ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CliffordAlgebra.submodule_map_involute_eq_comap
theorem submodule_map_involute_eq_comap (p : Submodule R (CliffordAlgebra Q)) :
    p.map (involute : CliffordAlgebra Q →ₐ[R] CliffordAlgebra Q).toLinearMap =
      p.comap (involute : CliffordAlgebra Q →ₐ[R] CliffordAlgebra Q).toLinearMap 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (p : Submodule R (CliffordAlgebra Q))
Goal: Submodule.map CliffordAlgebra.involute.toLinearMap p = Submodule.comap CliffordAlgebra.involute.toLinearMap p
-- CliffordAlgebra.ι_range_map_involute
@[simp]
theorem ι_range_map_involute :
    (ι Q).range.map (involute : CliffordAlgebra Q →ₐ[R] CliffordAlgebra Q).toLinearMap =
      LinearMap.range (ι Q) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M)
Goal: Submodule.map CliffordAlgebra.involute.toLinearMap (LinearMap.range (CliffordAlgebra.ι Q)) =
  LinearMap.range (CliffordAlgebra.ι Q)
-- CliffordAlgebra.ι_range_comap_involute
@[simp]
theorem ι_range_comap_involute :
    (ι Q).range.comap (involute : CliffordAlgebra Q →ₐ[R] CliffordAlgebra Q).toLinearMap =
      LinearMap.range (ι Q) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M)
Goal: Submodule.comap CliffordAlgebra.involute.toLinearMap (LinearMap.range (CliffordAlgebra.ι Q)) =
  LinearMap.range (CliffordAlgebra.ι Q)
-- CliffordAlgebra.evenOdd_map_involute
@[simp]
theorem evenOdd_map_involute (n : ZMod 2) :
    (evenOdd Q n).map (involute : CliffordAlgebra Q →ₐ[R] CliffordAlgebra Q).toLinearMap =
      evenOdd Q n 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (n : ZMod 2)
Goal: Submodule.map CliffordAlgebra.involute.toLinearMap (CliffordAlgebra.evenOdd Q n) = CliffordAlgebra.evenOdd Q n
-- CliffordAlgebra.evenOdd_comap_involute
@[simp]
theorem evenOdd_comap_involute (n : ZMod 2) :
    (evenOdd Q n).comap (involute : CliffordAlgebra Q →ₐ[R] CliffordAlgebra Q).toLinearMap =
      evenOdd Q n 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (n : ZMod 2)
Goal: Submodule.comap CliffordAlgebra.involute.toLinearMap (CliffordAlgebra.evenOdd Q n) = CliffordAlgebra.evenOdd Q n
-- CliffordAlgebra.submodule_map_reverse_eq_comap
theorem submodule_map_reverse_eq_comap (p : Submodule R (CliffordAlgebra Q)) :
    p.map (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) =
      p.comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (p : Submodule R (CliffordAlgebra Q))
Goal: Submodule.map CliffordAlgebra.reverse p = Submodule.comap CliffordAlgebra.reverse p
-- CliffordAlgebra.ι_range_map_reverse
@[simp]
theorem ι_range_map_reverse :
    (ι Q).range.map (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q)
      = LinearMap.range (ι Q) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M)
Goal: Submodule.map CliffordAlgebra.reverse (LinearMap.range (CliffordAlgebra.ι Q)) = LinearMap.range (CliffordAlgebra.ι Q)
-- CliffordAlgebra.ι_range_comap_reverse
@[simp]
theorem ι_range_comap_reverse :
    (ι Q).range.comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q)
      = LinearMap.range (ι Q) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M)
Goal: Submodule.comap CliffordAlgebra.reverse (LinearMap.range (CliffordAlgebra.ι Q)) = LinearMap.range (CliffordAlgebra.ι Q)
-- CliffordAlgebra.submodule_map_mul_reverse
theorem submodule_map_mul_reverse (p q : Submodule R (CliffordAlgebra Q)) :
    (p * q).map (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) =
      q.map (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) *
        p.map (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (p q : Submodule R (CliffordAlgebra Q))
Goal: Submodule.map CliffordAlgebra.reverse (p * q) =
  Submodule.map CliffordAlgebra.reverse q * Submodule.map CliffordAlgebra.reverse p
-- CliffordAlgebra.submodule_comap_mul_reverse
theorem submodule_comap_mul_reverse (p q : Submodule R (CliffordAlgebra Q)) :
    (p * q).comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) =
