ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AlternatingMap.domDomCongrₗ_toAddEquiv
@[simp]
theorem domDomCongrₗ_toAddEquiv (σ : ι ≃ ι') :
    (↑(domDomCongrₗ S σ : M [⋀^ι]→ₗ[R] N ≃ₗ[S] _) : M [⋀^ι]→ₗ[R] N ≃+ _) =
      domDomCongrEquiv σ 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (σ : ι ≃ ι')
Goal: ↑(AlternatingMap.domDomCongrₗ S σ) = AlternatingMap.domDomCongrEquiv σ
-- AlternatingMap.domDomCongr_eq_iff
@[simp]
theorem domDomCongr_eq_iff (σ : ι ≃ ι') (f g : M [⋀^ι]→ₗ[R] N) :
    f.domDomCongr σ = g.domDomCongr σ ↔ f = g 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') (f g : M [⋀^ι]→ₗ[R] N)
Goal: AlternatingMap.domDomCongr σ f = AlternatingMap.domDomCongr σ g ↔ f = g
-- AlternatingMap.domDomCongr_eq_zero_iff
@[simp]
theorem domDomCongr_eq_zero_iff (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N) :
    f.domDomCongr σ = 0 ↔ f = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N)
Goal: AlternatingMap.domDomCongr σ f = 0 ↔ f = 0
-- AlternatingMap.domDomCongr_perm
theorem domDomCongr_perm [Fintype ι] [DecidableEq ι] (σ : Equiv.Perm ι) :
    g.domDomCongr σ = Equiv.Perm.sign σ • g 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N') [Fintype ι] [DecidableEq ι] (σ : Equiv.Perm ι)
Goal: AlternatingMap.domDomCongr σ g = Equiv.Perm.sign σ • g
-- AlternatingMap.coe_domDomCongr
@[norm_cast]
theorem coe_domDomCongr (σ : ι ≃ ι') :
    ↑(f.domDomCongr σ) = (f : MultilinearMap R (fun _ : ι => M) N).domDomCongr σ 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (f : M [⋀^ι]→ₗ[R] N) (σ : ι ≃ ι')
Goal: ↑(AlternatingMap.domDomCongr σ f) = MultilinearMap.domDomCongr σ ↑f
-- AlternatingMap.map_linearDependent
theorem map_linearDependent {K : Type*} [Ring K] {M : Type*} [AddCommGroup M] [Module K M]
    {N : Type*} [AddCommGroup N] [Module K N] [NoZeroSMulDivisors K N] (f : M [⋀^ι]→ₗ[K] N)
    (v : ι → M) (h : ¬LinearIndependent K v) : f v = 0 
Arguments: {ι : Type u_7} {K : Type u_12} [Ring K] {M : Type u_13} [AddCommGroup M] [Module K M] {N : Type u_14} [AddCommGroup N] [Module K N] [NoZeroSMulDivisors K N] (f : M [⋀^ι]→ₗ[K] N) (v : ι → M) (h : ¬LinearIndependent K v)
Goal: f v = 0
-- AlternatingMap.map_vecCons_add
theorem map_vecCons_add {n : ℕ} (f : M [⋀^Fin n.succ]→ₗ[R] N) (m : Fin n → M) (x y : M) :
    f (Matrix.vecCons (x + y) m) = f (Matrix.vecCons x m) + f (Matrix.vecCons y m) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {n : ℕ} (f : M [⋀^Fin n.succ]→ₗ[R] N) (m : Fin n → M) (x y : M)
Goal: f (Matrix.vecCons (x + y) m) = f (Matrix.vecCons x m) + f (Matrix.vecCons y m)
-- AlternatingMap.map_vecCons_smul
theorem map_vecCons_smul {n : ℕ} (f : M [⋀^Fin n.succ]→ₗ[R] N) (m : Fin n → M) (c : R)
    (x : M) : f (Matrix.vecCons (c • x) m) = c • f (Matrix.vecCons x m) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {n : ℕ} (f : M [⋀^Fin n.succ]→ₗ[R] N) (m : Fin n → M) (c : R) (x : M)
Goal: f (Matrix.vecCons (c • x) m) = c • f (Matrix.vecCons x m)
-- _private.Mathlib.LinearAlgebra.Alternating.Basic.0.MultilinearMap.alternization_map_eq_zero_of_eq_aux
private theorem alternization_map_eq_zero_of_eq_aux (m : MultilinearMap R (fun _ : ι => M) N')
    (v : ι → M) (i j : ι) (i_ne_j : i ≠ j) (hv : v i = v j) :
