ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Matrix.PosSemidef.toLinearMap₂'_zero_iff
theorem PosSemidef.toLinearMap₂'_zero_iff [DecidableEq n]
    {A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
    Matrix.toLinearMap₂' 𝕜 A (star x) x = 0 ↔ Matrix.toLin' A x = 0 
Arguments: {n : Type u_2} {𝕜 : Type u_4} [Fintype n] [RCLike 𝕜] [DecidableEq n] {A : Matrix n n 𝕜} (hA : A.PosSemidef) (x : n → 𝕜)
Goal: (((Matrix.toLinearMap₂' 𝕜) A) (star x)) x = 0 ↔ (Matrix.toLin' A) x = 0
-- Matrix.PosDef.isHermitian
theorem isHermitian {M : Matrix n n R} (hM : M.PosDef) : M.IsHermitian 
Arguments: {n : Type u_2} {R : Type u_3} [Fintype n] [CommRing R] [PartialOrder R] [StarRing R] {M : Matrix n n R} (hM : M.PosDef)
Goal: M.IsHermitian
-- Matrix.PosDef.re_dotProduct_pos
theorem re_dotProduct_pos {M : Matrix n n 𝕜} (hM : M.PosDef) {x : n → 𝕜} (hx : x ≠ 0) :
    0 < RCLike.re (dotProduct (star x) (M *ᵥ x)) 
Arguments: {n : Type u_2} {𝕜 : Type u_4} [Fintype n] [RCLike 𝕜] {M : Matrix n n 𝕜} (hM : M.PosDef) {x : n → 𝕜} (hx : x ≠ 0)
Goal: 0 < RCLike.re (Matrix.dotProduct (star x) (M.mulVec x))
-- Matrix.PosDef.posSemidef
theorem posSemidef {M : Matrix n n R} (hM : M.PosDef) : M.PosSemidef 
Arguments: {n : Type u_2} {R : Type u_3} [Fintype n] [CommRing R] [PartialOrder R] [StarRing R] {M : Matrix n n R} (hM : M.PosDef)
Goal: M.PosSemidef
-- Matrix.PosDef.transpose
theorem transpose {M : Matrix n n R} (hM : M.PosDef) : Mᵀ.PosDef 
Arguments: {n : Type u_2} {R : Type u_3} [Fintype n] [CommRing R] [PartialOrder R] [StarRing R] {M : Matrix n n R} (hM : M.PosDef)
Goal: M.transpose.PosDef
-- Matrix.PosDef.add_posSemidef
protected lemma add_posSemidef {A : Matrix m m R} {B : Matrix m m R}
    (hA : A.PosDef) (hB : B.PosSemidef) : (A + B).PosDef 
Arguments: {m : Type u_1} {R : Type u_3} [Fintype m] [CommRing R] [PartialOrder R] [StarRing R] [StarOrderedRing R] {A B : Matrix m m R} (hA : A.PosDef) (hB : B.PosSemidef)
Goal: (A + B).PosDef
-- Matrix.PosDef.posSemidef_add
protected lemma posSemidef_add {A : Matrix m m R} {B : Matrix m m R}
    (hA : A.PosSemidef) (hB : B.PosDef) : (A + B).PosDef 
Arguments: {m : Type u_1} {R : Type u_3} [Fintype m] [CommRing R] [PartialOrder R] [StarRing R] [StarOrderedRing R] {A B : Matrix m m R} (hA : A.PosSemidef) (hB : B.PosDef)
Goal: (A + B).PosDef
-- Matrix.PosDef.add
protected lemma add {A : Matrix m m R} {B : Matrix m m R}
    (hA : A.PosDef) (hB : B.PosDef) : (A + B).PosDef 
Arguments: {m : Type u_1} {R : Type u_3} [Fintype m] [CommRing R] [PartialOrder R] [StarRing R] [StarOrderedRing R] {A B : Matrix m m R} (hA : A.PosDef) (hB : B.PosDef)
Goal: (A + B).PosDef
-- Matrix.PosDef.of_toQuadraticForm'
theorem of_toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.IsSymm)
    (hMq : M.toQuadraticMap'.PosDef) : M.PosDef 
