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 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Real.Basic | {
"line": 289,
"column": 33
} | {
"line": 289,
"column": 58
} | {
"line": 291,
"column": 0
} | [
{
"pp": "⊢ mk 0 = 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"abs",
"Real.ofCauchy_zero",
"congrArg",
"IsAbsoluteValue.abs_isAbsoluteValue",
"Rat",
"CauSeq.Completion.Cauchy",
"Rat.linearOr... | [] | rw [← ofCauchy_zero]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Real.Basic | {
"line": 429,
"column": 4
} | {
"line": 429,
"column": 28
} | {
"line": 430,
"column": 2
} | [
{
"pp": "case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ y✝¹ ≤ y✝¹ ⊔ y✝",
"ppTerm": "?h.h",
"assigned": true,
"usedConstants": [
"Rat",
"Rat.linearOrder",
"CauSeq.le_sup_left",
"Rat.instField",
"Rat.instIsStrictOrderedRing"
],
"usedFVars": [
"y✝¹",
"y✝"
... | [] | exact CauSeq.le_sup_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 318,
"column": 30
} | {
"line": 318,
"column": 47
} | {
"line": 319,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | [] | simpa using hsp i | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 318,
"column": 30
} | {
"line": 318,
"column": 47
} | {
"line": 319,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | [] | simpa using hsp i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 318,
"column": 30
} | {
"line": 318,
"column": 47
} | {
"line": 319,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | [] | simpa using hsp i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 180,
"column": 4
} | {
"line": 192,
"column": 64
} | {
"line": 194,
"column": 0
} | [
{
"pp": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA X Y : M\nhx : IsUnit X.det\nhy : IsUnit Y.det\nh : SemiconjBy A X Y\nn : ℕ\n⊢ SemiconjBy A (X ^ -[n+1]) (Y ^ -[n+1])",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Units.val",
... | [] | have hx' : IsUnit (X ^ n.succ).det := by
rw [det_pow]
exact hx.pow n.succ
have hy' : IsUnit (Y ^ n.succ).det := by
rw [det_pow]
exact hy.pow n.succ
rw [zpow_negSucc, zpow_negSucc, nonsing_inv_apply _ hx', nonsing_inv_apply _ hy', SemiconjBy]
refine (isRegular_of_isLeftRegular_det hy'... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 180,
"column": 4
} | {
"line": 192,
"column": 64
} | {
"line": 194,
"column": 0
} | [
{
"pp": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA X Y : M\nhx : IsUnit X.det\nhy : IsUnit Y.det\nh : SemiconjBy A X Y\nn : ℕ\n⊢ SemiconjBy A (X ^ -[n+1]) (Y ^ -[n+1])",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Units.val",
... | [] | have hx' : IsUnit (X ^ n.succ).det := by
rw [det_pow]
exact hx.pow n.succ
have hy' : IsUnit (Y ^ n.succ).det := by
rw [det_pow]
exact hy.pow n.succ
rw [zpow_negSucc, zpow_negSucc, nonsing_inv_apply _ hx', nonsing_inv_apply _ hy', SemiconjBy]
refine (isRegular_of_isLeftRegular_det hy'... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Eigenspace.Matrix | {
"line": 106,
"column": 39
} | {
"line": 106,
"column": 55
} | {
"line": 107,
"column": 4
} | [
{
"pp": "R : Type u_1\nn : Type u_2\nM : Type u_3\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Fintype n\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nd : n → R\nμ : R\nb : Basis n R M\ninst✝ : IsDomain R\nx : M\nhx : x ∈ maxGenEigenspace ((toLin b b) (diagonal d)) μ\nk : ℕ\nhk : (((toLin b b) (diago... | [] | simp [one_eq_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 510,
"column": 4
} | {
"line": 511,
"column": 56
} | {
"line": 512,
"column": 2
} | [
{
"pp": "n : Type u_2\ninst✝⁴ : Fintype n\nK : Type u_5\ninst✝³ : Field K\ninst✝² : PartialOrder K\ninst✝¹ : StarRing K\ninst✝ : DecidableEq n\nM : Matrix n n K\nhM : M.PosDef\nh : ¬IsUnit M\n⊢ ∃ a, a ≠ 0 ∧ M *ᵥ a = 0",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"... | [] | obtain ⟨a, b, ha⟩ := Function.not_injective_iff.mp <| mulVec_injective_iff_isUnit.not.mpr h
exact ⟨a - b, by simp [sub_eq_zero, ha, mulVec_sub]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 510,
"column": 4
} | {
"line": 511,
"column": 56
} | {
"line": 512,
"column": 2
} | [
{
"pp": "n : Type u_2\ninst✝⁴ : Fintype n\nK : Type u_5\ninst✝³ : Field K\ninst✝² : PartialOrder K\ninst✝¹ : StarRing K\ninst✝ : DecidableEq n\nM : Matrix n n K\nhM : M.PosDef\nh : ¬IsUnit M\n⊢ ∃ a, a ≠ 0 ∧ M *ᵥ a = 0",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"... | [] | obtain ⟨a, b, ha⟩ := Function.not_injective_iff.mp <| mulVec_injective_iff_isUnit.not.mpr h
exact ⟨a - b, by simp [sub_eq_zero, ha, mulVec_sub]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 132,
"column": 32
} | {
"line": 132,
"column": 40
} | {
"line": 132,
"column": 41
} | [
{
"pp": "S : Type u_1\ninst✝ : Semiring S\nn : ℕ\nk : S\n⊢ eval k (ascPochhammer S n * (X + ↑n)) = eval k (ascPochhammer S n) * (k + ↑n)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.eval",
"NonAssocSemiring.toAddC... | [
"S : Type u_1\ninst✝ : Semiring S\nn : ℕ\nk : S\n⊢ eval k (ascPochhammer S n * X + ascPochhammer S n * ↑n) = eval k (ascPochhammer S n) * (k + ↑n)"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 200,
"column": 34
} | {
"line": 200,
"column": 42
} | {
"line": 200,
"column": 43
} | [
{
"pp": "case succ\nS : Type u_1\ninst✝² : Semiring S\ninst✝¹ : PartialOrder S\ninst✝ : IsStrictOrderedRing S\ns : S\nh : 0 < s\nn : ℕ\nih : 0 < eval s (ascPochhammer S n)\n⊢ 0 < eval s (ascPochhammer S n * (X + ↑n))",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Distrib.leftDistri... | [
"case succ\nS : Type u_1\ninst✝² : Semiring S\ninst✝¹ : PartialOrder S\ninst✝ : IsStrictOrderedRing S\ns : S\nh : 0 < s\nn : ℕ\nih : 0 < eval s (ascPochhammer S n)\n⊢ 0 < eval s (ascPochhammer S n * X + ascPochhammer S n * ↑n)"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 368,
"column": 8
} | {
"line": 368,
"column": 22
} | {
"line": 368,
"column": 23
} | [
{
"pp": "case succ\nn : ℕ\nih : descPochhammer ℤ n = (ascPochhammer ℤ n).comp (X - ↑n + 1)\n⊢ descPochhammer ℤ (n + 1) = (ascPochhammer ℤ (n + 1)).comp (X - ↑(n + 1) + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Polynomial.instOne",
... | [
"case succ\nn : ℕ\nih : descPochhammer ℤ n = (ascPochhammer ℤ n).comp (X - ↑n + 1)\n⊢ descPochhammer ℤ (n + 1) = (ascPochhammer ℤ (n + 1)).comp (X - (↑n + 1) + 1)"
] | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 376,
"column": 8
} | {
"line": 376,
"column": 22
} | {
"line": 376,
"column": 23
} | [
{
"pp": "case succ\nR : Type u\ninst✝ : Ring R\nr : R\nn : ℕ\nih : eval r (descPochhammer R n) = eval (r - ↑n + 1) (ascPochhammer R n)\n⊢ eval r (descPochhammer R (n + 1)) = eval (r - ↑(n + 1) + 1) (ascPochhammer R (n + 1))",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case succ\nR : Type u\ninst✝ : Ring R\nr : R\nn : ℕ\nih : eval r (descPochhammer R n) = eval (r - ↑n + 1) (ascPochhammer R n)\n⊢ eval r (descPochhammer R (n + 1)) = eval (r - (↑n + 1) + 1) (ascPochhammer R (n + 1))"
] | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 395,
"column": 34
} | {
"line": 395,
"column": 42
} | {
"line": 395,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nr : R\nk : ℕ\n⊢ eval (-r) (ascPochhammer R k * (X + ↑k)) = (-1) ^ (k + 1) * eval r (descPochhammer R (k + 1))",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.eval",
"NegZeroClass.... | [
"R : Type u\ninst✝ : Ring R\nr : R\nk : ℕ\n⊢ eval (-r) (ascPochhammer R k * X + ascPochhammer R k * ↑k) = (-1) ^ (k + 1) * eval r (descPochhammer R (k + 1))"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 409,
"column": 65
} | {
"line": 409,
"column": 74
} | {
"line": 409,
"column": 75
} | [
{
"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) * 0 = ↑((n - k) * 0)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul"... | [
"case pos\nR : Type u\ninst✝ : Ring R\nn k : ℕ\nih : eval (↑n) (descPochhammer R k) = ↑(n.descFactorial k)\nh : n < k\n⊢ 0 = ↑((n - k) * 0)"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 409,
"column": 75
} | {
"line": 409,
"column": 84
} | {
"line": 409,
"column": 85
} | [
{
"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⊢ 0 = ↑((n - k) * 0)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"MulZeroClass.toMul",
... | [
"case pos\nR : Type u\ninst✝ : Ring R\nn k : ℕ\nih : eval (↑n) (descPochhammer R k) = ↑(n.descFactorial k)\nh : n < k\n⊢ 0 = ↑0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 523,
"column": 14
} | {
"line": 523,
"column": 54
} | {
"line": 523,
"column": 55
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionSemiring K\ninst✝ : CharZero K\na b : ℕ\n⊢ ↑(b ! * a.choose b) = eval (↑(a - (b - 1))) (ascPochhammer K b)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"K : Type u_1\ninst✝¹ : DivisionSemiring K\ninst✝ : CharZero K\na b : ℕ\n⊢ ↑(a.descFactorial b) = eval (↑(a - (b - 1))) (ascPochhammer K b)"
] | ← descFactorial_eq_factorial_mul_choose, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 533,
"column": 14
} | {
"line": 533,
"column": 54
} | {
"line": 533,
"column": 55
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : CharZero K\na b : ℕ\n⊢ ↑(b ! * a.choose b) = eval (↑a) (descPochhammer K b)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.choose",
... | [
"K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : CharZero K\na b : ℕ\n⊢ ↑(a.descFactorial b) = eval (↑a) (descPochhammer K b)"
] | ← descFactorial_eq_factorial_mul_choose, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Lagrange | {
"line": 171,
"column": 25
} | {
"line": 171,
"column": 36
} | {
"line": 171,
"column": 36
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\nx : F\n⊢ degree 0 = ⊥",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithBot",
"congrArg",
"id",
"Bot.bot",
"Polynomial.degree",
"Polynomial.degree_zero",
"Field.toSemifield",
"Polynom... | [
"F : Type u_1\ninst✝ : Field F\nx : F\n⊢ ⊥ = ⊥"
] | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Lagrange | {
"line": 237,
"column": 38
} | {
"line": 237,
"column": 54
} | {
"line": 237,
"column": 54
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhij : i ≠ j\nhj : j ∈ s\n⊢ ∏ j_1 ∈ s.erase i, eval (v j) (basisDivisor (v i) (v j_1)) = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",... | [
"F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhij : i ≠ j\nhj : j ∈ s\n⊢ ∃ a ∈ s.erase i, eval (v j) (basisDivisor (v i) (v a)) = 0"
] | prod_eq_zero_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Lagrange | {
"line": 408,
"column": 4
} | {
"line": 420,
"column": 97
} | {
"line": 421,
"column": 2
} | [
{
"pp": "case refine_1\nF : 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⊢ (s.sup fun b ↦ ((interpolate (insert b (t \\ s)) v) r * Lagrange.basis s v b).degree) < ↑(#t)",
"ppTerm": "?refine_1",
"assigned... | [] | simp_rw [Nat.cast_withBot, Finset.sup_lt_iff (WithBot.bot_lt_coe #t), degree_mul]
intro i hi
have hs : 1 ≤ #s := Nonempty.card_pos ⟨_, hi⟩
have hst' : #s ≤ #t := card_le_card hst
have H : #t = 1 + (#t - #s) + (#s - 1) := by
rw [add_assoc, tsub_add_tsub_cancel hst' hs, ← add_tsub_assoc_of_le (hs.tr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Lagrange | {
"line": 408,
"column": 4
} | {
"line": 420,
"column": 97
} | {
"line": 421,
"column": 2
} | [
{
"pp": "case refine_1\nF : 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⊢ (s.sup fun b ↦ ((interpolate (insert b (t \\ s)) v) r * Lagrange.basis s v b).degree) < ↑(#t)",
"ppTerm": "?refine_1",
"assigned... | [] | simp_rw [Nat.cast_withBot, Finset.sup_lt_iff (WithBot.bot_lt_coe #t), degree_mul]
intro i hi
have hs : 1 ≤ #s := Nonempty.card_pos ⟨_, hi⟩
have hst' : #s ≤ #t := card_le_card hst
have H : #t = 1 + (#t - #s) + (#s - 1) := by
rw [add_assoc, tsub_add_tsub_cancel hst' hs, ← add_tsub_assoc_of_le (hs.tr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 50
} | {
"line": 114,
"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 = ... | [] | simpa using! Subsingleton.elim (⟨ν, hν⟩ : E) 0 | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.LinearAlgebra.Lagrange | {
"line": 601,
"column": 4
} | {
"line": 601,
"column": 40
} | {
"line": 602,
"column": 4
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ns : Finset ι\nv : ι → R\ninst✝ : DecidableEq ι\ni : ι\nt : Finset ι\nhit : i ∉ t\nIH : derivative (nodal t v) = ∑ i ∈ t, nodal (t.erase i) v\n⊢ ∑ i_1 ∈ t, (X - C (v i)) * nodal (t.erase i_1) v = ∑ x ∈ t, nodal ((insert i t).erase x) v",
... | [
"case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ns : Finset ι\nv : ι → R\ninst✝ : DecidableEq ι\ni : ι\nt : Finset ι\nhit : i ∉ t\nIH : derivative (nodal t v) = ∑ i ∈ t, nodal (t.erase i) v\nj : ι\nhjt : j ∈ t\n⊢ (X - C (v i)) * nodal (t.erase j) v = nodal ((insert i t).erase j) v"
] | refine sum_congr rfl fun j hjt => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Charpoly.ToMatrix | {
"line": 79,
"column": 41
} | {
"line": 79,
"column": 91
} | {
"line": 79,
"column": 91
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : Module.Finite R M₁\ninst✝⁴ : Free R M₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : Module.Finite R M₂\ninst✝ : Free R M₂\ne : M₁ ≃ₗ[R] M₂\nφ : End R M₁\nb : Basis (Cho... | [
"R : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : Module.Finite R M₁\ninst✝⁴ : Free R M₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : Module.Finite R M₂\ninst✝ : Free R M₂\ne : M₁ ≃ₗ[R] M₂\nφ : End R M₁\nb : Basis (ChooseBasisInde... | ← LinearMap.charpoly_toMatrix (e.conj φ) (b.map e) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.Weight | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 60
} | {
"line": 314,
"column": 0
} | [
{
"pp": "case succ\nσ : Type u_5\ns : Set ℕ\nn : ℕ\nhn : n + 1 ≠ 0\n⊢ ⇑degree ⁻¹' ((n + 1) • s) = (n + 1) • ⇑degree ⁻¹' s",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"False",
"Nat.instMulZeroClass",
"Nat.recAux",
"instHSMul",
