mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.Matrix.PosDef

Goal state before:
pp: n : Type u_2
R : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : StarRing R
inst✝² : StarOrderedRing R
inst✝¹ : DecidableEq n
inst✝ : NoZeroDivisors R
d : n → R
h : ∀ (i : n), 0 < d i
x : n →₀ R
hx : x ≠ 0
⊢ 0 < x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * diagonal d i j * xj

Tactic:
refine Finsupp.sum_pos' (fun _ _ ↦ Finsupp.sum_nonneg ?_) ?_

Goal state after:
case refine_1
n : Type u_2
R : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : StarRing R
inst✝² : StarOrderedRing R
inst✝¹ : DecidableEq n
inst✝ : NoZeroDivisors R
d : n → R
h : ∀ (i : n), 0 < d i
x : n →₀ R
hx : x ≠ 0
x✝¹ : n
x✝ : x✝¹ ∈ x.support
⊢ ∀ i ∈ x.support, 0 ≤ star (x x✝¹) * diagonal d x✝¹ i * x i

case refine_2
n : Type u_2
R : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : StarRing R
inst✝² : StarOrderedRing R
inst✝¹ : DecidableEq n
inst✝ : NoZeroDivisors R
d : n → R
h : ∀ (i : n), 0 < d i
x : n →₀ R
hx : x ≠ 0
⊢ ∃ i ∈ x.support, 0 < x.sum fun j xj ↦ star (x i) * diagonal d i j * xj
Module: Mathlib.LinearAlgebra.Matrix.PosDef

Goal state before:
pp: case insert
m : Type u_1
R : Type u_3
inst✝³ : Ring R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
ι : Type u_5
inst✝ : AddLeftMono R
A : ι → Matrix m m R
i : ι
hi : Finset ι
hins : i ∉ hi
H : hi.Nonempty → (∀ i ∈ hi, (A i).PosDef) → (∑ i ∈ hi, A i).PosDef
hs : (insert i hi).Nonempty
hA : ∀ i_1 ∈ insert i hi, (A i_1).PosDef
⊢ (∑ i ∈ insert i hi, A i).PosDef

Tactic:
rw [Finset.sum_insert hins]
      by_cases h : ¬ hi.Nonempty
      · simp_all
      · exact PosDef.add (hA _ <| Finset.mem_insert_self i hi) <|
          H (not_not.mp h) fun _ _hi => hA _ (Finset.mem_insert_of_mem _hi)

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.PosDef

Goal state before:
pp: case mp
m : Type u_1
n : Type u_2
R' : Type u_4
inst✝⁷ : CommRing R'
inst✝⁶ : PartialOrder R'
inst✝⁵ : StarRing R'
inst✝⁴ : Fintype m
inst✝³ : StarOrderedRing R'
inst✝² : Finite n
inst✝¹ : DecidableEq m
A : Matrix m m R'
B : Matrix m n R'
D : Matrix n n R'
hA : A.PosDef
inst✝ : Invertible A
this✝ : Fintype n
h : (D - Bᴴ * A⁻¹ * B).IsHermitian ∧ ∀ (x : m ⊕ n → R'), 0 ≤ star x ⬝ᵥ fromBlocks A B Bᴴ D *ᵥ x
x : n → R'
this : 0 ≤ star x ᵥ* (D - Bᴴ * A⁻¹ * B) ⬝ᵥ x
⊢ 0 ≤ star x ⬝ᵥ (D - Bᴴ * A⁻¹ * B) *ᵥ x

Tactic:
← dotProduct_mulVec

Goal state after:
case mp
m : Type u_1
n : Type u_2
R' : Type u_4
inst✝⁷ : CommRing R'
inst✝⁶ : PartialOrder R'
inst✝⁵ : StarRing R'
inst✝⁴ : Fintype m
inst✝³ : StarOrderedRing R'
inst✝² : Finite n
inst✝¹ : DecidableEq m
A : Matrix m m R'
B : Matrix m n R'
D : Matrix n n R'
hA : A.PosDef
inst✝ : Invertible A
this✝ : Fintype n
h : (D - Bᴴ * A⁻¹ * B).IsHermitian ∧ ∀ (x : m ⊕ n → R'), 0 ≤ star x ⬝ᵥ fromBlocks A B Bᴴ D *ᵥ x
x : n → R'
this : 0 ≤ star x ⬝ᵥ (D - Bᴴ * A⁻¹ * B) *ᵥ x
⊢ 0 ≤ star x ⬝ᵥ (D - Bᴴ * A⁻¹ * B) *ᵥ x
Module: Mathlib.LinearAlgebra.Matrix.PosDef