    (∑ σ : Perm ι, Equiv.Perm.sign σ • m.domDomCongr σ) v = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} [Fintype ι] [DecidableEq ι] (m : MultilinearMap R (fun x => M) N') (v : ι → M) (i j : ι) (i_ne_j : i ≠ j) (hv : v i = v j)
Goal: (∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • MultilinearMap.domDomCongr σ m) v = 0
-- MultilinearMap.alternatization_def
theorem alternatization_def (m : MultilinearMap R (fun _ : ι => M) N') :
    ⇑(alternatization m) = (∑ σ : Perm ι, Equiv.Perm.sign σ • m.domDomCongr σ : _) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} [Fintype ι] [DecidableEq ι] (m : MultilinearMap R (fun x => M) N')
Goal: ⇑(MultilinearMap.alternatization m) = ⇑(∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • MultilinearMap.domDomCongr σ m)
-- MultilinearMap.alternatization_coe
theorem alternatization_coe (m : MultilinearMap R (fun _ : ι => M) N') :
    ↑(alternatization m) = (∑ σ : Perm ι, Equiv.Perm.sign σ • m.domDomCongr σ : _) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} [Fintype ι] [DecidableEq ι] (m : MultilinearMap R (fun x => M) N')
Goal: ↑(MultilinearMap.alternatization m) = ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • MultilinearMap.domDomCongr σ m
-- MultilinearMap.alternatization_apply
theorem alternatization_apply (m : MultilinearMap R (fun _ : ι => M) N') (v : ι → M) :
    alternatization m v = ∑ σ : Perm ι, Equiv.Perm.sign σ • m.domDomCongr σ v 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} [Fintype ι] [DecidableEq ι] (m : MultilinearMap R (fun x => M) N') (v : ι → M)
Goal: (MultilinearMap.alternatization m) v = ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • (MultilinearMap.domDomCongr σ m) v
-- AlternatingMap.coe_alternatization
theorem coe_alternatization [DecidableEq ι] [Fintype ι] (a : M [⋀^ι]→ₗ[R] N') :
    MultilinearMap.alternatization (a : MultilinearMap R (fun _ => M) N')
    = Nat.factorial (Fintype.card ι) • a 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} [DecidableEq ι] [Fintype ι] (a : M [⋀^ι]→ₗ[R] N')
Goal: MultilinearMap.alternatization ↑a = (Fintype.card ι).factorial • a
-- LinearMap.compMultilinearMap_alternatization
theorem compMultilinearMap_alternatization (g : N' →ₗ[R] N'₂)
    (f : MultilinearMap R (fun _ : ι => M) N') :
    MultilinearMap.alternatization (g.compMultilinearMap f)
      = g.compAlternatingMap (MultilinearMap.alternatization f) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} {N'₂ : Type u_10} [AddCommGroup N'₂] [Module R N'₂] [DecidableEq ι] [Fintype ι] (g : N' →ₗ[R] N'₂) (f : MultilinearMap R (fun x => M) N')
Goal: MultilinearMap.alternatization (g.compMultilinearMap f) = g.compAlternatingMap (MultilinearMap.alternatization f)
-- Basis.ext_alternating
theorem Basis.ext_alternating {f g : N₁ [⋀^ι]→ₗ[R'] N₂} (e : Basis ι₁ R' N₁)
    (h : ∀ v : ι → ι₁, Function.Injective v → (f fun i => e (v i)) = g fun i => e (v i)) :
    f = g 
Arguments: {ι : Type u_7} {ι₁ : Type u_10} [Finite ι] {R' : Type u_11} {N₁ : Type u_12} {N₂ : Type u_13} [CommSemiring R'] [AddCommMonoid N₁] [AddCommMonoid N₂] [Module R' N₁] [Module R' N₂] {f g : N₁ [⋀^ι]→ₗ[R'] N₂} (e : Basis ι₁ R' N₁) (h : ∀ (v : ι → ι₁), Function.Injective v → (f fun i => e (v i)) = g fun i => e (v i))
Goal: f = g
