ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SameRay.exists_nonneg_right
theorem exists_nonneg_right (h : SameRay R x y) (hy : y ≠ 0) : ∃ r : R, 0 ≤ r ∧ x = r • y 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {x y : M} (h : SameRay R x y) (hy : y ≠ 0)
Goal: ∃ r, 0 ≤ r ∧ x = r • y
-- SameRay.exists_eq_smul_add
theorem exists_eq_smul_add (h : SameRay R v₁ v₂) :
    ∃ a b : R, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ v₁ = a • (v₁ + v₂) ∧ v₂ = b • (v₁ + v₂) 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {v₁ v₂ : M} (h : SameRay R v₁ v₂)
Goal: ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ v₁ = a • (v₁ + v₂) ∧ v₂ = b • (v₁ + v₂)
-- SameRay.exists_eq_smul
theorem exists_eq_smul (h : SameRay R v₁ v₂) :
    ∃ (u : M) (a b : R), 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ v₁ = a • u ∧ v₂ = b • u 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {v₁ v₂ : M} (h : SameRay R v₁ v₂)
Goal: ∃ u a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ v₁ = a • u ∧ v₂ = b • u
-- exists_pos_left_iff_sameRay
theorem exists_pos_left_iff_sameRay (hx : x ≠ 0) (hy : y ≠ 0) :
    (∃ r : R, 0 < r ∧ r • x = y) ↔ SameRay R x y 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {x y : M} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: (∃ r, 0 < r ∧ r • x = y) ↔ SameRay R x y
-- exists_pos_left_iff_sameRay_and_ne_zero
theorem exists_pos_left_iff_sameRay_and_ne_zero (hx : x ≠ 0) :
    (∃ r : R, 0 < r ∧ r • x = y) ↔ SameRay R x y ∧ y ≠ 0 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {x y : M} (hx : x ≠ 0)
Goal: (∃ r, 0 < r ∧ r • x = y) ↔ SameRay R x y ∧ y ≠ 0
-- exists_nonneg_left_iff_sameRay
theorem exists_nonneg_left_iff_sameRay (hx : x ≠ 0) :
    (∃ r : R, 0 ≤ r ∧ r • x = y) ↔ SameRay R x y 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {x y : M} (hx : x ≠ 0)
Goal: (∃ r, 0 ≤ r ∧ r • x = y) ↔ SameRay R x y
-- exists_pos_right_iff_sameRay
theorem exists_pos_right_iff_sameRay (hx : x ≠ 0) (hy : y ≠ 0) :
    (∃ r : R, 0 < r ∧ x = r • y) ↔ SameRay R x y 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {x y : M} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: (∃ r, 0 < r ∧ x = r • y) ↔ SameRay R x y
-- exists_pos_right_iff_sameRay_and_ne_zero
theorem exists_pos_right_iff_sameRay_and_ne_zero (hy : y ≠ 0) :
    (∃ r : R, 0 < r ∧ x = r • y) ↔ SameRay R x y ∧ x ≠ 0 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {x y : M} (hy : y ≠ 0)
Goal: (∃ r, 0 < r ∧ x = r • y) ↔ SameRay R x y ∧ x ≠ 0
-- exists_nonneg_right_iff_sameRay
theorem exists_nonneg_right_iff_sameRay (hy : y ≠ 0) :
    (∃ r : R, 0 ≤ r ∧ x = r • y) ↔ SameRay R x y 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {x y : M} (hy : y ≠ 0)
Goal: (∃ r, 0 ≤ r ∧ x = r • y) ↔ SameRay R x y
-- algebraMap_galRestrictHom_apply
@[simp]
lemma algebraMap_galRestrictHom_apply (σ : L →ₐ[K] L) (x : B) :
    algebraMap B L (galRestrictHom A K L B σ x) = σ (algebraMap B L x) 
Arguments: (A : Type u_1) (K : Type u_2) (L : Type u_3) (B : Type u_4) [CommRing A] [CommRing B] [Algebra A B] [Field K] [Field L] [Algebra A K] [IsFractionRing A K] [Algebra B L] [Algebra K L] [Algebra A L] [IsScalarTower A B L] [IsScalarTower A K L] [IsIntegralClosure B A L] [FiniteDimensional K L] (σ : L →ₐ[K] L) (x : B)