Goal state before:
pp: case mpr.ha
m : Type u_1
n : Type u_2
R' : Type u_4
inst✝⁷ : CommRing R'
inst✝⁶ : PartialOrder R'
inst✝⁵ : StarRing R'
inst✝⁴ : Fintype m
inst✝³ : StarOrderedRing R'
inst✝² : Finite n
inst✝¹ : DecidableEq m
A : Matrix m m R'
B : Matrix m n R'
D : Matrix n n R'
hA : A.PosDef
inst✝ : Invertible A
this : Fintype n
h : (D - Bᴴ * A⁻¹ * B).PosSemidef
x : m ⊕ n → R'
⊢ 0 ≤ star (x ∘ Sum.inl + (A⁻¹ * B) *ᵥ x ∘ Sum.inr) ᵥ* A ⬝ᵥ (x ∘ Sum.inl + (A⁻¹ * B) *ᵥ x ∘ Sum.inr)

Tactic:
← dotProduct_mulVec

Goal state after:
case mpr.ha
m : Type u_1
n : Type u_2
R' : Type u_4
inst✝⁷ : CommRing R'
inst✝⁶ : PartialOrder R'
inst✝⁵ : StarRing R'
inst✝⁴ : Fintype m
inst✝³ : StarOrderedRing R'
inst✝² : Finite n
inst✝¹ : DecidableEq m
A : Matrix m m R'
B : Matrix m n R'
D : Matrix n n R'
hA : A.PosDef
inst✝ : Invertible A
this : Fintype n
h : (D - Bᴴ * A⁻¹ * B).PosSemidef
x : m ⊕ n → R'
⊢ 0 ≤ star (x ∘ Sum.inl + (A⁻¹ * B) *ᵥ x ∘ Sum.inr) ⬝ᵥ A *ᵥ (x ∘ Sum.inl + (A⁻¹ * B) *ᵥ x ∘ Sum.inr)
Module: Mathlib.LinearAlgebra.Matrix.ZMatrix

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝⁶ : Fintype ι
inst✝⁵ : DecidableEq ι
inst✝⁴ : CommRing R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : StarRing R
inst✝ : TrivialStar R
A : Matrix ι ι R
d : ι → R
hA : (diagonal d * A).PosDef
hD : ∀ (i : ι), 0 < d i
μ ρ : R
hμ : ∀ (i j : ι), A i j ≤ if i = j then μ else 0
hρ : HasEigenvalue (toLin' A) ρ
contra : 2 * μ ≤ ρ
N : Matrix ι ι R := μ • 1 - A
N_def : N = μ • 1 - A
hN : ∀ (i j : ι), 0 ≤ N i j
⊢ ∃ v, N *ᵥ v = (μ - ρ) • v ∧ v ≠ 0

Tactic:
replace hρ : HasEigenvalue (toLin' N) (μ - ρ) := by simpa [N_def, hasEigenvalue_sub'_iff]

Goal state after:
ι : Type u_1
R : Type u_2
inst✝⁶ : Fintype ι
inst✝⁵ : DecidableEq ι
inst✝⁴ : CommRing R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : StarRing R
inst✝ : TrivialStar R
A : Matrix ι ι R
d : ι → R
hA : (diagonal d * A).PosDef
hD : ∀ (i : ι), 0 < d i
μ ρ : R
hμ : ∀ (i j : ι), A i j ≤ if i = j then μ else 0
contra : 2 * μ ≤ ρ
N : Matrix ι ι R := μ • 1 - A
N_def : N = μ • 1 - A
hN : ∀ (i j : ι), 0 ≤ N i j
hρ : HasEigenvalue (toLin' N) (μ - ρ)
⊢ ∃ v, N *ᵥ v = (μ - ρ) • v ∧ v ≠ 0
Module: Mathlib.LinearAlgebra.Eigenspace.Minpoly