-- AlternatingMap.curryLeft_apply_apply
@[simps]
def curryLeft {n : ℕ} (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') :
    M'' →ₗ[R'] M'' [⋀^Fin n]→ₗ[R'] N'' 
Arguments: {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ} (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') (m : M'') (v : Fin n → M'')
Goal: (f.curryLeft m) v = f (Matrix.vecCons m v)
-- AlternatingMap.curryLeft_zero
@[simp]
theorem curryLeft_zero {n : ℕ} : curryLeft (0 : M'' [⋀^Fin n.succ]→ₗ[R'] N'') = 0 
Arguments: {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ}
Goal: AlternatingMap.curryLeft 0 = 0
-- AlternatingMap.curryLeft_add
@[simp]
theorem curryLeft_add {n : ℕ} (f g : M'' [⋀^Fin n.succ]→ₗ[R'] N'') :
    curryLeft (f + g) = curryLeft f + curryLeft g 
Arguments: {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ} (f g : M'' [⋀^Fin n.succ]→ₗ[R'] N'')
Goal: (f + g).curryLeft = f.curryLeft + g.curryLeft
-- AlternatingMap.curryLeft_smul
@[simp]
theorem curryLeft_smul {n : ℕ} (r : R') (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') :
    curryLeft (r • f) = r • curryLeft f 
Arguments: {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ} (r : R') (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'')
Goal: (r • f).curryLeft = r • f.curryLeft
-- AlternatingMap.curryLeftLinearMap_apply
@[simps]
def curryLeftLinearMap {n : ℕ} :
    (M'' [⋀^Fin n.succ]→ₗ[R'] N'') →ₗ[R'] M'' →ₗ[R'] M'' [⋀^Fin n]→ₗ[R'] N'' 
Arguments: {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ} (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'')
Goal: AlternatingMap.curryLeftLinearMap f = f.curryLeft
-- AlternatingMap.curryLeft_same
@[simp]
theorem curryLeft_same {n : ℕ} (f : M'' [⋀^Fin n.succ.succ]→ₗ[R'] N'') (m : M'') :
    (f.curryLeft m).curryLeft m = 0 
Arguments: {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ} (f : M'' [⋀^Fin n.succ.succ]→ₗ[R'] N'') (m : M'')
Goal: (f.curryLeft m).curryLeft m = 0
-- AlternatingMap.curryLeft_compAlternatingMap
@[simp]
theorem curryLeft_compAlternatingMap {n : ℕ} (g : N'' →ₗ[R'] N₂'')
    (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') (m : M'') :
    (g.compAlternatingMap f).curryLeft m = g.compAlternatingMap (f.curryLeft m) 
Arguments: {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} {N₂'' : Type u_14} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [AddCommMonoid N₂''] [Module R' M''] [Module R' N''] [Module R' N₂''] {n : ℕ} (g : N'' →ₗ[R'] N₂'') (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') (m : M'')
Goal: (g.compAlternatingMap f).curryLeft m = g.compAlternatingMap (f.curryLeft m)
-- AlternatingMap.curryLeft_compLinearMap
@[simp]
theorem curryLeft_compLinearMap {n : ℕ} (g : M₂'' →ₗ[R'] M'')
    (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') (m : M₂'') :
    (f.compLinearMap g).curryLeft m = (f.curryLeft (g m)).compLinearMap g 
Arguments: {R' : Type u_10} {M'' : Type u_11} {M₂'' : Type u_12} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid M₂''] [AddCommMonoid N''] [Module R' M''] [Module R' M₂''] [Module R' N''] {n : ℕ} (g : M₂'' →ₗ[R'] M'') (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') (m : M₂'')
Goal: (f.compLinearMap g).curryLeft m = (f.curryLeft (g m)).compLinearMap g
