module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 440,
"column": 4
} | {
"line": 440,
"column": 21
} | {
"line": 441,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\nx : M\nthis : ∀ (x : M), -x ∈ range ⇑P.root → x ∈ range ⇑P.root\nh : - -x ∈ range ⇑P.root\n⊢ -x ∈ range ⇑P.roo... | [] | exact this (-x) h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 1026,
"column": 21
} | {
"line": 1026,
"column": 64
} | {
"line": 1026,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nih : (∏ x ∈ Finset.range k, (2 * ↑x + 1)) * eval 1 ((⇑derivative)^[k] (T R n)) = ∏ x ∈ Finset.range k, (↑n ^ 2 - ↑x ^ 2)\n⊢ (↑n ^ 2 - ↑k ^ 2) * ∏ x ∈ Finset.range k, (↑n ^ 2 - ↑x ^ 2) = ∏ x ∈ insert k (Finset.range k), (↑n ^ 2 - ↑x ^ 2)",
"ppTerm": "?... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nih : (∏ x ∈ Finset.range k, (2 * ↑x + 1)) * eval 1 ((⇑derivative)^[k] (T R n)) = ∏ x ∈ Finset.range k, (↑n ^ 2 - ↑x ^ 2)\n⊢ (↑n ^ 2 - ↑k ^ 2) * ∏ x ∈ Finset.range k, (↑n ^ 2 - ↑x ^ 2) =\n (↑n ^ 2 - ↑k ^ 2) * ∏ x ∈ Finset.range k, (↑n ^ 2 - ↑x ^ 2)"
] | Finset.prod_insert Finset.notMem_range_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.NumDen | {
"line": 110,
"column": 4
} | {
"line": 118,
"column": 47
} | {
"line": 120,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsInteger A x\n⊢ IsUnit ↑(den A x)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"IsLo... | [] | have ⟨v, h⟩ := h
apply IsRelPrime.isUnit_of_dvd (num_den_reduced A x).symm
use v
apply_fun algebraMap A K
· simp only [map_mul, h]
rw [mul_comm, ← div_eq_iff]
· simp only [mk'_num_den']
simp
exact FaithfulSMul.algebraMap_injective A K | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.NumDen | {
"line": 110,
"column": 4
} | {
"line": 118,
"column": 47
} | {
"line": 120,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : IsInteger A x\n⊢ IsUnit ↑(den A x)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"IsLo... | [] | have ⟨v, h⟩ := h
apply IsRelPrime.isUnit_of_dvd (num_den_reduced A x).symm
use v
apply_fun algebraMap A K
· simp only [map_mul, h]
rw [mul_comm, ← div_eq_iff]
· simp only [mk'_num_den']
simp
exact FaithfulSMul.algebraMap_injective A K | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 84
} | {
"line": 200,
"column": 2
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module L M\ninst✝⁴ : Module L N\ninst✝³ : Module K M\ninst✝² : Module K N\ninst✝¹ : IsScalarTower K L M\np : M →ₗ[L] N →ₗ[L] L\ni... | [] | exact hp (b j) (by simpa [b] using hv₁ j.2) (bN i) (by simpa [bN] using hw₁ i.2) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.FiniteLength | {
"line": 93,
"column": 2
} | {
"line": 106,
"column": 13
} | {
"line": 108,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : Ring R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\n⊢ [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsFiniteLength R M,\n ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m].TFAE",
"ppTerm": "?m... | [] | rw [isFiniteLength_iff_isNoetherian_isArtinian]
obtain ⟨s, hs⟩ := IsSemisimpleModule.exists_sSupIndep_sSup_simples_eq_top R M
tfae_have 1 ↔ 2 := ⟨fun _ ↦ inferInstance, fun _ ↦ inferInstance⟩
tfae_have 2 → 5 := fun _ ↦ ⟨s, WellFoundedGT.finite_of_sSupIndep hs.1, hs⟩
tfae_have 3 → 5 := fun _ ↦ ⟨s, WellFoundedLT.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FiniteLength | {
"line": 93,
"column": 2
} | {
"line": 106,
"column": 13
} | {
"line": 108,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : Ring R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\n⊢ [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsFiniteLength R M,\n ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m].TFAE",
"ppTerm": "?m... | [] | rw [isFiniteLength_iff_isNoetherian_isArtinian]
obtain ⟨s, hs⟩ := IsSemisimpleModule.exists_sSupIndep_sSup_simples_eq_top R M
tfae_have 1 ↔ 2 := ⟨fun _ ↦ inferInstance, fun _ ↦ inferInstance⟩
tfae_have 2 → 5 := fun _ ↦ ⟨s, WellFoundedGT.finite_of_sSupIndep hs.1, hs⟩
tfae_have 3 → 5 := fun _ ↦ ⟨s, WellFoundedLT.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 162,
"column": 6
} | {
"line": 162,
"column": 71
} | {
"line": 163,
"column": 4
} | [
{
"pp": "case h\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na✝ b✝ : ValueGroup A K\nb : K\ne f : A\nhf : (algebraMap A K) (f * e) * b = (algebraMap A K) 1 * b\nhb : ¬b = 0\n⊢ 1 = f * e",
"ppT... | [] | exact IsFractionRing.injective _ _ (mul_right_cancel₀ hb hf).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 184,
"column": 10
} | {
"line": 184,
"column": 12
} | {
"line": 184,
"column": 12
} | [
{
"pp": "A : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na✝ : ValueGroup A K\na : K\nha : a = 0\n⊢ Quot.mk (⇑(MulAction.orbitRel Aˣ K)) a = 0",
"ppTerm": "?m.247",
"assigned": true,
"usedC... | [
"A : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\na✝ : ValueGroup A K\na : K\nha : a = 0\n⊢ Quot.mk (⇑(MulAction.orbitRel Aˣ K)) 0 = 0"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 86,
"column": 2
} | {
"line": 94,
"column": 41
} | {
"line": 95,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : CommRing S\ninst✝³ : Module S M\nJ : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : IsScalarTower R S M\nhIJ : Ideal.map (algebraMap R S) I ≤ J\ninst✝ : IsHausdorff J M\nx... | [
"case hxy\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : CommRing S\ninst✝³ : Module S M\nJ : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : IsScalarTower R S M\nhIJ : Ideal.map (algebraMap R S) I ≤ J\ninst✝ : IsHausdorff J M\nx : M\nh : ... | · rw [← AddSubgroup.toAddSubmonoid_le]
simp only [Submodule.smul_toAddSubmonoid, Submodule.top_toAddSubmonoid]
rw [AddSubmonoid.smul_le]
intro r hr m hm
rw [← algebraMap_smul S r m]
apply AddSubmonoid.smul_mem_smul ?_ hm
have := Ideal.mem_map_of_mem (algebraMap R S) hr
simp only [Ideal.map_p... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 249,
"column": 8
} | {
"line": 249,
"column": 44
} | {
"line": 250,
"column": 6
} | [
{
"pp": "case left\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\nx y : A\nh :\n (algebraMap (↥(valuation A K).integer) K)\n ((have this := { toFun := fun a ↦ ⟨(algebraMap A K) a, ⋯⟩, map_mul'... | [] | exact IsFractionRing.injective _ _ h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Valuation.Integers | {
"line": 443,
"column": 2
} | {
"line": 448,
"column": 82
} | {
"line": 450,
"column": 0
} | [
{