Goal state before:
pp: K : Type v
V : Type w
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
q : K[X]
hq : q.degree = 1
⊢ q.leadingCoeff ≠ 0

Tactic:
rw [Ne, leadingCoeff_eq_zero_iff_deg_eq_bot, hq]

Goal state after:
K : Type v
V : Type w
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
q : K[X]
hq : q.degree = 1
⊢ ¬1 = ⊥
Module: Mathlib.LinearAlgebra.Eigenspace.Minpoly

Goal state before:
pp: K : Type v
V : Type w
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
q : K[X]
hq : q.degree = 1
⊢ LinearMap.ker ((aeval f) (C q.leadingCoeff * X + C (q.coeff 0))) = LinearMap.ker ((aeval f) q)

Tactic:
rwa [← eq_X_add_C_of_degree_eq_one]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Pochhammer

Goal state before:
pp: S : Type u
inst✝ : Semiring S
n : ℕ
⊢ ascPochhammer S (n + 1) = ascPochhammer S n * (X + ↑n)

Tactic:
suffices h : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + (n : ℕ[X])) by
    apply_fun Polynomial.map (algebraMap ℕ S) at h
    simpa only [ascPochhammer_map, Polynomial.map_mul, Polynomial.map_add, map_X,
      Polynomial.map_natCast] using h
  induction n with
  | zero => simp
  | succ n ih =>
    conv_lhs =>
      rw [ascPochhammer_succ_left, ih, mul_comp, ← mul_assoc, ← ascPochhammer_succ_left, add_comp,
          X_comp, natCast_comp, add_assoc, add_comm (1 : ℕ[X]), ← Nat.cast_succ]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Pochhammer

Goal state before:
pp: S : Type u_1
inst✝² : Ring S
inst✝¹ : PartialOrder S
inst✝ : IsStrictOrderedRing S
n : ℕ
s : S
h : ↑n - 1 < s
⊢ 0 < eval s (descPochhammer S n)

Tactic:
← sub_pos,

Goal state after:
S : Type u_1
inst✝² : Ring S
inst✝¹ : PartialOrder S
inst✝ : IsStrictOrderedRing S
n : ℕ
s : S
h : 0 < s - (↑n - 1)
⊢ 0 < eval s (descPochhammer S n)
Module: Mathlib.LinearAlgebra.Vandermonde

Goal state before:
pp: case inr
R : Type u_1
inst✝ : CommRing R
n : ℕ
α : Type u_3
v w : α → R
i : α
hw : w i = 0
j : Fin (n + 1)
hlt : j < last n
⊢ v i ^ ↑j * 0 = 0

pp: case inr
R : Type u_1
inst✝ : CommRing R
n : ℕ
α : Type u_3
v w : α → R
i : α
hw : w i = 0
j : Fin (n + 1)
hlt : j < last n
⊢ ↑j.rev ≠ 0

Tactic:
mul_zero

Goal state after:
case inr
R : Type u_1
inst✝ : CommRing R
n : ℕ
α : Type u_3
v w : α → R
i : α
hw : w i = 0
j : Fin (n + 1)
hlt : j < last n
⊢ 0 = 0

case inr
R : Type u_1
inst✝ : CommRing R
n : ℕ
α : Type u_3
v w : α → R
i : α
hw : w i = 0
j : Fin (n + 1)
hlt : j < last n
⊢ ↑j.rev ≠ 0
Module: Mathlib.LinearAlgebra.Vandermonde

Goal state before:
pp: K : Type u_2
inst✝ : Field K
n : ℕ
v w : Fin (n + 1) → K
i j : Fin (n + 1)
hw : w i ≠ 0
⊢ v i ^ ↑j * w i ^ (↑j.rev + ↑j) = v i ^ ↑j * w i ^ n

