module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Lie.TraceForm | {
"line": 99,
"column": 74
} | {
"line": 102,
"column": 63
} | {
"line": 104,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nx : L\n⊢ ⁅x, traceForm R L M⁆ = 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConsta... | [] | by
ext y z
rw [LieHom.lie_apply, LinearMap.sub_apply, Module.Dual.lie_apply, LinearMap.zero_apply,
LinearMap.zero_apply, traceForm_apply_lie_apply', sub_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Sequence | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 21
} | {
"line": 93,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nS : Sequence R\nm : ℕ\nhCoeff : ∀ i < m, IsUnit (↑S i).leadingCoeff\n⊢ span R (↑S '' Set.Iio m) = degreeLT R m",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Polynomial.degreeLT",
"Semiring.toModule",
"Polynomial.Sequence.elems'",
... | [
"case h₁\nR : Type u_1\ninst✝ : Ring R\nS : Sequence R\nm : ℕ\nhCoeff : ∀ i < m, IsUnit (↑S i).leadingCoeff\n⊢ ↑S '' Set.Iio m ⊆ ↑(degreeLT R m)",
"case h₂\nR : Type u_1\ninst✝ : Ring R\nS : Sequence R\nm : ℕ\nhCoeff : ∀ i < m, IsUnit (↑S i).leadingCoeff\n⊢ degreeLT R m ≤ span R (↑S '' Set.Iio m)"
] | apply span_eq_of_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 543,
"column": 4
} | {
"line": 546,
"column": 15
} | {
"line": 547,
"column": 4
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα β : Weight K (↥H) L\nhyp : coroot α = coro... | [
"K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα β : Weight K (↥H) L\nhyp : coroot α = coroot β\nhα : ¬... | have hβ : β.IsNonZero := by
contrapose hα
simp only [← coroot_eq_zero_iff] at hα ⊢
rwa [hyp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.BaseChange | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 26
} | {
"line": 60,
"column": 2
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nL : Type u_3\ninst✝³ : Finite m\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : Field L\ne : m ≃ n\nK : Subfield L\nA : Matrix m n L\nB : Matrix n m L\nh_mem : ∀ (i : n) (j : m), B i j ∈ K\ni : m\nj : n\nhAB : Bᵀ * Aᵀ = 1\n⊢ A i j ∈ K",
"ppTerm": "?m.50",
"assig... | [
"m : Type u_1\nn : Type u_2\nL : Type u_3\ninst✝³ : Finite m\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : Field L\ne : m ≃ n\nK : Subfield L\nA : Matrix m n L\nB : Matrix n m L\nh_mem : ∀ (i : n) (j : m), B i j ∈ K\ni : m\nj : n\nhAB : Bᵀ * Aᵀ = 1\n⊢ Aᵀ j i ∈ K"
] | rw [← A.transpose_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 234,
"column": 10
} | {
"line": 234,
"column": 24
} | {
"line": 234,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℕ\nih1 : (T R (↑n + 1)).leadingCoeff = 2 ^ ((↑n + 1).natAbs - 1)\nih2 : (T R ↑n).leadingCoeff = 2 ^ ((↑n).natAbs - 1)\n⊢ (C 2).leadingCoeff = 2",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℕ\nih1 : (T R (↑n + 1)).leadingCoeff = 2 ^ ((↑n + 1).natAbs - 1)\nih2 : (T R ↑n).leadingCoeff = 2 ^ ((↑n).natAbs - 1)\n⊢ 2 = 2"
] | leadingCoeff_C | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 743,
"column": 28
} | {
"line": 743,
"column": 65
} | {
"line": 743,
"column": 65
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\n⊢ traceForm K (↥H) L = ∑ α ∈ LieSubalgebra.r... | [
"K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\n⊢ ∑ χ with χ.IsNonZero,\n finrank K ↥(genWeightSpac... | traceForm_eq_sum_finrank_nsmul' K H L | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.NumDen | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 47
} | {
"line": 120,
"column": 0
} | [
{
"pp": "case h.inj\nA : 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\nv : A\nh : (algebraMap A K) v = x\n⊢ Function.Injective ⇑(algebraMap A K)",
"ppTerm"... | [] | exact FaithfulSMul.algebraMap_injective A K | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.RationalRoot | {
"line": 129,
"column": 14
} | {
"line": 129,
"column": 32
} | {
"line": 129,
"column": 32
} | [
{
"pp": "case left\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = ↑(den A r) * inv\nd_ne_zero : (algebraMap ... | [
"case left\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = ↑(den A r) * inv\nd_ne_zero : (algebraMap A K) ↑(den A... | ← mk'_num_den' A r | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 433,
"column": 19
} | {
"line": 433,
"column": 33
} | {
"line": 433,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℕ\n⊢ (C 2).leadingCoeff = 2",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"CommSemiring.to... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℕ\n⊢ 2 = 2"
] | leadingCoeff_C | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 845,
"column": 76
} | {
"line": 846,
"column": 59
} | {
"line": 848,
"column": 0
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing R'\ninst✝ : Algebra R R'\nx : R\nn : ℤ\n⊢ (algebraMap R R') (eval x (C R n)) = eval ((algebraMap R R') x) (C R' n)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Po... | [] | by
rw [← aeval_algebraMap_apply_eq_algebraMap_eval, aeval_C] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 1007,
"column": 67
} | {
"line": 1007,
"column": 75
} | {
"line": 1007,
"column": 75
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nh : eval 0 ((⇑derivative)^[k + 2] (T R n)) = 0 - (↑n ^ 2 - ↑k ^ 2) * eval 0 ((⇑derivative)^[k] (T R n))\n⊢ eval 0 ((⇑derivative)^[k + 2] (T R n)) = -(↑n ^ 2 - ↑k ^ 2) * eval 0 ((⇑derivative)^[k] (T R n))",
"ppTerm": "?m.103",
"assigned": true,
... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nh : eval 0 ((⇑derivative)^[k + 2] (T R n)) = -((↑n ^ 2 - ↑k ^ 2) * eval 0 ((⇑derivative)^[k] (T R n)))\n⊢ eval 0 ((⇑derivative)^[k + 2] (T R n)) = -(↑n ^ 2 - ↑k ^ 2) * eval 0 ((⇑derivative)^[k] (T R n))"
] | zero_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 1014,
"column": 67
} | {
"line": 1014,
"column": 75
} | {
"line": 1014,
"column": 75
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nh : eval 0 ((⇑derivative)^[k + 2] (U R n)) = 0 - ((↑n + 1) ^ 2 - (↑k + 1) ^ 2) * eval 0 ((⇑derivative)^[k] (U R n))\n⊢ eval 0 ((⇑derivative)^[k + 2] (U R n)) = -((↑n + 1) ^ 2 - (↑k + 1) ^ 2) * eval 0 ((⇑derivative)^[k] (U R n))",
"ppTerm": "?m.115",
... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nh : eval 0 ((⇑derivative)^[k + 2] (U R n)) = -(((↑n + 1) ^ 2 - (↑k + 1) ^ 2) * eval 0 ((⇑derivative)^[k] (U R n)))\n⊢ eval 0 ((⇑derivative)^[k + 2] (U R n)) = -((↑n + 1) ^ 2 - (↑k + 1) ^ 2) * eval 0 ((⇑derivative)^[k] (U R n))"
] | zero_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 797,
"column": 2
} | {
"line": 797,
"column": 6
} | {
"line": 798,
"column": 2
} | [
{