Arguments: {n : Type u_2} [Fintype n] [DecidableEq n] {M : Matrix n n ℝ} (hM : M.IsSymm) (hMq : M.toQuadraticMap'.PosDef)
Goal: M.PosDef
-- Matrix.PosDef.toQuadraticForm'
theorem toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) :
    M.toQuadraticMap'.PosDef 
Arguments: {n : Type u_2} [Fintype n] [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef)
Goal: M.toQuadraticMap'.PosDef
-- Matrix.PosDef.eigenvalues_pos
lemma eigenvalues_pos [DecidableEq n] {A : Matrix n n 𝕜}
    (hA : Matrix.PosDef A) (i : n) : 0 < hA.1.eigenvalues i 
Arguments: {n : Type u_2} {𝕜 : Type u_4} [Fintype n] [RCLike 𝕜] [DecidableEq n] {A : Matrix n n 𝕜} (hA : A.PosDef) (i : n)
Goal: 0 < ⋯.eigenvalues i
-- Matrix.PosDef.det_pos
theorem det_pos [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) : 0 < det M 
Arguments: {n : Type u_2} {𝕜 : Type u_4} [Fintype n] [RCLike 𝕜] [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef)
Goal: 0 < M.det
-- QuadraticForm.posDef_of_toMatrix'
theorem posDef_of_toMatrix' [DecidableEq n] {Q : QuadraticForm ℝ (n → ℝ)}
    (hQ : Q.toMatrix'.PosDef) : Q.PosDef 
Arguments: {n : Type u_1} [Fintype n] [DecidableEq n] {Q : QuadraticForm ℝ (n → ℝ)} (hQ : (QuadraticMap.toMatrix' Q).PosDef)
Goal: QuadraticMap.PosDef Q
-- QuadraticForm.posDef_toMatrix'
theorem posDef_toMatrix' [DecidableEq n] {Q : QuadraticForm ℝ (n → ℝ)} (hQ : Q.PosDef) :
    Q.toMatrix'.PosDef 
Arguments: {n : Type u_1} [Fintype n] [DecidableEq n] {Q : QuadraticForm ℝ (n → ℝ)} (hQ : QuadraticMap.PosDef Q)
Goal: (QuadraticMap.toMatrix' Q).PosDef
-- Basis.ofRankEqZero_apply
@[simp]
theorem Basis.ofRankEqZero_apply [Module.Free K V] {ι : Type*} [IsEmpty ι]
    (hV : Module.rank K V = 0) (i : ι) : Basis.ofRankEqZero hV i = 0 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Module.Free K V] {ι : Type u_1} [IsEmpty ι] (hV : Module.rank K V = 0) (i : ι)
Goal: (Basis.ofRankEqZero hV) i = 0
-- le_rank_iff_exists_linearIndependent
theorem le_rank_iff_exists_linearIndependent [Module.Free K V] {c : Cardinal} :
    c ≤ Module.rank K V ↔ ∃ s : Set V, #s = c ∧ LinearIndependent K ((↑) : s → V) 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Module.Free K V] {c : Cardinal.{v}}
Goal: c ≤ Module.rank K V ↔ ∃ s, #↑s = c ∧ LinearIndependent (ι := { x // x ∈ s }) K Subtype.val
-- le_rank_iff_exists_linearIndependent_finset
theorem le_rank_iff_exists_linearIndependent_finset
    [Module.Free K V] {n : ℕ} : ↑n ≤ Module.rank K V ↔
    ∃ s : Finset V, s.card = n ∧ LinearIndependent K ((↑) : ↥(s : Set V) → V) 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Module.Free K V] {n : ℕ}
Goal: ↑n ≤ Module.rank K V ↔ ∃ s, s.card = n ∧ LinearIndependent (ι := { x // x ∈ ↑s }) K Subtype.val
-- rank_le_one_iff
theorem rank_le_one_iff [Module.Free K V] :
    Module.rank K V ≤ 1 ↔ ∃ v₀ : V, ∀ v, ∃ r : K, r • v₀ = v 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Module.Free K V]
