ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Bimod.AssociatorBimod.inv_hom_id
theorem inv_hom_id : inv P Q L ≫ hom P Q L = 𝟙 _ 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)] {R S T U : Mon_ C} (P : Bimod R S) (Q : Bimod S T) (L : Bimod T U)
Goal: Bimod.AssociatorBimod.inv P Q L ≫ Bimod.AssociatorBimod.hom P Q L = 𝟙 (P.tensorBimod (Q.tensorBimod L)).X
-- Bimod.LeftUnitorBimod.hom_inv_id
theorem hom_inv_id : hom P ≫ inv P = 𝟙 _ 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] {R S : Mon_ C} (P : Bimod R S)
Goal: Bimod.LeftUnitorBimod.hom P ≫ Bimod.LeftUnitorBimod.inv P = 𝟙 (Bimod.TensorBimod.X (Bimod.regular R) P)
-- Bimod.LeftUnitorBimod.inv_hom_id
theorem inv_hom_id : inv P ≫ hom P = 𝟙 _ 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] {R S : Mon_ C} (P : Bimod R S)
Goal: Bimod.LeftUnitorBimod.inv P ≫ Bimod.LeftUnitorBimod.hom P = 𝟙 P.X
-- Bimod.LeftUnitorBimod.hom_left_act_hom'
theorem hom_left_act_hom' :
    ((regular R).tensorBimod P).actLeft ≫ hom P = (R.X ◁ hom P) ≫ P.actLeft 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] {R S : Mon_ C} (P : Bimod R S) [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)]
Goal: ((Bimod.regular R).tensorBimod P).actLeft ≫ Bimod.LeftUnitorBimod.hom P = R.X ◁ Bimod.LeftUnitorBimod.hom P ≫ P.actLeft
-- Bimod.LeftUnitorBimod.hom_right_act_hom'
theorem hom_right_act_hom' :
    ((regular R).tensorBimod P).actRight ≫ hom P = (hom P ▷ S.X) ≫ P.actRight 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] {R S : Mon_ C} (P : Bimod R S) [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)]
Goal: ((Bimod.regular R).tensorBimod P).actRight ≫ Bimod.LeftUnitorBimod.hom P =
  Bimod.LeftUnitorBimod.hom P ▷ S.X ≫ P.actRight
-- Bimod.RightUnitorBimod.hom_inv_id
theorem hom_inv_id : hom P ≫ inv P = 𝟙 _ 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] {R S : Mon_ C} (P : Bimod R S)
Goal: Bimod.RightUnitorBimod.hom P ≫ Bimod.RightUnitorBimod.inv P = 𝟙 (Bimod.TensorBimod.X P (Bimod.regular S))
-- Bimod.RightUnitorBimod.inv_hom_id
theorem inv_hom_id : inv P ≫ hom P = 𝟙 _ 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] {R S : Mon_ C} (P : Bimod R S)
Goal: Bimod.RightUnitorBimod.inv P ≫ Bimod.RightUnitorBimod.hom P = 𝟙 P.X
-- Bimod.RightUnitorBimod.hom_left_act_hom'
theorem hom_left_act_hom' :
    (P.tensorBimod (regular S)).actLeft ≫ hom P = (R.X ◁ hom P) ≫ P.actLeft 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] {R S : Mon_ C} (P : Bimod R S) [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)]
Goal: (P.tensorBimod (Bimod.regular S)).actLeft ≫ Bimod.RightUnitorBimod.hom P =
  R.X ◁ Bimod.RightUnitorBimod.hom P ≫ P.actLeft
-- Bimod.RightUnitorBimod.hom_right_act_hom'
theorem hom_right_act_hom' :
    (P.tensorBimod (regular S)).actRight ≫ hom P = (hom P ▷ S.X) ≫ P.actRight 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] {R S : Mon_ C} (P : Bimod R S) [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)]
