ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MvPolynomial.ACounit_X
@[simp]
theorem ACounit_X (b : B) : ACounit A B (X b) = b 
Arguments: (A : Type u_1) {B : Type u_2} [CommSemiring A] [CommSemiring B] [Algebra A B] (b : B)
Goal: (MvPolynomial.ACounit A B) (MvPolynomial.X b) = b
-- MvPolynomial.ACounit_C
theorem ACounit_C (a : A) : ACounit A B (C a) = algebraMap A B a 
Arguments: {A : Type u_1} (B : Type u_2) [CommSemiring A] [CommSemiring B] [Algebra A B] (a : A)
Goal: (MvPolynomial.ACounit A B) (MvPolynomial.C a) = (algebraMap A B) a
-- MvPolynomial.ACounit_surjective
theorem ACounit_surjective : Surjective (ACounit A B) 
Arguments: (A : Type u_1) (B : Type u_2) [CommSemiring A] [CommSemiring B] [Algebra A B]
Goal: Function.Surjective ⇑(MvPolynomial.ACounit A B)
-- MvPolynomial.counit_surjective
theorem counit_surjective : Surjective (counit R) 
Arguments: (R : Type u_3) [CommRing R]
Goal: Function.Surjective ⇑(MvPolynomial.counit R)
-- MvPolynomial.counitNat_surjective
theorem counitNat_surjective : Surjective (counitNat A) 
Arguments: (A : Type u_1) [CommSemiring A]
Goal: Function.Surjective ⇑(MvPolynomial.counitNat A)
-- MvPolynomial.counit_C
theorem counit_C (n : ℤ) : counit R (C n) = n 
Arguments: (R : Type u_3) [CommRing R] (n : ℤ)
Goal: (MvPolynomial.counit R) (MvPolynomial.C n) = ↑n
-- MvPolynomial.counitNat_C
theorem counitNat_C (n : ℕ) : counitNat A (C n) = n 
Arguments: (A : Type u_1) [CommSemiring A] (n : ℕ)
Goal: (MvPolynomial.counitNat A) (MvPolynomial.C n) = ↑n
-- MvPolynomial.counit_X
@[simp]
theorem counit_X (r : R) : counit R (X r) = r 
Arguments: {R : Type u_3} [CommRing R] (r : R)
Goal: (MvPolynomial.counit R) (MvPolynomial.X r) = r
-- MvPolynomial.counitNat_X
@[simp]
theorem counitNat_X (a : A) : counitNat A (X a) = a 
Arguments: {A : Type u_1} [CommSemiring A] (a : A)
Goal: (MvPolynomial.counitNat A) (MvPolynomial.X a) = a
-- MvPolynomial.cardinal_mk_eq_max_lift
@[simp]
theorem cardinal_mk_eq_max_lift [Nonempty σ] [Nontrivial R] :
    #(MvPolynomial σ R) = max (max (Cardinal.lift.{u} #R) <| Cardinal.lift.{v} #σ) ℵ₀ 
Arguments: {σ : Type u} {R : Type v} [CommSemiring R] [Nonempty σ] [Nontrivial R]
Goal: #(MvPolynomial σ R) = max (max (Cardinal.lift.{u, v} #R) (Cardinal.lift.{v, u} #σ)) ℵ₀
-- MvPolynomial.cardinal_mk_eq_lift
@[simp]
theorem cardinal_mk_eq_lift [IsEmpty σ] : #(MvPolynomial σ R) = Cardinal.lift.{u} #R 
Arguments: {σ : Type u} {R : Type v} [CommSemiring R] [IsEmpty σ]
Goal: #(MvPolynomial σ R) = Cardinal.lift.{u, v} #R
-- MvPolynomial.cardinal_lift_mk_le_max
theorem cardinal_lift_mk_le_max {σ : Type u} {R : Type v} [CommSemiring R] : #(MvPolynomial σ R) ≤
    max (max (Cardinal.lift.{u} #R) <| Cardinal.lift.{v} #σ) ℵ₀ 
Arguments: {σ : Type u} {R : Type v} [CommSemiring R]
