Results about monadic operations on Array, in terms of SatisfiesM. #
The pure versions of these theorems are proved in Batteries.Data.Array.Lemmas directly,
in order to minimize dependence on SatisfiesM.
theorem
Array.SatisfiesM_foldlM
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
[LawfulMonad m]
{as : Array α}
{init : β}
{motive : Nat → β → Prop}
{f : β → α → m β}
(h0 : motive 0 init)
(hf : ∀ (i : Fin as.size) (b : β), motive (↑i) b → SatisfiesM (motive (↑i + 1)) (f b as[i]))
:
SatisfiesM (motive as.size) (foldlM f init as)
theorem
Array.SatisfiesM_mapM
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
[LawfulMonad m]
{as : Array α}
{f : α → m β}
{motive : Nat → Prop}
{p : Fin as.size → β → Prop}
(h0 : motive 0)
(hs : ∀ (i : Fin as.size), motive ↑i → SatisfiesM (fun (x : β) => p i x ∧ motive (↑i + 1)) (f as[i]))
:
theorem
Array.size_mapM
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
[LawfulMonad m]
(f : α → m β)
(as : Array α)
:
theorem
Array.SatisfiesM_anyM
{m : Type → Type u_1}
{α : Type u_2}
{start stop : Nat}
[Monad m]
[LawfulMonad m]
{p : α → m Bool}
{as : Array α}
(hstart : start ≤ min stop as.size)
(tru : Prop)
(fal : Nat → Prop)
(h0 : fal start)
(hp :
∀ (i : Fin as.size), ↑i < stop → fal ↑i → SatisfiesM (fun (x : Bool) => bif x then tru else fal (↑i + 1)) (p as[i]))
:
SatisfiesM (fun (res : Bool) => bif res then tru else fal (min stop as.size)) (anyM p as start stop)
theorem
Array.SatisfiesM_foldrM
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
[LawfulMonad m]
{as : Array α}
{init : β}
{motive : Nat → β → Prop}
{f : α → β → m β}
(h0 : motive as.size init)
(hf : ∀ (i : Fin as.size) (b : β), motive (↑i + 1) b → SatisfiesM (motive ↑i) (f as[i] b))
:
SatisfiesM (motive 0) (foldrM f init as)
theorem
Array.SatisfiesM_mapFinIdxM
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
[LawfulMonad m]
{as : Array α}
{f : (i : Nat) → α → i < as.size → m β}
{motive : Nat → Prop}
{p : (i : Nat) → β → i < as.size → Prop}
(h0 : motive 0)
(hs : ∀ (i : Nat) (h : i < as.size), motive i → SatisfiesM (fun (x : β) => p i x h ∧ motive (i + 1)) (f i as[i] h))
:
theorem
Array.SatisfiesM_mapIdxM
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
[LawfulMonad m]
{as : Array α}
{f : Nat → α → m β}
{p : (i : Nat) → β → i < as.size → Prop}
{motive : Nat → Prop}
(h0 : motive 0)
(hs : ∀ (i : Nat) (h : i < as.size), motive i → SatisfiesM (fun (x : β) => p i x h ∧ motive (i + 1)) (f i as[i]))
:
theorem
Array.size_mapFinIdxM
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
[LawfulMonad m]
(as : Array α)
(f : (i : Nat) → α → i < as.size → m β)
:
SatisfiesM (fun (arr : Array β) => arr.size = as.size) (as.mapFinIdxM f)
theorem
Array.size_modifyM
{m : Type u_1 → Type u_2}
{α : Type u_1}
[Monad m]
[LawfulMonad m]
(as : Array α)
(i : Nat)
(f : α → m α)
: