proposition: Lemma process_eval_rec5_loss_draw_ge ht (res : estate) l : Hiv l -> ht_valid ht -> \smin_(i <- l) f1 i <= draw -> lossdraw <= process_eval_rec5 f2 eloss edraw ht res l.
proof: Proof. elim: l ht res => [|i l IH] ht res HHiv Hv; first by rewrite big_nil. rewrite big_cons [process_eval_rec5 _ _ _ _ _ _]/= => H. 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 _ _ _ _). have [E2|/ltnW E2] := leqP (f1 i) (\smin_(j <- l) f1 j). rewrite smin_lel // in H. apply: leq_trans (ge_process_eval_rec5 _ _ _ _ _ _). rewrite (esflip_le drawwin). by apply: leq_trans E1 (H5loss_draw_le HHiv _ _). rewrite smin_ler // in H. by apply: IH => //; apply: (H5loss_draw_table HHiv). have [E2|E2] := leqP edraw (f2 _ _ _ _). have [E3|/ltnW E3] := leqP (f1 i) (\smin_(j <- l) f1 j). rewrite smin_lel // in H. apply: leq_trans (ge_process_eval_rec5 _ _ _ _ _ _). rewrite (esflip_le drawwin). by apply: leq_trans _ (H5loss_draw_le HHiv _ _). rewrite smin_ler // in H. by apply: IH => //; apply: (H5loss_draw_table HHiv). have [E3|E3] := leqP (f2 eloss edraw ht i) eloss. by rewrite (le_elossE E3). apply: leq_trans (ge_process_eval_rec5 _ _ _ _ _ _). rewrite (esflip_le drawwin). by apply: leq_trans (ltnW E2) _. Qed.
imports: HB(structures) mathcomp(all_ssreflect) mathcomp(all_algebra)