Tactic:
rev_add_cast

Goal state after:
K : Type u_2
inst✝ : Field K
n : ℕ
v w : Fin (n + 1) → K
i j : Fin (n + 1)
hw : w i ≠ 0
⊢ v i ^ ↑j * w i ^ n = v i ^ ↑j * w i ^ n
Module: Mathlib.LinearAlgebra.Vandermonde

Goal state before:
pp: case inr
R : Type u_1
inst✝ : CommRing R
n : ℕ
v w : Fin (n + 1) → R
i : Fin (n + 1)
hw : w i = 0
j : Fin (n + 1)
hlt : j < last n
⊢ v i ^ ↑j * 0 = 0

pp: case inr
R : Type u_1
inst✝ : CommRing R
n : ℕ
v w : Fin (n + 1) → R
i : Fin (n + 1)
hw : w i = 0
j : Fin (n + 1)
hlt : j < last n
⊢ ↑j.rev ≠ 0

Tactic:
mul_zero

Goal state after:
case inr
R : Type u_1
inst✝ : CommRing R
n : ℕ
v w : Fin (n + 1) → R
i : Fin (n + 1)
hw : w i = 0
j : Fin (n + 1)
hlt : j < last n
⊢ 0 = 0

case inr
R : Type u_1
inst✝ : CommRing R
n : ℕ
v w : Fin (n + 1) → R
i : Fin (n + 1)
hw : w i = 0
j : Fin (n + 1)
hlt : j < last n
⊢ ↑j.rev ≠ 0
Module: Mathlib.LinearAlgebra.Lagrange

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
i : ι
v r : ι → F
hvs : Set.InjOn v ↑s
hi : i ∈ s
j : ι
H : j ∈ s.erase i
⊢ r j * 0 = 0

Tactic:
mul_zero

Goal state after:
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
i : ι
v r : ι → F
hvs : Set.InjOn v ↑s
hi : i ∈ s
j : ι
H : j ∈ s.erase i
⊢ 0 = 0
Module: Mathlib.Algebra.Lie.CartanCriterion

Goal state before:
pp: ι : Type u_4
R : Type u_5
K : Type u_6
inst✝³ : Finite ι
inst✝² : CommRing R
inst✝¹ : Field K
inst✝ : Algebra R K
E : Submodule R K
a : ι → K
ha : ∀ (i : ι), a i ∈ E
f : ↥E →+ R
this :
  ∀ (ij kl : ι × ι),
    a ij.1 - a ij.2 = a kl.1 - a kl.2 →
      (algebraMap R K) (f ⟨a ij.1, ⋯⟩) - (algebraMap R K) (f ⟨a ij.2, ⋯⟩) =
        (algebraMap R K) (f ⟨a kl.1, ⋯⟩) - (algebraMap R K) (f ⟨a kl.2, ⋯⟩)
⊢ ∃ r, ∀ (i j : ι), Polynomial.eval (a i - a j) r = (algebraMap R K) (f ⟨a i, ⋯⟩) - (algebraMap R K) (f ⟨a j, ⋯⟩)

Tactic:
by
    obtain ⟨r, hr⟩ := (Polynomial.exists_eval_eq_iff _ _).mpr this
    exact ⟨r, fun i j ↦ hr (i, j)⟩

Goal state after:
Module: Mathlib.LinearAlgebra.Lagrange

Goal state before:
pp: case pos
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s t : Finset ι
v r : ι → F
hvt : Set.InjOn v ↑t
hs : s.Nonempty
hst : s ⊆ t
i : ι
hi : i ∈ t
hi' : i ∈ s
j : ι
hj : j ∈ s.erase i
hij : j ≠ i
right✝ : j ∈ s
⊢ eval (v i) ((interpolate (insert j (t \ s)) v) r) * 0 = 0

Tactic:
mul_zero

Goal state after:
case pos
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s t : Finset ι
v r : ι → F
hvt : Set.InjOn v ↑t
hs : s.Nonempty
hst : s ⊆ t
i : ι
hi : i ∈ t
hi' : i ∈ s
j : ι
hj : j ∈ s.erase i
hij : j ≠ i
right✝ : j ∈ s
⊢ 0 = 0
Module: Mathlib.LinearAlgebra.Lagrange

