Local Notation "<[nRA]>" := action_by : action_scope.

(* Specifications: *)
Lemma actbyE x a : x \in R -> a \in A -> <[nRA]>%act x a = to x a.
Lemma astab_actby S : 'C(S | <[nRA]>) = 'C_A(R :&: S | to).
Lemma astabs_actby S : 'N(S | <[nRA]>) = 'N_A(R :&: S | to).
Lemma acts_actby (B : {set aT}) S :
[acts B, on S | <[nRA]>] = (B \subset A) && [acts B, on R :&: S | to].
Lemma subact_is_action : is_action subact_dom subact.
Canonical subaction := Action subact_is_action.
Lemma astab_subact S : 'C(S | subaction) = subact_dom :&: 'C(val @: S | to).
Lemma astabs_subact S : 'N(S | subaction) = subact_dom :&: 'N(val @: S | to).
Lemma afix_subact A :
A \subset subact_dom -> 'Fix_subaction(A) = val @^-1: 'Fix_to(A).