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From Coq Require Export ssreflect.
From stdpp Require Export sorting.
From iris.algebra Require Export big_op.
From Coq.ZArith Require Import Znumtheory.
From stdpp Require Import gmap list.
From iris.program_logic Require Import weakestpre.
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From iris.bi Require Import bi.
From iris.proofmode Require Import proofmode.
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From stdpp Require Import natmap.
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From refinedc.lithium Require Import Z_bitblast.
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From RecordUpdate Require Export RecordSet.
Export RecordSetNotations.

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Set Default Proof Using "Type".
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Export Unset Program Cases.
Export Set Keyed Unification.

(* We always annotate hints with locality ([Global] or [Local]). This enforces
that at least global hints are annotated. *)
Export Set Warnings "+deprecated-hint-without-locality".
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(* ensure that set from RecordUpdate simplifies when it is applied to a concrete value *)
Arguments set _ _ _ _ _ !_ /.

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Typeclasses Opaque is_Some.
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(* This is necessary since otherwise keyed unification unfolds these definitions *)
Global Opaque rotate_nat_add rotate_nat_sub.

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Typeclasses Opaque Z.divide Z.modulo Z.div Z.shiftl Z.shiftr.
Arguments min : simpl nomatch.

Arguments Z.testbit : simpl never.
Arguments Z.shiftl : simpl never.
Arguments Z.shiftr : simpl never.
Arguments N.shiftl : simpl never.
Arguments N.shiftr : simpl never.
Arguments Pos.shiftl : simpl never.
Arguments Pos.shiftr : simpl never.
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Global Opaque Z.shiftl Z.shiftr.
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Notation "'[@{' A '}' x ; y ; .. ; z ]" :=  (@cons A x (@cons A y .. (@cons A z (@nil A)) ..)) (only parsing) : list_scope.
Notation "'[@{' A '}' x ]" := (@cons A x nil) (only parsing) : list_scope.
Notation "'[@{' A '}' ]" := (@nil A) (only parsing) : list_scope.

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(** More automation for modular arithmetics. *)
Ltac Zify.zify_post_hook ::= Z.to_euclidean_division_equations.

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(** * tactics *)
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Lemma rel_to_eq {A} (R : A  A  Prop) `{!Reflexive R} x y:
  x = y  R x y.
Proof. move => ->. reflexivity. Qed.

Ltac fast_reflexivity :=
  notypeclasses refine (rel_to_eq _ _ _ _); [solve [refine _] |];
  lazymatch goal with
  | |- ?x = ?y => lazymatch x with | y => exact: (eq_refl x) end
  end.

Ltac get_head e :=
  lazymatch e with
  | ?h _ => get_head constr:(h)
  | _    => constr:(e)
  end.

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(* Checks that a term is closed using a trick by Jason Gross. *)
Ltac check_closed t :=
  assert_succeeds (
    let x := fresh "x" in
    exfalso; pose t as x; revert x;
    repeat match goal with H : _ |- _ => clear H end;
    lazymatch goal with H : _ |- _ => fail | _ => idtac end
  ).

Tactic Notation "reduce_closed" constr(x) :=
  check_closed x;
  let r := eval vm_compute in x in
  change_no_check x with r in *
.

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(* from https://mattermost.mpi-sws.org/iris/pl/dcktjjjpsiycmrieyh74bzoagr *)
Ltac solve_sep_entails :=
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  try (apply (anti_symm _));
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  iIntros;
  repeat iMatchHyp (fun H P =>
    lazymatch P with
    | (_  _)%I => iDestruct H as "[??]"
    | ( _, _)%I => iDestruct H as (?) "?"
    end);
  eauto with iFrame.

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Lemma tac_evar_safe_vm_compute {A} (x y : A):
  ( z, y = z  x = z) 
  x = y.
Proof. naive_solver. Qed.
(* vm_compute likes to mess with evars that it find in the goal so
evar_safe_vm_compute hides them. *)
Ltac evar_safe_vm_compute :=
  apply tac_evar_safe_vm_compute;
  let H := fresh in
  intros ? H;
  vm_compute;
  apply H.

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(*
The following tactics are currently not used.

(* TODO: This tactic is quite inefficient (it calls unification for
every subterm in the goal and hyps). Can we do something about this? *)
Tactic Notation "select" "subterm" open_constr(pat) tactic3(tac) :=
  match goal with
  | |- context H [?x]       => unify x pat; tac x
  | _ : context H [?x] |- _ => unify x pat; tac x
  end.

Tactic Notation "reduce" "pattern" open_constr(pat) :=
  repeat select subterm pat (fun x => reduce_closed x).

(* TODO: This tactic is quite inefficient (it calls unification for
every subterm in the goal and hyps). Can we do something about this? *)
Tactic Notation "select" "closed" "subterm" "of" "type" constr(T) tactic3(tac) :=
  match goal with
  | |- context H [?x]       => let ty := type of x in unify ty T; check_closed x; tac x
  | _ : context H [?x] |- _ => let ty := type of x in unify ty T; check_closed x; tac x
  end.

Ltac evalZ :=
  repeat select closed subterm of type Z (fun x => progress reduce_closed x).
 *)

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(** * typeclasses *)
Inductive TCOneIsSome {A} : option A  option A  Prop :=
| tc_one_is_some_left n1 o2 : TCOneIsSome (Some n1) o2
| tc_one_is_some_right o1 n2 : TCOneIsSome o1 (Some n2).
Existing Class TCOneIsSome.
Global Existing Instance tc_one_is_some_left.
Global Existing Instance tc_one_is_some_right.