Goal: Module.rank K V ≤ 1 ↔ ∃ v₀, ∀ (v : V), ∃ r, r • v₀ = v
-- rank_eq_one_iff
theorem rank_eq_one_iff [Module.Free K V] :
    Module.rank K V = 1 ↔ ∃ v₀ : V, v₀ ≠ 0 ∧ ∀ v, ∃ r : K, r • v₀ = v 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Module.Free K V]
Goal: Module.rank K V = 1 ↔ ∃ v₀, v₀ ≠ 0 ∧ ∀ (v : V), ∃ r, r • v₀ = v
-- rank_submodule_le_one_iff
theorem rank_submodule_le_one_iff (s : Submodule K V) [Module.Free K s] :
    Module.rank K s ≤ 1 ↔ ∃ v₀ ∈ s, s ≤ K ∙ v₀ 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] (s : Submodule K V) [Module.Free K ↥s]
Goal: Module.rank K ↥s ≤ 1 ↔ ∃ v₀ ∈ s, s ≤ Submodule.span K {v₀}
-- rank_submodule_eq_one_iff
theorem rank_submodule_eq_one_iff (s : Submodule K V) [Module.Free K s] :
    Module.rank K s = 1 ↔ ∃ v₀ ∈ s, v₀ ≠ 0 ∧ s ≤ K ∙ v₀ 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] (s : Submodule K V) [Module.Free K ↥s]
Goal: Module.rank K ↥s = 1 ↔ ∃ v₀ ∈ s, v₀ ≠ 0 ∧ s ≤ Submodule.span K {v₀}
-- rank_submodule_le_one_iff'
theorem rank_submodule_le_one_iff' (s : Submodule K V) [Module.Free K s] :
    Module.rank K s ≤ 1 ↔ ∃ v₀, s ≤ K ∙ v₀ 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] (s : Submodule K V) [Module.Free K ↥s]
Goal: Module.rank K ↥s ≤ 1 ↔ ∃ v₀, s ≤ Submodule.span K {v₀}
-- Submodule.rank_le_one_iff_isPrincipal
theorem Submodule.rank_le_one_iff_isPrincipal (W : Submodule K V) [Module.Free K W] :
    Module.rank K W ≤ 1 ↔ W.IsPrincipal 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] (W : Submodule K V) [Module.Free K ↥W]
Goal: Module.rank K ↥W ≤ 1 ↔ W.IsPrincipal
-- Module.rank_le_one_iff_top_isPrincipal
theorem Module.rank_le_one_iff_top_isPrincipal [Module.Free K V] :
    Module.rank K V ≤ 1 ↔ (⊤ : Submodule K V).IsPrincipal 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Module.Free K V]
Goal: Module.rank K V ≤ 1 ↔ ⊤.IsPrincipal
-- finrank_eq_one_iff
theorem finrank_eq_one_iff [Module.Free K V] (ι : Type*) [Unique ι] :
    finrank K V = 1 ↔ Nonempty (Basis ι K V) 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Module.Free K V] (ι : Type u_1) [Unique ι]
Goal: FiniteDimensional.finrank K V = 1 ↔ Nonempty (Basis ι K V)
-- finrank_eq_one_iff'
theorem finrank_eq_one_iff' [Module.Free K V] :
    finrank K V = 1 ↔ ∃ v ≠ 0, ∀ w : V, ∃ c : K, c • v = w 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Module.Free K V]
Goal: FiniteDimensional.finrank K V = 1 ↔ ∃ v, v ≠ 0 ∧ ∀ (w : V), ∃ c, c • v = w
-- finrank_le_one_iff
theorem finrank_le_one_iff [Module.Free K V] [Module.Finite K V] :
    finrank K V ≤ 1 ↔ ∃ v : V, ∀ w : V, ∃ c : K, c • v = w 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V]
Goal: FiniteDimensional.finrank K V ≤ 1 ↔ ∃ v, ∀ (w : V), ∃ c, c • v = w
-- Submodule.finrank_le_one_iff_isPrincipal
theorem Submodule.finrank_le_one_iff_isPrincipal