... | [] | induction n <;> simp_all [succ_nsmul, degree_preimage_add] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 24
} | {
"line": 141,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nm : M\na b : MvPolynomial σ R\nha : a ∈ {x | IsWeightedHomogeneous w x m}\nhb : b ∈ {x | IsWeightedHomogeneous w x m}\nc : σ →₀ ℕ\nhc : coeff c (a + b) ≠ 0\n⊢ (weight w) c = m",
"ppTerm": "?m.39",... | [
"R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nm : M\na b : MvPolynomial σ R\nha : a ∈ {x | IsWeightedHomogeneous w x m}\nhb : b ∈ {x | IsWeightedHomogeneous w x m}\nc : σ →₀ ℕ\nhc : coeff c a + coeff c b ≠ 0\n⊢ (weight w) c = m"
] | rw [coeff_add] at hc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 309,
"column": 54
} | {
"line": 309,
"column": 69
} | {
"line": 310,
"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\nm : M\nmotive : (p : MvPolynomial σ R) → IsWeightedHomogeneous w p m → Prop\nzero : motive 0 ⋯\nadd :\n ∀ (p q : MvPolynomial σ R) (hp : IsWeightedHomogeneous w p m) (hq : IsWeightedHomogeneous w q m... | [] | simpa using h 1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 309,
"column": 54
} | {
"line": 309,
"column": 69
} | {
"line": 310,
"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\nm : M\nmotive : (p : MvPolynomial σ R) → IsWeightedHomogeneous w p m → Prop\nzero : motive 0 ⋯\nadd :\n ∀ (p q : MvPolynomial σ R) (hp : IsWeightedHomogeneous w p m) (hq : IsWeightedHomogeneous w q m... | [] | simpa using h 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 309,
"column": 54
} | {
"line": 309,
"column": 69
} | {
"line": 310,
"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\nm : M\nmotive : (p : MvPolynomial σ R) → IsWeightedHomogeneous w p m → Prop\nzero : motive 0 ⋯\nadd :\n ∀ (p q : MvPolynomial σ R) (hp : IsWeightedHomogeneous w p m) (hq : IsWeightedHomogeneous w q m... | [] | simpa using h 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 24
} | {
"line": 102,
"column": 4
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝ : CommSemiring R\nn : ℕ\na b : MvPolynomial σ R\nha : a ∈ {x | x.IsHomogeneous n}\nhb : b ∈ {x | x.IsHomogeneous n}\nc : σ →₀ ℕ\nhc : coeff c (a + b) ≠ 0\n⊢ (weight 1) c = n",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants":... | [
"σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝ : CommSemiring R\nn : ℕ\na b : MvPolynomial σ R\nha : a ∈ {x | x.IsHomogeneous n}\nhb : b ∈ {x | x.IsHomogeneous n}\nc : σ →₀ ℕ\nhc : coeff c a + coeff c b ≠ 0\n⊢ (weight 1) c = n"
] | rw [coeff_add] at hc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 590,
"column": 2
} | {
"line": 596,
"column": 37
} | {
"line": 598,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\nσ : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : LinearOrder M\ninst✝¹ : OrderBot M\ninst✝ : CanonicallyOrderedAdd M\nw : σ → M\nhw : NonTorsionWeight w\np : MvPolynomial σ R\n⊢ weightedTotalDegree w p = 0 ↔ ∀ m ∈ p.support, ∀ (x : σ), m x = 0",
... | [] | rw [← isWeightedHomogeneous_zero_iff_weightedTotalDegree_eq_zero, IsWeightedHomogeneous]
apply forall_congr'
intro m
rw [mem_support_iff]
apply forall_congr'
intro _
exact weightedDegree_eq_zero_iff hw | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 590,
"column": 2
} | {
"line": 596,
"column": 37
} | {
"line": 598,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\nσ : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : LinearOrder M\ninst✝¹ : OrderBot M\ninst✝ : CanonicallyOrderedAdd M\nw : σ → M\nhw : NonTorsionWeight w\np : MvPolynomial σ R\n⊢ weightedTotalDegree w p = 0 ↔ ∀ m ∈ p.support, ∀ (x : σ), m x = 0",
... | [] | rw [← isWeightedHomogeneous_zero_iff_weightedTotalDegree_eq_zero, IsWeightedHomogeneous]
apply forall_congr'
intro m
rw [mem_support_iff]
apply forall_congr'
intro _
exact weightedDegree_eq_zero_iff hw | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 267,
"column": 6
} | {
"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... | [
"case h₁\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 : Basis ... | · rintro kl - H
rw [this, if_neg H, map_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 31
} | {
"line": 144,
"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\n⊢ finrank K ↥(φ.maxGenEigenspace 0) = (charpoly φ).natTrailingDegree",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule... | [
"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\n⊢ finrank K ↥V = (charpoly φ).natTrailingDegree"
] | set V := φ.maxGenEigenspace 0 | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 451,
"column": 24
} | {
"line": 451,
"column": 43
} | {
"line": 453,
"column": 0
} | [
{
"pp": "case a\nR : Type u_1\nL : Type u_2\nM : Type u_3\nι : Type u_5\nι' : Type u_6\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : AddCommGroup L\ninst✝⁹ : Module R L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁶ : Fintype ι\ninst✝⁵ : Fintype ι'\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'... | [] | apply nilRankAux_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 451,
"column": 24
} | {
"line": 451,
"column": 43
} | {
"line": 453,
"column": 0
} | [
{
"pp": "case a\nR : Type u_1\nL : Type u_2\nM : Type u_3\nι : Type u_5\nι' : Type u_6\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : AddCommGroup L\ninst✝⁹ : Module R L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁶ : Fintype ι\ninst✝⁵ : Fintype ι'\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'... | [] | apply nilRankAux_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 550,
"column": 4
} | {
"line": 550,
"column": 17
} | {
"line": 551,
"column": 4
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup L\ninst✝⁷ : Module R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁴ : Free R M\ninst✝³ : Module.Finite R M\ninst✝² : Module.Finite R L\ninst✝¹ : Free R L\ninst✝ : IsDomain R\nh : ↑(finrank... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup L\ninst✝⁷ : Module R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁴ : Free R M\ninst✝³ : Module.Finite R M\ninst✝² : Module.Finite R L\ninst✝¹ : Free R L\ninst✝ : IsDomain R\nh : ↑(finrank R M) ≤ #R\n... | by_contra! h₀ | Mathlib.Tactic.ByContra._aux_Mathlib_Tactic_ByContra___macroRules_Mathlib_Tactic_ByContra_byContra!_1 | Mathlib.Tactic.ByContra.byContra! |
Mathlib.LinearAlgebra.SymplecticGroup | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 10
} | {
"line": 121,
"column": 0
} | [
{
"pp": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nh : A ∈ symplecticGroup l R\n⊢ -A ∈ symplecticGroup l R",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"inst... | [] | rw [mem_iff] at h ⊢
simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SymplecticGroup | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 10
} | {
"line": 121,
"column": 0
} | [
{
"pp": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nh : A ∈ symplecticGroup l R\n⊢ -A ∈ symplecticGroup l R",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"inst... | [] | rw [mem_iff] at h ⊢
simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.CartanExists | {
"line": 193,
"column": 34
} | {
"line": 196,
"column": 72
} | {
"line": 198,
"column": 2
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} :=... | [] | by
simp_rw [r, ← lieCharpoly_natDegree K E x' u] at this ⊢
rw [(lieCharpoly_monic K E x' u).eq_X_pow_iff_natDegree_le_natTrailingDegree]
exact le_natTrailingDegree (lieCharpoly_monic K E x' u).ne_zero this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Derivation.Lie | {
"line": 59,
"column": 13
} | {
"line": 59,
"column": 73
} | {
"line": 59,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nA : Type u_2\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nD1 D2 : Derivation R A A\na✝ : A\nr : R\nd e : Derivation R A A\na : A\n⊢ ⁅d, r • e⁆ a = (r • ⁅d, e⁆) a",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Derivation",
"Algebra.to_... | [] | simp only [commutator_apply, map_smul, smul_sub, smul_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Basis | {
"line": 323,
"column": 49
} | {
"line": 323,
"column": 60
} | {
"line": 323,
"column": 61
} | [
{
"pp": "case smul\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁴ : Finite ι\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nb : Basis ι R L\ninst✝ : Fintype ι\ni : ι\nx : L\nt : R\nu : L\nhu : u ∈ b.cartan\nhv' : ∀ (hx : u ∈ b.cartan), (b.baseSupp i) ⟨u, hx⟩ • b.e i = ⁅u, b.e i⁆\nhx : t •... | [
"case smul\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁴ : Finite ι\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nb : Basis ι R L\ninst✝ : Fintype ι\ni : ι\nx : L\nt : R\nu : L\nhu : u ∈ b.cartan\nhv' : ∀ (hx : u ∈ b.cartan), (b.baseSupp i) ⟨u, hx⟩ • b.e i = ⁅u, b.e i⁆\nhx : t • u ∈ b.carta... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Basis | {
"line": 322,
"column": 4
} | {
"line": 323,
"column": 75
} | {
"line": 325,
"column": 0
} | [
{
"pp": "case smul\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁴ : Finite ι\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nb : Basis ι R L\ninst✝ : Fintype ι\ni : ι\nx : L\nt : R\nu : L\nhu : u ∈ Submodule.span R (range b.h)\nhv' : ∀ (hx : u ∈ b.cartan), (b.baseSupp i) ⟨u, hx⟩ • b.e i = ... | [] | rw [← coe_cartan_eq_span, LieSubalgebra.mem_toSubmodule] at hu
rw [← SetLike.mk_smul_mk _ t u hu, map_smul, smul_assoc, hv', smul_lie] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Basis | {
"line": 322,
"column": 4
} | {
"line": 323,
"column": 75
} | {
"line": 325,
"column": 0
} | [
{
"pp": "case smul\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁴ : Finite ι\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nb : Basis ι R L\ninst✝ : Fintype ι\ni : ι\nx : L\nt : R\nu : L\nhu : u ∈ Submodule.span R (range b.h)\nhv' : ∀ (hx : u ∈ b.cartan), (b.baseSupp i) ⟨u, hx⟩ • b.e i = ... | [] | rw [← coe_cartan_eq_span, LieSubalgebra.mem_toSubmodule] at hu
rw [← SetLike.mk_smul_mk _ t u hu, map_smul, smul_assoc, hv', smul_lie] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Basis | {
"line": 337,
"column": 2
} | {
"line": 374,
"column": 57
} | {
"line": 376,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁶ : Finite ι\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nb : Basis ι R L\ninst✝² : Fintype ι\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\n⊢ b.borelUpper ≤ ⨆ n, ⨆ (_ : n ≠ 0), rootSpace b.cartan (∑ i, n i • ⇑(b.baseSupp i))",
"ppTerm... | [] | classical
intro x hx
replace hx : x ∈ lieSpan R L (range b.e) := by simpa [borelUpper] using hx
induction hx using lieSpan_induction with
| mem u hu =>
obtain ⟨i, rfl⟩ := hu
apply LieSubmodule.mem_iSup_of_mem (Pi.single i 1)
simp only [ne_eq, Pi.single_eq_zero_iff, one_ne_zero, not_false_eq_true, ns... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Algebra.Lie.Basis | {
"line": 337,
"column": 2
} | {
"line": 374,
"column": 57
} | {
"line": 376,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁶ : Finite ι\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nb : Basis ι R L\ninst✝² : Fintype ι\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\n⊢ b.borelUpper ≤ ⨆ n, ⨆ (_ : n ≠ 0), rootSpace b.cartan (∑ i, n i • ⇑(b.baseSupp i))",
"ppTerm... | [] | classical
intro x hx
replace hx : x ∈ lieSpan R L (range b.e) := by simpa [borelUpper] using hx
induction hx using lieSpan_induction with
| mem u hu =>
obtain ⟨i, rfl⟩ := hu
apply LieSubmodule.mem_iSup_of_mem (Pi.single i 1)
simp only [ne_eq, Pi.single_eq_zero_iff, one_ne_zero, not_false_eq_true, ns... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Basis | {
"line": 337,
"column": 2
} | {
"line": 374,
"column": 57
} | {
"line": 376,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁶ : Finite ι\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nb : Basis ι R L\ninst✝² : Fintype ι\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\n⊢ b.borelUpper ≤ ⨆ n, ⨆ (_ : n ≠ 0), rootSpace b.cartan (∑ i, n i • ⇑(b.baseSupp i))",
"ppTerm... | [] | classical
intro x hx
replace hx : x ∈ lieSpan R L (range b.e) := by simpa [borelUpper] using hx
induction hx using lieSpan_induction with
| mem u hu =>
obtain ⟨i, rfl⟩ := hu
apply LieSubmodule.mem_iSup_of_mem (Pi.single i 1)
simp only [ne_eq, Pi.single_eq_zero_iff, one_ne_zero, not_false_eq_true, ns... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.CartanExists | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 26
} | {
"line": 353,
"column": 26
} | [
{
"pp": "case h.refine_1\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, ... | [
"case h.refine_1\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⋯\nEy : ↑{x | ∃ y ∈ U, engel K y = x} := ⋯\nhUle... | apply Nat.find_min hz' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 38
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case h\nk : Type u_1\ninst✝¹⁰ : Field k\nL : Type u_2\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra k L\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : LieRingModule L V\ninst✝⁴ : LieModule k L V\ninst✝³ : CharZero k\ninst✝² : Module.Finite k V\ninst✝¹ : IsTriangularizable k L V\nA : L... | [
"case h.refine_1\nk : Type u_1\ninst✝¹⁰ : Field k\nL : Type u_2\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra k L\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : LieRingModule L V\ninst✝⁴ : LieModule k L V\ninst✝³ : CharZero k\ninst✝² : Module.Finite k V\ninst✝¹ : IsTriangularizable k L V\nA : LieI... | refine nontrivial_of_ne ⟨v, ?_⟩ 0 ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 200,
"column": 50
} | {
"line": 200,
"column": 61
} | {
"line": 200,
"column": 62
} | [
{
"pp": "k : Type u_1\ninst✝¹⁰ : Field k\nL : Type u_2\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra k L\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : LieRingModule L V\ninst✝⁴ : LieModule k L V\ninst✝³ : CharZero k\ninst✝² : Module.Finite k V\ninst✝¹ : IsTriangularizable k L V\nA : LieIdeal ... | [