Inductive TCOneIsSome3 {A} : option A  option A  option A  Prop :=
| tc_one_is_some3_left n1 o2 o3 : TCOneIsSome3 (Some n1) o2 o3
| tc_one_is_some3_middle o1 n2 o3 : TCOneIsSome3 o1 (Some n2) o3
| tc_one_is_some3_right o1 o2 n3 : TCOneIsSome3 o1 o2 (Some n3).
Existing Class TCOneIsSome3.
Global Existing Instance tc_one_is_some3_left.
Global Existing Instance tc_one_is_some3_middle.
Global Existing Instance tc_one_is_some3_right.

Definition exists_dec_unique {A} (x : A) (P : _  Prop) : ( y, P y  P x)  Decision (P x)  Decision ( y, P y).
Proof.
  intros Hx Hdec.
  refine (cast_if (decide (P x))).
  - abstract by eexists _.
  - abstract naive_solver.
Defined.

(** * bool *)
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Definition Z_of_bool (b : bool) : Z :=
  if b then 1 else 0.
Typeclasses Opaque Z_of_bool.

Lemma Z_of_bool_true b: Z_of_bool b  0  b = true.
Proof. destruct b; naive_solver. Qed.

Lemma Z_of_bool_false b: Z_of_bool b = 0  b = false.
Proof. destruct b; naive_solver. Qed.

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Lemma Is_true_eq (b : bool) : b  b = true.
Proof. by case: b. Qed.
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Lemma bool_decide_eq_x_x_true {A} (x : A) `{!Decision (x = x)} :
  bool_decide (x = x) = true.
Proof. by case_bool_decide.  Qed.
Lemma if_bool_decide_eq_branches {A} P `{!Decision P} (x : A) :
  (if bool_decide P then x else x) = x.
Proof. by case_bool_decide. Qed.
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Lemma negb_bool_decide_eq {A} (x y : A) `{!EqDecision A} :
  negb (bool_decide (x = y)) = bool_decide (x  y).
Proof. by repeat case_bool_decide. Qed.
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(** * apply_dfun *)
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(* TODO: does something like this exist in Iris? *)
Definition apply_dfun {A B} (f : A -d> B) (x : A) : B := f x.
Arguments apply_dfun : simpl never.

Global Instance apply_dfun_ne A B n:
  Proper ((dist n) ==> (=) ==> (dist n)) (@apply_dfun A B).
Proof. intros ?? H1 ?? ->. rewrite /apply_dfun. apply H1. Qed.

Global Instance apply_dfun_proper A B :
  Proper (() ==> (=) ==> ()) (@apply_dfun A B).
Proof. intros ?? H1 ?? ->. rewrite /apply_dfun. apply H1. Qed.

Global Instance discrete_fn_proper A B `{LeibnizEquiv A} (f : A -d> B):
  Proper (() ==> ()) f.
Proof. by intros ?? ->%leibniz_equiv. Qed.

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(** * list *)
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Lemma zip_fmap_r {A B C} (l1 : list A) (l2 : list B) (f : B  C) :
  zip l1 (f <$> l2) = (λ x, (x.1, f x.2)) <$>  zip l1 l2.
Proof. rewrite zip_with_fmap_r zip_with_zip. by apply: list_fmap_ext => // [[]]. Qed.

Lemma zip_with_nil_inv' {A B C : Type} (f : A  B  C) (l1 : list A) (l2 : list B) :
  length l1 = length l2  zip_with f l1 l2 = []  l1 = []  l2 = [].
Proof.
  move => Hlen /zip_with_nil_inv [] H; rewrite H /= in Hlen;
  split => //; by apply nil_length_inv.
Qed.

Lemma take_elem_of {A} (x : A) n l:
  x  take n l   i, (i < n)%nat  l !! i = Some x.
Proof.
  rewrite elem_of_list_lookup. f_equiv => i.
  destruct (decide (i < n)%nat);[rewrite lookup_take | rewrite lookup_take_ge]; naive_solver lia.
Qed.

Lemma list_find_Some' A (l : list A) x P `{! x, Decision (P x)}:
  list_find P l = Some x  l !! x.1 = Some x.2  P x.2   y, y  take x.1 l  ¬P y.
Proof.
  destruct x => /=. rewrite list_find_Some. do 2 f_equiv. setoid_rewrite take_elem_of.
  split => ?; [naive_solver..|].
  move => j ? ?. have [|??]:= lookup_lt_is_Some_2 l j. { by apply: lookup_lt_Some. }
  set_solver.
Qed.

Lemma replicate_O {A} (x : A) n:
  n = 0%nat -> replicate n x = [].
Proof. by move => ->. Qed.

Global Instance set_unfold_replicate A (x y : A) n:
  SetUnfoldElemOf x (replicate n y) (x = y  n  0%nat).
Proof. constructor. apply elem_of_replicate. Qed.

Lemma list_elem_of_insert1 {A} (l : list A) i (x1 x2 : A) :
  x1  <[i:=x2]> l  x1 = x2  x1  l.
Proof.
  destruct (decide (i < length l)%nat). 2: rewrite list_insert_ge; naive_solver lia.
  move => /(elem_of_list_lookup_1 _ _)[i' ].
  destruct (decide (i = i')); subst.
  - rewrite list_lookup_insert // => -[]. naive_solver.
  - rewrite list_lookup_insert_ne // elem_of_list_lookup. naive_solver.
Qed.

Lemma list_elem_of_insert2 {A} (l : list A) i (x1 x2 x3 : A) :
  l !! i = Some x3  x1  l  x1  <[i:=x2]> l  x1 = x3.
Proof.
  destruct (decide (i < length l)%nat). 2: rewrite list_insert_ge; naive_solver lia.
  move => ? /(elem_of_list_lookup_1 _ _)[i' ?].
  destruct (decide (i = i')); simplify_eq; first naive_solver.
  left. apply elem_of_list_lookup. exists i'. by rewrite list_lookup_insert_ne.
Qed.
Lemma list_elem_of_insert2' {A} (l : list A) i (x1 x2 x3 : A) :
  l !! i = Some x3  x1  l  x1  x3  x1  <[i:=x2]> l.
Proof. move => ???. efeed pose proof (list_elem_of_insert2 (A:=A)) as Hi; naive_solver. Qed.