    (W : Submodule K V) [Module.Free K W] [Module.Finite K W] :
    finrank K W ≤ 1 ↔ W.IsPrincipal 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] (W : Submodule K V) [Module.Free K ↥W] [Module.Finite K ↥W]
Goal: FiniteDimensional.finrank K ↥W ≤ 1 ↔ W.IsPrincipal
-- Module.finrank_le_one_iff_top_isPrincipal
theorem Module.finrank_le_one_iff_top_isPrincipal [Module.Free K V] [Module.Finite K V] :
    finrank K V ≤ 1 ↔ (⊤ : Submodule K V).IsPrincipal 
Arguments: {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V]
Goal: FiniteDimensional.finrank K V ≤ 1 ↔ ⊤.IsPrincipal
-- lift_cardinal_mk_eq_lift_cardinal_mk_field_pow_lift_rank
theorem lift_cardinal_mk_eq_lift_cardinal_mk_field_pow_lift_rank [Module.Free K V]
    [Module.Finite K V] : lift.{u} #V = lift.{v} #K ^ lift.{u} (Module.rank K V) 
Arguments: (K : Type u) (V : Type v) [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V]
Goal: Cardinal.lift.{u, v} #V = Cardinal.lift.{v, u} #K ^ Cardinal.lift.{u, v} (Module.rank K V)
-- cardinal_mk_eq_cardinal_mk_field_pow_rank
theorem cardinal_mk_eq_cardinal_mk_field_pow_rank (K V : Type u) [Ring K] [StrongRankCondition K]
    [AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V] :
    #V = #K ^ Module.rank K V 
Arguments: (K V : Type u) [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V]
Goal: #V = #K ^ Module.rank K V
-- cardinal_lt_aleph0_of_finiteDimensional
theorem cardinal_lt_aleph0_of_finiteDimensional [Finite K] [Module.Free K V] [Module.Finite K V] :
    #V < ℵ₀ 
Arguments: (K : Type u) (V : Type v) [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Finite K] [Module.Free K V] [Module.Finite K V]
Goal: #V < ℵ₀
-- Subalgebra.eq_bot_of_rank_le_one
theorem eq_bot_of_rank_le_one (h : Module.rank F S ≤ 1) [Module.Free F S] : S = ⊥ 
Arguments: {F : Type u_1} {E : Type u_2} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E] {S : Subalgebra F E} (h : Module.rank F ↥S ≤ 1) [Module.Free F ↥S]
Goal: S = ⊥
-- Subalgebra.eq_bot_of_finrank_one
theorem eq_bot_of_finrank_one (h : finrank F S = 1) [Module.Free F S] : S = ⊥ 
Arguments: {F : Type u_1} {E : Type u_2} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E] {S : Subalgebra F E} (h : FiniteDimensional.finrank F ↥S = 1) [Module.Free F ↥S]
Goal: S = ⊥
-- Subalgebra.rank_eq_one_iff
@[simp]
theorem rank_eq_one_iff [Nontrivial E] [Module.Free F S] : Module.rank F S = 1 ↔ S = ⊥ 
Arguments: {F : Type u_1} {E : Type u_2} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E] {S : Subalgebra F E} [Nontrivial E] [Module.Free F ↥S]
Goal: Module.rank F ↥S = 1 ↔ S = ⊥
-- Subalgebra.finrank_eq_one_iff
@[simp]
theorem finrank_eq_one_iff [Nontrivial E] [Module.Free F S] : finrank F S = 1 ↔ S = ⊥ 
Arguments: {F : Type u_1} {E : Type u_2} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E] {S : Subalgebra F E} [Nontrivial E] [Module.Free F ↥S]
