maybe.bend source

maybe.bend on the hub · documented module

# bend-mathlib/maybe.bend: Maybe monad and map laws over &2.import Base# Left identity: binding a pure value applies the function.law maybe_pure_bind:  for ~A: Data  for ~B: Data  for -f: A -> Maybe<&2, B>  for -x: A  {Maybe.bind(&2, A, B, Maybe.pure(&2, A, x), f) == f(x) : Maybe<&2, B>}def maybe_pure_bind(A, B, f, x):  {==}# Right identity: binding pure is the identity.law maybe_bind_pure:  for ~A: Data  for m: Maybe<&2, A>  {Maybe.bind(&2, A, A, m, x => Maybe.pure(&2, A, x)) == m : Maybe<&2, A>}def maybe_bind_pure(A, m):  match m:    case None{}:      {==}    case Some{x}:      {==}# Bind is associative.law maybe_bind_assoc:  for ~A: Data  for ~B: Data  for ~C: Data  for -f: A -> Maybe<&2, B>  for -g: B -> Maybe<&2, C>  for m: Maybe<&2, A>  {Maybe.bind(&2, B, C, Maybe.bind(&2, A, B, m, f), g) == Maybe.bind(&2, A, C, m, x => Maybe.bind(&2, B, C, f(x), g)) : Maybe<&2, C>}def maybe_bind_assoc(A, B, C, f, g, m):  match m:    case None{}:      {==}    case Some{x}:      {==}# Mapping a pure value is pure of the mapped value.law maybe_map_pure:  for ~A: Data  for ~B: Data  for -f: A -> B  for -x: A  {Maybe.map(&2, A, B, f, Maybe.pure(&2, A, x)) == Maybe.pure(&2, B, f(x)) : Maybe<&2, B>}def maybe_map_pure(A, B, f, x):  {==}# Mapping a composition maps the composition.law maybe_map_compose:  for ~A: Data  for ~B: Data  for ~C: Data  for -f: A -> B  for -g: B -> C  for m: Maybe<&2, A>  {Maybe.map(&2, B, C, g, Maybe.map(&2, A, B, f, m)) == Maybe.map(&2, A, C, x => g(f(x)), m) : Maybe<&2, C>}def maybe_map_compose(A, B, C, f, g, m):  match m:    case None{}:      {==}    case Some{x}:      {==}# --- generated: _sym twins (tools/mathlib/twins.ts), do not edit ---# Left identity: binding a pure value applies the function, reversed to rewrite toward the simple side.law maybe_pure_bind_sym:  for ~A: Data  for ~B: Data  for -f: A -> Maybe<&2, B>  for -x: A  {f(x) == Maybe.bind(&2, A, B, Maybe.pure(&2, A, x), f) : Maybe<&2, B>}def maybe_pure_bind_sym(A, B, f, x):  Equal.sym(Maybe<&2, B>, Maybe.bind(&2, A, B, Maybe.pure(&2, A, x), f), f(x), maybe_pure_bind(~A, ~B, f, x))# Right identity: binding pure is the identity, reversed to rewrite toward the simple side.law maybe_bind_pure_sym:  for ~A: Data  for m: Maybe<&2, A>  {m == Maybe.bind(&2, A, A, m, x => Maybe.pure(&2, A, x)) : Maybe<&2, A>}def maybe_bind_pure_sym(A, m):  Equal.sym(Maybe<&2, A>, Maybe.bind(&2, A, A, m, x => Maybe.pure(&2, A, x)), m, maybe_bind_pure(~A, m))# Bind is associative, reversed to rewrite toward the simple side.law maybe_bind_assoc_sym:  for ~A: Data  for ~B: Data  for ~C: Data  for -f: A -> Maybe<&2, B>  for -g: B -> Maybe<&2, C>  for m: Maybe<&2, A>  {Maybe.bind(&2, A, C, m, x => Maybe.bind(&2, B, C, f(x), g)) == Maybe.bind(&2, B, C, Maybe.bind(&2, A, B, m, f), g) : Maybe<&2, C>}def maybe_bind_assoc_sym(A, B, C, f, g, m):  Equal.sym(Maybe<&2, C>, Maybe.bind(&2, B, C, Maybe.bind(&2, A, B, m, f), g), Maybe.bind(&2, A, C, m, x => Maybe.bind(&2, B, C, f(x), g)), maybe_bind_assoc(~A, ~B, ~C, f, g, m))# Mapping a pure value is pure of the mapped value, reversed to rewrite toward the simple side.law maybe_map_pure_sym:  for ~A: Data  for ~B: Data  for -f: A -> B  for -x: A  {Maybe.pure(&2, B, f(x)) == Maybe.map(&2, A, B, f, Maybe.pure(&2, A, x)) : Maybe<&2, B>}def maybe_map_pure_sym(A, B, f, x):  Equal.sym(Maybe<&2, B>, Maybe.map(&2, A, B, f, Maybe.pure(&2, A, x)), Maybe.pure(&2, B, f(x)), maybe_map_pure(~A, ~B, f, x))# Mapping a composition maps the composition, reversed to rewrite toward the simple side.law maybe_map_compose_sym:  for ~A: Data  for ~B: Data  for ~C: Data  for -f: A -> B  for -g: B -> C  for m: Maybe<&2, A>  {Maybe.map(&2, A, C, x => g(f(x)), m) == Maybe.map(&2, B, C, g, Maybe.map(&2, A, B, f, m)) : Maybe<&2, C>}def maybe_map_compose_sym(A, B, C, f, g, m):  Equal.sym(Maybe<&2, C>, Maybe.map(&2, B, C, g, Maybe.map(&2, A, B, f, m)), Maybe.map(&2, A, C, x => g(f(x)), m), maybe_map_compose(~A, ~B, ~C, f, g, m))