proposition: Lemma process_eval_rec5_draw_win_le ht (res : estate) l : Hiv l -> ht_valid ht -> \smin_(i <- l) f1 i <= draw -> edraw <= process_eval_rec5 f2 edraw ewin 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 edraw ewin ht i by rewrite E. have -> : ht1 = f2 edraw ewin 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 edraw). by apply: leq_trans E1 (H5draw_win_le HHiv _ _). rewrite smin_ler // in H. by apply: IH => //; apply: (H5draw_win_table HHiv). have [E2|E2] := leqP ewin (f2 edraw ewin ht i). have [E3|/ltnW E3] := leqP (f1 i) (\smin_(j <- l) f1 j). rewrite smin_lel // in H. rewrite leqNgt in E2; case/negP: E2. by apply: leq_ltn_trans (H5draw_win_le HHiv _ _) _. rewrite smin_ler // in H. by apply: IH => //; apply: (H5draw_win_table HHiv). have [E3|E3] := leqP (f2 _ _ _ i) draw. by case: f2 E3 => // h; case; case: (l). have [E4|E4] := leqP (f1 i) draw. rewrite ltnNge in E3; case/negP: E3. by apply: (H5draw_win_le HHiv). rewrite (H5draw_win_win HHiv) // in E2. apply: ge_winE. by case: f1 E4. Qed.
imports: HB(structures) mathcomp(all_ssreflect) mathcomp(all_algebra)