Goal: #(MvPolynomial σ R) ≤ max (max (Cardinal.lift.{u, v} #R) (Cardinal.lift.{v, u} #σ)) ℵ₀
-- MvPolynomial.cardinal_mk_eq_max
theorem cardinal_mk_eq_max [Nonempty σ] [Nontrivial R] :
    #(MvPolynomial σ R) = max (max #R #σ) ℵ₀ 
Arguments: {σ R : Type u} [CommSemiring R] [Nonempty σ] [Nontrivial R]
Goal: #(MvPolynomial σ R) = max (max #R #σ) ℵ₀
-- MvPolynomial.cardinal_mk_le_max
theorem cardinal_mk_le_max : #(MvPolynomial σ R) ≤ max (max #R #σ) ℵ₀ 
Arguments: {σ R : Type u} [CommSemiring R]
Goal: #(MvPolynomial σ R) ≤ max (max #R #σ) ℵ₀
-- Submodule.coe_set_neg
@[simp]
theorem coe_set_neg (S : Submodule R M) : ↑(-S) = -(S : Set M) 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommGroup M] [Module R M] (S : Submodule R M)
Goal: ↑(-S) = -↑S
-- Submodule.neg_toAddSubmonoid
@[simp]
theorem neg_toAddSubmonoid (S : Submodule R M) : (-S).toAddSubmonoid = -S.toAddSubmonoid 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommGroup M] [Module R M] (S : Submodule R M)
Goal: (-S).toAddSubmonoid = -S.toAddSubmonoid
-- Submodule.mem_neg
@[simp]
theorem mem_neg {g : M} {S : Submodule R M} : g ∈ -S ↔ -g ∈ S 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommGroup M] [Module R M] {g : M} {S : Submodule R M}
Goal: g ∈ -S ↔ -g ∈ S
-- Submodule.neg_le_neg
@[simp]
theorem neg_le_neg (S T : Submodule R M) : -S ≤ -T ↔ S ≤ T 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommGroup M] [Module R M] (S T : Submodule R M)
Goal: -S ≤ -T ↔ S ≤ T
-- Submodule.neg_le
theorem neg_le (S T : Submodule R M) : -S ≤ T ↔ S ≤ -T 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommGroup M] [Module R M] (S T : Submodule R M)
Goal: -S ≤ T ↔ S ≤ -T
-- Submodule.closure_neg
theorem closure_neg (s : Set M) : span R (-s) = -span R s 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommGroup M] [Module R M] (s : Set M)
Goal: Submodule.span R (-s) = -Submodule.span R s
-- Submodule.neg_inf
@[simp]
theorem neg_inf (S T : Submodule R M) : -(S ⊓ T) = -S ⊓ -T 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommGroup M] [Module R M] (S T : Submodule R M)
Goal: -(S ⊓ T) = -S ⊓ -T
-- Submodule.neg_sup
@[simp]
theorem neg_sup (S T : Submodule R M) : -(S ⊔ T) = -S ⊔ -T 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommGroup M] [Module R M] (S T : Submodule R M)
Goal: -(S ⊔ T) = -S ⊔ -T
-- Submodule.neg_bot
@[simp]
theorem neg_bot : -(⊥ : Submodule R M) = ⊥ 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommGroup M] [Module R M]
Goal: -⊥ = ⊥
-- Submodule.neg_top
@[simp]
theorem neg_top : -(⊤ : Submodule R M) = ⊤ 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommGroup M] [Module R M]
Goal: -⊤ = ⊤
-- Submodule.neg_iInf
@[simp]
theorem neg_iInf {ι : Sort*} (S : ι → Submodule R M) : (-⨅ i, S i) = ⨅ i, -S i 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommGroup M] [Module R M] {ι : Sort u_4} (S : ι → Submodule R M)