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s t : Finset ι
v r : ι → F
hvt : Set.InjOn v ↑t
hs : s.Nonempty
hst : s ⊆ t
⊢ (interpolate t v) r = ∑ i ∈ s, (interpolate (insert i (t \ s)) v) r * Lagrange.basis s v i

Tactic:
symm
  refine eq_interpolate_of_eval_eq _ hvt (lt_of_le_of_lt (degree_sum_le _ _) ?_) fun i hi => ?_
  · 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.trans hst'),
        Nat.succ_add_sub_one, zero_add]
    rw [degree_basis (Set.InjOn.mono hst hvt) hi, H, WithBot.coe_add, Nat.cast_withBot,
      WithBot.add_lt_add_iff_right (@WithBot.coe_ne_bot _ (#s - 1))]
    convert!
      degree_interpolate_lt _
        (hvt.mono (coe_subset.mpr (insert_subset_iff.mpr ⟨hst hi, sdiff_subset⟩)))
    rw [card_insert_of_notMem (notMem_sdiff_of_mem_right hi), card_sdiff_of_subset hst, add_comm]
  · simp_rw [eval_finsetSum, eval_mul]
    by_cases hi' : i ∈ s
    · rw [← add_sum_erase _ _ hi', eval_basis_self (hvt.mono hst) hi',
        eval_interpolate_at_node _
          (hvt.mono (coe_subset.mpr (insert_subset_iff.mpr ⟨hi, sdiff_subset⟩)))
          (mem_insert_self _ _),
        mul_one, add_eq_left]
      refine sum_eq_zero fun j hj => ?_
      rcases mem_erase.mp hj with ⟨hij, _⟩
      rw [eval_basis_of_ne hij hi', mul_zero]
    · have H : (∑ j ∈ s, eval (v i) (Lagrange.basis s v j)) = 1 := by
        rw [← eval_finsetSum, sum_basis (hvt.mono hst) hs, eval_one]
      rw [← mul_one (r i), ← H, mul_sum]
      refine sum_congr rfl fun j hj => ?_
      congr
      exact
        eval_interpolate_at_node _ (hvt.mono (insert_subset_iff.mpr ⟨hst hj, sdiff_subset⟩))
          (mem_insert.mpr (Or.inr (mem_sdiff.mpr ⟨hi, hi'⟩)))

Goal state after:
Module: Mathlib.LinearAlgebra.Lagrange

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
ι : Type u_2
s : Finset ι
v : ι → R
inst✝ : Nontrivial R
⊢ nodal s v ≠ 0

Tactic:
by
  rcases s.eq_empty_or_nonempty with (rfl | h)
  · exact one_ne_zero
  · apply ne_zero_of_natDegree_gt (n := 0)
    simp only [natDegree_nodal, h.card_pos]

Goal state after:
Module: Mathlib.LinearAlgebra.Lagrange

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
ι : Type u_2
s : Finset ι
v : ι → R
i : ι
hi : i ∈ s
⊢ eval (v i) (nodal s v) = 0

Tactic:
eval_nodal

Goal state after:
R : Type u_1
inst✝ : CommRing R
ι : Type u_2
s : Finset ι
v : ι → R
i : ι
hi : i ∈ s
⊢ ∏ i_1 ∈ s, (v i - v i_1) = 0
Module: Mathlib.Algebra.MvPolynomial.Monad

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_3
inst✝ : CommSemiring R
f : σ → MvPolynomial τ R
d : σ →₀ ℕ
r : R
⊢ (bind₁ f) ((monomial d) r) = C r * ∏ i ∈ d.support, f i ^ d i

Tactic:
by
  simp only [monomial_eq, map_mul, bind₁_C_right, Finsupp.prod, map_prod,
    map_pow, bind₁_X_right]

Goal state after:
Module: Mathlib.LinearAlgebra.Lagrange

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
ι : Type u_2
s : Finset ι
v : ι → R
inst✝¹ : Nontrivial R
inst✝ : NoZeroDivisors R
x : R
hx : ∀ i ∈ s, x ≠ v i
⊢ ∀ a ∈ s, x - v a ≠ 0