Goal: (algebraMap B L) (((galRestrictHom A K L B) σ) x) = σ ((algebraMap B L) x)
-- galRestrictHom_symm_algebraMap_apply
@[simp, nolint unusedHavesSuffices] -- false positive from unfolding galRestrictHom
lemma galRestrictHom_symm_algebraMap_apply (σ : B →ₐ[A] B) (x : B) :
    (galRestrictHom A K L B).symm σ (algebraMap B L x) = algebraMap B L (σ x) 
Arguments: (A : Type u_1) (K : Type u_2) (L : Type u_3) (B : Type u_4) [CommRing A] [CommRing B] [Algebra A B] [Field K] [Field L] [Algebra A K] [IsFractionRing A K] [Algebra B L] [Algebra K L] [Algebra A L] [IsScalarTower A B L] [IsScalarTower A K L] [IsIntegralClosure B A L] [FiniteDimensional K L] (σ : B →ₐ[A] B) (x : B)
Goal: ((galRestrictHom A K L B).symm σ) ((algebraMap B L) x) = (algebraMap B L) (σ x)
-- coe_galRestrict_apply
lemma coe_galRestrict_apply (σ : L ≃ₐ[K] L) :
    (galRestrict A K L B σ : B →ₐ[A] B) = galRestrictHom A K L B σ 
Arguments: (A : Type u_1) {K : Type u_2} {L : Type u_3} (B : Type u_4) [CommRing A] [CommRing B] [Algebra A B] [Field K] [Field L] [Algebra A K] [IsFractionRing A K] [Algebra B L] [Algebra K L] [Algebra A L] [IsScalarTower A B L] [IsScalarTower A K L] [IsIntegralClosure B A L] [FiniteDimensional K L] (σ : L ≃ₐ[K] L)
Goal: ↑((galRestrict A K L B) σ) = (galRestrictHom A K L B) ↑σ
-- galRestrict_apply
lemma galRestrict_apply (σ : L ≃ₐ[K] L) (x : B) :
    galRestrict A K L B σ x = galRestrictHom A K L B σ x 
Arguments: (A : Type u_1) {K : Type u_2} {L : Type u_3} {B : Type u_4} [CommRing A] [CommRing B] [Algebra A B] [Field K] [Field L] [Algebra A K] [IsFractionRing A K] [Algebra B L] [Algebra K L] [Algebra A L] [IsScalarTower A B L] [IsScalarTower A K L] [IsIntegralClosure B A L] [FiniteDimensional K L] (σ : L ≃ₐ[K] L) (x : B)
Goal: ((galRestrict A K L B) σ) x = ((galRestrictHom A K L B) ↑σ) x
-- algebraMap_galRestrict_apply
lemma algebraMap_galRestrict_apply (σ : L ≃ₐ[K] L) (x : B) :
    algebraMap B L (galRestrict A K L B σ x) = σ (algebraMap B L x) 
Arguments: (A : Type u_1) {K : Type u_2} {L : Type u_3} {B : Type u_4} [CommRing A] [CommRing B] [Algebra A B] [Field K] [Field L] [Algebra A K] [IsFractionRing A K] [Algebra B L] [Algebra K L] [Algebra A L] [IsScalarTower A B L] [IsScalarTower A K L] [IsIntegralClosure B A L] [FiniteDimensional K L] (σ : L ≃ₐ[K] L) (x : B)
Goal: (algebraMap B L) (((galRestrict A K L B) σ) x) = σ ((algebraMap B L) x)
-- prod_galRestrict_eq_norm
lemma prod_galRestrict_eq_norm [IsGalois K L] [IsIntegrallyClosed A] (x : B) :
    (∏ σ : L ≃ₐ[K] L, galRestrict A K L B σ x) =
    algebraMap A B (IsIntegralClosure.mk' (R := A) A (Algebra.norm K <| algebraMap B L x)
      (Algebra.isIntegral_norm K (IsIntegralClosure.isIntegral A L x).algebraMap)) 
Arguments: (A : Type u_1) (K : Type u_2) (L : Type u_3) (B : Type u_4) [CommRing A] [CommRing B] [Algebra A B] [Field K] [Field L] [Algebra A K] [IsFractionRing A K] [Algebra B L] [Algebra K L] [Algebra A L] [IsScalarTower A B L] [IsScalarTower A K L] [IsIntegralClosure B A L] [FiniteDimensional K L] [IsGalois K L] [IsIntegrallyClosed A] (x : B)
Goal: ∏ σ : L ≃ₐ[K] L, ((galRestrict A K L B) σ) x =
  (algebraMap A B) (IsIntegralClosure.mk' A ((Algebra.norm K) ((algebraMap B L) x)) ⋯)
-- Algebra.map_intTraceAux