Goal: (P.tensorBimod (Bimod.regular S)).actRight ≫ Bimod.RightUnitorBimod.hom P =
  Bimod.RightUnitorBimod.hom P ▷ S.X ≫ P.actRight
-- Bimod.whiskerLeft_id_bimod
theorem whiskerLeft_id_bimod {X Y Z : Mon_ C} {M : Bimod X Y} {N : Bimod Y Z} :
    whiskerLeft M (𝟙 N) = 𝟙 (M.tensorBimod N) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)] {X Y Z : Mon_ C} {M : Bimod X Y} {N : Bimod Y Z}
Goal: M.whiskerLeft (𝟙 N) = 𝟙 (M.tensorBimod N)
-- Bimod.id_whiskerRight_bimod
theorem id_whiskerRight_bimod {X Y Z : Mon_ C} {M : Bimod X Y} {N : Bimod Y Z} :
    whiskerRight (𝟙 M) N = 𝟙 (M.tensorBimod N) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)] {X Y Z : Mon_ C} {M : Bimod X Y} {N : Bimod Y Z}
Goal: Bimod.whiskerRight (𝟙 M) N = 𝟙 (M.tensorBimod N)
-- Bimod.whiskerLeft_comp_bimod
theorem whiskerLeft_comp_bimod {X Y Z : Mon_ C} (M : Bimod X Y) {N P Q : Bimod Y Z} (f : N ⟶ P)
    (g : P ⟶ Q) : whiskerLeft M (f ≫ g) = whiskerLeft M f ≫ whiskerLeft M g 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)] {X Y Z : Mon_ C} (M : Bimod X Y) {N P Q : Bimod Y Z} (f : N ⟶ P) (g : P ⟶ Q)
Goal: M.whiskerLeft (f ≫ g) = M.whiskerLeft f ≫ M.whiskerLeft g
-- Bimod.id_whiskerLeft_bimod
theorem id_whiskerLeft_bimod {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N) :
    whiskerLeft (regular X) f = (leftUnitorBimod M).hom ≫ f ≫ (leftUnitorBimod N).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)] {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N)
Goal: (Bimod.regular X).whiskerLeft f = M.leftUnitorBimod.hom ≫ f ≫ N.leftUnitorBimod.inv
-- Bimod.comp_whiskerLeft_bimod
theorem comp_whiskerLeft_bimod {W X Y Z : Mon_ C} (M : Bimod W X) (N : Bimod X Y)
    {P P' : Bimod Y Z} (f : P ⟶ P') :
    whiskerLeft (M.tensorBimod N) f =
      (associatorBimod M N P).hom ≫
        whiskerLeft M (whiskerLeft N f) ≫ (associatorBimod M N P').inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)] {W X Y Z : Mon_ C} (M : Bimod W X) (N : Bimod X Y) {P P' : Bimod Y Z} (f : P ⟶ P')
Goal: (M.tensorBimod N).whiskerLeft f =
  (M.associatorBimod N P).hom ≫ M.whiskerLeft (N.whiskerLeft f) ≫ (M.associatorBimod N P').inv
-- Bimod.comp_whiskerRight_bimod
theorem comp_whiskerRight_bimod {X Y Z : Mon_ C} {M N P : Bimod X Y} (f : M ⟶ N) (g : N ⟶ P)
    (Q : Bimod Y Z) : whiskerRight (f ≫ g) Q = whiskerRight f Q ≫ whiskerRight g Q 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)] {X Y Z : Mon_ C} {M N P : Bimod X Y} (f : M ⟶ N) (g : N ⟶ P) (Q : Bimod Y Z)
Goal: Bimod.whiskerRight (f ≫ g) Q = Bimod.whiskerRight f Q ≫ Bimod.whiskerRight g Q
-- Bimod.whiskerRight_id_bimod
theorem whiskerRight_id_bimod {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N) :
    whiskerRight f (regular Y) = (rightUnitorBimod M).hom ≫ f ≫ (rightUnitorBimod N).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)] {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N)
Goal: Bimod.whiskerRight f (Bimod.regular Y) = M.rightUnitorBimod.hom ≫ f ≫ N.rightUnitorBimod.inv
