Library mathcomp.ssreflect.ssrnat

Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrfun ssrbool eqtype.
Require Import BinNat.
Require BinPos Ndec.
Require Export Ring.


Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.


Delimit Scope coq_nat_scope with coq_nat.

Notation "m + n" := (plus m n) : coq_nat_scope.
Notation "m - n" := (minus m n) : coq_nat_scope.
Notation "m * n" := (mult m n) : coq_nat_scope.
Notation "m <= n" := (le m n) : coq_nat_scope.
Notation "m < n" := (lt m n) : coq_nat_scope.
Notation "m >= n" := (ge m n) : coq_nat_scope.
Notation "m > n" := (gt m n) : coq_nat_scope.


Delimit Scope N_scope with num.
Delimit Scope nat_scope with N.
Delimit Scope nat_rec_scope with Nrec.


Notation succn := Datatypes.S.
Notation predn := Peano.pred.

Notation "n .+1" := (succn n) (at level 2, left associativity,
  format "n .+1") : nat_scope.
Notation "n .+2" := n.+1.+1 (at level 2, left associativity,
  format "n .+2") : nat_scope.
Notation "n .+3" := n.+2.+1 (at level 2, left associativity,
  format "n .+3") : nat_scope.
Notation "n .+4" := n.+2.+2 (at level 2, left associativity,
  format "n .+4") : nat_scope.

Notation "n .-1" := (predn n) (at level 2, left associativity,
  format "n .-1") : nat_scope.
Notation "n .-2" := n.-1.-1 (at level 2, left associativity,
  format "n .-2") : nat_scope.

Lemma succnK : cancel succn predn. Proof. by []. Qed.
Lemma succn_inj : injective succn. Proof. by move=> n m []. Qed.


Reserved Notation "n .*2" (at level 2, format "n .*2").
Reserved Notation "n ./2" (at level 2, format "n ./2").


Fixpoint eqn m n {struct m} :=
  match m, n with
  | 0, 0 => true
  | m'.+1, n'.+1 => eqn m' n'
  | _, _ => false
  end.

Lemma eqnP : Equality.axiom eqn.
Proof.
move=> n m; apply: (iffP idP) => [|<-]; last by elim n.
by elim: n m => [|n IHn] [|m] //= /IHn->.
Qed.

Canonical nat_eqMixin := EqMixin eqnP.
Canonical nat_eqType := Eval hnf in EqType nat nat_eqMixin.

Arguments eqnP [x y].
Prenex Implicits eqnP.

Lemma eqnE : eqn = eq_op. Proof. by []. Qed.

Lemma eqSS m n : (m.+1 == n.+1) = (m == n). Proof. by []. Qed.

Lemma nat_irrelevance (x y : nat) (E E' : x = y) : E = E'.
Proof. exact: eq_irrelevance. Qed.


Definition addn_rec := plus.
Notation "m + n" := (addn_rec m n) : nat_rec_scope.

Definition addn := nosimpl addn_rec.
Notation "m + n" := (addn m n) : nat_scope.

Lemma addnE : addn = addn_rec. Proof. by []. Qed.

Lemma plusE : plus = addn. Proof. by []. Qed.

Lemma add0n : left_id 0 addn. Proof. by []. Qed.
Lemma addSn m n : m.+1 + n = (m + n).+1. Proof. by []. Qed.
Lemma add1n n : 1 + n = n.+1. Proof. by []. Qed.

Lemma addn0 : right_id 0 addn. Proof. by move=> n; apply/eqP; elim: n. Qed.

Lemma addnS m n : m + n.+1 = (m + n).+1. Proof. by elim: m. Qed.

Lemma addSnnS m n : m.+1 + n = m + n.+1. Proof. by rewrite addnS. Qed.

Lemma addnCA : left_commutative addn.
Proof. by move=> m n p; elim: m => //= m; rewrite addnS => <-. Qed.

Lemma addnC : commutative addn.
Proof. by move=> m n; rewrite -{1}[n]addn0 addnCA addn0. Qed.

Lemma addn1 n : n + 1 = n.+1. Proof. by rewrite addnC. Qed.

Lemma addnA : associative addn.
Proof. by move=> m n p; rewrite (addnC n) addnCA addnC. Qed.

Lemma addnAC : right_commutative addn.
Proof. by move=> m n p; rewrite -!addnA (addnC n). Qed.

Lemma addnACA : interchange addn addn.
Proof. by move=> m n p q; rewrite -!addnA (addnCA n). Qed.

Lemma addn_eq0 m n : (m + n == 0) = (m == 0) && (n == 0).
Proof. by case: m; case: n. Qed.

Lemma eqn_add2l p m n : (p + m == p + n) = (m == n).
Proof. by elim: p. Qed.

Lemma eqn_add2r p m n : (m + p == n + p) = (m == n).
Proof. by rewrite -!(addnC p) eqn_add2l. Qed.

Lemma addnI : right_injective addn.
Proof. by move=> p m n Heq; apply: eqP; rewrite -(eqn_add2l p) Heq eqxx. Qed.

Lemma addIn : left_injective addn.
Proof. move=> p m n; rewrite -!(addnC p); apply addnI. Qed.

Lemma addn2 m : m + 2 = m.+2. Proof. by rewrite addnC. Qed.
Lemma add2n m : 2 + m = m.+2. Proof. by []. Qed.
Lemma addn3 m : m + 3 = m.+3. Proof. by rewrite addnC. Qed.
Lemma add3n m : 3 + m = m.+3. Proof. by []. Qed.
Lemma addn4 m : m + 4 = m.+4. Proof. by rewrite addnC. Qed.
Lemma add4n m : 4 + m = m.+4. Proof. by []. Qed.


Definition subn_rec := minus.
Notation "m - n" := (subn_rec m n) : nat_rec_scope.

Definition subn := nosimpl subn_rec.
Notation "m - n" := (subn m n) : nat_scope.

Lemma subnE : subn = subn_rec. Proof. by []. Qed.
Lemma minusE : minus = subn. Proof. by []. Qed.

Lemma sub0n : left_zero 0 subn. Proof. by []. Qed.
Lemma subn0 : right_id 0 subn. Proof. by case. Qed.
Lemma subnn : self_inverse 0 subn. Proof. by elim. Qed.

Lemma subSS n m : m.+1 - n.+1 = m - n. Proof. by []. Qed.
Lemma subn1 n : n - 1 = n.-1. Proof. by case: n => [|[]]. Qed.
Lemma subn2 n : (n - 2)%N = n.-2. Proof. by case: n => [|[|[]]]. Qed.

Lemma subnDl p m n : (p + m) - (p + n) = m - n.
Proof. by elim: p. Qed.

Lemma subnDr p m n : (m + p) - (n + p) = m - n.
Proof. by rewrite -!(addnC p) subnDl. Qed.

Lemma addKn n : cancel (addn n) (subn^~ n).
Proof. by move=> m; rewrite /= -{2}[n]addn0 subnDl subn0. Qed.

Lemma addnK n : cancel (addn^~ n) (subn^~ n).
Proof. by move=> m; rewrite /= (addnC m) addKn. Qed.

Lemma subSnn n : n.+1 - n = 1.
Proof. exact (addnK n 1). Qed.

Lemma subnDA m n p : n - (m + p) = (n - m) - p.
Proof. by elim: m n => [|m IHm] []. Qed.

Lemma subnAC : right_commutative subn.
Proof. by move=> m n p; rewrite -!subnDA addnC. Qed.

Lemma subnS m n : m - n.+1 = (m - n).-1.
Proof. by rewrite -addn1 subnDA subn1. Qed.

Lemma subSKn m n : (m.+1 - n).-1 = m - n.
Proof. by rewrite -subnS. Qed.


Definition leq m n := m - n == 0.

Notation "m <= n" := (leq m n) : nat_scope.
Notation "m < n" := (m.+1 <= n) : nat_scope.
Notation "m >= n" := (n <= m) (only parsing) : nat_scope.
Notation "m > n" := (n < m) (only parsing) : nat_scope.

Definition geq := [rel m n | m >= n].
Definition ltn := [rel m n | m < n].
Definition gtn := [rel m n | m > n].

Notation "m <= n <= p" := ((m <= n) && (n <= p)) : nat_scope.
Notation "m < n <= p" := ((m < n) && (n <= p)) : nat_scope.
Notation "m <= n < p" := ((m <= n) && (n < p)) : nat_scope.
Notation "m < n < p" := ((m < n) && (n < p)) : nat_scope.