lemma Algebra.map_intTraceAux [IsIntegrallyClosed A] (x : B) :
    algebraMap A K (Algebra.intTraceAux A K L B x) = Algebra.trace K L (algebraMap B L x) 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u_3} {B : Type u_4} [CommRing A] [CommRing B] [Algebra A B] [Field K] [Field L] [Algebra A K] [IsFractionRing A K] [Algebra B L] [Algebra K L] [Algebra A L] [IsScalarTower A B L] [IsScalarTower A K L] [IsIntegralClosure B A L] [FiniteDimensional K L] [IsIntegrallyClosed A] (x : B)
Goal: (algebraMap A K) ((Algebra.intTraceAux A K L B) x) = (Algebra.trace K L) ((algebraMap B L) x)
-- Algebra.algebraMap_intTrace
lemma Algebra.algebraMap_intTrace (x : B) :
    algebraMap A K (Algebra.intTrace A B x) = Algebra.trace K L (algebraMap B L x) 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u_3} {B : Type u_4} [CommRing A] [CommRing B] [Algebra A B] [Field K] [Field L] [Algebra A K] [IsFractionRing A K] [Algebra B L] [Algebra K L] [Algebra A L] [IsScalarTower A B L] [IsScalarTower A K L] [IsIntegralClosure B A L] [FiniteDimensional K L] [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B] [Module.Finite A B] [NoZeroSMulDivisors A B] (x : B)
Goal: (algebraMap A K) ((Algebra.intTrace A B) x) = (Algebra.trace K L) ((algebraMap B L) x)
-- Algebra.algebraMap_intTrace_fractionRing
lemma Algebra.algebraMap_intTrace_fractionRing (x : B) :
    algebraMap A (FractionRing A) (Algebra.intTrace A B x) =
      Algebra.trace (FractionRing A) (FractionRing B) (algebraMap B _ x) 
Arguments: {A : Type u_1} {B : Type u_4} [CommRing A] [CommRing B] [Algebra A B] [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B] [Module.Finite A B] [NoZeroSMulDivisors A B] (x : B)
Goal: (algebraMap A (FractionRing A)) ((Algebra.intTrace A B) x) =
  (Algebra.trace (FractionRing A) (FractionRing B)) ((algebraMap B (FractionRing B)) x)
-- Algebra.intTrace_eq_trace
lemma Algebra.intTrace_eq_trace [Module.Free A B] : Algebra.intTrace A B = Algebra.trace A B 
Arguments: (A : Type u_1) (B : Type u_4) [CommRing A] [CommRing B] [Algebra A B] [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B] [Module.Finite A B] [NoZeroSMulDivisors A B] [Module.Free A B]
Goal: Algebra.intTrace A B = Algebra.trace A B
-- Algebra.intTrace_eq_of_isLocalization
lemma Algebra.intTrace_eq_of_isLocalization
    (x : B) :
    algebraMap A Aₘ (Algebra.intTrace A B x) = Algebra.intTrace Aₘ Bₘ (algebraMap B Bₘ x) 
Arguments: (A : Type u_1) (B : Type u_4) [CommRing A] [CommRing B] [Algebra A B] {Aₘ : Type u_5} {Bₘ : Type u_6} [CommRing Aₘ] [CommRing Bₘ] [Algebra Aₘ Bₘ] [Algebra A Aₘ] [Algebra B Bₘ] [Algebra A Bₘ] [IsScalarTower A Aₘ Bₘ] [IsScalarTower A B Bₘ] (M : Submonoid A) [IsLocalization M Aₘ] [IsLocalization (Algebra.algebraMapSubmonoid B M) Bₘ] [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B] [Module.Finite A B] [NoZeroSMulDivisors A B] [IsDomain Aₘ] [IsIntegrallyClosed Aₘ] [IsDomain Bₘ] [IsIntegrallyClosed Bₘ] [NoZeroSMulDivisors Aₘ Bₘ] [Module.Finite Aₘ Bₘ] (x : B)
Goal: (algebraMap A Aₘ) ((Algebra.intTrace A B) x) = (Algebra.intTrace Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Algebra.map_intNormAux
lemma Algebra.map_intNormAux [IsIntegrallyClosed A] [Algebra.IsSeparable K L] (x : B) :
