ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Multiset.trop_sum
theorem Multiset.trop_sum [AddCommMonoid R] (s : Multiset R) :
    trop s.sum = Multiset.prod (s.map trop) 
Arguments: {R : Type u_1} [AddCommMonoid R] (s : Multiset R)
Goal: Tropical.trop s.sum = (Multiset.map Tropical.trop s).prod
-- trop_sum
theorem trop_sum [AddCommMonoid R] (s : Finset S) (f : S → R) :
    trop (∑ i ∈ s, f i) = ∏ i ∈ s, trop (f i) 
Arguments: {R : Type u_1} {S : Type u_2} [AddCommMonoid R] (s : Finset S) (f : S → R)
Goal: Tropical.trop (∑ i ∈ s, f i) = ∏ i ∈ s, Tropical.trop (f i)
-- List.untrop_prod
theorem List.untrop_prod [AddMonoid R] (l : List (Tropical R)) :
    untrop l.prod = List.sum (l.map untrop) 
Arguments: {R : Type u_1} [AddMonoid R] (l : List (Tropical R))
Goal: Tropical.untrop l.prod = (List.map Tropical.untrop l).sum
-- Multiset.untrop_prod
theorem Multiset.untrop_prod [AddCommMonoid R] (s : Multiset (Tropical R)) :
    untrop s.prod = Multiset.sum (s.map untrop) 
Arguments: {R : Type u_1} [AddCommMonoid R] (s : Multiset (Tropical R))
Goal: Tropical.untrop s.prod = (Multiset.map Tropical.untrop s).sum
-- untrop_prod
theorem untrop_prod [AddCommMonoid R] (s : Finset S) (f : S → Tropical R) :
    untrop (∏ i ∈ s, f i) = ∑ i ∈ s, untrop (f i) 
Arguments: {R : Type u_1} {S : Type u_2} [AddCommMonoid R] (s : Finset S) (f : S → Tropical R)
Goal: Tropical.untrop (∏ i ∈ s, f i) = ∑ i ∈ s, Tropical.untrop (f i)
-- List.trop_minimum
theorem List.trop_minimum [LinearOrder R] (l : List R) :
    trop l.minimum = List.sum (l.map (trop ∘ WithTop.some)) 
Arguments: {R : Type u_1} [LinearOrder R] (l : List R)
Goal: Tropical.trop l.minimum = (List.map (Tropical.trop ∘ WithTop.some) l).sum
-- Multiset.trop_inf
theorem Multiset.trop_inf [LinearOrder R] [OrderTop R] (s : Multiset R) :
    trop s.inf = Multiset.sum (s.map trop) 
Arguments: {R : Type u_1} [LinearOrder R] [OrderTop R] (s : Multiset R)
Goal: Tropical.trop s.inf = (Multiset.map Tropical.trop s).sum
-- Finset.trop_inf
theorem Finset.trop_inf [LinearOrder R] [OrderTop R] (s : Finset S) (f : S → R) :
    trop (s.inf f) = ∑ i ∈ s, trop (f i) 
Arguments: {R : Type u_1} {S : Type u_2} [LinearOrder R] [OrderTop R] (s : Finset S) (f : S → R)
Goal: Tropical.trop (s.inf f) = ∑ i ∈ s, Tropical.trop (f i)
-- trop_sInf_image
theorem trop_sInf_image [ConditionallyCompleteLinearOrder R] (s : Finset S) (f : S → WithTop R) :
    trop (sInf (f '' s)) = ∑ i ∈ s, trop (f i) 
Arguments: {R : Type u_1} {S : Type u_2} [ConditionallyCompleteLinearOrder R] (s : Finset S) (f : S → WithTop R)
Goal: Tropical.trop (sInf (f '' ↑s)) = ∑ i ∈ s, Tropical.trop (f i)
-- trop_iInf
theorem trop_iInf [ConditionallyCompleteLinearOrder R] [Fintype S] (f : S → WithTop R) :
    trop (⨅ i : S, f i) = ∑ i : S, trop (f i) 
Arguments: {R : Type u_1} {S : Type u_2} [ConditionallyCompleteLinearOrder R] [Fintype S] (f : S → WithTop R)
Goal: Tropical.trop (⨅ i, f i) = ∑ i : S, Tropical.trop (f i)
-- Multiset.untrop_sum
theorem Multiset.untrop_sum [LinearOrder R] [OrderTop R] (s : Multiset (Tropical R)) :