Tactic:
sub_ne_zero

Goal state after:
R : Type u_1
inst✝² : CommRing R
ι : Type u_2
s : Finset ι
v : ι → R
inst✝¹ : Nontrivial R
inst✝ : NoZeroDivisors R
x : R
hx : ∀ i ∈ s, x ≠ v i
⊢ ∀ a ∈ s, x ≠ v a
Module: Mathlib.LinearAlgebra.Charpoly.Basic

Goal state before:
pp: R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
t : R
⊢ ((scalar (ChooseBasisIndex R M)) t - (toMatrix (chooseBasis R M) (chooseBasis R M)) f).det =
    ((toMatrix (chooseBasis R M) (chooseBasis R M)) ((algebraMap R (M →ₗ[R] M)) t - f)).det

Tactic:
map_sub,

Goal state after:
R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
t : R
⊢ ((scalar (ChooseBasisIndex R M)) t - (toMatrix (chooseBasis R M) (chooseBasis R M)) f).det =
    ((toMatrix (chooseBasis R M) (chooseBasis R M)) ((algebraMap R (M →ₗ[R] M)) t) -
        (toMatrix (chooseBasis R M) (chooseBasis R M)) f).det
Module: Mathlib.LinearAlgebra.Charpoly.Basic

Goal state before:
pp: R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module.Free R M
inst✝¹ : Module.Finite R M
f : M →ₗ[R] M
inst✝ : Nontrivial R
hf : Function.Injective ⇑f
h✝ : (minpoly R f).coeff 0 = 0
P : R[X]
hP : minpoly R f = X * 0
h : P = 0
⊢ False

Tactic:
mul_zero

Goal state after:
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module.Free R M
inst✝¹ : Module.Finite R M
f : M →ₗ[R] M
inst✝ : Nontrivial R
hf : Function.Injective ⇑f
h✝ : (minpoly R f).coeff 0 = 0
P : R[X]
hP : minpoly R f = 0
h : P = 0
⊢ False
Module: Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
m n : M
⊢ ∀ m_1 ∈ weightedHomogeneousSubmodule R w m,
    ∀ n_1 ∈ weightedHomogeneousSubmodule R w n, m_1 * n_1 ∈ weightedHomogeneousSubmodule R w (m + n)

Tactic:
intro φ hφ ψ hψ c hc

Goal state after:
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
m n : M
φ : MvPolynomial σ R
hφ : φ ∈ weightedHomogeneousSubmodule R w m
ψ : MvPolynomial σ R
hψ : ψ ∈ weightedHomogeneousSubmodule R w n
c : σ →₀ ℕ
hc : coeff c (φ * ψ) ≠ 0
⊢ (weight w) c = m + n
Module: Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
m n : M
φ : MvPolynomial σ R
hφ : φ ∈ weightedHomogeneousSubmodule R w m
ψ : MvPolynomial σ R
hψ : ψ ∈ weightedHomogeneousSubmodule R w n
c : σ →₀ ℕ
hc : ∑ x ∈ antidiagonal c, coeff x.1 φ * coeff x.2 ψ ≠ 0
d e : σ →₀ ℕ
hde : (d, e) ∈ antidiagonal c
H : coeff (d, e).1 φ * coeff (d, e).2 ψ ≠ 0
aux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0
⊢ (weight w) ((d, e).1 + (d, e).2) = m + n

Tactic:
← hφ aux.1,

Goal state after:
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
m n : M
φ : MvPolynomial σ R
hφ : φ ∈ weightedHomogeneousSubmodule R w m
ψ : MvPolynomial σ R
hψ : ψ ∈ weightedHomogeneousSubmodule R w n
c : σ →₀ ℕ
hc : ∑ x ∈ antidiagonal c, coeff x.1 φ * coeff x.2 ψ ≠ 0
d e : σ →₀ ℕ
hde : (d, e) ∈ antidiagonal c
H : coeff (d, e).1 φ * coeff (d, e).2 ψ ≠ 0
aux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0
⊢ (weight w) ((d, e).1 + (d, e).2) = (weight w) d + n
Module: Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
d : σ →₀ ℕ
r : R
m : M
hm : (weight w) d = m
⊢ IsWeightedHomogeneous w ((monomial d) r) m

