intros.
intros.
set (x_1 := cA_1 m one x) in |- *.
set (xh0 := top m zero x) in |- *.
set (xh0_1 := cA_1 m one xh0) in |- *.
assert
((expf m x_1 z \/ expf m xh0_1 z) /\ (expf m x_1 t \/ expf m xh0_1 t) \/
expf m z t /\ ~ expf m x_1 z /\ ~ expf m xh0_1 z).
unfold x_1 in |- *; unfold xh0_1 in |- *; unfold xh0 in |- *.
apply expf_transfert.
tauto.
apply succ_exd with zero.
tauto.
tauto.
fold x0 in |- *; fold xb0 in |- *.
tauto.
fold xb0 in |- *.
fold x0 in |- *.
tauto.
apply expf_B0_CS_2_aux.
tauto.
tauto.
fold x0 in |- *.
fold xb0 in |- *.
tauto.
fold x_1 in |- *.
fold xh0 in |- *.
fold xh0_1 in |- *.
tauto.
Qed.
(* FROM WHERE, BY COMBINING: *)

Theorem expf_B0_CS:forall (m:fmap)(x z t:dart),
inv_hmap m -> succ m zero x ->
let x0 := cA m zero x in
let xb0 := bottom m zero x in
let x_1 := cA_1 m one x in
let xh0 := top m zero x in
let xh0_1 := cA_1 m one xh0 in
(if expf_dec m x0 xb0
then
betweenf m x_1 z xb0 /\ betweenf m x_1 t xb0
\/ betweenf m xh0_1 z x0 /\ betweenf m xh0_1 t x0
\/ ~ expf m x_1 z /\ expf m z t
else
expf m xb0 z /\ expf m x0 t
\/ expf m xb0 t /\ expf m x0 z
\/ expf m z t)
-> expf (B m zero x) z t.