Lemma ltnS m n : (m < n.+1) = (m <= n). Proof. by []. Qed.
Lemma leq0n n : 0 <= n. Proof. by []. Qed.
Lemma ltn0Sn n : 0 < n.+1. Proof. by []. Qed.
Lemma ltn0 n : n < 0 = false. Proof. by []. Qed.
Lemma leqnn n : n <= n. Proof. by elim: n. Qed.
Hint Resolve leqnn.
Lemma ltnSn n : n < n.+1. Proof. by []. Qed.
Lemma eq_leq m n : m = n -> m <= n. Proof. by move->. Qed.
Lemma leqnSn n : n <= n.+1. Proof. by elim: n. Qed.
Hint Resolve leqnSn.
Lemma leq_pred n : n.-1 <= n. Proof. by case: n => /=. Qed.
Lemma leqSpred n : n <= n.-1.+1. Proof. by case: n => /=. Qed.

Lemma ltn_predK m n : m < n -> n.-1.+1 = n.
Proof. by case: n. Qed.

Lemma prednK n : 0 < n -> n.-1.+1 = n.
Proof. exact: ltn_predK. Qed.

Lemma leqNgt m n : (m <= n) = ~~ (n < m).
Proof. by elim: m n => [|m IHm] []. Qed.

Lemma ltnNge m n : (m < n) = ~~ (n <= m).
Proof. by rewrite leqNgt. Qed.

Lemma ltnn n : n < n = false.
Proof. by rewrite ltnNge leqnn. Qed.

Lemma leqn0 n : (n <= 0) = (n == 0). Proof. by case: n. Qed.
Lemma lt0n n : (0 < n) = (n != 0). Proof. by case: n. Qed.
Lemma lt0n_neq0 n : 0 < n -> n != 0. Proof. by case: n. Qed.
Lemma eqn0Ngt n : (n == 0) = ~~ (n > 0). Proof. by case: n. Qed.
Lemma neq0_lt0n n : (n == 0) = false -> 0 < n. Proof. by case: n. Qed.
Hint Resolve lt0n_neq0 neq0_lt0n.

Lemma eqn_leq m n : (m == n) = (m <= n <= m).
Proof. by elim: m n => [|m IHm] []. Qed.

Lemma anti_leq : antisymmetric leq.
Proof. by move=> m n; rewrite -eqn_leq => /eqP. Qed.

Lemma neq_ltn m n : (m != n) = (m < n) || (n < m).
Proof. by rewrite eqn_leq negb_and orbC -!ltnNge. Qed.

Lemma gtn_eqF m n : m < n -> n == m = false.
Proof. by rewrite eqn_leq (leqNgt n) => ->. Qed.

Lemma ltn_eqF m n : m < n -> m == n = false.
Proof. by move/gtn_eqF; rewrite eq_sym. Qed.

Lemma leq_eqVlt m n : (m <= n) = (m == n) || (m < n).
Proof. by elim: m n => [|m IHm] []. Qed.

Lemma ltn_neqAle m n : (m < n) = (m != n) && (m <= n).
Proof. by rewrite ltnNge leq_eqVlt negb_or -leqNgt eq_sym. Qed.

Lemma leq_trans n m p : m <= n -> n <= p -> m <= p.
Proof. by elim: n m p => [|i IHn] [|m] [|p] //; apply: IHn m p. Qed.

Lemma leq_ltn_trans n m p : m <= n -> n < p -> m < p.
Proof. by move=> Hmn; apply: leq_trans. Qed.

Lemma ltnW m n : m < n -> m <= n.
Proof. exact: leq_trans. Qed.
Hint Resolve ltnW.

Lemma leqW m n : m <= n -> m <= n.+1.
Proof. by move=> le_mn; apply: ltnW. Qed.

Lemma ltn_trans n m p : m < n -> n < p -> m < p.
Proof. by move=> lt_mn /ltnW; apply: leq_trans. Qed.

Lemma leq_total m n : (m <= n) || (m >= n).
Proof. by rewrite -implyNb -ltnNge; apply/implyP; apply: ltnW. Qed.


Lemma leP m n : reflect (m <= n)%coq_nat (m <= n).
Proof.
apply: (iffP idP); last by elim: n / => // n _ /leq_trans->.
elim: n => [|n IHn]; first by case: m.
by rewrite leq_eqVlt ltnS => /predU1P[<- // | /IHn]; right.
Qed.
Arguments leP [m n].

Lemma le_irrelevance m n le_mn1 le_mn2 : le_mn1 = le_mn2 :> (m <= n)%coq_nat.
Proof.
elim: {n}n.+1 {-1}n (erefl n.+1) => // n IHn _ [<-] in le_mn1 le_mn2 *.
pose def_n2 := erefl n; transitivity (eq_ind _ _ le_mn2 _ def_n2) => //.
move def_n1: {1 4 5 7}n le_mn1 le_mn2 def_n2 => n1 le_mn1.
case: n1 / le_mn1 def_n1 => [|n1 le_mn1] def_n1 [|n2 le_mn2] def_n2.
- by rewrite [def_n2]eq_axiomK.
- by move/leP: (le_mn2); rewrite -{1}def_n2 ltnn.
- by move/leP: (le_mn1); rewrite {1}def_n2 ltnn.
case: def_n2 (def_n2) => ->{n2} def_n2 in le_mn2 *.
by rewrite [def_n2]eq_axiomK /=; congr le_S; apply: IHn.
Qed.

Lemma ltP m n : reflect (m < n)%coq_nat (m < n).
Proof. exact leP. Qed.
Arguments ltP [m n].

Lemma lt_irrelevance m n lt_mn1 lt_mn2 : lt_mn1 = lt_mn2 :> (m < n)%coq_nat.
Proof. exact: (@le_irrelevance m.+1). Qed.


CoInductive leq_xor_gtn m n : bool -> bool -> Set :=
  | LeqNotGtn of m <= n : leq_xor_gtn m n true false
  | GtnNotLeq of n < m : leq_xor_gtn m n false true.

Lemma leqP m n : leq_xor_gtn m n (m <= n) (n < m).
Proof.
by rewrite ltnNge; case le_mn: (m <= n); constructor; rewrite // ltnNge le_mn.
Qed.

CoInductive ltn_xor_geq m n : bool -> bool -> Set :=
  | LtnNotGeq of m < n : ltn_xor_geq m n false true
  | GeqNotLtn of n <= m : ltn_xor_geq m n true false.

Lemma ltnP m n : ltn_xor_geq m n (n <= m) (m < n).
Proof. by rewrite -(ltnS n); case: leqP; constructor. Qed.

CoInductive eqn0_xor_gt0 n : bool -> bool -> Set :=
  | Eq0NotPos of n = 0 : eqn0_xor_gt0 n true false
  | PosNotEq0 of n > 0 : eqn0_xor_gt0 n false true.

Lemma posnP n : eqn0_xor_gt0 n (n == 0) (0 < n).
Proof. by case: n; constructor. Qed.

CoInductive compare_nat m n :
   bool -> bool -> bool -> bool -> bool -> bool -> Set :=
  | CompareNatLt of m < n : compare_nat m n true false true false false false
  | CompareNatGt of m > n : compare_nat m n false true false true false false
  | CompareNatEq of m = n : compare_nat m n true true false false true true.

Lemma ltngtP m n : compare_nat m n (m <= n) (n <= m) (m < n)
                                   (n < m) (n == m) (m == n).
Proof.
rewrite !ltn_neqAle [_ == m]eq_sym; case: ltnP => [mn|].
  by rewrite ltnW // gtn_eqF //; constructor.
rewrite leq_eqVlt; case: ltnP; rewrite ?(orbT, orbF) => //= lt_nm eq_mn.
  by rewrite ltn_eqF //; constructor.
by rewrite eq_mn; constructor; apply/eqP.
Qed.


Lemma leq_add2l p m n : (p + m <= p + n) = (m <= n).
Proof. by elim: p. Qed.

Lemma ltn_add2l p m n : (p + m < p + n) = (m < n).
Proof. by rewrite -addnS; apply: leq_add2l. Qed.

Lemma leq_add2r p m n : (m + p <= n + p) = (m <= n).
Proof. by rewrite -!(addnC p); apply: leq_add2l. Qed.

Lemma ltn_add2r p m n : (m + p < n + p) = (m < n).
Proof. exact: leq_add2r p m.+1 n. Qed.

Lemma leq_add m1 m2 n1 n2 : m1 <= n1 -> m2 <= n2 -> m1 + m2 <= n1 + n2.
Proof.
by move=> le_mn1 le_mn2; rewrite (@leq_trans (m1 + n2)) ?leq_add2l ?leq_add2r.
Qed.

Lemma leq_addr m n : n <= n + m.
Proof. by rewrite -{1}[n]addn0 leq_add2l. Qed.

Lemma leq_addl m n : n <= m + n.
Proof. by rewrite addnC leq_addr. Qed.