Lemma list_fmap_ext' {A B} f (g : A  B) (l1 l2 : list A) :
    ( x, x  l1  f x = g x)  l1 = l2  f <$> l1 = g <$> l2.
Proof. intros ? <-. induction l1; f_equal/=; set_solver. Qed.


Lemma imap_fst_NoDup {A B C} l (f : nat  A  B) (g : nat  C):
  Inj eq eq g 
  NoDup (imap (λ i o, (g i, f i o)) l).*1.
Proof.
  move => ?. rewrite fmap_imap (imap_ext _ (λ i o, g i)%nat) // imap_seq_0.
    by apply NoDup_fmap, NoDup_ListNoDup, seq_NoDup.
Qed.
Global Instance set_unfold_imap A B f (l : list A) (x : B):
  SetUnfoldElemOf x (imap f l) ( i y, x = f i y  l !! i = Some y).
Proof.
  constructor.
  elim: l f => /=. set_solver. set_unfold. move => ? ? IH f.
  rewrite IH {IH}. split. case.
  - move => ->. set_solver.
  - move => [n [v [-> ?]]]. exists (S n), v => /=. set_solver.
  - move => [[|n] /= [v [-> ?]]]; simplify_eq; [by left | right].
    naive_solver.
Qed.

Lemma list_insert_fold {A} l i (x : A) :
  list_insert i x l = <[i := x]> l.
Proof. done. Qed.

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Lemma default_last_cons {A} (x1 x2 y : A) l :
  default x1 (last (y :: l)) = default x2 (last (y :: l)).
Proof. elim: l y => //=. Qed.
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Lemma list_lookup_length_default_last {A} (x : A) (l : list A) :
  (x :: l) !! length l = Some (default x (last l)).
Proof. elim: l x => //= a l IH x. rewrite IH. f_equal. destruct l => //=. apply default_last_cons. Qed.

Lemma filter_nil_inv {A} P `{! x, Decision (P x)} (l : list A):
  filter P l = []  ( x : A, x  l  ¬ P x).
Proof.
  elim: l. 1: by rewrite filter_nil; set_solver.
  move => x l IH. rewrite filter_cons. case_decide; set_solver.
Qed.

Lemma length_filter_gt {A} P `{! x, Decision (P x)} (l : list A) x:
  x  l  P x 
  (0 < length (filter P l))%nat.
Proof. elim; move => *; rewrite filter_cons; case_decide; naive_solver lia. Qed.

Lemma length_filter_insert {A} P `{! x, Decision (P x)} (l : list A) i x x':
  l !! i = Some x' 
  length (filter P (<[i:=x]>l)) =
  (length (filter P l) + (if bool_decide (P x) then 1 else 0) - (if bool_decide (P x') then 1 else 0))%nat.
Proof.
  elim: i l. move => [] //=??[->]. rewrite !filter_cons. by repeat (case_decide; case_bool_decide) => //=; lia.
  move => i IH [|? l]//=?. rewrite !filter_cons. case_decide => //=; rewrite IH // -minus_Sn_m //.
  repeat case_bool_decide => //; try lia. feed pose proof (length_filter_gt P l x') => //; try lia.
    by apply: elem_of_list_lookup_2.
Qed.

Lemma length_filter_replicate_True {A} P `{! x, Decision (P x)} (x : A) len:
  P x  length (filter P (replicate len x)) = len.
Proof. elim: len => //=???. rewrite filter_cons. case_decide => //=. f_equal. naive_solver. Qed.

Lemma reshape_app {A} (ln1 ln2 : list nat) (l : list A) :
  reshape (ln1 ++ ln2) l = reshape ln1 (take (sum_list ln1) l) ++ reshape ln2 (drop (sum_list ln1) l).
Proof. elim: ln1 l => //= n ln1 IH l. rewrite take_take skipn_firstn_comm IH drop_drop. repeat f_equal; lia. Qed.
Lemma omap_app {A B} (f : A  option B) (s1 s2 : list A):
  omap f (s1 ++ s2) = omap f s1 ++ omap f s2.
Proof. elim: s1 => //. csimpl => ?? ->. case_match; naive_solver. Qed.
Lemma sum_list_with_take {A} f (l : list A) i:
   (sum_list_with f (take i l)  sum_list_with f l)%nat.
Proof. elim: i l => /=. lia. move => ? IH [|? l2] => //=. move: (IH l2). lia.  Qed.

Lemma omap_length_eq {A B C} (f : A  option B) (g : A  option C) (l : list A):
  ( i x, l !! i = Some x  const () <$> (f x) = const () <$> (g x)) 
  length (omap f l) = length (omap g l).
Proof.
  elim: l => //; csimpl => x ? IH Hx. move: (Hx O x ltac:(done)) => /=?.
  do 2 case_match => //=; rewrite IH // => i ??; by apply: (Hx (S i)).
Qed.

Lemma join_length {A} (l : list (list A)) :
  length (mjoin l) = sum_list (length <$> l).
Proof. elim: l => // ?? IH; csimpl. rewrite app_length IH //. Qed.

Lemma sum_list_eq l1 l2:
  Forall2 eq l1 l2 
  sum_list l1 = sum_list l2.
Proof. by elim => // ???? -> /= ? ->. Qed.