"pp": "R : Type u_1\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\na : ℕ → ℕ\nha : StrictMono a\nf : (n : ℕ) → M →ₗ[R] N ⧸ I ^ a n • ⊤\nhf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m\nm n :... | [
"R : Type u_1\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\na : ℕ → ℕ\nha : StrictMono a\nf : (n : ℕ) → M →ₗ[R] N ⧸ I ^ a n • ⊤\nhf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m\nm n : ℕ\nhle : m ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 12
} | {
"line": 171,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nϖ : R\nhϖ : Irreducible ϖ\nhR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x\np : R\nhp : Irreducible p\n⊢ ∀ n < 1, n = 0",
"ppTerm": "?m.130",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"of_decide_eq_true",
"PartialOrder.toPreo... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 6
} | {
"line": 226,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associates.mk fx.prod = Associates.mk x\n⊢... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associates.mk fx.prod = Associates.mk x\n⊢ Multiset.ma... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 43
} | {
"line": 253,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : R[X]\n⊢ p ∈ Subring.closure (insert X {f | f.degree ≤ 0})",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Polynomial.induction_on",
"Nat.instMulZeroClass",
"WithBot",
"Subring.instSetLike",
... | [
"case refine_1\nR : Type u_1\ninst✝ : Ring R\np : R[X]\n⊢ ∀ (a : R), C a ∈ Subring.closure (insert X {f | f.degree ≤ 0})",
"case refine_2\nR : Type u_1\ninst✝ : Ring R\np : R[X]\n⊢ ∀ (p q : R[X]),\n p ∈ Subring.closure (insert X {f | f.degree ≤ 0}) →\n q ∈ Subring.closure (insert X {f | f.degree ≤ 0}) → p... | refine Polynomial.induction_on p ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 62
} | {
"line": 287,
"column": 2
} | [
{
"pp": "R : 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] ⧸ P) S... | [
"R : 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] ⧸ P) Sₘ\ninst✝ :\n... | let φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ M.le_comap_map | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 502,
"column": 4
} | {
"line": 504,
"column": 42
} | {
"line": 505,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nx y : R\nh : (addVal R) x = (addVal R) y\nhx : ¬x = 0\nhy : ¬y = 0\nϖ : R\nhϖ : Irreducible ϖ\nm : ℕ\nα : Rˣ\nhx' : x = ↑α * ϖ ^ m\nn : ℕ\nβ : Rˣ\nhy' : y = ↑β * ϖ ^ n\n⊢ Associated x y",
"ppTerm": ... | [
"case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nx y : R\nhx : ¬x = 0\nhy : ¬y = 0\nϖ : R\nhϖ : Irreducible ϖ\nm : ℕ\nα : Rˣ\nhx' : x = ↑α * ϖ ^ m\nn : ℕ\nβ : Rˣ\nhy' : y = ↑β * ϖ ^ n\nh : ↑m * (addVal R) ϖ = ↑n * (addVal R) ϖ\n⊢ Associated (ϖ ^ m) (ϖ ^ n)"
] | simp only [hx', AddValuation.map_mul, addVal_eq_zero_of_unit, AddValuation.map_pow,
nsmul_eq_mul, zero_add, hy', associated_unit_mul_right_iff,
associated_unit_mul_left_iff] at h ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Valuation.Basic | {
"line": 848,
"column": 22
} | {
"line": 848,
"column": 37
} | {
"line": 848,
"column": 38
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : LinearOrderedCommGroupWithZero Γ'₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ''₀\ninst✝ : Ring R\nv : Valuation R Γ₀\nw : Valuation R Γ... | [
"K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : LinearOrderedCommGroupWithZero Γ'₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ''₀\ninst✝ : Ring R\nv : Valuation R Γ₀\nw : Valuation R Γ'₀\nu : Valu... | hx.choose_spec, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.DiscreteValuationRing.TFAE | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 14
} | {
"line": 120,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDedekindDomain R\nne_bot : ¬maximalIdeal R = ⊥\na : R\nha₁ : a ∈ maximalIdeal R\nha₂ : a ≠ 0\nhle : Ideal.span {a} ≤ maximalIdeal R\nthis✝ : (Ideal.span {a}).radical = maximalIdeal R\nthis : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}\nn... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDedekindDomain R\nne_bot : ¬maximalIdeal R = ⊥\na : R\nha₁ : a ∈ maximalIdeal R\nha₂ : a ≠ 0\nhle : Ideal.span {a} ≤ maximalIdeal R\nthis✝ : (Ideal.span {a}).radical = maximalIdeal R\nthis : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}\nn : ℕ\nhn : N... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.DiscreteValuationRing.TFAE | {
"line": 151,
"column": 6
} | {
"line": 151,
"column": 16
} | {
"line": 151,
"column": 16
} | [
{
"pp": "case neg.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDedekindDomain R\nne_bot : ¬maximalIdeal R = ⊥\na : R\nha₁ : a ∈ maximalIdeal R\nha₂ : a ≠ 0\nhle : Ideal.span {a} ≤ maximalIdeal R\nthis✝¹ : (Ideal.span {a}).radical = maximalIdeal R\nthis✝ : ∃ n, maximalIdeal R ^ n ≤ Ide... | [
"case neg.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDedekindDomain R\nne_bot : ¬maximalIdeal R = ⊥\na : R\nha₁ : a ∈ maximalIdeal R\nha₂ : a ≠ 0\nhle : Ideal.span {a} ≤ maximalIdeal R\nthis✝¹ : (Ideal.span {a}).radical = maximalIdeal R\nthis✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}\... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 579,
"column": 2
} | {
"line": 579,
"column": 29
} | {
"line": 580,
"column": 2
} | [
{
"pp": "case intro\nR : Type u_1\ninst✝² : CommRing R\nι : Type u_2\ninst✝¹ : _root_.Finite ι\ninst✝ : IsJacobsonRing R\nval✝ : Fintype ι\n⊢ IsJacobsonRing (MvPolynomial ι R)",
"ppTerm": "?intro",
"assigned": true,
"usedConstants": [
"Fintype.card",
"Equiv",
"Fintype.equivFin",
... | [
"case intro\nR : Type u_1\ninst✝² : CommRing R\nι : Type u_2\ninst✝¹ : _root_.Finite ι\ninst✝ : IsJacobsonRing R\nval✝ : Fintype ι\ne : ι ≃ Fin (Fintype.card ι) := Fintype.equivFin ι\n⊢ IsJacobsonRing (MvPolynomial ι R)"
] | let e := Fintype.equivFin ι | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Valuation.Basic | {
"line": 888,
"column": 2
} | {
"line": 888,
"column": 46
} | {
"line": 890,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : Ring R\nv : Valuation R Γ₀\nh : v.IsEquiv v\nx y : R\nw✝¹ : (ofClass v) x ≠ 0\nw✝ : (ofClass v) y ≠ 0\n⊢ h.orderMonoidIso ↑(valueGroup.mk (ofClass v) x y w✝¹ w✝) =\n (OrderMonoidIso.refl (ofClass v).ValueGroup... | [] | · simp [orderMonoidIso, valueGroup₀Fun_spec] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Valuation.Basic | {
"line": 896,
"column": 2
} | {
"line": 896,
"column": 46
} | {
"line": 898,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_3\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : LinearOrderedCommGroupWithZero Γ'₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ''₀\ninst✝ : Ring R\nv : Valuation R Γ₀\nw : Valuation R Γ'₀\nu : Valuation R Γ''₀\nh : v.IsEquiv w\n... | [] | · simp [orderMonoidIso, valueGroup₀Fun_spec] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 328,