"k : Type u_1\ninst✝¹⁰ : Field k\nL : Type u_2\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra k L\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : LieRingModule L V\ninst✝⁴ : LieModule k L V\ninst✝³ : CharZero k\ninst✝² : Module.Finite k V\ninst✝¹ : IsTriangularizable k L V\nA : LieIdeal k L\nhA✝ : I... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 420,
"column": 8
} | {
"line": 420,
"column": 31
} | {
"line": 420,
"column": 31
} | [
{
"pp": "case mk\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nval✝ : Cycle α\ns : List α\nhn✝ : Cycle.Nodup (Quot.mk (⇑(IsRotated.setoid α)) s)\nht : Cycle.Nontrivial (Quot.mk (⇑(IsRotated.setoid α)) s)\nx : α\nhl : 2 ≤ s.length\nhn : s.Nodup\nhx : x ∈ s\n⊢ s.formPerm.toCycle ⋯ ... | [
"case mk\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nval✝ : Cycle α\ns : List α\nhn✝ : Cycle.Nodup (Quot.mk (⇑(IsRotated.setoid α)) s)\nht : Cycle.Nontrivial (Quot.mk (⇑(IsRotated.setoid α)) s)\nx : α\nhl : 2 ≤ s.length\nhn : s.Nodup\nhx : x ∈ s\n⊢ ↑(s.formPerm.toList x) = ↑s",
"... | toCycle_eq_toList _ _ x | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 38
} | {
"line": 250,
"column": 2
} | [
{
"pp": "case h\nk : Type u_1\ninst✝¹¹ : Field k\nL : Type u_2\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra k L\nV : Type u_3\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : LieRingModule L V\ninst✝⁵ : LieModule k L V\ninst✝⁴ : CharZero k\ninst✝³ : Module.Finite k V\ninst✝² : Nontrivial V\ninst✝¹ : IsSolva... | [
"case h.refine_1\nk : Type u_1\ninst✝¹¹ : Field k\nL : Type u_2\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra k L\nV : Type u_3\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : LieRingModule L V\ninst✝⁵ : LieModule k L V\ninst✝⁴ : CharZero k\ninst✝³ : Module.Finite k V\ninst✝² : Nontrivial V\ninst✝¹ : IsSolvable... | refine nontrivial_of_ne ⟨v, ?_⟩ 0 ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 32
} | {
"line": 177,
"column": 4
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq r : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhr : r ∣ q\nhq : q ≠ 0\ni : ℕ := (normalizedFactors r).card\nhi : normalizedFactor... | [
"M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq r : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhr : r ∣ q\nhq : q ≠ 0\ni : ℕ := (normalizedFactors r).card\nhi : normalizedFactors r = Multis... | simp only [Finset.mem_image] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 142,
"column": 2
} | {
"line": 143,
"column": 52
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : FaithfulSMul R P\nI : FractionalIdeal S P\nreg : IsSMulRegular P I.den\nx✝ : P\nhx : x✝ ∈ ↑I\n⊢ (DistribSMul.toLinearMap R P I.den) x✝ ∈ Submodule.map (Algebra.linearMap R... | [
"case refine_2\nR : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : FaithfulSMul R P\nI : FractionalIdeal S P\nreg : IsSMulRegular P I.den\nx✝¹ x✝ : ↥↑I\nhxy : ((DistribSMul.toLinearMap R P I.den).restrict ⋯) x✝¹ = ((DistribSMul.toLinearMap R P I.den)... | · rw [← den_mul_self_eq_num]
exact Submodule.smul_mem_pointwise_smul _ _ _ hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 92
} | {
"line": 332,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : CommMonoidWithZero M\ninst✝³ : IsCancelMulZero M\nN : Type u_2\ninst✝² : CommMonoidWithZero N\ninst✝¹ : UniqueFactorizationMonoid N\ninst✝ : UniqueFactorizationMonoid M\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic ... | [
"M : Type u_1\ninst✝⁴ : CommMonoidWithZero M\ninst✝³ : IsCancelMulZero M\nN : Type u_2\ninst✝² : CommMonoidWithZero N\ninst✝¹ : UniqueFactorizationMonoid N\ninst✝ : UniqueFactorizationMonoid M\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ emulti... | refine le_antisymm (emultiplicity_prime_le_emultiplicity_image_by_factor_orderIso hp d) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 264,
"column": 73
} | {
"line": 267,
"column": 40
} | {
"line": 269,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : IsLocalization S P\n⊢ canonicalEquiv S P P = RingEquiv.refl (FractionalIdeal S P)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"FractionalIdeal.canonicalEq... | [] | by
rw [← canonicalEquiv_trans_canonicalEquiv S P P]
convert! (canonicalEquiv S P P).symm_trans_self
exact (canonicalEquiv_symm S P P).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 589,
"column": 62
} | {
"line": 589,
"column": 90
} | {
"line": 589,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : IsLocalization S P\nI : FractionalIdeal S P\n⊢ (R ∙ (algebraMap R P) ↑I.den) * ↑I = ↑↑I.num",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"S... | [
"R : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : IsLocalization S P\nI : FractionalIdeal S P\n⊢ (algebraMap R P) ↑I.den • ↑I = ↑↑I.num"
] | Submodule.span_singleton_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 334,
"column": 6
} | {
"line": 334,
"column": 84
} | {
"line": 335,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI✝ : Ideal A\nI : FractionalIdeal A⁰ (FractionRing A)\nhI : I ≠ ⊥\na : A\nJ : Ideal A\nha : a ≠ 0\nhJ : I = spanSingleton ... | [] | exact ⟨spanSingleton A⁰ (algebraMap _ _ a) * (J : FractionalIdeal A⁰ _)⁻¹, h₂⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 36
} | {
"line": 272,
"column": 2
} | [
{
"pp": "case neg\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : IsDedekindDomain A\nI : Ideal A\nι : Type u_4\ninst✝ : Nonempty ι\nJ : ι → Ideal A\nhI : ¬I = 0\nH : ⨅ i, I * J i ≤ I\n⊢ ⨅ i, I * J i ≤ I * ⨅ i, J i",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Ideal.... | [
"case neg\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : IsDedekindDomain A\nI : Ideal A\nι : Type u_4\ninst✝ : Nonempty ι\nJ : ι → Ideal A\nhI : ¬I = 0\nH : ⨅ i, I * J i ≤ I\nK : Ideal A\nhK : ⨅ i, I * J i = I * K\n⊢ ⨅ i, I * J i ≤ I * ⨅ i, J i"
] | obtain ⟨K, hK⟩ := dvd_iff_le.mpr H | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 204,
"column": 46
} | {
"line": 204,
"column": 83
} | {
"line": 204,
"column": 83
} | [
{
"pp": "K : Type u_1\nV : Type u_2\nV₂ : Type u_4\ninst✝⁵ : CommRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : IsNoetherianRing K\nf : V →ₗ[K] V₂\n⊢ f.HasNoetherianRange ↔ f.HasFiniteRange",
"ppTerm": "?m.43",
"assigned": true,
"usedCon... | [
"K : Type u_1\nV : Type u_2\nV₂ : Type u_4\ninst✝⁵ : CommRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : IsNoetherianRing K\nf : V →ₗ[K] V₂\n⊢ f.HasFiniteRange ↔ f.HasFiniteRange"
] | hasNoetherianRange_iff_hasFiniteRange | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 231,
"column": 36
} | {
"line": 231,
"column": 73
} | {
"line": 231,
"column": 73
} | [
{
"pp": "K : Type u_1\nV : Type u_2\nV₂ : Type u_4\ninst✝⁵ : CommRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : IsNoetherianRing K\nu v : V →ₗ[K] V₂\n⊢ (u - v).HasNoetherianRange ↔ (u - v).HasFiniteRange",
"ppTerm": "?m.51",