Tactic:
intro c

Goal state after:
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
d : σ →₀ ℕ
r : R
m : M
hm : (weight w) d = m
c : σ →₀ ℕ
⊢ coeff c ((monomial d) r) ≠ 0 → (weight w) c = m
Module: Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : Semiring A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : GradedRing 𝒜
I : Ideal A
h : IsHomogeneous 𝒜 I
x : A
hx : x ∈ I
⊢ x ∈ homogeneousCore' 𝒜 I

Tactic:
← DirectSum.sum_support_decompose 𝒜 x

Goal state after:
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : Semiring A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : GradedRing 𝒜
I : Ideal A
h : IsHomogeneous 𝒜 I
x : A
hx : x ∈ I
⊢ ∑ i ∈ DFinsupp.support ((decompose 𝒜) x), ↑(((decompose 𝒜) x) i) ∈ homogeneousCore' 𝒜 I
Module: Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal

Goal state before:
pp: ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁷ : Semiring A
inst✝⁶ : DecidableEq ι
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : PartialOrder ι
inst✝³ : CanonicallyOrderedAdd ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
x : A
hx : x ∈ 𝒜₊.toAddSubmonoid
⊢ x ∈ ⨆ i, ⨆ (_ : i > 0), AddSubmonoid.ofClass (𝒜 i)

Tactic:
← DirectSum.sum_support_decompose 𝒜 x

Goal state after:
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁷ : Semiring A
inst✝⁶ : DecidableEq ι
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : PartialOrder ι
inst✝³ : CanonicallyOrderedAdd ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
x : A
hx : x ∈ 𝒜₊.toAddSubmonoid
⊢ ∑ i ∈ DFinsupp.support ((decompose 𝒜) x), ↑(((decompose 𝒜) x) i) ∈ ⨆ i, ⨆ (_ : i > 0), AddSubmonoid.ofClass (𝒜 i)
Module: Mathlib.LinearAlgebra.Eigenspace.Zero

Goal state before:
pp: K : Type u_2
M : Type u_3
inst✝³ : Field K
inst✝² : AddCommGroup M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
φ : End K M
tfae_1_iff_2 : φ.HasEigenvalue 0 ↔ (minpoly K φ).IsRoot 0
tfae_2_to_3 : (minpoly K φ).IsRoot 0 → constantCoeff (charpoly φ) = 0
h : ((toMatrix (chooseBasis K M) (chooseBasis K M)) φ).charpoly.coeff 0 = 0
⊢ (-1) ^ Fintype.card (ChooseBasisIndex K M) * 0 = 0

Tactic:
mul_zero

Goal state after:
K : Type u_2
M : Type u_3
inst✝³ : Field K
inst✝² : AddCommGroup M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
φ : End K M
tfae_1_iff_2 : φ.HasEigenvalue 0 ↔ (minpoly K φ).IsRoot 0
tfae_2_to_3 : (minpoly K φ).IsRoot 0 → constantCoeff (charpoly φ) = 0
h : ((toMatrix (chooseBasis K M) (chooseBasis K M)) φ).charpoly.coeff 0 = 0
⊢ 0 = 0
Module: Mathlib.Algebra.Module.LinearMap.Polynomial

Goal state before:
pp: case e_f.hX
R : Type u_1
L : Type u_2
M : Type u_3
ι : Type u_5
ιM : Type u_7
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup L
inst✝⁸ : Module R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
φ : L →ₗ[R] End R M
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ιM
inst✝³ : DecidableEq ι
inst✝² : DecidableEq ιM
b : Basis ι R L
bₘ : Basis ιM R M
A : Type u_8
inst✝¹ : CommRing A
inst✝ : Algebra R A
ij kl : ιM × ιM
⊢ ((basis A bₘ).end.repr ((basis A bₘ).end kl)) ij = if kl = ij then 1 else 0