"column": 69
} | {
"line": 334,
"column": 6
} | {
"line": 336,
"column": 0
} | [
{
"pp": "ι : 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\ninst✝ : Fintype ι\ni : ι\n⊢ (∏ a, (P.RootForm (P.root a)) (P.root a)) • P.coroot i ∈ (P.Polarization.domRestr... | [] | by
obtain ⟨c, hc⟩ := Finset.dvd_prod_of_mem (fun a ↦ P.RootForm (P.root a) (P.root a))
(Finset.mem_univ i)
rw [hc, mul_comm, mul_smul, rootForm_self_smul_coroot]
refine LinearMap.mem_range.mpr ?_
use ⟨c • 2 • P.root i, by aesop⟩
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 78,
"column": 17
} | {
"line": 78,
"column": 35
} | {
"line": 78,
"column": 36
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup M\ninst✝ : Module K M\np : ι → Submodule K M\nh₁ : ∀ (i : ι), p i ≠ ⊤\nj : ι\ns : Finset ι\nhj : j ∉ s\nhs : s.Nonempty\nh₂ : ↑s.card + 1 < ENat.card K\nhj' : ↑(p j) ∪ ⋃ i ∈ s, ↑(p i) = univ\nx : M\nhx : x ∈ p j\nhx₀ : x ... | [
"ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup M\ninst✝ : Module K M\np : ι → Submodule K M\nh₁ : ∀ (i : ι), p i ≠ ⊤\nj : ι\ns : Finset ι\nhj : j ∉ s\nhs : s.Nonempty\nh₂ : ↑s.card + 1 < ENat.card K\nhj' : ↑(p j) ∪ ⋃ i ∈ s, ↑(p i) = univ\nx : M\nhx : x ∈ p j\nhx₀ : x ≠ 0\ny : M\n... | this.encard_image, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 87,
"column": 52
} | {
"line": 87,
"column": 92
} | {
"line": 87,
"column": 92
} | [
{
"pp": "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 : Weight K (↥H) L\nhα : ¬α.IsZ... | [
"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 : Weight K (↥H) L\nhα : ¬α.IsZero\nx' : L ... | (chainLength_aux α β hα h.1).choose_spec | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 98,
"column": 27
} | {
"line": 102,
"column": 34
} | {
"line": 104,
"column": 0
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup M\ninst✝² : Module K M\ninst✝¹ : Finite ι\ninst✝ : Infinite K\nf : ι → Dual K M\nh : ∀ (i : ι), ∃ x, (f i) x ≠ 0\n⊢ ∃ x, ∀ (i : ι), (f i) x ≠ 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"... | [] | by
let p i := LinearMap.ker (f i)
replace h i : p i ≠ ⊤ := by specialize h i; aesop
obtain ⟨x, hx⟩ := Submodule.exists_forall_notMem_of_forall_ne_top p h
exact ⟨x, by simpa [p] using hx⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 240,
"column": 8
} | {
"line": 240,
"column": 12
} | {
"line": 240,
"column": 13
} | [
{
"pp": "case a\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.IsNonZ... | [
"case a\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.IsNonZero\nβ' : We... | hβ', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 77
} | {
"line": 247,
"column": 2
} | [
{
"pp": "K : 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.IsNonZero\nβ' ... | [
"K : 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.IsNonZero\nβ' : Weight K (... | have : (β' : H → K) = -n • (-α) + β := by rwa [neg_smul, smul_neg, neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 6
} | {
"line": 263,
"column": 2
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα : Weight K (↥H) L\ninst✝ : Nontrivial L\n... | [
"K : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα : Weight K (↥H) L\ninst✝ : Nontrivial L\nhα : α.IsNon... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 374,
"column": 35
} | {
"line": 381,
"column": 21
} | {
"line": 383,
"column": 0
} | [
{
"pp": "K : 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\nhβ : β.IsNonZero\n⊢ (... | [] | by
intro e
have : β (coroot α) = 0 := by
by_cases hα : α.IsZero
· simp [coroot_eq_zero_iff.mpr hα]
simpa [root_apply_coroot hα, mul_two] using congr_fun (sub_eq_zero.mp e) (coroot α)
have : reflectRoot α β = β := by ext; simp [reflectRoot, this]
exact hβ (this ▸ e) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.WeylGroup | {
"line": 138,
"column": 4
} | {
"line": 139,
"column": 22
} | {
"line": 140,
"column": 4
} | [
{
"pp": "case refine_1\nι : 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\ni : ι\n⊢ (MulOpposite.op ∘ P.coreflection) i ∈ ↑((Equiv.coweightHom P).restrict P.weylGroup).ra... | [
"case refine_1\nι : 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\ni : ι\n⊢ ∃ x ∈ P.weylGroup, (Equiv.coweightHom P) x = (MulOpposite.op ∘ P.coreflection) i"
] | simp only [MonoidHom.restrict_range, Subgroup.coe_map, mem_image,
SetLike.mem_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.WeylGroup | {
"line": 166,
"column": 4
} | {
"line": 167,
"column": 22
} | {
"line": 168,
"column": 4
} | [
{
"pp": "case refine_1\nι : 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\ni : ι\n⊢ P.reflectionPerm i ∈ ↑P.weylGroupToPerm.range",
"ppTerm": "?refine_1",
"assign... | [
"case refine_1\nι : 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\ni : ι\n⊢ ∃ x ∈ P.weylGroup, (Equiv.indexHom P) x = P.reflectionPerm i"
] | simp only [MonoidHom.restrict_range, Subgroup.coe_map, mem_image,
SetLike.mem_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 450,
"column": 19
} | {
"line": 450,
"column": 58
} | {
"line": 452,
"column": 0
} | [
{
"pp": "case inr\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 : Weight K (↥H) L\nhα : α ... | [] | simpa using DFunLike.congr_fun h.symm x | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RepresentationTheory.Basic | {
"line": 476,
"column": 2
} | {
"line": 476,
"column": 38
} | {
"line": 477,
"column": 2
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Group G\nH : Type u_4\ninst✝ : MulAction G H\ng : G\nf : k[H]\nh : H\n⊢ (((ofMulAction k G H) g) f).coeff h = f.coeff (g⁻¹ • h)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
... | [
"k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Group G\nH : Type u_4\ninst✝ : MulAction G H\ng : G\nf : k[H]\nh : H\n⊢ (((ofMulAction k G H) g) f).coeff (g • g⁻¹ • h) = f.coeff (g⁻¹ • h)"
] | conv_lhs => rw [← smul_inv_smul g h] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.LinearAlgebra.RootSystem.Irreducible | {
"line": 130,
"column": 73
} | {
"line": 135,
"column": 16
} | {
"line": 136,
"column": 4
} | [
{
"pp": "ι : 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\ninst✝ : Nontrivial M\nh : IsSimpleOrder ↥P.weylGroupRootRep.invtSubmodule\nq : Submodule R M\nhq₁ : ∀ (i : ι)... | [] | by
let q' : P.weylGroupRootRep.invtSubmodule :=
⟨q, (Representation.mem_invtSubmodule P.weylGroupRootRep).mpr this⟩
suffices q' = ⊤ by simpa [q']
apply (IsSimpleOrder.eq_bot_or_eq_top _).resolve_left
simpa [q'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Int.ConditionallyCompleteOrder | {
"line": 40,
"column": 4
} | {
"line": 41,
"column": 49
} | {
"line": 42,
"column": 2
} | [
{
