proposition: Lemma process_eval_rec5_loss_draw_le ht (res : estate) l : Hiv l -> ht_valid ht -> draw <= res -> draw <= \smin_(i <- l) f1 i -> process_eval_rec5 f2 eloss edraw ht res l <= edraw.
proof: Proof. elim: l ht res => [|i l IH] ht res HHiv Hv. by rewrite big_nil (esflip_le _ edraw). rewrite big_cons [process_eval_rec5 _ _ _ _ _ _]/= => H1 H2. case E : f2 => [ht1 res1]. have -> : res1 = f2 eloss edraw ht i by rewrite E. have -> : ht1 = f2 eloss edraw ht i by rewrite E. have HHiv1 := Hiv_cons HHiv. have [E1|E1] := leqP res (f2 _ _ _ _). apply: IH => //; first by apply: (H5loss_draw_table HHiv). by apply: leq_trans H2 (ge_sminr _ _). rewrite ifT; last first. apply (H5loss_draw_ge HHiv) => //. apply: leq_trans H2 (ge_sminl _ _). apply: IH => //; first by apply: (H5loss_draw_table HHiv). by apply: (H5loss_draw_ge HHiv) => //; apply: leq_trans H2 (ge_sminl _ _). by apply: leq_trans H2 (ge_sminr _ _). Qed.
imports: HB(structures) mathcomp(all_ssreflect) mathcomp(all_algebra)