Definition expfo(m:fmap)(x:dart)(z t:dart):=
  expf (L (B m zero x) zero x (A m zero x)) z t.

(* Specifications: *)
Lemma expfo_expf:forall(m:fmap)(x:dart)(z t:dart), inv_hmap m -> succ m zero x -> (expfo m x z t <-> expf m z t).
Lemma eqc_B_CN: forall(m:fmap)(k:dim)(x z t:dart), inv_hmap m -> succ m k x -> let xk:= A m k x in let m0:= B m k x in eqc m z t -> (eqc m0 z t \/ eqc m0 z x /\ eqc m0 xk t \/ eqc m0 z xk /\ eqc m0 x t).
Lemma eqc_B_CS: forall(m:fmap)(k:dim)(x z t:dart), inv_hmap m -> succ m k x -> let xk:= A m k x in let m0:= B m k x in (eqc m0 z t \/ eqc m0 z x /\ eqc m0 xk t \/ eqc m0 z xk /\ eqc m0 x t) -> eqc m z t.
Theorem eqc_B: forall(m:fmap)(k:dim)(x z t:dart), inv_hmap m -> succ m k x -> let xk:= A m k x in let m0:= B m k x in (eqc m z t <-> (eqc m0 z t \/ eqc m0 z x /\ eqc m0 xk t \/ eqc m0 z xk /\ eqc m0 x t)).