"pp": "Γ₀ : Type v\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nK : Type u_1\ninst✝ : Field K\nv : Valuation K Γ₀\nI : Ideal ↥v.integer\nx : ↥v.integer\nhx : x ∈ I\n⊢ v.leIdeal (v ↑x) ≤ I",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"div_le_one_of_le₀",
"ZeroMemClass.coe_e... | [] | rcases eq_or_ne x 0 with rfl | hx0
· simp
intro y hy
have : v ((y : K) / x) ≤ 1 := by simpa using div_le_one_of_le₀ hy zero_le
convert! I.smul_mem ⟨_, this⟩ hx using 1
simp [Subtype.ext_iff, div_mul_cancel₀ _ (ZeroMemClass.coe_eq_zero.not.mpr hx0)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.Integers | {
"line": 443,
"column": 2
} | {
"line": 448,
"column": 82
} | {
"line": 450,
"column": 0
} | [
{
"pp": "Γ₀ : Type v\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nK : Type u_1\ninst✝ : Field K\nv : Valuation K Γ₀\nI : Ideal ↥v.integer\nx : ↥v.integer\nhx : x ∈ I\n⊢ v.leIdeal (v ↑x) ≤ I",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"div_le_one_of_le₀",
"ZeroMemClass.coe_e... | [] | rcases eq_or_ne x 0 with rfl | hx0
· simp
intro y hy
have : v ((y : K) / x) ≤ 1 := by simpa using div_le_one_of_le₀ hy zero_le
convert! I.smul_mem ⟨_, this⟩ hx using 1
simp [Subtype.ext_iff, div_mul_cancel₀ _ (ZeroMemClass.coe_eq_zero.not.mpr hx0)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 409,
"column": 6
} | {
"line": 409,
"column": 62
} | {
"line": 409,
"column": 62
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nx✝¹ : AdicCompletion I M\nh : ∀ (n : ℕ), x✝¹ ≡ 0 [SMOD I ^ n • ⊤]\nn : ℕ\nr : R\nhr : r ∈ I ^... | [
"case refine_1.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nx✝¹ : AdicCompletion I M\nh : ∀ (n : ℕ), x✝¹ ≡ 0 [SMOD I ^ n • ⊤]\nn : ℕ\nr : R\nhr : r ∈ I ^ n\nx : Ad... | induction x.val n using Quotient.inductionOn' with | _ a
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 467,
"column": 2
} | {
"line": 468,
"column": 35
} | {
"line": 470,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : AdicCauchySequence I M → ℕ → M := Subtype.val\n⊢ AddCommGroup (AdicCauchySequence I M)",
"ppTerm": "... | [] | apply Subtype.val_injective.addCommGroup f rfl (fun _ _ ↦ rfl) (fun _ ↦ rfl) (fun _ _ ↦ rfl)
(fun _ _ ↦ rfl) (fun _ _ ↦ rfl) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 197,
"column": 12
} | {
"line": 197,
"column": 14
} | {
"line": 197,
"column": 14
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsCancelMulZero R\nhR : HasUnitMulPowIrreducibleFactorization R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x ≠ 0... | [
"case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsCancelMulZero R\nhR : HasUnitMulPowIrreducibleFactorization R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x ≠ 0\nq : R\nhq ... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 84
} | {
"line": 116,
"column": 0
} | [
{
"pp": "case right\nR : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nH : IsJacobsonRing R\nf : R →+* S\nhf : Function.Surjective ⇑f\np : Ideal S\nhp : p.IsPrime\nthis : p = Ideal.map f (comap f p).jacobson\n⊢ p = sInf (Ideal.map f '' {J | comap f p ≤ J ∧ J.IsMaximal})",
"ppTerm": "?righ... | [] | exact this.trans (map_sInf hf fun J ⟨hJ, _⟩ => le_trans (Ideal.ker_le_comap f) hJ) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 519,
"column": 4
} | {
"line": 520,
"column": 50
} | {
"line": 521,
"column": 4
} | [
{
"pp": "R✝ : Type u\ninst✝⁵ : CommRing R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : IsDiscreteValuationRing R✝\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nI : Ideal R\nx : R := generator I\n⊢ (fun n ↦ ENat.recTopCoe ⊥ (fun n ↦ maximalIdeal R ^ n) (OrderDual.ofDual n))\n ... | [
"R✝ : Type u\ninst✝⁵ : CommRing R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : IsDiscreteValuationRing R✝\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nI : Ideal R\nx : R := generator I\n⊢ ENat.recTopCoe ⊥ (fun n ↦ maximalIdeal R ^ n) ((addVal R) x) = span {x}"
] | suffices (addVal R x).recTopCoe ⊥ (fun n ↦ maximalIdeal R ^ n) = span {x} by
rwa [Ideal.span_singleton_generator] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 526,
"column": 32
} | {
"line": 526,
"column": 51
} | {
"line": 526,
"column": 52
} | [
{
"pp": "case neg\nR✝ : Type u\ninst✝⁵ : CommRing R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : IsDiscreteValuationRing R✝\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nI : Ideal R\nx : R := generator I\nhx0 : ¬x = 0\nϖ : R\nhϖ : Irreducible ϖ\nn : ℕ\nu : Rˣ\nhu : x = ↑u * ϖ ^... | [
"case neg\nR✝ : Type u\ninst✝⁵ : CommRing R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : IsDiscreteValuationRing R✝\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nI : Ideal R\nx : R := generator I\nhx0 : ¬x = 0\nϖ : R\nhϖ : Irreducible ϖ\nn : ℕ\nu : Rˣ\nhu : x = ↑u * ϖ ^ n\n⊢ span {... | hϖ.maximalIdeal_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 534,
"column": 56
} | {
"line": 534,
"column": 75
} | {
"line": 535,
"column": 8
} | [
{
"pp": "case coe\nR✝ : Type u\ninst✝⁵ : CommRing R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : IsDiscreteValuationRing R✝\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nk : ℕ\nϖ : R\nhϖ : Irreducible ϖ\n⊢ (addVal R) (generator (maximalIdeal R ^ k)) = ↑k",
"ppTerm": "?coe",... | [
"case coe\nR✝ : Type u\ninst✝⁵ : CommRing R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : IsDiscreteValuationRing R✝\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nk : ℕ\nϖ : R\nhϖ : Irreducible ϖ\n⊢ (addVal R) (generator (span {ϖ} ^ k)) = ↑k"
] | hϖ.maximalIdeal_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 548,
"column": 40
} | {
"line": 548,
"column": 59
} | {
"line": 548,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nn : ℕ\nϖ : R\nhϖ : Irreducible ϖ\n⊢ maximalIdeal R ^ n = span {ϖ ^ n}",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"IsScalarTower.right",
... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nn : ℕ\nϖ : R\nhϖ : Irreducible ϖ\n⊢ span {ϖ} ^ n = span {ϖ ^ n}"
] | hϖ.maximalIdeal_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 311,
"column": 6
} | {
"line": 311,
"column": 95
} | {
"line": 312,
"column": 6
} | [
{
"pp": "case refine_1.inr\nR : Type u_1\ninst✝⁶ : CommRing R\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nP : Ideal R[X]\npX : R[X]\nhpX : pX ∈ P\ninst✝³ : Algebra (R ⧸ comap C P) Rₘ\ninst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ\ninst✝¹ : A... | [