-- Bimod.whiskerRight_comp_bimod
theorem whiskerRight_comp_bimod {W X Y Z : Mon_ C} {M M' : Bimod W X} (f : M ⟶ M') (N : Bimod X Y)
    (P : Bimod Y Z) :
    whiskerRight f (N.tensorBimod P) =
      (associatorBimod M N P).inv ≫
        whiskerRight (whiskerRight f N) P ≫ (associatorBimod M' N P).hom 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)] {W X Y Z : Mon_ C} {M M' : Bimod W X} (f : M ⟶ M') (N : Bimod X Y) (P : Bimod Y Z)
Goal: Bimod.whiskerRight f (N.tensorBimod P) =
  (M.associatorBimod N P).inv ≫ Bimod.whiskerRight (Bimod.whiskerRight f N) P ≫ (M'.associatorBimod N P).hom
-- Bimod.whisker_assoc_bimod
theorem whisker_assoc_bimod {W X Y Z : Mon_ C} (M : Bimod W X) {N N' : Bimod X Y} (f : N ⟶ N')
    (P : Bimod Y Z) :
    whiskerRight (whiskerLeft M f) P =
      (associatorBimod M N P).hom ≫
        whiskerLeft M (whiskerRight f P) ≫ (associatorBimod M N' P).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)] {W X Y Z : Mon_ C} (M : Bimod W X) {N N' : Bimod X Y} (f : N ⟶ N') (P : Bimod Y Z)
Goal: Bimod.whiskerRight (M.whiskerLeft f) P =
  (M.associatorBimod N P).hom ≫ M.whiskerLeft (Bimod.whiskerRight f P) ≫ (M.associatorBimod N' P).inv
-- Bimod.whisker_exchange_bimod
theorem whisker_exchange_bimod {X Y Z : Mon_ C} {M N : Bimod X Y} {P Q : Bimod Y Z} (f : M ⟶ N)
    (g : P ⟶ Q) : whiskerLeft M g ≫ whiskerRight f Q =
      whiskerRight f P ≫ whiskerLeft N g 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)] {X Y Z : Mon_ C} {M N : Bimod X Y} {P Q : Bimod Y Z} (f : M ⟶ N) (g : P ⟶ Q)
Goal: M.whiskerLeft g ≫ Bimod.whiskerRight f Q = Bimod.whiskerRight f P ≫ N.whiskerLeft g
-- Bimod.pentagon_bimod
theorem pentagon_bimod {V W X Y Z : Mon_ C} (M : Bimod V W) (N : Bimod W X) (P : Bimod X Y)
    (Q : Bimod Y Z) :
    whiskerRight (associatorBimod M N P).hom Q ≫
      (associatorBimod M (N.tensorBimod P) Q).hom ≫
        whiskerLeft M (associatorBimod N P Q).hom =
      (associatorBimod (M.tensorBimod N) P Q).hom ≫
        (associatorBimod M N (P.tensorBimod Q)).hom 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)] {V W X Y Z : Mon_ C} (M : Bimod V W) (N : Bimod W X) (P : Bimod X Y) (Q : Bimod Y Z)
Goal: Bimod.whiskerRight (M.associatorBimod N P).hom Q ≫
    (M.associatorBimod (N.tensorBimod P) Q).hom ≫ M.whiskerLeft (N.associatorBimod P Q).hom =
  ((M.tensorBimod N).associatorBimod P Q).hom ≫ (M.associatorBimod N (P.tensorBimod Q)).hom
-- Bimod.triangle_bimod
theorem triangle_bimod {X Y Z : Mon_ C} (M : Bimod X Y) (N : Bimod Y Z) :
    (associatorBimod M (regular Y) N).hom ≫ whiskerLeft M (leftUnitorBimod N).hom =
      whiskerRight (rightUnitorBimod M).hom N 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasCoequalizers C] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)] [(X : C) →
    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
      (CategoryTheory.MonoidalCategory.tensorRight X)] {X Y Z : Mon_ C} (M : Bimod X Y) (N : Bimod Y Z)