-- AlternatingMap.constLinearEquivOfIsEmpty_symm_apply
@[simps]
def constLinearEquivOfIsEmpty [IsEmpty ι] : N'' ≃ₗ[R'] (M'' [⋀^ι]→ₗ[R'] N'') 
Arguments: {ι : Type u_7} {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] [IsEmpty ι] (f : M'' [⋀^ι]→ₗ[R'] N'')
Goal: AlternatingMap.constLinearEquivOfIsEmpty.symm f = f 0
-- AlternatingMap.constLinearEquivOfIsEmpty_apply
@[simps]
def constLinearEquivOfIsEmpty [IsEmpty ι] : N'' ≃ₗ[R'] (M'' [⋀^ι]→ₗ[R'] N'') 
Arguments: {ι : Type u_7} {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] [IsEmpty ι] (m : N'')
Goal: AlternatingMap.constLinearEquivOfIsEmpty m = AlternatingMap.constOfIsEmpty R' M'' ι m
-- AlgebraicClosure.toSplittingField_evalXSelf
theorem toSplittingField_evalXSelf {s : Finset (MonicIrreducible k)} {f} (hf : f ∈ s) :
    toSplittingField k s (evalXSelf k f) = 0 
Arguments: (k : Type u) [Field k] {s : Finset (AlgebraicClosure.MonicIrreducible k)} {f : AlgebraicClosure.MonicIrreducible k} (hf : f ∈ s)
Goal: (AlgebraicClosure.toSplittingField k s) (AlgebraicClosure.evalXSelf k f) = 0
-- AlgebraicClosure.spanEval_ne_top
theorem spanEval_ne_top : spanEval k ≠ ⊤ 
Arguments: (k : Type u) [Field k]
Goal: AlgebraicClosure.spanEval k ≠ ⊤
-- AlgebraicClosure.le_maxIdeal
theorem le_maxIdeal : spanEval k ≤ maxIdeal k 
Arguments: (k : Type u) [Field k]
Goal: AlgebraicClosure.spanEval k ≤ AlgebraicClosure.maxIdeal k
-- AlgebraicClosure.AdjoinMonic.algebraMap
theorem AdjoinMonic.algebraMap : algebraMap k (AdjoinMonic k) = (Ideal.Quotient.mk _).comp
    (C : k →+* MvPolynomial (MonicIrreducible k) k) 
Arguments: (k : Type u) [Field k]
Goal: algebraMap k (AlgebraicClosure.AdjoinMonic k) = (Ideal.Quotient.mk (AlgebraicClosure.maxIdeal k)).comp MvPolynomial.C
-- AlgebraicClosure.AdjoinMonic.isIntegral
theorem AdjoinMonic.isIntegral (z : AdjoinMonic k) : IsIntegral k z 
Arguments: (k : Type u) [Field k] (z : AlgebraicClosure.AdjoinMonic k)
Goal: IsIntegral k z
-- AlgebraicClosure.AdjoinMonic.exists_root
theorem AdjoinMonic.exists_root {f : k[X]} (hfm : f.Monic) (hfi : Irreducible f) :
    ∃ x : AdjoinMonic k, f.eval₂ (toAdjoinMonic k) x = 0 
Arguments: (k : Type u) [Field k] {f : k[X]} (hfm : f.Monic) (hfi : Irreducible f)
Goal: ∃ x, Polynomial.eval₂ (AlgebraicClosure.toAdjoinMonic k) x f = 0
-- AlgebraicClosure.Step.zero
theorem Step.zero : Step k 0 = k 
Arguments: (k : Type u) [Field k]
Goal: AlgebraicClosure.Step k 0 = k
-- AlgebraicClosure.Step.succ
theorem Step.succ (n : ℕ) : Step k (n + 1) = AdjoinMonic (Step k n) 
Arguments: (k : Type u) [Field k] (n : ℕ)
Goal: AlgebraicClosure.Step k (n + 1) = AlgebraicClosure.AdjoinMonic (AlgebraicClosure.Step k n)
-- AlgebraicClosure.toStepSucc.exists_root
theorem toStepSucc.exists_root {n} {f : Polynomial (Step k n)} (hfm : f.Monic)
    (hfi : Irreducible f) : ∃ x : Step k (n + 1), f.eval₂ (toStepSucc k n) x = 0 
Arguments: (k : Type u) [Field k] {n : ℕ} {f : (AlgebraicClosure.Step k n)[X]} (hfm : f.Monic) (hfi : Irreducible f)
Goal: ∃ x, Polynomial.eval₂ (AlgebraicClosure.toStepSucc k n) x f = 0
-- _private.Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure.0.AlgebraicClosure.toStepOfLE'.succ
private theorem toStepOfLE'.succ (m n : ℕ) (h : m ≤ n) :