"case refine_1.inr\nR : Type u_1\ninst✝⁶ : CommRing R\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nP : Ideal R[X]\npX : R[X]\nhpX : pX ∈ P\ninst✝³ : Algebra (R ⧸ comap C P) Rₘ\ninst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ\ninst✝¹ : Algebra (R[X]... | refine ⟨X - C (algebraMap _ _ ((Ideal.Quotient.mk P') (p.coeff 0))), monic_X_sub_C _, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Valuation.Basic | {
"line": 617,
"column": 2
} | {
"line": 617,
"column": 96
} | {
"line": 618,
"column": 2
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝³ : DivisionRing K\nΓ₀✝ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀✝\nv✝ : Valuation R Γ₀✝\nΓ₀ : Type u_7\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nhv : v.IsNontrivial\nx :... | [
"case right\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝³ : DivisionRing K\nΓ₀✝ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀✝\nv✝ : Valuation R Γ₀✝\nΓ₀ : Type u_7\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nhv : v.IsNontrivial\nx :... | use (Units.mk0 (v x) h0), (MonoidWithZeroHom.ofClass v).mem_valueMonoid (Set.mem_range_self x) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.DedekindDomain.Dvr | {
"line": 135,
"column": 2
} | {
"line": 137,
"column": 53
} | {
"line": 139,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\ninst✝⁴ : IsDedekindDomain A\nP : Ideal A\nhP : P ≠ ⊥\npP : P.IsPrime\nAₘ : Type u_2\ninst✝³ : CommRing Aₘ\ninst✝² : IsDomain Aₘ\ninst✝¹ : Algebra A Aₘ\ninst✝ : IsLocalization.AtPrime Aₘ P\nthis✝ : IsNoetherianRing Aₘ := isNoetherianRing P.primeCom... | [] | exact
((IsDiscreteValuationRing.TFAE Aₘ hnf).out 0 2).mpr
(IsLocalization.AtPrime.isDedekindDomain A P _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Flat.TorsionFree | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 14
} | {
"line": 103,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\nhtors : torsion R M = ⊥\n⊢ ∀ ⦃I : Ideal R⦄, I.FG → Function.Injective ⇑(lift (lsmul R M ∘ₗ Submodule.subtype I))",
"ppTerm": "?m.32",
"assigned": true,
"us... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\nhtors : torsion R M = ⊥\nI : Ideal R\nhFG : I.FG\n⊢ Function.Injective ⇑(lift (lsmul R M ∘ₗ Submodule.subtype I))"
] | rintro I hFG | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 292,
"column": 2
} | {
"line": 293,
"column": 25
} | {
"line": 295,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\n⊢ Submodule.map (↑P.toPerfPair) (P.rootSpan R) = span R (range P.root')",
"ppTerm": "?m.72",
"assigned... | [] | rw [rootSpan, Submodule.map_span, ← image_univ, ← image_comp, image_univ, LinearEquiv.coe_coe,
toPerfPair_comp_root] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 292,
"column": 2
} | {
"line": 293,
"column": 25
} | {
"line": 295,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\n⊢ Submodule.map (↑P.toPerfPair) (P.rootSpan R) = span R (range P.root')",
"ppTerm": "?m.72",
"assigned... | [] | rw [rootSpan, Submodule.map_span, ← image_univ, ← image_comp, image_univ, LinearEquiv.coe_coe,
toPerfPair_comp_root] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 292,
"column": 2
} | {
"line": 293,
"column": 25
} | {
"line": 295,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\n⊢ Submodule.map (↑P.toPerfPair) (P.rootSpan R) = span R (range P.root')",
"ppTerm": "?m.72",
"assigned... | [] | rw [rootSpan, Submodule.map_span, ← image_univ, ← image_comp, image_univ, LinearEquiv.coe_coe,
toPerfPair_comp_root] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 98
} | {
"line": 118,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Fintype ι\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : CommRing R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsAnisotropic\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R N\ni j k : ι\nm n ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Fintype ι\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : CommRing R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsAnisotropic\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R N\ni j k : ι\nm n : R\nhk : m ... | have h₄ : P.pairing j i * lsq i = P.pairing i j * lsq j := B.pairing_mul_eq_pairing_mul_swap i j | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 50
} | {
"line": 206,
"column": 2
} | [
{
"pp": "case h\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : AddCommGroup N\ninst✝¹³ : CommRing R\ninst✝¹² : Module R M\ninst✝¹¹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹⁰ : CommRing S\ninst✝⁹ : LinearOrder S\ninst✝⁸ : IsStri... | [] | exact ⟨Finset.mem_univ i, mul_self_pos.mpr hi⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 220,
"column": 4
} | {
"line": 222,
"column": 71
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case left\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : AddCommGroup N\ninst✝¹³ : CommRing R\ninst✝¹² : Module R M\ninst✝¹¹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹⁰ : CommRing S\ninst✝⁹ : LinearOrder S\ninst✝⁸ : IsS... | [] | intro x
contrapose!
exact fun hx ↦ ⟨x, (posRootForm_posForm_pos_of_ne_zero P S hx).ne'⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 220,
"column": 4
} | {
"line": 222,
"column": 71
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case left\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : AddCommGroup N\ninst✝¹³ : CommRing R\ninst✝¹² : Module R M\ninst✝¹¹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹⁰ : CommRing S\ninst✝⁹ : LinearOrder S\ninst✝⁸ : IsS... | [] | intro x
contrapose!
exact fun hx ↦ ⟨x, (posRootForm_posForm_pos_of_ne_zero P S hx).ne'⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 220,
"column": 4
} | {
"line": 222,
"column": 71
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case right\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : AddCommGroup N\ninst✝¹³ : CommRing R\ninst✝¹² : Module R M\ninst✝¹¹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹⁰ : CommRing S\ninst✝⁹ : LinearOrder S\ninst✝⁸ : Is... | [] | intro x
contrapose!
exact fun hx ↦ ⟨x, (posRootForm_posForm_pos_of_ne_zero P S hx).ne'⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 220,
"column": 4
} | {
"line": 222,
"column": 71
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case right\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : AddCommGroup N\ninst✝¹³ : CommRing R\ninst✝¹² : Module R M\ninst✝¹¹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹⁰ : CommRing S\ninst✝⁹ : LinearOrder S\ninst✝⁸ : Is... | [] | intro x
contrapose!