Goal: -⨅ i, S i = ⨅ i, -S i
-- Submodule.neg_iSup
@[simp]
theorem neg_iSup {ι : Sort*} (S : ι → Submodule R M) : (-⨆ i, S i) = ⨆ i, -S i 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommGroup M] [Module R M] {ι : Sort u_4} (S : ι → Submodule R M)
Goal: -⨆ i, S i = ⨆ i, -S i
-- Submodule.neg_eq_self
@[simp]
theorem neg_eq_self [Ring R] [AddCommGroup M] [Module R M] (p : Submodule R M) : -p = p 
Arguments: {R : Type u_2} {M : Type u_3} [Ring R] [AddCommGroup M] [Module R M] (p : Submodule R M)
Goal: -p = p
-- Submodule.add_eq_sup
@[simp]
theorem add_eq_sup (p q : Submodule R M) : p + q = p ⊔ q 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] (p q : Submodule R M)
Goal: p + q = p ⊔ q
-- Submodule.zero_eq_bot
@[simp, nolint simpNF]
theorem zero_eq_bot : (0 : Submodule R M) = ⊥ 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: 0 = ⊥
-- Submodule.coe_pointwise_smul
@[simp]
theorem coe_pointwise_smul (a : α) (S : Submodule R M) : ↑(a • S) = a • (S : Set M) 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid α] [DistribMulAction α M] [SMulCommClass α R M] (a : α) (S : Submodule R M)
Goal: ↑(a • S) = a • ↑S
-- Submodule.pointwise_smul_toAddSubmonoid
@[simp]
theorem pointwise_smul_toAddSubmonoid (a : α) (S : Submodule R M) :
    (a • S).toAddSubmonoid = a • S.toAddSubmonoid 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid α] [DistribMulAction α M] [SMulCommClass α R M] (a : α) (S : Submodule R M)
Goal: (a • S).toAddSubmonoid = a • S.toAddSubmonoid
-- Submodule.pointwise_smul_toAddSubgroup
@[simp]
theorem pointwise_smul_toAddSubgroup {R M : Type*} [Ring R] [AddCommGroup M] [DistribMulAction α M]
    [Module R M] [SMulCommClass α R M] (a : α) (S : Submodule R M) :
    (a • S).toAddSubgroup = a • S.toAddSubgroup 
Arguments: {α : Type u_1} [Monoid α] {R : Type u_4} {M : Type u_5} [Ring R] [AddCommGroup M] [DistribMulAction α M] [Module R M] [SMulCommClass α R M] (a : α) (S : Submodule R M)
Goal: (a • S).toAddSubgroup = a • S.toAddSubgroup
-- Submodule.smul_mem_pointwise_smul
theorem smul_mem_pointwise_smul (m : M) (a : α) (S : Submodule R M) : m ∈ S → a • m ∈ a • S 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid α] [DistribMulAction α M] [SMulCommClass α R M] (m : M) (a : α) (S : Submodule R M)
Goal: m ∈ S → a • m ∈ a • S
-- Submodule.smul_bot'
@[simp]
theorem smul_bot' (a : α) : a • (⊥ : Submodule R M) = ⊥ 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid α] [DistribMulAction α M] [SMulCommClass α R M] (a : α)
Goal: a • ⊥ = ⊥
-- Submodule.smul_sup'
theorem smul_sup' (a : α) (S T : Submodule R M) : a • (S ⊔ T) = a • S ⊔ a • T 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid α] [DistribMulAction α M] [SMulCommClass α R M] (a : α) (S T : Submodule R M)
Goal: a • (S ⊔ T) = a • S ⊔ a • T
-- Submodule.smul_span