      q.comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) *
        p.comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (p q : Submodule R (CliffordAlgebra Q))
Goal: Submodule.comap CliffordAlgebra.reverse (p * q) =
  Submodule.comap CliffordAlgebra.reverse q * Submodule.comap CliffordAlgebra.reverse p
-- CliffordAlgebra.submodule_map_pow_reverse
theorem submodule_map_pow_reverse (p : Submodule R (CliffordAlgebra Q)) (n : ℕ) :
    (p ^ n).map (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) =
      p.map (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) ^ n 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (p : Submodule R (CliffordAlgebra Q)) (n : ℕ)
Goal: Submodule.map CliffordAlgebra.reverse (p ^ n) = Submodule.map CliffordAlgebra.reverse p ^ n
-- CliffordAlgebra.submodule_comap_pow_reverse
theorem submodule_comap_pow_reverse (p : Submodule R (CliffordAlgebra Q)) (n : ℕ) :
    (p ^ n).comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) =
      p.comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) ^ n 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (p : Submodule R (CliffordAlgebra Q)) (n : ℕ)
Goal: Submodule.comap CliffordAlgebra.reverse (p ^ n) = Submodule.comap CliffordAlgebra.reverse p ^ n
-- CliffordAlgebra.evenOdd_map_reverse
@[simp]
theorem evenOdd_map_reverse (n : ZMod 2) :
    (evenOdd Q n).map (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) = evenOdd Q n 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (n : ZMod 2)
Goal: Submodule.map CliffordAlgebra.reverse (CliffordAlgebra.evenOdd Q n) = CliffordAlgebra.evenOdd Q n
-- CliffordAlgebra.evenOdd_comap_reverse
@[simp]
theorem evenOdd_comap_reverse (n : ZMod 2) :
    (evenOdd Q n).comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) = evenOdd Q n 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (n : ZMod 2)
Goal: Submodule.comap CliffordAlgebra.reverse (CliffordAlgebra.evenOdd Q n) = CliffordAlgebra.evenOdd Q n
-- CliffordAlgebra.involute_mem_evenOdd_iff
@[simp]
theorem involute_mem_evenOdd_iff {x : CliffordAlgebra Q} {n : ZMod 2} :
    involute x ∈ evenOdd Q n ↔ x ∈ evenOdd Q n 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) {x : CliffordAlgebra Q} {n : ZMod 2}
Goal: CliffordAlgebra.involute x ∈ CliffordAlgebra.evenOdd Q n ↔ x ∈ CliffordAlgebra.evenOdd Q n
-- CliffordAlgebra.reverse_mem_evenOdd_iff
@[simp]
theorem reverse_mem_evenOdd_iff {x : CliffordAlgebra Q} {n : ZMod 2} :
    reverse x ∈ evenOdd Q n ↔ x ∈ evenOdd Q n 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) {x : CliffordAlgebra Q} {n : ZMod 2}
Goal: CliffordAlgebra.reverse x ∈ CliffordAlgebra.evenOdd Q n ↔ x ∈ CliffordAlgebra.evenOdd Q n
-- CliffordAlgebra.involute_eq_of_mem_even
theorem involute_eq_of_mem_even {x : CliffordAlgebra Q} (h : x ∈ evenOdd Q 0) : involute x = x 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {x : CliffordAlgebra Q} (h : x ∈ CliffordAlgebra.evenOdd Q 0)
Goal: CliffordAlgebra.involute x = x
-- CliffordAlgebra.involute_eq_of_mem_odd
theorem involute_eq_of_mem_odd {x : CliffordAlgebra Q} (h : x ∈ evenOdd Q 1) : involute x = -x 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {x : CliffordAlgebra Q} (h : x ∈ CliffordAlgebra.evenOdd Q 1)
Goal: CliffordAlgebra.involute x = -x
-- QuadraticForm.equivalent_sum_squares
theorem equivalent_sum_squares {M : Type*} [AddCommGroup M] [Module ℂ M] [FiniteDimensional ℂ M]
    (Q : QuadraticForm ℂ M) (hQ : (associated (R := ℂ) Q).SeparatingLeft) :
    Equivalent Q (weightedSumSquares ℂ (1 : Fin (FiniteDimensional.finrank ℂ M) → ℂ)) 
Arguments: {M : Type u_2} [AddCommGroup M] [Module ℂ M] [FiniteDimensional ℂ M] (Q : QuadraticForm ℂ M) (hQ : LinearMap.SeparatingLeft (QuadraticMap.associated Q))
Goal: QuadraticMap.Equivalent Q (QuadraticMap.weightedSumSquares ℂ 1)
-- QuadraticForm.complex_equivalent
theorem complex_equivalent {M : Type*} [AddCommGroup M] [Module ℂ M] [FiniteDimensional ℂ M]