    algebraMap A K (Algebra.intNormAux A K L B x) = Algebra.norm K (algebraMap B L x) 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u_3} {B : Type u_4} [CommRing A] [CommRing B] [Algebra A B] [Field K] [Field L] [Algebra A K] [IsFractionRing A K] [Algebra B L] [Algebra K L] [Algebra A L] [IsScalarTower A B L] [IsScalarTower A K L] [IsIntegralClosure B A L] [FiniteDimensional K L] [IsIntegrallyClosed A] [Algebra.IsSeparable K L] (x : B)
Goal: (algebraMap A K) ((Algebra.intNormAux A K L B) x) = (Algebra.norm K) ((algebraMap B L) x)
-- Algebra.algebraMap_intNorm
lemma Algebra.algebraMap_intNorm (x : B) :
    algebraMap A K (Algebra.intNorm A B x) = Algebra.norm K (algebraMap B L x) 
Arguments: {A : Type u_1} {K : Type u_2} {L : Type u_3} {B : Type u_4} [CommRing A] [CommRing B] [Algebra A B] [Field K] [Field L] [Algebra A K] [IsFractionRing A K] [Algebra B L] [Algebra K L] [Algebra A L] [IsScalarTower A B L] [IsScalarTower A K L] [IsIntegralClosure B A L] [FiniteDimensional K L] [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B] [Module.Finite A B] [NoZeroSMulDivisors A B] [Algebra.IsSeparable (FractionRing A) (FractionRing B)] (x : B)
Goal: (algebraMap A K) ((Algebra.intNorm A B) x) = (Algebra.norm K) ((algebraMap B L) x)
-- Algebra.algebraMap_intNorm_fractionRing
@[simp]
lemma Algebra.algebraMap_intNorm_fractionRing (x : B) :
    algebraMap A (FractionRing A) (Algebra.intNorm A B x) =
      Algebra.norm (FractionRing A) (algebraMap B (FractionRing B) x) 
Arguments: {A : Type u_1} {B : Type u_4} [CommRing A] [CommRing B] [Algebra A B] [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B] [Module.Finite A B] [NoZeroSMulDivisors A B] [Algebra.IsSeparable (FractionRing A) (FractionRing B)] (x : B)
Goal: (algebraMap A (FractionRing A)) ((Algebra.intNorm A B) x) =
  (Algebra.norm (FractionRing A)) ((algebraMap B (FractionRing B)) x)
-- Algebra.intNorm_eq_norm
lemma Algebra.intNorm_eq_norm [Module.Free A B] : Algebra.intNorm A B = Algebra.norm A 
Arguments: (A : Type u_1) (B : Type u_4) [CommRing A] [CommRing B] [Algebra A B] [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B] [Module.Finite A B] [NoZeroSMulDivisors A B] [Algebra.IsSeparable (FractionRing A) (FractionRing B)] [Module.Free A B]
Goal: Algebra.intNorm A B = Algebra.norm A
-- Algebra.intNorm_zero
@[simp]
lemma Algebra.intNorm_zero : Algebra.intNorm A B 0 = 0 
Arguments: (A : Type u_1) (B : Type u_4) [CommRing A] [CommRing B] [Algebra A B] [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B] [Module.Finite A B] [NoZeroSMulDivisors A B] [Algebra.IsSeparable (FractionRing A) (FractionRing B)]
Goal: (Algebra.intNorm A B) 0 = 0
-- Algebra.intNorm_eq_zero
@[simp]
lemma Algebra.intNorm_eq_zero {x : B} : Algebra.intNorm A B x = 0 ↔ x = 0 
Arguments: {A : Type u_1} {B : Type u_4} [CommRing A] [CommRing B] [Algebra A B] [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B] [Module.Finite A B] [NoZeroSMulDivisors A B] [Algebra.IsSeparable (FractionRing A) (FractionRing B)] {x : B}
Goal: (Algebra.intNorm A B) x = 0 ↔ x = 0
-- Algebra.intNorm_ne_zero
lemma Algebra.intNorm_ne_zero {x : B} : Algebra.intNorm A B x ≠ 0 ↔ x ≠ 0 
Arguments: {A : Type u_1} {B : Type u_4} [CommRing A] [CommRing B] [Algebra A B] [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B] [Module.Finite A B] [NoZeroSMulDivisors A B] [Algebra.IsSeparable (FractionRing A) (FractionRing B)] {x : B}