exact fun hx ↦ ⟨x, (posRootForm_posForm_pos_of_ne_zero P S hx).ne'⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 76,
"column": 2
} | {
"line": 77,
"column": 25
} | {
"line": 79,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ni j : ι\ninst✝¹ : Nontrivial R\ninst✝ : P.IsReduced\nh : i ≠ j → P.root i = -P.root j\nh' : i ≠ j\n... | [] | · rw [h h']
exact ⟨1, 1, by simp⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 103,
"column": 4
} | {
"line": 104,
"column": 37
} | {
"line": 105,
"column": 2
} | [
{
"pp": "case refine_1\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : CharZero R\ninst✝¹ : IsAddTorsionFree M\ninst✝ : P.IsReduced\ni j : ι\nh : P.root i +... | [] | rw [hij, ← two_smul (R := ℕ)] at h
exact P.nsmul_notMem_range_root h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 103,
"column": 4
} | {
"line": 104,
"column": 37
} | {
"line": 105,
"column": 2
} | [
{
"pp": "case refine_1\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : CharZero R\ninst✝¹ : IsAddTorsionFree M\ninst✝ : P.IsReduced\ni j : ι\nh : P.root i +... | [] | rw [hij, ← two_smul (R := ℕ)] at h
exact P.nsmul_notMem_range_root h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 407,
"column": 4
} | {
"line": 407,
"column": 77
} | {
"line": 408,
"column": 4
} | [
{
"pp": "case inr.inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : Module R M\ninst✝¹³ : AddCommGroup N\ninst✝¹² : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : LinearOrder S\ninst✝⁹ : IsStrictOrderedRing ... | [
"case inr.inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : Module R M\ninst✝¹³ : AddCommGroup N\ninst✝¹² : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : LinearOrder S\ninst✝⁹ : IsStrictOrderedRing S\ninst✝⁸ : ... | rw [le_iff_eq_or_lt, le_iff_eq_or_lt, or_iff_right hij, or_iff_right hji] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 345,
"column": 6
} | {
"line": 345,
"column": 96
} | {
"line": 346,
"column": 6
} | [
{
"pp": "case inr.inr.refine_1\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH✝ : LieSubalgebra K L\ninst✝¹ : H✝.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H✝) L\nα β : Weight K (↥H... | [
"case inr.inr.refine_1\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH✝ : LieSubalgebra K L\ninst✝¹ : H✝.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H✝) L\nα β : Weight K (↥H✝) L\nhα : α... | simp only [tsub_le_iff_right, le_add_iff_nonneg_right, Nat.cast_nonneg, neg_sub, true_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 368,
"column": 4
} | {
"line": 368,
"column": 49
} | {
"line": 369,
"column": 4
} | [
{
"pp": "case pos\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα✝ β✝ α β : Weight K (↥H) L\nhα : ... | [
"case neg\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα✝ β✝ α β : Weight K (↥H) L\nhα : ¬α.IsZero\n⊢... | · simpa [hα.eq] using β.genWeightSpace_ne_bot | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.Irreducible | {
"line": 112,
"column": 20
} | {
"line": 112,
"column": 39
} | {
"line": 112,
"column": 39
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\nK : Type u_5\ninst✝⁴ : Field K\ninst✝³ : NeZero 2\ninst✝² : Module K M\ninst✝¹ : Module K N\nP : RootPairing ι K M N\ninst✝ : P.IsRootSystem\nq : ↥P.invtRootSubmodule\nQ : Submodule K M := ↑q\nS : Submodule K M ... | [
"ι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\nK : Type u_5\ninst✝⁴ : Field K\ninst✝³ : NeZero 2\ninst✝² : Module K M\ninst✝¹ : Module K N\nP : RootPairing ι K M N\ninst✝ : P.IsRootSystem\nq : ↥P.invtRootSubmodule\nQ : Submodule K M := ↑q\nS : Submodule K M := span K (⇑... | rw [this, add_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.Irreducible | {
"line": 240,
"column": 30
} | {
"line": 240,
"column": 69
} | {
"line": 241,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : NeZero 2\ninst✝ : P.IsIrreducible\nB : P.InvariantForm\ni j : ι\ncontra : ∀ (k : ι), (B.form (P.root... | [] | by simpa [span_orbit_eq_top] using this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Hom | {
"line": 641,
"column": 4
} | {
"line": 641,
"column": 48
} | {
"line": 642,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nι₂ : Type u_5\nM₂ : Type u_6\nN₂ : Type u_7\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : AddCommGroup N₂\ninst✝ : Module ... | [] | exact LinearEquiv.bijective (P.reflection i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 173,
"column": 2
} | {
"line": 175,
"column": 10
} | {
"line": 177,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : P.IsCrystallographic\ni j : ι\ninst✝ : IsDomain R\nB : P.Inv... | [] | have : Module.IsReflexive R M := .of_isPerfPair P.toLinearMap
have := P.pairingIn_pairingIn_mem_set_of_length_eq len_eq
simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 173,
"column": 2
} | {
"line": 175,
"column": 10
} | {
"line": 177,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : P.IsCrystallographic\ni j : ι\ninst✝ : IsDomain R\nB : P.Inv... | [] | have : Module.IsReflexive R M := .of_isPerfPair P.toLinearMap
have := P.pairingIn_pairingIn_mem_set_of_length_eq len_eq
simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 247,
"column": 4
} | {
"line": 248,
"column": 90
} | {
"line": 250,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\... | [] | simp only [chainTopCoeff_of_not_linearIndependent h, chainTopCoeff_of_not_linearIndependent h',
chainBotCoeff_of_not_linearIndependent h, chainBotCoeff_of_not_linearIndependent h'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 262,
"column": 4
} | {
"line": 263,
"column": 90
} | {
"line": 265,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\... | [] | simp only [chainTopCoeff_of_not_linearIndependent h, chainTopCoeff_of_not_linearIndependent h',
chainBotCoeff_of_not_linearIndependent h, chainBotCoeff_of_not_linearIndependent h'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 45
} | {
"line": 296,
"column": 2
} | [
{
"pp": "case refine_1\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j... | [
"case refine_2\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\nthis✝ ... | · simpa using this (P.chainTopCoeff i (-j)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 398,
"column": 4
} | {
"line": 400,
"column": 53
} | {
"line": 401,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\... | [] | simp only [chainBotIdx, reduceDIte, h]
exact (P.root_sub_nsmul_mem_range_iff_le_chainBotCoeff h).mpr
(le_refl <| P.chainBotCoeff i j) |>.choose_spec | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 398,
"column": 4
} | {
"line": 400,
"column": 53
} | {
"line": 401,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\... | [] | simp only [chainBotIdx, reduceDIte, h]
exact (P.root_sub_nsmul_mem_range_iff_le_chainBotCoeff h).mpr