Goal: (M.associatorBimod (Bimod.regular Y) N).hom ≫ M.whiskerLeft N.leftUnitorBimod.hom =
  Bimod.whiskerRight M.rightUnitorBimod.hom N
-- MonovaryOn.sum_smul_sum_le_card_smul_sum
theorem MonovaryOn.sum_smul_sum_le_card_smul_sum (hfg : MonovaryOn f g s) :
    ((∑ i ∈ s, f i) • ∑ i ∈ s, g i) ≤ s.card • ∑ i ∈ s, f i • g i 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β] {s : Finset ι} {f : ι → α} {g : ι → β} (hfg : MonovaryOn f g ↑s)
Goal: (∑ i ∈ s, f i) • ∑ i ∈ s, g i ≤ s.card • ∑ i ∈ s, f i • g i
-- AntivaryOn.card_smul_sum_le_sum_smul_sum
theorem AntivaryOn.card_smul_sum_le_sum_smul_sum (hfg : AntivaryOn f g s) :
    (s.card • ∑ i ∈ s, f i • g i) ≤ (∑ i ∈ s, f i) • ∑ i ∈ s, g i 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β] {s : Finset ι} {f : ι → α} {g : ι → β} (hfg : AntivaryOn f g ↑s)
Goal: s.card • ∑ i ∈ s, f i • g i ≤ (∑ i ∈ s, f i) • ∑ i ∈ s, g i
-- Monovary.sum_smul_sum_le_card_smul_sum
theorem Monovary.sum_smul_sum_le_card_smul_sum (hfg : Monovary f g) :
    ((∑ i, f i) • ∑ i, g i) ≤ Fintype.card ι • ∑ i, f i • g i 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β] {f : ι → α} {g : ι → β} [Fintype ι] (hfg : Monovary f g)
Goal: (∑ i : ι, f i) • ∑ i : ι, g i ≤ Fintype.card ι • ∑ i : ι, f i • g i
-- Antivary.card_smul_sum_le_sum_smul_sum
theorem Antivary.card_smul_sum_le_sum_smul_sum (hfg : Antivary f g) :
    (Fintype.card ι • ∑ i, f i • g i) ≤ (∑ i, f i) • ∑ i, g i 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β] {f : ι → α} {g : ι → β} [Fintype ι] (hfg : Antivary f g)
Goal: Fintype.card ι • ∑ i : ι, f i • g i ≤ (∑ i : ι, f i) • ∑ i : ι, g i
-- MonovaryOn.sum_mul_sum_le_card_mul_sum
theorem MonovaryOn.sum_mul_sum_le_card_mul_sum (hfg : MonovaryOn f g s) :
    ((∑ i ∈ s, f i) * ∑ i ∈ s, g i) ≤ s.card * ∑ i ∈ s, f i * g i 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedRing α] {s : Finset ι} {f g : ι → α} (hfg : MonovaryOn f g ↑s)
Goal: (∑ i ∈ s, f i) * ∑ i ∈ s, g i ≤ ↑s.card * ∑ i ∈ s, f i * g i
-- AntivaryOn.card_mul_sum_le_sum_mul_sum
theorem AntivaryOn.card_mul_sum_le_sum_mul_sum (hfg : AntivaryOn f g s) :
    ((s.card : α) * ∑ i ∈ s, f i * g i) ≤ (∑ i ∈ s, f i) * ∑ i ∈ s, g i 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedRing α] {s : Finset ι} {f g : ι → α} (hfg : AntivaryOn f g ↑s)
Goal: ↑s.card * ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i) * ∑ i ∈ s, g i
-- sq_sum_le_card_mul_sum_sq
theorem sq_sum_le_card_mul_sum_sq : (∑ i ∈ s, f i) ^ 2 ≤ s.card * ∑ i ∈ s, f i ^ 2 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedRing α] {s : Finset ι} {f : ι → α}
Goal: (∑ i ∈ s, f i) ^ 2 ≤ ↑s.card * ∑ i ∈ s, f i ^ 2
-- Monovary.sum_mul_sum_le_card_mul_sum
theorem Monovary.sum_mul_sum_le_card_mul_sum (hfg : Monovary f g) :
    ((∑ i, f i) * ∑ i, g i) ≤ Fintype.card ι * ∑ i, f i * g i 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedRing α] {f g : ι → α} [Fintype ι] (hfg : Monovary f g)
Goal: (∑ i : ι, f i) * ∑ i : ι, g i ≤ ↑(Fintype.card ι) * ∑ i : ι, f i * g i
-- Antivary.card_mul_sum_le_sum_mul_sum
