# 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.
A boolean equality over N
Notation Peqb := Peqb (only parsing). Notation Neqb := Neqb (only parsing).
Notation Peqb_correct := Peqb_refl (only parsing).
Lemma Peqb_complete : forall p p', Peqb p p' = true -> p = p'.
Lemma Peqb_Pcompare : forall p p', Peqb p p' = true -> Pcompare p p' Eq = Eq.
Lemma Pcompare_Peqb : forall p p', Pcompare p p' Eq = Eq -> Peqb p p' = true.
Lemma Neqb_correct : forall n, Neqb n n = true.
Lemma Neqb_Ncompare : forall n n', Neqb n n' = true -> Ncompare n n' = Eq.
Lemma Ncompare_Neqb : forall n n', Ncompare n n' = Eq -> Neqb n n' = true.
Lemma Neqb_complete : forall a a', Neqb a a' = true -> a = a'.
Lemma Neqb_comm : forall a a', Neqb a a' = Neqb a' a.
Lemma Nxor_eq_true :
forall a a', Nxor a a' = N0 -> Neqb a a' = true.
Lemma Nxor_eq_false :
forall a a' p, Nxor a a' = Npos p -> Neqb a a' = false.
Lemma Nodd_not_double :
forall a,
Nodd a -> forall a0, Neqb (Ndouble a0) a = false.
Lemma Nnot_div2_not_double :
forall a a0,
Neqb (Ndiv2 a) a0 = false -> Neqb a (Ndouble a0) = false.
Lemma Neven_not_double_plus_one :
forall a,
Neven a -> forall a0, Neqb (Ndouble_plus_one a0) a = false.
Lemma Nnot_div2_not_double_plus_one :
forall a a0,
Neqb (Ndiv2 a) a0 = false -> Neqb (Ndouble_plus_one a0) a = false.
Lemma Nbit0_neq :
forall a a',
Nbit0 a = false -> Nbit0 a' = true -> Neqb a a' = false.
Lemma Ndiv2_eq :
forall a a', Neqb a a' = true -> Neqb (Ndiv2 a) (Ndiv2 a') = true.
Lemma Ndiv2_neq :
forall a a',
Neqb (Ndiv2 a) (Ndiv2 a') = false -> Neqb a a' = false.
Lemma Ndiv2_bit_eq :
forall a a',
Nbit0 a = Nbit0 a' -> Ndiv2 a = Ndiv2 a' -> a = a'.
Lemma Ndiv2_bit_neq :
forall a a',
Neqb a a' = false ->
Nbit0 a = Nbit0 a' -> Neqb (Ndiv2 a) (Ndiv2 a') = false.
Lemma Nneq_elim :
forall a a',
Neqb a a' = false ->
Nbit0 a = negb (Nbit0 a') \/
Neqb (Ndiv2 a) (Ndiv2 a') = false.
Lemma Ndouble_or_double_plus_un :
forall a,
{a0 : N | a = Ndouble a0} + {a1 : N | a = Ndouble_plus_one a1}.
A boolean order on N
Definition Nleb (a b:N) := leb (nat_of_N a) (nat_of_N b).
Lemma Nleb_Nle : forall a b, Nleb a b = true <-> Nle a b.
Lemma Nleb_refl : forall a, Nleb a a = true.
Lemma Nleb_antisym :
forall a b, Nleb a b = true -> Nleb b a = true -> a = b.
Lemma Nleb_trans :
forall a b c, Nleb a b = true -> Nleb b c = true -> Nleb a c = true.
Lemma Nleb_ltb_trans :
forall a b c,
Nleb a b = true -> Nleb c b = false -> Nleb c a = false.
Lemma Nltb_leb_trans :
forall a b c,
Nleb b a = false -> Nleb b c = true -> Nleb c a = false.
Lemma Nltb_trans :
forall a b c,
Nleb b a = false -> Nleb c b = false -> Nleb c a = false.
Lemma Nltb_leb_weak : forall a b:N, Nleb b a = false -> Nleb a b = true.
Lemma Nleb_double_mono :
forall a b,
Nleb a b = true -> Nleb (Ndouble a) (Ndouble b) = true.
Lemma Nleb_double_plus_one_mono :
forall a b,
Nleb a b = true ->
Nleb (Ndouble_plus_one a) (Ndouble_plus_one b) = true.
Lemma Nleb_double_mono_conv :
forall a b,
Nleb (Ndouble a) (Ndouble b) = true -> Nleb a b = true.
Lemma Nleb_double_plus_one_mono_conv :
forall a b,
Nleb (Ndouble_plus_one a) (Ndouble_plus_one b) = true ->
Nleb a b = true.
Lemma Nltb_double_mono :
forall a b,
Nleb a b = false -> Nleb (Ndouble a) (Ndouble b) = false.
Lemma Nltb_double_plus_one_mono :
forall a b,
Nleb a b = false ->
Nleb (Ndouble_plus_one a) (Ndouble_plus_one b) = false.
Lemma Nltb_double_mono_conv :
forall a b,
Nleb (Ndouble a) (Ndouble b) = false -> Nleb a b = false.
Lemma Nltb_double_plus_one_mono_conv :
forall a b,
Nleb (Ndouble_plus_one a) (Ndouble_plus_one b) = false ->
Nleb a b = false.
Lemma Nltb_Ncompare : forall a b,
Nleb a b = false <-> Ncompare a b = Gt.
Lemma Ncompare_Gt_Nltb : forall a b,
Ncompare a b = Gt -> Nleb a b = false.
Lemma Ncompare_Lt_Nltb : forall a b,
Ncompare a b = Lt -> Nleb b a = false.
Definition Nmin' (a b:N) := if Nleb a b then a else b.
Lemma Nmin_Nmin' : forall a b, Nmin a b = Nmin' a b.
Lemma Nmin_choice : forall a b, {Nmin a b = a} + {Nmin a b = b}.
Lemma Nmin_le_1 : forall a b, Nleb (Nmin a b) a = true.
Lemma Nmin_le_2 : forall a b, Nleb (Nmin a b) b = true.
Lemma Nmin_le_3 :
forall a b c, Nleb a (Nmin b c) = true -> Nleb a b = true.
Lemma Nmin_le_4 :
forall a b c, Nleb a (Nmin b c) = true -> Nleb a c = true.
Lemma Nmin_le_5 :
forall a b c,
Nleb a b = true -> Nleb a c = true -> Nleb a (Nmin b c) = true.
Lemma Nmin_lt_3 :
forall a b c, Nleb (Nmin b c) a = false -> Nleb b a = false.
Lemma Nmin_lt_4 :
forall a b c, Nleb (Nmin b c) a = false -> Nleb c a = false.