"pp": "x✝ : Set ℤ\nhn : x✝.Nonempty\nhb : BddAbove x✝\n⊢ IsLUB x✝ (if h : x✝.Nonempty ∧ BddAbove x✝ then ↑((Classical.choose ⋯).greatestOfBdd ⋯ ⋯) else 0)",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Int.instLinearOrder",
"PartialOrder.to... | [] | rw [dif_pos ⟨hn, hb⟩]
exact (isGreatest_coe_greatestOfBdd ..).isLUB | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Int.ConditionallyCompleteOrder | {
"line": 40,
"column": 4
} | {
"line": 41,
"column": 49
} | {
"line": 42,
"column": 2
} | [
{
"pp": "x✝ : Set ℤ\nhn : x✝.Nonempty\nhb : BddAbove x✝\n⊢ IsLUB x✝ (if h : x✝.Nonempty ∧ BddAbove x✝ then ↑((Classical.choose ⋯).greatestOfBdd ⋯ ⋯) else 0)",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Int.instLinearOrder",
"PartialOrder.to... | [] | rw [dif_pos ⟨hn, hb⟩]
exact (isGreatest_coe_greatestOfBdd ..).isLUB | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Hom | {
"line": 544,
"column": 8
} | {
"line": 544,
"column": 84
} | {
"line": 544,
"column": 85
} | [
{
"pp": "case h.right\nι : 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\ni... | [] | rw [← Hom.coweightMap_mul, f.inv_val, Hom.coweightMap_one, LinearMap.id_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 15
} | {
"line": 175,
"column": 2
} | [
{
"pp": "case h\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\nb : P.Base\ninst✝² : Finite ι\ninst✝¹ : IsAddTorsionFree M\ninst✝ : IsAddTorsionFree N\ni : ι\nh : i ... | [
"case h\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\nb : P.Base\ninst✝² : Finite ι\ninst✝¹ : IsAddTorsionFree M\ninst✝ : IsAddTorsionFree N\ni : ι\nh : i ∈ b.support\... | rw [hg] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 369,
"column": 2
} | {
"line": 369,
"column": 12
} | {
"line": 370,
"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\nb : P.Base\ni : ι\n⊢ ∀ m ∈ AddSubmonoid.closure (⇑P.root '' ↑b.support),\n ∃ f, Function.support f ⊆ ↑b.sup... | [
"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\nb : P.Base\ni : ι\nm : M\nhm : m ∈ AddSubmonoid.closure (⇑P.root '' ↑b.support)\n⊢ ∃ f, Function.support f ⊆ ↑b.support ∧ ... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 126,
"column": 16
} | {
"line": 126,
"column": 27
} | {
"line": 126,
"column": 28
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Field R\ninst✝⁴ : CharZero R\ninst✝³ : Module R M\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\nf : M →+ ℚ\nhf : ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Field R\ninst✝⁴ : CharZero R\ninst✝³ : Module R M\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\nf : M →+ ℚ\nhf : ∀ (i : ι), f... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 96
} | {
"line": 119,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\nhf : IsCauSeq abv f\ni : ℕ\nh : ∀ j ≥ i, abv (f j - f i) < 1\nR : ℕ → α := Nat.rec (abv (f 0)) fun i c ↦ max c (abv (f i.... | [] | simpa using (abv_add abv _ _).trans_lt <| add_lt_add_of_le_of_lt (this i _ le_rfl) (h _ hij) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 96
} | {
"line": 119,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\nhf : IsCauSeq abv f\ni : ℕ\nh : ∀ j ≥ i, abv (f j - f i) < 1\nR : ℕ → α := Nat.rec (abv (f 0)) fun i c ↦ max c (abv (f i.... | [] | simpa using (abv_add abv _ _).trans_lt <| add_lt_add_of_le_of_lt (this i _ le_rfl) (h _ hij) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 96
} | {
"line": 119,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\nhf : IsCauSeq abv f\ni : ℕ\nh : ∀ j ≥ i, abv (f j - f i) < 1\nR : ℕ → α := Nat.rec (abv (f 0)) fun i c ↦ max c (abv (f i.... | [] | simpa using (abv_add abv _ _).trans_lt <| add_lt_add_of_le_of_lt (this i _ le_rfl) (h _ hij) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 424,
"column": 4
} | {
"line": 424,
"column": 80
} | {
"line": 425,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nε : α\nε0 : 0 < ε\nx✝ : ℕ\nH : ∀ j ≥ x✝, abv (↑(f - g) j) < ε / 2 ∧ ∀ k ≥ j, abv (↑f k - ↑f j) < ε / 2\nj : ℕ\n... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nε : α\nε0 : 0 < ε\nx✝ : ℕ\nH : ∀ j ≥ x✝, abv (↑(f - g) j) < ε / 2 ∧ ∀ k ≥ j, abv (↑f k - ↑f j) < ε / 2\nj : ℕ\nij : j ≥ x✝\... | have := lt_of_le_of_lt (abv_add abv (f j - g j) _) (add_lt_add h₁ (h₂ _ jk)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 624,
"column": 57
} | {
"line": 624,
"column": 89
} | {
"line": 625,
"column": 6
} | [
{
"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 : ι\... | [] | simp [hm, Fin.le_def, Fin.is_le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 618,
"column": 32
} | {
"line": 618,
"column": 40
} | {
"line": 618,
"column": 40
} | [
{
"pp": "case inr.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nf : CauSeq α abs\nh✝ : ¬f.LimZero\nK : α\nK0 : K > 0\nhK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|\ni : ℕ\nhi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K\nh : ↑f i ≤ 0\nj : ℕ\nij : j ≥ i\nthis : K ≤ |↑f j|... | [
"case inr.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nf : CauSeq α abs\nh✝ : ¬f.LimZero\nK : α\nK0 : K > 0\nhK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|\ni : ℕ\nhi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K\nh : ↑f i ≤ 0\nj : ℕ\nij : j ≥ i\nthis : K ≤ |↑f j|\nh₁ : K ≤ -... | zero_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 690,
"column": 40
} | {
"line": 692,
"column": 37
} | {
"line": 692,
"column": 37
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nf : CauSeq α abs\nK : α\nH : ∀ (i : ℕ), |↑f i| < K\ni : ℕ\nx✝ : i ≥ 0\n⊢ 1 ≤ ↑(const (K + 1) - f) i",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"C... | [] | by
rw [sub_apply, const_apply, le_sub_iff_add_le', add_le_add_iff_right]
exact le_of_lt (abs_lt.1 (H _)).2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 800,
"column": 4
} | {
"line": 805,
"column": 30
} | {
"line": 806,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ j ≥ i, ε ≤ ↑(a - b) j\n⊢ a ⊓ b ≈ b",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mp... | [] | intro _ _
refine ⟨i, fun j hj => ?_⟩
dsimp
rw [← min_sub_sub_right]
rwa [sub_self, min_eq_right, abs_zero]
exact ε0.le.trans (h _ hj) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 800,
"column": 4
} | {
"line": 805,
"column": 30
} | {
"line": 806,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ j ≥ i, ε ≤ ↑(a - b) j\n⊢ a ⊓ b ≈ b",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mp... | [] | intro _ _
refine ⟨i, fun j hj => ?_⟩
dsimp
rw [← min_sub_sub_right]
rwa [sub_self, min_eq_right, abs_zero]
exact ε0.le.trans (h _ hj) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 18
} | {
"line": 173,
"column": 2
} | [
{
"pp": "case mp\nn : Type u_2\nR : Type u_4\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nv : n → R\na✝ : Nontrivial R\ni : n\nh : 0 < star (v i) * v i\nhv : v i = 0\n⊢ False",