Goal: FiniteDimensional.finrank F ↥S = 1 ↔ S = ⊥
-- Subalgebra.bot_eq_top_iff_rank_eq_one
theorem bot_eq_top_iff_rank_eq_one [Nontrivial E] [Module.Free F E] :
    (⊥ : Subalgebra F E) = ⊤ ↔ Module.rank F E = 1 
Arguments: {F : Type u_1} {E : Type u_2} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E] [Nontrivial E] [Module.Free F E]
Goal: ⊥ = ⊤ ↔ Module.rank F E = 1
-- Subalgebra.bot_eq_top_iff_finrank_eq_one
theorem bot_eq_top_iff_finrank_eq_one [Nontrivial E] [Module.Free F E] :
    (⊥ : Subalgebra F E) = ⊤ ↔ finrank F E = 1 
Arguments: {F : Type u_1} {E : Type u_2} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E] [Nontrivial E] [Module.Free F E]
Goal: ⊥ = ⊤ ↔ FiniteDimensional.finrank F E = 1
-- lift_rank_mul_lift_rank
theorem lift_rank_mul_lift_rank :
    Cardinal.lift.{w} (Module.rank F K) * Cardinal.lift.{v} (Module.rank K A) =
      Cardinal.lift.{v} (Module.rank F A) 
Arguments: (F : Type u) (K : Type v) (A : Type w) [Ring F] [Ring K] [AddCommGroup A] [Module F K] [Module K A] [Module F A] [IsScalarTower F K A] [StrongRankCondition F] [StrongRankCondition K] [Module.Free F K] [Module.Free K A]
Goal: Cardinal.lift.{w, v} (Module.rank F K) * Cardinal.lift.{v, w} (Module.rank K A) = Cardinal.lift.{v, w} (Module.rank F A)
-- rank_mul_rank
theorem rank_mul_rank (A : Type v) [AddCommGroup A]
    [Module K A] [Module F A] [IsScalarTower F K A] [Module.Free K A] :
    Module.rank F K * Module.rank K A = Module.rank F A 
Arguments: (F : Type u) (K : Type v) [Ring F] [Ring K] [Module F K] [StrongRankCondition F] [StrongRankCondition K] [Module.Free F K] (A : Type v) [AddCommGroup A] [Module K A] [Module F A] [IsScalarTower F K A] [Module.Free K A]
Goal: Module.rank F K * Module.rank K A = Module.rank F A
-- FiniteDimensional.finrank_mul_finrank
theorem FiniteDimensional.finrank_mul_finrank : finrank F K * finrank K A = finrank F A 
Arguments: (F : Type u) (K : Type v) (A : Type w) [Ring F] [Ring K] [AddCommGroup A] [Module F K] [Module K A] [Module F A] [IsScalarTower F K A] [StrongRankCondition F] [StrongRankCondition K] [Module.Free F K] [Module.Free K A]
Goal: FiniteDimensional.finrank F K * FiniteDimensional.finrank K A = FiniteDimensional.finrank F A
-- Module.Free.rank_eq_card_chooseBasisIndex
theorem rank_eq_card_chooseBasisIndex : Module.rank R M = #(ChooseBasisIndex R M) 
Arguments: (R : Type u) (M : Type v) [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M]
Goal: Module.rank R M = #(Module.Free.ChooseBasisIndex R M)
-- FiniteDimensional.finrank_eq_card_chooseBasisIndex
theorem _root_.FiniteDimensional.finrank_eq_card_chooseBasisIndex [Module.Finite R M] :
    finrank R M = Fintype.card (ChooseBasisIndex R M) 
Arguments: (R : Type u) (M : Type v) [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M]