"assigned": true... | [
"K : Type u_1\nV : Type u_2\nV₂ : Type u_4\ninst✝⁵ : CommRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : IsNoetherianRing K\nu v : V →ₗ[K] V₂\n⊢ (u - v).HasFiniteRange ↔ (u - v).HasFiniteRange"
] | hasNoetherianRange_iff_hasFiniteRange | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 415,
"column": 4
} | {
"line": 415,
"column": 29
} | {
"line": 416,
"column": 4
} | [
{
"pp": "case a\nT : Type u_4\ninst✝¹ : CommRing T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nthis : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0\na : Ideal T\n⊢ count a (normalizedFactors (I ⊔ J)) ≤ count a (normalizedFactors I ∩ normalizedFactors J)",
"ppTerm": "?a✝",
... | [
"case a\nT : Type u_4\ninst✝¹ : CommRing T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nthis : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0\na : Ideal T\n⊢ count a (normalizedFactors (I ⊔ J)) ≤ min (count a (normalizedFactors I)) (count a (normalizedFactors J))"
] | rw [Multiset.count_inter] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 29
} | {
"line": 200,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R M\nl : M →ₗ[R] N\nhl : Function.Surjective ⇑l\ns : Finset N\nhs : Submodule.span R ↑s = ⊤\nhs' : (linearCombination R Subtype.va... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R M\nl : M →ₗ[R] N\nhl : Function.Surjective ⇑l\ns : Finset N\nhs : Submodule.span R ↑s = ⊤\nhs' : (linearCombination R Subtype.val).ker.FG\nH... | obtain ⟨y, hy⟩ := H (l x) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 637,
"column": 4
} | {
"line": 637,
"column": 28
} | {
"line": 638,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Field K\ninst✝ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nf : R ⧸ I →+* A ⧸ J\nhf : Function.Surjective ⇑f\n⊢ Monotone fun X ↦ ⟨comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)), ⋯⟩",
... | [
"R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Field K\ninst✝ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nf : R ⧸ I →+* A ⧸ J\nhf : Function.Surjective ⇑f\nX : Ideal R\nhX : X ∣ I\nY : Ideal R\nhY : Y ∣ I\nh : ⟨X, hX⟩ ≤ ⟨Y, hY⟩\n⊢ (fun X ↦ ⟨comap (Ideal.Quotient.mk... | rintro ⟨X, hX⟩ ⟨Y, hY⟩ h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 87
} | {
"line": 219,
"column": 4
} | [
{
"pp": "case hs1\nR : Type u_1\nM : Type u_3\nN : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : FinitePresentation R N\nl : M →ₗ[R] N\nhl : Function.Surjective ⇑l\ninst✝ : FinitePresentation R ↥l.ker\n⊢ (Submodule.map l ⊤).FG",
... | [] | rw [Submodule.map_top, LinearMap.range_eq_top.mpr hl]; exact Module.Finite.fg_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 87
} | {
"line": 219,
"column": 4
} | [
{
"pp": "case hs1\nR : Type u_1\nM : Type u_3\nN : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : FinitePresentation R N\nl : M →ₗ[R] N\nhl : Function.Surjective ⇑l\ninst✝ : FinitePresentation R ↥l.ker\n⊢ (Submodule.map l ⊤).FG",
... | [] | rw [Submodule.map_top, LinearMap.range_eq_top.mpr hl]; exact Module.Finite.fg_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 803,
"column": 47
} | {
"line": 806,
"column": 31
} | {
"line": 808,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn : ℕ\nh : a * b ∈ I ^ n\n⊢ a ∈ I ^ n ∨ b ∈ I",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
... | [] | by
rw [mul_comm] at h
rw [or_comm]
exact IsPrime.mul_mem_pow _ h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 374,
"column": 2
} | {
"line": 380,
"column": 16
} | {
"line": 381,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_4\nN' : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\ninst✝² : AddCommGroup N'\ninst✝¹ : Module R N'\nS : Submonoid R\nf : N →ₗ[R] N'\ninst✝ : IsLocalizedModule S f\ng : M →ₗ[R] N'\nσ : F... | [
"R : Type u_1\nM : Type u_2\nN : Type u_4\nN' : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\ninst✝² : AddCommGroup N'\ninst✝¹ : Module R N'\nS : Submonoid R\nf : N →ₗ[R] N'\ninst✝ : IsLocalizedModule S f\ng : M →ₗ[R] N'\nσ : Finset M\nhσ ... | have hi : f ∘ₗ Finsupp.linearCombination R i = (s₀ • g) ∘ₗ π := by
ext j
simp only [LinearMap.coe_comp, Function.comp_apply, Finsupp.lsingle_apply,
linearCombination_single, one_smul, LinearMap.map_smul_of_tower, ← hs, LinearMap.smul_apply,
i, s₀, π]
rw [← mul_smul, Finset.prod_erase_mul]
ex... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1026,
"column": 6
} | {
"line": 1026,
"column": 36
} | {
"line": 1027,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : NormalizationMonoid R\na b : R\nha : a ∈ normalizedFactors b\nhb : ¬b = 0\nthis : Prime (span {a})\nc : Ideal R\nhc : c ∈ normalizedFactors (span {b})\nhc' : Associated (span {a}) c\n⊢ span {a} ∈ normalized... | [] | rwa [associated_iff_eq.mp hc'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.MvPolynomial.Derivation | {
"line": 95,
"column": 83
} | {
"line": 95,
"column": 94
} | {
"line": 96,
"column": 8
} | [
{
"pp": "case monomial\nσ : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Module R A\ninst✝¹ : Module (MvPolynomial σ R) A\ninst✝ : IsScalarTower R (MvPolynomial σ R) A\nD : MvPolynomial σ R →ₗ[R] A\nh₁ : D 1 = 0\nH : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 ... | [
"case monomial\nσ : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Module R A\ninst✝¹ : Module (MvPolynomial σ R) A\ninst✝ : IsScalarTower R (MvPolynomial σ R) A\nD : MvPolynomial σ R →ₗ[R] A\nh₁ : D 1 = 0\nH : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (mo... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.PDeriv | {
"line": 83,
"column": 31
} | {
"line": 83,
"column": 40
} | {
"line": 83,
"column": 41
} | [
{
"pp": "case pos\nR : Type u\nσ : Type v\ninst✝ : CommSemiring R\ni : σ\nm : σ →₀ ℕ\nr : R\nh : m i = 0\n⊢ (monomial (single i 1 + (m - single i 1))) (1 * (r * 0)) = (monomial m) (0 • r)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",... | [
"case pos\nR : Type u\nσ : Type v\ninst✝ : CommSemiring R\ni : σ\nm : σ →₀ ℕ\nr : R\nh : m i = 0\n⊢ (monomial (single i 1 + (m - single i 1))) 0 = (monomial m) (0 • r)"
] | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 490,
"column": 8
} | {
"line": 491,
"column": 53
} | {
"line": 492,
"column": 6
} | [
{
"pp": "case refine_2\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 : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\n... | [] | have : ↑(-β) ∈ q := by rw [Weight.toLinear_neg]; exact q.neg_mem hβ_mem
exact le_iSup₂_of_le _ (hne (-β) this) le_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 490,
"column": 8
} | {
"line": 491,
"column": 53
} | {
"line": 492,
"column": 6
} | [
{
"pp": "case refine_2\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 : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\n... | [] | have : ↑(-β) ∈ q := by rw [Weight.toLinear_neg]; exact q.neg_mem hβ_mem
exact le_iSup₂_of_le _ (hne (-β) this) le_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 485,