Goal: (Algebra.intNorm A B) x ≠ 0 ↔ x ≠ 0
-- Algebra.intNorm_eq_of_isLocalization
lemma Algebra.intNorm_eq_of_isLocalization (x : B) :
    algebraMap A Aₘ (Algebra.intNorm A B x) = Algebra.intNorm Aₘ Bₘ (algebraMap B Bₘ x) 
Arguments: {A : Type u_1} {B : Type u_4} [CommRing A] [CommRing B] [Algebra A B] {Aₘ : Type u_5} {Bₘ : Type u_6} [CommRing Aₘ] [CommRing Bₘ] [Algebra Aₘ Bₘ] [Algebra A Aₘ] [Algebra B Bₘ] [Algebra A Bₘ] [IsScalarTower A Aₘ Bₘ] [IsScalarTower A B Bₘ] (M : Submonoid A) [IsLocalization M Aₘ] [IsLocalization (Algebra.algebraMapSubmonoid B M) Bₘ] [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B] [Module.Finite A B] [NoZeroSMulDivisors A B] [Algebra.IsSeparable (FractionRing A) (FractionRing B)] [IsDomain Aₘ] [IsIntegrallyClosed Aₘ] [IsDomain Bₘ] [IsIntegrallyClosed Bₘ] [NoZeroSMulDivisors Aₘ Bₘ] [Module.Finite Aₘ Bₘ] [Algebra.IsSeparable (FractionRing Aₘ) (FractionRing Bₘ)] (x : B)
Goal: (algebraMap A Aₘ) ((Algebra.intNorm A B) x) = (Algebra.intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Algebra.algebraMap_intNorm_of_isGalois
lemma Algebra.algebraMap_intNorm_of_isGalois
    [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B]
    [Module.Finite A B] [NoZeroSMulDivisors A B] [IsGalois (FractionRing A) (FractionRing B)]
    {x : B} :
    algebraMap A B (Algebra.intNorm A B x) = ∏ σ : B ≃ₐ[A] B, σ x 
Arguments: (A : Type u_1) (B : Type u_4) [CommRing A] [CommRing B] [Algebra A B] [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B] [Module.Finite A B] [NoZeroSMulDivisors A B] [IsGalois (FractionRing A) (FractionRing B)] {x : B}
Goal: (algebraMap A B) ((Algebra.intNorm A B) x) = ∏ σ : B ≃ₐ[A] B, σ x
-- LinearMap.charpoly_def
theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly 
Arguments: {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
Goal: f.charpoly = ((LinearMap.toMatrix (Module.Free.chooseBasis R M) (Module.Free.chooseBasis R M)) f).charpoly
-- LinearMap.charpoly_monic
theorem charpoly_monic : f.charpoly.Monic 
Arguments: {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
Goal: f.charpoly.Monic
-- LinearMap.charpoly_natDegree
lemma charpoly_natDegree [StrongRankCondition R] : natDegree (charpoly f) = finrank R M 
Arguments: {R : Type u} {M : Type v} [CommRing R] [Nontrivial R] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M) [StrongRankCondition R]
Goal: f.charpoly.natDegree = FiniteDimensional.finrank R M
-- LinearMap.aeval_self_charpoly
theorem aeval_self_charpoly : aeval f f.charpoly = 0 
Arguments: {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
Goal: (Polynomial.aeval f) f.charpoly = 0
-- LinearMap.isIntegral
theorem isIntegral : IsIntegral R f 
Arguments: {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
Goal: IsIntegral R f
-- LinearMap.minpoly_dvd_charpoly
theorem minpoly_dvd_charpoly {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M]
    [FiniteDimensional K M] (f : M →ₗ[K] M) : minpoly K f ∣ f.charpoly 
Arguments: {K : Type u} {M : Type v} [Field K] [AddCommGroup M] [Module K M] [FiniteDimensional K M] (f : M →ₗ[K] M)
Goal: minpoly K f ∣ f.charpoly
-- LinearMap.aeval_eq_aeval_mod_charpoly