(le_refl <| P.chainBotCoeff i j) |>.choose_spec | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 426,
"column": 4
} | {
"line": 432,
"column": 46
} | {
"line": 434,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni : ι\nf : ι → ℤ\nhf₀ : Function.support f ⊆ ↑b.support\nhf₂ : P.ro... | [] | refine (Finset.sum_neg' (fun i _ ↦ neg.le i) ?_).ne
by_contra! contra
replace contra (j : ι) : f j = 0 := by
by_cases hj : j ∈ f.support
· exact le_antisymm (neg.le j) (contra j (hf₀ hj))
· simpa using hj
exact P.ne_zero i <| by simp [hf₂, contra] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 426,
"column": 4
} | {
"line": 432,
"column": 46
} | {
"line": 434,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni : ι\nf : ι → ℤ\nhf₀ : Function.support f ⊆ ↑b.support\nhf₂ : P.ro... | [] | refine (Finset.sum_neg' (fun i _ ↦ neg.le i) ?_).ne
by_contra! contra
replace contra (j : ι) : f j = 0 := by
by_cases hj : j ∈ f.support
· exact le_antisymm (neg.le j) (contra j (hf₀ hj))
· simpa using hj
exact P.ne_zero i <| by simp [hf₂, contra] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 592,
"column": 2
} | {
"line": 592,
"column": 56
} | {
"line": 593,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN✝ : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N✝\ninst✝⁴ : Module R N✝\nP : RootPairing ι R M N✝\nb : P.Base\ninst✝³ : CharZero R\ninst✝² : Finite ι\ninst✝¹ : IsDomain R\ninst✝ : P.IsCrystallographic\ni ... | [] | induction N using Int.induction_on generalizing i with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 684,
"column": 2
} | {
"line": 684,
"column": 17
} | {
"line": 685,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁴ : CharZero R\ninst✝³ : Finite ι\ninst✝² : IsDomain R\ninst✝¹ : P.IsCrystallographic\ninst✝... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁴ : CharZero R\ninst✝³ : Finite ι\ninst✝² : IsDomain R\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsReduc... | intro i j hi hj | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Data.Real.Basic | {
"line": 202,
"column": 21
} | {
"line": 202,
"column": 75
} | {
"line": 203,
"column": 2
} | [
{
"pp": "x a b c : ℝ\n⊢ a * b * c = a * (b * c)",
"ppTerm": "?m.133",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Real",
"Real.cauchy",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Real.ext_cauchy",
"abs",
"congrArg",
"IsAbsoluteVal... | [] | by apply ext_cauchy; simp only [cauchy_mul, mul_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 226,
"column": 47
} | {
"line": 226,
"column": 62
} | {
"line": 226,
"column": 63
} | [
{
"pp": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nh : IsUnit A.det\nm n : ℕ\n⊢ A ^ (↑m * ↑n) = A ^ (m * n)",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"HMul.hMul",
"congrArg",... | [
"n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nh : IsUnit A.det\nm n : ℕ\n⊢ A ^ (↑m * ↑n) = A ^ ↑(m * n)"
] | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 339,
"column": 4
} | {
"line": 340,
"column": 15
} | {
"line": 342,
"column": 0
} | [
{
"pp": "case neg\nn : Type u_2\nR' : Type u_4\ninst✝⁴ : CommRing R'\ninst✝³ : PartialOrder R'\ninst✝² : StarRing R'\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nM : Matrix n n R'\nhM : M.PosSemidef\nh : ¬IsUnit M.det\n⊢ M⁻¹.PosSemidef",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"... | [] | rw [nonsing_inv_apply_not_isUnit _ h]
exact .zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 339,
"column": 4
} | {
"line": 340,
"column": 15
} | {
"line": 342,
"column": 0
} | [
{
"pp": "case neg\nn : Type u_2\nR' : Type u_4\ninst✝⁴ : CommRing R'\ninst✝³ : PartialOrder R'\ninst✝² : StarRing R'\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nM : Matrix n n R'\nhM : M.PosSemidef\nh : ¬IsUnit M.det\n⊢ M⁻¹.PosSemidef",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"... | [] | rw [nonsing_inv_apply_not_isUnit _ h]
exact .zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 63
} | {
"line": 477,
"column": 0
} | [
{
"pp": "n : Type u_2\ninst✝⁵ : Fintype n\nR : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : TrivialStar R\ninst✝ : DecidableEq n\nM : Matrix n n R\nhM : M.IsSymm\nhMq : QuadraticMap.PosDef M.toQuadraticForm'\nx : n → R\nhx : x ≠ 0\n⊢ 0 < star x ⬝ᵥ M *ᵥ x",
"ppTerm": ... | [] | simpa [toQuadraticForm', toLinearMap₂'_apply'] using hMq x hx | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 87
} | {
"line": 236,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\ninst... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\ninst✝¹ : P.IsRed... | have hq₀ : q ≠ ⊥ := q.ne_bot_iff.mpr ⟨P.root i, subset_span <| by simpa, P.ne_zero i⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Eigenspace.Minpoly | {
"line": 82,
"column": 4
} | {
"line": 89,
"column": 87
} | {
"line": 90,
"column": 2
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : End R M\nμ : R\ninst✝¹ : IsDomain R\ninst✝ : Module.Finite R M\nh : (minpoly R f).IsRoot μ\nq : R[X]\nhq : minpoly R f = (X - C μ) * q\n⊢ ∃ v, ((aeval f) q) v ≠ 0",
"ppTerm": "?m.51",
"assigned": true... | [] | by_contra! h_contra
have := minpoly.min R f
((monic_X_sub_C μ).of_mul_monic_left (hq ▸ minpoly.monic (Algebra.IsIntegral.isIntegral f)))
(LinearMap.ext h_contra)
rw [hq, degree_mul, degree_X_sub_C, degree_eq_natDegree] at this
· norm_cast at this; grind
· rintro rfl
exact minpoly.ne_ze... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Eigenspace.Minpoly | {
"line": 82,
"column": 4
} | {
"line": 89,
"column": 87
} | {
"line": 90,
"column": 2
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : End R M\nμ : R\ninst✝¹ : IsDomain R\ninst✝ : Module.Finite R M\nh : (minpoly R f).IsRoot μ\nq : R[X]\nhq : minpoly R f = (X - C μ) * q\n⊢ ∃ v, ((aeval f) q) v ≠ 0",
"ppTerm": "?m.51",
"assigned": true... | [] | by_contra! h_contra
have := minpoly.min R f
((monic_X_sub_C μ).of_mul_monic_left (hq ▸ minpoly.monic (Algebra.IsIntegral.isIntegral f)))
(LinearMap.ext h_contra)
rw [hq, degree_mul, degree_X_sub_C, degree_eq_natDegree] at this
· norm_cast at this; grind
· rintro rfl
exact minpoly.ne_ze... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Basis.Base | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 68
} | {
"line": 79,
"column": 69
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁷ : Fintype ι\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\nb : Basis ι H\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : IsKilling K L\nthis✝ : H.IsCartanSubalgeb... | [
"case refine_1\nι : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁷ : Fintype ι\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\nb : Basis ι H\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : IsKilling K L\nthis✝ : H.IsCartanSubal... | refine hs.symm.imp (fun ⟨n, hn₀, hn⟩ ↦ ?_) (fun ⟨n, hn₀, hn⟩ ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 80