Lemma reshape_join {A} szs (ls : list (list A)) :
  Forall2 (λ l sz, length l = sz) ls szs 
  reshape szs (mjoin ls) = ls.
Proof.
  revert ls. induction szs as [|sz szs IH]; simpl; intros ls; [by inversion 1|].
  intros (?&?&?&?&?)%Forall2_cons_inv_r; simplify_eq/=. rewrite take_app drop_app. f_equal.
  naive_solver.
Qed.

Lemma lookup_eq_app_r {A} (l1 l2 suffix : list A) (i : nat) :
  length l1 = length l2 
  l1 !! i = l2 !! i  (l1 ++ suffix) !! i = (l2 ++ suffix) !! i.
Proof.
  move => Hlen. destruct (l1 !! i) as [v|] eqn:HEq.
  + rewrite lookup_app_l; last by eapply lookup_lt_Some.
    rewrite lookup_app_l; first by rewrite -HEq.
    apply lookup_lt_Some in HEq. by rewrite -Hlen.
  + rewrite lookup_app_r; last by apply lookup_ge_None.
    apply lookup_ge_None in HEq. rewrite Hlen in HEq.
    apply lookup_ge_None in HEq. rewrite HEq.
    split => [_|]//. rewrite lookup_app_r; first by rewrite Hlen.
    by apply lookup_ge_None.
Qed.
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Lemma StronglySorted_lookup_le {A} R (l : list A) i j x y:
  StronglySorted R l  l !! i = Some x  l !! j = Some y  (i  j)%nat  x = y  R x y.
Proof.
  move => Hsorted. elim: Hsorted i j => // z {}l ? IH Hall [|?] [|?]//=???; simplify_eq; try lia.
  - by left.
  - right. by apply: (Forall_lookup_1 _ _ _ _ Hall).
  - apply: IH => //. lia.
Qed.

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Lemma StronglySorted_lookup_lt {A} R (l : list A) i j x y:
  StronglySorted R l  l !! i = Some x  l !! j = Some y  ¬ R y x  x  y  (i < j)%nat.
Proof.
  move => Hs Hi Hj HR Hneq. elim: Hs j i Hj Hi => // z {}l _ IH /Forall_forall Hall.
  case => /=.
  - case; first naive_solver. move => n [?]/= /(elem_of_list_lookup_2 _ _ _)?; subst. naive_solver.
  - move => n. case; first lia. move => n2 /= ??. apply lt_n_S. naive_solver.
Qed.

(** * vec *)
Lemma vec_cast {A} n (v : vec A n) m:
  n = m   v' : vec A m, vec_to_list v = vec_to_list v'.
Proof.
  elim: v m => [|??? IH] [|m']// ?.
  - by eexists vnil.
  - have [|??]:= IH m'. { lia. }
    eexists (vcons _ _) => /=. by f_equal.
Qed.
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(** * map *)
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Section theorems.
Context `{FinMap K M}.
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Lemma partial_alter_self_alt' {A} (m : M A) i f:
  m !! i = f (m !! i)  partial_alter f i m = m.
Proof using Type*.
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|];
    rewrite ?lookup_partial_alter ?lookup_partial_alter_ne.
Qed.
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Lemma partial_alter_to_insert {A} x (m : M A) f k:
    f (m !! k) = Some x 
    partial_alter f k m = <[k := x]> m.
Proof using Type*. move => ?. by apply: partial_alter_ext => ? <-. Qed.

End theorems.

(** * option  *)
Lemma apply_default {A B} (f : A  B) (d : A) (o : option A) :
  f (default d o) = default (f d) (f <$> o).
Proof. by destruct o. Qed.

(** * list_subequiv *)
Definition list_subequiv {A} (ignored : list nat) (l1 l2 : list A) : Prop :=
   i, length l1 = length l2  (i  ignored  l1 !! i = l2 !! i).
Section list_subequiv.
  Context {A : Type}.
  Implicit Type l : list A.

  Global Instance list_subequiv_equiv ig : Equivalence (list_subequiv (A:=A) ig).
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  Proof.
    unfold list_subequiv. split => //.
    - move => ?? H i. move: (H i) => [-> ?]. split; first done. symmetry. naive_solver.
    - move => ??? H1 H2 i. move: (H1 i) => [-> H1i]. move: (H2 i) => [-> H2i].
      split; first done. etrans; first exact: H1i. naive_solver.
  Qed.

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  Lemma list_subequiv_nil_l l ig:
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    list_subequiv ig [] l  l = [].
  Proof. split => Hs; destruct l => //. by move: (Hs 0) => [??]. Qed.

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  Lemma list_subequiv_nil_r l ig:
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    list_subequiv ig l []  l = [].
  Proof. split => Hs; destruct l => //. by move: (Hs 0) => [??]. Qed.

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  Lemma list_subequiv_insert_in_l l1 l2 j x ig:
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    j  ig 
    list_subequiv ig (<[j := x]>l1) l2  list_subequiv ig l1 l2.
  Proof.
    unfold list_subequiv. move => ?. split => Hs i; move: (Hs i) => [<- H].
    - split; first by rewrite insert_length. move => ?.
      rewrite -H; last done. rewrite list_lookup_insert_ne; naive_solver.
    - split; first by rewrite insert_length. move => ?.
      rewrite list_lookup_insert_ne; naive_solver.
  Qed.

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  Lemma list_subequiv_insert_in_r l1 l2 j x ig:
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    j  ig 
    list_subequiv ig l1 (<[j := x]>l2)  list_subequiv ig l1 l2.
  Proof.
    move => ?.
    rewrite (symmetry_iff (list_subequiv _)) [in X in _  X](symmetry_iff (list_subequiv _)).
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      by apply list_subequiv_insert_in_l.
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  Qed.