Lemma ltn_addr m n p : m < n -> m < n + p.
Proof. by move/leq_trans=> -> //; apply: leq_addr. Qed.

Lemma ltn_addl m n p : m < n -> m < p + n.
Proof. by move/leq_trans=> -> //; apply: leq_addl. Qed.

Lemma addn_gt0 m n : (0 < m + n) = (0 < m) || (0 < n).
Proof. by rewrite !lt0n -negb_and addn_eq0. Qed.

Lemma subn_gt0 m n : (0 < n - m) = (m < n).
Proof. by elim: m n => [|m IHm] [|n] //; apply: IHm n. Qed.

Lemma subn_eq0 m n : (m - n == 0) = (m <= n).
Proof. by []. Qed.

Lemma leq_subLR m n p : (m - n <= p) = (m <= n + p).
Proof. by rewrite -subn_eq0 -subnDA. Qed.

Lemma leq_subr m n : n - m <= n.
Proof. by rewrite leq_subLR leq_addl. Qed.

Lemma subnKC m n : m <= n -> m + (n - m) = n.
Proof. by elim: m n => [|m IHm] [|n] // /(IHm n) {2}<-. Qed.

Lemma subnK m n : m <= n -> (n - m) + m = n.
Proof. by rewrite addnC; apply: subnKC. Qed.

Lemma addnBA m n p : p <= n -> m + (n - p) = m + n - p.
Proof. by move=> le_pn; rewrite -{2}(subnK le_pn) addnA addnK. Qed.

Lemma subnBA m n p : p <= n -> m - (n - p) = m + p - n.
Proof. by move=> le_pn; rewrite -{2}(subnK le_pn) subnDr. Qed.

Lemma subKn m n : m <= n -> n - (n - m) = m.
Proof. by move/subnBA->; rewrite addKn. Qed.

Lemma subSn m n : m <= n -> n.+1 - m = (n - m).+1.
Proof. by rewrite -add1n => /addnBA <-. Qed.

Lemma subnSK m n : m < n -> (n - m.+1).+1 = n - m.
Proof. by move/subSn. Qed.

Lemma leq_sub2r p m n : m <= n -> m - p <= n - p.
Proof.
by move=> le_mn; rewrite leq_subLR (leq_trans le_mn) // -leq_subLR.
Qed.

Lemma leq_sub2l p m n : m <= n -> p - n <= p - m.
Proof.
rewrite -(leq_add2r (p - m)) leq_subLR.
by apply: leq_trans; rewrite -leq_subLR.
Qed.

Lemma leq_sub m1 m2 n1 n2 : m1 <= m2 -> n2 <= n1 -> m1 - n1 <= m2 - n2.
Proof. by move/(leq_sub2r n1)=> le_m12 /(leq_sub2l m2); apply: leq_trans. Qed.

Lemma ltn_sub2r p m n : p < n -> m < n -> m - p < n - p.
Proof. by move/subnSK <-; apply: (@leq_sub2r p.+1). Qed.

Lemma ltn_sub2l p m n : m < p -> m < n -> p - n < p - m.
Proof. by move/subnSK <-; apply: leq_sub2l. Qed.

Lemma ltn_subRL m n p : (n < p - m) = (m + n < p).
Proof. by rewrite !ltnNge leq_subLR. Qed.

Lemma subn_if_gt T m n F (E : T) :
  (if m.+1 - n is m'.+1 then F m' else E) = (if n <= m then F (m - n) else E).
Proof.
by case: leqP => [le_nm | /eqnP-> //]; rewrite -{1}(subnK le_nm) -addSn addnK.
Qed.


Definition maxn m n := if m < n then n else m.

Definition minn m n := if m < n then m else n.

Lemma max0n : left_id 0 maxn. Proof. by case. Qed.
Lemma maxn0 : right_id 0 maxn. Proof. by []. Qed.

Lemma maxnC : commutative maxn.
Proof. by move=> m n; rewrite /maxn; case ltngtP. Qed.

Lemma maxnE m n : maxn m n = m + (n - m).
Proof. by rewrite /maxn addnC; case: leqP => [/eqnP->|/ltnW/subnK]. Qed.

Lemma maxnAC : right_commutative maxn.
Proof. by move=> m n p; rewrite !maxnE -!addnA !subnDA -!maxnE maxnC. Qed.

Lemma maxnA : associative maxn.
Proof. by move=> m n p; rewrite !(maxnC m) maxnAC. Qed.

Lemma maxnCA : left_commutative maxn.
Proof. by move=> m n p; rewrite !maxnA (maxnC m). Qed.

Lemma maxnACA : interchange maxn maxn.
Proof. by move=> m n p q; rewrite -!maxnA (maxnCA n). Qed.

Lemma maxn_idPl {m n} : reflect (maxn m n = m) (m >= n).
Proof. by rewrite -subn_eq0 -(eqn_add2l m) addn0 -maxnE; apply: eqP. Qed.

Lemma maxn_idPr {m n} : reflect (maxn m n = n) (m <= n).
Proof. by rewrite maxnC; apply: maxn_idPl. Qed.

Lemma maxnn : idempotent maxn.
Proof. by move=> n; apply/maxn_idPl. Qed.

Lemma leq_max m n1 n2 : (m <= maxn n1 n2) = (m <= n1) || (m <= n2).
Proof.
without loss le_n21: n1 n2 / n2 <= n1.
  by case/orP: (leq_total n2 n1) => le_n12; last rewrite maxnC orbC; apply.
by rewrite (maxn_idPl le_n21) orb_idr // => /leq_trans->.
Qed.
Lemma leq_maxl m n : m <= maxn m n. Proof. by rewrite leq_max leqnn. Qed.
Lemma leq_maxr m n : n <= maxn m n. Proof. by rewrite maxnC leq_maxl. Qed.

Lemma gtn_max m n1 n2 : (m > maxn n1 n2) = (m > n1) && (m > n2).
Proof. by rewrite !ltnNge leq_max negb_or. Qed.

Lemma geq_max m n1 n2 : (m >= maxn n1 n2) = (m >= n1) && (m >= n2).
Proof. by rewrite -ltnS gtn_max. Qed.

Lemma maxnSS m n : maxn m.+1 n.+1 = (maxn m n).+1.
Proof. by rewrite !maxnE. Qed.

Lemma addn_maxl : left_distributive addn maxn.
Proof. by move=> m1 m2 n; rewrite !maxnE subnDr addnAC. Qed.

Lemma addn_maxr : right_distributive addn maxn.
Proof. by move=> m n1 n2; rewrite !(addnC m) addn_maxl. Qed.

Lemma min0n : left_zero 0 minn. Proof. by case. Qed.
Lemma minn0 : right_zero 0 minn. Proof. by []. Qed.

Lemma minnC : commutative minn.
Proof. by move=> m n; rewrite /minn; case ltngtP. Qed.

Lemma addn_min_max m n : minn m n + maxn m n = m + n.
Proof. by rewrite /minn /maxn; case: ltngtP => // [_|->] //; apply: addnC. Qed.

Lemma minnE m n : minn m n = m - (m - n).
Proof. by rewrite -(subnDl n) -maxnE -addn_min_max addnK minnC. Qed.

Lemma minnAC : right_commutative minn.
Proof.
by move=> m n p; rewrite !minnE -subnDA subnAC -maxnE maxnC maxnE subnAC subnDA.
Qed.

Lemma minnA : associative minn.
Proof. by move=> m n p; rewrite minnC minnAC (minnC n). Qed.

Lemma minnCA : left_commutative minn.
Proof. by move=> m n p; rewrite !minnA (minnC n). Qed.

Lemma minnACA : interchange minn minn.
Proof. by move=> m n p q; rewrite -!minnA (minnCA n). Qed.

Lemma minn_idPl {m n} : reflect (minn m n = m) (m <= n).
Proof.
rewrite (sameP maxn_idPr eqP) -(eqn_add2l m) eq_sym -addn_min_max eqn_add2r.
exact: eqP.
Qed.

Lemma minn_idPr {m n} : reflect (minn m n = n) (m >= n).
Proof. by rewrite minnC; apply: minn_idPl. Qed.

Lemma minnn : idempotent minn.
Proof. by move=> n; apply/minn_idPl. Qed.

Lemma leq_min m n1 n2 : (m <= minn n1 n2) = (m <= n1) && (m <= n2).
Proof.
wlog le_n21: n1 n2 / n2 <= n1.
  by case/orP: (leq_total n2 n1) => ?; last rewrite minnC andbC; auto.
by rewrite /minn ltnNge le_n21 /= andbC; case: leqP => // /leq_trans->.
Qed.

Lemma gtn_min m n1 n2 : (m > minn n1 n2) = (m > n1) || (m > n2).
Proof. by rewrite !ltnNge leq_min negb_and. Qed.

