Require Export Undecidability.Shared.Libs.PSL.Base Lia Arith PArith.
From Undecidability.L Require Export MoreList.


#[global]
Instance le_preorder : PreOrder le.
Proof.
  constructor. all:cbv. all:intros;.
Qed.


#[global]
Instance S_le_proper : Proper (le le) S.
Proof.
  cbv. fold plus. intros. .
Qed.


#[global]
Instance plus_le_proper : Proper (le le le) plus.
Proof.
  cbv. fold plus. intros. .
Qed.


#[global]
Instance mult_le_proper : Proper (le le le) mult.
Proof.
  cbv. intros.
  apply mult_le_compat. all:eauto.
Qed.


#[global]
Instance pow_le_proper : Proper (le eq le) Nat.pow.
Proof.
  cbv - [Nat.pow]. intros. subst. apply Nat.pow_le_mono_l. easy.
Qed.


#[global]
Instance max_le_proper : Proper (le le le) max.
Proof.
repeat intro. repeat eapply Nat.max_case_strong;.
Qed.


#[global]
Instance min_le_proper : Proper (le le le) min.
Proof.
repeat intro. repeat eapply Nat.min_case_strong;.
Qed.


#[global]
Instance Nat_log2_le_Proper : Proper (le le) Nat.log2.
Proof.
  repeat intro. apply Nat.log2_le_mono. assumption.
Qed.


#[global]
Instance Pos_to_nat_le_Proper : Proper (Pos.le le) Pos.to_nat.
Proof.
  repeat intro. apply Pos2Nat.inj_le. assumption.
Qed.


#[global]
Instance Pos_add_le_Proper : Proper (Pos.le Pos.le Pos.le) Pos.add.
Proof.
  repeat intro. eapply Pos.add_le_mono. 3:eauto. all:eauto.
Qed.


Lemma nth_error_Some_lt A (H:list A) a x : nth_error H a = Some x a < |H|.
Proof.
  intros eq. revert H eq. induction a;intros;destruct H;cbn in *;inv eq. . apply IHa in . .
Qed.


Definition maxP (P: Prop) m := P m ( m', P m' m' m).

Lemma sumn_le_bound l c :
  ( n, n l n c) sumn l length l * c.
Proof.
  induction l;cbn. easy.
  intros H.
  rewrite IHl, H. all:now eauto.
Qed.


Lemma sumn_map_le_pointwise X (xs:list X) f1 f2:
  ( x, x xs x x) sumn (map xs) sumn (map xs).
Proof.
  intros Hle.
  induction xs. easy.
  cbn. rewrite Hle. 2:easy. rewrite IHxs. easy. intros. eauto.
Qed.