    untrop s.sum = Multiset.inf (s.map untrop) 
Arguments: {R : Type u_1} [LinearOrder R] [OrderTop R] (s : Multiset (Tropical R))
Goal: Tropical.untrop s.sum = (Multiset.map Tropical.untrop s).inf
-- Finset.untrop_sum'
theorem Finset.untrop_sum' [LinearOrder R] [OrderTop R] (s : Finset S) (f : S → Tropical R) :
    untrop (∑ i ∈ s, f i) = s.inf (untrop ∘ f) 
Arguments: {R : Type u_1} {S : Type u_2} [LinearOrder R] [OrderTop R] (s : Finset S) (f : S → Tropical R)
Goal: Tropical.untrop (∑ i ∈ s, f i) = s.inf (Tropical.untrop ∘ f)
-- untrop_sum_eq_sInf_image
theorem untrop_sum_eq_sInf_image [ConditionallyCompleteLinearOrder R] (s : Finset S)
    (f : S → Tropical (WithTop R)) : untrop (∑ i ∈ s, f i) = sInf (untrop ∘ f '' s) 
Arguments: {R : Type u_1} {S : Type u_2} [ConditionallyCompleteLinearOrder R] (s : Finset S) (f : S → Tropical (WithTop R))
Goal: Tropical.untrop (∑ i ∈ s, f i) = sInf (Tropical.untrop ∘ f '' ↑s)
-- untrop_sum
theorem untrop_sum [ConditionallyCompleteLinearOrder R] [Fintype S] (f : S → Tropical (WithTop R)) :
    untrop (∑ i : S, f i) = ⨅ i : S, untrop (f i) 
Arguments: {R : Type u_1} {S : Type u_2} [ConditionallyCompleteLinearOrder R] [Fintype S] (f : S → Tropical (WithTop R))
Goal: Tropical.untrop (∑ i : S, f i) = ⨅ i, Tropical.untrop (f i)
-- Finset.untrop_sum
theorem Finset.untrop_sum [ConditionallyCompleteLinearOrder R] (s : Finset S)
    (f : S → Tropical (WithTop R)) : untrop (∑ i ∈ s, f i) = ⨅ i : s, untrop (f i) 
Arguments: {R : Type u_1} {S : Type u_2} [ConditionallyCompleteLinearOrder R] (s : Finset S) (f : S → Tropical (WithTop R))
Goal: Tropical.untrop (∑ i ∈ s, f i) = ⨅ i, Tropical.untrop (f ↑i)
-- HomologicalComplex.mapBifunctorMapHomotopy.ιMapBifunctor_hom₁
@[reassoc]
lemma ιMapBifunctor_hom₁ (i₁ i₁' : I₁) (i₂ : I₂) (j j' : J)
    (h : ComplexShape.π c₁ c₂ c (i₁', i₂) = j) (h' : c₁.prev i₁' = i₁) :
    ιMapBifunctor K₁ K₂ F c i₁' i₂ j h ≫ hom₁ h₁ f₂ F c j j' = ComplexShape.ε₁ c₁ c₂ c (i₁, i₂) •
      (F.map (h₁.hom i₁' i₁)).app (K₂.X i₂) ≫ (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫
        ιMapBifunctorOrZero L₁ L₂ F c _ _ j' 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} {D : Type u_3} {I₁ : Type u_4} {I₂ : Type u_5} {J : Type u_6} [CategoryTheory.Category.{u_9, u_1} C₁] [CategoryTheory.Category.{u_8, u_2} C₂] [CategoryTheory.Category.{u_7, u_3} D] [CategoryTheory.Preadditive C₁] [CategoryTheory.Preadditive C₂] [CategoryTheory.Preadditive D] {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} {K₁ L₁ : HomologicalComplex C₁ c₁} {f₁ f₁' : K₁ ⟶ L₁} (h₁ : Homotopy f₁ f₁') {K₂ L₂ : HomologicalComplex C₂ c₂} (f₂ : K₂ ⟶ L₂) (F : C₁ ⥤ C₂ ⥤ D) [F.Additive] [∀ (X₁ : C₁), (F.obj X₁).Additive] (c : ComplexShape J) [DecidableEq J] [TotalComplexShape c₁ c₂ c] [K₁.HasMapBifunctor K₂ F c] [L₁.HasMapBifunctor L₂ F c] (i₁ i₁' : I₁) (i₂ : I₂) (j j' : J) (h : c₁.π c₂ c (i₁', i₂) = j) (h' : c₁.prev i₁' = i₁)
Goal: K₁.ιMapBifunctor K₂ F c i₁' i₂ j h ≫ HomologicalComplex.mapBifunctorMapHomotopy.hom₁ h₁ f₂ F c j j' =
  c₁.ε₁ c₂ c (i₁, i₂) •