    toStepOfLE' k m (Nat.succ n) (h.trans n.le_succ) =
    (toStepSucc k n) ∘ toStepOfLE' k m n h 
Arguments: (k : Type u) [Field k] (m n : ℕ) (h : m ≤ n)
Goal: AlgebraicClosure.toStepOfLE' k m n.succ ⋯ = ⇑(AlgebraicClosure.toStepSucc k n) ∘ AlgebraicClosure.toStepOfLE' k m n h
-- AlgebraicClosure.coe_toStepOfLE
@[simp]
theorem coe_toStepOfLE (m n : ℕ) (h : m ≤ n) :
    (toStepOfLE k m n h : Step k m → Step k n) = Nat.leRecOn h @fun n => toStepSucc k n 
Arguments: (k : Type u) [Field k] (m n : ℕ) (h : m ≤ n)
Goal: ⇑(AlgebraicClosure.toStepOfLE k m n h) = fun a => Nat.leRecOn h (fun n => ⇑(AlgebraicClosure.toStepSucc k n)) a
-- _private.Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure.0.AlgebraicClosure.toStepOfLE.succ
private theorem toStepOfLE.succ (n : ℕ) (h : 0 ≤ n) :
    toStepOfLE k 0 (n + 1) (h.trans n.le_succ) =
    (toStepSucc k n).comp (toStepOfLE k 0 n h) 
Arguments: (k : Type u) [Field k] (n : ℕ) (h : 0 ≤ n)
Goal: AlgebraicClosure.toStepOfLE k 0 (n + 1) ⋯ = (AlgebraicClosure.toStepSucc k n).comp (AlgebraicClosure.toStepOfLE k 0 n h)
-- AlgebraicClosure.Step.isIntegral
theorem Step.isIntegral (n) : ∀ z : Step k n, IsIntegral k z 
Arguments: (k : Type u) [Field k] (n : ℕ) (z : AlgebraicClosure.Step k n)
Goal: IsIntegral k z
-- AlgebraicClosureAux.ofStep_succ
theorem ofStep_succ (n : ℕ) : (ofStep k (n + 1)).comp (toStepSucc k n) = ofStep k n 
Arguments: (k : Type u) [Field k] (n : ℕ)
Goal: (AlgebraicClosureAux.ofStep k (n + 1)).comp (AlgebraicClosure.toStepSucc k n) = AlgebraicClosureAux.ofStep k n
-- AlgebraicClosureAux.exists_ofStep
theorem exists_ofStep (z : AlgebraicClosureAux k) : ∃ n x, ofStep k n x = z 
Arguments: (k : Type u) [Field k] (z : AlgebraicClosureAux k)
Goal: ∃ n x, (AlgebraicClosureAux.ofStep k n) x = z
-- AlgebraicClosureAux.exists_root
theorem exists_root {f : Polynomial (AlgebraicClosureAux k)}
    (hfm : f.Monic) (hfi : Irreducible f) : ∃ x : AlgebraicClosureAux k, f.eval x = 0 
Arguments: (k : Type u) [Field k] {f : (AlgebraicClosureAux k)[X]} (hfm : f.Monic) (hfi : Irreducible f)
Goal: ∃ x, Polynomial.eval x f = 0
-- AlgebraicClosureAux.instIsAlgClosed
@[local instance] theorem instIsAlgClosed : IsAlgClosed (AlgebraicClosureAux k) 
Arguments: (k : Type u) [Field k]
Goal: IsAlgClosed (AlgebraicClosureAux k)
-- AlgebraicClosureAux.isAlgClosure
@[local instance] theorem isAlgClosure : IsAlgClosure k (AlgebraicClosureAux k) 
Arguments: (k : Type u) [Field k]
Goal: IsAlgClosure k (AlgebraicClosureAux k)
-- Polynomial.IsSplittingField.adjoin_rootSet'
class IsSplittingField (f : K[X]) : Prop where
  splits' : Splits (algebraMap K L) f
  adjoin_rootSet' : Algebra.adjoin K (f.rootSet L : Set L) = ⊤
Arguments: {K : Type v} {L : Type w} [Field K] [Field L] [Algebra K L] {f : K[X]} [self : Polynomial.IsSplittingField K L f]
Goal: Algebra.adjoin K (f.rootSet L) = ⊤
-- Polynomial.IsSplittingField.splits'
class IsSplittingField (f : K[X]) : Prop where
  splits' : Splits (algebraMap K L) f
  adjoin_rootSet' : Algebra.adjoin K (f.rootSet L : Set L) = ⊤
Arguments: {K : Type v} {L : Type w} [Field K] [Field L] [Algebra K L] {f : K[X]} [self : Polynomial.IsSplittingField K L f]
Goal: Polynomial.Splits (algebraMap K L) f
-- Polynomial.IsSplittingField.splits
