The Coq Proof Assistant

# Library Coq.NArith.Nnat

Require Import BinPos BinNat PeanoNat Pnat.

# Conversions from N to nat

Module N2Nat.
N.to_nat is a bijection between N and nat, with Pos.of_nat as reciprocal. See Nat2N.id below for the dual equation.
Lemma id a : N.of_nat (N.to_nat a) = a.
N.to_nat is hence injective
Lemma inj a a' : N.to_nat a = N.to_nat a' -> a = a'.
Lemma inj_iff a a' : N.to_nat a = N.to_nat a' <-> a = a'.
Interaction of this translation and usual operations.
Lemma inj_0 : N.to_nat 0 = 0.
Lemma inj_double a : N.to_nat (N.double a) = 2*(N.to_nat a).
Lemma inj_succ_double a : N.to_nat (N.succ_double a) = S (2*(N.to_nat a)).
Lemma inj_succ a : N.to_nat (N.succ a) = S (N.to_nat a).
Lemma inj_add a a' :
N.to_nat (a + a') = N.to_nat a + N.to_nat a'.
Lemma inj_mul a a' :
N.to_nat (a * a') = N.to_nat a * N.to_nat a'.
Lemma inj_sub a a' :
N.to_nat (a - a') = N.to_nat a - N.to_nat a'.
Lemma inj_pred a : N.to_nat (N.pred a) = Nat.pred (N.to_nat a).
Lemma inj_div2 a : N.to_nat (N.div2 a) = Nat.div2 (N.to_nat a).
Lemma inj_compare a a' :
(a ?= a')%N = (N.to_nat a ?= N.to_nat a').
Lemma inj_div n m :
N.to_nat (n / m) = N.to_nat n / N.to_nat m.
Lemma inj_mod a a' :
N.to_nat (a mod a') = N.to_nat a mod N.to_nat a'.
Lemma inj_pow a a' :
N.to_nat (a ^ a') = N.to_nat a ^ N.to_nat a'.
Lemma inj_max a a' :
N.to_nat (N.max a a') = Nat.max (N.to_nat a) (N.to_nat a').
Lemma inj_min a a' :
N.to_nat (N.min a a') = Nat.min (N.to_nat a) (N.to_nat a').
Lemma inj_iter a {A} (f:A->A) (x:A) :
N.iter a f x = Nat.iter (N.to_nat a) f x.
End N2Nat.
Global Hint Rewrite N2Nat.inj_div N2Nat.inj_mod N2Nat.inj_pow
N2Nat.inj_double N2Nat.inj_succ_double
N2Nat.inj_succ N2Nat.inj_add N2Nat.inj_mul N2Nat.inj_sub
N2Nat.inj_pred N2Nat.inj_div2 N2Nat.inj_max N2Nat.inj_min
N2Nat.id
: Nnat.

# Conversions from nat to N

Module Nat2N.
N.of_nat is an bijection between nat and N, with Pos.to_nat as reciprocal. See N2Nat.id above for the dual equation.
Lemma id n : N.to_nat (N.of_nat n) = n.
Global Hint Rewrite id : Nnat.
Ltac nat2N := apply N2Nat.inj; now autorewrite with Nnat.
N.of_nat is hence injective
Lemma inj n n' : N.of_nat n = N.of_nat n' -> n = n'.
Lemma inj_iff n n' : N.of_nat n = N.of_nat n' <-> n = n'.
Interaction of this translation and usual operations.
Lemma inj_double n : N.of_nat (2*n) = N.double (N.of_nat n).
Lemma inj_succ_double n : N.of_nat (S (2*n)) = N.succ_double (N.of_nat n).
Lemma inj_succ n : N.of_nat (S n) = N.succ (N.of_nat n).
Lemma inj_pred n : N.of_nat (Nat.pred n) = N.pred (N.of_nat n).
Lemma inj_add n n' : N.of_nat (n+n') = (N.of_nat n + N.of_nat n')%N.
Lemma inj_sub n n' : N.of_nat (n-n') = (N.of_nat n - N.of_nat n')%N.
Lemma inj_mul n n' : N.of_nat (n*n') = (N.of_nat n * N.of_nat n')%N.
Lemma inj_div2 n : N.of_nat (Nat.div2 n) = N.div2 (N.of_nat n).
Lemma inj_compare n n' :
(n ?= n') = (N.of_nat n ?= N.of_nat n')%N.
Lemma inj_div n n' :
N.of_nat (n / n') = (N.of_nat n / N.of_nat n')%N.
Lemma inj_mod n n' :
N.of_nat (n mod n') = (N.of_nat n mod N.of_nat n')%N.
Lemma inj_pow n n' :
N.of_nat (n ^ n') = (N.of_nat n ^ N.of_nat n')%N.
Lemma inj_min n n' :
N.of_nat (Nat.min n n') = N.min (N.of_nat n) (N.of_nat n').
Lemma inj_max n n' :
N.of_nat (Nat.max n n') = N.max (N.of_nat n) (N.of_nat n').
Lemma inj_iter n {A} (f:A->A) (x:A) :
Nat.iter n f x = N.iter (N.of_nat n) f x.
End Nat2N.
Global Hint Rewrite Nat2N.id : Nnat.
Compatibility notations
Notation nat_of_N_inj := N2Nat.inj (only parsing).
Notation N_of_nat_of_N := N2Nat.id (only parsing).
Notation nat_of_Ndouble := N2Nat.inj_double (only parsing).
Notation nat_of_Ndouble_plus_one := N2Nat.inj_succ_double (only parsing).
Notation nat_of_Nsucc := N2Nat.inj_succ (only parsing).
Notation nat_of_Nplus := N2Nat.inj_add (only parsing).
Notation nat_of_Nmult := N2Nat.inj_mul (only parsing).
Notation nat_of_Nminus := N2Nat.inj_sub (only parsing).
Notation nat_of_Npred := N2Nat.inj_pred (only parsing).
Notation nat_of_Ndiv2 := N2Nat.inj_div2 (only parsing).
Notation nat_of_Ncompare := N2Nat.inj_compare (only parsing).
Notation nat_of_Ndiv := N2Nat.inj_div (only parsing).
Notation nat_of_Nmod := N2Nat.inj_mod (only parsing).
Notation nat_of_Npow := N2Nat.inj_pow (only parsing).
Notation nat_of_Nmax := N2Nat.inj_max (only parsing).
Notation nat_of_Nmin := N2Nat.inj_min (only parsing).
Notation nat_of_N_of_nat := Nat2N.id (only parsing).
Notation N_of_nat_inj := Nat2N.inj (only parsing).
Notation N_of_double := Nat2N.inj_double (only parsing).
Notation N_of_double_plus_one := Nat2N.inj_succ_double (only parsing).
Notation N_of_S := Nat2N.inj_succ (only parsing).
Notation N_of_pred := Nat2N.inj_pred (only parsing).
Notation N_of_plus := Nat2N.inj_add (only parsing).
Notation N_of_minus := Nat2N.inj_sub (only parsing).
Notation N_of_mult := Nat2N.inj_mul (only parsing).
Notation N_of_div2 := Nat2N.inj_div2 (only parsing).
Notation N_of_nat_compare := Nat2N.inj_compare (only parsing).
Notation N_of_nat_div := Nat2N.inj_div (only parsing).
Notation N_of_nat_mod := Nat2N.inj_mod (only parsing).
Notation N_of_nat_pow := Nat2N.inj_pow (only parsing).
Notation N_of_min := Nat2N.inj_min (only parsing).
Notation N_of_max := Nat2N.inj_max (only parsing).