk : Fin n → ℕ\nx : Fin n → R\nhk : StrictMono k\nhx : ∀ (i : Fin n), x i ≠ 0\nhf : f.eraseLead... | rw [sum_eq_single, coeff_C_mul, coeff_X_pow_self, mul_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Monic | {
"line": 400,
"column": 2
} | {
"line": 400,
"column": 10
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Nontrivial S\nP : R[X]\nhmo : P.Monic\nf : R →+* S\n⊢ (Polynomial.map f P).degree = P.degree",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"False",
"WithBo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.Monic | {
"line": 400,
"column": 2
} | {
"line": 400,
"column": 10
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Nontrivial S\nP : R[X]\nhmo : P.Monic\nf : R →+* S\n⊢ (Polynomial.map f P).degree = P.degree",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"False",
"WithBo... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Monic | {
"line": 400,
"column": 2
} | {
"line": 400,
"column": 10
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Nontrivial S\nP : R[X]\nhmo : P.Monic\nf : R →+* S\n⊢ (Polynomial.map f P).degree = P.degree",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"False",
"WithBo... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Monic | {
"line": 399,
"column": 35
} | {
"line": 400,
"column": 10
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Nontrivial S\nP : R[X]\nhmo : P.Monic\nf : R →+* S\n⊢ (Polynomial.map f P).degree = P.degree",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"False",
"WithBo... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 439,
"column": 8
} | {
"line": 439,
"column": 29
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nn : ℕ\nhn : ∀ {f : R[X]}, #f.support = n → ∃ k x, ∃ (_ : StrictMono k) (_ : ∀ (i : Fin n), x i ≠ 0), f = ∑ i, C (x i) * X ^ k i\nf : R[X]\nh : #f.support = n + 1\nk : Fin n → ℕ\nx : Fin n → R\nhk : StrictMono k\nhx : ∀ (i : Fin n), x i ≠ 0\nhf : f.eraseLead =... | obtain ⟨i, rfl⟩ := hi | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 40
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Semiring R\nn : ℕ\nx : R[X]\nx_zero : x = 0\n⊢ x ∈ degreeLT R (n + 1) ↔ x ∈ degreeLE R ↑n",
"usedConstants": [
"Eq.mpr",
"Polynomial.degreeLT",
"Submodule",
"WithBot",
"Semiring.toModule",
"congrArg",
"Polynomial.degreeLE",
... | simp_rw [x_zero, Submodule.zero_mem] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 40
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Semiring R\nn : ℕ\nx : R[X]\nx_zero : x = 0\n⊢ x ∈ degreeLT R (n + 1) ↔ x ∈ degreeLE R ↑n",
"usedConstants": [
"Eq.mpr",
"Polynomial.degreeLT",
"Submodule",
"WithBot",
"Semiring.toModule",
"congrArg",
"Polynomial.degreeLE",
... | simp_rw [x_zero, Submodule.zero_mem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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