"ppTerm": "?mp",
"assigne... | [] | simp [hv] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.Hermitian | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 28
} | {
"line": 132,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nn : Type u_4\ninst✝ : Star α\nA : Matrix m m (Matrix n n α)\n⊢ ((comp m m n n α) A).IsHermitian ↔ ∀ (i j : m) (i' j' : n), star (A j i j' i') = A i j i' j'",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Matrix.comp",
"Eq.mpr",
"Equiv... | [
"α : Type u_1\nm : Type u_3\nn : Type u_4\ninst✝ : Star α\nA : Matrix m m (Matrix n n α)\n⊢ (∀ (a : m) (b : n) (a_1 : m) (b_1 : n), star (A a_1 a b_1 b) = A a a_1 b b_1) ↔\n ∀ (i j : m) (i' j' : n), star (A j i j' i') = A i j i' j'"
] | simp [IsHermitian.ext_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Eigenspace.Matrix | {
"line": 55,
"column": 10
} | {
"line": 55,
"column": 21
} | {
"line": 55,
"column": 22
} | [
{
"pp": "R : Type u_1\nn : Type u_2\nM : Type u_3\ninst✝⁷ : DecidableEq n\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : Nontrivial R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nd : n → R\nμ : R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nb : Basis n R M\nthis : ∀ (i : n), HasEigenvalue ((toLin b b)... | [
"R : Type u_1\nn : Type u_2\nM : Type u_3\ninst✝⁷ : DecidableEq n\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : Nontrivial R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nd : n → R\nμ : R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nb : Basis n R M\nthis : ∀ (i : n), HasEigenvalue ((toLin b b) (diagonal d... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 63
} | {
"line": 258,
"column": 0
} | [
{
"pp": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nha : IsUnit A.det\nz1 z2 : ℤ\n⊢ A ^ (z1 - z2) = A ^ z1 / A ^ z2",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
... | [] | rw [sub_eq_add_neg, zpow_add ha, zpow_neg ha, div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 63
} | {
"line": 258,
"column": 0
} | [
{
"pp": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nha : IsUnit A.det\nz1 z2 : ℤ\n⊢ A ^ (z1 - z2) = A ^ z1 / A ^ z2",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
... | [] | rw [sub_eq_add_neg, zpow_add ha, zpow_neg ha, div_eq_mul_inv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 63
} | {
"line": 258,
"column": 0
} | [
{
"pp": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nha : IsUnit A.det\nz1 z2 : ℤ\n⊢ A ^ (z1 - z2) = A ^ z1 / A ^ z2",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
... | [] | rw [sub_eq_add_neg, zpow_add ha, zpow_neg ha, div_eq_mul_inv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 611,
"column": 2
} | {
"line": 611,
"column": 52
} | {
"line": 613,
"column": 0
} | [
{
"pp": "n : Type u_1\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nQ : QuadraticForm ℝ (n → ℝ)\nhQ :\n (LinearMap.BilinMap.toQuadraticMap\n ((LinearMap.toMatrix₂' ℝ).invFun ((↑(LinearMap.toMatrix₂' ℝ)).toFun ((QuadraticMap.associatedHom ℝ) Q)))).PosDef\n⊢ Q.toMatrix'.PosDef",
"ppTerm": "?m.92",
"as... | [] | exact .of_toQuadraticForm' (isSymm_toMatrix' Q) hQ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Eigenspace.Minpoly | {
"line": 101,
"column": 2
} | {
"line": 105,
"column": 79
} | {
"line": 107,
"column": 0
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nf : End R M\ninst✝² : IsDomain R\ninst✝¹ : Module.Finite R M\ninst✝ : IsTorsionFree R M\n⊢ Set.Finite f.HasEigenvalue",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Module.End.instRi... | [] | have h : minpoly R f ≠ 0 := minpoly.ne_zero (Algebra.IsIntegral.isIntegral (R := R) f)
convert! (minpoly R f).rootSet_finite R
ext μ
change f.HasEigenvalue μ ↔ _
rw [hasEigenvalue_iff_isRoot, mem_rootSet_of_ne h, IsRoot, coe_aeval_eq_eval] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Eigenspace.Minpoly | {
"line": 101,
"column": 2
} | {
"line": 105,
"column": 79
} | {
"line": 107,
"column": 0
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nf : End R M\ninst✝² : IsDomain R\ninst✝¹ : Module.Finite R M\ninst✝ : IsTorsionFree R M\n⊢ Set.Finite f.HasEigenvalue",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Module.End.instRi... | [] | have h : minpoly R f ≠ 0 := minpoly.ne_zero (Algebra.IsIntegral.isIntegral (R := R) f)
convert! (minpoly R f).rootSet_finite R
ext μ
change f.HasEigenvalue μ ↔ _
rw [hasEigenvalue_iff_isRoot, mem_rootSet_of_ne h, IsRoot, coe_aeval_eq_eval] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 158,
"column": 2
} | {
"line": 159,
"column": 32
} | {
"line": 160,
"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 : P.IsCrystallographic\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : Finite ι\ni j : ... | [
"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 : P.IsCrystallographic\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : Finite ι\ni j : ↥b.support\n... | rw [b.cartanMatrix_eq_neg_chainTopCoeff hij, neg_eq_zero, Int.natCast_eq_zero,
P.chainTopCoeff_eq_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Lagrange | {
"line": 310,
"column": 53
} | {
"line": 310,
"column": 89
} | {
"line": 312,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\nv r : ι → F\n⊢ (interpolate ∅ v) r = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Polynomial.C",
"Semiring.toModule",
"Pi.addCommMonoid",
... | [] | by rw [interpolate_apply, sum_empty] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Lagrange | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 6
} | {
"line": 407,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\nv r : ι → F\nhvt : Set.InjOn v ↑t\nhs : s.Nonempty\nhst : s ⊆ t\n⊢ (interpolate t v) r = ∑ i ∈ s, (interpolate (insert i (t \\ s)) v) r * Lagrange.basis s v i",
"ppTerm": "?m.35",
"assigned": true,
"usedCon... | [
"F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\nv r : ι → F\nhvt : Set.InjOn v ↑t\nhs : s.Nonempty\nhst : s ⊆ t\n⊢ ∑ i ∈ s, (interpolate (insert i (t \\ s)) v) r * Lagrange.basis s v i = (interpolate t v) r"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.LinearAlgebra.Lagrange | {
"line": 472,
"column": 8
} | {
"line": 472,
"column": 47
} | {
"line": 473,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv r : ι → F\nhvs : Set.InjOn v ↑s\nk : ℕ\nhk : k < #s\ni : ι\nhi : i ∈ s\nhvs' : Set.InjOn v ↑(s.erase i)\n⊢ (⇑derivative)^[k] (∏ vj ∈ image v (s.erase i), (X - C vj)) =\n ↑k.factorial * ∑ t ∈ powersetCard (#s - (k + ... | [] | grind [iterate_derivative_prod_X_sub_C] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.LinearAlgebra.Lagrange | {
"line": 472,
"column": 8
} | {
"line": 472,
"column": 47
} | {