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  Lemma list_subequiv_insert_ne_l l1 l2 j x ig:
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    (j < length l1)%nat  j  ig 
    list_subequiv ig (<[j := x]>l1) l2  l2 !! j = Some x  list_subequiv (j :: ig) l1 l2.
  Proof.
    move => ??. unfold list_subequiv. split.
    - move => Hs. move: (Hs j) => [<- <-]//. rewrite list_lookup_insert //. split => // i.
      rewrite insert_length. split => // Hi. move: (Hs i) => [? <-];[|set_solver].
      rewrite list_lookup_insert_ne //. set_solver.
    - rewrite insert_length. move => [? Hs] i. split; first by move: (Hs 0) => [? _]//.
      case: (decide (i = j)) => [->|?].
      + by rewrite list_lookup_insert.
      + rewrite list_lookup_insert_ne//. move: (Hs i) => [? H]// ?. apply H. set_solver.
  Qed.

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  Lemma list_subequiv_app_r l1 l2 l3 ig:
    list_subequiv ig (l1 ++ l3) (l2 ++ l3)  list_subequiv ig l1 l2.
  Proof.
  rewrite /list_subequiv. split => H i; move: (H i) => [Hlen Hlookup].
  - rewrite app_length app_length in Hlen. split; first by lia.
    move => /Hlookup. apply lookup_eq_app_r. by lia.
  - split; first by rewrite app_length app_length Hlen.
    move => /Hlookup. apply lookup_eq_app_r. by lia.
  Qed.

  Lemma list_subequiv_fmap {B} ig l1 l2 (f : A  B):
    list_subequiv ig l1 l2 
    list_subequiv ig (f <$> l1) (f <$> l2).
  Proof.
    move => Hs i. move: (Hs 0%nat) => [Hlen _].
    do 2 rewrite fmap_length. split => // ?. rewrite !list_lookup_fmap.
    f_equal. move: (Hs i) => [_ ?]. naive_solver.
  Qed.

  Lemma list_insert_subequiv l1 l2 j x1 :
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    (j < length l1)%nat 
    <[j:=x1]>l1 = l2  l2 !! j = Some x1  list_subequiv [j] l1 l2.
  Proof.
    move => ?. split.
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    - move => <-. rewrite list_lookup_insert // list_subequiv_insert_in_r //. set_solver.
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    - move => [? Hsub]. apply list_eq => i. case: (decide (i = j)) => [->|?].
      + by rewrite list_lookup_insert.
      + rewrite list_lookup_insert_ne//. apply Hsub. set_solver.
  Qed.

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  Lemma list_subequiv_split l1 l2 i :
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    length l1 = length l2 
    list_subequiv [i] l1 l2 
    take i l1 = take i l2  drop (S i) l1 = drop (S i) l2.
  Proof.
    move => Hlen. split.
    - move => Hsub. split; apply list_eq => n; move: (Hsub n) => Hn; set_unfold.
      + destruct (decide (n < i)%nat).
        * rewrite !lookup_take; by naive_solver lia.
        * rewrite !lookup_ge_None_2 // take_length; lia.
      + rewrite !lookup_drop. apply Hsub. set_unfold. lia.
    - move => [Ht Hd] n. split; first done.
      move => ?. have ? : (n  i) by set_solver.
      destruct (decide (n < i)%nat).
      + by rewrite -(lookup_take l1 i) // -(lookup_take l2 i) // Ht.
      + have ->:(n = (S i) + (n - (S i)))%nat by lia.
        by rewrite -!lookup_drop Hd.
  Qed.
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End list_subequiv.
Typeclasses Opaque list_subequiv.
Global Opaque list_subequiv.
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(** * big_op *)
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Section big_op.
Context {PROP : bi}.
Implicit Types P Q : PROP.
Implicit Types Ps Qs : list PROP.
Implicit Types A : Type.

(** ** Big ops over lists *)
Section sep_list.
  Context {A : Type}.
  Implicit Types l : list A.
  Implicit Types Φ Ψ : nat  A  PROP.

  Lemma big_sepL_insert l i x Φ:
    (i < length l)%nat 
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    (([ list] kv  <[i:=x]> l, Φ k v)  Φ i x  ([ list] kv  l, if decide (k = i) then emp else Φ k v)).
  Proof.
    intros Hl.
    destruct (lookup_lt_is_Some_2 l i Hl) as [y Hy].
    rewrite big_sepL_delete; [| by apply list_lookup_insert].
    rewrite insert_take_drop // -{3}(take_drop_middle l i y) // !big_sepL_app /=.
    do 3 f_equiv. rewrite take_length. case_decide => //. lia.
  Qed.
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Lemma big_sepL_impl' {B} Φ (Ψ : _  B  _) (l1 : list A) (l2 : list B) :
    length l1 = length l2 
    ([ list] kx  l1, Φ k x) -
     ( k x1 x2, l1 !! k = Some x1 - l2 !! k = Some x2 - Φ k x1 - Ψ k x2) -
    [ list] kx  l2, Ψ k x.
  Proof.
    iIntros (Hlen) "Hl #Himpl".
    iInduction l1 as [|x1 l1] "IH" forall (Φ Ψ l2 Hlen); destruct l2 => //=; simpl in *.
    iDestruct "Hl" as "[Hx1 Hl]". iSplitL "Hx1". by iApply "Himpl".
    iApply ("IH" $! (Φ  S) (Ψ  S) l2 _ with "[] Hl").
    iIntros "!>" (k y1 y2 ?) "Hl2 /= ?".
      by iApply ("Himpl" with "[] [Hl2]").
      Unshelve. lia.
  Qed.
End sep_list.