} | {
"line": 170,
"column": 0
} | [
{
"pp": "a b : ℕ\n⊢ eval a (ascPochhammer ℕ b) = (a + b - 1).descFactorial b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"congrArg",
"ascPochhammer",
"HSub.hSub",
"Nat.ascFactorial",
"id",
"instSubNat",
... | [] | rw [ascPochhammer_nat_eq_ascFactorial, Nat.add_descFactorial_eq_ascFactorial'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 80
} | {
"line": 170,
"column": 0
} | [
{
"pp": "a b : ℕ\n⊢ eval a (ascPochhammer ℕ b) = (a + b - 1).descFactorial b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"congrArg",
"ascPochhammer",
"HSub.hSub",
"Nat.ascFactorial",
"id",
"instSubNat",
... | [] | rw [ascPochhammer_nat_eq_ascFactorial, Nat.add_descFactorial_eq_ascFactorial'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 80
} | {
"line": 170,
"column": 0
} | [
{
"pp": "a b : ℕ\n⊢ eval a (ascPochhammer ℕ b) = (a + b - 1).descFactorial b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"congrArg",
"ascPochhammer",
"HSub.hSub",
"Nat.ascFactorial",
"id",
"instSubNat",
... | [] | rw [ascPochhammer_nat_eq_ascFactorial, Nat.add_descFactorial_eq_ascFactorial'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 172,
"column": 2
} | {
"line": 173,
"column": 41
} | {
"line": 175,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝ : Semiring S\na b : ℕ\n⊢ eval (↑a) (ascPochhammer S b) = ↑((a + b - 1).descFactorial b)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"ascPochham... | [] | norm_cast
rw [ascPochhammer_nat_eq_descFactorial] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 172,
"column": 2
} | {
"line": 173,
"column": 41
} | {
"line": 175,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝ : Semiring S\na b : ℕ\n⊢ eval (↑a) (ascPochhammer S b) = ↑((a + b - 1).descFactorial b)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"ascPochham... | [] | norm_cast
rw [ascPochhammer_nat_eq_descFactorial] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 292,
"column": 6
} | {
"line": 292,
"column": 98
} | {
"line": 293,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\nP : RootPairing ι R M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallogr... | [] | ext; simp [g', this, cartanMatrixIn, Matrix.vecMul_eq_sum, b.support.sum_subtype (by tauto)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 292,
"column": 6
} | {
"line": 292,
"column": 98
} | {
"line": 293,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\nP : RootPairing ι R M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallogr... | [] | ext; simp [g', this, cartanMatrixIn, Matrix.vecMul_eq_sum, b.support.sum_subtype (by tauto)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 324,
"column": 4
} | {
"line": 324,
"column": 50
} | {
"line": 325,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nn : ℕ\nh : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - ↑n)\n⊢ descPochhammer R (n + 1) = descPochhammer R n * (X - ↑n)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Algebra.algebraMap",
"HEq.refl",
"descP... | [
"R : Type u\ninst✝ : Ring R\nn : ℕ\nh : map (algebraMap ℤ R) (descPochhammer ℤ (n + 1)) = map (algebraMap ℤ R) (descPochhammer ℤ n * (X - ↑n))\n⊢ descPochhammer R (n + 1) = descPochhammer R n * (X - ↑n)"
] | apply_fun Polynomial.map (algebraMap ℤ R) at h | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyFun___elabRules_Mathlib_Tactic_applyFun_1 | Mathlib.Tactic.applyFun |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 409,
"column": 10
} | {
"line": 409,
"column": 49
} | {
"line": 409,
"column": 50
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Ring R\nn k : ℕ\nih : eval (↑n) (descPochhammer R k) = ↑(n.descFactorial k)\nh : n < k\n⊢ (↑n - ↑k) * ↑(n.descFactorial k) = ↑((n - k) * n.descFactorial k)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NonAsso... | [
"case pos\nR : Type u\ninst✝ : Ring R\nn k : ℕ\nih : eval (↑n) (descPochhammer R k) = ↑(n.descFactorial k)\nh : n < k\n⊢ (↑n - ↑k) * ↑0 = ↑((n - k) * 0)"
] | Nat.descFactorial_eq_zero_iff_lt.mpr h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 504,
"column": 12
} | {
"line": 504,
"column": 14
} | {
"line": 504,
"column": 15
} | [
{
"pp": "case succ\nS : Type u_1\ninst✝² : Ring S\ninst✝¹ : PartialOrder S\ninst✝ : IsStrictOrderedRing S\nn : ℕ\nih : MonotoneOn (fun x ↦ eval x (descPochhammer S n)) (Set.Ici (↑n - 1))\na : S\n⊢ a ∈ Set.Ici (↑(n + 1) - 1) →\n ∀ ⦃b : S⦄,\n b ∈ Set.Ici (↑(n + 1) - 1) →\n a ≤ b → (fun x ↦ eval x (... | [
"case succ\nS : Type u_1\ninst✝² : Ring S\ninst✝¹ : PartialOrder S\ninst✝ : IsStrictOrderedRing S\nn : ℕ\nih : MonotoneOn (fun x ↦ eval x (descPochhammer S n)) (Set.Ici (↑n - 1))\na : S\nha : a ∈ Set.Ici (↑(n + 1) - 1)\n⊢ ∀ ⦃b : S⦄,\n b ∈ Set.Ici (↑(n + 1) - 1) →\n a ≤ b → (fun x ↦ eval x (descPochhammer S ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.Lagrange | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 62
} | {
"line": 185,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\nx y : F\nhxy : x ≠ y\n⊢ eval x (basisDivisor x y) = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.eval",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid... | [
"F : Type u_1\ninst✝ : Field F\nx y : F\nhxy : x ≠ y\n⊢ (x - y)⁻¹ * (x - y) = 1"
] | simp only [basisDivisor, eval_mul, eval_C, eval_sub, eval_X] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Vandermonde | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 45
} | {
"line": 198,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nn✝ n : ℕ\nih : ∀ (v w : Fin n → K), (projVandermonde v w).det = ∏ i, ∏ j > i, (v j * w i - v i * w j)\nv w : Fin (n + 1) → K\nh0 : w 0 ≠ 0\nr : K := v 0 / w 0\nhr : r = v 0 / w 0\nW : Matrix (Fin (n + 1)) (Fin (n + 1)) K :=\n of fun i ↦\n Fin.cons (projVandermonde v w... | [
"K : Type u_2\ninst✝ : Field K\nn✝ n : ℕ\nih : ∀ (v w : Fin n → K), (projVandermonde v w).det = ∏ i, ∏ j > i, (v j * w i - v i * w j)\nv w : Fin (n + 1) → K\nh0 : w 0 ≠ 0\nr : K := v 0 / w 0\nhr : r = v 0 / w 0\nW : Matrix (Fin (n + 1)) (Fin (n + 1)) K :=\n of fun i ↦\n Fin.cons (projVandermonde v w i 0) fun j ... | obtain ⟨j, rfl⟩ := j.eq_succ_of_ne_zero hj0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Lie.Basis.Prod | {
"line": 273,
"column": 6
} | {
"line": 273,
"column": 42
} | {
"line": 274,
"column": 6
} | [
{
"pp": "ι₁ : Type u_1\nι₂ : Type u_2\nL₁ : Type u_3\nL₂ : Type u_4\ninst✝¹¹ : Finite ι₁\ninst✝¹⁰ : Finite ι₂\neι : ι₁ ≃ ι₂\ninst✝⁹ : LieRing L₁\ninst✝⁸ : LieRing L₂\nK : Type u_5\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieAlgebra K L₁\ninst✝⁴ : FiniteDimensional K L₁\nH₁ : LieSubalgebra K L₁\nb₁ : Bas... | [