"column": 4
} | {
"line": 493,
"column": 60
} | {
"line": 494,
"column": 4
} | [
{
"pp": "case mp\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 : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀... | [
"case mp\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 : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀ (i : ↥LieSu... | have h_le : J.restr H ≤ ⨆ (χ : H → K) (_ : χ ≠ (α : Weight K H L)), genWeightSpace L χ := by
refine iSup_le fun ⟨β, hβ_mem, hβ_nz⟩ ↦ ?_
rw [sl2SubmoduleOfRoot_eq_sup]
refine sup_le (sup_le ?_ ?_) ?_
· exact le_iSup₂_of_le _ (hne β hβ_mem) le_rfl
· have : ↑(-β) ∈ q := by rw [Weight.toLinear... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 519,
"column": 2
} | {
"line": 521,
"column": 84
} | {
"line": 522,
"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 : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nI : LieIdeal K L\nhI : ∀ (α : ↥LieSubalgebra... | [
"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 : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nI : LieIdeal K L\nhI : ∀ (α : ↥LieSubalgebra.root), I.ro... | have h_eq : ∀ α : H.root, α ∈ J.rootSet ↔ α ∈ I.rootSet := fun α ↦ by
rw [mem_rootSet_invtSubmoduleToLieIdeal, rootSystem_root_apply]
exact ⟨I.mem_rootSet_of_mem_rootSpan, fun h ↦ Submodule.subset_span ⟨α, h, rfl⟩⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 23
} | {
"line": 217,
"column": 23
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nthis :\n aeval (Generators.localizationAway S r).val =\n (↑(mvPolynomialQuotientEquiv S r)).comp (Ideal.Quotient.mkₐ R (Ideal.span {C r * X () - 1}))\n⊢ RingHom.ker ((↑↑(m... | [
"R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nthis :\n aeval (Generators.localizationAway S r).val =\n (↑(mvPolynomialQuotientEquiv S r)).comp (Ideal.Quotient.mkₐ R (Ideal.span {C r * X () - 1}))\n⊢ Ideal.comap (↑(Ideal.Quotient.... | ← RingHom.comap_ker | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 381,
"column": 4
} | {
"line": 381,
"column": 25
} | {
"line": 382,
"column": 4
} | [
{
"pp": "case add\nR : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nι' : Type u_1\nσ' : Type u_2\nT : Type u_3\ninst✝¹ : CommRing T\ninst✝ : Algebra S T\nQ : Presentation S T ι' σ'\nP : Presentation R S ι σ\nq : MvPolynomial ι' S\nhq : ∃ a, (Q.aux P... | [
"case add\nR : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nι' : Type u_1\nσ' : Type u_2\nT : Type u_3\ninst✝¹ : CommRing T\ninst✝ : Algebra S T\nQ : Presentation S T ι' σ'\nP : Presentation R S ι σ\na b : MvPolynomial (ι' ⊕ ι) R\n⊢ ∃ a_1, (Q.aux P) a_1... | obtain ⟨b, rfl⟩ := hq | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Extension.Basic | {
"line": 372,
"column": 4
} | {
"line": 373,
"column": 48
} | {
"line": 374,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nP : Extension R S\nR' : Type ?u.16\nS' : Type ?u.18\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝² : CommRing R''\ninst✝¹ : Comm... | [] | have := smul_eq_zero_of_mem (P.σ (r + s) - (P.σ r + P.σ s) : P.Ring) (by simp) x
simpa only [sub_smul, add_smul, sub_eq_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Basic | {
"line": 372,
"column": 4
} | {
"line": 373,
"column": 48
} | {
"line": 374,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nP : Extension R S\nR' : Type ?u.16\nS' : Type ?u.18\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝² : CommRing R''\ninst✝¹ : Comm... | [] | have := smul_eq_zero_of_mem (P.σ (r + s) - (P.σ r + P.σ s) : P.Ring) (by simp) x
simpa only [sub_smul, add_smul, sub_eq_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Generators | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 24
} | {
"line": 114,
"column": 25
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Generators R S ι\nx y : S\ne : P.σ x = P.σ y\n⊢ x = y",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiri... | [
"R : Type u\nS : Type v\nι : Type w\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Generators R S ι\nx y : S\ne : P.σ x = P.σ y\n⊢ (aeval P.val) (P.σ x) = y"
] | ← P.aeval_val_σ x, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 172,
"column": 6
} | {
"line": 172,
"column": 25
} | {
"line": 172,
"column": 26
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝⁶ : CommRing R'\ninst✝⁵ : CommRing S'\ninst✝⁴ : Algebra R' S'\nP' : Extension R' S'\ninst✝³ : Algebra R R'\ninst✝² : Algebra S S'\ninst✝¹ : Algebra R S'\ninst✝ : I... | [
"R : Type u\nS : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝⁶ : CommRing R'\ninst✝⁵ : CommRing S'\ninst✝⁴ : Algebra R' S'\nP' : Extension R' S'\ninst✝³ : Algebra R R'\ninst✝² : Algebra S S'\ninst✝¹ : Algebra R S'\ninst✝ : IsScalarTower... | CotangentSpace.map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Generators | {
"line": 681,
"column": 4
} | {
"line": 682,
"column": 95
} | {
"line": 683,
"column": 4
} | [
{
"pp": "case a\nR : Type u\nS : Type v\nι : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nι' : Type u_3\nT : Type u_7\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nQ : Generators S T ι'\nP : Generators R S ι\nthis : DecidableEq (ι' →₀ ℕ... | [
"case e'_1\nR : Type u\nS : Type v\nι : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nι' : Type u_3\nT : Type u_7\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nQ : Generators S T ι'\nP : Generators R S ι\nthis : DecidableEq (ι' →₀ ℕ) := Clas... | convert_to monomial (e.symm (i, 0)) 1 * (Q.toComp P).toAlgHom.toRingHom
(∑ j ∈ (support x).map e.toEmbedding with j.1 = i, monomial j.2 (coeff (e.symm j) x)) ∈ _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convertTo_1 | Mathlib.Tactic.convertTo |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 69,
"column": 45
} | {
"line": 69,
"column": 56
} | {
"line": 69,
"column": 57
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\nM : Submonoid R\nS' : Type u_4\ninst✝⁵ : CommSemiring S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\nM : Submonoid R\nS' : Type u_4\ninst✝⁵ : CommSemiring S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset ... | smul_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 56
} | {
"line": 175,
"column": 0
} | [
{
"pp": "case h\nR : Type u\ninst✝¹¹ : CommSemiring R\nS : Submonoid R\nRₚ : Type v\ninst✝¹⁰ : CommSemiring Rₚ\ninst✝⁹ : Algebra R Rₚ\ninst✝⁸ : IsLocalization S Rₚ\nM : Type w\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nMₚ : Type t\ninst✝⁵ : AddCommMonoid Mₚ\ninst✝⁴ : Module R Mₚ\ninst✝³ : Module Rₚ Mₚ\nins... | [] | simpa using span_eq_top_of_isLocalizedModule Rₚ S f hT | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.Fin.Parity | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 32
} | {
"line": 49,
"column": 2
} | [
{
"pp": "n : ℕ\nk : Fin n\nh : Even ↑k\n⊢ Even k",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"NeZero.mk",
"Fin.pos",
"instOfNatNat",
"LT.lt.ne'",
"Nat.instPreorder",
"Nat",
"NeZero",
"OfNat.ofNat",