theorem Antivary.card_mul_sum_le_sum_mul_sum (hfg : Antivary f g) :
    ((Fintype.card ι : α) * ∑ i, f i * g i) ≤ (∑ i, f i) * ∑ i, g i 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedRing α] {f g : ι → α} [Fintype ι] (hfg : Antivary f g)
Goal: ↑(Fintype.card ι) * ∑ i : ι, f i * g i ≤ (∑ i : ι, f i) * ∑ i : ι, g i
-- sum_div_card_sq_le_sum_sq_div_card
theorem sum_div_card_sq_le_sum_sq_div_card :
    ((∑ i ∈ s, f i) / s.card) ^ 2 ≤ (∑ i ∈ s, f i ^ 2) / s.card 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedField α] {s : Finset ι} {f : ι → α}
Goal: ((∑ i ∈ s, f i) / ↑s.card) ^ 2 ≤ (∑ i ∈ s, f i ^ 2) / ↑s.card
-- NonemptyInterval.toProd_one
@[to_additive (attr := simp) toProd_zero]
theorem toProd_one : (1 : NonemptyInterval α).toProd = 1 
Arguments: {α : Type u_2} [Preorder α] [One α]
Goal: NonemptyInterval.toProd 1 = 1
-- NonemptyInterval.toProd_zero
@[to_additive (attr := simp) toProd_zero]
theorem toProd_one : (1 : NonemptyInterval α).toProd = 1 
Arguments: {α : Type u_2} [Preorder α] [Zero α]
Goal: NonemptyInterval.toProd 0 = 0
-- NonemptyInterval.fst_one
@[to_additive]
theorem fst_one : (1 : NonemptyInterval α).fst = 1 
Arguments: {α : Type u_2} [Preorder α] [One α]
Goal: (NonemptyInterval.toProd 1).1 = 1
-- NonemptyInterval.fst_zero
@[to_additive]
theorem fst_one : (1 : NonemptyInterval α).fst = 1 
Arguments: {α : Type u_2} [Preorder α] [Zero α]
Goal: (NonemptyInterval.toProd 0).1 = 0
-- NonemptyInterval.snd_one
@[to_additive]
theorem snd_one : (1 : NonemptyInterval α).snd = 1 
Arguments: {α : Type u_2} [Preorder α] [One α]
Goal: (NonemptyInterval.toProd 1).2 = 1
-- NonemptyInterval.snd_zero
@[to_additive]
theorem snd_one : (1 : NonemptyInterval α).snd = 1 
Arguments: {α : Type u_2} [Preorder α] [Zero α]
Goal: (NonemptyInterval.toProd 0).2 = 0
-- NonemptyInterval.coe_zero_interval
@[to_additive (attr := push_cast, simp)]
theorem coe_one_interval : ((1 : NonemptyInterval α) : Interval α) = 1 
Arguments: {α : Type u_2} [Preorder α] [Zero α]
Goal: ↑0 = 0
-- NonemptyInterval.coe_one_interval
@[to_additive (attr := push_cast, simp)]
theorem coe_one_interval : ((1 : NonemptyInterval α) : Interval α) = 1 
Arguments: {α : Type u_2} [Preorder α] [One α]
Goal: ↑1 = 1
-- NonemptyInterval.pure_one
@[to_additive (attr := simp)]
theorem pure_one : pure (1 : α) = 1 
Arguments: {α : Type u_2} [Preorder α] [One α]
Goal: NonemptyInterval.pure 1 = 1
-- NonemptyInterval.pure_zero
@[to_additive (attr := simp)]
theorem pure_one : pure (1 : α) = 1 
Arguments: {α : Type u_2} [Preorder α] [Zero α]
Goal: NonemptyInterval.pure 0 = 0
-- Interval.pure_one
@[to_additive (attr := simp)]
theorem pure_one : pure (1 : α) = 1 
Arguments: {α : Type u_2} [Preorder α] [One α]
Goal: Interval.pure 1 = 1
-- Interval.pure_zero
@[to_additive (attr := simp)]
theorem pure_one : pure (1 : α) = 1 
Arguments: {α : Type u_2} [Preorder α] [Zero α]
Goal: Interval.pure 0 = 0
-- Interval.zero_ne_bot
@[to_additive] lemma one_ne_bot : (1 : Interval α) ≠ ⊥ 