theorem splits (f : K[X]) [IsSplittingField K L f] : Splits (algebraMap K L) f 
Arguments: {K : Type v} (L : Type w) [Field K] [Field L] [Algebra K L] (f : K[X]) [Polynomial.IsSplittingField K L f]
Goal: Polynomial.Splits (algebraMap K L) f
-- Polynomial.IsSplittingField.adjoin_rootSet
theorem adjoin_rootSet (f : K[X]) [IsSplittingField K L f] :
    Algebra.adjoin K (f.rootSet L : Set L) = ⊤ 
Arguments: {K : Type v} (L : Type w) [Field K] [Field L] [Algebra K L] (f : K[X]) [Polynomial.IsSplittingField K L f]
Goal: Algebra.adjoin K (f.rootSet L) = ⊤
-- Polynomial.IsSplittingField.splits_iff
theorem splits_iff (f : K[X]) [IsSplittingField K L f] :
    Splits (RingHom.id K) f ↔ (⊤ : Subalgebra K L) = ⊥ 
Arguments: {K : Type v} (L : Type w) [Field K] [Field L] [Algebra K L] (f : K[X]) [Polynomial.IsSplittingField K L f]
Goal: Polynomial.Splits (RingHom.id K) f ↔ ⊤ = ⊥
-- Polynomial.IsSplittingField.mul
theorem mul (f g : F[X]) (hf : f ≠ 0) (hg : g ≠ 0) [IsSplittingField F K f]
    [IsSplittingField K L (g.map <| algebraMap F K)] : IsSplittingField F L (f * g) 
Arguments: {F : Type u} {K : Type v} (L : Type w) [Field K] [Field L] [Field F] [Algebra K L] [Algebra F K] [Algebra F L] [IsScalarTower F K L] (f g : F[X]) (hf : f ≠ 0) (hg : g ≠ 0) [Polynomial.IsSplittingField F K f] [Polynomial.IsSplittingField K L (Polynomial.map (algebraMap F K) g)]
Goal: Polynomial.IsSplittingField F L (f * g)
-- Polynomial.IsSplittingField.finiteDimensional
theorem finiteDimensional (f : K[X]) [IsSplittingField K L f] : FiniteDimensional K L 
Arguments: {K : Type v} (L : Type w) [Field K] [Field L] [Algebra K L] (f : K[X]) [Polynomial.IsSplittingField K L f]
Goal: FiniteDimensional K L
-- Polynomial.IsSplittingField.of_algEquiv
theorem of_algEquiv [Algebra K F] (p : K[X]) (f : F ≃ₐ[K] L) [IsSplittingField K F p] :
    IsSplittingField K L p 
Arguments: {F : Type u} {K : Type v} (L : Type w) [Field K] [Field L] [Field F] [Algebra K L] [Algebra K F] (p : K[X]) (f : F ≃ₐ[K] L) [Polynomial.IsSplittingField K F p]
Goal: Polynomial.IsSplittingField K L p
-- Polynomial.IsSplittingField.adjoin_rootSet_eq_range
theorem adjoin_rootSet_eq_range [Algebra K F] (f : K[X]) [IsSplittingField K L f] (i : L →ₐ[K] F) :
    Algebra.adjoin K (rootSet f F) = i.range 
Arguments: {F : Type u} {K : Type v} (L : Type w) [Field K] [Field L] [Field F] [Algebra K L] [Algebra K F] (f : K[X]) [Polynomial.IsSplittingField K L f] (i : L →ₐ[K] F)
Goal: Algebra.adjoin K (f.rootSet F) = i.range
-- IntermediateField.splits_of_splits
theorem IntermediateField.splits_of_splits (h : p.Splits (algebraMap K L))
    (hF : ∀ x ∈ p.rootSet L, x ∈ F) : p.Splits (algebraMap K F) 
Arguments: {K : Type v} {L : Type w} [Field K] [Field L] [Algebra K L] {p : K[X]} {F : IntermediateField K L} (h : Polynomial.Splits (algebraMap K L) p) (hF : ∀ x ∈ p.rootSet L, x ∈ F)
Goal: Polynomial.Splits (algebraMap K ↥F) p
-- IsIntegral.mem_intermediateField_of_minpoly_splits
theorem IsIntegral.mem_intermediateField_of_minpoly_splits {x : L} (int : IsIntegral K x)
    {F : IntermediateField K L} (h : Splits (algebraMap K F) (minpoly K x)) : x ∈ F 
Arguments: {K : Type v} {L : Type w} [Field K] [Field L] [Algebra K L] {x : L} (int : IsIntegral K x) {F : IntermediateField K L} (h : Polynomial.Splits (algebraMap K ↥F) (minpoly K x))