"line": 473,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv r : ι → F\nhvs : Set.InjOn v ↑s\nk : ℕ\nhk : k < #s\ni : ι\nhi : i ∈ s\nhvs' : Set.InjOn v ↑(s.erase i)\n⊢ (⇑derivative)^[k] (∏ vj ∈ image v (s.erase i), (X - C vj)) =\n ↑k.factorial * ∑ t ∈ powersetCard (#s - (k + ... | [] | grind [iterate_derivative_prod_X_sub_C] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Lagrange | {
"line": 472,
"column": 8
} | {
"line": 472,
"column": 47
} | {
"line": 473,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv r : ι → F\nhvs : Set.InjOn v ↑s\nk : ℕ\nhk : k < #s\ni : ι\nhi : i ∈ s\nhvs' : Set.InjOn v ↑(s.erase i)\n⊢ (⇑derivative)^[k] (∏ vj ∈ image v (s.erase i), (X - C vj)) =\n ↑k.factorial * ∑ t ∈ powersetCard (#s - (k + ... | [] | grind [iterate_derivative_prod_X_sub_C] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Lagrange | {
"line": 618,
"column": 4
} | {
"line": 619,
"column": 42
} | {
"line": 620,
"column": 4
} | [
{
"pp": "case h_deg_eq\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nG : Subgroup Rˣ\ninst✝ : Fintype ↥G\nh : degree 1 < (X ^ Fintype.card ↥G).degree\n⊢ (nodal (↑G).toFinset Units.val).degree = (X ^ Fintype.card ↥G - 1).degree",
"ppTerm": "?h_deg_eq",
"assigned": true,
"usedConstants": [
... | [
"case h_deg_eq\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nG : Subgroup Rˣ\ninst✝ : Fintype ↥G\nh : degree 1 < (X ^ Fintype.card ↥G).degree\n⊢ Fintype.card ↑↑G = Fintype.card ↥G"
] | rw [degree_sub_eq_left_of_degree_lt h, degree_nodal, Set.toFinset_card, degree_pow, degree_X,
nsmul_eq_mul, mul_one, Nat.cast_inj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 289,
"column": 68
} | {
"line": 289,
"column": 78
} | {
"line": 289,
"column": 79
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\nS : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* MvPolynomial σ S\nd : σ →₀ ℕ\n⊢ f 1 * (monomial d) 1 = (monomial d) 1",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"NonAs... | [
"σ : Type u_1\nR : Type u_3\nS : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* MvPolynomial σ S\nd : σ →₀ ℕ\n⊢ 1 * (monomial d) 1 = (monomial d) 1"
] | f.map_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 321,
"column": 4
} | {
"line": 321,
"column": 31
} | {
"line": 322,
"column": 4
} | [
{
"pp": "case calc_2\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\nj : τ\n⊢ (∃ a ∈ φ.support, ∃ a_1 ∈ a.support, j ∈ (f a_1).vars) → ∃ a ∈ φ.vars, j ∈ (f a).vars",
"ppTerm": "?calc_2",
"assigned": true,
"use... | [
"case calc_2\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\nj : τ\nd : σ →₀ ℕ\nhd : d ∈ φ.support\ni : σ\nhi : i ∈ d.support\nhj : j ∈ (f i).vars\n⊢ ∃ a ∈ φ.vars, j ∈ (f a).vars"
] | rintro ⟨d, hd, ⟨i, hi, hj⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.LinearAlgebra.Charpoly.Basic | {
"line": 116,
"column": 2
} | {
"line": 130,
"column": 70
} | {
"line": 132,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Module.Free R M\ninst✝¹ : Module.Finite R M\nf : M →ₗ[R] M\ninst✝ : Nontrivial R\nhf : Function.Injective ⇑f\n⊢ (minpoly R f).coeff 0 ≠ 0",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants... | [] | intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine degree_lt_degree_mul_X fun h => ?_
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
rwa [Monic.de... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Charpoly.Basic | {
"line": 116,
"column": 2
} | {
"line": 130,
"column": 70
} | {
"line": 132,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Module.Free R M\ninst✝¹ : Module.Finite R M\nf : M →ₗ[R] M\ninst✝ : Nontrivial R\nhf : Function.Injective ⇑f\n⊢ (minpoly R f).coeff 0 ≠ 0",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants... | [] | intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine degree_lt_degree_mul_X fun h => ?_
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP
have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
rwa [Monic.de... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 100
} | {
"line": 165,
"column": 2
} | [
{
"pp": "L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = ... | [] | rw [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, aeval_apply_of_mem_apply_eq_smul (hsv i), hq, hyv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 375,
"column": 2
} | {
"line": 376,
"column": 7
} | {
"line": 378,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : AddCommMonoid M\nw : σ → M\nn : M\nφ : MvPolynomial σ R\ninst✝ : DecidableEq M\n⊢ (weightedHomogeneousComponent w n) φ = ∑ d ∈ φ.support with (weight w) d = n, (monomial d) (coeff d φ)",
"ppTerm": "?m.42",
"assigned": t... | [] | simp [weightedHomogeneousComponent, MvPolynomial, coeff, Finsupp.filter_eq_sum, support, monomial]
congr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 375,
"column": 2
} | {
"line": 376,
"column": 7
} | {
"line": 378,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : AddCommMonoid M\nw : σ → M\nn : M\nφ : MvPolynomial σ R\ninst✝ : DecidableEq M\n⊢ (weightedHomogeneousComponent w n) φ = ∑ d ∈ φ.support with (weight w) d = n, (monomial d) (coeff d φ)",
"ppTerm": "?m.42",
"assigned": t... | [] | simp [weightedHomogeneousComponent, MvPolynomial, coeff, Finsupp.filter_eq_sum, support, monomial]
congr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 419,
"column": 2
} | {
"line": 419,
"column": 12
} | {
"line": 420,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nφ : MvPolynomial σ R\n⊢ (Function.support fun m ↦ (weightedHomogeneousComponent w m) φ) ⊆ (fun d ↦ (weight w) d) '' ↑φ.support",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nφ : MvPolynomial σ R\nm : M\nhm : m ∈ Function.support fun m ↦ (weightedHomogeneousComponent w m) φ\n⊢ m ∈ (fun d ↦ (weight w) d) '' ↑φ.support"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 268,
"column": 8
} | {
"line": 268,
"column": 37
} | {
"line": 269,
"column": 6
} | [
{
"pp": "case h₀\nR : Type u_1\nL : Type u_2\nM : Type u_3\nι : Type u_5\nιM : Type u_7\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup L\ninst✝⁸ : Module R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ιM\ninst✝³ : DecidableEq ι\ninst✝² : DecidableEq ιM... | [] | rw [this, if_neg H, map_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 271,
"column": 39
} | {
"line": 271,
"column": 84
} | {
"line": 272,
"column": 6
} | [
{
"pp": "case e_f.hX\nR : Type u_1\nL : Type u_2\nM : Type u_3\nι : Type u_5\nιM : Type u_7\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup L\ninst✝⁸ : Module R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ιM\ninst✝³ : DecidableEq ι\ninst✝² : DecidableE... | [