Goal: FiniteDimensional.finrank R M = Fintype.card (Module.Free.ChooseBasisIndex R M)
-- Module.Free.rank_eq_mk_of_infinite_lt
lemma rank_eq_mk_of_infinite_lt [Infinite R] (h_lt : lift.{v} #R < lift.{u} #M) :
    Module.rank R M = #M 
Arguments: (R : Type u) (M : Type v) [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] [Infinite R] (h_lt : Cardinal.lift.{v, u} #R < Cardinal.lift.{u, v} #M)
Goal: Module.rank R M = #M
-- nonempty_linearEquiv_of_lift_rank_eq
theorem nonempty_linearEquiv_of_lift_rank_eq
    (cnd : Cardinal.lift.{v'} (Module.rank R M) = Cardinal.lift.{v} (Module.rank R M')) :
    Nonempty (M ≃ₗ[R] M') 
Arguments: {R : Type u} {M : Type v} {M' : Type v'} [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] [AddCommGroup M'] [Module R M'] [Module.Free R M'] (cnd : Cardinal.lift.{v', v} (Module.rank R M) = Cardinal.lift.{v, v'} (Module.rank R M'))
Goal: Nonempty (M ≃ₗ[R] M')
-- nonempty_linearEquiv_of_rank_eq
theorem nonempty_linearEquiv_of_rank_eq (cond : Module.rank R M = Module.rank R M₁) :
    Nonempty (M ≃ₗ[R] M₁) 
Arguments: {R : Type u} {M M₁ : Type v} [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] [AddCommGroup M₁] [Module R M₁] [Module.Free R M₁] (cond : Module.rank R M = Module.rank R M₁)
Goal: Nonempty (M ≃ₗ[R] M₁)
-- LinearEquiv.nonempty_equiv_iff_lift_rank_eq
theorem LinearEquiv.nonempty_equiv_iff_lift_rank_eq : Nonempty (M ≃ₗ[R] M') ↔
    Cardinal.lift.{v'} (Module.rank R M) = Cardinal.lift.{v} (Module.rank R M') 
Arguments: {R : Type u} {M : Type v} {M' : Type v'} [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] [AddCommGroup M'] [Module R M'] [Module.Free R M']
Goal: Nonempty (M ≃ₗ[R] M') ↔ Cardinal.lift.{v', v} (Module.rank R M) = Cardinal.lift.{v, v'} (Module.rank R M')
-- LinearEquiv.nonempty_equiv_iff_rank_eq
theorem LinearEquiv.nonempty_equiv_iff_rank_eq :
    Nonempty (M ≃ₗ[R] M₁) ↔ Module.rank R M = Module.rank R M₁ 
Arguments: {R : Type u} {M M₁ : Type v} [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] [AddCommGroup M₁] [Module R M₁] [Module.Free R M₁]
Goal: Nonempty (M ≃ₗ[R] M₁) ↔ Module.rank R M = Module.rank R M₁
-- FiniteDimensional.nonempty_linearEquiv_of_finrank_eq
theorem FiniteDimensional.nonempty_linearEquiv_of_finrank_eq
    [Module.Finite R M] [Module.Finite R M'] (cond : finrank R M = finrank R M') :
    Nonempty (M ≃ₗ[R] M') 
Arguments: {R : Type u} {M : Type v} {M' : Type v'} [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] [AddCommGroup M'] [Module R M'] [Module.Free R M'] [Module.Finite R M] [Module.Finite R M'] (cond : FiniteDimensional.finrank R M = FiniteDimensional.finrank R M')
Goal: Nonempty (M ≃ₗ[R] M')
-- FiniteDimensional.nonempty_linearEquiv_iff_finrank_eq
theorem FiniteDimensional.nonempty_linearEquiv_iff_finrank_eq [Module.Finite R M]
    [Module.Finite R M'] : Nonempty (M ≃ₗ[R] M') ↔ finrank R M = finrank R M' 