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) 
Arguments: {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M) (p : R[X])
Goal: (Polynomial.aeval f) p = (Polynomial.aeval f) (p %ₘ f.charpoly)
-- LinearMap.pow_eq_aeval_mod_charpoly
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) 
Arguments: {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M) (k : ℕ)
Goal: f ^ k = (Polynomial.aeval f) (Polynomial.X ^ k %ₘ f.charpoly)
-- LinearMap.minpoly_coeff_zero_of_injective
theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
    (minpoly R f).coeff 0 ≠ 0 
Arguments: {R : Type u} {M : Type v} [CommRing R] [Nontrivial R] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] {f : M →ₗ[R] M} (hf : Function.Injective ⇑f)
Goal: (minpoly R f).coeff 0 ≠ 0
-- lineMap_mono_left
theorem lineMap_mono_left (ha : a ≤ a') (hr : r ≤ 1) : lineMap a b r ≤ lineMap a' b r 
Arguments: {k : Type u_1} {E : Type u_2} [OrderedRing k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a a' b : E} {r : k} (ha : a ≤ a') (hr : r ≤ 1)
Goal: (AffineMap.lineMap a b) r ≤ (AffineMap.lineMap a' b) r
-- lineMap_strict_mono_left
theorem lineMap_strict_mono_left (ha : a < a') (hr : r < 1) : lineMap a b r < lineMap a' b r 
Arguments: {k : Type u_1} {E : Type u_2} [OrderedRing k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a a' b : E} {r : k} (ha : a < a') (hr : r < 1)
Goal: (AffineMap.lineMap a b) r < (AffineMap.lineMap a' b) r
-- lineMap_mono_right
theorem lineMap_mono_right (hb : b ≤ b') (hr : 0 ≤ r) : lineMap a b r ≤ lineMap a b' r 
Arguments: {k : Type u_1} {E : Type u_2} [OrderedRing k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b b' : E} {r : k} (hb : b ≤ b') (hr : 0 ≤ r)
Goal: (AffineMap.lineMap a b) r ≤ (AffineMap.lineMap a b') r
-- lineMap_strict_mono_right
theorem lineMap_strict_mono_right (hb : b < b') (hr : 0 < r) : lineMap a b r < lineMap a b' r 
Arguments: {k : Type u_1} {E : Type u_2} [OrderedRing k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b b' : E} {r : k} (hb : b < b') (hr : 0 < r)
Goal: (AffineMap.lineMap a b) r < (AffineMap.lineMap a b') r
-- lineMap_mono_endpoints
theorem lineMap_mono_endpoints (ha : a ≤ a') (hb : b ≤ b') (h₀ : 0 ≤ r) (h₁ : r ≤ 1) :
    lineMap a b r ≤ lineMap a' b' r 
Arguments: {k : Type u_1} {E : Type u_2} [OrderedRing k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a a' b b' : E} {r : k} (ha : a ≤ a') (hb : b ≤ b') (h₀ : 0 ≤ r) (h₁ : r ≤ 1)
Goal: (AffineMap.lineMap a b) r ≤ (AffineMap.lineMap a' b') r
-- lineMap_strict_mono_endpoints
theorem lineMap_strict_mono_endpoints (ha : a < a') (hb : b < b') (h₀ : 0 ≤ r) (h₁ : r ≤ 1) :
    lineMap a b r < lineMap a' b' r 
Arguments: {k : Type u_1} {E : Type u_2} [OrderedRing k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a a' b b' : E} {r : k} (ha : a < a') (hb : b < b') (h₀ : 0 ≤ r) (h₁ : r ≤ 1)
Goal: (AffineMap.lineMap a b) r < (AffineMap.lineMap a' b') r
-- lineMap_lt_lineMap_iff_of_lt
theorem lineMap_lt_lineMap_iff_of_lt (h : r < r') : lineMap a b r < lineMap a b r' ↔ a < b 
Arguments: {k : Type u_1} {E : Type u_2} [OrderedRing k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E} {r r' : k} (h : r < r')
Goal: (AffineMap.lineMap a b) r < (AffineMap.lineMap a b) r' ↔ a < b
-- left_lt_lineMap_iff_lt