  Lemma big_sepL2_impl' {A B C D} (Φ : _  _  _  PROP) (Ψ : _  _  _  _) (l1 : list A) (l2 : list B) (l1' : list C) (l2' : list D)  `{!BiAffine PROP} :
    length l1 = length l1'  length l2 = length l2' 
    ([ list] kx;y  l1; l2, Φ k x y) -
     ( k x1 x2 y1 y2, l1 !! k = Some x1 - l2 !! k = Some x2 - l1' !! k = Some y1 -  l2' !! k = Some y2 - Φ k x1 x2 - Ψ k y1 y2) -
    [ list] kx;y  l1';l2', Ψ k x y.
  Proof.
    iIntros (Hlen1 Hlen2) "Hl #Himpl".
    rewrite !big_sepL2_alt. iDestruct "Hl" as (Hl1) "Hl".
    iSplit. { iPureIntro. congruence. }
    iApply (big_sepL_impl' with "Hl"). { rewrite !zip_with_length. lia. }
    iIntros "!>" (k [x1 x2] [y1 y2]).
    rewrite !lookup_zip_with_Some.
    iDestruct 1 as %(?&?&?&?).
    iDestruct 1 as %(?&?&?&?). simplify_eq. destruct_and!.
    by iApply "Himpl".
  Qed.
End big_op.

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(** * power_of_two and factor2  *)
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Definition is_power_of_two (n : nat) :=  m : nat, n = (2^m)%nat.
Arguments is_power_of_two : simpl never.
Typeclasses Opaque is_power_of_two.

Fixpoint Pos_factor2 (p : positive) : nat :=
  match p with
  | xO p => S (Pos_factor2 p)
  | _ => 0%nat
  end.

Definition factor2' (n : nat) : option nat :=
  match N.of_nat n with
  | N0 => None
  | Npos p => Some (Pos_factor2 p)
  end.
Definition factor2 (n : nat) (def : nat) : nat :=
  default def (factor2' n).

Definition keep_factor2 (n : nat) (def : nat) : nat :=
  default def (pow 2 <$> factor2' n).

Lemma Pos_pow_add_r a b c:
  (a ^ (b + c) = a ^ b * a ^ c)%positive.
Proof. zify. rewrite Z.pow_add_r; lia. Qed.

Lemma Pos_factor2_mult_xI a b:
  Pos_factor2 (a~1 * b) = Pos_factor2 b.
Proof.
  move: a. elim b => // p IH a. rewrite /= -/Pos.mul. f_equal.
  rewrite Pos.mul_xO_r [X in Pos_factor2 (_ + xO X) = _]Pos.mul_comm.
  rewrite -Pos.mul_xI_r Pos.mul_comm. apply IH.
Qed.

Lemma Pos_factor2_mult a b:
  Pos_factor2 (a * b) = (Pos_factor2 a + Pos_factor2 b)%nat.
Proof.
  elim: a b => // p IH b.
  - by rewrite Pos_factor2_mult_xI.
  - by rewrite Pos.mul_comm Pos.mul_xO_r /= Pos.mul_comm IH.
Qed.

Lemma Pos_factor2_pow n:
  Pos_factor2 (2^n)%positive = Pos.to_nat n.
Proof.
  elim: n => // p IH; rewrite ?Pos.xI_succ_xO -(Pos.add_diag p) -?Pos.add_succ_r -?Pos.add_1_r !Pos_pow_add_r !Pos_factor2_mult !IH /=; lia.
Qed.

Lemma Zdivide_mult_l n1 n2 a :
  ((n1 * n2 | a)  (n1 | a))%Z.
Proof. move => [? ->]. by apply Z.divide_mul_r, Z.divide_mul_l. Qed.

Lemma Zdivide_nat_pow a b c:
  ((b  c)%nat  ((a ^ b)%nat | (a ^ c)%nat))%Z.
Proof.
  move => ?. apply: (Zdivide_mult_l _ (a^(c - b))%nat).
  by rewrite -Nat2Z.inj_mul -Nat.pow_add_r le_plus_minus_r.
Qed.

Lemma Pos_factor2_divide p :
  ((2 ^ Pos_factor2 p)%nat | Z.pos p)%Z.
Proof.
  elim: p => //=. by move => *; apply Z.divide_1_l.
  move => p IH. rewrite -plus_n_O Pos2Z.inj_xO Nat2Z.inj_add Z.add_diag. by apply Z.mul_divide_mono_l.
Qed.

Lemma factor2_divide n x :
  ((2 ^ factor2 n x)%nat | n)%Z.
Proof.
  rewrite /factor2/factor2'. rewrite -(nat_N_Z n). case_match => /=; first by apply Z.divide_0_r.
  apply Pos_factor2_divide.
Qed.

Lemma factor2'_pow n:
  factor2' (2^n)%nat = Some n.
Proof.
  rewrite /factor2'. destruct (N.of_nat (2 ^ n)) eqn:H.
  - exfalso. elim: n H => // /=. lia.
  - f_equal. move: p H. induction n as [|n IH].
    + move => p /= Hp. destruct p => //.
    + move => p Hp. destruct p.
      * exfalso. zify. rewrite Nat.pow_succ_r' in Hp. lia.
      * rewrite /=. f_equal. apply IH.
        zify. rewrite Nat.pow_succ_r' in Hp. lia.
      * exfalso. zify. rewrite Nat.pow_succ_r' in Hp. lia.
Qed.

Lemma factor2_pow n x:
  factor2 (2^n)%nat x = n.
Proof. by rewrite /factor2 factor2'_pow. Qed.

Lemma keep_factor2_0 n:
  keep_factor2 0 n = n.
Proof. done. Qed.

Lemma keep_factor2_mult n m o:
  n  0  m  0 
  keep_factor2 (m * n) o = keep_factor2 m o * keep_factor2 n o.
Proof.
  rewrite /keep_factor2 /factor2' => ??. destruct n,m => //=.
  rewrite -Nat.pow_add_r -Pos_factor2_mult. do 2 f_equal. lia.
Qed.