Arguments: {R : Type u} {M : Type v} {M' : Type v'} [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] [AddCommGroup M'] [Module R M'] [Module.Free R M'] [Module.Finite R M] [Module.Finite R M']
Goal: Nonempty (M ≃ₗ[R] M') ↔ FiniteDimensional.finrank R M = FiniteDimensional.finrank R M'
-- Module.rank_lt_alpeh0_iff
lemma Module.rank_lt_alpeh0_iff :
    Module.rank R M < ℵ₀ ↔ Module.Finite R M 
Arguments: {R : Type u} {M : Type v} [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M]
Goal: Module.rank R M < ℵ₀ ↔ Module.Finite R M
-- FiniteDimensional.finrank_of_not_finite
theorem FiniteDimensional.finrank_of_not_finite
    (h : ¬Module.Finite R M) :
    finrank R M = 0 
Arguments: {R : Type u} {M : Type v} [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] (h : ¬Module.Finite R M)
Goal: FiniteDimensional.finrank R M = 0
-- Module.finite_of_finrank_pos
theorem Module.finite_of_finrank_pos (h : 0 < finrank R M) :
    Module.Finite R M 
Arguments: {R : Type u} {M : Type v} [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] (h : 0 < FiniteDimensional.finrank R M)
Goal: Module.Finite R M
-- Module.finite_of_finrank_eq_succ
theorem Module.finite_of_finrank_eq_succ {n : ℕ}
    (hn : finrank R M = n.succ) : Module.Finite R M 
Arguments: {R : Type u} {M : Type v} [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] {n : ℕ} (hn : FiniteDimensional.finrank R M = n.succ)
Goal: Module.Finite R M
-- Module.finite_iff_of_rank_eq_nsmul
theorem Module.finite_iff_of_rank_eq_nsmul {W} [AddCommGroup W]
    [Module R W] [Module.Free R W] {n : ℕ} (hn : n ≠ 0)
    (hVW : Module.rank R M = n • Module.rank R W) :
    Module.Finite R M ↔ Module.Finite R W 
Arguments: {R : Type u} {M : Type v} [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] {W : Type v} [AddCommGroup W] [Module R W] [Module.Free R W] {n : ℕ} (hn : n ≠ 0) (hVW : Module.rank R M = n • Module.rank R W)
Goal: Module.Finite R M ↔ Module.Finite R W
-- FiniteDimensional.basisUnique_repr_eq_zero_iff
@[simp]
theorem basisUnique_repr_eq_zero_iff {ι : Type*} [Unique ι]
    {h : finrank R M = 1} {v : M} {i : ι} :
    (basisUnique ι h).repr v i = 0 ↔ v = 0 
Arguments: {R : Type u} {M : Type v} [Ring R] [StrongRankCondition R] [AddCommGroup M] [Module R M] [Module.Free R M] {ι : Type u_1} [Unique ι] {h : FiniteDimensional.finrank R M = 1} {v : M} {i : ι}
Goal: ((FiniteDimensional.basisUnique ι h).repr v) i = 0 ↔ v = 0
-- ExteriorAlgebra.ι_sq_zero
-- @[simp] -- Porting note (#10618): simp can prove this
theorem ι_sq_zero (m : M) : ι R m * ι R m = 0 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (m : M)
Goal: (ExteriorAlgebra.ι R) m * (ExteriorAlgebra.ι R) m = 0
-- ExteriorAlgebra.comp_ι_sq_zero
theorem comp_ι_sq_zero (g : ExteriorAlgebra R M →ₐ[R] A) (m : M) : g (ι R m) * g (ι R m) = 0 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] (g : ExteriorAlgebra R M →ₐ[R] A) (m : M)
