Section CR.
Open Scope recurrence_scope.
Inductive CR :=
| liftBRToCR: BR -> CR
| recurCR: R -> (R -> R -> R) -> CR -> CR
.
(** Interpret a CR as nested BRs as described in section 2 - Chains of Recurrences *)
Fixpoint CR_to_BR (cr: CR): BR :=
match cr with
| liftBRToCR br =>  br
| recurCR r0 bop cr' => mkBR r0 bop (evalAtIx (CR_to_BR cr'))
end.
Definition evalCR(cr: CR) (n: nat): R := (CR_to_BR cr) # n.
(** Define what it means to evaluate a CR *)
Global Instance loopVariantCR : LoopVariant CR R :=
{
evalAtIx (cr: CR) (n: nat) := evalCR cr n
}.
Hint Resolve creval_lift_br_to_cr_inverses: Recurrence.
Hint Rewrite @evalCR_zero: Recurrence.
Hint Rewrite @evalCR_step: Recurrence.
(** Now that we know what happens at 0 and inductive case, we're done *)
Opaque evalCR.
Hint Resolve add_const_to_CR : Recurrence.
Hint Rewrite @add_const_to_CR : Recurrence.

(** Lemma 17 *)
Lemma mul_const_to_CR:
forall `{Ring R} (r c0: R) (cr: CR) (n: nat),
r * ((recurCR c0 mult cr) # n) =
((recurCR (c0 * r) mult cr) # n).