File: Mathlib/Algebra/Tropical/BigOperators.lean

full_name: List.trop_sum
code: theorem List.trop_sum [AddMonoid R] (l : List R) : trop l.sum = List.prod (l.map trop) 
goal: Tropical.trop l.sum = (List.map Tropical.trop l).prod

full_name: Multiset.trop_sum
code: theorem Multiset.trop_sum [AddCommMonoid R] (s : Multiset R) :
    trop s.sum = Multiset.prod (s.map trop) 
goal: Tropical.trop s.sum = (Multiset.map Tropical.trop s).prod

full_name: trop_sum
code: theorem trop_sum [AddCommMonoid R] (s : Finset S) (f : S → R) :
    trop (∑ i ∈ s, f i) = ∏ i ∈ s, trop (f i) 
goal: Tropical.trop (∑ i ∈ s, f i) = ∏ i ∈ s, Tropical.trop (f i)

full_name: List.untrop_prod
code: theorem List.untrop_prod [AddMonoid R] (l : List (Tropical R)) :
    untrop l.prod = List.sum (l.map untrop) 
goal: Tropical.untrop l.prod = (List.map Tropical.untrop l).sum

full_name: Multiset.untrop_prod
code: theorem Multiset.untrop_prod [AddCommMonoid R] (s : Multiset (Tropical R)) :
    untrop s.prod = Multiset.sum (s.map untrop) 
goal: Tropical.untrop s.prod = (Multiset.map Tropical.untrop s).sum

full_name: untrop_prod
code: theorem untrop_prod [AddCommMonoid R] (s : Finset S) (f : S → Tropical R) :
    untrop (∏ i ∈ s, f i) = ∑ i ∈ s, untrop (f i) 
goal: Tropical.untrop (∏ i ∈ s, f i) = ∑ i ∈ s, Tropical.untrop (f i)

full_name: List.trop_minimum
code: theorem List.trop_minimum [LinearOrder R] (l : List R) :
    trop l.minimum = List.sum (l.map (trop ∘ WithTop.some)) 
goal: Tropical.trop l.minimum = (List.map (Tropical.trop ∘ WithTop.some) l).sum

full_name: Multiset.trop_inf
code: theorem Multiset.trop_inf [LinearOrder R] [OrderTop R] (s : Multiset R) :
    trop s.inf = Multiset.sum (s.map trop) 
goal: Tropical.trop s.inf = (Multiset.map Tropical.trop s).sum

full_name: Finset.trop_inf
code: theorem Finset.trop_inf [LinearOrder R] [OrderTop R] (s : Finset S) (f : S → R) :
    trop (s.inf f) = ∑ i ∈ s, trop (f i) 
goal: Tropical.trop (s.inf f) = ∑ i ∈ s, Tropical.trop (f i)

full_name: trop_sInf_image
code: theorem trop_sInf_image [ConditionallyCompleteLinearOrder R] (s : Finset S) (f : S → WithTop R) :
    trop (sInf (f '' s)) = ∑ i ∈ s, trop (f i) 
goal: Tropical.trop (sInf (f '' ↑s)) = ∑ i ∈ s, Tropical.trop (f i)

full_name: trop_iInf
code: theorem trop_iInf [ConditionallyCompleteLinearOrder R] [Fintype S] (f : S → WithTop R) :
    trop (⨅ i : S, f i) = ∑ i : S, trop (f i) 
goal: Tropical.trop (⨅ i, f i) = ∑ i : S, Tropical.trop (f i)

full_name: Multiset.untrop_sum
code: theorem Multiset.untrop_sum [LinearOrder R] [OrderTop R] (s : Multiset (Tropical R)) :
    untrop s.sum = Multiset.inf (s.map untrop) 
goal: Tropical.untrop s.sum = (Multiset.map Tropical.untrop s).inf

full_name: Finset.untrop_sum'
code: theorem Finset.untrop_sum' [LinearOrder R] [OrderTop R] (s : Finset S) (f : S → Tropical R) :
    untrop (∑ i ∈ s, f i) = s.inf (untrop ∘ f) 
goal: Tropical.untrop (∑ i ∈ s, f i) = s.inf (Tropical.untrop ∘ f)

full_name: untrop_sum_eq_sInf_image
code: 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) 
goal: Tropical.untrop (∑ i ∈ s, f i) = sInf (Tropical.untrop ∘ f '' ↑s)

full_name: untrop_sum
code: theorem untrop_sum [ConditionallyCompleteLinearOrder R] [Fintype S] (f : S → Tropical (WithTop R)) :
    untrop (∑ i : S, f i) = ⨅ i : S, untrop (f i) 
goal: Tropical.untrop (∑ i : S, f i) = ⨅ i, Tropical.untrop (f i)

full_name: Finset.untrop_sum
code: /-- Note we cannot use `i ∈ s` instead of `i : s` here
as it is simply not true on conditionally complete lattices! -/
theorem Finset.untrop_sum [ConditionallyCompleteLinearOrder R] (s : Finset S)
    (f : S → Tropical (WithTop R)) : untrop (∑ i ∈ s, f i) = ⨅ i : s, untrop (f i) 
goal: Tropical.untrop (∑ i ∈ s, f i) = ⨅ i, Tropical.untrop (f ↑i)