Lemma geq_min m n1 n2 : (m >= minn n1 n2) = (m >= n1) || (m >= n2).
Proof. by rewrite -ltnS gtn_min. Qed.

Lemma geq_minl m n : minn m n <= m. Proof. by rewrite geq_min leqnn. Qed.
Lemma geq_minr m n : minn m n <= n. Proof. by rewrite minnC geq_minl. Qed.

Lemma addn_minr : right_distributive addn minn.
Proof. by move=> m1 m2 n; rewrite !minnE subnDl addnBA ?leq_subr. Qed.

Lemma addn_minl : left_distributive addn minn.
Proof. by move=> m1 m2 n; rewrite -!(addnC n) addn_minr. Qed.

Lemma minnSS m n : minn m.+1 n.+1 = (minn m n).+1.
Proof. by rewrite -(addn_minr 1). Qed.

Lemma maxnK m n : minn (maxn m n) m = m.
Proof. exact/minn_idPr/leq_maxl. Qed.

Lemma maxKn m n : minn n (maxn m n) = n.
Proof. exact/minn_idPl/leq_maxr. Qed.

Lemma minnK m n : maxn (minn m n) m = m.
Proof. exact/maxn_idPr/geq_minl. Qed.

Lemma minKn m n : maxn n (minn m n) = n.
Proof. exact/maxn_idPl/geq_minr. Qed.

Lemma maxn_minl : left_distributive maxn minn.
Proof.
move=> m1 m2 n; wlog le_m21: m1 m2 / m2 <= m1.
  move=> IH; case/orP: (leq_total m2 m1) => /IH //.
  by rewrite minnC [in R in _ = R]minnC.
rewrite (minn_idPr le_m21); apply/esym/minn_idPr.
by rewrite geq_max leq_maxr leq_max le_m21.
Qed.

Lemma maxn_minr : right_distributive maxn minn.
Proof. by move=> m n1 n2; rewrite !(maxnC m) maxn_minl. Qed.

Lemma minn_maxl : left_distributive minn maxn.
Proof.
by move=> m1 m2 n; rewrite maxn_minr !maxn_minl -minnA maxnn (maxnC _ n) !maxnK.
Qed.

Lemma minn_maxr : right_distributive minn maxn.
Proof. by move=> m n1 n2; rewrite !(minnC m) minn_maxl. Qed.


Section ExMinn.

Variable P : pred nat.
Hypothesis exP : exists n, P n.

Inductive acc_nat i : Prop := AccNat0 of P i | AccNatS of acc_nat i.+1.

Lemma find_ex_minn : {m | P m & forall n, P n -> n >= m}.
Proof.
have: forall n, P n -> n >= 0 by [].
have: acc_nat 0.
  case exP => n; rewrite -(addn0 n); elim: n 0 => [|n IHn] j; first by left.
  by rewrite addSnnS; right; apply: IHn.
move: 0; fix find_ex_minn 2 => m IHm m_lb; case Pm: (P m); first by exists m.
apply: find_ex_minn m.+1 _ _ => [|n Pn]; first by case: IHm; rewrite ?Pm.
by rewrite ltn_neqAle m_lb //; case: eqP Pm => // -> /idP[].
Qed.

Definition ex_minn := s2val find_ex_minn.

Inductive ex_minn_spec : nat -> Type :=
  ExMinnSpec m of P m & (forall n, P n -> n >= m) : ex_minn_spec m.

Lemma ex_minnP : ex_minn_spec ex_minn.
Proof. by rewrite /ex_minn; case: find_ex_minn. Qed.

End ExMinn.

Section ExMaxn.

Variables (P : pred nat) (m : nat).
Hypotheses (exP : exists i, P i) (ubP : forall i, P i -> i <= m).

Lemma ex_maxn_subproof : exists i, P (m - i).
Proof. by case: exP => i Pi; exists (m - i); rewrite subKn ?ubP. Qed.

Definition ex_maxn := m - ex_minn ex_maxn_subproof.

CoInductive ex_maxn_spec : nat -> Type :=
  ExMaxnSpec i of P i & (forall j, P j -> j <= i) : ex_maxn_spec i.

Lemma ex_maxnP : ex_maxn_spec ex_maxn.
Proof.
rewrite /ex_maxn; case: ex_minnP => i Pmi min_i; split=> // j Pj.
have le_i_mj: i <= m - j by rewrite min_i // subKn // ubP.
rewrite -subn_eq0 subnBA ?(leq_trans le_i_mj) ?leq_subr //.
by rewrite addnC -subnBA ?ubP.
Qed.

End ExMaxn.

Lemma eq_ex_minn P Q exP exQ : P =1 Q -> @ex_minn P exP = @ex_minn Q exQ.
Proof.
move=> eqPQ; case: ex_minnP => m1 Pm1 m1_lb; case: ex_minnP => m2 Pm2 m2_lb.
by apply/eqP; rewrite eqn_leq m1_lb (m2_lb, eqPQ) // -eqPQ.
Qed.

Lemma eq_ex_maxn (P Q : pred nat) m n exP ubP exQ ubQ :
  P =1 Q -> @ex_maxn P m exP ubP = @ex_maxn Q n exQ ubQ.
Proof.
move=> eqPQ; case: ex_maxnP => i Pi max_i; case: ex_maxnP => j Pj max_j.
by apply/eqP; rewrite eqn_leq max_i ?eqPQ // max_j -?eqPQ.
Qed.

Section Iteration.

Variable T : Type.
Implicit Types m n : nat.
Implicit Types x y : T.

Definition iter n f x :=
  let fix loop m := if m is i.+1 then f (loop i) else x in loop n.

Definition iteri n f x :=
  let fix loop m := if m is i.+1 then f i (loop i) else x in loop n.

Definition iterop n op x :=
  let f i y := if i is 0 then x else op x y in iteri n f.

Lemma iterSr n f x : iter n.+1 f x = iter n f (f x).
Proof. by elim: n => //= n <-. Qed.

Lemma iterS n f x : iter n.+1 f x = f (iter n f x). Proof. by []. Qed.

Lemma iter_add n m f x : iter (n + m) f x = iter n f (iter m f x).
Proof. by elim: n => //= n ->. Qed.

Lemma iteriS n f x : iteri n.+1 f x = f n (iteri n f x).
Proof. by []. Qed.

Lemma iteropS idx n op x : iterop n.+1 op x idx = iter n (op x) x.
Proof. by elim: n => //= n ->. Qed.

Lemma eq_iter f f' : f =1 f' -> forall n, iter n f =1 iter n f'.
Proof. by move=> eq_f n x; elim: n => //= n ->; rewrite eq_f. Qed.

Lemma eq_iteri f f' : f =2 f' -> forall n, iteri n f =1 iteri n f'.
Proof. by move=> eq_f n x; elim: n => //= n ->; rewrite eq_f. Qed.

Lemma eq_iterop n op op' : op =2 op' -> iterop n op =2 iterop n op'.
Proof. by move=> eq_op x; apply: eq_iteri; case. Qed.

End Iteration.

Lemma iter_succn m n : iter n succn m = m + n.
Proof. by elim: n => //= n ->. Qed.

Lemma iter_succn_0 n : iter n succn 0 = n.
Proof. exact: iter_succn. Qed.

Lemma iter_predn m n : iter n predn m = m - n.
Proof. by elim: n m => /= [|n IHn] m; rewrite ?subn0 // IHn subnS. Qed.


Definition muln_rec := mult.
Notation "m * n" := (muln_rec m n) : nat_rec_scope.

Definition muln := nosimpl muln_rec.
Notation "m * n" := (muln m n) : nat_scope.

Lemma multE : mult = muln. Proof. by []. Qed.
Lemma mulnE : muln = muln_rec. Proof. by []. Qed.

Lemma mul0n : left_zero 0 muln. Proof. by []. Qed.
Lemma muln0 : right_zero 0 muln. Proof. by elim. Qed.
Lemma mul1n : left_id 1 muln. Proof. exact: addn0. Qed.
Lemma mulSn m n : m.+1 * n = n + m * n. Proof. by []. Qed.
Lemma mulSnr m n : m.+1 * n = m * n + n. Proof. exact: addnC. Qed.

Lemma mulnS m n : m * n.+1 = m + m * n.
Proof. by elim: m => // m; rewrite !mulSn !addSn addnCA => ->. Qed.
Lemma mulnSr m n : m * n.+1 = m * n + m.
Proof. by rewrite addnC mulnS. Qed.

Lemma iter_addn m n p : iter n (addn m) p = m * n + p.
Proof. by elim: n => /= [|n ->]; rewrite ?muln0 // mulnS addnA. Qed.

Lemma iter_addn_0 m n : iter n (addn m) 0 = m * n.
Proof. by rewrite iter_addn addn0. Qed.