Arguments: {α : Type u_2} [Preorder α] [Zero α]
Goal: 0 ≠ ⊥
-- Interval.one_ne_bot
@[to_additive] lemma one_ne_bot : (1 : Interval α) ≠ ⊥ 
Arguments: {α : Type u_2} [Preorder α] [One α]
Goal: 1 ≠ ⊥
-- Interval.bot_ne_zero
@[to_additive] lemma bot_ne_one : (⊥ : Interval α) ≠ 1 
Arguments: {α : Type u_2} [Preorder α] [Zero α]
Goal: ⊥ ≠ 0
-- Interval.bot_ne_one
@[to_additive] lemma bot_ne_one : (⊥ : Interval α) ≠ 1 
Arguments: {α : Type u_2} [Preorder α] [One α]
Goal: ⊥ ≠ 1
-- NonemptyInterval.coe_zero
@[to_additive (attr := simp)]
theorem coe_one : ((1 : NonemptyInterval α) : Set α) = 1 
Arguments: {α : Type u_2} [PartialOrder α] [Zero α]
Goal: ↑0 = 0
-- NonemptyInterval.coe_one
@[to_additive (attr := simp)]
theorem coe_one : ((1 : NonemptyInterval α) : Set α) = 1 
Arguments: {α : Type u_2} [PartialOrder α] [One α]
Goal: ↑1 = 1
-- NonemptyInterval.one_mem_one
@[to_additive]
theorem one_mem_one : (1 : α) ∈ (1 : NonemptyInterval α) 
Arguments: {α : Type u_2} [PartialOrder α] [One α]
Goal: 1 ∈ 1
-- NonemptyInterval.zero_mem_zero
@[to_additive]
theorem one_mem_one : (1 : α) ∈ (1 : NonemptyInterval α) 
Arguments: {α : Type u_2} [PartialOrder α] [Zero α]
Goal: 0 ∈ 0
-- Interval.coe_one
@[to_additive (attr := simp)]
theorem coe_one : ((1 : Interval α) : Set α) = 1 
Arguments: {α : Type u_2} [PartialOrder α] [One α]
Goal: ↑1 = 1
-- Interval.coe_zero
@[to_additive (attr := simp)]
theorem coe_one : ((1 : Interval α) : Set α) = 1 
Arguments: {α : Type u_2} [PartialOrder α] [Zero α]
Goal: ↑0 = 0
-- Interval.zero_mem_zero
@[to_additive]
theorem one_mem_one : (1 : α) ∈ (1 : Interval α) 
Arguments: {α : Type u_2} [PartialOrder α] [Zero α]
Goal: 0 ∈ 0
-- Interval.one_mem_one
@[to_additive]
theorem one_mem_one : (1 : α) ∈ (1 : Interval α) 
Arguments: {α : Type u_2} [PartialOrder α] [One α]
Goal: 1 ∈ 1
-- NonemptyInterval.toProd_mul
@[to_additive (attr := simp) toProd_add]
theorem toProd_mul : (s * t).toProd = s.toProd * t.toProd 
Arguments: {α : Type u_2} [Preorder α] [Mul α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (s t : NonemptyInterval α)
Goal: (s * t).toProd = s.toProd * t.toProd
-- NonemptyInterval.toProd_add
@[to_additive (attr := simp) toProd_add]
theorem toProd_mul : (s * t).toProd = s.toProd * t.toProd 
Arguments: {α : Type u_2} [Preorder α] [Add α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (s t : NonemptyInterval α)
Goal: (s + t).toProd = s.toProd + t.toProd
-- NonemptyInterval.fst_add
@[to_additive]
theorem fst_mul : (s * t).fst = s.fst * t.fst 
Arguments: {α : Type u_2} [Preorder α] [Add α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (s t : NonemptyInterval α)
Goal: (s + t).toProd.1 = s.toProd.1 + t.toProd.1
-- NonemptyInterval.fst_mul
@[to_additive]
theorem fst_mul : (s * t).fst = s.fst * t.fst 
Arguments: {α : Type u_2} [Preorder α] [Mul α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (s t : NonemptyInterval α)
Goal: (s * t).toProd.1 = s.toProd.1 * t.toProd.1