Goal: x ∈ F
-- MvPolynomial.quotient_mk_comp_C_injective
theorem quotient_mk_comp_C_injective (I : Ideal (MvPolynomial σ K)) (hI : I ≠ ⊤) :
    Function.Injective ((Ideal.Quotient.mk I).comp MvPolynomial.C) 
Arguments: (σ : Type u) (K : Type v) [Field K] (I : Ideal (MvPolynomial σ K)) (hI : I ≠ ⊤)
Goal: Function.Injective ⇑((Ideal.Quotient.mk I).comp MvPolynomial.C)
-- MvPolynomial.rank_mvPolynomial
theorem rank_mvPolynomial : Module.rank K (MvPolynomial σ K) = Cardinal.mk (σ →₀ ℕ) 
Arguments: {σ K : Type u} [Field K]
Goal: Module.rank K (MvPolynomial σ K) = Cardinal.mk (σ →₀ ℕ)
-- LinearMap.isOrtho_def
theorem isOrtho_def {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M} {x y} : B.IsOrtho x y ↔ B x y = 0 
Arguments: {R : Type u_1} {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_5} {M₁ : Type u_6} {M₂ : Type u_7} [CommSemiring R] [CommSemiring R₁] [AddCommMonoid M₁] [Module R₁ M₁] [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] [AddCommMonoid M] [Module R M] {I₁ : R₁ →+* R} {I₂ : R₂ →+* R} {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M} {x : M₁} {y : M₂}
Goal: B.IsOrtho x y ↔ (B x) y = 0
-- LinearMap.isOrtho_zero_left
theorem isOrtho_zero_left (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x) : IsOrtho B (0 : M₁) x 
Arguments: {R : Type u_1} {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_5} {M₁ : Type u_6} {M₂ : Type u_7} [CommSemiring R] [CommSemiring R₁] [AddCommMonoid M₁] [Module R₁ M₁] [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] [AddCommMonoid M] [Module R M] {I₁ : R₁ →+* R} {I₂ : R₂ →+* R} (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x : M₂)
Goal: B.IsOrtho 0 x
-- LinearMap.isOrtho_zero_right
theorem isOrtho_zero_right (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x) : IsOrtho B x (0 : M₂) 
Arguments: {R : Type u_1} {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_5} {M₁ : Type u_6} {M₂ : Type u_7} [CommSemiring R] [CommSemiring R₁] [AddCommMonoid M₁] [Module R₁ M₁] [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] [AddCommMonoid M] [Module R M] {I₁ : R₁ →+* R} {I₂ : R₂ →+* R} (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x : M₁)
Goal: B.IsOrtho x 0
-- LinearMap.isOrtho_flip
theorem isOrtho_flip {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M} {x y} : B.IsOrtho x y ↔ B.flip.IsOrtho y x 
Arguments: {R : Type u_1} {R₁ : Type u_2} {M : Type u_5} {M₁ : Type u_6} [CommSemiring R] [CommSemiring R₁] [AddCommMonoid M₁] [Module R₁ M₁] [AddCommMonoid M] [Module R M] {I₁ I₁' : R₁ →+* R} {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M} {x y : M₁}
Goal: B.IsOrtho x y ↔ B.flip.IsOrtho y x
-- LinearMap.isOrthoᵢ_def
theorem isOrthoᵢ_def {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M} {v : n → M₁} :
    B.IsOrthoᵢ v ↔ ∀ i j : n, i ≠ j → B (v i) (v j) = 0 
Arguments: {R : Type u_1} {R₁ : Type u_2} {M : Type u_5} {M₁ : Type u_6} {n : Type u_19} [CommSemiring R] [CommSemiring R₁] [AddCommMonoid M₁] [Module R₁ M₁] [AddCommMonoid M] [Module R M] {I₁ I₁' : R₁ →+* R} {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M} {v : n → M₁}
Goal: B.IsOrthoᵢ v ↔ ∀ (i j : n), i ≠ j → (B (v i)) (v j) = 0