Lemma muln1 : right_id 1 muln.
Proof. by move=> n; rewrite mulnSr muln0. Qed.

Lemma mulnC : commutative muln.
Proof.
by move=> m n; elim: m => [|m]; rewrite (muln0, mulnS) // mulSn => ->.
Qed.

Lemma mulnDl : left_distributive muln addn.
Proof. by move=> m1 m2 n; elim: m1 => //= m1 IHm; rewrite -addnA -IHm. Qed.

Lemma mulnDr : right_distributive muln addn.
Proof. by move=> m n1 n2; rewrite !(mulnC m) mulnDl. Qed.

Lemma mulnBl : left_distributive muln subn.
Proof.
move=> m n [|p]; first by rewrite !muln0.
by elim: m n => // [m IHm] [|n] //; rewrite mulSn subnDl -IHm.
Qed.

Lemma mulnBr : right_distributive muln subn.
Proof. by move=> m n p; rewrite !(mulnC m) mulnBl. Qed.

Lemma mulnA : associative muln.
Proof. by move=> m n p; elim: m => //= m; rewrite mulSn mulnDl => ->. Qed.

Lemma mulnCA : left_commutative muln.
Proof. by move=> m n1 n2; rewrite !mulnA (mulnC m). Qed.

Lemma mulnAC : right_commutative muln.
Proof. by move=> m n p; rewrite -!mulnA (mulnC n). Qed.

Lemma mulnACA : interchange muln muln.
Proof. by move=> m n p q; rewrite -!mulnA (mulnCA n). Qed.

Lemma muln_eq0 m n : (m * n == 0) = (m == 0) || (n == 0).
Proof. by case: m n => // m [|n] //=; rewrite muln0. Qed.

Lemma muln_eq1 m n : (m * n == 1) = (m == 1) && (n == 1).
Proof. by case: m n => [|[|m]] [|[|n]] //; rewrite muln0. Qed.

Lemma muln_gt0 m n : (0 < m * n) = (0 < m) && (0 < n).
Proof. by case: m n => // m [|n] //=; rewrite muln0. Qed.

Lemma leq_pmull m n : n > 0 -> m <= n * m.
Proof. by move/prednK <-; apply: leq_addr. Qed.

Lemma leq_pmulr m n : n > 0 -> m <= m * n.
Proof. by move/leq_pmull; rewrite mulnC. Qed.

Lemma leq_mul2l m n1 n2 : (m * n1 <= m * n2) = (m == 0) || (n1 <= n2).
Proof. by rewrite {1}/leq -mulnBr muln_eq0. Qed.

Lemma leq_mul2r m n1 n2 : (n1 * m <= n2 * m) = (m == 0) || (n1 <= n2).
Proof. by rewrite -!(mulnC m) leq_mul2l. Qed.

Lemma leq_mul m1 m2 n1 n2 : m1 <= n1 -> m2 <= n2 -> m1 * m2 <= n1 * n2.
Proof.
move=> le_mn1 le_mn2; apply (@leq_trans (m1 * n2)).
  by rewrite leq_mul2l le_mn2 orbT.
by rewrite leq_mul2r le_mn1 orbT.
Qed.

Lemma eqn_mul2l m n1 n2 : (m * n1 == m * n2) = (m == 0) || (n1 == n2).
Proof. by rewrite eqn_leq !leq_mul2l -orb_andr -eqn_leq. Qed.

Lemma eqn_mul2r m n1 n2 : (n1 * m == n2 * m) = (m == 0) || (n1 == n2).
Proof. by rewrite eqn_leq !leq_mul2r -orb_andr -eqn_leq. Qed.

Lemma leq_pmul2l m n1 n2 : 0 < m -> (m * n1 <= m * n2) = (n1 <= n2).
Proof. by move/prednK=> <-; rewrite leq_mul2l. Qed.
Arguments leq_pmul2l [m n1 n2].

Lemma leq_pmul2r m n1 n2 : 0 < m -> (n1 * m <= n2 * m) = (n1 <= n2).
Proof. by move/prednK <-; rewrite leq_mul2r. Qed.
Arguments leq_pmul2r [m n1 n2].

Lemma eqn_pmul2l m n1 n2 : 0 < m -> (m * n1 == m * n2) = (n1 == n2).
Proof. by move/prednK <-; rewrite eqn_mul2l. Qed.
Arguments eqn_pmul2l [m n1 n2].

Lemma eqn_pmul2r m n1 n2 : 0 < m -> (n1 * m == n2 * m) = (n1 == n2).
Proof. by move/prednK <-; rewrite eqn_mul2r. Qed.
Arguments eqn_pmul2r [m n1 n2].

Lemma ltn_mul2l m n1 n2 : (m * n1 < m * n2) = (0 < m) && (n1 < n2).
Proof. by rewrite lt0n !ltnNge leq_mul2l negb_or. Qed.

Lemma ltn_mul2r m n1 n2 : (n1 * m < n2 * m) = (0 < m) && (n1 < n2).
Proof. by rewrite lt0n !ltnNge leq_mul2r negb_or. Qed.

Lemma ltn_pmul2l m n1 n2 : 0 < m -> (m * n1 < m * n2) = (n1 < n2).
Proof. by move/prednK <-; rewrite ltn_mul2l. Qed.
Arguments ltn_pmul2l [m n1 n2].

Lemma ltn_pmul2r m n1 n2 : 0 < m -> (n1 * m < n2 * m) = (n1 < n2).
Proof. by move/prednK <-; rewrite ltn_mul2r. Qed.
Arguments ltn_pmul2r [m n1 n2].

Lemma ltn_Pmull m n : 1 < n -> 0 < m -> m < n * m.
Proof. by move=> lt1n m_gt0; rewrite -{1}[m]mul1n ltn_pmul2r. Qed.

Lemma ltn_Pmulr m n : 1 < n -> 0 < m -> m < m * n.
Proof. by move=> lt1n m_gt0; rewrite mulnC ltn_Pmull. Qed.

Lemma ltn_mul m1 m2 n1 n2 : m1 < n1 -> m2 < n2 -> m1 * m2 < n1 * n2.
Proof.
move=> lt_mn1 lt_mn2; apply (@leq_ltn_trans (m1 * n2)).
  by rewrite leq_mul2l orbC ltnW.
by rewrite ltn_pmul2r // (leq_trans _ lt_mn2).
Qed.

Lemma maxn_mulr : right_distributive muln maxn.
Proof. by case=> // m n1 n2; rewrite /maxn (fun_if (muln _)) ltn_pmul2l. Qed.

Lemma maxn_mull : left_distributive muln maxn.
Proof. by move=> m1 m2 n; rewrite -!(mulnC n) maxn_mulr. Qed.

Lemma minn_mulr : right_distributive muln minn.
Proof. by case=> // m n1 n2; rewrite /minn (fun_if (muln _)) ltn_pmul2l. Qed.

Lemma minn_mull : left_distributive muln minn.
Proof. by move=> m1 m2 n; rewrite -!(mulnC n) minn_mulr. Qed.


Definition expn_rec m n := iterop n muln m 1.
Notation "m ^ n" := (expn_rec m n) : nat_rec_scope.
Definition expn := nosimpl expn_rec.
Notation "m ^ n" := (expn m n) : nat_scope.

Lemma expnE : expn = expn_rec. Proof. by []. Qed.

Lemma expn0 m : m ^ 0 = 1. Proof. by []. Qed.
Lemma expn1 m : m ^ 1 = m. Proof. by []. Qed.
Lemma expnS m n : m ^ n.+1 = m * m ^ n. Proof. by case: n; rewrite ?muln1. Qed.
Lemma expnSr m n : m ^ n.+1 = m ^ n * m. Proof. by rewrite mulnC expnS. Qed.

Lemma iter_muln m n p : iter n (muln m) p = m ^ n * p.
Proof. by elim: n => /= [|n ->]; rewrite ?mul1n // expnS mulnA. Qed.

Lemma iter_muln_1 m n : iter n (muln m) 1 = m ^ n.
Proof. by rewrite iter_muln muln1. Qed.

Lemma exp0n n : 0 < n -> 0 ^ n = 0. Proof. by case: n => [|[]]. Qed.

Lemma exp1n n : 1 ^ n = 1.
Proof. by elim: n => // n; rewrite expnS mul1n. Qed.

Lemma expnD m n1 n2 : m ^ (n1 + n2) = m ^ n1 * m ^ n2.
Proof. by elim: n1 => [|n1 IHn]; rewrite !(mul1n, expnS) // IHn mulnA. Qed.

Lemma expnMn m1 m2 n : (m1 * m2) ^ n = m1 ^ n * m2 ^ n.
Proof. by elim: n => // n IHn; rewrite !expnS IHn -!mulnA (mulnCA m2). Qed.