"MulZ... | [
"n : ℕ\nk : Fin n\nh : Even ↑k\nthis : NeZero n\n⊢ Even k"
] | have : NeZero n := ⟨k.pos.ne'⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Fin.Parity | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 32
} | {
"line": 58,
"column": 2
} | [
{
"pp": "n : ℕ\nhn : Odd n\nk : Fin n\n⊢ Even k",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"NeZero.mk",
"Fin.pos",
"instOfNatNat",
"LT.lt.ne'",
"Nat.instPreorder",
"Nat",
"NeZero",
"OfNat.ofNat",
"MulZe... | [
"n : ℕ\nhn : Odd n\nk : Fin n\nthis : NeZero n\n⊢ Even k"
] | have : NeZero n := ⟨k.pos.ne'⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.ExteriorPower.Pairing | {
"line": 119,
"column": 8
} | {
"line": 119,
"column": 23
} | {
"line": 119,
"column": 23
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : LinearOrder ι\nx : ι → M\nf : ι → Module.Dual R M\nh₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0\nn : ℕ\na b : Fin n ↪o ι\nh : a ≠ b\nσ : Equiv.Perm (Fin n)\nx✝ : σ ∈ Finset.univ\nh' : ¬∏ x_... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : LinearOrder ι\nx : ι → M\nf : ι → Module.Dual R M\nh₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0\nn : ℕ\na b : Fin n ↪o ι\nh : a ≠ b\nσ : Equiv.Perm (Fin n)\nx✝ : σ ∈ Finset.univ\nh' : ¬∏ x_1, (f (a x_1... | ← a.map_rel_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 60,
"column": 2
} | {
"line": 62,
"column": 55
} | {
"line": 63,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns : ↑(powersetCard I n)\n⊢ (Matrix.of fun i j ↦ (b.coord ((ofFinEmbEquiv.symm s) j)) ((⇑b ∘ ⇑(ofFinEmbEquiv.symm s)) i)).det = 1",
"ppTerm": "?... | [
"R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns : ↑(powersetCard I n)\n⊢ (Matrix.of fun i j ↦ (b.coord ((ofFinEmbEquiv.symm s) j)) (b ((ofFinEmbEquiv.symm s) i))) = 1"
] | suffices Matrix.of (fun i j => b.coord (powersetCard.ofFinEmbEquiv.symm s j)
(b (powersetCard.ofFinEmbEquiv.symm s i))) = 1 by
simp_rw [Function.comp_apply, this, Matrix.det_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.RingTheory.LocalRing.Module | {
"line": 184,
"column": 2
} | {
"line": 214,
"column": 72
} | {
"line": 215,
"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 : IsLocalRing R\ninst✝ : FinitePresentation R M\nι : Type u_5\nv : ι → M\nhli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)\nhsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1)... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : FinitePresentation R M\nι : Type u_5\nv : ι → M\nhli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)\nhsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤\n... | let iequiv : (ι →₀ R) ≃ₗ[R] M := by
refine LinearEquiv.ofBijective i ⟨?_, hi⟩
-- By Nakayama's lemma, it suffices to show that `k ⊗ ker(i) = 0`.
rw [← LinearMap.ker_eq_bot, ← Submodule.subsingleton_iff_eq_bot,
← IsLocalRing.subsingleton_tensorProduct (R := R), subsingleton_iff_forall_eq 0]
have : ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 437,
"column": 36
} | {
"line": 437,
"column": 44
} | {
"line": 438,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\nM N : Submodule R S\nH : M.LinearDisjoint N\nM' : Submodule R S\nh : M' ≤ M\ninst✝ : Flat R ↥N\ni : ↥M' ⊗[R] ↥N →ₗ[R] S := M.mulMap N ∘ₗ LinearMap.rTensor (↥N) (inclusion h)\nhi : Function.Injective ⇑i\nx✝¹ : ↥M'\nx✝ : ... | [] | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 448,
"column": 36
} | {
"line": 448,
"column": 44
} | {
"line": 449,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\nM N : Submodule R S\nH : M.LinearDisjoint N\nN' : Submodule R S\nh : N' ≤ N\ninst✝ : Flat R ↥M\ni : ↥M ⊗[R] ↥N' →ₗ[R] S := M.mulMap N ∘ₗ LinearMap.lTensor (↥M) (inclusion h)\nhi : Function.Injective ⇑i\nx✝¹ : ↥M\nx✝ : ↥... | [] | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.PiTensorProduct | {
"line": 178,
"column": 35
} | {
"line": 178,
"column": 82
} | {
"line": 179,
"column": 6
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid... | [] | simp [FreeAddMonoid.lift_eval_of, C0 r' f i hf] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.PiTensorProduct | {
"line": 178,
"column": 35
} | {
"line": 178,
"column": 82
} | {
"line": 179,
"column": 6
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid... | [] | simp [FreeAddMonoid.lift_eval_of, C0 r' f i hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.PiTensorProduct | {
"line": 178,
"column": 35
} | {
"line": 178,
"column": 82
} | {
"line": 179,
"column": 6
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid... | [] | simp [FreeAddMonoid.lift_eval_of, C0 r' f i hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.MinimalPrime.Colon | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 47
} | {
"line": 54,
"column": 2
} | [
{
"pp": "case neg\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\nI : Ideal R\nx : M\ninst✝ : IsNoetherianRing R\nhx : x ∉ N\nann : Ideal R := N.colon {x}\nhI : I ∈ ann.minimalPrimes\nkey : ∃ n, n ≠ 0 ∧ ∃ J, I ^ n * J ≤ ann ∧ ¬J ≤ I\n⊢ ∃ x'... | [
"case neg\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\nI : Ideal R\nx : M\ninst✝ : IsNoetherianRing R\nhx : x ∉ N\nann : Ideal R := N.colon {x}\nhI : I ∈ ann.minimalPrimes\nkey : ∃ n, n ≠ 0 ∧ ∃ J, I ^ n * J ≤ ann ∧ ¬J ≤ I\nhn0 : Nat.find key... | obtain ⟨hn0, J, hJ, hJI⟩ := Nat.find_spec key | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.PiTensorProduct | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 51
} | {
"line": 353,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝² : CommSemiring R\ns : ι → Type u_7\ninst✝¹ : (i : ι) → AddCommMonoid (s i)\ninst✝ : (i : ι) → Module R (s i)\nx : ⨂[R] (i : ι), s i\np : FreeAddMonoid (R × ((i : ι) → s i))\nh : (List.map (fun x ↦ x.1 • ⨂ₜ[R] (i : ι), x.2 i) (FreeAddMonoid.toList p)).sum = x\na : R\n⊢... | [] | simp [Function.comp_def, mul_smul, List.smul_sum] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 40
} | {
"line": 142,
"column": 2
} | [
{
"pp": "A : Type u\ninst✝³ : CommRing A\ninst✝² : IsNoetherianRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nx✝ : Module.Finite A M\nmotive : (N : Type v) → [inst : AddCommGroup N] → [inst_1 : Module A N] → [Module.Finite A N] → Prop\nsubsingleton :\n ∀ (N : Type v) [inst : AddCommGroup N] [... | [] | exact equiv _ _ Submodule.topEquiv H | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.ClassGroup.Basic | {
"line": 347,
"column": 72
} | {
"line": 350,
"column": 59
} | {
"line": 352,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\n⊢ Function.Surjective ⇑mk0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"ClassGroup.mk0_integralRep",
"Units.val",
"Eq.mpr",
"MonoidHom.range",
"FractionRing.field"... | [] | by
rintro ⟨I⟩
refine ⟨⟨ClassGroup.integralRep I.1, ClassGroup.integralRep_mem_nonZeroDivisors I.ne_zero⟩, ?_⟩
rw [ClassGroup.mk0_integralRep, ClassGroup.Quot_mk_eq_mk] | [anonymous] | Lean.Parser.Term.byTactic |
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