    (F.map (h₁.hom i₁' i₁)).app (K₂.X i₂) ≫ (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫ L₁.ιMapBifunctorOrZero L₂ F c i₁ i₂ j'
-- HomologicalComplex.mapBifunctorMapHomotopy.zero₁
lemma zero₁ (j j' : J) (h : ¬ c.Rel j' j) :
    hom₁ h₁ f₂ F c j j' = 0 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} {D : Type u_3} {I₁ : Type u_4} {I₂ : Type u_5} {J : Type u_6} [CategoryTheory.Category.{u_8, u_1} C₁] [CategoryTheory.Category.{u_9, u_2} C₂] [CategoryTheory.Category.{u_7, u_3} D] [CategoryTheory.Preadditive C₁] [CategoryTheory.Preadditive C₂] [CategoryTheory.Preadditive D] {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} {K₁ L₁ : HomologicalComplex C₁ c₁} {f₁ f₁' : K₁ ⟶ L₁} (h₁ : Homotopy f₁ f₁') {K₂ L₂ : HomologicalComplex C₂ c₂} (f₂ : K₂ ⟶ L₂) (F : C₁ ⥤ C₂ ⥤ D) [F.Additive] [∀ (X₁ : C₁), (F.obj X₁).Additive] (c : ComplexShape J) [DecidableEq J] [TotalComplexShape c₁ c₂ c] [K₁.HasMapBifunctor K₂ F c] [L₁.HasMapBifunctor L₂ F c] (j j' : J) (h : ¬c.Rel j' j)
Goal: HomologicalComplex.mapBifunctorMapHomotopy.hom₁ h₁ f₂ F c j j' = 0
-- HomologicalComplex.mapBifunctorMapHomotopy.comm₁_aux
lemma comm₁_aux {i₁ i₁' : I₁} (hi₁ : c₁.Rel i₁ i₁') {i₂ i₂' : I₂} (hi₂ : c₂.Rel i₂ i₂') (j : J)
    (hj : ComplexShape.π c₁ c₂ c (i₁', i₂) = j) :
    ComplexShape.ε₁ c₁ c₂ c (i₁, i₂) • (F.map (h₁.hom i₁' i₁)).app (K₂.X i₂) ≫
      (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫
        (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).d₂ c i₁ i₂ j =
    -(((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₂ c i₁' i₂ (c.next j) ≫
      hom₁ h₁ f₂ F c (c.next j) j 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} {D : Type u_3} {I₁ : Type u_4} {I₂ : Type u_5} {J : Type u_6} [CategoryTheory.Category.{u_9, u_1} C₁] [CategoryTheory.Category.{u_8, u_2} C₂] [CategoryTheory.Category.{u_7, u_3} D] [CategoryTheory.Preadditive C₁] [CategoryTheory.Preadditive C₂] [CategoryTheory.Preadditive D] {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} {K₁ L₁ : HomologicalComplex C₁ c₁} {f₁ f₁' : K₁ ⟶ L₁} (h₁ : Homotopy f₁ f₁') {K₂ L₂ : HomologicalComplex C₂ c₂} (f₂ : K₂ ⟶ L₂) (F : C₁ ⥤ C₂ ⥤ D) [F.Additive] [∀ (X₁ : C₁), (F.obj X₁).Additive] (c : ComplexShape J) [DecidableEq J] [TotalComplexShape c₁ c₂ c] [K₁.HasMapBifunctor K₂ F c] [L₁.HasMapBifunctor L₂ F c] {i₁ i₁' : I₁} (hi₁ : c₁.Rel i₁ i₁') {i₂ i₂' : I₂} (hi₂ : c₂.Rel i₂ i₂') (j : J) (hj : c₁.π c₂ c (i₁', i₂) = j)
Goal: c₁.ε₁ c₂ c (i₁, i₂) •
    (F.map (h₁.hom i₁' i₁)).app (K₂.X i₂) ≫
      (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫ (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).d₂ c i₁ i₂ j =
  -(((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₂ c i₁' i₂ (c.next j) ≫
      HomologicalComplex.mapBifunctorMapHomotopy.hom₁ h₁ f₂ F c (c.next j) j
-- HomologicalComplex.mapBifunctorMapHomotopy.comm₁
lemma comm₁ (j : J) :
    (mapBifunctorMap f₁ f₂ F c).f j =
    (mapBifunctor K₁ K₂ F c).d j (c.next j) ≫
          mapBifunctorMapHomotopy.hom₁ h₁ f₂ F c (c.next j) j +
        mapBifunctorMapHomotopy.hom₁ h₁ f₂ F c j (c.prev j) ≫
          (mapBifunctor L₁ L₂ F c).d (c.prev j) j +
