Library Coq.NArith.Ndec


Require Import Bool.
Require Import Sumbool.
Require Import Arith.
Require Import BinPos.
Require Import BinNat.
Require Import Pnat.
Require Import Nnat.
Require Import Ndigits.

Local Open Scope N_scope.

Obsolete results about boolean comparisons over N, kept for compatibility with IntMap and SMC.

Notation Peqb := Pos.eqb (compat "8.3").
Notation Neqb := N.eqb (compat "8.3").
Notation Peqb_correct := Pos.eqb_refl (compat "8.3").
Notation Neqb_correct := N.eqb_refl (compat "8.3").
Notation Neqb_comm := N.eqb_sym (compat "8.3").

Lemma Peqb_complete p p' : Pos.eqb p p' = true -> p = p'.

Lemma Peqb_Pcompare p p' : Pos.eqb p p' = true -> Pos.compare p p' = Eq.

Lemma Pcompare_Peqb p p' : Pos.compare p p' = Eq -> Pos.eqb p p' = true.

Lemma Neqb_Ncompare n n' : N.eqb n n' = true -> N.compare n n' = Eq.

Lemma Ncompare_Neqb n n' : N.compare n n' = Eq -> N.eqb n n' = true.

Lemma Neqb_complete n n' : N.eqb n n' = true -> n = n'.

Lemma Nxor_eq_true n n' : N.lxor n n' = 0 -> N.eqb n n' = true.

Ltac eqb2eq := rewrite <- ?not_true_iff_false in *; rewrite ?N.eqb_eq in *.

Lemma Nxor_eq_false n n' p :
  N.lxor n n' = N.pos p -> N.eqb n n' = false.

Lemma Nodd_not_double a :
  Nodd a -> forall a0, N.eqb (N.double a0) a = false.

Lemma Nnot_div2_not_double a a0 :
  N.eqb (N.div2 a) a0 = false -> N.eqb a (N.double a0) = false.

Lemma Neven_not_double_plus_one a :
  Neven a -> forall a0, N.eqb (N.succ_double a0) a = false.

Lemma Nnot_div2_not_double_plus_one a a0 :
  N.eqb (N.div2 a) a0 = false -> N.eqb (N.succ_double a0) a = false.

Lemma Nbit0_neq a a' :
  N.odd a = false -> N.odd a' = true -> N.eqb a a' = false.

Lemma Ndiv2_eq a a' :
  N.eqb a a' = true -> N.eqb (N.div2 a) (N.div2 a') = true.

Lemma Ndiv2_neq a a' :
  N.eqb (N.div2 a) (N.div2 a') = false -> N.eqb a a' = false.

Lemma Ndiv2_bit_eq a a' :
  N.odd a = N.odd a' -> N.div2 a = N.div2 a' -> a = a'.

Lemma Ndiv2_bit_neq a a' :
  N.eqb a a' = false ->
   N.odd a = N.odd a' -> N.eqb (N.div2 a) (N.div2 a') = false.

Lemma Nneq_elim a a' :
   N.eqb a a' = false ->
   N.odd a = negb (N.odd a') \/
   N.eqb (N.div2 a) (N.div2 a') = false.

Lemma Ndouble_or_double_plus_un a :
   {a0 : N | a = N.double a0} + {a1 : N | a = N.succ_double a1}.

An inefficient boolean order on N. Please use N.leb instead now.

Definition Nleb (a b:N) := leb (N.to_nat a) (N.to_nat b).

Lemma Nleb_alt a b : Nleb a b = N.leb a b.

Lemma Nleb_Nle a b : Nleb a b = true <-> a <= b.

Lemma Nleb_refl a : Nleb a a = true.

Lemma Nleb_antisym a b : Nleb a b = true -> Nleb b a = true -> a = b.

Lemma Nleb_trans a b c : Nleb a b = true -> Nleb b c = true -> Nleb a c = true.

Lemma Nleb_ltb_trans a b c :
  Nleb a b = true -> Nleb c b = false -> Nleb c a = false.

Lemma Nltb_leb_trans a b c :
  Nleb b a = false -> Nleb b c = true -> Nleb c a = false.

Lemma Nltb_trans a b c :
  Nleb b a = false -> Nleb c b = false -> Nleb c a = false.

Lemma Nltb_leb_weak a b : Nleb b a = false -> Nleb a b = true.

Lemma Nleb_double_mono a b :
  Nleb a b = true -> Nleb (N.double a) (N.double b) = true.

Lemma Nleb_double_plus_one_mono a b :
  Nleb a b = true ->
   Nleb (N.succ_double a) (N.succ_double b) = true.

Lemma Nleb_double_mono_conv a b :
  Nleb (N.double a) (N.double b) = true -> Nleb a b = true.

Lemma Nleb_double_plus_one_mono_conv a b :
  Nleb (N.succ_double a) (N.succ_double b) = true ->
   Nleb a b = true.

Lemma Nltb_double_mono a b :
   Nleb a b = false -> Nleb (N.double a) (N.double b) = false.

Lemma Nltb_double_plus_one_mono a b :
  Nleb a b = false ->
   Nleb (N.succ_double a) (N.succ_double b) = false.

Lemma Nltb_double_mono_conv a b :
  Nleb (N.double a) (N.double b) = false -> Nleb a b = false.

Lemma Nltb_double_plus_one_mono_conv a b :
  Nleb (N.succ_double a) (N.succ_double b) = false ->
   Nleb a b = false.



Lemma Nltb_Ncompare a b : Nleb a b = false <-> N.compare a b = Gt.

Lemma Ncompare_Gt_Nltb a b : N.compare a b = Gt -> Nleb a b = false.

Lemma Ncompare_Lt_Nltb a b : N.compare a b = Lt -> Nleb b a = false.


Notation Nmin_choice := N.min_dec (compat "8.3").

Lemma Nmin_le_1 a b : Nleb (N.min a b) a = true.

Lemma Nmin_le_2 a b : Nleb (N.min a b) b = true.

Lemma Nmin_le_3 a b c : Nleb a (N.min b c) = true -> Nleb a b = true.

Lemma Nmin_le_4 a b c : Nleb a (N.min b c) = true -> Nleb a c = true.

Lemma Nmin_le_5 a b c :
   Nleb a b = true -> Nleb a c = true -> Nleb a (N.min b c) = true.

Lemma Nmin_lt_3 a b c : Nleb (N.min b c) a = false -> Nleb b a = false.

Lemma Nmin_lt_4 a b c : Nleb (N.min b c) a = false -> Nleb c a = false.