theorem smul_span (a : α) (s : Set M) : a • span R s = span R (a • s) 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid α] [DistribMulAction α M] [SMulCommClass α R M] (a : α) (s : Set M)
Goal: a • Submodule.span R s = Submodule.span R (a • s)
-- Submodule.span_smul
theorem span_smul (a : α) (s : Set M) : span R (a • s) = a • span R s 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid α] [DistribMulAction α M] [SMulCommClass α R M] (a : α) (s : Set M)
Goal: Submodule.span R (a • s) = a • Submodule.span R s
-- Submodule.smul_le_self_of_tower
@[simp]
theorem smul_le_self_of_tower {α : Type*} [Semiring α] [Module α R] [Module α M]
    [SMulCommClass α R M] [IsScalarTower α R M] (a : α) (S : Submodule R M) : a • S ≤ S 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] {α : Type u_4} [Semiring α] [Module α R] [Module α M] [SMulCommClass α R M] [IsScalarTower α R M] (a : α) (S : Submodule R M)
Goal: a • S ≤ S
-- Submodule.mem_set_smul_def
lemma mem_set_smul_def (x : M) :
    x ∈ s • N ↔
  x ∈ sInf { p : Submodule R M | ∀ ⦃r : S⦄ {n : M}, r ∈ s → n ∈ N → r • n ∈ p } 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] (s : Set S) (N : Submodule R M) (x : M)
Goal: x ∈ s • N ↔ x ∈ sInf {p | ∀ ⦃r : S⦄ {n : M}, r ∈ s → n ∈ N → r • n ∈ p}
-- Submodule.mem_set_smul_of_mem_mem
@[aesop safe]
lemma mem_set_smul_of_mem_mem {r : S} {m : M} (mem1 : r ∈ s) (mem2 : m ∈ N) :
    r • m ∈ s • N 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] {s : Set S} {N : Submodule R M} {r : S} {m : M} (mem1 : r ∈ s) (mem2 : m ∈ N)
Goal: r • m ∈ s • N
-- Submodule.set_smul_le
lemma set_smul_le (p : Submodule R M)
    (closed_under_smul : ∀ ⦃r : S⦄ ⦃n : M⦄, r ∈ s → n ∈ N → r • n ∈ p) :
    s • N ≤ p 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] (s : Set S) (N p : Submodule R M) (closed_under_smul : ∀ ⦃r : S⦄ ⦃n : M⦄, r ∈ s → n ∈ N → r • n ∈ p)
Goal: s • N ≤ p
-- Submodule.set_smul_le_iff
lemma set_smul_le_iff (p : Submodule R M) :
    s • N ≤ p ↔
    ∀ ⦃r : S⦄ ⦃n : M⦄, r ∈ s → n ∈ N → r • n ∈ p 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] (s : Set S) (N p : Submodule R M)
Goal: s • N ≤ p ↔ ∀ ⦃r : S⦄ ⦃n : M⦄, r ∈ s → n ∈ N → r • n ∈ p
-- Submodule.set_smul_eq_of_le
lemma set_smul_eq_of_le (p : Submodule R M)
    (closed_under_smul : ∀ ⦃r : S⦄ ⦃n : M⦄, r ∈ s → n ∈ N → r • n ∈ p)
    (le : p ≤ s • N) :
    s • N = p 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] (s : Set S) (N p : Submodule R M) (closed_under_smul : ∀ ⦃r : S⦄ ⦃n : M⦄, r ∈ s → n ∈ N → r • n ∈ p) (le : p ≤ s • N)
Goal: s • N = p
-- Submodule.set_smul_mono_right
@[deprecated smul_mono_right (since := "2024-03-31")]
theorem set_smul_mono_right {p q : Submodule R M} (le : p ≤ q) :
    s • p ≤ s • q 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] (s : Set S) {p q : Submodule R M} (le : p ≤ q)