      (mapBifunctorMap f₁' f₂ F c).f j 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} {D : Type u_3} {I₁ : Type u_4} {I₂ : Type u_5} {J : Type u_6} [CategoryTheory.Category.{u_8, u_1} C₁] [CategoryTheory.Category.{u_9, u_2} C₂] [CategoryTheory.Category.{u_7, u_3} D] [CategoryTheory.Preadditive C₁] [CategoryTheory.Preadditive C₂] [CategoryTheory.Preadditive D] {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} {K₁ L₁ : HomologicalComplex C₁ c₁} {f₁ f₁' : K₁ ⟶ L₁} (h₁ : Homotopy f₁ f₁') {K₂ L₂ : HomologicalComplex C₂ c₂} (f₂ : K₂ ⟶ L₂) (F : C₁ ⥤ C₂ ⥤ D) [F.Additive] [∀ (X₁ : C₁), (F.obj X₁).Additive] (c : ComplexShape J) [DecidableEq J] [TotalComplexShape c₁ c₂ c] [K₁.HasMapBifunctor K₂ F c] [L₁.HasMapBifunctor L₂ F c] (j : J)
Goal: (HomologicalComplex.mapBifunctorMap f₁ f₂ F c).f j =
  (K₁.mapBifunctor K₂ F c).d j (c.next j) ≫ HomologicalComplex.mapBifunctorMapHomotopy.hom₁ h₁ f₂ F c (c.next j) j +
      HomologicalComplex.mapBifunctorMapHomotopy.hom₁ h₁ f₂ F c j (c.prev j) ≫ (L₁.mapBifunctor L₂ F c).d (c.prev j) j +
    (HomologicalComplex.mapBifunctorMap f₁' f₂ F c).f j
-- SMulWithZero.zero_smul
class SMulWithZero [Zero R] [Zero M] extends SMulZeroClass R M where
    zero_smul : ∀ m : M, (0 : R) • m = 0
Arguments: {R : Type u_1} {M : Type u_3} [Zero R] [Zero M] [self : SMulWithZero R M] (m : M)
Goal: 0 • m = 0
-- zero_smul
@[simp]
theorem zero_smul (m : M) : (0 : R) • m = 0 
Arguments: (R : Type u_1) {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] (m : M)
Goal: 0 • m = 0
-- smul_eq_zero_of_left
lemma smul_eq_zero_of_left (h : a = 0) (b : M) : a • b = 0 
Arguments: {R : Type u_1} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] {a : R} (h : a = 0) (b : M)
Goal: a • b = 0
-- left_ne_zero_of_smul
lemma left_ne_zero_of_smul : a • b ≠ 0 → a ≠ 0 
Arguments: {R : Type u_1} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] {a : R} {b : M}
Goal: a • b ≠ 0 → a ≠ 0
-- MulActionWithZero.zero_smul
class MulActionWithZero extends MulAction R M where
  -- these fields are copied from `SMulWithZero`, as `extends` behaves poorly
    smul_zero : ∀ r : R, r • (0 : M) = 0
    zero_smul : ∀ m : M, (0 : R) • m = 0
Arguments: {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [self : MulActionWithZero R M] (m : M)
Goal: 0 • m = 0
-- MulActionWithZero.smul_zero
class MulActionWithZero extends MulAction R M where
  -- these fields are copied from `SMulWithZero`, as `extends` behaves poorly
    smul_zero : ∀ r : R, r • (0 : M) = 0
    zero_smul : ∀ m : M, (0 : R) • m = 0
Arguments: {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [self : MulActionWithZero R M] (r : R)
Goal: r • 0 = 0
-- MulActionWithZero.subsingleton
protected lemma MulActionWithZero.subsingleton
    [MulActionWithZero R M] [Subsingleton R] : Subsingleton M 
Arguments: (R : Type u_1) (M : Type u_3) [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [Subsingleton R]
Goal: Subsingleton M
-- MulActionWithZero.nontrivial
protected lemma MulActionWithZero.nontrivial
    [MulActionWithZero R M] [Nontrivial M] : Nontrivial R 