    (Q₁ Q₂ : QuadraticForm ℂ M) (hQ₁ : (associated (R := ℂ) Q₁).SeparatingLeft)
    (hQ₂ : (associated (R := ℂ) Q₂).SeparatingLeft) : Equivalent Q₁ Q₂ 
Arguments: {M : Type u_2} [AddCommGroup M] [Module ℂ M] [FiniteDimensional ℂ M] (Q₁ Q₂ : QuadraticForm ℂ M) (hQ₁ : LinearMap.SeparatingLeft (QuadraticMap.associated Q₁)) (hQ₂ : LinearMap.SeparatingLeft (QuadraticMap.associated Q₂))
Goal: QuadraticMap.Equivalent Q₁ Q₂
-- Ideal.smul_eq_mul
protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J 
Arguments: {R : Type u} [CommSemiring R] (I J : Ideal R)
Goal: I • J = I * J
-- Module.mem_annihilator
theorem _root_.Module.mem_annihilator {r} : r ∈ Module.annihilator R M ↔ ∀ m : M, r • m = 0 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {r : R}
Goal: r ∈ Module.annihilator R M ↔ ∀ (m : M), r • m = 0
-- LinearMap.annihilator_le_of_injective
theorem _root_.LinearMap.annihilator_le_of_injective (f : M →ₗ[R] M') (hf : Function.Injective f) :
    Module.annihilator R M' ≤ Module.annihilator R M 
Arguments: {R : Type u} {M : Type v} {M' : Type u_1} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (f : M →ₗ[R] M') (hf : Function.Injective ⇑f)
Goal: Module.annihilator R M' ≤ Module.annihilator R M
-- LinearMap.annihilator_le_of_surjective
theorem _root_.LinearMap.annihilator_le_of_surjective (f : M →ₗ[R] M')
    (hf : Function.Surjective f) : Module.annihilator R M ≤ Module.annihilator R M' 
Arguments: {R : Type u} {M : Type v} {M' : Type u_1} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (f : M →ₗ[R] M') (hf : Function.Surjective ⇑f)
Goal: Module.annihilator R M ≤ Module.annihilator R M'
-- LinearEquiv.annihilator_eq
theorem _root_.LinearEquiv.annihilator_eq (e : M ≃ₗ[R] M') :
    Module.annihilator R M = Module.annihilator R M' 
Arguments: {R : Type u} {M : Type v} {M' : Type u_1} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (e : M ≃ₗ[R] M')
Goal: Module.annihilator R M = Module.annihilator R M'
-- Submodule.annihilator_top
theorem annihilator_top : (⊤ : Submodule R M).annihilator = Module.annihilator R M 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M]
Goal: ⊤.annihilator = Module.annihilator R M
-- Submodule.mem_annihilator
theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M} {r : R}
Goal: r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = 0
-- Submodule.mem_annihilator'
theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M} {r : R}
Goal: r ∈ N.annihilator ↔ N ≤ Submodule.comap (r • LinearMap.id) ⊥
-- Submodule.mem_annihilator_span
theorem mem_annihilator_span (s : Set M) (r : R) :
    r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (s : Set M) (r : R)
Goal: r ∈ (Submodule.span R s).annihilator ↔ ∀ (n : ↑s), r • ↑n = 0
-- Submodule.mem_annihilator_span_singleton
theorem mem_annihilator_span_singleton (g : M) (r : R) :
    r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (g : M) (r : R)
Goal: r ∈ (Submodule.span R {g}).annihilator ↔ r • g = 0
-- Submodule.annihilator_bot
theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M]
Goal: ⊥.annihilator = ⊤
-- Submodule.annihilator_eq_top_iff
theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M}
Goal: N.annihilator = ⊤ ↔ N = ⊥
-- Submodule.annihilator_mono
theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {N P : Submodule R M} (h : N ≤ P)
Goal: P.annihilator ≤ N.annihilator
-- Submodule.annihilator_iSup
theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) :
    annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (ι : Sort w) (f : ι → Submodule R M)
Goal: (⨆ i, f i).annihilator = ⨅ i, (f i).annihilator
-- Submodule.smul_mem_smul
theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {I : Ideal R} {N : Submodule R M} {r : R} {n : M} (hr : r ∈ I) (hn : n ∈ N)
Goal: r • n ∈ I • N
-- Submodule.smul_le
theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {I : Ideal R} {N P : Submodule R M}