"case e_f.hX\nR : Type u_1\nL : Type u_2\nM : Type u_3\nι : Type u_5\nιM : Type u_7\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup L\ninst✝⁸ : Module R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ιM\ninst✝³ : DecidableEq ι\ninst✝² : DecidableEq ιM\nb : Ba... | TensorProduct.AlgebraTensorModule.lift_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 452,
"column": 6
} | {
"line": 452,
"column": 47
} | {
"line": 453,
"column": 6
} | [
{
"pp": "R : Type u_5\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nN : ℕ\nIH : ∀ {F : MvPolynomial (Fin N) R} {n : ℕ}, F.IsHomogeneous n → F ≠ 0 → ↑n ≤ #R → ∃ r, (eval r) F ≠ 0\nF : MvPolynomial (Fin (N + 1)) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhnR : ↑n ≤ #R\nhdeg : ((finSuccEquiv R N) F).natDegree < n ... | [
"R : Type u_5\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nN : ℕ\nIH : ∀ {F : MvPolynomial (Fin N) R} {n : ℕ}, F.IsHomogeneous n → F ≠ 0 → ↑n ≤ #R → ∃ r, (eval r) F ≠ 0\nF : MvPolynomial (Fin (N + 1)) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhnR : ↑n ≤ #R\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\ni : ℕ\n... | refine lt_of_le_of_lt natDegree_map_le ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 179,
"column": 2
} | {
"line": 189,
"column": 31
} | {
"line": 190,
"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)\nhVW : IsCompl V W\nhφV : ∀ x ∈ V, φ x ∈ V\nhφW : ∀ x ∈ W, φ ... | [
"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)\nhVW : IsCompl V W\nhφV : ∀ x ∈ V, φ x ∈ V\nhφW : ∀ x ∈ W, φ x ∈ W\nF : ↥... | have hG : natTrailingDegree (charpoly G) = 0 := by
apply Polynomial.natTrailingDegree_eq_zero_of_constantCoeff_ne_zero
apply ((not_hasEigenvalue_zero_tfae G).out 2 5).mpr
intro x hx
apply Subtype.ext
suffices x.1 ∈ V ⊓ W by rwa [hVW.inf_eq_bot, Submodule.mem_bot] at this
suffices x.1 ∈ V from ⟨t... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 399,
"column": 13
} | {
"line": 399,
"column": 32
} | {
"line": 399,
"column": 33
} | [
{
"pp": "m : Type um\nR : Type uR\ninst✝¹ : Field R\ninst✝ : DecidableEq m\nw : m → R\nw' : { i // w i ≠ 0 } → m → R := fun i ↦ diagonal w ↑i\nh : LinearIndependent R w'\nhrw : insert 0 (range (diagonal w).col) = insert 0 (range w')\n⊢ Module.rank R ↥(span R (range (diagonal w).col)) = lift.{uR, um} #{ i // w i... | [
"m : Type um\nR : Type uR\ninst✝¹ : Field R\ninst✝ : DecidableEq m\nw : m → R\nw' : { i // w i ≠ 0 } → m → R := fun i ↦ diagonal w ↑i\nh : LinearIndependent R w'\nhrw : insert 0 (range (diagonal w).col) = insert 0 (range w')\n⊢ Module.rank R ↥(span R (insert 0 (range (diagonal w).col))) = lift.{uR, um} #{ i // w i ... | ← span_insert_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Classical | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 15
} | {
"line": 131,
"column": 0
} | [
{
"pp": "n : Type u_1\nR : Type u₂\ninst✝² : CommRing R\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\ni j k : n\nr : R\n⊢ (singleSubSingle i j) r + (singleSubSingle j k) r = (singleSubSingle i k) r",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"AddMemClass.toAddCommSemigroup",
... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Classical | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 15
} | {
"line": 131,
"column": 0
} | [
{
"pp": "n : Type u_1\nR : Type u₂\ninst✝² : CommRing R\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\ni j k : n\nr : R\n⊢ (singleSubSingle i j) r + (singleSubSingle j k) r = (singleSubSingle i k) r",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"AddMemClass.toAddCommSemigroup",
... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Classical | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 15
} | {
"line": 136,
"column": 0
} | [
{
"pp": "n : Type u_1\nR : Type u₂\ninst✝² : CommRing R\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\ni j k : n\nr : R\n⊢ (singleSubSingle i k) r - (singleSubSingle i j) r = (singleSubSingle j k) r",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"instSMulOfMu... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Classical | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 15
} | {
"line": 136,
"column": 0
} | [
{
"pp": "n : Type u_1\nR : Type u₂\ninst✝² : CommRing R\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\ni j k : n\nr : R\n⊢ (singleSubSingle i k) r - (singleSubSingle i j) r = (singleSubSingle j k) r",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"instSMulOfMu... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Classical | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 15
} | {
"line": 141,
"column": 0
} | [
{
"pp": "n : Type u_1\nR : Type u₂\ninst✝² : CommRing R\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\ni j k : n\nr : R\n⊢ (singleSubSingle i k) r - (singleSubSingle j k) r = (singleSubSingle i j) r",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"sub_sub_sub_cancel_right",
"Semir... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Classical | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 15
} | {
"line": 141,
"column": 0
} | [
{
"pp": "n : Type u_1\nR : Type u₂\ninst✝² : CommRing R\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\ni j k : n\nr : R\n⊢ (singleSubSingle i k) r - (singleSubSingle j k) r = (singleSubSingle i j) r",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"sub_sub_sub_cancel_right",
"Semir... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 499,
"column": 6
} | {
"line": 499,
"column": 23
} | {
"line": 499,
"column": 24
} | [
{
"pp": "case intro\nm : Type um\nn : Type un\nR : Type uR\ninst✝² : Fintype n\ninst✝¹ : Field R\ninst✝ : Finite m\nA : Matrix m n R\nval✝ : Fintype m\n⊢ A.rank = finrank R ↥(span R (range A.row))",
"ppTerm": "?intro",
"assigned": true,
"usedConstants": [
"Matrix.rank_transpose",
"Eq.mpr... | [
"case intro\nm : Type um\nn : Type un\nR : Type uR\ninst✝² : Fintype n\ninst✝¹ : Field R\ninst✝ : Finite m\nA : Matrix m n R\nval✝ : Fintype m\n⊢ Aᵀ.rank = finrank R ↥(span R (range A.row))"
] | ← rank_transpose, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Classical | {
"line": 221,
"column": 4
} | {
"line": 222,
"column": 52
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case inr.inr\np : Type u_2\nq : Type u_3\nR : Type u₂\ninst✝⁴ : DecidableEq p\ninst✝³ : DecidableEq q\ninst✝² : CommRing R\ninst✝¹ : Fintype p\ninst✝ : Fintype q\ni : R\nhi : i * i = -1\nx y : q\n⊢ ((Pso p q R i)ᵀ * indefiniteDiagonal p q R * Pso p q R i) (Sum.inr x) (Sum.inr y) = 1 (Sum.inr x) (Sum.in... | [] | by_cases h : x = y <;>
simp [Pso, indefiniteDiagonal, h, hi, one_apply] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Lie.Classical | {
"line": 221,
"column": 4
} | {
"line": 222,
"column": 52
