Specification of foldr_with_index_aux.
Equations
- list.foldr_with_index_aux_spec f start b as = list.foldr (function.uncurry f) b (list.enum_from start as)
theorem
list.foldr_with_index_aux_spec_cons
{α : Type u}
{β : Type v}
(f : ℕ → α → β → β)
(start : ℕ)
(b : β)
(a : α)
(as : list α) :
list.foldr_with_index_aux_spec f start b (a :: as) = f start a (list.foldr_with_index_aux_spec f (start + 1) b as)
theorem
list.foldr_with_index_aux_eq_foldr_with_index_aux_spec
{α : Type u}
{β : Type v}
(f : ℕ → α → β → β)
(start : ℕ)
(b : β)
(as : list α) :
list.foldr_with_index_aux f start b as = list.foldr_with_index_aux_spec f start b as
theorem
list.foldr_with_index_eq_foldr_enum
{α : Type u}
{β : Type v}
(f : ℕ → α → β → β)
(b : β)
(as : list α) :
list.foldr_with_index f b as = list.foldr (function.uncurry f) b as.enum
theorem
list.indexes_values_eq_filter_enum
{α : Type u}
(p : α → Prop)
[decidable_pred p]
(as : list α) :
list.indexes_values p as = list.filter (p ∘ prod.snd) as.enum
theorem
list.find_indexes_eq_map_indexes_values
{α : Type u}
(p : α → Prop)
[decidable_pred p]
(as : list α) :
list.find_indexes p as = list.map prod.fst (list.indexes_values p as)
Specification of foldl_with_index_aux.
Equations
- list.foldl_with_index_aux_spec f start a bs = list.foldl (λ (a : α) (p : ℕ × β), f p.fst a p.snd) a (list.enum_from start bs)
theorem
list.foldl_with_index_aux_spec_cons
{α : Type u}
{β : Type v}
(f : ℕ → α → β → α)
(start : ℕ)
(a : α)
(b : β)
(bs : list β) :
list.foldl_with_index_aux_spec f start a (b :: bs) = list.foldl_with_index_aux_spec f (start + 1) (f start a b) bs
theorem
list.foldl_with_index_aux_eq_foldl_with_index_aux_spec
{α : Type u}
{β : Type v}
(f : ℕ → α → β → α)
(start : ℕ)
(a : α)
(bs : list β) :
list.foldl_with_index_aux f start a bs = list.foldl_with_index_aux_spec f start a bs
theorem
list.foldl_with_index_eq_foldl_enum
{α : Type u}
{β : Type v}
(f : ℕ → α → β → α)
(a : α)
(bs : list β) :
list.foldl_with_index f a bs = list.foldl (λ (a : α) (p : ℕ × β), f p.fst a p.snd) a bs.enum
theorem
list.mfoldr_with_index_eq_mfoldr_enum
{m : Type u → Type v}
[monad m]
{α : Type u_1}
{β : Type u}
(f : ℕ → α → β → m β)
(b : β)
(as : list α) :
list.mfoldr_with_index f b as = list.mfoldr (function.uncurry f) b as.enum
theorem
list.mfoldl_with_index_eq_mfoldl_enum
{m : Type u → Type v}
[monad m]
[is_lawful_monad m]
{α : Type u_1}
{β : Type u}
(f : ℕ → β → α → m β)
(b : β)
(as : list α) :
list.mfoldl_with_index f b as = list.mfoldl (λ (b : β) (p : ℕ × α), f p.fst b p.snd) b as.enum
def
list.mmap_with_index_aux_spec
{m : Type u → Type v}
[applicative m]
{α : Type u_1}
{β : Type u} :
Specification of mmap_with_index_aux.
Equations
- list.mmap_with_index_aux_spec f start as = list.traverse (function.uncurry f) (list.enum_from start as)
theorem
list.mmap_with_index_aux_spec_cons
{m : Type u → Type v}
[applicative m]
{α : Type u_1}
{β : Type u}
(f : ℕ → α → m β)
(start : ℕ)
(a : α)
(as : list α) :
list.mmap_with_index_aux_spec f start (a :: as) = list.cons <$> f start a <*> list.mmap_with_index_aux_spec f (start + 1) as
theorem
list.mmap_with_index_aux_eq_mmap_with_index_aux_spec
{m : Type u → Type v}
[applicative m]
{α : Type u_1}
{β : Type u}
(f : ℕ → α → m β)
(start : ℕ)
(as : list α) :
list.mmap_with_index_aux f start as = list.mmap_with_index_aux_spec f start as
theorem
list.mmap_with_index_eq_mmap_enum
{m : Type u → Type v}
[applicative m]
{α : Type u_1}
{β : Type u}
(f : ℕ → α → m β)
(as : list α) :
list.mmap_with_index f as = list.traverse (function.uncurry f) as.enum
theorem
list.mmap_with_index'_aux_eq_mmap_with_index_aux
{m : Type u → Type v}
[applicative m]
[is_lawful_applicative m]
{α : Type u_1}
(f : ℕ → α → m punit)
(start : ℕ)
(as : list α) :
list.mmap_with_index'_aux f start as = list.mmap_with_index_aux f start as *> has_pure.pure punit.star
theorem
list.mmap_with_index'_eq_mmap_with_index
{m : Type u → Type v}
[applicative m]
[is_lawful_applicative m]
{α : Type u_1}
(f : ℕ → α → m punit)
(as : list α) :