Goal: I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P
-- Submodule.coe_set_smul
@[simp, norm_cast]
lemma coe_set_smul : (I : Set R) • N = I • N 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {I : Ideal R} {N : Submodule R M}
Goal: ↑I • N = I • N
-- Submodule.smul_induction_on
@[elab_as_elim]
theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (smul : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
    (add : ∀ x y, p x → p y → p (x + y)) : p x 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {I : Ideal R} {N : Submodule R M} {p : M → Prop} {x : M} (H : x ∈ I • N) (smul : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (add : ∀ (x y : M), p x → p y → p (x + y))
Goal: p x
-- Submodule.smul_induction_on'
@[elab_as_elim]
theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop}
    (smul : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn))
    (add : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {I : Ideal R} {N : Submodule R M} {x : M} (hx : x ∈ I • N) {p : (x : M) → x ∈ I • N → Prop} (smul : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) ⋯) (add : ∀ (x : M) (hx : x ∈ I • N) (y : M) (hy : y ∈ I • N), p x hx → p y hy → p (x + y) ⋯)
Goal: p x hx
-- Submodule.mem_smul_span_singleton
theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} :
    x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {I : Ideal R} {m x : M}
Goal: x ∈ I • Submodule.span R {m} ↔ ∃ y ∈ I, y • m = x
-- Submodule.smul_le_right
theorem smul_le_right : I • N ≤ N 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {I : Ideal R} {N : Submodule R M}
Goal: I • N ≤ N
-- Submodule.smul_mono
theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {I J : Ideal R} {N P : Submodule R M} (hij : I ≤ J) (hnp : N ≤ P)
Goal: I • N ≤ J • P
-- Submodule.smul_mono_left
theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {I J : Ideal R} {N : Submodule R M} (h : I ≤ J)
Goal: I • N ≤ J • N
-- Submodule.smul_mono_right
@[deprecated smul_mono_right (since := "2024-03-31")]
protected theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {I : Ideal R} {N P : Submodule R M} (h : N ≤ P)
Goal: I • N ≤ I • P
-- Submodule.map_le_smul_top
theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) :
    Submodule.map f I ≤ I • (⊤ : Submodule R M) 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (I : Ideal R) (f : R →ₗ[R] M)
Goal: Submodule.map f I ≤ I • ⊤
-- Submodule.annihilator_smul
@[simp]
theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (N : Submodule R M)
Goal: N.annihilator • N = ⊥
-- Submodule.annihilator_mul
@[simp]
theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ 
Arguments: {R : Type u} [CommSemiring R] (I : Ideal R)
Goal: Submodule.annihilator I * I = ⊥
-- Submodule.mul_annihilator
@[simp]
theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ 
Arguments: {R : Type u} [CommSemiring R] (I : Ideal R)
Goal: I * Submodule.annihilator I = ⊥
-- Submodule.smul_bot
@[simp]
theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (I : Ideal R)
Goal: I • ⊥ = ⊥
-- Submodule.bot_smul
@[simp]
theorem bot_smul : (⊥ : Ideal R) • N = ⊥ 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (N : Submodule R M)
Goal: ⊥ • N = ⊥
-- Submodule.top_smul
@[simp]
theorem top_smul : (⊤ : Ideal R) • N = N 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (N : Submodule R M)
Goal: ⊤ • N = N
-- Submodule.smul_sup
theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (I : Ideal R) (N P : Submodule R M)
Goal: I • (N ⊔ P) = I • N ⊔ I • P
-- Submodule.sup_smul
theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (I J : Ideal R) (N : Submodule R M)
Goal: (I ⊔ J) • N = I • N ⊔ J • N
-- Submodule.smul_assoc
protected theorem smul_assoc : (I • J) • N = I • J • N 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (I J : Ideal R) (N : Submodule R M)
Goal: (I • J) • N = I • J • N
-- Submodule.smul_inf_le
@[deprecated smul_inf_le (since := "2024-03-31")]
