Hypothesis prog_is_wf : wf_prog max_arity prog.

(* Specifications: *)
Lemma activation_bound_spec :
forall proof_tree, wf proof_tree -> forall p, In p (activations proof_tree) ->
let (i, s) := rule_subst_of_cbv_function p in
term_size (subst s (rhs_of_rule (nth i prog rule_default))) <= activation_bound prog (term_size (proj_left p)).
Lemma activation_bound_spec ( subst_default : variable -> value) :
forall proof_tree, wf proof_tree -> forall p, In p (activations proof_tree) ->
let (i, s) := rule_subst_of_cbv_update subst_default p in
term_size (subst s (rhs_of_rule (nth i prog rule_default))) <= activation_bound prog (term_size (proj_left p)).