Arguments: (R : Type u_1) (M : Type u_3) [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [Nontrivial M]
Goal: Nontrivial R
-- ite_zero_smul
lemma ite_zero_smul (a : R) (b : M) : (if p then a else 0 : R) • b = if p then a • b else 0 
Arguments: {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] (p : Prop) [Decidable p] (a : R) (b : M)
Goal: (if p then a else 0) • b = if p then a • b else 0
-- boole_smul
lemma boole_smul (a : M) : (if p then 1 else 0 : R) • a = if p then a else 0 
Arguments: {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] (p : Prop) [Decidable p] (a : M)
Goal: (if p then 1 else 0) • a = if p then a else 0
-- Pi.single_apply_smul
lemma Pi.single_apply_smul {ι : Type*} [DecidableEq ι] (x : M) (i j : ι) :
    (Pi.single i 1 : ι → R) j • x = (Pi.single i x : ι → M) j 
Arguments: {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] {ι : Type u_5} [DecidableEq ι] (x : M) (i j : ι)
Goal: Pi.single i 1 j • x = Pi.single i x j
-- smul_inv₀
theorem smul_inv₀ [SMulCommClass α β β] [IsScalarTower α β β] (c : α) (x : β) :
    (c • x)⁻¹ = c⁻¹ • x⁻¹ 
Arguments: {α : Type u_5} {β : Type u_6} [GroupWithZero α] [GroupWithZero β] [MulActionWithZero α β] [SMulCommClass α β β] [IsScalarTower α β β] (c : α) (x : β)
Goal: (c • x)⁻¹ = c⁻¹ • x⁻¹
-- smulMonoidWithZeroHom_apply
@[simps]
def smulMonoidWithZeroHom {α β : Type*} [MonoidWithZero α] [MulZeroOneClass β]
    [MulActionWithZero α β] [IsScalarTower α β β] [SMulCommClass α β β] : α × β →*₀ β 
Arguments: {α : Type u_5} {β : Type u_6} [MonoidWithZero α] [MulZeroOneClass β] [MulActionWithZero α β] [IsScalarTower α β β] [SMulCommClass α β β]
Goal: ∀ (a : α × β), smulMonoidWithZeroHom a = (↑smulMonoidHom).toFun a
-- finsuppLEquivDirectSum_single
@[simp]
theorem finsuppLEquivDirectSum_single (i : ι) (m : M) :
    finsuppLEquivDirectSum R M ι (Finsupp.single i m) = DirectSum.lof R ι _ i m 
Arguments: (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] (ι : Type u_1) [DecidableEq ι] (i : ι) (m : M)
Goal: (finsuppLEquivDirectSum R M ι) (Finsupp.single i m) = (DirectSum.lof R ι (fun i => M) i) m
-- finsuppLEquivDirectSum_symm_lof
@[simp]
theorem finsuppLEquivDirectSum_symm_lof (i : ι) (m : M) :
    (finsuppLEquivDirectSum R M ι).symm (DirectSum.lof R ι _ i m) = Finsupp.single i m 
Arguments: (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] (ι : Type u_1) [DecidableEq ι] (i : ι) (m : M)
Goal: (finsuppLEquivDirectSum R M ι).symm ((DirectSum.lof R ι (fun x => M) i) m) = Finsupp.single i m
-- Tropical.trop_injective
theorem trop_injective : Function.Injective (trop : R → Tropical R) 
Arguments: {R : Type u}
Goal: Function.Injective Tropical.trop
-- Tropical.untrop_injective
theorem untrop_injective : Function.Injective (untrop : Tropical R → R) 
Arguments: {R : Type u}
Goal: Function.Injective Tropical.untrop
-- Tropical.trop_inj_iff
@[simp]
theorem trop_inj_iff (x y : R) : trop x = trop y ↔ x = y 
Arguments: {R : Type u} (x y : R)
Goal: Tropical.trop x = Tropical.trop y ↔ x = y
-- Tropical.untrop_inj_iff
@[simp]