Goal: s • p ≤ s • q
-- Submodule.set_smul_mono_left
lemma set_smul_mono_left {s t : Set S} (le : s ≤ t) :
    s • N ≤ t • N 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] (N : Submodule R M) {s t : Set S} (le : s ≤ t)
Goal: s • N ≤ t • N
-- Submodule.set_smul_le_of_le_le
lemma set_smul_le_of_le_le {s t : Set S} {p q : Submodule R M}
    (le_set : s ≤ t) (le_submodule : p ≤ q) : s • p ≤ t • q 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] {s t : Set S} {p q : Submodule R M} (le_set : s ≤ t) (le_submodule : p ≤ q)
Goal: s • p ≤ t • q
-- Submodule.set_smul_eq_iSup
lemma set_smul_eq_iSup [SMulCommClass S R M] (s : Set S) (N : Submodule R M) :
    s • N = ⨆ (a ∈ s), a • N 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] [SMulCommClass S R M] (s : Set S) (N : Submodule R M)
Goal: s • N = ⨆ a ∈ s, a • N
-- Submodule.set_smul_span
theorem set_smul_span [SMulCommClass S R M] (s : Set S) (t : Set M) :
    s • span R t = span R (s • t) 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] [SMulCommClass S R M] (s : Set S) (t : Set M)
Goal: s • Submodule.span R t = Submodule.span R (s • t)
-- Submodule.span_set_smul
theorem span_set_smul [SMulCommClass S R M] (s : Set S) (t : Set M) :
    span R (s • t) = s • span R t 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] [SMulCommClass S R M] (s : Set S) (t : Set M)
Goal: Submodule.span R (s • t) = s • Submodule.span R t
-- Submodule.set_smul_inductionOn
@[elab_as_elim]
lemma set_smul_inductionOn {motive : (x : M) → (_ : x ∈ s • N) → Prop}
    (x : M)
    (hx : x ∈ s • N)
    (smul₀ : ∀ ⦃r : S⦄ ⦃n : M⦄ (mem₁ : r ∈ s) (mem₂ : n ∈ N),
      motive (r • n) (mem_set_smul_of_mem_mem mem₁ mem₂))
    (smul₁ : ∀ (r : R) ⦃m : M⦄ (mem : m ∈ s • N) ,
      motive m mem → motive (r • m) (Submodule.smul_mem _ r mem)) --
    (add : ∀ ⦃m₁ m₂ : M⦄ (mem₁ : m₁ ∈ s • N) (mem₂ : m₂ ∈ s • N),
      motive m₁ mem₁ → motive m₂ mem₂ → motive (m₁ + m₂) (Submodule.add_mem _ mem₁ mem₂))
    (zero : motive 0 (Submodule.zero_mem _)) :
    motive x hx 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] {s : Set S} {N : Submodule R M} {motive : (x : M) → x ∈ s • N → Prop} (x : M) (hx : x ∈ s • N) (smul₀ : ∀ ⦃r : S⦄ ⦃n : M⦄ (mem₁ : r ∈ s) (mem₂ : n ∈ N), motive (r • n) ⋯) (smul₁ : ∀ (r : R) ⦃m : M⦄ (mem : m ∈ s • N), motive m mem → motive (r • m) ⋯) (add : ∀ ⦃m₁ m₂ : M⦄ (mem₁ : m₁ ∈ s • N) (mem₂ : m₂ ∈ s • N), motive m₁ mem₁ → motive m₂ mem₂ → motive (m₁ + m₂) ⋯) (zero : motive 0 ⋯)
Goal: motive x hx
-- Submodule.set_smul_eq_map
lemma set_smul_eq_map [SMulCommClass R R N] :
    sR • N =
    Submodule.map
      (N.subtype.comp (Finsupp.lsum R <| DistribMulAction.toLinearMap _ _))
      (Finsupp.supported N R sR) 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] (sR : Set R) (N : Submodule R M) [SMulCommClass R R ↥N]