Lemma keep_factor2_neq_0 n x:
  n  0 
  keep_factor2 n x  0.
Proof. move => ?. destruct n => //. rewrite /keep_factor2 /=. by apply Nat.pow_nonzero. Qed.

Lemma keep_factor2_is_power_of_two n x:
  is_power_of_two n 
  keep_factor2 n x = n.
Proof. move => [? ->]. by rewrite /keep_factor2 factor2'_pow. Qed.

Lemma keep_factor2_leq (n m : nat):
  is_power_of_two n  (n | m) 
  n  keep_factor2 m n.
Proof.
  move => ? [o ->]. destruct (decide (o = 0)); first by subst; rewrite keep_factor2_0; lia.
  destruct (decide (n = 0)); first lia.
  rewrite keep_factor2_mult // (keep_factor2_is_power_of_two n) //.
  have ?: keep_factor2 o n  0 by apply keep_factor2_neq_0.
  destruct (keep_factor2 o n); lia.
Qed.

Lemma keep_factor2_min_eq (n m : nat):
  is_power_of_two n  (n | m) 
  (n `min` keep_factor2 m n) = n.
Proof. move => ??. apply: Nat.min_l. by apply: keep_factor2_leq. Qed.

Lemma keep_factor2_min_1 n:
  1 `min` keep_factor2 n 1 = 1.
Proof.
  rewrite /keep_factor2 /factor2'. destruct (N.of_nat n) => // /=.
  apply Nat.min_l. generalize (Pos_factor2 p) => k. induction k as [|k IH] => //.
  rewrite Nat.pow_succ_r'. lia.
Qed.

Lemma keep_factor2_twice n m:
  (keep_factor2 n (keep_factor2 n m)) = (keep_factor2 n m).
Proof. by destruct n. Qed.

(* Lemma mult_is_one_Z n m : 0 ≤ n → 0 ≤ m → n * m = 1 → n = 1 ∧ m = 1. *)
(* Proof. *)
(*   intros Hn Hm. *)
(*   move: (Z_of_nat_complete n Hn) => [i ->]. *)
(*   move: (Z_of_nat_complete m Hm) => [j ->]. *)
(*   move => HZ. assert (i * j = 1)%nat as H by lia. *)
(*   apply mult_is_one in H. lia. *)
(* Qed. *)

(* Lemma mult_is_mult_of_pow2_Z n1 n2 (m : nat): *)
(*   0 ≤ n1 → 0 ≤ n2 → n1 * n2 = 2 ^ m → ∃ (m1 m2 : nat), n1 = 2 ^ m1 ∧ n2 = 2 ^ m2. *)
(* Proof. *)
(*   revert n1 n2. induction m as [|m IH] => n1 n2 Hn1 Hn2. *)
(*   - rewrite Z.pow_0_r. move => /(mult_is_one_Z _ _ Hn1 Hn2) [-> ->]. by exists 0%nat, 0%nat. *)
(*   - assert (Z.of_nat (S m) = Z.succ m) as -> by lia. rewrite Z.pow_succ_r; last by lia. *)
(*     move => H. assert (2 | n1 * n2) as Hdiv. { rewrite H. apply Z.divide_mul_l, Z.divide_refl. } *)
(*     apply prime_mult in Hdiv; last by apply prime_2. *)
(*     destruct Hdiv as [Hdiv|Hdiv]; destruct Hdiv as [k ->]. *)
(*     + assert (k * 2 * n2 = 2 * (k * n2)) as Htmp by lia; rewrite Htmp in H; clear Htmp. *)
(*       apply Z.mul_cancel_l in H => //. apply IH in H; [ .. | lia | lia ]. *)
(*       destruct H as [m1 [m2 [-> ->]]]. exists (S m1), m2. split => //. *)
(*       rewrite Z.mul_comm -Z.pow_succ_r; last by lia. f_equal. lia. *)
(*     + assert (n1 * (k * 2) = 2 * (n1 * k)) as Htmp by lia; rewrite Htmp in H; clear Htmp. *)
(*       apply Z.mul_cancel_l in H => //. apply IH in H; [ .. | lia | lia ]. *)
(*       destruct H as [m1 [m2 [-> ->]]]. exists m1, (S m2). split => //. *)
(*       rewrite Z.mul_comm -Z.pow_succ_r; last by lia. f_equal. lia. *)
(* Qed. *)

Lemma divide_mult_2 n1 n2 : divide 2 (n1 * n2)  divide 2 n1  divide 2 n2.
  move => /Nat2Z_divide. rewrite Nat2Z.inj_mul. move => /(prime_mult _ prime_2).
  move => [H|H]; [left | right]; apply Z2Nat_divide in H; try lia.
  - rewrite Nat2Z.id in H. assert (Z.to_nat 2 = 2) as Heq by lia. by rewrite Heq in H.
  - rewrite Nat2Z.id in H. assert (Z.to_nat 2 = 2) as Heq by lia. by rewrite Heq in H.
Qed.