Lemma expnM m n1 n2 : m ^ (n1 * n2) = (m ^ n1) ^ n2.
Proof.
elim: n1 => [|n1 IHn]; first by rewrite exp1n.
by rewrite expnD expnS expnMn IHn.
Qed.

Lemma expnAC m n1 n2 : (m ^ n1) ^ n2 = (m ^ n2) ^ n1.
Proof. by rewrite -!expnM mulnC. Qed.

Lemma expn_gt0 m n : (0 < m ^ n) = (0 < m) || (n == 0).
Proof.
by case: m => [|m]; elim: n => //= n IHn; rewrite expnS // addn_gt0 IHn.
Qed.

Lemma expn_eq0 m e : (m ^ e == 0) = (m == 0) && (e > 0).
Proof. by rewrite !eqn0Ngt expn_gt0 negb_or -lt0n. Qed.

Lemma ltn_expl m n : 1 < m -> n < m ^ n.
Proof.
move=> m_gt1; elim: n => //= n; rewrite -(leq_pmul2l (ltnW m_gt1)) expnS.
by apply: leq_trans; apply: ltn_Pmull.
Qed.

Lemma leq_exp2l m n1 n2 : 1 < m -> (m ^ n1 <= m ^ n2) = (n1 <= n2).
Proof.
move=> m_gt1; elim: n1 n2 => [|n1 IHn] [|n2] //; last 1 first.
- by rewrite !expnS leq_pmul2l ?IHn // ltnW.
- by rewrite expn_gt0 ltnW.
by rewrite leqNgt (leq_trans m_gt1) // expnS leq_pmulr // expn_gt0 ltnW.
Qed.

Lemma ltn_exp2l m n1 n2 : 1 < m -> (m ^ n1 < m ^ n2) = (n1 < n2).
Proof. by move=> m_gt1; rewrite !ltnNge leq_exp2l. Qed.

Lemma eqn_exp2l m n1 n2 : 1 < m -> (m ^ n1 == m ^ n2) = (n1 == n2).
Proof. by move=> m_gt1; rewrite !eqn_leq !leq_exp2l. Qed.

Lemma expnI m : 1 < m -> injective (expn m).
Proof. by move=> m_gt1 e1 e2 /eqP; rewrite eqn_exp2l // => /eqP. Qed.

Lemma leq_pexp2l m n1 n2 : 0 < m -> n1 <= n2 -> m ^ n1 <= m ^ n2.
Proof. by case: m => [|[|m]] // _; [rewrite !exp1n | rewrite leq_exp2l]. Qed.

Lemma ltn_pexp2l m n1 n2 : 0 < m -> m ^ n1 < m ^ n2 -> n1 < n2.
Proof. by case: m => [|[|m]] // _; [rewrite !exp1n | rewrite ltn_exp2l]. Qed.

Lemma ltn_exp2r m n e : e > 0 -> (m ^ e < n ^ e) = (m < n).
Proof.
move=> e_gt0; apply/idP/idP=> [|ltmn].
  rewrite !ltnNge; apply: contra => lemn.
  by elim: e {e_gt0} => // e IHe; rewrite !expnS leq_mul.
by elim: e e_gt0 => // [[|e] IHe] _; rewrite ?expn1 // ltn_mul // IHe.
Qed.

Lemma leq_exp2r m n e : e > 0 -> (m ^ e <= n ^ e) = (m <= n).
Proof. by move=> e_gt0; rewrite leqNgt ltn_exp2r // -leqNgt. Qed.

Lemma eqn_exp2r m n e : e > 0 -> (m ^ e == n ^ e) = (m == n).
Proof. by move=> e_gt0; rewrite !eqn_leq !leq_exp2r. Qed.

Lemma expIn e : e > 0 -> injective (expn^~ e).
Proof. by move=> e_gt1 m n /eqP; rewrite eqn_exp2r // => /eqP. Qed.


Fixpoint fact_rec n := if n is n'.+1 then n * fact_rec n' else 1.

Definition factorial := nosimpl fact_rec.

Notation "n `!" := (factorial n) (at level 2, format "n `!") : nat_scope.

Lemma factE : factorial = fact_rec. Proof. by []. Qed.

Lemma fact0 : 0`! = 1. Proof. by []. Qed.

Lemma factS n : (n.+1)`! = n.+1 * n`!. Proof. by []. Qed.

Lemma fact_gt0 n : n`! > 0.
Proof. by elim: n => //= n IHn; rewrite muln_gt0. Qed.


Coercion nat_of_bool (b : bool) := if b then 1 else 0.

Lemma leq_b1 (b : bool) : b <= 1. Proof. by case: b. Qed.

Lemma addn_negb (b : bool) : ~~ b + b = 1. Proof. by case: b. Qed.

Lemma eqb0 (b : bool) : (b == 0 :> nat) = ~~ b. Proof. by case: b. Qed.

Lemma eqb1 (b : bool) : (b == 1 :> nat) = b. Proof. by case: b. Qed.

Lemma lt0b (b : bool) : (b > 0) = b. Proof. by case: b. Qed.

Lemma sub1b (b : bool) : 1 - b = ~~ b. Proof. by case: b. Qed.

Lemma mulnb (b1 b2 : bool) : b1 * b2 = b1 && b2.
Proof. by case: b1; case: b2. Qed.

Lemma mulnbl (b : bool) n : b * n = (if b then n else 0).
Proof. by case: b; rewrite ?mul1n. Qed.

Lemma mulnbr (b : bool) n : n * b = (if b then n else 0).
Proof. by rewrite mulnC mulnbl. Qed.

Fixpoint odd n := if n is n'.+1 then ~~ odd n' else false.

Lemma oddb (b : bool) : odd b = b. Proof. by case: b. Qed.

Lemma odd_add m n : odd (m + n) = odd m (+) odd n.
Proof. by elim: m => [|m IHn] //=; rewrite -addTb IHn addbA addTb. Qed.

Lemma odd_sub m n : n <= m -> odd (m - n) = odd m (+) odd n.
Proof.
by move=> le_nm; apply: (@canRL bool) (addbK _) _; rewrite -odd_add subnK.
Qed.

Lemma odd_opp i m : odd m = false -> i <= m -> odd (m - i) = odd i.
Proof. by move=> oddm le_im; rewrite (odd_sub (le_im)) oddm. Qed.

Lemma odd_mul m n : odd (m * n) = odd m && odd n.
Proof. by elim: m => //= m IHm; rewrite odd_add -addTb andb_addl -IHm. Qed.

Lemma odd_exp m n : odd (m ^ n) = (n == 0) || odd m.
Proof. by elim: n => // n IHn; rewrite expnS odd_mul {}IHn orbC; case odd. Qed.


Fixpoint double_rec n := if n is n'.+1 then n'.*2%Nrec.+2 else 0
where "n .*2" := (double_rec n) : nat_rec_scope.

Definition double := nosimpl double_rec.
Notation "n .*2" := (double n) : nat_scope.

Lemma doubleE : double = double_rec. Proof. by []. Qed.

Lemma double0 : 0.*2 = 0. Proof. by []. Qed.

Lemma doubleS n : n.+1.*2 = n.*2.+2. Proof. by []. Qed.

Lemma addnn n : n + n = n.*2.
Proof. by apply: eqP; elim: n => // n IHn; rewrite addnS. Qed.

Lemma mul2n m : 2 * m = m.*2.
Proof. by rewrite mulSn mul1n addnn. Qed.

Lemma muln2 m : m * 2 = m.*2.
Proof. by rewrite mulnC mul2n. Qed.

Lemma doubleD m n : (m + n).*2 = m.*2 + n.*2.
Proof. by rewrite -!addnn -!addnA (addnCA n). Qed.

Lemma doubleB m n : (m - n).*2 = m.*2 - n.*2.
Proof. by elim: m n => [|m IHm] []. Qed.

Lemma leq_double m n : (m.*2 <= n.*2) = (m <= n).
Proof. by rewrite /leq -doubleB; case (m - n). Qed.

Lemma ltn_double m n : (m.*2 < n.*2) = (m < n).
Proof. by rewrite 2!ltnNge leq_double. Qed.

Lemma ltn_Sdouble m n : (m.*2.+1 < n.*2) = (m < n).
Proof. by rewrite -doubleS leq_double. Qed.

Lemma leq_Sdouble m n : (m.*2 <= n.*2.+1) = (m <= n).
Proof. by rewrite leqNgt ltn_Sdouble -leqNgt. Qed.

Lemma odd_double n : odd n.*2 = false.
Proof. by rewrite -addnn odd_add addbb. Qed.