Goal: sR • N = Submodule.map (N.subtype ∘ₗ (Finsupp.lsum R) (DistribMulAction.toLinearMap R ↥N)) (Finsupp.supported (↥N) R sR)
-- Submodule.mem_set_smul
lemma mem_set_smul (x : M) [SMulCommClass R R N] :
    x ∈ sR • N ↔ ∃ (c : R →₀ N), (c.support : Set R) ⊆ sR ∧ x = c.sum fun r m ↦ r • m 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] (sR : Set R) (N : Submodule R M) (x : M) [SMulCommClass R R ↥N]
Goal: x ∈ sR • N ↔ ∃ c, ↑c.support ⊆ sR ∧ x = ↑(c.sum fun r m => r • m)
-- Submodule.empty_set_smul
@[simp] lemma empty_set_smul : (∅ : Set S) • N = ⊥ 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] (N : Submodule R M)
Goal: ∅ • N = ⊥
-- Submodule.set_smul_bot
@[simp] lemma set_smul_bot : s • (⊥ : Submodule R M) = ⊥ 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] (s : Set S)
Goal: s • ⊥ = ⊥
-- Submodule.singleton_set_smul
lemma singleton_set_smul [SMulCommClass S R M] (r : S) : ({r} : Set S) • N = r • N 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] (N : Submodule R M) [SMulCommClass S R M] (r : S)
Goal: {r} • N = r • N
-- Submodule.mem_singleton_set_smul
lemma mem_singleton_set_smul [SMulCommClass R S M] (r : S) (x : M) :
    x ∈ ({r} : Set S) • N ↔ ∃ (m : M), m ∈ N ∧ x = r • m 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] (N : Submodule R M) [SMulCommClass R S M] (r : S) (x : M)
Goal: x ∈ {r} • N ↔ ∃ m ∈ N, x = r • m
-- Submodule.sup_set_smul
lemma sup_set_smul (s t : Set S) :
    (s ⊔ t) • N = s • N ⊔ t • N 
Arguments: {R : Type u_2} {M : Type u_3} [Semiring R] {S : Type u_4} [Monoid S] [AddCommMonoid M] [Module R M] [DistribMulAction S M] (N : Submodule R M) (s t : Set S)
Goal: (s ⊔ t) • N = s • N ⊔ t • N
-- Submodule.coe_span_smul
lemma coe_span_smul {R' M' : Type*} [CommSemiring R'] [AddCommMonoid M'] [Module R' M']
    (s : Set R') (N : Submodule R' M') :
    (Ideal.span s : Set R') • N = s • N 
Arguments: {R' : Type u_5} {M' : Type u_6} [CommSemiring R'] [AddCommMonoid M'] [Module R' M'] (s : Set R') (N : Submodule R' M')
Goal: ↑(Ideal.span s) • N = s • N
-- MvPolynomial.supported_eq_range_rename
theorem supported_eq_range_rename (s : Set σ) : supported R s = (rename ((↑) : s → σ)).range 
Arguments: {σ : Type u_1} {R : Type u} [CommSemiring R] (s : Set σ)
Goal: MvPolynomial.supported R s = (MvPolynomial.rename Subtype.val).range
-- MvPolynomial.supportedEquivMvPolynomial_symm_C
@[simp]
theorem supportedEquivMvPolynomial_symm_C (s : Set σ) (x : R) :
    (supportedEquivMvPolynomial s).symm (C x) = algebraMap R (supported R s) x 
Arguments: {σ : Type u_1} {R : Type u} [CommSemiring R] (s : Set σ) (x : R)
Goal: (MvPolynomial.supportedEquivMvPolynomial s).symm (MvPolynomial.C x) = (algebraMap R ↥(MvPolynomial.supported R s)) x
-- MvPolynomial.supportedEquivMvPolynomial_symm_X
@[simp]
theorem supportedEquivMvPolynomial_symm_X (s : Set σ) (i : s) :