theorem untrop_inj_iff (x y : Tropical R) : untrop x = untrop y ↔ x = y 
Arguments: {R : Type u} (x y : Tropical R)
Goal: Tropical.untrop x = Tropical.untrop y ↔ x = y
-- Tropical.trop_untrop
@[simp]
theorem trop_untrop (x : Tropical R) : trop (untrop x) = x 
Arguments: {R : Type u} (x : Tropical R)
Goal: Tropical.trop (Tropical.untrop x) = x
-- Tropical.untrop_trop
@[simp]
theorem untrop_trop (x : R) : untrop (trop x) = x 
Arguments: {R : Type u} (x : R)
Goal: Tropical.untrop (Tropical.trop x) = x
-- Tropical.leftInverse_trop
theorem leftInverse_trop : Function.LeftInverse (trop : R → Tropical R) untrop 
Arguments: {R : Type u}
Goal: Function.LeftInverse Tropical.trop Tropical.untrop
-- Tropical.rightInverse_trop
theorem rightInverse_trop : Function.RightInverse (trop : R → Tropical R) untrop 
Arguments: {R : Type u}
Goal: Function.RightInverse Tropical.trop Tropical.untrop
-- Tropical.tropEquiv_coe_fn
@[simp]
theorem tropEquiv_coe_fn : (tropEquiv : R → Tropical R) = trop 
Arguments: {R : Type u}
Goal: ⇑Tropical.tropEquiv = Tropical.trop
-- Tropical.tropEquiv_symm_coe_fn
@[simp]
theorem tropEquiv_symm_coe_fn : (tropEquiv.symm : Tropical R → R) = untrop 
Arguments: {R : Type u}
Goal: ⇑Tropical.tropEquiv.symm = Tropical.untrop
-- Tropical.trop_eq_iff_eq_untrop
theorem trop_eq_iff_eq_untrop {x : R} {y} : trop x = y ↔ x = untrop y 
Arguments: {R : Type u} {x : R} {y : Tropical R}
Goal: Tropical.trop x = y ↔ x = Tropical.untrop y
-- Tropical.untrop_eq_iff_eq_trop
theorem untrop_eq_iff_eq_trop {x} {y : R} : untrop x = y ↔ x = trop y 
Arguments: {R : Type u} {x : Tropical R} {y : R}
Goal: Tropical.untrop x = y ↔ x = Tropical.trop y
-- Tropical.injective_trop
theorem injective_trop : Function.Injective (trop : R → Tropical R) 
Arguments: {R : Type u}
Goal: Function.Injective Tropical.trop
-- Tropical.injective_untrop
theorem injective_untrop : Function.Injective (untrop : Tropical R → R) 
Arguments: {R : Type u}
Goal: Function.Injective Tropical.untrop
-- Tropical.surjective_trop
theorem surjective_trop : Function.Surjective (trop : R → Tropical R) 
Arguments: {R : Type u}
Goal: Function.Surjective Tropical.trop
-- Tropical.surjective_untrop
theorem surjective_untrop : Function.Surjective (untrop : Tropical R → R) 
Arguments: {R : Type u}
Goal: Function.Surjective Tropical.untrop
-- Tropical.untrop_le_iff
@[simp]
theorem untrop_le_iff [LE R] {x y : Tropical R} : untrop x ≤ untrop y ↔ x ≤ y 
Arguments: {R : Type u} [LE R] {x y : Tropical R}
Goal: Tropical.untrop x ≤ Tropical.untrop y ↔ x ≤ y
-- Tropical.untrop_lt_iff
@[simp]
theorem untrop_lt_iff [LT R] {x y : Tropical R} : untrop x < untrop y ↔ x < y 
Arguments: {R : Type u} [LT R] {x y : Tropical R}
Goal: Tropical.untrop x < Tropical.untrop y ↔ x < y
-- Tropical.tropOrderIso_coe_fn
@[simp]
theorem tropOrderIso_coe_fn [Preorder R] : (tropOrderIso : R → Tropical R) = trop 
Arguments: {R : Type u} [Preorder R]
Goal: ⇑Tropical.tropOrderIso = Tropical.trop
-- Tropical.tropOrderIso_symm_coe_fn
@[simp]
theorem tropOrderIso_symm_coe_fn [Preorder R] : (tropOrderIso.symm : Tropical R → R) = untrop 