Lemma double_gt0 n : (0 < n.*2) = (0 < n).
Proof. by case: n. Qed.

Lemma double_eq0 n : (n.*2 == 0) = (n == 0).
Proof. by case: n. Qed.

Lemma doubleMl m n : (m * n).*2 = m.*2 * n.
Proof. by rewrite -!mul2n mulnA. Qed.

Lemma doubleMr m n : (m * n).*2 = m * n.*2.
Proof. by rewrite -!muln2 mulnA. Qed.


Fixpoint half (n : nat) : nat := if n is n'.+1 then uphalf n' else n
with uphalf (n : nat) : nat := if n is n'.+1 then n'./2.+1 else n
where "n ./2" := (half n) : nat_scope.

Lemma doubleK : cancel double half.
Proof. by elim=> //= n ->. Qed.

Definition half_double := doubleK.
Definition double_inj := can_inj doubleK.

Lemma uphalf_double n : uphalf n.*2 = n.
Proof. by elim: n => //= n ->. Qed.

Lemma uphalf_half n : uphalf n = odd n + n./2.
Proof. by elim: n => //= n ->; rewrite addnA addn_negb. Qed.

Lemma odd_double_half n : odd n + n./2.*2 = n.
Proof.
by elim: n => //= n {3}<-; rewrite uphalf_half doubleD; case (odd n).
Qed.

Lemma half_bit_double n (b : bool) : (b + n.*2)./2 = n.
Proof. by case: b; rewrite /= (half_double, uphalf_double). Qed.

Lemma halfD m n : (m + n)./2 = (odd m && odd n) + (m./2 + n./2).
Proof.
rewrite -{1}[n]odd_double_half addnCA -{1}[m]odd_double_half -addnA -doubleD.
by do 2!case: odd; rewrite /= ?add0n ?half_double ?uphalf_double.
Qed.

Lemma half_leq m n : m <= n -> m./2 <= n./2.
Proof. by move/subnK <-; rewrite halfD addnA leq_addl. Qed.

Lemma half_gt0 n : (0 < n./2) = (1 < n).
Proof. by case: n => [|[]]. Qed.

Lemma odd_geq m n : odd n -> (m <= n) = (m./2.*2 <= n).
Proof.
move=> odd_n; rewrite -{1}[m]odd_double_half -[n]odd_double_half odd_n.
by case: (odd m); rewrite // leq_Sdouble ltnS leq_double.
Qed.

Lemma odd_ltn m n : odd n -> (n < m) = (n < m./2.*2).
Proof. by move=> odd_n; rewrite !ltnNge odd_geq. Qed.

Lemma odd_gt0 n : odd n -> n > 0. Proof. by case: n. Qed.

Lemma odd_gt2 n : odd n -> n > 1 -> n > 2.
Proof. by move=> odd_n n_gt1; rewrite odd_geq. Qed.


Lemma mulnn m : m * m = m ^ 2.
Proof. by rewrite !expnS muln1. Qed.

Lemma sqrnD m n : (m + n) ^ 2 = m ^ 2 + n ^ 2 + 2 * (m * n).
Proof.
rewrite -!mulnn mul2n mulnDr !mulnDl (mulnC n) -!addnA.
by congr (_ + _); rewrite addnA addnn addnC.
Qed.

Lemma sqrn_sub m n : n <= m -> (m - n) ^ 2 = m ^ 2 + n ^ 2 - 2 * (m * n).
Proof.
move/subnK=> def_m; rewrite -{2}def_m sqrnD -addnA addnAC.
by rewrite -2!addnA addnn -mul2n -mulnDr -mulnDl def_m addnK.
Qed.

Lemma sqrnD_sub m n : n <= m -> (m + n) ^ 2 - 4 * (m * n) = (m - n) ^ 2.
Proof.
move=> le_nm; rewrite -[4]/(2 * 2) -mulnA mul2n -addnn subnDA.
by rewrite sqrnD addnK sqrn_sub.
Qed.

Lemma subn_sqr m n : m ^ 2 - n ^ 2 = (m - n) * (m + n).
Proof. by rewrite mulnBl !mulnDr addnC (mulnC m) subnDl !mulnn. Qed.

Lemma ltn_sqr m n : (m ^ 2 < n ^ 2) = (m < n).
Proof. by rewrite ltn_exp2r. Qed.

Lemma leq_sqr m n : (m ^ 2 <= n ^ 2) = (m <= n).
Proof. by rewrite leq_exp2r. Qed.

Lemma sqrn_gt0 n : (0 < n ^ 2) = (0 < n).
Proof. exact: (ltn_sqr 0). Qed.

Lemma eqn_sqr m n : (m ^ 2 == n ^ 2) = (m == n).
Proof. by rewrite eqn_exp2r. Qed.

Lemma sqrn_inj : injective (expn ^~ 2).
Proof. exact: expIn. Qed.


Definition leqif m n C := ((m <= n) * ((m == n) = C))%type.

Notation "m <= n ?= 'iff' C" := (leqif m n C) : nat_scope.

Coercion leq_of_leqif m n C (H : m <= n ?= iff C) := H.1 : m <= n.

Lemma leqifP m n C : reflect (m <= n ?= iff C) (if C then m == n else m < n).
Proof.
rewrite ltn_neqAle; apply: (iffP idP) => [|lte]; last by rewrite !lte; case C.
by case C => [/eqP-> | /andP[/negPf]]; split=> //; apply: eqxx.
Qed.

Lemma leqif_refl m C : reflect (m <= m ?= iff C) C.
Proof. by apply: (iffP idP) => [-> | <-] //; split; rewrite ?eqxx. Qed.

Lemma leqif_trans m1 m2 m3 C12 C23 :
  m1 <= m2 ?= iff C12 -> m2 <= m3 ?= iff C23 -> m1 <= m3 ?= iff C12 && C23.
Proof.
move=> ltm12 ltm23; apply/leqifP; rewrite -ltm12.
case eqm12: (m1 == m2).
  by rewrite (eqP eqm12) ltn_neqAle !ltm23 andbT; case C23.
by rewrite (@leq_trans m2) ?ltm23 // ltn_neqAle eqm12 ltm12.
Qed.

Lemma mono_leqif f : {mono f : m n / m <= n} ->
  forall m n C, (f m <= f n ?= iff C) = (m <= n ?= iff C).
Proof. by move=> f_mono m n C; rewrite /leqif !eqn_leq !f_mono. Qed.

Lemma leqif_geq m n : m <= n -> m <= n ?= iff (m >= n).
Proof. by move=> lemn; split=> //; rewrite eqn_leq lemn. Qed.

Lemma leqif_eq m n : m <= n -> m <= n ?= iff (m == n).
Proof. by []. Qed.

Lemma geq_leqif a b C : a <= b ?= iff C -> (b <= a) = C.
Proof. by case=> le_ab; rewrite eqn_leq le_ab. Qed.

Lemma ltn_leqif a b C : a <= b ?= iff C -> (a < b) = ~~ C.
Proof. by move=> le_ab; rewrite ltnNge (geq_leqif le_ab). Qed.

Lemma leqif_add m1 n1 C1 m2 n2 C2 :
    m1 <= n1 ?= iff C1 -> m2 <= n2 ?= iff C2 ->
  m1 + m2 <= n1 + n2 ?= iff C1 && C2.
Proof.
rewrite -(mono_leqif (leq_add2r m2)) -(mono_leqif (leq_add2l n1) m2).
exact: leqif_trans.
Qed.

Lemma leqif_mul m1 n1 C1 m2 n2 C2 :
    m1 <= n1 ?= iff C1 -> m2 <= n2 ?= iff C2 ->
  m1 * m2 <= n1 * n2 ?= iff (n1 * n2 == 0) || (C1 && C2).
Proof.
case: n1 => [|n1] le1; first by case: m1 le1 => [|m1] [_ <-] //.
case: n2 m2 => [|n2] [|m2] /=; try by case=> // _ <-; rewrite !muln0 ?andbF.
have /leq_pmul2l-/mono_leqif<-: 0 < n1.+1 by [].
by apply: leqif_trans; have /leq_pmul2r-/mono_leqif->: 0 < m2.+1.
Qed.

Lemma nat_Cauchy m n : 2 * (m * n) <= m ^ 2 + n ^ 2 ?= iff (m == n).
Proof.
without loss le_nm: m n / n <= m.
  by case: (leqP m n); auto; rewrite eq_sym addnC (mulnC m); auto.
apply/leqifP; case: ifP => [/eqP-> | ne_mn]; first by rewrite mulnn addnn mul2n.
by rewrite -subn_gt0 -sqrn_sub // sqrn_gt0 subn_gt0 ltn_neqAle eq_sym ne_mn.
Qed.

