# 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.