Arguments: {R : Type u} [Preorder R]
Goal: ⇑Tropical.tropOrderIso.symm = Tropical.untrop
-- Tropical.trop_monotone
theorem trop_monotone [Preorder R] : Monotone (trop : R → Tropical R) 
Arguments: {R : Type u} [Preorder R]
Goal: Monotone Tropical.trop
-- Tropical.untrop_monotone
theorem untrop_monotone [Preorder R] : Monotone (untrop : Tropical R → R) 
Arguments: {R : Type u} [Preorder R]
Goal: Monotone Tropical.untrop
-- Tropical.untrop_zero
@[simp]
theorem untrop_zero [Top R] : untrop (0 : Tropical R) = ⊤ 
Arguments: {R : Type u} [Top R]
Goal: Tropical.untrop 0 = ⊤
-- Tropical.trop_top
@[simp]
theorem trop_top [Top R] : trop (⊤ : R) = 0 
Arguments: {R : Type u} [Top R]
Goal: Tropical.trop ⊤ = 0
-- Tropical.trop_coe_ne_zero
@[simp]
theorem trop_coe_ne_zero (x : R) : trop (x : WithTop R) ≠ 0 
Arguments: {R : Type u} (x : R)
Goal: Tropical.trop ↑x ≠ 0
-- Tropical.zero_ne_trop_coe
@[simp]
theorem zero_ne_trop_coe (x : R) : (0 : Tropical (WithTop R)) ≠ trop x 
Arguments: {R : Type u} (x : R)
Goal: 0 ≠ Tropical.trop ↑x
-- Tropical.le_zero
@[simp]
theorem le_zero [LE R] [OrderTop R] (x : Tropical R) : x ≤ 0 
Arguments: {R : Type u} [LE R] [OrderTop R] (x : Tropical R)
Goal: x ≤ 0
-- Tropical.untrop_add
@[simp]
theorem untrop_add (x y : Tropical R) : untrop (x + y) = min (untrop x) (untrop y) 
Arguments: {R : Type u} [LinearOrder R] (x y : Tropical R)
Goal: Tropical.untrop (x + y) = min (Tropical.untrop x) (Tropical.untrop y)
-- Tropical.trop_min
@[simp]
theorem trop_min (x y : R) : trop (min x y) = trop x + trop y 
Arguments: {R : Type u} [LinearOrder R] (x y : R)
Goal: Tropical.trop (min x y) = Tropical.trop x + Tropical.trop y
-- Tropical.trop_inf
@[simp]
theorem trop_inf (x y : R) : trop (x ⊓ y) = trop x + trop y 
Arguments: {R : Type u} [LinearOrder R] (x y : R)
Goal: Tropical.trop (x ⊓ y) = Tropical.trop x + Tropical.trop y
-- Tropical.trop_add_def
theorem trop_add_def (x y : Tropical R) : x + y = trop (min (untrop x) (untrop y)) 
Arguments: {R : Type u} [LinearOrder R] (x y : Tropical R)
Goal: x + y = Tropical.trop (min (Tropical.untrop x) (Tropical.untrop y))
-- Tropical.untrop_sup
@[simp]
theorem untrop_sup (x y : Tropical R) : untrop (x ⊔ y) = untrop x ⊔ untrop y 
Arguments: {R : Type u} [LinearOrder R] (x y : Tropical R)
Goal: Tropical.untrop (x ⊔ y) = Tropical.untrop x ⊔ Tropical.untrop y
-- Tropical.untrop_max
@[simp]
theorem untrop_max (x y : Tropical R) : untrop (max x y) = max (untrop x) (untrop y) 
Arguments: {R : Type u} [LinearOrder R] (x y : Tropical R)
Goal: Tropical.untrop (max x y) = max (Tropical.untrop x) (Tropical.untrop y)
-- Tropical.min_eq_add
@[simp]
theorem min_eq_add : (min : Tropical R → Tropical R → Tropical R) = (· + ·) 
Arguments: {R : Type u} [LinearOrder R]
Goal: min = fun x x_1 => x + x_1
-- Tropical.inf_eq_add
@[simp]
theorem inf_eq_add : ((· ⊓ ·) : Tropical R → Tropical R → Tropical R) = (· + ·) 
Arguments: {R : Type u} [LinearOrder R]
Goal: (fun x x_1 => x ⊓ x_1) = fun x x_1 => x + x_1
-- Tropical.trop_max_def
theorem trop_max_def (x y : Tropical R) : max x y = trop (max (untrop x) (untrop y)) 