Lemma nat_AGM2 m n : 4 * (m * n) <= (m + n) ^ 2 ?= iff (m == n).
Proof.
rewrite -[4]/(2 * 2) -mulnA mul2n -addnn sqrnD; apply/leqifP.
by rewrite ltn_add2r eqn_add2r ltn_neqAle !nat_Cauchy; case: ifP => ->.
Qed.


Module NatTrec.


Fixpoint add m n := if m is m'.+1 then m' + n.+1 else n
where "n + m" := (add n m) : nat_scope.

Fixpoint add_mul m n s := if m is m'.+1 then add_mul m' n (n + s) else s.

Definition mul m n := if m is m'.+1 then add_mul m' n n else 0.

Notation "n * m" := (mul n m) : nat_scope.

Fixpoint mul_exp m n p := if n is n'.+1 then mul_exp m n' (m * p) else p.

Definition exp m n := if n is n'.+1 then mul_exp m n' m else 1.

Notation "n ^ m" := (exp n m) : nat_scope.

Local Notation oddn := odd.
Fixpoint odd n := if n is n'.+2 then odd n' else eqn n 1.

Local Notation doublen := double.
Definition double n := if n is n'.+1 then n' + n.+1 else 0.
Notation "n .*2" := (double n) : nat_scope.

Lemma addE : add =2 addn.
Proof. by elim=> //= n IHn m; rewrite IHn addSnnS. Qed.

Lemma doubleE : double =1 doublen.
Proof. by case=> // n; rewrite -addnn -addE. Qed.

Lemma add_mulE n m s : add_mul n m s = addn (muln n m) s.
Proof. by elim: n => //= n IHn in m s *; rewrite IHn addE addnCA addnA. Qed.

Lemma mulE : mul =2 muln.
Proof. by case=> //= n m; rewrite add_mulE addnC. Qed.

Lemma mul_expE m n p : mul_exp m n p = muln (expn m n) p.
Proof.
by elim: n => [|n IHn] in p *; rewrite ?mul1n //= expnS IHn mulE mulnCA mulnA.
Qed.

Lemma expE : exp =2 expn.
Proof. by move=> m [|n] //=; rewrite mul_expE expnS mulnC. Qed.

Lemma oddE : odd =1 oddn.
Proof.
move=> n; rewrite -{1}[n]odd_double_half addnC.
by elim: n./2 => //=; case (oddn n).
Qed.

Definition trecE := (addE, (doubleE, oddE), (mulE, add_mulE, (expE, mul_expE))).

End NatTrec.

Notation natTrecE := NatTrec.trecE.

Lemma eq_binP : Equality.axiom Ndec.Neqb.
Proof.
move=> p q; apply: (iffP idP) => [|<-]; last by case: p => //; elim.
by case: q; case: p => //; elim=> [p IHp|p IHp|] [q|q|] //=; case/IHp=> ->.
Qed.

Canonical bin_nat_eqMixin := EqMixin eq_binP.
Canonical bin_nat_eqType := Eval hnf in EqType N bin_nat_eqMixin.

Section NumberInterpretation.

Import BinPos.

Section Trec.

Import NatTrec.

Fixpoint nat_of_pos p0 :=
  match p0 with
  | xO p => (nat_of_pos p).*2
  | xI p => (nat_of_pos p).*2.+1
  | xH => 1
  end.

End Trec.

Local Coercion nat_of_pos : positive >-> nat.

Coercion nat_of_bin b := if b is Npos p then p : nat else 0.

Fixpoint pos_of_nat n0 m0 :=
  match n0, m0 with
  | n.+1, m.+2 => pos_of_nat n m
  | n.+1, 1 => xO (pos_of_nat n n)
  | n.+1, 0 => xI (pos_of_nat n n)
  | 0, _ => xH
  end.

Definition bin_of_nat n0 := if n0 is n.+1 then Npos (pos_of_nat n n) else 0%num.

Lemma bin_of_natK : cancel bin_of_nat nat_of_bin.
Proof.
have sub2nn n : n.*2 - n = n by rewrite -addnn addKn.
case=> //= n; rewrite -{3}[n]sub2nn.
by elim: n {2 4}n => // m IHm [|[|n]] //=; rewrite IHm // natTrecE sub2nn.
Qed.

Lemma nat_of_binK : cancel nat_of_bin bin_of_nat.
Proof.
case=> //=; elim=> //= p; case: (nat_of_pos p) => //= n [<-].
  by rewrite natTrecE !addnS {2}addnn; elim: {1 3}n.
by rewrite natTrecE addnS /= addnS {2}addnn; elim: {1 3}n.
Qed.

Lemma nat_of_succ_gt0 p : Psucc p = p.+1 :> nat.
Proof. by elim: p => //= p ->; rewrite !natTrecE. Qed.

Lemma nat_of_addn_gt0 p q : (p + q)%positive = p + q :> nat.
Proof.
apply: @fst _ (Pplus_carry p q = (p + q).+1 :> nat) _.
elim: p q => [p IHp|p IHp|] [q|q|] //=; rewrite !natTrecE //;
  by rewrite ?IHp ?nat_of_succ_gt0 ?(doubleS, doubleD, addn1, addnS).
Qed.

Lemma nat_of_add_bin b1 b2 : (b1 + b2)%num = b1 + b2 :> nat.
Proof. by case: b1 b2 => [|p] [|q] //=; apply: nat_of_addn_gt0. Qed.

Lemma nat_of_mul_bin b1 b2 : (b1 * b2)%num = b1 * b2 :> nat.
Proof.
case: b1 b2 => [|p] [|q] //=; elim: p => [p IHp|p IHp|] /=;
  by rewrite ?(mul1n, nat_of_addn_gt0, mulSn) //= !natTrecE IHp doubleMl.
Qed.

Lemma nat_of_exp_bin n (b : N) : n ^ b = pow_N 1 muln n b.
Proof.
by case: b; last (elim=> //= p <-; rewrite natTrecE mulnn -expnM muln2 ?expnS).
Qed.

End NumberInterpretation.


Record number : Type := Num {bin_of_number :> N}.

Definition extend_number (nn : number) m := Num (nn * 1000 + bin_of_nat m).

Coercion extend_number : number >-> Funclass.

Canonical number_subType := [newType for bin_of_number].
Definition number_eqMixin := Eval hnf in [eqMixin of number by <:].
Canonical number_eqType := Eval hnf in EqType number number_eqMixin.

Notation "[ 'Num' 'of' e ]" := (Num (bin_of_nat e))
  (at level 0, format "[ 'Num' 'of' e ]") : nat_scope.


Lemma nat_semi_ring : semi_ring_theory 0 1 addn muln (@eq _).
Proof. exact: mk_srt add0n addnC addnA mul1n mul0n mulnC mulnA mulnDl. Qed.

Lemma nat_semi_morph :
  semi_morph 0 1 addn muln (@eq _) 0%num 1%num Nplus Nmult pred1 nat_of_bin.
Proof.
by move: nat_of_add_bin nat_of_mul_bin; split=> //= m n; move/eqP->.
Qed.

Lemma nat_power_theory : power_theory 1 muln (@eq _) nat_of_bin expn.
Proof. by split; apply: nat_of_exp_bin. Qed.


Fixpoint pop_succn e := if e is e'.+1 then fun n => pop_succn e' n.+1 else id.

Ltac pop_succn e := eval lazy beta iota delta [pop_succn] in (pop_succn e 1).

Ltac nat_litteral e :=
  match pop_succn e with
  | ?n.+1 => constr: (bin_of_nat n)
  | _ => NotConstant
  end.

Ltac succn_to_add :=
  match goal with
  | |- context G [?e.+1] =>
    let x := fresh "NatLit0" in
    match pop_succn e with
    | ?n.+1 => pose x := n.+1; let G' := context G [x] in change G'
    | _ ?e' ?n => pose x := n; let G' := context G [x + e'] in change G'
    end; succn_to_add; rewrite {}/x
  | _ => idtac
  end.

Add Ring nat_ring_ssr : nat_semi_ring (morphism nat_semi_morph,
   constants [nat_litteral], preprocess [succn_to_add],
   power_tac nat_power_theory [nat_litteral]).


Ltac nat_norm :=
  succn_to_add; rewrite ?add0n ?addn0 -?addnA ?(addSn, addnS, add0n, addn0).

Ltac nat_congr := first
 [ apply: (congr1 succn _)
 | apply: (congr1 predn _)
 | apply: (congr1 (addn _) _)
 | apply: (congr1 (subn _) _)
 | apply: (congr1 (addn^~ _) _)
 | match goal with |- (?X1 + ?X2 = ?X3) =>
     symmetry;
     rewrite -1?(addnC X1) -?(addnCA X1);
     apply: (congr1 (addn X1) _);
     symmetry
   end ].