theorem left_lt_lineMap_iff_lt (h : 0 < r) : a < lineMap a b r ↔ a < b 
Arguments: {k : Type u_1} {E : Type u_2} [OrderedRing k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E} {r : k} (h : 0 < r)
Goal: a < (AffineMap.lineMap a b) r ↔ a < b
-- lineMap_lt_left_iff_lt
theorem lineMap_lt_left_iff_lt (h : 0 < r) : lineMap a b r < a ↔ b < a 
Arguments: {k : Type u_1} {E : Type u_2} [OrderedRing k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E} {r : k} (h : 0 < r)
Goal: (AffineMap.lineMap a b) r < a ↔ b < a
-- lineMap_lt_right_iff_lt
theorem lineMap_lt_right_iff_lt (h : r < 1) : lineMap a b r < b ↔ a < b 
Arguments: {k : Type u_1} {E : Type u_2} [OrderedRing k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E} {r : k} (h : r < 1)
Goal: (AffineMap.lineMap a b) r < b ↔ a < b
-- right_lt_lineMap_iff_lt
theorem right_lt_lineMap_iff_lt (h : r < 1) : b < lineMap a b r ↔ b < a 
Arguments: {k : Type u_1} {E : Type u_2} [OrderedRing k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E} {r : k} (h : r < 1)
Goal: b < (AffineMap.lineMap a b) r ↔ b < a
-- midpoint_le_midpoint
theorem midpoint_le_midpoint (ha : a ≤ a') (hb : b ≤ b') : midpoint k a b ≤ midpoint k a' b' 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedRing k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] [Invertible 2] {a a' b b' : E} (ha : a ≤ a') (hb : b ≤ b')
Goal: midpoint k a b ≤ midpoint k a' b'
-- lineMap_le_lineMap_iff_of_lt
theorem lineMap_le_lineMap_iff_of_lt (h : r < r') : lineMap a b r ≤ lineMap a b r' ↔ a ≤ b 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E} {r r' : k} (h : r < r')
Goal: (AffineMap.lineMap a b) r ≤ (AffineMap.lineMap a b) r' ↔ a ≤ b
-- left_le_lineMap_iff_le
theorem left_le_lineMap_iff_le (h : 0 < r) : a ≤ lineMap a b r ↔ a ≤ b 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E} {r : k} (h : 0 < r)
Goal: a ≤ (AffineMap.lineMap a b) r ↔ a ≤ b
-- left_le_midpoint
@[simp]
theorem left_le_midpoint : a ≤ midpoint k a b ↔ a ≤ b 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E}
Goal: a ≤ midpoint k a b ↔ a ≤ b
-- lineMap_le_left_iff_le
theorem lineMap_le_left_iff_le (h : 0 < r) : lineMap a b r ≤ a ↔ b ≤ a 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E} {r : k} (h : 0 < r)
Goal: (AffineMap.lineMap a b) r ≤ a ↔ b ≤ a
-- midpoint_le_left
@[simp]
theorem midpoint_le_left : midpoint k a b ≤ a ↔ b ≤ a 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E}
Goal: midpoint k a b ≤ a ↔ b ≤ a
-- lineMap_le_right_iff_le
theorem lineMap_le_right_iff_le (h : r < 1) : lineMap a b r ≤ b ↔ a ≤ b 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E} {r : k} (h : r < 1)
Goal: (AffineMap.lineMap a b) r ≤ b ↔ a ≤ b
-- midpoint_le_right
@[simp]
theorem midpoint_le_right : midpoint k a b ≤ b ↔ a ≤ b 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E}
Goal: midpoint k a b ≤ b ↔ a ≤ b
-- right_le_lineMap_iff_le
theorem right_le_lineMap_iff_le (h : r < 1) : b ≤ lineMap a b r ↔ b ≤ a 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E} {r : k} (h : r < 1)
Goal: b ≤ (AffineMap.lineMap a b) r ↔ b ≤ a
-- right_le_midpoint
@[simp]
theorem right_le_midpoint : b ≤ midpoint k a b ↔ b ≤ a 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {a b : E}
Goal: b ≤ midpoint k a b ↔ b ≤ a