Arguments: {R : Type u} [LinearOrder R] (x y : Tropical R)
Goal: max x y = Tropical.trop (max (Tropical.untrop x) (Tropical.untrop y))
-- Tropical.trop_sup_def
theorem trop_sup_def (x y : Tropical R) : x ⊔ y = trop (untrop x ⊔ untrop y) 
Arguments: {R : Type u} [LinearOrder R] (x y : Tropical R)
Goal: x ⊔ y = Tropical.trop (Tropical.untrop x ⊔ Tropical.untrop y)
-- Tropical.add_eq_left
@[simp]
theorem add_eq_left ⦃x y : Tropical R⦄ (h : x ≤ y) : x + y = x 
Arguments: {R : Type u} [LinearOrder R] ⦃x y : Tropical R⦄ (h : x ≤ y)
Goal: x + y = x
-- Tropical.add_eq_right
@[simp]
theorem add_eq_right ⦃x y : Tropical R⦄ (h : y ≤ x) : x + y = y 
Arguments: {R : Type u} [LinearOrder R] ⦃x y : Tropical R⦄ (h : y ≤ x)
Goal: x + y = y
-- Tropical.add_eq_left_iff
theorem add_eq_left_iff {x y : Tropical R} : x + y = x ↔ x ≤ y 
Arguments: {R : Type u} [LinearOrder R] {x y : Tropical R}
Goal: x + y = x ↔ x ≤ y
-- Tropical.add_eq_right_iff
theorem add_eq_right_iff {x y : Tropical R} : x + y = y ↔ y ≤ x 
Arguments: {R : Type u} [LinearOrder R] {x y : Tropical R}
Goal: x + y = y ↔ y ≤ x
-- Tropical.add_self
theorem add_self (x : Tropical R) : x + x = x 
Arguments: {R : Type u} [LinearOrder R] (x : Tropical R)
Goal: x + x = x
-- Tropical.add_eq_iff
theorem add_eq_iff {x y z : Tropical R} : x + y = z ↔ x = z ∧ x ≤ y ∨ y = z ∧ y ≤ x 
Arguments: {R : Type u} [LinearOrder R] {x y z : Tropical R}
Goal: x + y = z ↔ x = z ∧ x ≤ y ∨ y = z ∧ y ≤ x
-- Tropical.add_eq_zero_iff
@[simp]
theorem add_eq_zero_iff {a b : Tropical (WithTop R)} : a + b = 0 ↔ a = 0 ∧ b = 0 
Arguments: {R : Type u} [LinearOrder R] {a b : Tropical (WithTop R)}
Goal: a + b = 0 ↔ a = 0 ∧ b = 0
-- Tropical.trop_add
@[simp]
theorem trop_add [Add R] (x y : R) : trop (x + y) = trop x * trop y 
Arguments: {R : Type u} [Add R] (x y : R)
Goal: Tropical.trop (x + y) = Tropical.trop x * Tropical.trop y
-- Tropical.untrop_mul
@[simp]
theorem untrop_mul [Add R] (x y : Tropical R) : untrop (x * y) = untrop x + untrop y 
Arguments: {R : Type u} [Add R] (x y : Tropical R)
Goal: Tropical.untrop (x * y) = Tropical.untrop x + Tropical.untrop y
-- Tropical.trop_mul_def
theorem trop_mul_def [Add R] (x y : Tropical R) : x * y = trop (untrop x + untrop y) 
Arguments: {R : Type u} [Add R] (x y : Tropical R)
Goal: x * y = Tropical.trop (Tropical.untrop x + Tropical.untrop y)
-- Tropical.trop_zero
@[simp]
theorem trop_zero [Zero R] : trop (0 : R) = 1 
Arguments: {R : Type u} [Zero R]
Goal: Tropical.trop 0 = 1
-- Tropical.untrop_one
@[simp]
theorem untrop_one [Zero R] : untrop (1 : Tropical R) = 0 
Arguments: {R : Type u} [Zero R]
Goal: Tropical.untrop 1 = 0
-- Tropical.untrop_inv
@[simp]
theorem untrop_inv [Neg R] (x : Tropical R) : untrop x⁻¹ = -untrop x 
Arguments: {R : Type u} [Neg R] (x : Tropical R)
Goal: Tropical.untrop x⁻¹ = -Tropical.untrop x
-- Tropical.untrop_div
@[simp]
theorem untrop_div [Sub R] (x y : Tropical R) : untrop (x / y) = untrop x - untrop y 
Arguments: {R : Type u} [Sub R] (x y : Tropical R)
Goal: Tropical.untrop (x / y) = Tropical.untrop x - Tropical.untrop y