"ι₁ : Type u_1\nι₂ : Type u_2\nL₁ : Type u_3\nL₂ : Type u_4\ninst✝¹¹ : Finite ι₁\ninst✝¹⁰ : Finite ι₂\neι : ι₁ ≃ ι₂\ninst✝⁹ : LieRing L₁\ninst✝⁸ : LieRing L₂\nK : Type u_5\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieAlgebra K L₁\ninst✝⁴ : FiniteDimensional K L₁\nH₁ : LieSubalgebra K L₁\nb₁ : Basis ι₁ H₁\nin... | rw [disjoint_iff, _root_.eq_bot_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 297,
"column": 21
} | {
"line": 297,
"column": 31
} | {
"line": 297,
"column": 31
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq τ\nf : σ → MvPolynomial τ R\nφ : MvPolynomial σ R\n⊢ ((bind₁ f) φ).vars = ((bind₁ f) (∑ x ∈ φ.support, (monomial x) (coeff x φ))).vars",
"ppTerm": "?m.199",
"assigned": true,
"usedConstants": [
"Eq.... | [
"σ : Type u_1\nτ : Type u_2\nR : Type u_3\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq τ\nf : σ → MvPolynomial τ R\nφ : MvPolynomial σ R\n⊢ ((bind₁ f) φ).vars = ((bind₁ f) φ).vars"
] | ← φ.as_sum | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 295,
"column": 7
} | {
"line": 304,
"column": 52
} | {
"line": 306,
"column": 2
} | [] | [
"case calc_1\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq τ\nf : σ → MvPolynomial τ R\nφ : MvPolynomial σ R\n⊢ (φ.support.biUnion fun d ↦ (C (coeff d φ) * ∏ i ∈ d.support, f i ^ d i).vars) ⊆\n φ.support.biUnion fun d ↦ d.support.biUnion fun i ↦ (f i).vars",
"case calc... | (bind₁ f φ).vars
_ = (φ.support.sum fun x : σ →₀ ℕ => (bind₁ f) (monomial x (coeff x φ))).vars := by
rw [← map_sum, ← φ.as_sum]
_ ≤ φ.support.biUnion fun i : σ →₀ ℕ => ((bind₁ f) (monomial i (coeff i φ))).vars :=
(vars_sum_subset _ _)
_ = φ.support.biUnion fun d : σ →₀ ℕ => vars (C (coeff d φ) *... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Submodule | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 48
} | {
"line": 80,
"column": 4
} | [
{
"pp": "ιA : Type u_1\nιM : Type u_2\nσA : Type u_3\nσM : Type u_4\nA : Type u_5\nM : Type u_6\ninst✝¹³ : Semiring A\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module A M\n𝒜 : ιA → σA\nℳ : ιM → σM\ninst✝¹⁰ : DecidableEq ιA\ninst✝⁹ : AddMonoid ιA\ninst✝⁸ : SetLike σA A\ninst✝⁷ : AddSubmonoidClass σA A\ninst✝⁶ : Gra... | [
"ιA : Type u_1\nιM : Type u_2\nσA : Type u_3\nσM : Type u_4\nA : Type u_5\nM : Type u_6\ninst✝¹³ : Semiring A\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module A M\n𝒜 : ιA → σA\nℳ : ιM → σM\ninst✝¹⁰ : DecidableEq ιA\ninst✝⁹ : AddMonoid ιA\ninst✝⁸ : SetLike σA A\ninst✝⁷ : AddSubmonoidClass σA A\ninst✝⁶ : GradedRing 𝒜\n... | rintro ⟨p, hp⟩ ⟨q, hq⟩ (h : (p : Set M) = q) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.LinearAlgebra.Lagrange | {
"line": 695,
"column": 4
} | {
"line": 696,
"column": 46
} | {
"line": 698,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\nx : F\nhvs : Set.InjOn v ↑s\nhx : ∀ i ∈ s, x ≠ v i\nhs : s.Nonempty\n⊢ eval x (nodal s v) * ∑ i ∈ s, nodalWeight s v i * (x - v i)⁻¹ = 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [] | simpa only [Pi.one_apply, interpolate_one hvs hs, eval_one, mul_one] using
(eval_interpolate_not_at_node 1 hx).symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Lagrange | {
"line": 695,
"column": 4
} | {
"line": 696,
"column": 46
} | {
"line": 698,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\nx : F\nhvs : Set.InjOn v ↑s\nhx : ∀ i ∈ s, x ≠ v i\nhs : s.Nonempty\n⊢ eval x (nodal s v) * ∑ i ∈ s, nodalWeight s v i * (x - v i)⁻¹ = 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [] | simpa only [Pi.one_apply, interpolate_one hvs hs, eval_one, mul_one] using
(eval_interpolate_not_at_node 1 hx).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Lagrange | {
"line": 695,
"column": 4
} | {
"line": 696,
"column": 46
} | {
"line": 698,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\nx : F\nhvs : Set.InjOn v ↑s\nhx : ∀ i ∈ s, x ≠ v i\nhs : s.Nonempty\n⊢ eval x (nodal s v) * ∑ i ∈ s, nodalWeight s v i * (x - v i)⁻¹ = 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [] | simpa only [Pi.one_apply, interpolate_one hvs hs, eval_one, mul_one] using
(eval_interpolate_not_at_node 1 hx).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 71
} | {
"line": 182,
"column": 2
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : LieRingModule L M\ninst✝³ : Module R M\ninst✝² : LieModule R L M\ninst✝¹ : IsNoetherian R M\ninst✝ : Free R M\nh : tra... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : LieRingModule L M\ninst✝³ : Module R M\ninst✝² : LieModule R L M\ninst✝¹ : IsNoetherian R M\ninst✝ : Free R M\nh : traceForm R L M... | have _i : FaithfulSMul R A := FaithfulSMul.trans R (FractionRing R) A | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal | {
"line": 459,
"column": 6
} | {
"line": 459,
"column": 16
} | {
"line": 459,
"column": 17
} | [
{
"pp": "case e'_3\nι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\ncoe_mono : Monotone toIdeal\nx : Ideal A\n⊢ x ∈ {J | IsHomogeneous 𝒜 J ∧ J ≤ I} ... | [
"case e'_3\nι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\ncoe_mono : Monotone toIdeal\nx : Ideal A\n⊢ x ∈ {J | IsHomogeneous 𝒜 J ∧ J ≤ I} ↔ ∃ x_1 ∈ {b... | mem_image, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 198,
"column": 14
} | {
"line": 198,
"column": 46
} | {
"line": 199,
"column": 4
} | [
{
"pp": "case refine_1.zero\nσ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\n⊢ homogeneousSubmodule σ R 1 ^ 0 ≤ homogeneousSubmodule σ R 0",
"ppTerm": "?refine_1.zero",
"assigned": true,
"usedConstants": [
"Submodule",
"MulOne.toOne",
"Nat.instMulZeroClass",
"AddMonoidAlg... | [] | simp [homogeneousSubmodule_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 198,
"column": 14
} | {
"line": 198,
"column": 46
} | {
"line": 199,
"column": 4
} | [
{
"pp": "case refine_1.zero\nσ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\n⊢ homogeneousSubmodule σ R 1 ^ 0 ≤ homogeneousSubmodule σ R 0",
"ppTerm": "?refine_1.zero",
"assigned": true,
"usedConstants": [
"Submodule",
"MulOne.toOne",
"Nat.instMulZeroClass",
"AddMonoidAlg... | [] | simp [homogeneousSubmodule_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 198,
"column": 14
} | {
"line": 198,
"column": 46
} | {
"line": 199,
"column": 4
} | [
{
"pp": "case refine_1.zero\nσ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\n⊢ homogeneousSubmodule σ R 1 ^ 0 ≤ homogeneousSubmodule σ R 0",
"ppTerm": "?refine_1.zero",
"assigned": true,
"usedConstants": [
"Submodule",
"MulOne.toOne",
"Nat.instMulZeroClass",
"AddMonoidAlg... | [] | simp [homogeneousSubmodule_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 257,
"column": 2
} | {
"line": 259,
"column": 47
} | {
"line": 261,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\nφ : MvPolynomial σ R\nm : ℕ\nhφ : φ.IsHomogeneous m\nn : ℕ\n⊢ (φ ^ n).IsHomogeneous (m * n)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"i... | [] | rw [show φ ^ n = ∏ _i ∈ Finset.range n, φ by simp]
rw [show m * n = ∑ _i ∈ Finset.range n, m by simp [mul_comm]]