Goal: g ((ExteriorAlgebra.ι R) m) * g ((ExteriorAlgebra.ι R) m) = 0
-- ExteriorAlgebra.lift_symm_apply
@[simps! symm_apply]
def lift : { f : M →ₗ[R] A // ∀ m, f m * f m = 0 } ≃ (ExteriorAlgebra R M →ₐ[R] A) 
Arguments: (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A]
Goal: ∀ (a : ExteriorAlgebra R M →ₐ[R] A), (ExteriorAlgebra.lift R).symm a = ⟨↑((CliffordAlgebra.lift 0).symm a), ⋯⟩
-- ExteriorAlgebra.ι_comp_lift
@[simp]
theorem ι_comp_lift (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = 0) :
    (lift R ⟨f, cond⟩).toLinearMap.comp (ι R) = f 
Arguments: (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = 0)
Goal: ((ExteriorAlgebra.lift R) ⟨f, cond⟩).toLinearMap ∘ₗ ExteriorAlgebra.ι R = f
-- ExteriorAlgebra.lift_ι_apply
@[simp]
theorem lift_ι_apply (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = 0) (x) :
    lift R ⟨f, cond⟩ (ι R x) = f x 
Arguments: (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = 0) (x : M)
Goal: ((ExteriorAlgebra.lift R) ⟨f, cond⟩) ((ExteriorAlgebra.ι R) x) = f x
-- ExteriorAlgebra.lift_unique
@[simp]
theorem lift_unique (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = 0) (g : ExteriorAlgebra R M →ₐ[R] A) :
    g.toLinearMap.comp (ι R) = f ↔ g = lift R ⟨f, cond⟩ 
Arguments: (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = 0) (g : ExteriorAlgebra R M →ₐ[R] A)
Goal: g.toLinearMap ∘ₗ ExteriorAlgebra.ι R = f ↔ g = (ExteriorAlgebra.lift R) ⟨f, cond⟩
-- ExteriorAlgebra.lift_comp_ι
@[simp]
theorem lift_comp_ι (g : ExteriorAlgebra R M →ₐ[R] A) :
    lift R ⟨g.toLinearMap.comp (ι R), comp_ι_sq_zero _⟩ = g 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] (g : ExteriorAlgebra R M →ₐ[R] A)
Goal: (ExteriorAlgebra.lift R) ⟨g.toLinearMap ∘ₗ ExteriorAlgebra.ι R, ⋯⟩ = g
-- ExteriorAlgebra.hom_ext
@[ext]
theorem hom_ext {f g : ExteriorAlgebra R M →ₐ[R] A}
    (h : f.toLinearMap.comp (ι R) = g.toLinearMap.comp (ι R)) : f = g 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] {f g : ExteriorAlgebra R M →ₐ[R] A} (h : f.toLinearMap ∘ₗ ExteriorAlgebra.ι R = g.toLinearMap ∘ₗ ExteriorAlgebra.ι R)
Goal: f = g
-- ExteriorAlgebra.induction
@[elab_as_elim]
theorem induction {C : ExteriorAlgebra R M → Prop}
    (algebraMap : ∀ r, C (algebraMap R (ExteriorAlgebra R M) r)) (ι : ∀ x, C (ι R x))
    (mul : ∀ a b, C a → C b → C (a * b)) (add : ∀ a b, C a → C b → C (a + b))
    (a : ExteriorAlgebra R M) : C a 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {C : ExteriorAlgebra R M → Prop} (algebraMap : ∀ (r : R), C ((algebraMap R (ExteriorAlgebra R M)) r)) (ι : ∀ (x : M), C ((ExteriorAlgebra.ι R) x)) (mul : ∀ (a b : ExteriorAlgebra R M), C a → C b → C (a * b)) (add : ∀ (a b : ExteriorAlgebra R M), C a → C b → C (a + b)) (a : ExteriorAlgebra R M)
Goal: C a