Lemma is_power_of_two_mult n1 n2:
  (is_power_of_two (n1 * n2))  (is_power_of_two n1  is_power_of_two n2).
Proof.
  rewrite /is_power_of_two. split.
  - move => [m Hm]. move: n1 n2 Hm. elim: m.
    + move => /= ?? /mult_is_one [->->]. split; by exists 0.
    + move => n IH n1 n2 H. rewrite Nat.pow_succ_r' in H.
      assert (divide 2 (n1 * n2)) as Hdiv. { exists (2 ^ n). lia. }
      apply divide_mult_2 in Hdiv as [[k ->]|[k ->]].
      * assert (k * n2 = 2 ^ n) as Hkn2 by lia.
        apply IH in Hkn2 as [[m ->] Hn2]. split => //.
        exists (S m). by rewrite mult_comm -Nat.pow_succ_r'.
      * assert (n1 * k = 2 ^ n) as Hn1k by lia.
        apply IH in Hn1k as [Hn1 [m ->]]. split => //.
        exists (S m). by rewrite mult_comm -Nat.pow_succ_r'.
  - move => [[m1 ->] [m2 ->]]. exists (m1 + m2). by rewrite Nat.pow_add_r.
Qed.

Lemma Z_distr_mul_sub_1 a b:
  (a * b - b = (a - 1) * b)%Z.
Proof. nia. Qed.

Lemma mult_le_compat_r_1 m p:
  (1  m)%nat  (p  m * p)%nat.
Proof. nia. Qed.

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(** * shifts *)
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Section shifts.
Local Open Scope Z_scope.
Lemma Z_shiftl_le_mono_l n a b:
  0  n 
  a  b 
  a  n  b  n.
Proof.
  move => ??. rewrite !Z.shiftl_mul_pow2 //.
  apply Z.mul_le_mono_nonneg_r => //. by apply: Z.pow_nonneg.
Qed.
Lemma Z_shiftr_le_mono_l n a b:
  0  n 
  a  b 
  a  n  b  n.
Proof.
  move => ??. rewrite !Z.shiftr_div_pow2 //.
  apply: Z.div_le_mono => //. by apply: Z.pow_pos_nonneg.
Qed.
Lemma Z_shiftl_lt_mono_l n a b:
  0  n 
  a < b 
  a  n < b  n.
Proof.
  move => ??.
  rewrite !Z.shiftl_mul_pow2 //. apply Z.mul_lt_mono_pos_r; [|done].
    by apply: Z.pow_pos_nonneg.
Qed.
Lemma Z_shiftr_lt_mono_l n a b:
  0  n 
  a < b 
  Z.land b (Z.ones n) = 0 
  a  n < b  n.
Proof.
  move => ???.
  have ?:= Z.pow_pos_nonneg 2 n.
  rewrite !Z.shiftr_div_pow2 //.
  apply: Z.div_lt_upper_bound; [lia|].
  rewrite -Z_div_exact_2 //; [lia|]. rewrite -Z.land_ones => //.
Qed.
Lemma Z_shiftr_shiftl_0 a n :
  0  n 
  (a  n)  n = a.
Proof. move => ?. by rewrite Z.shiftr_shiftl_l // Z.sub_diag Z.shiftl_0_r. Qed.
Lemma Z_shiftl_shiftr_0 a n :
  0  n 
  Z.land a (Z.ones n) = 0 
  (a  n)  n = a.
Proof.
  move => ? Hland.
  rewrite -Z.ldiff_ones_r //.
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  bitblast.
  move/Z.bits_inj_iff' in Hland. move: (Hland i ltac:(done)).
  by rewrite_testbit.
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Qed.
Lemma Z_shiftl_distr_add a b c:
  0  c 
  (a + b)  c = (a  c + b  c).
Proof. move => ?. rewrite !Z.shiftl_mul_pow2 //. lia. Qed.
End shifts.

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(** Z.lnot (bitwise negation) for unsigned integers with [bits] bits. *)
Definition Z_lunot (bits n : Z) :=
  (Z.lnot n `mod` 2 ^ bits)%Z.
Typeclasses Opaque Z_lunot.

Lemma Z_lunot_spec bits n k:
  (0  k < bits)%Z  Z.testbit (Z_lunot bits n) k = negb (Z.testbit n k).
Proof.
  move => [? ?].
  by rewrite Z.mod_pow2_bits_low ?Z.lnot_spec.
Qed.

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Lemma Z_lunot_spec_high bits n k:
  (0  bits  k)%Z  Z.testbit (Z_lunot bits n) k = false.
Proof.
  move => ?. by rewrite Z.mod_pow2_bits_high.
Qed.

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Lemma Z_lunot_range bits n:
  (0  bits  0  Z_lunot bits n < 2 ^ bits)%Z.
Proof.
  move => ?.
  apply: Z.mod_pos_bound.
  by apply: Z.pow_pos_nonneg.
Qed.
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(* bits for `- n` *)
Lemma Z_bits_opp_z n i :
  (0  i)%Z 
  (n `mod` 2 ^ i = 0)%Z 
  Z.testbit (- n) i = Z.testbit n i.
Proof.
  intros.
  rewrite !Z.testbit_eqb; [|lia..].
  have ? : (0 < 2 ^ i)%Z by apply Z.pow_pos_nonneg; lia.
  rewrite Z.div_opp_l_z; lia.
Qed.

Lemma Z_bits_opp_nz n i :
  (0  i)%Z 
  (n `mod` 2 ^ i  0)%Z 
  Z.testbit (-n) i = negb (Z.testbit n i).
Proof.
  intros.
  rewrite !Z.testbit_eqb; [|lia..].
  rewrite Z.div_opp_l_nz; lia.
Qed.

Lemma Z_mod_pow2_zero_iff n k :
  (0  k)%Z 
  (n `mod` 2 ^ k = 0)%Z   i, (0  i < k)%Z  Z.testbit n i = false.
Proof.
  intros.
  rewrite -Z.land_ones; last lia.
  split.
  - move => Hz i ?.
    have <- : Z.testbit (Z.land n (Z.ones k)) i = false
      by rewrite Hz Z.bits_0.
    rewrite Z.land_spec Z.ones_spec_low //.
    by simplify_bool_eq.
  - move => Hf.
    bitblast.
    by rewrite Hf.
Qed.