apply IsHomogeneous.prod _ _ _ (fun _ _ ↦ hφ) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 257,
"column": 2
} | {
"line": 259,
"column": 47
} | {
"line": 261,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\nφ : MvPolynomial σ R\nm : ℕ\nhφ : φ.IsHomogeneous m\nn : ℕ\n⊢ (φ ^ n).IsHomogeneous (m * n)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"i... | [] | rw [show φ ^ n = ∏ _i ∈ Finset.range n, φ by simp]
rw [show m * n = ∑ _i ∈ Finset.range n, m by simp [mul_comm]]
apply IsHomogeneous.prod _ _ _ (fun _ _ ↦ hφ) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 343,
"column": 2
} | {
"line": 347,
"column": 37
} | {
"line": 349,
"column": 0
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\ninst✝ : CommSemiring R\nφ : MvPolynomial σ R\nn : ℕ\nf : σ → τ\nhf : Function.Injective f\n⊢ ((rename f) φ).IsHomogeneous n ↔ φ.IsHomogeneous n",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr... | [] | refine ⟨fun h d hd ↦ ?_, rename_isHomogeneous⟩
convert! ← @h (d.mapDomain f) _
· simp only [weight_apply, Pi.one_apply, smul_eq_mul, mul_one]
exact Finsupp.sum_mapDomain_index_inj (h := fun _ ↦ id) hf
· rwa [coeff_rename_mapDomain f hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 343,
"column": 2
} | {
"line": 347,
"column": 37
} | {
"line": 349,
"column": 0
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\ninst✝ : CommSemiring R\nφ : MvPolynomial σ R\nn : ℕ\nf : σ → τ\nhf : Function.Injective f\n⊢ ((rename f) φ).IsHomogeneous n ↔ φ.IsHomogeneous n",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr... | [] | refine ⟨fun h d hd ↦ ?_, rename_isHomogeneous⟩
convert! ← @h (d.mapDomain f) _
· simp only [weight_apply, Pi.one_apply, smul_eq_mul, mul_one]
exact Finsupp.sum_mapDomain_index_inj (h := fun _ ↦ id) hf
· rwa [coeff_rename_mapDomain f hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 17
} | {
"line": 397,
"column": 2
} | [
{
"pp": "R : Type u_3\ninst✝ : CommSemiring R\nN : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\ni : ℕ\nhi : n - i ≠ 0\n⊢ 0 ≠ n - i",
"ppTerm": "?m.136",
"assigned": true,
"us... | [] | exact hi.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 57
} | {
"line": 150,
"column": 2
} | [
{
"pp": "K : Type u_2\nM : Type u_3\ninst✝³ : Field K\ninst✝² : AddCommGroup M\ninst✝¹ : Module K M\ninst✝ : Module.Finite K M\nφ : End K M\nV : Submodule K M := φ.maxGenEigenspace 0\nhV : V = ⨆ n, ker (φ ^ n)\nW : Submodule K M := ⨅ n, range (φ ^ n)\n⊢ IsCompl (⨆ n, ker (φ ^ n)) W",
"ppTerm": "?m.115",
... | [] | exact LinearMap.isCompl_iSup_ker_pow_iInf_range_pow φ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 148,
"column": 13
} | {
"line": 148,
"column": 25
} | {
"line": 148,
"column": 26
} | [
{
"pp": "m : Type um\nR : Type uR\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\ninst✝¹ : DecidableEq m\ninst✝ : StrongRankCondition R\nh : LinearIndependent R (col 1)\n⊢ Module.rank R ↥(span R (range (col 1))) = lift.{uR, um} #m",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"m : Type um\nR : Type uR\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\ninst✝¹ : DecidableEq m\ninst✝ : StrongRankCondition R\nh : LinearIndependent R (col 1)\n⊢ #↑(range (col 1)) = lift.{uR, um} #m"
] | rank_span h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 46
} | {
"line": 234,
"column": 2
} | [
{
"pp": "m : Type um\nn : Type un\ninst✝¹ : Fintype n\nR : Type u_1\ninst✝ : CommRing R\nc : R\nB : Matrix m n R\nhc : c ∈ nonZeroDivisors R\nhc' : IsSMulRegular R c\nhreg : IsSMulRegular (m → R) c\nf : (m → R) →ₗ[R] m → R := (LinearMap.lsmul R (m → R)) c\nhcomp : (c • B).mulVecLin = f ∘ₗ B.mulVecLin\n⊢ (c • B)... | [
"m : Type um\nn : Type un\ninst✝¹ : Fintype n\nR : Type u_1\ninst✝ : CommRing R\nc : R\nB : Matrix m n R\nhc : c ∈ nonZeroDivisors R\nhc' : IsSMulRegular R c\nhreg : IsSMulRegular (m → R) c\nf : (m → R) →ₗ[R] m → R := (LinearMap.lsmul R (m → R)) c\nhcomp : (c • B).mulVecLin = f ∘ₗ B.mulVecLin\n⊢ finrank R ↥(Submodu... | rw [rank, rank, hcomp, LinearMap.range_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 439,
"column": 15
} | {
"line": 439,
"column": 21
} | {
"line": 439,
"column": 22
} | [
{
"pp": "m : Type um\nR : Type uR\ninst✝² : Field R\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix m m R\nL L' : List (TransvectionStruct m R)\nD : m → R\nhM0 : M = (List.map toMatrix L).prod * diagonal D * (List.map toMatrix L').prod\nE : m → R := fun i ↦ if D i = 0 then 1 else (D i)⁻¹\nE_def : E = fun... | [
"m : Type um\nR : Type uR\ninst✝² : Field R\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix m m R\nL L' : List (TransvectionStruct m R)\nD : m → R\nhM0 : M = (List.map toMatrix L).prod * diagonal D * (List.map toMatrix L').prod\nE : m → R := fun i ↦ if D i = 0 then 1 else (D i)⁻¹\nE_def : E = fun i ↦ if D i ... | U_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 85
} | {
"line": 452,
"column": 2
} | [
{
"pp": "m : Type um\nR : Type uR\ninst✝² : Field R\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix m m R\nL L' : List (TransvectionStruct m R)\nD : m → R\nhM0 : M = (List.map toMatrix L).prod * diagonal D * (List.map toMatrix L').prod\nE : m → R := fun i ↦ if D i = 0 then 1 else (D i)⁻¹\nE_def : E = fun... | [
"m : Type um\nR : Type uR\ninst✝² : Field R\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix m m R\nL L' : List (TransvectionStruct m R)\nD : m → R\nhM0 : M = (List.map toMatrix L).prod * diagonal D * (List.map toMatrix L').prod\nE : m → R := fun i ↦ if D i = 0 then 1 else (D i)⁻¹\nE_def : E = fun i ↦ if D i ... | refine ⟨V, U, e, (isUnit_iff_isUnit_det _).2 hVdet, isUnit_prod_comp_inverse _, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Lie.Cochain | {
"line": 106,
"column": 32
} | {
"line": 106,
"column": 47
} | {
"line": 106,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nL : Type u_2\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nf : oneCochain R L M\nx : L\nx✝¹ : R\nx✝ : L\n⊢ ⁅x, f (x✝¹ • x✝)⁆ - ⁅x✝¹ • x✝, f x⁆ - f ⁅x, x✝¹ ... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Cochain | {
"line": 106,
"column": 32
} | {
"line": 106,
"column": 47
} | {
"line": 106,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nL : Type u_2\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nf : oneCochain R L M\nx : L\nx✝¹ : R\nx✝ : L\n⊢ ⁅x, f (x✝¹ • x✝)⁆ - ⁅x✝¹ • x✝, f x⁆ - f ⁅x, x✝¹ ... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Cochain | {
"line": 106,
"column": 32
} | {
"line": 106,
"column": 47
} | {
"line": 106,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nL : Type u_2\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nf : oneCochain R L M\nx : L\nx✝¹ : R\nx✝ : L\n⊢ ⁅x, f (x✝¹ • x✝)⁆ - ⁅x✝¹ • x✝, f x⁆ - f ⁅x, x✝¹ ... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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