    (↑((supportedEquivMvPolynomial s).symm (X i : MvPolynomial s R)) : MvPolynomial σ R) = X ↑i 
Arguments: {σ : Type u_1} {R : Type u} [CommSemiring R] (s : Set σ) (i : ↑s)
Goal: ↑((MvPolynomial.supportedEquivMvPolynomial s).symm (MvPolynomial.X i)) = MvPolynomial.X ↑i
-- MvPolynomial.mem_supported
theorem mem_supported : p ∈ supported R s ↔ ↑p.vars ⊆ s 
Arguments: {σ : Type u_1} {R : Type u} [CommSemiring R] {p : MvPolynomial σ R} {s : Set σ}
Goal: p ∈ MvPolynomial.supported R s ↔ ↑p.vars ⊆ s
-- MvPolynomial.supported_eq_vars_subset
theorem supported_eq_vars_subset : (supported R s : Set (MvPolynomial σ R)) = { p | ↑p.vars ⊆ s } 
Arguments: {σ : Type u_1} {R : Type u} [CommSemiring R] {s : Set σ}
Goal: ↑(MvPolynomial.supported R s) = {p | ↑p.vars ⊆ s}
-- MvPolynomial.mem_supported_vars
@[simp]
theorem mem_supported_vars (p : MvPolynomial σ R) : p ∈ supported R (↑p.vars : Set σ) 
Arguments: {σ : Type u_1} {R : Type u} [CommSemiring R] (p : MvPolynomial σ R)
Goal: p ∈ MvPolynomial.supported R ↑p.vars
-- MvPolynomial.supported_eq_adjoin_X
theorem supported_eq_adjoin_X : supported R s = Algebra.adjoin R (X '' s) 
Arguments: {σ : Type u_1} {R : Type u} [CommSemiring R] (s : Set σ)
Goal: MvPolynomial.supported R s = Algebra.adjoin R (MvPolynomial.X '' s)
-- MvPolynomial.supported_univ
@[simp]
theorem supported_univ : supported R (Set.univ : Set σ) = ⊤ 
Arguments: {σ : Type u_1} {R : Type u} [CommSemiring R]
Goal: MvPolynomial.supported R Set.univ = ⊤
-- MvPolynomial.supported_empty
@[simp]
theorem supported_empty : supported R (∅ : Set σ) = ⊥ 
Arguments: {σ : Type u_1} {R : Type u} [CommSemiring R]
Goal: MvPolynomial.supported R ∅ = ⊥
-- MvPolynomial.supported_mono
theorem supported_mono (st : s ⊆ t) : supported R s ≤ supported R t 
Arguments: {σ : Type u_1} {R : Type u} [CommSemiring R] {s t : Set σ} (st : s ⊆ t)
Goal: MvPolynomial.supported R s ≤ MvPolynomial.supported R t
-- MvPolynomial.X_mem_supported
@[simp]
theorem X_mem_supported [Nontrivial R] {i : σ} : X i ∈ supported R s ↔ i ∈ s 
Arguments: {σ : Type u_1} {R : Type u} [CommSemiring R] {s : Set σ} [Nontrivial R] {i : σ}
Goal: MvPolynomial.X i ∈ MvPolynomial.supported R s ↔ i ∈ s
-- MvPolynomial.supported_le_supported_iff
@[simp]
theorem supported_le_supported_iff [Nontrivial R] : supported R s ≤ supported R t ↔ s ⊆ t 
Arguments: {σ : Type u_1} {R : Type u} [CommSemiring R] {s t : Set σ} [Nontrivial R]
Goal: MvPolynomial.supported R s ≤ MvPolynomial.supported R t ↔ s ⊆ t
-- MvPolynomial.supported_strictMono
theorem supported_strictMono [Nontrivial R] :
    StrictMono (supported R : Set σ → Subalgebra R (MvPolynomial σ R)) 
Arguments: {σ : Type u_1} {R : Type u} [CommSemiring R] [Nontrivial R]
Goal: StrictMono (MvPolynomial.supported R)
-- MvPolynomial.exists_restrict_to_vars