protected theorem smul_inf_le (M₁ M₂ : Submodule R M) :
    I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (I : Ideal R) (M₁ M₂ : Submodule R M)
Goal: I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂
-- Submodule.smul_iSup
theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {ι : Sort u_4} {I : Ideal R} {t : ι → Submodule R M}
Goal: I • iSup t = ⨆ i, I • t i
-- Submodule.smul_iInf_le
@[deprecated smul_iInf_le (since := "2024-03-31")]
protected theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} :
    I • iInf t ≤ ⨅ i, I • t i 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {ι : Sort u_4} {I : Ideal R} {t : ι → Submodule R M}
Goal: I • iInf t ≤ ⨅ i, I • t i
-- Submodule.span_smul_span
theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (S : Set R) (T : Set M)
Goal: Ideal.span S • Submodule.span R T = Submodule.span R (⋃ s ∈ S, ⋃ t ∈ T, {s • t})
-- Submodule.ideal_span_singleton_smul
theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) :
    (Ideal.span {r} : Ideal R) • N = r • N 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (r : R) (N : Submodule R M)
Goal: Ideal.span {r} • N = r • N
-- Submodule.mem_of_span_top_of_smul_mem
theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M)
    (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ (r : ↑s), ↑r • x ∈ M')
Goal: x ∈ M'
-- Submodule.mem_of_span_eq_top_of_smul_pow_mem
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
    (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ (r : ↑s), ∃ n, ↑r ^ n • x ∈ M')
Goal: x ∈ M'
-- Submodule.map_smul''
@[simp]
theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (I : Ideal R) (N : Submodule R M) {M' : Type w} [AddCommMonoid M'] [Module R M'] (f : M →ₗ[R] M')
Goal: Submodule.map f (I • N) = I • Submodule.map f N
-- Submodule.map_pointwise_smul
@[simp]
theorem map_pointwise_smul (r : R) (N : Submodule R M) (f : M →ₗ[R] M') :
    (r • N).map f = r • N.map f 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {M' : Type w} [AddCommMonoid M'] [Module R M'] (r : R) (N : Submodule R M) (f : M →ₗ[R] M')
Goal: Submodule.map f (r • N) = r • Submodule.map f N
-- Submodule.mem_smul_span
theorem mem_smul_span {s : Set M} {x : M} :
    x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {I : Ideal R} {s : Set M} {x : M}
Goal: x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ a ∈ I, ⋃ b ∈ s, {a • b})
-- Submodule.mem_ideal_smul_span_iff_exists_sum
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (I : Ideal R) {ι : Type u_4} (f : ι → M) (x : M)
Goal: x ∈ I • Submodule.span R (Set.range f) ↔ ∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (a.sum fun i c => c • f i) = x
-- Submodule.mem_ideal_smul_span_iff_exists_sum'
theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) :
    x ∈ I • span R (f '' s) ↔
    ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (I : Ideal R) {ι : Type u_4} (s : Set ι) (f : ι → M) (x : M)
Goal: x ∈ I • Submodule.span R (f '' s) ↔ ∃ a, ∃ (_ : ∀ (i : ↑s), a i ∈ I), (a.sum fun i c => c • f ↑i) = x
-- Submodule.mem_smul_top_iff
theorem mem_smul_top_iff (N : Submodule R M) (x : N) :
    x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (I : Ideal R) (N : Submodule R M) (x : ↥N)
Goal: x ∈ I • ⊤ ↔ ↑x ∈ I • N
-- Submodule.smul_comap_le_comap_smul
@[simp]
theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) :
    I • S.comap f ≤ (I • S).comap f 
Arguments: {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {M' : Type w} [AddCommMonoid M'] [Module R M'] (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R)
Goal: I • Submodule.comap f S ≤ Submodule.comap f (I • S)
-- Ideal.add_eq_sup
@[simp]
theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J 
Arguments: {R : Type u} [Semiring R] {I J : Ideal R}
Goal: I + J = I ⊔ J
-- Ideal.zero_eq_bot
@[simp, nolint simpNF]
theorem zero_eq_bot : (0 : Ideal R) = ⊥ 
Arguments: {R : Type u} [Semiring R]
Goal: 0 = ⊥