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case inr.inr\np : Type u_2\nq : Type u_3\nR : Type u₂\ninst✝⁴ : DecidableEq p\ninst✝³ : DecidableEq q\ninst✝² : CommRing R\ninst✝¹ : Fintype p\ninst✝ : Fintype q\ni : R\nhi : i * i = -1\nx y : q\n⊢ ((Pso p q R i)ᵀ * indefiniteDiagonal p q R * Pso p q R i) (Sum.inr x) (Sum.inr y) = 1 (Sum.inr x) (Sum.in... | [] | by_cases h : x = y <;>
simp [Pso, indefiniteDiagonal, h, hi, one_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Classical | {
"line": 221,
"column": 4
} | {
"line": 222,
"column": 52
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case inr.inr\np : Type u_2\nq : Type u_3\nR : Type u₂\ninst✝⁴ : DecidableEq p\ninst✝³ : DecidableEq q\ninst✝² : CommRing R\ninst✝¹ : Fintype p\ninst✝ : Fintype q\ni : R\nhi : i * i = -1\nx y : q\n⊢ ((Pso p q R i)ᵀ * indefiniteDiagonal p q R * Pso p q R i) (Sum.inr x) (Sum.inr y) = 1 (Sum.inr x) (Sum.in... | [] | by_cases h : x = y <;>
simp [Pso, indefiniteDiagonal, h, hi, one_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Classical | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 6
} | {
"line": 378,
"column": 2
} | [
{
"pp": "n : Type u_1\np : Type u_2\nq : Type u_3\nl : Type u_4\nR : Type u₂\ninst✝⁸ : DecidableEq p\ninst✝⁷ : DecidableEq q\ninst✝⁶ : DecidableEq l\ninst✝⁵ : CommRing R\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype p\ninst✝² : Fintype q\ninst✝¹ : Fintype l\ninst✝ : Invertible 2\n⊢ ↥(skewAdjointMatricesLieSubalgebr... | [
"n : Type u_1\np : Type u_2\nq : Type u_3\nl : Type u_4\nR : Type u₂\ninst✝⁸ : DecidableEq p\ninst✝⁷ : DecidableEq q\ninst✝⁶ : DecidableEq l\ninst✝⁵ : CommRing R\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype p\ninst✝² : Fintype q\ninst✝¹ : Fintype l\ninst✝ : Invertible 2\n⊢ ↥(so' (Unit ⊕ l) l R) ≃ₗ⁅R⁆ ↥(skewAdjointMatr... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Lie.Derivation.BaseChange | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 33
} | {
"line": 92,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁴ : CommRing R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nL : Type u_3\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nx✝³ x✝² : LieDerivation R L L\nz x✝¹ x✝ : A ⊗[R] L\nhx :\n { toFun := ⇑(LinearMap.lTensor A ↑⁅x✝³, x✝²⁆), map_add' := ⋯, map_smul' := ⋯, ... | [] | simp_all [sub_add_sub_comm] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Derivation.BaseChange | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 33
} | {
"line": 92,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁴ : CommRing R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nL : Type u_3\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nx✝³ x✝² : LieDerivation R L L\nz x✝¹ x✝ : A ⊗[R] L\nhx :\n { toFun := ⇑(LinearMap.lTensor A ↑⁅x✝³, x✝²⁆), map_add' := ⋯, map_smul' := ⋯, ... | [] | simp_all [sub_add_sub_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Derivation.BaseChange | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 33
} | {
"line": 92,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁴ : CommRing R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nL : Type u_3\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nx✝³ x✝² : LieDerivation R L L\nz x✝¹ x✝ : A ⊗[R] L\nhx :\n { toFun := ⇑(LinearMap.lTensor A ↑⁅x✝³, x✝²⁆), map_add' := ⋯, map_smul' := ⋯, ... | [] | simp_all [sub_add_sub_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 79,
"column": 6
} | {
"line": 79,
"column": 16
} | {
"line": 80,
"column": 6
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nA : Type u_3\nV : Type u_4\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : IsPrincipalIdealRing R\ninst✝¹⁷ : IsDomain R\ninst✝¹⁶ : CharZero R\ninst✝¹⁵ : LieRing L\ninst✝¹⁴ : LieAlgebra R L\ninst✝¹³ : LieRing A\ninst✝¹² : LieAlgebra R A\ninst✝¹¹ : Bracket L A\ninst✝¹⁰ : Bracket A L\ninst✝⁹ ... | [
"R : Type u_1\nL : Type u_2\nA : Type u_3\nV : Type u_4\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : IsPrincipalIdealRing R\ninst✝¹⁷ : IsDomain R\ninst✝¹⁶ : CharZero R\ninst✝¹⁵ : LieRing L\ninst✝¹⁴ : LieAlgebra R L\ninst✝¹³ : LieRing A\ninst✝¹² : LieAlgebra R A\ninst✝¹¹ : Bracket L A\ninst✝¹⁰ : Bracket A L\ninst✝⁹ : AddCommGro... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 83,
"column": 6
} | {
"line": 85,
"column": 100
} | {
"line": 86,
"column": 6
} | [
{
"pp": "case inr\nR : Type u_1\nL : Type u_2\nA : Type u_3\nV : Type u_4\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : IsPrincipalIdealRing R\ninst✝¹⁷ : IsDomain R\ninst✝¹⁶ : CharZero R\ninst✝¹⁵ : LieRing L\ninst✝¹⁴ : LieAlgebra R L\ninst✝¹³ : LieRing A\ninst✝¹² : LieAlgebra R A\ninst✝¹¹ : Bracket L A\ninst✝¹⁰ : Bracket A ... | [
"case inr\nR : Type u_1\nL : Type u_2\nA : Type u_3\nV : Type u_4\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : IsPrincipalIdealRing R\ninst✝¹⁷ : IsDomain R\ninst✝¹⁶ : CharZero R\ninst✝¹⁵ : LieRing L\ninst✝¹⁴ : LieAlgebra R L\ninst✝¹³ : LieRing A\ninst✝¹² : LieAlgebra R A\ninst✝¹¹ : Bracket L A\ninst✝¹⁰ : Bracket A L\ninst✝⁹ : ... | rw [T, LinearMap.sub_apply, pow_succ', Module.End.mul_apply, LieModule.toEnd_apply_apply,
LieModule.toEnd_apply_apply, LinearMap.smul_apply, Module.End.one_apply, leibniz_lie,
lie_swap_lie w z, H, H, lie_add, lie_smul, add_sub_assoc, add_sub_assoc, sub_self, add_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Loop | {
"line": 114,
"column": 22
} | {
"line": 114,
"column": 57
} | {
"line": 116,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nL : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : AddCommGroup A\ninst✝¹ : DistribSMul A R\ninst✝ : SMulCommClass A R R\nΦ : LinearMap.BilinForm R L\nr : R\nx : loopAlgebra R A L\n⊢ { toFun := fun g ↦ ((toFinsupp R A L) g).sum fun a v ↦... | [] | ext; simp [-smul_eq_mul, smul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Loop | {
"line": 114,
"column": 22
} | {
"line": 114,
"column": 57
} | {
"line": 116,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nL : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : AddCommGroup A\ninst✝¹ : DistribSMul A R\ninst✝ : SMulCommClass A R R\nΦ : LinearMap.BilinForm R L\nr : R\nx : loopAlgebra R A L\n⊢ { toFun := fun g ↦ ((toFinsupp R A L) g).sum fun a v ↦... | [] | ext; simp [-smul_eq_mul, smul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Cartan | {
"line": 177,
"column": 34
} | {
"line": 177,
"column": 40
} | {
"line": 179,
"column": 0
} | [
{
"pp": "⊢ A 1 = !![2]",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"of_decide_eq_true",
"Matrix.decidableEq",
"Matrix",
"Matrix.of",
"Int.instDecidableEq",
"id",
"Equiv",
"instOfNatNat",
"Int",
"Ca... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
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