theorem exists_restrict_to_vars (R : Type*) [CommRing R] {F : MvPolynomial σ ℤ}
    (hF : ↑F.vars ⊆ s) : ∃ f : (s → R) → R, ∀ x : σ → R, f (x ∘ (↑) : s → R) = aeval x F 
Arguments: {σ : Type u_1} {s : Set σ} (R : Type u_3) [CommRing R] {F : MvPolynomial σ ℤ} (hF : ↑F.vars ⊆ s)
Goal: ∃ f, ∀ (x : σ → R), f (x ∘ Subtype.val) = (MvPolynomial.aeval x) F
-- MvPolynomial.pderiv_def
theorem pderiv_def [DecidableEq σ] (i : σ) : pderiv i = mkDerivation R (Pi.single i 1) 
Arguments: {R : Type u} {σ : Type v} [CommSemiring R] [DecidableEq σ] (i : σ)
Goal: MvPolynomial.pderiv i = MvPolynomial.mkDerivation R (Pi.single i 1)
-- MvPolynomial.pderiv_monomial
@[simp]
theorem pderiv_monomial {i : σ} :
    pderiv i (monomial s a) = monomial (s - single i 1) (a * s i) 
Arguments: {R : Type u} {σ : Type v} {a : R} {s : σ →₀ ℕ} [CommSemiring R] {i : σ}
Goal: (MvPolynomial.pderiv i) ((MvPolynomial.monomial s) a) = (MvPolynomial.monomial (s - Finsupp.single i 1)) (a * ↑(s i))
-- MvPolynomial.pderiv_C
theorem pderiv_C {i : σ} : pderiv i (C a) = 0 
Arguments: {R : Type u} {σ : Type v} {a : R} [CommSemiring R] {i : σ}
Goal: (MvPolynomial.pderiv i) (MvPolynomial.C a) = 0
-- MvPolynomial.pderiv_one
theorem pderiv_one {i : σ} : pderiv i (1 : MvPolynomial σ R) = 0 
Arguments: {R : Type u} {σ : Type v} [CommSemiring R] {i : σ}
Goal: (MvPolynomial.pderiv i) 1 = 0
-- MvPolynomial.pderiv_X
@[simp]
theorem pderiv_X [DecidableEq σ] (i j : σ) :
    pderiv i (X j : MvPolynomial σ R) = Pi.single (f := fun j => _) i 1 j 
Arguments: {R : Type u} {σ : Type v} [CommSemiring R] [DecidableEq σ] (i j : σ)
Goal: (MvPolynomial.pderiv i) (MvPolynomial.X j) = Pi.single i 1 j
-- MvPolynomial.pderiv_X_self
@[simp]
theorem pderiv_X_self (i : σ) : pderiv i (X i : MvPolynomial σ R) = 1 
Arguments: {R : Type u} {σ : Type v} [CommSemiring R] (i : σ)
Goal: (MvPolynomial.pderiv i) (MvPolynomial.X i) = 1
-- MvPolynomial.pderiv_X_of_ne
@[simp]
theorem pderiv_X_of_ne {i j : σ} (h : j ≠ i) : pderiv i (X j : MvPolynomial σ R) = 0 
Arguments: {R : Type u} {σ : Type v} [CommSemiring R] {i j : σ} (h : j ≠ i)
Goal: (MvPolynomial.pderiv i) (MvPolynomial.X j) = 0
-- MvPolynomial.pderiv_eq_zero_of_not_mem_vars
theorem pderiv_eq_zero_of_not_mem_vars {i : σ} {f : MvPolynomial σ R} (h : i ∉ f.vars) :
    pderiv i f = 0 
Arguments: {R : Type u} {σ : Type v} [CommSemiring R] {i : σ} {f : MvPolynomial σ R} (h : i ∉ f.vars)
Goal: (MvPolynomial.pderiv i) f = 0
-- MvPolynomial.pderiv_monomial_single
theorem pderiv_monomial_single {i : σ} {n : ℕ} : pderiv i (monomial (single i n) a) =
    monomial (single i (n - 1)) (a * n) 
Arguments: {R : Type u} {σ : Type v} {a : R} [CommSemiring R] {i : σ} {n : ℕ}
Goal: (MvPolynomial.pderiv i) ((MvPolynomial.monomial (Finsupp.single i n)) a) =
  (MvPolynomial.monomial (Finsupp.single i (n - 1))) (a * ↑n)