Tactic:
Basis.repr_self_apply

Goal state after:
case e_f.hX
R : Type u_1
L : Type u_2
M : Type u_3
ι : Type u_5
ιM : Type u_7
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup L
inst✝⁸ : Module R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
φ : L →ₗ[R] End R M
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ιM
inst✝³ : DecidableEq ι
inst✝² : DecidableEq ιM
b : Basis ι R L
bₘ : Basis ιM R M
A : Type u_8
inst✝¹ : CommRing A
inst✝ : Algebra R A
ij kl : ιM × ιM
⊢ (if kl = ij then 1 else 0) = if kl = ij then 1 else 0
Module: Mathlib.RingTheory.MvPolynomial.Homogeneous

Goal state before:
pp: R : Type u_5
inst✝¹ : CommRing R
inst✝ : IsDomain R
N : ℕ
IH : ∀ {F : MvPolynomial (Fin N) R} {n : ℕ}, F.IsHomogeneous n → F ≠ 0 → ↑n ≤ #R → ∃ r, (eval r) F ≠ 0
F : MvPolynomial (Fin (N + 1)) R
n : ℕ
hF : F.IsHomogeneous n
hnR : ↑n ≤ #R
hdeg : ((finSuccEquiv R N) F).natDegree < n + 1
hF₀ : ∀ (i : ℕ), ((finSuccEquiv R N) F).coeff i = 0
⊢ ∀ (n : ℕ), ((finSuccEquiv R N) F).coeff n = ((finSuccEquiv R N) 0).coeff n

Tactic:
by simpa using! hF₀

Goal state after:
Module: Mathlib.Algebra.Module.LinearMap.Polynomial

Goal state before:
pp: R : Type u_1
L : Type u_2
M : Type u_3
ι : Type u_5
inst✝⁹ : CommRing R
inst✝⁸ : AddCommGroup L
inst✝⁷ : Module R L
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
φ : L →ₗ[R] End R M
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : Free R M
inst✝¹ : Module.Finite R M
b : Basis ι R L
inst✝ : Nontrivial R
⊢ (φ.polyCharpoly b).coeff (φ.nilRankAux b) ≠ 0

Tactic:
by
  apply Polynomial.trailingCoeff_nonzero_iff_nonzero.mpr
  apply polyCharpoly_ne_zero

Goal state after:
Module: Mathlib.Algebra.Lie.Matrix

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
T S : Module.End R (n → R)
f : ((n → R) →ₗ[R] n → R) ≃ₗ[R] Matrix n n R := LinearMap.toMatrix'
h : ∀ (T S : Module.End R (n → R)), f (T ∘ₗ S) = f T * f S
⊢ f (T ∘ₗ S - S ∘ₗ T) = f T * f S - f S * f T

Tactic:
map_sub,

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
T S : Module.End R (n → R)
f : ((n → R) →ₗ[R] n → R) ≃ₗ[R] Matrix n n R := LinearMap.toMatrix'
h : ∀ (T S : Module.End R (n → R)), f (T ∘ₗ S) = f T * f S
⊢ f (T ∘ₗ S) - f (S ∘ₗ T) = f T * f S - f S * f T
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
B : BilinForm R M
N : Type w
inst✝¹ : AddCommGroup N
inst✝ : Module R N
e : N ≃ₗ[R] M
f : Module.End R M
⊢ e.lieConj.toLinearEquiv.symm f ∈ (skewAdjointLieSubalgebra (LinearMap.compl₁₂ B ↑e ↑e)).toSubmodule ↔
    f ∈ skewAdjointLieSubalgebra B

Tactic:
exact (LinearMap.isPairSelfAdjoint_equiv (B := -B) (F := B) e f).symm

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Diagonal

Goal state before:
pp: m : Type u_1
inst✝² : Fintype m
K : Type u
inst✝¹ : Semifield K
inst✝ : DecidableEq m
w : m → K
⊢ (toLin' (diagonal w)).ker = ⨆ i ∈ {i | w i = 0}, (LinearMap.single K (fun x ↦ K) i).range

Tactic:
← comap_bot

Goal state after:
m : Type u_1
inst✝² : Fintype m
K : Type u
inst✝¹ : Semifield K
inst✝ : DecidableEq m
w : m → K
⊢ comap (toLin' (diagonal w)) ⊥ = ⨆ i ∈ {i | w i = 0}, (LinearMap.single K (fun x ↦ K) i).range